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PONTIFICIA ACADEMIA SCIENTIARVM
CITTA DFI VATICANO
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
9
SEMAINE D' ETUDE
SUR
LE ROLE DE L'ANALYSE ECONOMETRIOUE
DANS LA FORMULATION DE PLANS
DE DEVELOPPEMENT
7-13 Octobre 1963
PONTIFICIA
ACADEMIA
SCIENTIARVM
NORTH - HoLLaND PusLISHING COMPANY, AMSTERDAM
Ranp McNairy & Company CHICAGO
1 0G 5
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
~q
S TUDY WEEK
ON
THE ECONOMETRIC APPROACH
TO DEVELOPMENT PLANNING
sctober 7 - 10, 1900
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ACADE
VWCIFN 1am
reat 21 ‘ 6 65
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NORTH - HOLLAND PUBLISHING COMPANY, AMSTERDAM
RAND McCNAaiLYy & Company. CHICAGO
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© Copyright 1965 — PONTIFICIA ACADE-MIA
SCIENTIARVM - CITTA DEL VATICANO
Adresse :
“PONTIFICIA
CASINA Pio IV
ACADEMIA SCIENTIARVM?”
CITTA DEL VATICANO
LE ROLE _L . aL YSL _11A10METRIQUE
DANS LA FORMULATION DIF PLANS DE
DEVELOPPEMT
Les économies modernes sont exirêmement complexes el ..
théorie et l’expérience démontrent que le libre jeu des choix
individuels n’assure pas, comme on le croyait par le passé, ds
résultats favorables pour la collectivité.
Ceci posé s'ensuit la nécessité de préétablir des instruments
appropriés de connaissance et de contrôle, et de fixer les obiecifs
vers lesquels l’activité économique doit être orientée.
De ces exigences a pris son départ l’Econométrie, qui embloie
la méthode statistico-mathématique tant dans 1’étude théo-‘ique
des phénomènes économiques que pour formuler des
directives de politique économique et des plans de développement.
L'Econométrie est une discipline relativement récente et,
au sujet de sa structure, de ses méthodes et de ses finalités,
s’est engagée une discussion animée entre les savants spécia
listes du monde entier.
L'intérêt de cette discussion est accru du fait que l’on voit
se répandre le recours à des plans à développement e. des
politiques de contrôle de la conioneti
A
La méthode économétrique représente un progrès considérable
sur les systèmes non mathématiques d’étude des phénomènes
qui se rattachent à l’activité économique.
Elle a permis de créer les structures d’une nouvelle discipline
ayant toutes les caractéristiques des sciences naturelles
traditionnelles, parce que, bien qu’elle traite une matière
substantiellement diverse de celle des disciplines physiques et
biologiques, elle suit des procédés logiques et techniques qui la
rendent strictement analogue à celles-ci.
La Semaine d’Etude que l’Académie Pontificale des Sciences
a tenu en son siège dans les Jardins du Vatican et qui a
réuni quelques-uns parmi les plus illustres spécialistes du
monde en Econométrie, à cherché à étudier la contribution que
l’analyse économétrique a apportée ou peut apporter à la connaissance
des problèmes du développement et des fluctuations
économiques.
PIETRO SALVIUCCI
Chancelier de l’Académie
LA SEMAINE D’ ETUDE
SUR
LE ROLE DE L’ANALYSE ECONOMETRIQUE
DANS LA FORMULATION DE PLANS DE
DEVELOPPEMENT
Le but des « Semaines d’Etude » de l’Académie Pontificale des
sciences a été ainsi défini par son premier Président. S.E. le Rév.me
Père AGOSTINO GEMELLI O.F.M.:
« Tandis qu’on fixait, après sa fondation, les travaux de l’Académie,
un problème se présenta bien vite avec évidence: les sciences
posent chaque jour des problèmes nouveaux qui donnent lieu d’ordinaire
à divers essais de solution, souvent contradictoires. Il arrive
ainsi constamment que parmi les représentants les plus autorisés
d’une science et, en particulier, entre ceux qui se sont consacrés à
l'étude d’une même question, on rencontre des opinions opposées.
De pareilles divergences se maintiennent parfois pendant de longues
périodes et constituent à la fois une grave difficulté pour l’enseignement
des sciences et fréquemment aussi un obstacle considérable à
leur développement. D'ailleurs, l’expérience montre que les méthodes
actuellement pratiquées dans la discussion des problèmes
scientifiques n’ont qu’une efficacité limitée au point de vue de
‘établissement d’une unité de doctrine. Il serait hautement souhai-‘able
de promouvoir tout ce qui pourrait favoriser une entente sur
es points en discussion.
« Un tel procédé semble devoir être particulièrement utile sous
ce rapport: savoir établir des contacts personnels prolongés entre
quelques représentants d’opinions différentes au sujet d’une aues-‘on
déterminée ».
XIV PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
“
Dans ce but, l’Académie Pontificale des Sciences a réalisé une
nouvelle « Semaine d'Etude » ayant pour titre: « Le rôle de l’analyse
économétrique dans la formulation de plans de développement
» (1).
Bien que ces derniers temps un travail intense ait été fourni sur
les divers aspects de ce problème, il restait cependant quelques questions
de détail a résoudre, et de nouvelles questions s’étaient de plus
posées pendant ces dernières années.
Etant donné qu’on n’avait pas encore provoqué un débat approfondi
a ce sujet et que le moment semblait propice pour le faire,
l’Académie Pontificale des Sciences s’est proposée de réunir un nombre
restreint de savants, spécialistes de la question. Son but était
de recueillir, au cours d’une discussion approfondie, les synthèses des
nombreuses recherches effectuées dans ce domaine; de formuler clairement
l’état des différents problèmes qui s’y rapportent; et par là
(1) Cette « Semaine d’Etude » sur « Le role de l’analyse économétrique
dans la formulation de plans de développement » est la septième de la série.
La première « Semaine d’Etude » a eu lieu du 6 au 13 juin 1949; elle
a été dédiée au « Problème biologique du Cancer », et a été présidée
par 1’Académicien Pontifical S. E. Pierro Ronponi, Professeur de Pathologie
générale et expérimentale à l’Université de Milan, y ont participé
personnellement 15 savants tandis que 3 autres ont envoyé des mémoires.
Les comptes-rendus de la « Semaine d’Etude » ont été publiés dans le
7ème volume des « Scripta Varia » de l’Académie; ils représentent un volume
de 364 pages.
La deuxième « Semaine d'Etude » a eu lieu du 19 au 26 novembre 1951;
elle a été dédiée au « Problème des Microséismes », et a été présidée par
l’Académicien Pontifical S. E. FRANCEsco VERCELLI, Directeur de l’Institut
Thalassographique et de l'Observatoire Géophysique de Trieste; y ont participé
personnellement 15 savants tandis que 4 autres ont envoyé des mémoires.
Les comptes-rendus de la « Semaine d’Etude » ont été publiés
dans le 12ème volume des « Scripta Varia » de l’Académie; ils forment un
volume de 466 pages.
La troisième « Semaine d’Etude » a eu lieu du 24 avril au 2 mai 1955;
elle a été dédiée au « Problème des Oligoéléments dans la vie végétale et
animale », et a été présidée par l’Académicien Pontifical S. E. José MARIA
ALBAREDA HERRERA. Directeur de l'Institut de Pédologie et de Phvsiologie
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XV
de pouvoir fixer les directives de recherche les plus logiques, les
plus persuasives et les plus prometteuses, étant donné l’état actuel
le la science.
À cet effet ont été invités par l’Académie des experts qualifiés
zn économétrie, en économie, en économie politique et en statistique,
qui, grâce à leurs études spécifiques, ont contribué à eclaircir le
:ôle fondamental de l’analyse statistico-mathématique dans la formulation
de plans de developpement
La présidence de cette « Semaine d’Etude » sur « Le rôle de
l'analyse économétrique dans la formulation de plans de dévelopsement
» a été confiée par le Président de l’Académie Pontificale
des Sciences, S.E. le Rév.me Monseigneur GEorGES LEMAÎTRE, à
"Académicien Pontifical S.E. MARCELLO BOLDRINI, Professeur de
Statistique à l’Université de Rome, et l’organisation générale au
chancelier de l’Académie Pontificale des Sciences Prof. PIETRO SAL-"UCC.
végétale de l’Université de Madrid, Secrétaire Général du Conseil Supérieur
des Recherches Scientifiques d’Espagne; y ont participé personnellement
[9 savants tandis qu’un autre a envoyé un mémoire. Les comptes-rendus
de la « Semaine d’Etude » ont été publiés dans le 14ème volume des
« Scripta Varia » de l’Académie; ils forment un volume de 630 pages.
La quatrième « Semaine d’Etude » a eu lieu de 20 au 28 mai 1957;
2lle a été dédiée au « Probleme des Populations stellaires » et a été présidée
par 1'Académicien Pontifical Surnuméraire le Rév.me Père DANIEL J. K.
D'CONNELL, Directeur de la « Specola Vaticana » de Castelgandolfo; y ont
participé personnellement 21 savants. Les comptes-rendus de la « Semaine
d'Etude » ont été publiés dans le 16ème volume des « Scripta Varia » de
l'Académie; ils forment un volume de 615 pages.
La cinquième « Semaine d’Etude » a eu lieu du 23 au 31 octobre 1961;
lle a été dédiée au « Problème des macromolécules d'intérêt biologique avec
référence spéciale aux nucléoprotéides », et a été présidée par l’Académicien
Pontifical S.E. ARNE TrseLIUS, Professeur de Biochimie à l’Université de
Uppsala; y ont participé personnellement 28 savants. Les comptes-rendus de
à « Semaine d'Etude » ont été publiés dans le 22ème volume des « Scripta
Varia » de l’Académie; ils forment un volume de 514 pages.
La sixième « Semaine d’Etude » a eu lieu du 1 au 6 octobre 1962; elle
L été dediée an « Prohlème du ravonnement cosmique dans l’espace inter-
XV]
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2€
Ont été invités a la réunion les savants suivants:
Prof. Dr. MAURICE ALLAIS, Professeur d’Economie générale a
l’Ecole Nationale Supérieure des Mines et d’Economie théorique à
l’Institut de Statistique à l’Université de Paris - Paris (France).
S.E. Prof. MARCELLO BOLDRINI, Académicien Pontifical, Professeur
de Statistique à l’Université de Rome - Rome (Italie).
Prof. Dr. ROBERT DORFMAN, Professeur d’Economie politique à
la Harvard University - Cambridge, Mass. (U.S.A.).
Prof. Dr. FRANKLIN M. FISHER, Professeur associé d’Economie
politique au Massachusetts Institute of Technology - Cambridge,
Mass. (U.S.A.).
Prof. Dr. RagNAR FriscH, Professeur d'Economie a 1'Université
de Oslo et Directeur de l’Institut des Recherches économiques à la
même Université - Osio (Norvège).
Prof. Dr. TryovE HAAVELMO, Professeur d’Economie politique à
l’Université de Oslo - Oslo (Norvège).
Prof, Dr. WALTER Isarp, Professeur d'Economie politique a
l’Université de Pennsylvanie et Président honoraire de I’ Association
de Science Régionale - Philadelphia, Penn. (U.S.A.).
Prof. Dr. D. GALE JoHNSON, Professeur d’Economie politique
et Doyen de la Division des Sciences sociales à l’Université de
Chicago - Chicago, Ill. (U.S.A.).
planétaire », et devait être présidée par l’Académicien Pontifical S.E. VIcToR
Francis Hess, Professeur émérite de Physique à l’Université Fordham de
New York; malheureusement l’Académicien VIcTor FRANCIS HEss n’a pas
pu, en raison de son état de santé, être présent et la Semaine d’Etude a
été présidée par S.E. l’Académicien GEORGES LEMAÎTRE, Professeur de Mécanique
et de Méthodologie mathématique à l’Université de Louvain et
Président de l’Académie. V ont participé personnellement 24 savants, Les
comptes-rendus de la « Semaine d’Etude » ont été publiés dans le 25ème
volume des « Scripta Varia » de l’Académie; ils forment un volume de
574 pages.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XVII
Prof. Dr. TjaLLiNG CHARLES KooPMANS, Professeur d’Economie
politique à la Yale University et Directeur de la Cowles Foundation
‘or Research in Economics - New Haven, Conn. (U.S.A.}
Prof. Dr. WassiLy W. LeoNTIEF, Professeur d'Economie politique
à la Harvard University et Directeur du Harvard Economic
Research Project - Cambridge, Mass. (U.S.A).
Prof. Dr. PRASANTA CHANDRA MAHALANOBIS, Directeur honoraire
de l’Institut Indien de Statistique, Conseiller honoraire pour la Staistique
du Gouvernment Indien, Membre de la Commission de Staistique
des Nations Unies - New Delhi (Inde).
Prof. Dr. Epmonp MaALINVAUD, Directeur de l’Ecole Nationale
de la Statistique et de l’Administration Economique - Paris (France).
Prof. Dr. MrcHIo MoRISHIMA, Professeur d’Economie politique
à l’Université d’Osaka - Kobe (Japon).
Prof. Dr. Luict PaSINETTI, Professeur d'Economie politique,
Membre et chargé de cours d'Economie politiaue au King’s College
Cambridge (Grande-Bretagne).
Prof. Dr. FRICH SCHNEIDER, Professeur d’Economie politique et
Directeur de l’Institut für Weltwirtschaft à l’Université de Kiel
Kiel (Allemagne).
Prof. Dr. JouN RICHARD NICHOLSON STONE, Professeur de Finance
et de Comptabilité à l’Université de Cambridge, Membre du
King’s College - Cambridge (Grande-Bretagne).
Prof. Dr. HENRY THEIL, Professeur d’Econométrie et Directeur
de l’Institut d’Econométrie à l’Ecole néerlandaise d’Economie
Rotterdam (Pavs-Bas).
Prof. Dr. JAN TINBERGEN, Professeur d’Economie à l’Ecole néer
landaise d'Economie de Rotterdam - Rotterdam (Pavs-Bas)
Prof. Dr. HERMAN O. A. Worp, Professeur de Statistique à l’Université
d’'Upn: le, Membre de l’Académie des Sciences de Suède
Tbbsala ©
XVIII PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - Z'
Tous les invités, à l’exception du Prof. JAN TINBERGEN, qui n’a
pas pu intervenir, ont participé à la Réunion.
Le Président de la « Semaine d’Etude » S.E. l’Académicien Pontifical
MARCELLO BOLDRINI à appelé à faire partie du Secrétariat
scientifique le Prof. ANTONINO GIANNONE, le Prof. GIANCARLO Maz-ZOCCHI,
le Dr. PIERO GIARDA, le Dr, PIERCARLO NiCoLA, le Dr. Gra-COMO
VACIAGO.
Le « Règlement des Semaines d’Etude » prescrivant que le
nombre des Participants doit être rigoureusement limité, a malheureusement
empêché d’inviter d’autres illustres savants.
Ont aussi participé à la réunion: en qualité d’interpréte et
chef de Secrétariat Mme VALENTINE PRÉOBRAJENSKI; en qualité de
sténographes polyglottes de séance Mlles Maura Barocco et
PAMELA SUTTON; en qualité de sténo-dactylographes polyglottes
chargées des Procès-verbaux: Mlle VALERIA CRAJA, Mlle JOSEPHINE
Lucas et Mme PAULETTE ROSSALDI; en qualité de technicien pour
l’enregistrement et la projection, Mr MAURO ERCOLE, assisté par des
opérateurs de Radio-Vatican. Le Bureau de Presse était confié
au Dr. FRANCESCO SALVIUCCI, Coadjuteur du Chancelier de l’Académie.
Le Comité de Réception pour les Dames dirigé par Mme HELENE
Lorri, était composé de la Comtesse KARINA CALVI DI COENZO,
la Comtesse ISABELLA CALVI DI COENZO et Mlle MARIA Luisa LorTTI.
Le dimanche 13 octobre tous les Participants ont été reçus en
Audience Solennelle par le Souverain Pontife qui leur adressa un
discours et après l’Audience a eu lieu, au Siège de l’Académie Pon-
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XIX
‘ificale des Sciences, une séance extraordinaire de l’Académie, à
aquelle ont été invités également les Participants à la « Semaine
l’Etude ».
Tous les participants à la Semaine d'Etude ont reçu toutes les
communications à l'avance. Durant la Semaine d’Etude chaque
rommunication a été discutée tant par un groupe de spécialistes (!)
qu’en une session plénière, exception faite pour deux communicaions
(STONE et FRISCH) qui onté tét directement discutées en session
plénière. Les communications ont été mises ici dans une succession
aussi rapprochée que possible à leur ordre de présentation et de
discussion.
Les séances se tenaient deux fois par jour, le matin de g h. --à
12 1. 30 et l’après-midi de 16 h. à T0
La réussite de la « Semaine d’Etude » a pleinement satisfait les
llustres Participants qui, à la fin de leurs travaux, ont tenu à
exprimer au Saint Père leur profonde gratitude et leur très sincère
admiration pour cette manifestation scientifique si réussie, en envoyant
à l’Auguste Pontife, animateur et mécène de l’Académie.
le télégramme suivant
« Sa Sainteté le Souverain Pontife Paul VI, Cité du Vatican. —
Les participants de la Semaine d'étude sur le rôle de l’analyse économetrique
dans la formulation des plans de developpement et l'étude
les fluctuations économiques prient Sa Sainteté de daigner accepter
"expression de leur respect et de leur gratitude. Grâce au milieu
‘déal offert bar son Académie des Sciences, ils ont eu la bossibilité
f1) t# groupe: STONE (Président), DorFMAN, JOHNSON, KooPMANS, MAHA--ANOBIS,
MALINVAUD, MORISHIMA, PASINETTI, SCHNEIDER. 2° groupe: LEonrirr
(Président). ArLAIs Fisurr Friscen Haavrrmo Tearn Turin Waorn
a
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
de discuter dans une atmosphère de sérénité et d’indépendance
intellectuelle des problèmes de grande actualité. Ils espèrent que
leurs travaux ajouteront, serait-ce dans une mesure modeste, au progrès
de la Science et à l'amélioration de la condition humaine. — AL-LAIS,
BOLDRINI, DORFMAN, FISHER, FRrISCH, HAAVELMO, ISARD,
JOHNSON, Ko0OPMANS, LEONTIEF, MAHALANOBIS, MALINVAUD, MORI-SHIMA,
PASINETTI, SCHNEIDER, STONE, THEIL, WOLD ».
A ce télégramme d’hommage et de remerciement, le Saint-Père
a daigné répondre par le message suivant, signé par Son Eminence
le Cardinal Secrétaire d’Etat:
« Professeur Salviucci, Chancelier Académie Pontificale des Sciences
- Casina di Pio IV - Citta del Vaticano. — Sa Sainteté trés touchée
délicat Message participants Semaine étude sur rôle analyse économétrique
dans formulations plans développements se réjouit contribution
illustres savants progrès de la Science et amélioration conditton
humaine renouvelle sentiments paternelle bienveillance dans
récent discours invoque sur continuation recherches distingués Académiciens
abondantes faveurs divines, — Card. CICOGNANI ».
Dans les pages qui suivent, après le compte-rendu de l’Audience
du Saint-Père et le « Règlement des Semaines d’Etude », sont imprimés
les rapports originaux présentés à la Réunion, et les discussions
qui les ont suivis, mis en ordre et publiés par les soins du Dr. PIERO
GIARDA et du Dr. Giacomo VACIAGO.
Les « Conclusions » de la « Semaine d’Etude » se trouvent a
la fin du présent volume
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
XXI
Pendant la Semaine, les Participants ont visité les Musées Pio-Clementino,
Chiaramonti, Etrusque, Egyptien; le Braccio Nuovo;
es Galeries des tapisseries, des cartes géographiques; les Chambres
st les Loggia de Raphaël, la Chapelle de Fra Angelico, la Chapelle
Sixtine, l’Appartement Borgia et la Pinacothèque Vaticane avec
assistance du Prof. Comm. FirIPPo MaG1 et du Dr. DEOCLECIO
REDIG DE CAMPOS de la Direction Générale des Monuments de la
Cité du Vatican.
Les Participants ont aussi visité la Bibliothéque Apostolique Va-:icane
et les Archives Secrétes Vaticanes sous la conduite des
Rév.mes Préfets Mons. MARTINO GIUSTI et Père ALPHONSE Races, S.I.
et la Station de Radio-Vatican qui leur fut présentée par le Directeur,
e Rév.me Père ANTONIO STEFANIZZI
Enfin, le soir du même samedi 12 octobre, un dîner d’adieu a é
offert par l’Académie, selon \a coutume. aux savants n° ‘‘cisent
la « Semaine d’Etudes
1
42
. AUDIENCE
-
DISCOURS DU SAINT-PERE
Le matin de dimanche 13 octobre, le Saint-Père a accordé dans
la Salle du Consistoire du Palais Apostolique Vatican, une Audience
Solennelle à l’Académie Pontificale des Sciences à l’occasion de la
« Semaine d'Etude » sur « Le rôle de l’analyse économétrique dans
la formulation de plans de développement » tenue par l’Académie
même. Ont participé aussi à l’Audience de nombreux hauts personnages.
Etaient présents Leurs Eminences les Cardinaux: EUGÈNE TISSE-RANT,
Président honoraire de l’Académie; AMLETO GIOVANNI C1CO-GNANI,
Secrétaire d’Etat, GIUSEPPE PIZZARDO et ANSELMO ALBAREDA,
Académiciens honoraires; GIACOMo LUIGI CoPELLO, JOSEPH FRINGS,
PAUL EMILE LÉGER, FRANZISKUS KôNIG, JosEPH LEFEBRE. BERNARD
JAN ALFRINK, EFREM FORNI.
De nombreux Académiciens Pontificaux sont intervenus, et spéialement
Leurs Excellences: le Rév.me Monseigneur GEorGES LE-MAÎTRE
Président, José MARIA ALBAREDA-HERRERA, MARCELLO Boz-ORINI,
GIOVAMBATTISTA BONINO, HERMANN ALEXANDER BRUCK, CAR-L0S
CHAGAS FILHO, GUSTAVO COLONNETTI, EDWARD Josep Conway,
EDUARDO CRUZ-COKE, GEORGE CHARLES DE HEVESY, JoHN CAREW
ECCLES, JOSÉ GARCIA-SINERIZ, ALESSANDRO GHIGI, GIORDANO GIACO-MELLO,
WALTER RUDOLF HEss, CORNEILLE HEYMANS, CYRIL NORMAN
FMINSELWOOD, BERNARDO ALBERTO Houssay, ALBERTO HURTADO,
LoUIS LEPRINCE-RINGUET, DOMENICO MAROTTA, SAN-IcHIRO PAULO
MIZUSHIMA, ANTONIO PENSA, ENRICO PISTOLESI, MANUEL SANDOVAL-VALLARTA,
GEORGE SPERI-SPERTI. HIDEKY YUKawa: les Académi-
XXVI PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ©
ciens Pontificaux Surnuméraires: Rév. P. DANIEL JosEPH Kerry
O’CONNELL S.I. et Mgr. MarTINO GIUsTI; le Chancelier de 1’Académie
Prof. Dr. PIETRO SALviucct et le Coadjuteur du Chancelier
Dr. FRANCESCO SALVIUCCI.
Parmi le groupe des Académiciens assistaient les savants spécialistes
« Participants » à la Semaine d’Etude sur « Le rôle de l’analyse
économétrique dans la formulation de plans de développement »
MM. les Professeurs: MAURICE ALLAIS, ROBERT DORFMANN,
FRANKLIN M. FISHER, RAGNAR FRISCH, TRYGVE HAAVELMO, WALTER
IsarD, D. GALE JOHNSON, TJALLING CHARLES KooPMANS, WassILY
W. LEONTIEF, PRASANTA CHANDRA MAHALANOBIS, EDMOND MALINvAUD,
MicH10 MORISHIMA, LUIGI PASINETTI, ERICH SCHNEIDER, JOHN
RICHARD NICHOLSON STONE, HENRY THEIL, HERMAN O.A. WoLD et
les Secrétaires Scientifiques: Prof. Dr. ANTONINO GIANNONE, Prof.
Dr. GIANCARLO MAZzoccH1, Dr. PIERO GIARDA, Dr. GIACOMO VACIAGO.
Etaient également présents: Son Excellence Rév.me Monseigneur
ANTONIo SAMORÉ Secrétaire de la Sacré Congrégation des Affaires
Ecclésiastiques Extraordinaires, Son Excellence Rév.me Monseigneur
ANGELO DELL’ACQUA Substitut de la Secrétairerie d’Etat; Son Excellence
Rév.me Monseigneur CARLO GRANO Nonce Apostolique en
Italie; un groupe d’Assesseurs et de Secrétaires des Sacrées Congrégations,
ainsi qu’un groupe d’Archevêques et d’Evêques, parmi lesquels
LL.EE. Rév.mes Nosseigneurs DiEco VENINI, PIETRO CANISIO
VAN LIERDE, PIETRO SIGISMONDI, PRIMO PRINCIPI, et autres personnalités
de la Curie et de I’Etat de la Cité du Vatican.
Au complet le Corps Diplomatique accrédité près le Saint-Siège,
dont les Membres furent reçus par le Gr. Uff. Dr. MArI0 BELARDO.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XXVII
Le Saint-Père a fait son entrée dans la Salle du Consistoire à
ro heures, accompagné par sa Noble Antichambre avec LL.EE. Nosseigneurs
FEDERICO CALLORI DI VIGNALE Majordome et Mario Na-SALLI
Rocca pr CorNELIANO Maitre de Chambre, par son Secrétaire
Particulier Monseigneur PASQUALE MACCHI, ses Camériers Secrets
Participants et sa Garde Noble.
Une déférente manifestation d’hommage a accueilli l’arrivée du
Saint-Pere.
Après avoir gagné le trône, le Saint-Père donna son assentiment
au Président LEMAÎTRE qui s’adressa alors au Souverain Pontife en
res termes:
« Très Saint-Père, il y a un an, Sa Sainteté le Pape Jean XXIII
Jaignait recevoir les Membres de I’Académie Pontificale des Sciences
réunis en Séance Plénière, ainsi que les participants à la Semaine
l'Etude sur ’’Les Rayons Cosmiques dans l’espace interplanétaire’’.
« Nous ne nous doutions pas que c’était la dernière fois que nous
pouvions recevoir ses paternels encouragements pour nos travaux et
ui exprimer notre profonde vénération.
« Je ne pouvais pas ne pas évoquer avec émotion ce souvenir au
moment où son vénéré Successeur nous accueille à son tour avec la
même paternelle bonté à la clôture de nos travaux de la Session
Plénière et de la Semaine d'Etude sur “Le rôle de l’analyse économétrique
dans la formulation des plans de développement”.
«Il nous est impossible d’exprimer adéquatement tout ce aue
nous devons déjà à Sa Sainteté le Pape Pau. VI.
« Nous savons qu’à côté de ses illustres prédécesseurs, il a eu
une part des plus actives dans l'érection et le développement de
Académie Pontificale des Sciences et nous connaissons la profonde
XXVIII
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
et affectueuse amitié qu’il témoignait pour son regretté Président, le
Père AGOSTINO GEMELLI.
«« Nous ne pouvons nous empêcher de penser que notre Académie,
par sa large ouverture à toutes les formes de la Science et à toutes
les personnalités du monde scientifique sans égard pour leur appartenance
philosophique ou religieuse, a pu faire présager et inaugurer
dans un domaine particulier ce large souffle de liberté et de respect
de toutes les valeurs humaines qui anime en ce moment l'Eglise et
qui, sous la direction éclairée de ses vénérés Pontifes, lui fait aborder
un des plus grands tournants de son histoire.
« Je prie Sa Sainteté de daigner agréer le respectueux hommage
de son Académie des Sciences ».
Le Saint-Père daigne répondre par le discours que nous reproduisons
plus loin.
A la fin de l’Audience le Souverain Pontife daigna remettre
au Prof. Dr. AAGE Bour, Professeur de Physique theorique à l’Université
de Copenhague, la grande médaille d’or qui porte le nom
auguste de Pie XI, Fondateur de l’Académie Pontificale, et adressait
au Prof. BoHR, qui était accompagné du Président et du Chancelier
de l’Académie des paroles de satisfaction et des félicitations.
Le Saint-Père s’entretint ensuite, après avoir reçu l'hommage des
Cardinaux, avec le Président LEMAÎTRE, les Académiciens Pontificaux,
le Chancelier SALvrucez et les savants « Participants » à la
« Semaine d’Etude », trouvant pour chacun d’aimables paroles de
félicitations et de souhaits, pour eux, leurs familles et leur activité
scientifique.
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. XL.
4
L’assistance exprima enfin ses remerciements au Saint-Père, sa
reconnaissance émue et sa profonde gratitude, et le plus chaleureux
rommage se manifesta de nouveau au moment où, l’Audience terminée,
le Souverain Pontife quitta la Salle du Consistoire
lessieur
Nous n'avons pas l'intention de vous faire un discours. Ce n’est
bas que Nous n’aurions bien des choses à vous dire: cette rencontre
avec l’Académie pontificale des Sciences éveille en effet dans Notre
âme toutes sortes de thèmes, de questions, de sentiments, qui mériteraient
que Nous leur donnions expression. Mais ce n’est pas le
moment. En ces jours, absorbés par le Concile et par les problèmes
qu'il soulève, le temps Nous manque. Ce ne sera donc qu’une brève
salutation que Nous vous adressons, salutation pleine de cordialité
bour les Personnes que Nous avons I- orand honneur de rencont-Tentlemen.
We do not propose to deliver a discourse. Not that we should
10t have plenty to say to you; this meeting with the Pontifical Academy
of Sciences in fact calls to mind all sorts of topics, questions,
feelings, which it would be worth while to express, but this is not
‘he moment. In these days, absorbed as they are by the Council
and the problems to which it gives rise, We have no time to spare.
This will be merely a brief greeting that we address to you, a greeting
full of cordiality for the persons that We have the great honour
of meeting, full of respect for this institution that We are happy
to see here once again.
As you have just said, Mr. President, an esteem of long standing
and a sincere friendship bind Us to vour Academv. We are ¢lad
XXXII PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
pleine de déférence pour Uinstitution que Nous avons I’ heureuse occasion
de revoir.
Comme vous venez de le dire, Monsieur le Président, une estime
datant de loin et une sincère amitié Nous lient à votre Académie.
Il Nous est agréable de refaire aujourd’hui connaissance avec elle
et de saluer d’abord en votre personne, Monsieur le Président, le
digne successeur du regretté et inoubliable Père Gemelli.
C’est une joie pour Nous de retrouver l’Académie, dans la plémtude
de ses effectifs, appliquée à poursuivre fidèlement ses activités
traditionnelles.
Et à ce propos, Nous Nous faisons un devoir de confirmer aux
anciens académiciens Nos sentiments dévoués, et de souhaiter une
joyeuse bienvenue à ceux des nouveaux que Nous n'avons pas eu
encore le plaisir de saluer comme membres de cette illustre société.
Nous voulons aussi exprimer Notre reconnaissance aux personnalités
qui ont accueilli l'invitation de Notre Académie et sont venues
prendre part à cette semaine d'études, y apportant la précieuse conto
be able to-day to renew acquainstance with it and to greet, first
of all, in you, Mr. President, the worthy successor of the lamented
and unforgettable Padre Gemelli.
It is for Us a joy to find the Academy, and all its members,
dedicated to the faithful carrying out of its traditional activities.
We take this opportunity to express to the veteran Academicians
our devoted esteem and to bid a happy welcome to those whon We
have not previously had the pleasure of greeting as members of this
illustrious society.
We wish also to express Our gratitude to those scientists who
have accepted the invitation of Our Academy and who have come
to take part in this semaine d’études, bringing to it the valuable
contribution of their learned research and honouring it with their
presence.
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. XXXIII
fribution de leurs savants travaux et la flatteuse adhésion de leur
brésence.
Nous voulons ainsi confirmer à ceux qui appartiennent a I’Académie
pontificale des Sciencse et a ceux qui participent a ses travaux
ou l'honorent de leur sympathie Notre haute estime pour cette insti-‘ution
et, en conséquence, la résolution qui Nous anime de lui accorder
l'appui et l'honneur capables d'assurer sa stabilité et de favoriser
son développement.
Elle est solennelle, à Nos yeux, la responsabilité qui Nous vient
du Pape fondateur de votre Académie; profonde, l'estime que Nous
nourrissons pour ceux qui en sont les membres et les promoteurs;
aiguë est en Nous la conscience de l'importance et des besoins de
la haute culture scientifique de notre temps; vivant et agissant dans
notre âme, le sentiment du devoir, de l'intérêt, et, dans un certain
sens, de la nécessité, pour l'Eglise catholique, d'entretenir les rapborts
les plus sincères avec le monde scientifique contemporain. Disons
enfin que Nous Nous sentons stimulé par la certitude aue not
To those who belong to the Pontifical Academy of Sciences, and
to those who participate in its work or honour it with their friendly
Interest, we wish to reaffirm our high esteem for this institution, and
che resolution we have taken to grant it the support and honour
which will ensure its stability and favour its development.
We have inherited a solemn responsibility from the Pope who
rounded your Academy, for whose members and promoters We
cherish a profound esteem, We have a keen appreciation of the
importance and the needs of modern science and a lively sense of
che duty, the interest, and in a way the necessity, for the Catholic
Church to maintain the most sincere relations with the contemporary
scientific world, Finally We may say that We feel ourselves stimulited
by the certainty that our religion not only does not oppose
any real objection to the studv of natural truths. but that without
XXXIV PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -~
religion, non seulement wn’ oppose aucune objection réelle à l’étude
des vérités naturelles, mais qu’elle peut, sans sortir des limites de
sa propre sphère, m franchir celles du domaine de la science proprement
dite, aider la recherche scientifique, honorer ses résultats, favoviser
leur meilleure utilisation pour le bien de l’humanité.
La religion que nous avons le bonheur de professer est, en effet,
la science supréme de la vie: elle est donc la plus haute et la plus
bienfaisante maitresse dans tous les domaines où la vie se manifeste.
Elle pourra sembler absente quand non seulement elle permet, mais
ordonne au savant de n’obéir qu’aux lois de la vérité; mais à y regarder
de près, elle sera encore près de lui pour l’encourager dans
sa difficile exploration, en lui assurant que la vérité existe, qu’elle
est intelligible, qu’elle est magnifique, qu’elle est divine; et pour
lui rappeler, à chaque pas, que la pensée est un instrument apte à
la conquête de la vérité et qu’il faut l’utiliser avec un tel respect
pour ses propres lois que l’on sente continuellement la référence à
une responsabilité qui l’engage et la transcende.
crossing the bounds of its proper sphere or transgressing those of
the domain of science properly so-called, it can promote scientific
research, honour its results and help them to be better used for the
good of humanity.
The religion which we have the happiness to profess is, in fact,
the supreme science of life. It is thus the highest and most beneficent
mentor in all those domains where life is manifested. It might seem
to be absent when it not merely permits, but directs, the scientist
to obey only the laws of truth. When, however, it is looked at more
closely, it will be seen to be still beside him, to encourage him in
his difficult task of exploration, assuring him that truth exists, that
it is intelligible, splendid, divine; and also to remind him at every
step that thought is an instrument for the conquest of truth and that
it should be used with such respect for its own laws that one feels
continually the transcendent responsibility that it imposes.
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. XXXV
C’est vous dire, Messieurs, avec quel sérieux et avec quelle faveur
Nous considérons cette institution dans laquelle Nous Nous
blaisons à voir une représentation du monde scientifique, auquel
Nous envoyons, à cette occasion, et par le moyen des interprètes
autorisés que vous êtes, Notre salut respecteux et Nos encourage
ments.
Ce salut peut être symbolisé par la médaille d’or « Pie ~
que Nous avons le plaisir de remettre au professeur Aage B3i
fils d’une nation dont Nous apprécions les insignes mérites, lu Du
nemark —, savant célèbre pour ses études sur la structure nucléaire
et sur l'analyse théorique des mouvements des noyaux atomique: .
Que la remise de cette récompense soit une marque d’ admiration ei
d'encouragement, tant pour la digne personne de ce jeune professeur,
que pour la noble phalange, devenue aujorud’hui une véri-‘able
armée, des savants engagés dans la moderne et merveilleuse
exploration du microcosme physique
Que. venant d “V-< maire sac. “otal
This will show vou, Gentlemen, how seriously and with what
favour We regard this institution, which We like to consider as
representative of the scientific world, to which We send through you,
‘ts authoritative interpreters. Our respectiful greetings and encouragement.
A symbol of this greeting is the Pius XI Gold Medal which We
have the pleasure of presenting to Professor Aage Bohr, son of Denmark,
a nation whose signal merits are appreciated by Us, a scientist
celebrated for these studies of nuclear structure and for the theoretical
analysis of the motions of atomic nuclei. May the granting of this
award be a token of respect and encouragement, both for the worthy
person of this young professor as well as for the noble company,
nowadays become a whole army, of scientists devoted to the exnloraion
of the marvels of the physical microcosm.
Coming from Our priestlv hands mav this award constitute
XXXVI PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
leureuse invitation, un appel évangélique à tous les responsables:
qu'ils ne fassent jamais de la science, ou plutôt de ses multiples
applications pratiques — en particulier de la science nucléaire et
de ses formidables emplois possibles —, un péril, un cauchemar, un
instrument de destruction pour la vie humaine. Déjà un autre de
Nos sages prédécesseurs, Pie XII, dès 1943, et encore en 1048,
mettait en garde, devant cette même Académie, contre la terrible et
menaçante possibilté que l'énergie atomique pût devenir fatale pour
l'humanité. Et récemment encore, le Pape Jean XXIII, d’heureuse
mémoire, dans son Encyclique « Pacem in terris », désormais célebre,
formait le voeu de la prohibition des armes atomiques.
Nous voulons faire Nôtre leur cri paternel et, avec tous les hommes
pleins de bonté et de sagesse qui sont dans le monde, souhaiter
que soit conjurée une telle menace au salut et à la paix de l’humanité.
Dans votre pacifique assemblée, grâce à Dieu, vous êtes loin
de ces perspectives si ténébreuses. Vous y parlez du « rôle de l’anawarm
invitation, an evangelical appeal, to all those in authority —
that they may never make science, or rather its multiple practical
applications, in particular those of nuclear science and its terrible
possibilities — that they may never make it a peril, a nightmare, an
instrument of destruction for human life. Another of Our wise predecessors,
Pius XII, already in 1943 and again in 1948, addressing
this same Academy, warned against the terrible and menacing possibility
that atomic energy might become tatal for humanity. And,
still more recently, Pope John XXIII, of happy memory, in his now
famous Encyclical « Pacem in Terris », expressed the wish that
atomic weapons be banned.
We wish to make Our own their fatherly appeal and to hope,
with all good and wise men everywhere in the world, that this threat
to the safety and peace of humanity may be averted.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XXXVI]
lyse économétrique dans la formulation des plans de développement».
C’est là le thème de votre semaine d’études, un thème
qui tend à rassembler les résultats modernes d’une branche scien-‘ifique
nouvelle, l’économétrie, et à les présenter à la politique économique,
pour l'aider à formuler ces plans de sécurité mieux assurée
et de plus grand développement qui peuvent tant apporter dour le
hien-étre et la paix des peuples.
Nous ne voulons pas aborder ce thème ni y ajouter de commenfaires.
Mais Nous sommes heureux que des personnes si éminente:
soient venues l’exposer devant cette Académie, et Nous les remerions
de cette haute contribution qu’ils apportent ainsi au progrès
de la science et à la bonne renommée de cette même Académie. Nous
tenons à vous exprimer Nos félicitations pour le choix, la manière
de traiter et les buts d’un thème aussi riche pour la recherche scienifique
que fécond en applications pratiques. Nous sommes sûr aussi
que ces vd: d’économétrie intégrées aux autres connaissanc.
In your peaceful assembly you are, thank God, far removed
from these sombre prospects. You will be speaking of « The
>conometric Approach to Development Planning ». This is the theme
of your study week, a theme which seeks to gather together the
atest results of a new branch of science, econometry, and to present
them to political economists in order to aid them in formulating
those plans for a more stable security and for greater development
which can contribute so much to the well-heing and peace of nations.
We do not intend to enter upon this theme or to comment on
t, but We are happy that such eminent men have come to treat of
t before this Academy, and We thank them for this important conribution
which they are making to the advance of science and to
he reputation of this Academv. We are happy to congratulate vou
XXXVIII PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
des phénomenes humains, y compris dans le domaine économique,
seront vraiment de grande utilité au progrès ordonné de la civilisahon
humaine.
Et, en vous saluant paternellement, Nous implorons sur vos personnes
et sur vos travaux la protection de Dieu, en vous donnant
à tous Notre Bénédiction Apostolique.
on the choice, the method of treatment and the aim of a theme as
fruitful for scientific research as it is rich in practical applications.
We are sure also that these econometric studies, integrated with
the rest of our knowledge of human phenomena, including those
in the field of economics, will truly prove of great utility in the
ordered progress of human civilisation.
We give you a fatherly greeting and beg the divine protection
for you and for your labours, bestowing on vou all Our Apostolic
Blessing.
5
SEMA.
LV
sw
_—
[TUDE
_
~
a. à
EUR REGLEMENT
Lorsque l’Académie Pontificale des Sciences fut fondée par le
Souverain Pontife Pie XI, de vénérée mémoire, par son « Motu
Proprio » du 28 octobre 1936 « In multis solaciis », cette initiative
suscita dans les milieux scientifiques un mouvement général de
sympathie et d’admiration. Cette institution unique au monde, qui
groupait en une même assemblée des représentants de toutes les
nations civilisées était appelée, en effet, à de hautes destinées dans
le développement de la pensée scientifique.
D'autre part, cette oeuvre de coopération fut accueillie avec un
véritable soulagement par tous ceux que plongeait dans le désarroi
le plus profond la période qui suivit la guerre 1914-18. On voyait,
en effet, s’altérer profondément les caractères d’objectivité et de
désintéressement propres au travail scientifique, et s'affirmer même
1ne tendance à asservir la science à des fins pragmatiques.
Tout au contraire, dans l’immortel « Motu Proprio » du 28 octopre
1026 J= Pane Pie XI Droclamait solennellement la dignité de
A general movement of sympathy and admiration was aroused in
scientific circles when, in 1936, the Pontifical Academy of Science was
founded by His Holiness Pope Pius XI, of venerable memory, by means
Of his « Motu Proprio » of October 28, « In Multis solaciis ». This institution,
the only one of its kind in the world, which brought the representatives
of all civilized nations into touch with each other, was, in fact
"alled upon to play a leading role in the development of scientific thought.
This work of cooperation was, moreover, welcomed with a sense of
real relief by all those who were plunged in a deen state of confusion in
‘he period following the 1914-18 war.
Signs of drastic changes were, in fact, discernible in the objective and
disinterested nature of scientific work and even a tendencv to make science
subiect to pragmatic aims
XLII
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
la recherche de la vérité pour elle-méme (*), et, élevant sa pensée
au-dessus de toute préoccupation utilitaire, affirmait qu’il ne demandait
rien d'autre aux nouveaux « Académiciens Pontificaux » que de
se consacrer, avec une ferveur toujours plus grande, au progrès de
la science et, par là, au culte de la vérité: « C’est Notre souhait
ardent et Notre ferme espérance que par cet Institut, à la fois
Nôtre et leur, les “Académiciens Pontificaux” contribuent toujours
plus et mieux au progrès des sciences, Nous ne leur demandons pas
autre chose; car en ce dessein généreux et ce noble labeur consiste
le service, qu’en faveur de la vérité, Nous attendons de leur part ».
La consécration pratique de cette idée, par la nomination d’un
certain nombre de non-catholiques parmi les nouveaux Académiciens
Pontificaux, a fait une profonde impression sur beaucoup d’esprits,
comme l'ont montré les réactions de la presse internationale de
l’époque et de nombreux témoignages individuels d’hommes de
science et des plus grands savants du monde.
(*) « Nobis autem in votis expectationeque est, fore ut “Pontificii Academici”
vel per hoc Nostrum suumque studiorum Institutum, ad scientiarum
progressionem fovendam amplius excelsiusque procedant; ac nihil praeterea
aliud petimus, quandoquidem hoc eximio praeclaroque labore famulatus ille
nititur servientium veritati, quem ab iisdem postulamus ».
In his immortal « Motu Proprio » of October 28, 1936, Pope Pius XI,
on the contrary, solemnly proclaimed the dignity of the search for truth
for its own sake (*) and, raising his thoughts above all preoccupations of
an utilization nature, asserted that all he asked of the new « Pontifical
Academy » and its members was that they should dedicate themselves,
with increasing fervour, to the furthering of the progress of science and,
consequently, to the cult of truth: « It is Our ardent wish and firm hope
that, by means of this Institute, which is both Ours and theirs, the
‘’Pontifical Academicians’’ will contribute to an increasingly great extent
to the progress of science. We ask nothing more than that from them
because the service in favour of truth that We expect from them consists
in this generous intention and noble work ».
By including a certain number of non-Catholics amongst the new
Pontifical Academicians, the practical application of this idea made a deep
impression on many persons, as is proved by the reaction of the interna-
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. XLII
Beaucoup de préjugés a l’égard de 1’Eglise ont été fortement
ébranlés par ce geste du Souverain Pontife qui a obligé à reconnaître
la place éminente réservée aux valeurs purement intellectuelles dans
l’Eglise Catholique.
Pour toutes ces raisons, la fondation de l’Académie Pontificale des
sciences a été hautement appréciée dans le monde scientifique et y
a fait naître de grands espoirs quant aux possibilités d’action d’une
institution si opportune.
Le Saint-Père Pie XII, qui avait collaboré avec son Prédécesseur
au projet et à la fondation de l’Académie et qui l’avait représenté
:omme Légat personnel lors de l'inauguration solennelle, ne s’est
pas borné à maintenir à son égard ses sentiments de haute estime
par sa présence à de solennelles séances académiques, où il daigna
prononcer ses discours d’une haute portée scientifique; il a tenu, en
outre, à lui donner un nouveau témoignage de son auguste satisfaction
en accordant à ses membres le titre d’Excellence par le Bref Apostolique
du 25 novembre 104€
Les sciences posent chaque jour des problèmes nouveaux qui
lonnent lieu d’ordinaire à divers essais de <olntion. souvent contra
tional press of the time and by the innumerable individual tributes paid
bv scientists and by the greatest scholars of the world.
Many prejudices against the Church were very deeply shaken by this
zesture on the part of the Sovereign Pontiff, since it called attention to
‘he lofty place reserved for purely intellectual values in the Catholic Church.
For all these reasons, the foundation of the Pontifical Academy of
~cience was greatly appreciated by the scientific world and aroused high
100es as to the prospects open to such a timely institution.
His Holiness Pope Pius XII, who had helped his predecessor to draw
1p the plan and to found the Academy, and who had represented Him as
tis personal Legate at the time of its solemn inauguration, did not confine
nimself to the expression of lofty sentiments when attending solemn academic
gatherings, where he deigned to make speeches of great scientific importance,
out he also afforded proof of his august satisfaction by granting the title
pf Excellency to the members of the Academy. bv an Apostolic Brief of
November 25, 19.10
XLIV PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
dictoires. Il arrive ainsi constamment que parmi les représentants les
plus autorisés d’une science, et, en particulier, parmi ceux qui se sont
consacrés à l'étude d’une même question, on rencontre des opinions
opposées, Pareilles divergences se maintiennent parfois durant de
longues périodes et constituent à la fois une grave difficulté pour
l’enseignement des sciences et fréquemment aussi un obstacle considérable
à leur développement.
Par ailleurs, l’expérience montre que les méthodes actuellement
pratiquées dans la discussion des problèmes scientifiques n’ont qu’une
efficacité limitée au point de vue de l’établissement d’une unité de
doctrine.
Il serait dès lors hautement souhaitable de promouvoir tout ce
qui pourrait favoriser un accord sur les points en discussion.
Un procédé semble devoir être particulièrement utile sous ce
rapport: à savoir, l’établissement de contacts personnels prolongés
entre quelques représentants d’opinions différentes au sujet d’une
question déterminée.
En effet, le contact personnel entre hommes de science constitue,
sans aucun doute, le moyen le plus efficace de résoudre les controverses
scientifiques.
Dans ce but, l’Académie Pontificale des Sciences a décidé d’organiser
de pareilles rencontres scientifiques. L’organisation de ces ren-Every
day science raises new problems, which usually give rise to
various, and often contradictory, solutions. Consequently it often happens
that amongst the most authoritative representatives of a given branch of
science, and particularly amongst those who are engaged in studying the
same question, one meets with contrasting opinions. Divergences of this
kind often exist over long periods of time and are a serious obstacle not only
to the teaching of science but also to its development.
Experience shows, moreover, that the methods at present in use in
the discussion of scientific problems have only a limited efficacy in so far
as concerns doctrinal unity.
It would, therefore, be highly desirable if everything that could favour
agreement on controversial points were to be promoted.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. XL,
contres qu’on a appelées « Semaines d'Etude » a été établie de la
manière suivante :
RÈGLEMENT DES SEMAINES D'ÉTUDE
I. - L’Académie invite quelques illustres savants, parmi ceux
qui, ayant étudié spécialement une question déterminée, sont arrivés
à des conclusions différentes, à se rencontrer à Rome, à son siège, la
«Casina di Pio IV », à l’intérieur de l’Etat de la Cité du Vatican,
afin d'y procéder en commun, en dehors de toute autre préoccupaion.
à un examen général de toutes les données du problème.
2. - Le but essentiel de ces discussions est de chercher à formuler
de façon précise les raisons qui sont à la base de la divergence des
opinions. Les savants conviés aux réunions s’engageraient d’avance
à concentrer leurs efforts dans cette direction.
3. - Un examen critique de ces raisons aboutira soit à un accord
sur une solution déterminée, soit à la constatation qu’à l’état actuel
des connaissance il est impossitMe d’Atablir une unité de doctrine
iu sujet du problème envis- ‘
One process that would seem to be particularly useful from this point
of view would be the establishment of prolonged personal contacts between
some of the representatives of different trends of thought on a given subject.
Personal contacts amongst scientists are. in fact. the most efficacious
neans of solving scientific controversies.
With this aim in mind, the Pontifical Academy of Science decided to
organize scientific meetings of this description. These meetings. known as
« Studv Weeks ». were nlanned on the following lines :
STANDING RULES FOR « STUDY WEEKS »
I. - The Academy invites a number of illustrious scholars — comprising
those who have especially studied a given question and have arrived at
lifferent conclusions — to meet in Rome at its headquarters, the « Casina
ii Pio IV », situated in the Vatican City, so as to make a joint examination.
iree from all other preoccupations, of all data concerning the problem.
2. - The chief aim of these discussions is to endeavour to formulate
preciselv the reasons which are at the root of the differences of opinion.
XLVI
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Dans ce dernier cas, les savants invités auront pour tâche:
a) de préciser les motifs pour lesquels un accord s’avère pré
sentement irréalisable;
b) de définir le genre de recherches qu’il serait souhaitable
d’entreprendre en vue de résoudre la question.
4. - L’invitation ne sera adressée par l’Académie qu’à un très
petit nombre de représentants de chaque science: ceux-ci seront
choisis parmi les personnalités étrangères à l’Académie, auxquels se
joindront, dans la discussion, les Académiciens versés dans la même
discipline, Cette invitation, de plus, ne se rapportera qu’à l’étude
d’une question déterminée, pour chaque science.
5. - Les discussions auront un caractère strictement privé; elles
prendront la forme de conversations particulières, sans autre assistance
que celle de quelques membres de l’Académie Pontificale des
Sciences particulièrement compétents dans la matière.
Des interprètes polyglottes, des sténographes, des rapporteurs,
etc., seront mis à la disposition des savants réunis.
6. - Les « Conclusions » des discussions seront publiées sous la
The scholars invited to these meetings undertake in advance to concentrate
their efforts on this.
3. À critical examination of these reasons should lead, either to
agreement on a given solution or else to the conclusion that, on the basis
of the information actually available, it is impossible to establish doctrinal
unity on the problem envisaged.
In the latter event the scholars concerned will be called upon:
a) to define the reasons why agreement appears to be impossible
for the present;
b) to specify the kind of research work it would be desirable to
undertake with a view to solving the problem.
4. - The invitation will be addressed by the Academy to only a small
number of representatives of each branch of science: these will be selected
from amongst those who are not connected with the Academy. They will
be joined during the discussions by Academicians versed in the same
discipline. This invitation, moreover, will apply onlv to the study of one
precise problem in each branch of science
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIOUE ETC.
Lai
forme d’une « Note Collective Finale » (à laquelle pourront éventuel
ement être jointes des annotations individuelles), mentionnant :
2) les points sur lesquels un accord aurait été réalisé;
b) les points sur lesquels un accord n’aurait pas paru réalisable;
:) les raisons pour lesquelles l’accord n’aurait pu être réalisé:
d) des suggestions relatives aux recherches paraissant les plus
aptes à résoudre les difficultés.
7. - Les « Conclusions » seront aussitôt imprimées et communiquées,
par les soins de l’Académie Pontificale des Sciences, à tous
es centres scientifiques qu’elles seraient de nature à intéresser
8. - Tous les frais de voyage et de séjour à Rome des personaalités
invitées seront à la charge de l’Académie Pontificale des
sciences. L’hospitalité sera assurée dans l’un des principaux hôtels
de Rome.
L'Académie se fera un plaisir d'offrir la même hospitalité aux
Spouses’ savants invités ‘ v“usion tontefais 7 A; 7
TOV”
5. - The debates will be strictly private and will take the form of
personal talks, in the presence only of a few members of the Pontifical
Academy of Science with special knowledge of the subject under discussion.
Polyglot interpreters, stenographers. reporters. etc. will be placed at
‘he disposal cf the participants.
6. - The « Conclusions » arrived at will be published in the form of
1 « Collective Note » (to which mav eventuallv be added individual notes)
mentioning:
a) the points on which agreement was reached;
b) the points on which it was impossible to reach agreement:
c) the reasons why it was not possible to reach agreement;
d) suggestions regarding the research work which appears most
suitable for arriving at a solution of the difficulties.
7. - The « Conclusions » reached will be immediately printed and
rransmitted, by the Pontifical Academy of Science, to all the scientific
entres which might be interested therein.
8 - All travelling expenses, and accommodation in one of the best hotels
n Rome of the persons invited to the meetings will be borne bv the Pon-‘ifical
Academy of Science.
The Academy will be pleased to offer similar accommodation to the
vives of the scholars who are invited. but not their travelling expenses
ALNTIFION L
J
=
OJ
NISCUSSIONS
Ul.
3
2x
Ju uk. “IFhyu UIARDA
ET DU DR. GIACOMO VACTAGCO
All participants to the Study-Week have received
all the papers in advance. At the Study-Week each
paper has been discussed both in a group of specialists
(*) and in a plenary session, except for two
papers (STONE and FRISCH) which have been discussed
directly in plenary session. The papers have
been put herve in a succession which is as close as
possible to the chronological order of presentation
and discussion.
(*) 1st group: StoNE (Chairman), DORFMAN, JOHNSON,
Koopmans, MAHALANOBIS, MALINVAUD, MORISHIMA,
PASINETTI, SCHNEIDER,
znd group: LEONTIEF (Chairman), ArLAIS, FISHER
FriscH, HaavELMO, IsArp, THEIL, Woup.
THE ANALYSIS OF ECONOMIC SYSTEMS
RICHARD {STONE
Cambridge University - Cambridge - Great Britain
MODELLING ECONOMIC SYSTEMS
t. THE BACKGROUND
It has long been a commonplace that an economy is a
system, an exceedingly complex probabilistic system. According
to the classical theory, the information flowing in this
system consists of price signals. Constrained principally by
the law and public opinion on the one hand and by the state
of technology on the other, individuals and groups respond
to these signals mainly in terms of self-interest. First, the productive
part of the system produces as efficiently as possible
what the consuming part wants to consume. Second, in every
part of the system strong tendencies exist to check any departure
from equilibrium. Third, not only is every firm efficiently
operated but new ideas and inventions are adopted as soon
as they become profitable and so the standard of living grows
as fast as human ingenuity can make it. In other words, the
system is efficient, stable and progressive. It gives the maximum
1 Stone - pag. I
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
scope to individual desires and initiatives and, like physical
or biological systems, is regulated by means of viable governors
set in motion by the system itself and not by any outside organ
of control. To its other virtues, therefore, a final one is added:
compatibility in the highest degree with human freedom.
There was a time, not so long ago, when the rulers of the
world wished to believe this story and, generally speaking, succeeded
in doing so. But the imperfections of laissez faire as
a mode of economic organisation are so glaring that it has
been either thrown out altogether, as in the socialist countries,
or modified out of all recognition by state intervention even
in countries devoted to the principle of free enterprise. So
angry have men been at the abuses, injustices and waste of
resources around them, so strong has been their desire for
change, that they have shown very little appreciation of the
good points of the system they were destroying and so have
made very little effort to incorporate them in the system that
was to take its place.
The purpose of this paper is to discuss how economic models
might help us to reconcile the advantages of central planning
with those of individual initiative. The basic ideas are
simple: first, whether we consider a private firm or a government
agency, sensible decisions cannot be reached unless there
is an adequate amount of information flowing within the system
and available in the right place at the right time; and second,
since some kinds of information are expensive, if not impossible,
to transfer from one decision centre to another, it makes
a great deal of difference which decisions are taken at which
centre.
As to the first point, private firms and government agencies
usually try to enrich the flow of information in their neighbourhood
by making special surveys, projections and so on.
But just because the economy is a system, because, that is, its
different parts are interdependent, the task is very difficult for
any institution acting in isolation. Without a centralised ser-[1]
Stone - pag.
2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
vice of consistent projections which concerns itself both with
social objectives and with practical possibilities, individual
forecasts are likely to be unduly conservative. For example, a
realisable plan might require the output of a particular industry
to double in a decade, but if no such plan exists the industry
in question can hardly be expected to try to double its output
in the next decade if in the past its market has been expand-‘ng
at a slower rate.
As to the second point, in operating a system that is not
Jeterministic, adjustments of one kind or another are constantly
needed at all stages. There are costs, not to say dangers, in
requiring all decisions to be referred to a central authority.
Accordingly, we should examine carefully the sources of information
in the system, the costs of transferring information,
and the rules needed to ensure that centralised and decentralised
decisions will not conflict.
- &
MODELS AND THEIR ENVIRONMEN-The
main discussion of this paper will centre round a computable
model of the economic system. Before we come to the
general features of such models and to the particular example
that I shall give, which is a revised version of that described
in [7], it may be useful if I try to set an economic model in
its environment, that is, relate it to the objectives it is intended
to serve, the administrative arrangements needed to make it
work and the general experience of economic life which it molifies
and by which in turn it is modified.
Following the treatment in [37], the essence of the situation
as I see it can be represented graphically as in diagram 1 below.
The combination of theories with facts gives rise to a model.
Since economics is largely concerned with quantities, the model
must be quantitative: and since we need to know not onlv how
"1 Stone - pag.
2
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
DIAGRAM .
: Model in its Environment
> Experience
the system would behave if left to itself but also what would be
the problems and results of trying to change it in various respects,
the model must be readily computable. Model-building
is a scientific activity but I do not think that a usable model
of an economic system can be built by a group of scientists
working in isolation, for the simple reason that, however wellendowed
the group may be, it cannot possibly know as much
as it needs to know about all aspects of the economy without
the cooperation of many people engaged in a variety of practical
tasks. The model-builders can of course build a prototype, but
they will be very unwise if they do not seek a great deal of
practical advice before they go fully into production.
The combination of a model with a set of objectives gives
rise to a policy. This is where the politician comes in: he formulates
the objectives. He may specify objectives in detail or
he may delegate his responsibility, as when, with whatever safe-1]
Stone - pag. 4
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
guards he thinks desirable, he accepts the principle of consumers’
sovereignty: this means that consumers may spend their
money as they please, and the economic system must try to
orovide the goods they want and not some other goods. Thus,
just as the initiative in model-building lies with the scientist,
so the initiative in formulating the objectives lies with the politician.
But neither works on his own; each is dependent on
social acceptance and the cooperation of others.
The combination of a policy with a set of administrative
procedures, or what I have called controls, gives rise to a plan.
This is where the administrator, both public and private, comes
in. But he too may delegate his authority and indeed in
many cases must do so if the cost of basing all decisions on
sufficient information is to be minimised. The relationship
between centralised and decentralised decisions has been the
subject of recent papers by MALINVAUD 7231 and hv KORNAI
and LIPTAK [20], [21].
When the plan is put into practice it comes up against
svents and this gives rise to our experience of the part of life
with which the plan deals. As a result every element that
led to the plan may to some extent be modified. It is inconceivable
that initially we should have succeeded in building a
perfect model, specifying a perfect set of objectives or designing
a perfect set of controls. The possibility of these modifications
s indicated in diagram 1 by the feedbacks from experience to
theories, facts, objectives and controls, from which the effects
f experience spread to the model, the policy and the plan.
Of course this is a very simplified presentation of a very
complicated process. In fact, each stage in the process throws
light on the preceding stages as well as on the succeeding ones,
and each part of the system interacts to a greater or less degree
with most of the other parts, so that actually there is much
more feedback between the boxes than is shown in the diagram.
For example, the model is to some extent determined bv the
‘11 Stone - pag.
5
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
objectives it is designed to serve, and the objectives are likely
to be modified in the light of calculations made with the model.
Again, the policy is bound to be affected by the structure of
the existing control mechanism, while this is likely to be modified
in the interests of the policy. And so on. In the next
chapter we shall examine these connections in greater detail.
+] Stone - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
“DELS, POLICIES AND PL.NS
MODELS
The purpose of building a model of anything is to understand
how the thing works and, if possible, to make it work
better. As BALL has argued [2] in reviewing the model of
sconomic growth that I and my colleagues are working on in
Cambridge, the great thing is to make a start and to follow
this up by intensive work on the less satisfactory components
of the initial version. It is a mistake to try to perfect all the
components at the outset. Such an attempt may easily result
in no model at all, for one never achieves initially a degree of
oerfection in the components that makes one entirely happy
about leaving them alone. Furthermore, as he says, the adequacy
of particular types of relationship can never be properly
judged in isolation but should be evaluated in terms of their
role in the complete model. Our motto. therefore, is solvitur
ambulando.
[he decisions to be taken in building a model can be
grouped under four main headings: a) the variables with which
‘he model is concerned; b) the relationships by means of which
‘he variables are connected and the precise form that these
relationships are to take; c) the statistical and other methods
oy which the parameters in the relationships are to be estimated;
and d) the methods of numerical analysis to be used in
calculating the parameters and the unknown elements in the
svstem. Let us now look at each heading in turn.
a) Variables. The basic concepts of economics are production.
consumption and accumulation. to which mav be
"11 Stone - pag. 7
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
added, since national economic systems are never isolated,
foreign trade. The variables relevant to these concepts can,
as everyone knows, be set out in a system of national accounts.
These accounts have the merit of all accounts: their entries
satisfy certain arithmetic and accounting identities.
The entries in accounts are values: sums of money. If an
entry relates to a product, whether final or intermediate, or to
a primary input, it can be decomposed into a quantity and a
price. Thus values, quantities and prices have their places in
an accounting system. Consistency requires, therefore, that we
set up at least the main variables in our model within an
accounting framework.
The technique of doing this is called social accounting and
consists essentially in subdividing the national accounts. For
example, in input-output analysis we subdivide the national
production account so as to provide a separate account for each
group of commodities that the economy produces. When we
start doing this we find that different parts of the economy
habitually use different systems of classification. Our system
of social accounts should accommodate these different classifications
and show how they are reconciled. In this way we
can establish behavioural or technical relationships for different
parts of the economy which reflect the categories habitually
used by each part. For example, private consumers’ expenditure
is usually expressed in terms ot a classification of goods
and services which follows the lines of a consumer’s shopping
list; and public consumers’ expenditure is usually expressed
in terms of various purposes, such as education, health and
defence. Neither of these classifications has a one-to-one correspondence
with the industrial classification of products, and
these products, in turn, are not in one-to-one correspondence
with the industries in which they are produced. Consequently,
if we wish to be able to establish relationships for the components
of each of these categories we must make sure that each
1] Stone - pag. 8
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
i
of them is properly represented in our accounting system. Some
‘thoughts on this subiect are given in F&! "561.
bY Relationships. The general nature of the relationships
connecting production, consumption, accumulation and foreign
rade was defined long ago by the great economists of the
second half of the nineteenth century, such as JEvoNs, WALRAS,
MARSHALL and PARETO. Their contribution was, precisely, to
formulate an economy as a system. These general ideas were
set out fully and explicitly by BowLEY [4], who remarked in
his introduction that « there seems to be no book in existence,
at least in English, that presents in a coherent form the mathematical
treatment of the theory of political economy which has
Seen developed during the past eighty years or more. » This
was said in 1924.
But these general ideas provide no more than a guide. In
trying to formulate the economy as a system, the great writers
of the past naturally made many simplifying assumptions. For
example, in the theory of consumers’ behaviour, they assumed
that an individual tries to maximise the utility of his consumption
subject to a fixed set of prices and a budget constraint.
In this theory individual preferences are assumed to be fixed.
[f, like these writers, we are concerned with formulating economic
relationships in a general way, this is a legitimate assumption
which leaves us free to concentrate on the constrainedmaximum
problem whose solution is the outcome of the theory.
But if we are interested in empirical demand analysis we
cannot make such an assumption since we know that individual
oreferences change, partly because the circumstances of the
individual change with age and partly because, with time,
new commodities appear and social attitudes and fashions
change. Thus is studying the pattern of consumers’ expenliture
over time, it may be more important to find a way of
allowing for changes in preferences than to have a very sophisticated
means of allowing for responses to income and prices
Stone - pag. o
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
The earlier writers were aware of such complications, but
their first concern was to realise their vision of the economy
as a system, not to work out its operating characteristics.
MARSHALL said in 1896: « the nineteenth century has in great
measure achieved qualitative analysis in economics; but it has
not gone farther. It has felt the necessity for quantitative analysis,
and has made some rough preliminary surveys of the
way in which it is to be achieved: but the achievement itself
stands over for you. »
During the last fifty years or so, the development hoped
for by MARSHALL has slowly gathered momentum and has been
accompanied by a great deal of useful theorising. Our first
task now is to perfect, in the light of the vast mass of quantitative
information available, the relationships sketched out
by our predecessors: production functions, consumption functions
and so on. But just as the simplifying assumptions of the
earlier theorists restrict particular relationships to the point
where they cannot be applied, so also do they sometimes
restrict the field of phenomena that should be considered in an
economic model. Thus our second task is to formulate new
relationships wherever experience shows them to be needed.
An example of this is the distribution of skills which accompanies
any technique of production. Labour is a factor
of production, and everybody recognises that there are many
different kinds of labour, which require different kinds of education
and training. Nevertheless, in production functions labour
is usually treated as homogeneous. This is acceptable as
a first approximation provided one it confident that in some
unspecified way the right kinds of skill will be produced and
flow to that part of the system where they are needed. But
will they? Granted that demand tends to create its own supply,
may we not find, in a period of rapid technical change, a
serious lag between the skills needed for the new techniques
and the skills in fact provided by the system of education and
training? The answer seems to be « ves ». in which case edu-1]
Stone - pag. 10
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
cation and training become a necessary part of an economic
model, not something which can be left on one side as a social
process irrelevant to economics.
Another important point to keep in mind about the relationships
in an economic model is whether they are to be used
lo answer questions or to ask them. For example, if the modelbuilder
knew enough about the alternative techniques of production
that will become available in the near future in different
industries, he could say what changes in labour productivity
could and should be brought about in order to reach
certain objectives. In practice, however, the model-builder
is most unlikely, at first, to know as much as this. But he can
work out on reasonable assumptions a set of changes in labour
productivity in different industries which would enable a given
vector of output to be produced with a given quantity of labour.
Then, instead of answering the question ‘what increase in
labour productivity is needed in each industry’ he could ask
the question ‘could the increase in labour productivity I have
worked out be achieved and, if not, what are the obstacles to
achieving it’. By bringing such questions out into the open and
showing what is needed if certain results are to be obtained,
a great deal can be learnt and the people responsible for action
can be provided with useful information. It may be that an
industry simply cannot do what a model initially suggests; but
it may equally happen that it could do as much or more if it
believed in other results of the model, for example the estimated
future demand for its products. In this case the modelbuilder
is simply providing agenda for industry discussions.
This may often be more useful than trying to anticipate the
outcome of these discussions, since the discussions themselves
may provide information to the model-builder more reliable
han he could possibly have reached on his own, and may thus
enable him to produce statements about the possible future of
the economy which are realistic as well as consistent.
The final point I want to make about relationships con
“11 Stone - pag. 11
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA ~
cerns time-horizons. Any discussion of how best to operate
the economy so as to achieve certain ends involves comparisons
over time and leads to a variational problem. In theory there
is no reason to stop at any particular point in the future and
so we are led to a consideration of an infinite time-horizon.
Valuable insights can be gained by this method, as in RAMSEY’s
theory of saving [34], but the weakness of the method from a
practical point of view is that it calls for knowledge that we
cannot possibly possess. This suggests that we should reformulate
the problem with a finite, indeed fairly short timehorizon,
a practice that is in fact adopted in all centrally planned
economies.
c) Estimation. Having decided on the variables that are
to enter into the model and on the forms of the relationships
by which they are connected, the next thing to do is to estimate
the parameters in these relationships. This may be a matter
of simple arithmetic, as when an input-output coefficient is
estimated by dividing the input of product j into product &
by the output of product 2. So simple a method, however, is
only resorted to when there is an extreme shortage of information;
more generally statistical methods, and in particular regression
analysis, are involved. In any case the estimation
procedure that must inevitably be followed at the outset can
be described as the econometric analysis of past observations.
The methods of estimation available to econometricians have
improved very considerably over the last generation. In particular
it has come to be realised from HAAVELMO’s original
paper on the subject [16] that the fact that an economy is a
system in which different influences may operate simultaneously
has a bearing on the appropriate method of estimating economic
relationships. This is called the problem of identification: in
relating the price of a commodity to the quantity of it sold, how
can we identify the parameter as a demand parameter, a supply
parameter or some mixture of the two; alternatively, how
can we arrange our estimation procedure so that the estimate
1] Stone - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
we obtain is, let us say, a demand parameter? This problem
1as been the subject of intensive study at the Cowles Commission
[11] [12] and more recently has been given an elegant
form by presentation by Hurwicz [18] and, in terms of sta-‘istical
estimation, by DURBIN [13].
Important as these developments are, they certainly do not
solve all the problems that the model-builder has to face. One
of the main problems arises from the fact that relationships
are constantly changing. I have already given consumers’
preferences as an example of this. Another example is offered
by the techniques of production: a past input-output table
will not be a good description of present intermediate technoogy;
a fortiori it will not be a good description of the intermediate
technology of the future. How then are we to proceed?
The best thing to do is undoubtedly to consult outside experts
about the way in which inputs have been changing, and to
get from them an estimate of future input structures in different
industries. But such information can only be expected from a
limited number of highly articulate industries. In other cases
ne can sometimes find time-series of input-output coefficients
and project these into the future. Often, however, this approach
is closed too, and there is nothing to be done but to project
nput-output coefficients by a general method of extrapolation.
A means of doing this is given in [36] and, in greater detail.
n [9].
Another practical problem arises from the fact that, in
formulating relationships, we usually begin by considering only
the more general influences which we believe to be at work.
For example, in formulating demand relationships we usually
begin by allowing for the effects of income, prices and changing
tastes on the demand for different commodities. We know
of course that other, more specific, influences are at work: an
abnormally cold winter or hot summer, the temporary rationing
of some commodity, a particular advertising campaign. Such
specific influences are numerous and difficult to take into ac-“Stone
- pag.
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
count, and their effect on the broad pattern of consumption is
not very noticeable; but in accounting for the detailed pattern
they may well be important. In such cases it may be useful
to follow a hierarchical principle. Thus we might start by
estimating how, in given circumstances, total consumption
would be divided among its main constituents, namely food,
clothing, household expenses etc. We might then try to
estimate how the expenditure on each of these groups would
be divided among the group’s components and what specific
influences, if any, might affect each component. And so on.
As we got into greater and greater detail, however, we should
expect any manageable system of demand relationships to
break down. At this point we should have reached a position
where our demand model was no longer specific enough and
where, accordingly, we needed outside help, as in the case
of the input-output relationships discussed above.
What I have just said brings out the fact that with the
information at our disposal there is a limit to the amount of
detail we can handle. It is sometimes argued that if only we
could increase the size of our models we should get correspondingly
better results. I do not think this is true. If, for example,
we were to multiply the number of industries we distinguish,
and were to represent each industry in the detail necessary to
operate it, we should have to make the relationships of our
model altogether more sophisticated. In practice we could not
do this. I suggest therefore that the proper way to introduce
great detail into a model of the economy is not to expand that
model beyond a certain point, but to set up separate submodels
for different industries, related to the general model
but established and operated by the industries themselves with
all the expert knowledge that this would make possible. The
final outcome would result from an iteration between the general
model and the industry models, somewhat on the following
lines. The general model would indicate the output levels required
and the distribution of these outputs over uses. The
[1] Stone - pag. 14
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pf
industry models would then convert these demands into specific
oroducts and decide how these products could best be produced.
The general model would then check the total primaryinput
requirements of the productive system and indicate
whether any primary inputs were in short supply and, if so,
to what extent they should be economised on. It would also
change its initial cost structures and indicate revised output
requirements. Such a treatment would allow for a manageable
expansion of the initial model and at the same time would
ensure that the information available was put to the best use.
By the various methods I have suggested we may hope to
increase the amount of information that the model can absorb.
But cases may still arise where we are unable to formulate
some of the relationships we need: some processes of real life
cannot be represented in the model. All we can do is to observe
certain influences and certain effects; we cannot relate the two.
We must then turn to the ‘black box’ technique as adapted
to economic and industrial problems by BEER [3].
Reduced to its simplest terms this idea can be formalised
as follows. In the real world there is some system, the economy
in our case, which produces a certain outcome: in a
circuit diagram we could represent this process by an arrow
eading from a box marked E to a box marked O. By studying
this system we make a model of it and use this model to formulate
a plan: this conceptual activity could be represented
oy an arrow leading from a box marked M to a box marked P.
The two sides of life are connected: the model is based on the
real world, and so an arrow loads from E to M; and the plan
influences the outcome, and so an arrow leads from P to O.
Now, the model consists of certain relationships and these
control the plan. If some of these relationships change in
‘he real world but not in the model, then the plan determined
by the model will always be wrong and the situation will get
out of control. We could avoid this happening if between the
model and the plan we could insert a device, the ‘black box’.
[1] Stone - pag. 15
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
which, being controlled by the economy and by the plan,
would correct the instructions given by the model, as illustrated
in diagram 2 below.
DIAGRAM 2
A Corrective Device for Economic Models
A familiar, if somewhat formal, example can be given to
illustrate this scheme. Suppose that the purpose of the model is
to tell us how much to produce of a certain commodity and,
under free market conditions, what price to charge. Suppose,
further, that market conditions are not always free but that at
certain times the government imposes price control. If we
set up the model in terms of a demand relationship and a supply
relationship, we may assume for purposes of argument that:
1] Stone - pag. 16
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1
the quantity demanded, y,, is a homogeneous linear stochastic
function of the price, y;, and of an exogenous demand variable,
x, ; the quantity supplied, y,, is a homogeneous linear stochastic
function of the price, y;, and of an exogenous supply variable,
x,; under free market conditions the quantity supplied is equal
0 the quantity demanded; and under price control the actual
price is equal to the price fixed by the government, x;. In
ts reduced form this system can be written as
IL. I,
In (IL.1), the a’s and b’s are demand and supply parameters,
the e’s are disturbances and À is a number which takes the
value 1 under free market conditions and o under conditions
of price control.
In terms of the diagram, the model consists of the first two
equations in (II.1) and the purpose of the black box is to
ensure the appropriate value of À at any time. If the model
was established under free market conditions, it will operate
initially with A =1. But if market conditions change to a state
of price control, the black box must find this out and switch
over to A=0. As indicated in the diagram, it does this by
continuously comparing the price calculated by the model with
the price actually charged in the real world. If they differ,
it changes the value of A. For under free market conditions
the actual price and the model price will be the same, but if
price control is introduced, the two prices will, in general,
diverge unless A =1 is changed to A=0. With this change, the
actual price and the model price will be the same as long as
price control persists. If, however, price control is abolished.
"11 Stone - pag.
.
wf
20)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -§
the two prices will, in general, again diverge unless A=o0 is
changed back to A=1.
Thus the purpose of the black box is to ensure that the
model stays in touch with reality. As I have said, the example
is a formal one, but there is no necessity in practice to restrict
the device marked B to a single, instantaneous decision.
d) Computability. With a large model, computability presents
a problem at every stage: in the preliminary processing
of data, in estimating the parameters and in reaching solutions
for the system of equations. I have already emphasised the
importance of making a start and later modifying and extending
the initial model; if to this is added the need to reach solutions
for a wide range of initial assumptions, it is clear that
some special steps will have to be taken if the computing
problems are to be kept under control. As explained in greater
detail in [7] [44], a number of straightforward techniques are
useful for this purpose.
First, the extensive use of matrix algebra, in addition to
its notational convenience, has the advantage that in the numerical
manipulation of matrices a fundamentally simple set
of operations is repeated a fixed number of times. This number
depends on the order of the matrices and can be regarded as
a parameter of the programme. Accordingly, if the number of
categories in some classification has to be increased, it is only
necessary to change a parameter and continue with the old
programme.
Second, great convenience lies in the computer’s facility
for handling iterative or relaxation processes. By requiring a
sequence of operations to be repeated until some numerical
condition is satisfied, the same master programme can be retained
when linear relationships, which would themselves admit of
direct analytical solution, are replaced by more complex relationships
which would not. To do this, a general method of
solution must be adopted at the outset.
‘1] Stone - pag. 18
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Third, the programme can be subdivided into independent
parts each of which can be contained, data and programme,
in the high-speed memory of the computer. This allows each
stage to be developed and tested independently so that, as
research progresses, it can be replaced by more complicated
versions without necessarily upsetting the remaining stages.
Fourth, the programme can be used to compile ‘ready
reckoners’ which make it possible to trace the effect of changes
n the initial conditions without the need for further runs. For
example, the matrix multiplier (I-A)-! is a ready reckoner
whose elements show the additional output of ÿ which will
‘ollow directly or indirectly from a unit increase in the final
demand for k. Analogous matrices can be compiled to show
the effect of unit changes in the elements of final demand on
complementary or competitive imports. on investment requirements,
and so on.
Finally, once an initial model has been set up, it is necessary
to establish an order of priorities in improving its various parts.
Sor this purpose, sensitivity analysis, that is the systematic
>xploration of the model to find out what is sensitive to what.
would probably be very useful.
POLICIES
Policies express what we want from a system after we have
reconciled competing objectives in the light of a model; they
provide therefore a set of consistent aims. As I have said, the
‘ormulation of objectives is primarily the responsibility of the
politician, just as the construction of models is primarily the
responsibility of the scientist. But the politician, like the
scientist, cannot perform his task unaided and simply hand
down a set of objectives to the model-builder without discussion.
Three reasons for this have been suggested by HrrcH in [17].
One is that there is seldom a clear-cut national objective whic!
"11 Stone - pag. TC
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
can be ‘given’ in all its details by a central authority; an
obvious example of this is the composition of consumers’ expenditure.
Another reason is that objectives are multiple and
conflicting: many of them can be satisfied in several ways
which have substantial, differential effects on others. Finally,
it is impossible to state the objectives that make up a policy
without knowing a great deal about feasibility and cost. In
each case the necessary knowledge can only come from the kind
of analysis that a model is designed to provide.
An important question at this stage is the range of problems
that the politician would like to see integrated. For example, if
the model is a national model with no regional dimension, it
can say nothing about locative problems. If there is to be a
regional aspect to economic policy, then the model must be
capable of providing solutions for the different regions. A regional
dimension to an economic model is useful because it
makes people think why they want particular things to take
place in particular localities and enables them to compare the
costs of the alternatives.
Thus we see that the politician needs the help of the modelbuilder
in formulating a policy, just as the model-builder needs
the help of many other people in building his model. We also
see that the objectives from which the policy will be shaped get
into a model in various ways.
An objective may be built into the model. This is the case
where it is agreed that private consumers should be left to decide
how they spend their money. In effect the policy maker says to
the model-builder: I shall tell you how much money to assume
available for private spending; your job is to find out how this
spending will be allocated given the shadow prices that emerge
from your model, and what indirect demands it will place on
the system. In the course of drawing up the policy, alternative
calculations are sure to be needed, but the maximisation of the
average consumer’s utility forms an integral part of the model.
Or, an objective may form a constraint on acceptable solu-[1]
Stone - pag. 20
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ov
tions of the model. For example, having fixed the terminal
conditions at the end of the transitional period, we may agree
to choose a path through that period by maximising consumption
subject to the terminal conditions and, of course, to the
operating conditions of the period itself. For any given set
of conditions we can use the model to work out a path. If the
path is unacceptable we must reconsider the conditions that determined
it.
Or again, the model-builder may require some numerical
data from the policy maker. For example, he will need to
know how much is to be spent on public health, education,
roads and other items of government consumption or investment
in social capital. He will also need to know the input
structure of these expenditures. Since all demands on resources
-ompete with each other, it will be possible to settle priorities,
including the level of total private consumption, only when
he implications of these proposed demands have been work->d
out.
ft will be noticed at this point that there is a need for a
aumber of government sub-models just like the industry submodels
advocated earlier. The operation of a health service
or a defence system is a complicated matter which cannot be
built into a general model. What is required is a separate
sub-model for each of these activities. Given an amount of
money to spend and a set of prices, each sub-model would be
used to decide how this money should be spent. From this a
cost structure would emerge for use in the general model. The
general model might show that some revision in the sums to
be spent would be necessary. As in the case of industry, there
would have to be iteration between the model and the submodels.
Social cost-benefit analysis should prove of great value
in this field of research [14] [15].
To sum up, the main features of policy-making for a system
are, in my opinion, as follows. First, the different objectives
of policy should, as far as possible, be considered together.
r_ =
Stone - pag. 27
24
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
The fact that in my examples I did not mention the maintenance
of full employment, say, or the relief of poverty is of no
significance; I was not trying to list all the aims, either present
or future, of economic policy. In practice we shall certainly
have to begin with something narrower than we wish. Thus
even if we succeed in getting policies for education and training
or of regional development into the general picture, it will be
some time before we can do the same for urban renewal, to
say nothing of mental health or crime. Nevertheless I believe
that our aims should ultimately cover the whole socio-economic
system.
Second, each objective must be expressed in sufficient detail
to enable alternative methods of meeting it to be considered,
and the demands which any of these methods places on the
system to be worked out. In this way we can begin to compare
objectives, and as a result make perhaps a better use of our
resources.
Third, the existence of conflicts of interest should be recognised
and as far as possible faced. For example, two towns
may compete with one another to attract, say, a new power
station or motorway. The choice between them can be greatly
improved by an analysis of its consequences.
Fourth, policy makers should see that they have adequate
information on which to base their decisions and should try
to assess costs and benefits wherever possible. For this purpose,
existing market prices are extremely useful; but they
are insufficient, because many of the things we value are not
priced on any market. Examples of this are uncongested roads
and quiet surroundings: only recently has a partial attempt
been made to put a price on road space by means of parking
meters; and the cost of noise to health and productivity has
so far received more attention from physiologists and psychologists
than from politicians or economists.
Two conclusions emerge from this summing up: there must
be close cooperation between the policy maker and the model-'1]
Stone - pag. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 2)
builder; and there must be a conscious effort to supplement
the market mechanism by calculating costs and prices for activities
which lie outside the market economy or which have
important aspects which the market does not value.
LLNS
A plan tells us how to set about achieving our policies given
the operating characteristics of the system. It can be identified
with administration or control. These words can in turn be
identified either with coercion, exemplified by the policeman,
or with a means of self-regulation, exemplified by the Watts
governor or the thermostat. The opposite of plan is no-plan.
or anarchy.
In theory laissez faire is not an example of no-plan; it is
a perfectly coherent plan for operating an economic system.
Îts strength lies in the fact that it is self-regulating. This is
achieved by placing decisions in a large number of centres
2ach of which is intent on maximising its advantage. An
examination of biological and ecological systems suggests that
‘hey owe their robustness to similar forms of control: the prelator-prey
relationship cannot be understood by identifying
‘he predator with a policeman. The fundamental objective to
laissez faire as a form of planning is that it works with limited
values and limited information: the values of the market place
and the information provided by current prices and by the
orices on a small number of forward markets.
The reaction to laissez faire has taken two forms: central
planning and government intervention in specific aspects of
>conomic life.
The first reaction has the merit that it places the determination
of policies squarely where this belongs, in the class of
political decisions. Its shortcomings which derive largely from
its political origins. lie in an exaggerated notion of the pos-Stone
- pag. 23
28
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
sibility of concentrating all information at some central point
and in believing that all decisions can sensibly be taken at this
point. The difficulties that follow this approach are increasingly
recognised in centrally planned economies and the natural
dialectical process may be expected to produce a better
balance between the centre and the periphery.
The second reaction has the merit of recognising the fact
that different decisions belong to different centres. Its shortcomings,
which derive largely from the partial abandonment
of one coherent political philosophy without the acceptance of
a new one, lie in an exaggerated notion of the usefulness of
modifying some part of a system while ignoring the others:
specific acts are justified in terms of the necessity to do something
in the area concerned and of the immediate effects
intended. The bad consequences of this kind of sporadic planning
are becoming every day more obvious, but it is not yet
realised that the more one tries to plan a system without studying
it as a whole, the less one is likely to succeed.
Thus it is a mistake to associate planning with collectivism
and antiplanning with private ownership. We should make
better progress if we thought in terms of good planning and
bad planning, that is, of functional and unfunctional design.
In other words, having agreed on what the system is supposed
to do, we should make sure that the operating controls
are designed so as to get it done. If we look at planning from
this point of view, we are likely to discover that the key problems
of economic organisation are the following.
First, the administrative machinery, public and private,
which has grown up historically will often exhibit cases where
a given function is duplicated and cases where a given function
is simply not performed or where the arrangements for performing
it are unsatisfactory. An example of this in Britain
is the large number of agencies, both public and private,
concerned in one way or another with the problems of redundancy
and retraining: obviously this part of the administrative
1] Stone - pag. 24
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machinery must be considerably modified and coordinated if
he labour force is to accept mobility and equip itself with
modern skills.
Second, even where the machinery exists, we may question
the criteria on which it operates, on which, that is, administra-‘ive
decisions are taken. It used to be generally believed that
optimal or near optimal decision-rules were inherent in the
modus operandi of private enterprise. Experience has shown,
however, that this is not always so, even if we agree to consider
only private costs and benefits in determining efficiency.
still less is it so if we consider social costs and benefits which
the market does not value. Many writers, such as LERNER [22],
have tried to formulate effective and mutually consistent rules
for economic decision-making. These ideas should be followed
1p and as far as possible put into practice.
Third, the application of any set of administrative criteria
presupposes an adequate flow of information to enable the controls
to work properly. We must therefore examine the flow
of information in the system and ensure that the necessary
information is available in the right place at the right time.
We must also keep in mind that if a system is to control itself
oy virtue of the information that flows in it, this information
must not be distorted by interference; if it is, the system will
work badly, and if the interference is insistent enough, the
system will break down altogether. A good deal of the information
available in any economy comes from the movement of
relative prices and costs; without such information, derived
either from the working of the market or, in the form of
shadow-prices, from the working of a model, the decentralisation
of decisions is virtually impossible. So we should make
‘he most of price signals as a source of information and should
seware of fixing or changing them arbitrarily in the interest of
some specific policy, or they will defeat their own purpose. The
stop-and-go measures adopted by Britain in recent years for
‘he sake of balancing the balance of payments provide a good
®
Stone - pag. 25
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - =
illustration of these dangers. The objection to these measures
is not that they are unduly authoritarian or unduly laissez faire,
but simply that they are inappropriate to a system. They do
indeed enable the country to muddle through, but at a considerable
cost in bad business planning at home and loss of
prestige abroad.
Fourth, if we want to do away with sporadic interference,
we must examine the robustness of the system in reacting to
unforeseen events and try to build stability into it wherever
we can. An example of such an administrative arrangement
is MEADE’s proposal [25] to vary social security contributions
automatically with the level of unemployment so as to offset
short-term fluctuations in purchasing power. These devices,
as PHILLIPS has emphasised recently [28] [29] [30], must
be very knowledgeably designed, since a good control mechanism
should keep a variable at a chosen level and not allow
it to fluctuate.
These problems are encountered in any system and therefore
in any economy, whatever its political complexion.
A wider recognition of this fact may be expected to bring the
operating characteristics of eastern and western economies
much closer together over the next generation.
I] Stone - pag. 26
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4
À
DUAL MODEL OF ECONOMIC GROWT.x
[. A NEW DEVELOPMENT
[ shall now exemplify the ideas on model-building expressed
in the first section of the preceding chapter in terms of the
model of the British economy on which I and my colleagues
are working. Four progress reports on this work have already
appeared in this series [7] [8] [9] [10]. In the first of these
we began by setting up a single model for a future state of
steady growth and, in our conclusions, hinted at the need for
an extension of the model which would help us to plot out a
path from the present to the future state. This extension is
now beginning to take shape.
In this short chapter, I shall outline the two parts of the
model and explain how they interact. In chapter IV, I shall
review in greater detail the structure of the first part, our original
model, and discuss a number of modifications we are bringing
to it. Finally, in chapter V, I shall describe the structure of
‘he second part, on which we have recently started work.
[n my exposition I shall try to bring out: the organic chacacter
of the model, that is, how its structure and relationships
tend to change as knowledge is accumulated; the receptiveness
of the model to observations from different sources, including
sstimates from industrial experts and others, which can improve
its realism; and the degree of consistency which we trv to
impose on the model F--"
1] Stone - pag. 2,
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
2. STEADY STATES AND TRANSIENT STATES
In planning for the future, we should recognise, first, that
what we do now affects to some extent what we can do in the
future and, second, that what we shall want to do in the
future, and what facilities we shall have to do it with, becomes
less and less clear as we try to imagine times which are more
and more remote. We cannot work forward into the future
without setting a target and we cannot derive this target from
the remote future because we have no information about it.
Accordingly, we have set up our model of growth in two parts,
and our solution comes from iterating between them. One part
is concerned with the rates at which the outputs of different
products might grow after a transitional period ending, say,
in 1970; this is the long-run or steady-state model. The other
part is concerned with the problem of adapting the economy
during the transitional period to meet the initial conditions
of the steady state of growth; this is the short-run or transient
model.
The structure of these two models and the relationship between
them can best be seen by concentrating on essentials and
leaving all detail for subsequent treatment. The two structures
are shown in diagram 3 below.
Each structure involves a building block for each period,
containing five components: assets, labour, output, investment
and consumption. We can describe the relationships between
these five components in two ways. On the one hand we may
say that given amounts of assets and of labour enable us to
produce a certain amount of final output which will be absorbed
either by investment or by consumption: this is the order
that appears in the transient model. On the other hand we
may say that investment and consumption add up to final output
and that, given the labour available, the production of this
output will require certain quantities of assets: this is the order
[1] Stone - pag. 28
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dL
DIAGRAM ”
model nf economic growth
a
Asser
Lane
| Ass
——
Latthat
appears in the steady-state model. In each case the building
block is the same; what differs is the nature and direction
of the arrows which show the relationships between the components.
In each model the building block is repeated for as many
years, or time-periods, as are necessary. In the transient model
these time periods stretch from a base year, in our case 1960,
‘0 the end of the transitional period, in our case 1969, and
culminate in a single component, the stock of assets, as a terminal
stock. This stock is shown in the right-hand top corner
of the upper half of the diagram, enclosed in a double square.
In the steady-state model, the time-periods stretch from the
first post-transitional year, in our case 1970, into the indefinite
P +7
Stone - pag. 20
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -
future. In terms of the highly simplified account I am now
giving, the purpose of the steady-state model is to determine
the minimum initial stock of assets required in the post-transitional
period. This initial stock is, of course, the same as the
terminal stock of assets of the transitional period and is shown
in the left-hand top corner of the lower half of the diagram,
also enclosed in a double square to indicate that it is the link
between the two models.
Our first step is to use the steady-state model to determine
the stock of assets that must exist at the beginning of 1970.
This is done on the assumptions that consumption is to take
a certain value in 1970 and to grow steadily at a given rate
from 1970 onwards. For this rate to be possible, there must
take place in 1970 a certain amount of investment which can
be thought of as determined by the stock of assets and work
in progress needed at the beginning of 1971, which in turn is
determined by the assets and work in progress needed at the
beginning of 1972, and so on. In fact the whole future of the
structure of assets is involved although, if we do not try to
push the rate of growth of output up to the technological ceiling
of the system, we may expect subsequent requirements to have
a rapidly diminishing influence on the assets needed at the
beginning of 1970.
Once we have determined these initial assets which, given
the labour force and the technology expected to be available
in 1970, will depend partly on the level of consumption in 1970
and partly on the rates at which its components are to grow,
we can pass to the transient model. We have a known stock
of assets in the base year and a required stock at the end of
the transitional period. Given the labour force and technology
expected to be available throughout the transitional period, we
now try to maximise consumption over this period subject to
the base-year stock of, and terminal requirements for, assets;
and subject also to some restriction on consumption, for
[1] Stone - pag. 30
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D
example that it should not fall below a certain level or that it
should not fall below the level of the preceding year.
The attempt to solve this problem will show whether or not
it is solvable. If it is not, we have either put an insufficient
restriction on consumption during the transitional period or we
have set our sights too high for the post-transitional period.
In any case we can make a modification and try again until we
obtain a solution.
Once we have a solution we can examine the whole timepath
of consumption. Before the base period consumption
moved in a known way; we have just calculated its path
through the transitional period; and we have arranged that it
shall be able to grow at such and such a rate after the transiional
period. Taken as a whole this path may be unacceptable
because, let us say, it is too uneven: it may show a certain
rate of growth up to the transitional period, followed by a
slower one which increases enormously at the end of the transtional
period to meet the initial conditions of the steady state,
‘ollowed by the long-run growth rate. An acceptable path
should, one might suppose — though this is a political question
— be reasonably smooth. One way of smoothing it would be
to reduce the level of consumption assumed for the beginning
of the steady state; in other words, to make the take-off into
steady growth start from a lower level. The purpose of iteration
between the two models is to reach an acceptable path.
In the following two chapters I shall show how the basic
simplifications adopted in the above account can be removed
doth in theory and in practice. There is, however, one important
point that must be mentioned here since it affects the
relationship between the two models. Briefly, it is this.
[ have so far assumed fixed relationships between the assets
available to an industry and its capacity to produce, and be--ween
the assets installed in an industry and the investment
necessary to instal them. Consequently, the investment of the
‘ransitional period could in principle take place at any con-11
Stone - pag. 3
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2!
venient time without affecting the composition of the stock of
assets at the beginning of the steady-state period. This has
enabled me to speak of the assets needed at the beginning of
1970 independently of the distribution of investment through
the 1960’s, and thus to make a sharp distinction between the
two models. In fact we know that the relationships I have
assumed to be fixed do change with time, and therefore that
we cannot calculate exactly the assets needed to provide a
given capacity at the beginning of the steady-state period until
we know the timing of investment through the transitional
period. Here again, as in the case of consumption, the problem
can only be solved by iterating between the two models.
3. WHY A DUAL MODEL?
Questions are sometimes raised about the method I have
just outlined. Why, it is asked, do we need two models?
Could we not work out a path which would carry consumption
into some preassigned rate of growth as soon as possible? My
answer is that at the present stage of the work it seems easier
to divide the problem into two. With a single model it would
be impossible to say in advance when the growth rate of the
path calculated by the model would approximate to the growth
rate preassigned for the steady state. In other words, it would
be impossible to say how long the transitional period would
last: the model would be a transitional model without any
time limit. Consequently, it would be necessary to take a
view on preferences, technology, and other variable factors,
for an indefinite time-span in the future.
Also, the desire for a single model is sometimes accompanied
by a belief that it must always be possible to move
smoothly into a faster rate of growth without ever falling below
the initial growth rate of the system. We may hope that this
1] Stone. - pag. 32
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. :
is true, but it does not seem sensible to found an analysis on
the assumption that it is. In any case, the dual model is
ntended to explore several possible paths to the steady state
and to indicate the good and bad points of each alternative.
However, as illustrated by this series of papers, our approach
to model-building is essentially organic. In time, the
dual aspect of the model may come to seem less important and
‘he two models may become more closely integrated and even
be merged into one. This is not a matter of principle but of
practice.
sinne - pag.
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
THE STEADY-STATE MODEL
I. INTRODUCTION
The structure of this model as it stood eighteen months ago
was explained in detail in [7]. A further addition relating to
labour skills and to the system of education and training on
which they depend, together with some preliminary results for
1970, were set out in [5]. Thanks to the generosity of the
Bank of England and other institutions in the City of London
mn response to our appeal for research funds, we are now startng
work on the financing side of the model. This means that
we shall study the flow of capital funds between different sectors
of the economy, the saving behaviour which adds to these
funds and the preferences of different sectors for holding par-‘icular
portfolios of assets and claims.
In this chapter I shall outline the model following the order
of section I of chapter IT above. I shall not go over every
point described in [3] [7] but shall try to show how our ideas
have developed, why we have abandoned certain details that
appeared in our original presentation and what our intentions
are for the future.
2. THE VARIABLES
The basic variables of the model are brought together in a
social accounting matrix, SAM for short, which at present
contains 253 accounts grouped into fifteen classes. The comp-{17
Stone - pag. 34
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
~
lete matrix for our base year, 1960, is given in detail in [8];
a summary version for 1962, showing the totals within each
class, is given in table I below.
For obvious statistical convenience we have keyed in our
main totals to the official estimates of national income and
expenditure [48]. At certain points, however, we have departed
from the treatment followed in these estimates. The
most important difference lies in the fact that we define consumers’
durables not as consumption goods but as fixed assets.
This means that in our treatment these goods are bought on
capital account and their consumption is measured by depreciation.
Nevertheless, our estimates of total private consumption
plus net investment in consumers’ durables are equal to
the official estimates of consumers’ expenditure.
As can be seen from table 1, the accounts in SAM are
simply a logical development of the four national accounts [42],
and can easily be reduced back to them by appropriate consolidation.
The use of fifteen classes of accounts instead of
four is largely dictated by the need to reconcile different classifications.
This can be illustrated by considering the four
classes which appear as the first four rows and columns in
table 1 and which, taken together, constitute the national account
for production.
Class I relates to commodities, that is to say products or
groups of products which are characteristic of British industries.
The entries in column 1 show the sources of these commodities:
£44,272 million come from British production and £2,458
million, to which must be added £134 million of customs duties,
come from abroad in the form of competitive imports. The
entries in row I show the uses to which these commodities are
put: £20,943 million go to industries as intermediate product;
£13,249 million go to private consumers; £1,761 million go
to public consumers; and so on until, as can be seen from the
entry in column 15, £5,128 million go to the rest of the world
,1] Stone - pag. 35
TABLE 1
A PROVISIONAL ACCOUNTING MATRIX FOR BRITAIN, 1962
(£ million)
[ype of
account
Production
accounts
[Income
nd outlay
accounts
Capital
-ansactions
accounts
All
iccounts
Description of class
. Commodities . . . . . .
. Industries . . . . . . ©
3. Consumers’ goods & services
. Government purposes .
5. Indirect taxes and subsidies
6. Institutional sectors
7. Commodities . . . . .
3. Industries, replacements .
9. Industries, extensions . .
‘0. Cons. goods, replacements
tr. Cons. goods, extensions . .
"2. Govt. purposes, replacements
13. Govt. purposes, extensions .
‘A. Institutional scactore
15. Rest of the world .
"otal outecoings .
Production accounts
À
0 20943 13249 1761
14272 0 0 C
0 0 0 0
0 0 0 0
134 638 2160 30
0 18855 852 92735
0
0
)
)
J
J
3
0
1200
586
0
C
0
0 a
0 2
763 0
658 0
0 ge
0 rm
n
1
f
XA
2458 — 2186 681 155
16864 44272 18383 —_—
Income
and outlay
accounts
A
0 0
0 0
0 18149
0 4858
0
3979
0
0
0
n
1
0
0
h
CG
|
|
0
9022
D
108
3979 926048
Capital transactions accounts
7
8-9 10 11 12 13 1/
81 1168 1969 661 1416 81 407
0 0 0 0 0 0 0
0 0 0 0 60 0 0
0 0 0 0 60 0 0
0
0
0
Nn
0 32 45 102 118 0 0 0
0 1} Nn 0 0 0 0 0
]
0
0
¢
0
9
0
0
n
0 0 0 0 0 86
0 0 6 0 0 C
0 0 0 CG 0 1428
0 0 0 0 0 0
0 0 0 0 0 876
br n 0 0 0 0
+ i 0 © 0 333
fi fn nn € n n
4
3
hy
N
}
Nn
7.
86 1200 2014 763 1534 81 407 2797
All
accounts
1g
5128
0
214
0
0
395
0
0
0
0
0
0
0
n
]
ARAT
Total
incomings
46864
44272
18363
4856
3279
26046
86
1200
2014
763
1534
81
407
2797
AG67
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
JY
as British exports. In our complete matrix, class I contains
at present thirty-one commodities.
Class 2 relates to the productive activity of the thirty-one
industries which produce these commodities. It can be seen
from row 2 that the whole output of these industries, £44,272
million, flows into class 1. The cost of producing this output
is shown in column 2, thus: nearly half the total cost, £20,943
million, relates to the intermediate inputs, namely raw materials,
semi-finished products and fuels, absorbed in production;
£638 million represent indirect taxes (less subsidies),
which we charge direct to industries; £18,855 million represent
factor incomes, namely wages, profits etc., paid out by the
industries; £1,786 million represent depreciation, of which
£1,200 million correspond to the value of assets estimated to
have been scrapped during the year and the balance, £586
million, is available to finance extensions, that is additions to
the stock of assets; - £136 million is not a real entry, but corresponds
to the residual error which appears in the official
accounts; finally, £2,186 million represent complementary imports,
that is to say imports of products which are either not
produced in Britain, like crude oil and raw cotton, or produced
there in relatively small quantities, like raw wool.
We have adopted this distinction between commodities and
industries because there is not a one-to-one correspondence between
the two concepts. Most of our basic data make use of
the distinction and so it is convenient to follow it in setting out
these data, although for input-output analysis we get rid of
it and make use of a table which shows the commodities needed
to produce commodities, as is explained in detail in [9].
Class 3 relates to consumers’ goods and services, or private
consumption. These goods and services are classified in SAM
under forty headings, corresponding broadly to a shopping
list. In this form they lend themselves better to demand analysis,
but they still have to be related to products in the industrial
classification: again, there is no one-to-one correspondence
1 | Stone - pag.
37
10
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28&
For example, in the consumer classification the category
clothing’ includes the principal products of five industries:
textiles, clothing, other manufacturing, transport and distribution.
For this reason we carry out our analysis of consumers’
demand in two parts: first, an analysis of this demand in terms
of consumers’ categories; and second, a conversion from these
categories into commodity categories and hence into demands
on industries.
Class 4 relates to government purposes. Like private consumers,
government departments habitually classify their expenditure
in a manner different from that used for industrial
products. But this time it is not a shopping list that is used
but a classification by purposes: health, education, defence
and so on. In order to form any view of future demands for
these purposes we must work with this classification, but in
order to relate these demands to demands on industries we
must convert them into a commodity classification.
It is for these reasons that we have four classes of production
accounts. As a consequence we have an elaborate accounting
system which many people may regard as tedious and
‘echnical. So, in principle, it is. But in principle does not
mean in practice, and a realistic representation of the actual
world inevitably requires such technicalities.
I shall not continue with a list of classes and the reasons
for adopting them; all such details can be found in [8].
Instead, I shall mention one part of the existing structure that
we hope to change and a number of directions in which we
nope to develop.
The change will come not in the structure itself but in the
numerical value of the entries. As I have mentioned, we divide
capital expenditures between replacements and extensions. In
SAM, replacements are calculated by reference to past capital
expenditures and to the life-spans of different kinds of asset,
which are assumed to be fixed. We know perfectly well, however,
that while there may be an average life-span for any
I] Stone - pag. 38
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
asset, the actual life will depend on economic circumstances.
A more realistic approach, therefore, would be to assume that
assets are scrapped when they cease to earn a return. We
should like to substitute this assumption for the one we are
making at present, and our current work on production functions
points to a way in which this might be done.
The first development I shall mention relates to the distribution
of incomes. At present, households are a single sector
in SAM, and so have a single income and expenditure account.
As a consequence, our model can say nothing about the distribution
of incomes by size; it can neither say what changes,
if any, in the present distribution might be expected in any
particular circumstances nor can it say what consequences
would follow if a particular change were to come about. In
order to study these problems we should have to subdivide
the income and expenditure account of households into separate
accounts for different income groups. This could certainly be
done; it is entirely a question of time and money. The first
problem would involve dividing the incoming side of the account
by income group and classifying the incomings by industry
of origin and type of income; this would be comparatively
difficult to do. The second problem would involve dividing
the outgoing side of the account by income group and detailing
the expenditure patterns of the different groups; this would not
present much difficulty. As a second step, however, it would
be necessary to work out how the average expenditure pattern
would change if the distribution of income changed; this could
be done by an extension of our present demand functions.
A second development on which, as I have said, we have
just started to work relates to problems of financing. In terms
of variables, this means in the first place a large extension of
our institutional capital accounts, class 14, so as to introduce
a flow-of-funds statement into the system. For this purpose
we intend to increase considerably the number of sectors in
this part of the accounting system so as to distinguish many
1] Stone - pag. 39
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
different kinds of financial intermediary, and to add a new
class of accounts relating to different kinds of financial claim.
In the second place it means a serious attempt to construct
sector balance sheets, an aspect of the social accounts which,
in Britain, has only recently been pioneered in the work of
MORGAN [26] and in the as yet unpublished study of the national
capital on which REVELL has been working at the Department
of Applied Economics.
A third development which seems to me desirable is to add
a regional dimension to the model. As our model stands at
present it is a purely national model and therefore has nothing
to say on the regional aspect of any economic activity. But
regional problems are obviously important. Again, it is a
matter of time and money.
3. THE RELATIONSHIPS
The relationships of the steady-state model, as they stand
at present, can be summarised conveniently in a flow diagram
which I have already made use of in [5] and which is reproduced
here as diagram 4. This consists of four interacting
circuits which enable the model to be built up gradually and the
complexities of the real world to be introduced bit by bit.
In my description I shall mention two further circuits which
do not appear in the diagram.
a) The circuit of real flows. If we begin at the beginning
of this circuit, we find two boxes labelled consumption demands
and rates of growth in demands. Consumption demands relate
to both private and public consumption and include, so as not
to complicate the diagram unduly, investment in consumer’s
durables, housing and social capital such as schools, hospitals
and roads. For the moment let us assume that we have succeeded
in calculating the level of these demands in 1970, have
1] Stone - pag. 40
DIAGRAM 4
The Steady-state Model
are
PQASUIE |
DEN
=
Yall
CL
NSU”
2ADE:
Jomestre
oricer
. The circuit of real Flows
—The price circuit
~The foreign trade circuit
<== The circurt of education and training
3
1
7
i
‘ry ov
Ld
+ amas
Consumptior
demands
rr
cational
rrbutior
tf past
-rulati
wth.
, €
i
PO
se *
ta rial
1stribut on
C8 aba a
rota.
‘ahour foree
Demand
inp, ohille
Frenp
D
>
-
"
J
-
7
Joa,
LT
vrgtional
‘'bulion
nulation
[| Educational
* distribution
of labour
Ps
Su, iy
af “pp
7
7
A
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2€
converted them into demands for industrial products and for
each product have calculated the growth rate which would
accompany a given growth rate in consumption as a whole.
The first relationship we have to consider connects this information
with the investment demands of industry. We can ignore
replacement demands since, until we improve our production
functions, these depend, as I have said, on past investment
and on the fixed life-spans assumed for different assets. We
need therefore a relationship connecting consumption demands.
and their rates of growth with industrial extensions.
To obtain this relationship, we first write the basic flow
equation for products in the form
(IV. 1)
q=Ag+v+e
where q, v and e denote respectively vectors of output, industrial
investment and consumption, and where A denotes a
current input-output coefficient matrix. Equation (IV. 1) states
that output is divided between intermediate demands, Ag, and
final demands, (v+e); and that final demands are divided
between investment demands v, and consumption demands, e.
Second, we write the relationship between investment demands
and the growth of output from one year to the next in
the form
(IV. 2)
v=KAg
where Ag denotes the excess of next year’s output over this
year’s output and K denotes a capital input-output coefficient
matrix.
Finally, we consider the case in which the components of
consumption are to grow exponentially. This can be expressed
in the form
IV. 3)
Ee= (I+7)e
I] Stone - pag. 42
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
where E denotes an operator which advances by one year the
variable to which it is applied, so that Ee denotes next year’s
consumption vector; I denotes the unit matrix; and y denotes
a diagonal matrix of the growth rates of the components oi
consumption.
It has been shown in [7] [43] that on these assumptions
(IV. 4)
U
5 KI y5
4!
This is the relationship we need. It has been shown in _«/ _
that the infinite sum in (IV. 4) will converge provided that the
largest element of » does not exceed the smallest latent roof
of K(I- A)-!. This problem is also discussed in [17].
If the components of consumption are assumed tc grow
linearly rather than exponentially, then (IV. 3) is replaced by
(IV. 5)
i.e = (I+ 07)
and (IV. 4) is replaced uy
(IV. 6)
v=K(l- A) ‘we
which is simply the first term of (IV. 4).
It has been shown in [7] that (IV. 4) is capable of two
generalization. First, if technology, as summarised in A and
K, is changing in a known way, the expression corresponding
to (IV. 4) can be derived. Second, if allowance is made for
different investment lags, then (IV. 2) must be rewritten. To
do this we need information about the work that must be done
on investment goods in the successive years of their construction.
If we have this information, then, again, we can rewrite
(IV. 4) in an appropriate way. The information is important
1] Stone - pag. 43
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
if the growth of the components of the output vector deviates
from linearity. The problem has therefore a particular bearing
on the transient model. So far however, we have not done any
work on the decomposition of K.
Having thus determined investment demands, the next thing
to do is to determine output levels. This follows immediately
from (IV. 4), since
g=(I—A)y*(v+e¢
TV. 7) — (I _ A)-t S [K(1— AF
The elements of g are gross outputs. Their units might
be physical, such as tons of steel or kilowatt-hours of electricity,
but in practice, because of the coarseness of our product classification,
they are values.
Our problem now is to pass from gross outputs to net
outputs, or values added, since these are the output measures
which influence most directly the industrial distribution of labour
and the industrial distribution of assets. To handle this.
problem we shall assume an initial vector, p say, of product
prices. How this vector is determined and modified in successive
rounds of calculations we shall see when we come to
‘he price circuit. For the moment let us take it as given.
If we use f to denote a vector of primary inputs per unit
of output, then, corresponding to (IV. 1), we can write
‘IV. 8
p=A'p+]
i
AV. 0)
;—d-A)p
1 Stone - pag. 44
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
where A’ is the transpose
vector of net outputs, then
x
if we use
vv to denote the
(IV. 10)
J
on premultiplying (IV. 9) by q. The elements of y are now ti.
be related to the labour and capital inputs they require.
In our original exposition [7] (where, incidentally, we did
not distinguish between q and y), we proposed to relate outputs
to primary inputs by a modified form of the CoBB-DoUGLAS
function. This modification, proposed by PITCHFORD in [31]
and by ARROW and others in [1], is designed to generalise
the CoB-DoucLas function so that the elasticity of substitution
between labour and capital, though still a constant, need
no longer be numerically equal to one. This type of function
can be written in the form
(IV. 1...
y.
— Lt
0 {. Vs
+
where the suffix s denotes the s’th element of each vector.
Thus y, denotes the net output of industry s, and /, and k,
denote respectively the inputs of labour and capital into industry
s. The three parameters a, b, and c, can be given an
economic connotation: a, is associated with the efficiency with
which labour and capital are used in industry s; b, is associated
with the shares of labour and capital in the net output of industry
s; and c, is associated with the substitution of labour
and capital in industry s. The elasticity of substitution between
labour and capital in industry s is equal to (1 +c,)7", and so,
as c,>o (IV.11), approaches the simple form of the CoBB-DoucLAS
function.
1] Stone - pag. 45
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 72}
In order to use (IV. 11) or a similar relationship to determine
the industrial distribution of the labour force and the
associated stocks of assets needed to produce a given vector
of net outputs, we must know how large the total labour force
will be and on what criterion it should be distributed. For the
size of the labour force in 1970 we already have official estimates.
For the criterion on which it should be distributed we
have the familiar condition that the marginal physical products
of labour and capital should bear a common ratio to one another
in every industry; this is equivalent to saying that we must
choose a distribution such that it could not be improved by
any redistribution.
On further reflection, however, we have decided not to use
relationships of the form of (IV. 11). Two reasons are perhaps
sufficient to explain this decision. First, technical progress
gets into (IV. 11) by allowing a, to increase with time. This
is not satisfactory because it implies that output will increase
over time for given inputs of labour and capital independently
of the amount of investment that is being carried out. But if
no investment is being carried ‘out the quality of the capital
stock cannot improve, and it is hard to see, therefore, how any
substantial amount of technical progress could take place.
Second, if c,<I it is possible to substitute capital for labour
‘ndefinitely and thus to produce any amount of output with
a given labour force simply by giving it more and more capital
to work with. But we know that this is not true. At any given
time new plant will embody about as much capital as can profitably
be used and it seems doubtful whether much more capital
would be used even if capital were a free good. The reason
is that, typically, techniques do not exist for using much more
capital. Accordingly, there is a limit to the substitution of
capital for labour and we could not, even if we wanted to,
increase output indefinitely with a given labour force.
Our latest ideas on this subject have been set out in [33]
and lead to a production function for industry s at time # which
[1] Stone - pag. 46
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIQUE ETC.
can be written, on the assumption that plants have a fixed lifespan,
in the form
(IV 14
. I2)
2)
y
s£-- ll
pa!
st
01
de (T4
a*
JF
x) À
In this equation, y, and /, denote respectively the net output
of, and labour employed by, industry s at time #; v*, is the
investment undertaken by industry s at time t expressed in
wage units, and r*, is the initial rate of return on plant which
comes into operation in industry s in year t; and a*, and b*,
are parameters. The integration spans a period of © years
equal to the fixed life spans of the plants.
We have already seen, when discussing SAM, that the
assumption of fixed life-spans is not satisfactory because actual
life-spans depend on economic conditions. We can allow for
this by integrating over an interval which is different from
year to year and which for each year is chosen so that a plant
1s assumed to be scrapped only when the real wage becomes
such that the value added by the plant would be less than the
wage bill, even if it were operating at full capacity. This can
be expressed by writing (IV. 12) as the functional
(IV. 1))
i
-
* ) dwhere
R,(#) is the set of values of 7 which satisfy the no-scrap
ping requirement at time /# in industry s.
We are only now assembling the data necessary to estimate
a*, b*, and R,(f). So in the preliminary estimates for 1970
given in [5] we had to adopt a simpler approach based on [32].
Let us now drop the subscript s. Let Ay denote the increase
in capacity net output in a particular industry in a particular
[1] Stone - pag.
47
3
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
year; let x denote the capacity added in the year as a result
of new plant coming into operation; and let s denote the capacity
subtracted through the scrapping of old plant. Then
(IV. 14)
Av=x-s
Correspondingly, let Al denote the increase in employment;
let » denote the labour added to man the new plants; and let #
denote the reduction in employment due to plant retirements.
Then
(IV. 15)
Finally, let p denote the net output price, which is equal to
the cost of labour and capital per unit of output, and let w
denote the wage rate. Then if we multiply (IV. 14) by b and
(IV. 15), by w and subtract, we obtain
(IV. 16) pAy—wÂl =(px—wn)—(ps—wr)
= (px —wn)
if plant is scrapped when it ceases to earn a return, that is
when ps=wr. |
Now let us define the initial rate of return, #*, as the gross
rate of return to capital embodied in new plant in the first year
of its operation. Then, denoting gross investment in new plant
by v¥,
(IV. 1
7)
y*
= (px - wn)/u*
(pÂv - wAÂl)/v
from (IV. 16), on the assumption that plant is scrapped when it
ceases to earn a return. Equation (IV. 17) can be rewritten
either as
dV. 15)
oF 3 gdh Le
'1] Stone - pag. 48
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC
which shows the value of the change in capacity net output,
py, divided into a part due to labour, wil, and a part due
to capital, »* v*; or as
(IV. 10)
A: — (pAy
AT
+
U*
Ww
which shows the change in employment, Al, as equal to the
value of the change in capacity net output, py, minus the part
due to capital, #* v*, all divided by the wage rate, w. As
described in [5], estimates can be made of the terms on the
right-hand side of (IV. 19) over a future period, and this enables
us to calculate Al for the different industry groups.
On this basis, the initial stock oi assets required in 1970
is determined by capital-output ratios. The allocation of labour
to the different industries (1) is such that the labour force expected
to be available is fully used and (2) implies that the
initial rates of return (or pay-off periods) are the same as,
or related to, those observed in an earlier period. The required
changes in labour productivity in the different industries emerge
from these calculations and average out to the productivity
implied by the total increase in output and the total labour
available.
The results obtained by this method are based on less information
than would be supplied by production functions. As
a consequence they are provisional and are certainly not demonstrably
achievable. They do, however, provide a ground
for discussion with individual industries until we have deve:
loped our production functions.
b) The price circuit. At the moment, prices enter explicity
into the model only as determinants of the composition of
private consumption, government consumption being treated
in a simpler wav indicated in section 4 below Many of the
1] Stone - pag. 40
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
effects of prices appear implicity, however, in the projection
of input-output coefficients, which will also be discussed in
section 4. There is, of course, nothing final about this arrangement;
it is simply a reflection of practical difficulties.
In terms of diagram 4, we see total consumers’ expenditure
and domestic prices coming together to determine consumption
demands. The level of consumers’ expenditure is fixed by
assumption and left unchanged through each run of the calculations,
but relative prices can, initially, only be guessed at.
However, as the model simulates the productive process, it
ouilds up a cost structure in each branch of activity and this
makes it possible to revise the initial estimates of prices, as follows.
Given an average wage rate as a unit of account and
the corresponding rates of profit implied by the efficient distribution
of labour and assets, we work out the future costs of
labour and capital per unit of output in each industry and
thus obtain values added per unit of output. To these we add
the cost of intermediate inputs and of indirect taxes and thus
obtain the cost, or price, of a unit of output in each industry.
[f these new domestic prices are different from those we had
assumed at the start, we must alter the figure for total consumers’
expenditure to correspond, and repeat the cycle of calculations
until our estimates of prices cease to change.
If in the model future consumption is to be sensitive to
future prices, two things are needed: 1) a set of price-sensitive
demand functions; and 2) a set of future prices. Let us now
see how each of these requirements is met. In the following
treatment I shall restrict myself to private consumption and I
shall find it convenient to set out the analysis on a per head
basis; to apply the results to the whole community all that is
needed is to multiply them by the population.
As explained in [7], our model of consumers’ behaviour is
a variant of the linear expenditure system which allows expli-1]
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citly for changes in tastes and habits. The version we have
used so far can be summarised in the following three equations:
‘IV. 20)
‘IV. 21)
and
‘IV. 22
In (IV, 20), p denotes a vector of commodity prices and #
denotes a diagonal matrix formed from this vector; e denotes
a vector of quantities of the different commodities demanded
per head of the population; p=p’e denotes total expenditure
per head; b and c¢ denote vectors of parameters restricted only
by the fact that ?b=1; and, as usual, ¢ and I denote respectively
the unit vector and the unit matrix. In (IV. 21) and
(IV. 22), © denotes a particular year; and the starred b's and
¢’s denote vectors of parameters restricted only by the fact that
’b* =1 and ¢'b** =o.
The second row of (IV. 20) makes possible a simple interpretation
of the elements of b and ¢. The elements of ¢ represent
the components of the average consumer’s basic standard
of living and are bought whatever values are taken by 7
and p. When these purchases have been paid for, the amount
of money left over is 0 - p’c, and this is allocated to the different
commodities in proportion to the elements of b.
An obvious criticism of (IV. 20) taken in isolation is that
the elements of b and c are unlikely to remain constant over
time. The simplest means of meeting this criticism is set out
in (IV. 21) and (IV. 22), where the elements of b and c are ali
made linear functions of time.
11
Stone - pag. 51
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The results of fitting the model consisting of (IV. 20),
(IV. 21) and (IV. 22) to annual data for Britain relating to
eight main commodity groups from 1900 to 1960 are given
in [45]. On the whole the fit is good, and in two cases where
comparisons were easy to make, namely food and clothing, we
found close agreement between the total expenditure elasticities
derived from the model and the corresponding estimates derived
independently from family budgets. A still better fit was
obtained with quadratic trends in b and ¢ [39]. But such
simple trends are certainly not ideal for projection purposes
and we are now working on more complicated varieties.
The model can be fitted simultaneously to time series and
budgets [39] and can be generalised to cover adaptive behaviour,
that is gradual responses to changes in circumstances
71 [35]
The model is decomposable and so can be applied hierarchically.
This means that we can start with an analysis of main
groups, then analyse separately the sub-groups of these main
groups, then the sub-groups of the sub-groups ,and so on [7].
At each stage we can check on the performance of the model.
This is necessary because we may expect that its performance
will get worse as we go into greater and greater detail unless
we are able to take the special features of individual markets
into account. As with other parts of the main model, we are
working at present on improving it, and are trying to obtain
outside comments on the projections it yields.
This brings me to the second requirement: estimates of
future prices. We start with an extrapolation of current pricetrends
and adjust the base-year expenditure to allow for this
change in prices. This adjustment is based on the constantutility
price-index implied by the model [19], which can be
written in the form p*,/p, where
‘IV. 25) ur
I
Stone - pag. 52
21 + (0 L'oC1) Ya
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55
Here the suffixes 0 and 1 denote respectively the base year and
the projection year; and y,, denotes a geometric index of price
ratios (year I in relation to year o) with the elements of b
appropriate to the projection year as weights. The value of p,
is then a multiple of p*,, the multiplier depending on the
increase in real consumption assumed between the base year
and the projection year.
Once we have reached the end of the circuit of real flows,
we obtain, as in (IV. 8), a vector, f, of primary inputs, or
value added, per unit of output. From this we can recalculate
the price vector from the relationship
(IV. 24,
p=(l
£
+
We must now recalculate p*, and continue until the price
vector in the projection year ceases to change. Only a single
cycle of calculations is shown in diagram 4.
c) The foreign trade circuit. The introduction of foreign
trade complicates two of the relationships given so far and
adds a new one.
First, in calculating investment demands we must allow for
exports and their expected rates of growth. If we denote the
export vector bv x and the rates of growth of its components
by s, then, in place of (IV. 4), we have
(IV. 2y)
LL
(I —A) 1) (70 e + so
Second, we must allow for the fact that part of the goods
needed by the economy will come from imports, and so the
demands on domestic production will be affected. We divide
imports, the elements of a vector m, into two categories: competitive
imports, m,, namely goods like steel, cars and clothing,
which are produced in large quantities in Britain as well as
1] Stone - pag. 53
er
;
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
being ‘imported; and complementary imports, m,, namely
goods like crude oil, raw cotton and wool, which are either
not produced in Britain at all or are produced there only in
relatively small quantities. We assume: that m, depends on
‘he amount of money available from export sales and income
received from abroad, after allowing for the necessary expenditure
on complementary imports and for the sums required
for gifts or lending abroad; and that the elements of m, are
proportional to those of g, the vector of output levels. If we denote
by B the balance of trade, that is the excess of the value of
exports, p'«, over the value of all imports, p*,m, + b*,m,,
where the p*’s denote vectors of foreign prices, then, following
the argument of [7],
m=m, +m,
‘IV. 26) = [a +p*,! a, (p'x-a', P*, q—B)] + a39
where the vector a, and a, contain the intercepts and slopes in
the linear equations assumed to connect the elements of p*, my
with their total, p*,"m,; and the elements of the vector a, are
the factors of proportionality relating the elements of m, to
the corresponding elements of gq. The only element of (IV. 26)
which is so far unknown is g. It can be worked out from a
revision of (IV. 1) which, with the complication of foreign
trade, can be expressed as
q=Ag+v+e+x—m,
(IV. 27) =(—A -playa, pl) [v+e+(I—p~ta,p) x
pa”
Third, we must allow for the fact that domestic prices are
now affected because some of the inputs into British production
come from abroad. Let each element of a vector % denote the
1]
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J
share of domestic production in the total supply of a competitive
product: then in place of (IV. 24) we have
{
‘IV.
28
) p
Wp + (1-0) p3} + à p5 + À
ATTA RY pt + dup)
+,
a
The account just given of the treatment of foreign trade is
only one of three variants proposed in [7]. All three, however,
are highly simplified because they represent an attempt to do
without foreign trading functions, on which foreign trade may
be supposed to depend. The establishment of such trading
functions is quite beyond our present scope; in time, however,
a co-operative venture may make it possible to extend the model
in this way. Looking forward to that time, the following sketch
may indicate the kind of information needed.
Let us begin by rewriting (IV. 27) to indicate a separate
flow equation for complementary imports, and let us now define
these imports as goods which cannot be produced domestically
Then we can write
(IV. 20,
Most of the symbols in (IV. 29) have already been definea
The new ones are: m*,, the elements of which are the comple
mentary imports flowing directlv into final demand: ana A*
11 Stone - pag. 55
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the elements a*,, of which represent the input of complementary
import j into a unit of output of R.
Suppose that we know v, e and m*, but not x or m. From
(IV. 29) we could calculate a provisional value of the vector
lg im,!. Given the price of each product in each trading region,
we could try to allocate the demand for each element of
ig | m, | over the sources of supply by means of a price-sensitive
variant of the linear expenditure system [38]. This means
that for the jth element of ¢ we should use
IV. 30) 9 = (¢;+ Cyp)+ pt by [ny — pi (¢;+ Cy py)
where g; is a vector whose elements are the amounts of commodity
j which come from domestic production or from one
of the possible foreign sources of supply. Initially 4, is unknown
and must be adjusted until ig; is equal to the jth element
of g. The matrix C is a symmetric matrix of parameters and
is of order equal to the number of:sources of supply. A method
of estimating the elements of this matrix is suggested in [38].
If we applied (IV. 30) to each commodity in each region
we should generate a complete set of imports and exports.
These would then have to be added and subtracted to give
fv+e+x-m im*} and the whole exercise would have to
be carried out again with this vector in place of the provisional
Vu te m*,{. This process would then be continued until it
converged.
At this point we can recombine the estimates to give a threedimensional
regional trading matrix: region by region by commodity.
From all this information we can construct a region by
region trading matrix, T say. The element #,, say, of T shows
the total exports of region 7 to region s, while the element #,,
shows the total exports of region s to region r. For simplicity,
11 Stone - pag. 56
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3
[ assume that all transactions in T are recorded in terms of a
single currency, the conversions to this currency being made
by means of a supposedly consistent set of exchange rates.
The balances of trade of the different regions are given by
the excesses of their total sales over their total purchases. If
these balances are the elements of a vector ¢*, say, then
(IV. 31)
=(T -T'):
It is to be expected that #*#jo, 0, …, of. On the assumption
that trade balances are determined by buying and selling,
let us consider how to adjust the rates of exchange between
the currencies so that with the new rates of exchange
{#*= to, o, ..., 0}.
Consider first one region, » say, and one commodity, j say.
From the demand equations we can set up a matrix of order
equal to the number of regions, D,; say, the elements of which
are the derivatives of the expenditure in region » on commodity
j obtained from region s with respect to the price of j in
any one of the regions. For commodity j we can do this for
each region. Now let us form D ~ wher
(IV. 22)
where » denotes the number of regions. Having done this for
one commodity, we can do it for every other. So let us define
Dp as
(IV. = |
where m denotes the number of commoditie.
.| Stone - pag. 57
30
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Now let the initial exchange rates be the elements of a
vector x, and the adjusted exchange rates be the elements of
a vector x,. Then
(IV. 34)
(Dp) (x=! x, — 1) =(T’ = Ti
that is
(IV. 35)
x, =x, [I+ Dp)! (IT - T)]i
The elements of Dp tell us how the total expenditure of region 7
on the products of region s changes with a uniform change in
the prices of each of the regions. The elements of x," x, -1
represent the changes in the exchange rates expressed as a proportion
of their initial levels. And the elements of (T”- T}
represent the corrections to the initial balances of trade required
to give t*={o, o, ..., 0}. Consequently the set of exchange
rates which will lead to a redistribution of purchases such that
all the trade balances balance is given by (IV. 35). Evidently
there is no difficulty if we wish to put #*=#** in place of
t*= lo, 0, ..., 0}.
For each trading region the initial prices would come out
of a model analogous to our main model. Having balanced
all the balances of trade by changing the exchange rates, we
should alter in each region the relative prices of goods obtained
from different sources of supply. When we returned to the
demand equations we should obtain a new set of demands.
These would balance the balances of trade but, in general,
would alter production levels in the different regions and so:
would alter costs and prices. And so on.
Perhaps enough has been said to show why, in the first
instance, we adopted a short cut to the problem of foreign
trade.
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0.
d) The circuit of education and training. The purpose of
this circuit is to try to match the future demand for skills, as
determined by future output levels and techniques in different
industries, with the future supply of skills, as determined by
the demographic characteristics of the population and the
system of education and training. The idea is a simple one:
to get away from the assumption of homogeneity in the labour
force which is usually made in empirical work on production
functions, and to accord to labour skills an importance equal
to that usually accorded to changes in capital equipment. If the
productivity of labour is to rise, new techniques must be
adopted, but this will not come about easily unless more people
are trained in the appropriate skills. Nowadays technological
change is particularly rapid, and a failure to realise this may
make it impossible to take full advantage of the improvements
that science is offering.
The work we have done so far on this subject is contained
in [5] [6]. It relates entirely to the demand for skills and
suggests that, in Britain at any rate, the supply of the higher
and, more particularly, the medium skills is not keeping pace
with demand. This indicates the need to increase the number
of technicians and craftsmen turned out by the system of education
and training. The craftsmen, in particular, must be
trained in the newer crafts; there are many crafts that are
dving and do not need replacement.
But given that we know how the demand for skills is changing,
how are we to change the supply to meet this demand?
Our intention here is to set up an activity model of the system
of education and training, including the very important element
of retraining. By this means we hope to be able to work out
the activity levels needed in different branches of the educational
system to provide in the future an educational distribution
of the population which, after allowing for ‘wastage’, that
is for skills not used in the productive process, will ensure an
1] Stone - pag. 59
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
appropriate educational distribution of labour and thus remove
the discrepancy, or error, between supply and demand.
These calculations will produce an educational programme
relating to the transitional period and beyond, to replace in
our model the rather crude educational projections with which
we are making do at present. In its turn, this educational programme
will produce a certain pattern of educational expenditure
in 1970 and thus affect our estimates of consumption demands,
since these include, as I have said before, all expenditure
on education, both private and public, current and capital.
e) The financial circuit. This circuit is not shown on the
diagram because our work on it is not advanced enough. Our
existing model, being set in an accounting framework, automatically
generates enough income, and therefore saving, to finance
Investment at home and abroad. This does not mean, however,
that the community will necessarily want to save the
amount implied nor that its various sectors will want to hold
the portfolios of new assets and claims that will emerge as a
consequence of what is happening on the real side of the
economy. For this reason, the relationships which we intend
to study first will be concerned with saving behaviour and
preferred portfolio patterns in different sectors of the economy.
By such means we may hope to discover whether a given programme
of growth is likely to lead to financing difficulties and,
if so, what steps would be needed to overcome them.
The only work we have published so far on financial relationships
is concerned with personal saving [46].
f) The circuit of research and development. This circuit
is also not shown on the diagram and, indeed, forms no part of
our immediate programme of work. It is, however, important in
connection with the increases in productivity which emerged
from the first circuit. If these increases cannot be met as
things stand, perhaps research and development could bring us
nearer to meeting them. Although it is difficult to find a rela-1]
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Ro
tionship between productivity and research and development,
it is fairly clear that one exists. Further study should show
where the main opportunities lie.
THE METHODS OF ESTIMATION
Like the relationships of the model, the methods of estimation
go through various stages of development. This is true
both of the initial values and of the parameters. The typical
procedure is to start either with existing estimates or with the
econometric analysis of past observations. The parameters in
both the consumption functions and the input-output relationships
vary systematically with time, and so the next step is
to calculate their values in 1970. These results are then scrutinised
with the help of casual empirical knowledge to see how
far they seem sensible, and some adjustments are made. Finally
they are discussed, whenever possible, with outside experts,
and further adjustments are made. So far we have not
got much beyond the early stages of this process; in particular.
our main attack on the last stage is only just beginning.
Let me now illustrate this sequence by outlining the methods
of estimation we have used in the order in which they occur
in the calculations.
a) Exogenous final demand.
This category can be divided into five components.
1) Private consumers’ expenditure. Here we began vy
applying the model consisting of (IV. 20), (IV. 21) and (IV. 24),
and a similar model which makes use of quadratic trends, to
eight major groups of expenditure. The parameters were estimated
from annual data covering the period 1goo-1g60 with
the exception of the years 1914-1919 and 1940-1947. This
was done by means of an iterative, two-stage, least-squares procedure.
as follows.
11 Stone - pag. 61
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
We begin by guessing values of b* and b** in (IV. 21),
which I shall here denote by b*, and b**,. We then form a
vector, y, say, of type nx I, where n denotes the number of
groups of expenditure; thus
(IV. 36) Yo = Pa € — (b* + 96°) v,
We also form a matrix, Y» say, of order n, where
(IV. 37) Ya = [1— (63 + 958) #1 pe
Apart from a random element, y, and Y, are connected by
the relationship
(IV. 38)
Yo = [Yq : OY]
I c*
; p**
If, denoting successive time-periods by 1, 2, ..., t, we now
define
(IV. 39)
VE} 15 Vas oer Vil
111d
(IV. 40
VV.
1Y,, Y,, ..., Y,!
we can write, apart from a random element
(IV. 41)
Xo
1] Stone - pag. 62
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55
where X—[Y ©Y1 and g=}c*:c**{ The least-squares estimator,
g,, of gis
(IV. 42)
oo (XX) N-Given
g,, we can form a vector, w, say, of type x
where
IV. 43)
Po [eg — (c] + BcT):
and a matrix, W say, of order n, where
(IV. 44)
Wa = [tq — Po (cf + Bet
Apart from a random element, w, and W, are connected
the relationship
TV. 45)
Te Nu. 6
UU
If we now define
(IV. 46)
w=lw,, Ww, ..., W;}
and
(IV. 47
W-}
VV
, W
7
…, W
we can write, apart from a random element,
‘IV. 48)
7h
g. 03
S - pag. 6
e
1 Ston
dy
56
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
where Z = [WOW] and h = {b* b**}, The least-squares
estimator, h,, of h is
(IV.49) h, =(Z', 27! Z'w
Given %;, we can return to (IV. 38), replace b*, and b**,
by b*, and b**, and calculate the next approximation to g,
namely g,= {}c*,:c**,{. If we continue in this way until the
estimates cease to change, we shall have reached a solution.
We can see from (IV. 44) than W;, is a scalar matrix, and
50 in estimating » the system of equations breaks down into a
set of single equations. From (IV. 37) we can see that Y , is not
a scalar matrix, and so in estimating g we are, in effect, obtaining
average values derived from all the equations. From
‘IV, 20) we can see that bp appears as a separate term on the
right-hand side, and so, since p’e=p, it follows from the addingup
theorem that the constraint ’b=1 is automatically satisfied
by the more complicated form in (IV. 21).
Further details of this procedure and of the results and
orojections obtained by it for the eight expenditure groups are
given in [39] [45]. We are at present working on combining
these estimates with those obtained from family budgets and
on analysing the components of each main group by the same
procedure.
Until this work is completed we have to use more rough and
ready methods. What we do is to estimate the levels of expenditure
on the components of each main group by reference
to their changing relative importance within the group; for
example, within the food group the proportion spent on bread
and cereals tends to fall with time, whereas the proportion spent
on meat, fruit and vegetables increases at a rate well above
the group’s average. We also try to allow subjectively for
the tempo of substitutions, such as an acceleration of the substitution
of electricity and oil for coal as domestic fuels.
1] Stone - pag. 64
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67
Once we have estimated the future values of these components,
which correspond to the forty categories into which we
divide consumers’ expenditure, we then have the further task
of converting them into demands for the principal products
of our thirty-one industries, demands for complementary imports
such as tea, cigars and wine bottled abroad, demands
for direct labour such as domestic servants, and payments of
certain indirect taxes. Here we have based our calculations
largely on our classification converter for 1960.
Eventually, in view of the increasing importance of consumers’
durables, for which the process of adaptation is relatively
slow, we hope to use the adaptive version of the demand model
described in [7] [35]7.
2) Public consumption. Here we have made use of the
trends suggested in the report of the National Economic Development
Council [50]. Eventually we hope to get a new
view on some of the components of public consumption
through the addition of new circuits to the model, such as the
circuit relating to education and training.
3) Public expenditure on social capital. These estimates
are rough and subjective. For example, investment in educational
buildings is assumed to rise in proportion to current
expenditure on education; the road-building programme is assumed
to treble between 1960 and 1970. Gross investment in
dwellings is similarly estimated at the present stage.
4) Exports. Here we have again based ourselves on the
work of the National Economic Development Council [50]
but have scaled down their annual growth rates for 1061-1066
in making our estimates for 1060-1940.
5) Industrial replacements. These are based on a study
of investment statistics and of the life-spans of different types
1] Stone - pag. 65
58
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
fe
of asset in different industries [10]. We have not yet reached
the stage of replacing this physical determination of scrapping
by an economic one.
b) Endogenous final demand.
This category can be divided into three components.
1) Industrial extensions. These are based on a calculation
of the type of (IV. 25) and thus involve projected input-output
matrices, both current and capital, as well as levels and growth
rates of exogenous final demand. The capital coefficients were
estimated with the help of the material relating to the years
[948-1960 brought together in [10], amended in many cases
in the light of outside information.
2) Investments in stocks. These are related to changes
in output levels on the basis of experience in the 1950’s.
3) Imports. These are based on a calculation of the type
of (IV. 26). The first step is to set a figure on the balance of
trade to be aimed at in the future. If this is subtracted from
export proceeds, the total value of imports is obtained.
Complementary intermediate imports are estimated by fixed
coefficients applied to the outputs of the various using industries,
in exactly the same way as any other form of intermediate
input. These coefficients are based on postwar trends
and at present are very rough, because the classification of
imports has only been carried out for the three years 1948,
1054 and 1960.
Individual groups of competitive imports, as explained in
section 3(c) above, are treated as linear functions of the total
amount of money available for competitive imports as a whole.
These coefficients are roughly assessed on the basis of recent
experience.
Considerable improvements could be made in all these
estimates simply by a more thorough analysis of existing
1] Stone - pag. 66
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69
foreign trade statistics. We hope to undertake this work before
long.
c) Intermediate demand.
The last element needed in calculating total output is intermediate
demand. With the help of a projected current inputoutput
matrix we can then bring all the estimates together as
in (IV. 27).
The methods we have used to bring up to date and project
the input-output coefficients are described in detail in [0].
Apart from complications arising from changes in classification,
the distinction between products and industries and similar
practical problems, our procedure can be outlined as follows.
First, we bring the coefficient matrix estimated directly for
1954 [47] up to 1960 by adjusting the rows and columns of the
corresponding transaction table to agree with marginal totals
of intermediate outputs and inputs. These adjustments are
based on the assumption that changes in coefficients are due
to three factors: 1) price changes; 2) substitution effects which
influence all the elements in a given row; and 3) fabrication
effects which influence all the elements in a given column. On
the further assumption that the second factor operates uniformly
along the rows of the matrix and that the third factor operates
uniformly along its columns, the problem and its solution can
be formulated as follows.
Let A, denote a known, initial matrix of input-output coefficients,
and let A denote the unknown matrix for period 1
which we wish to estimate. Let p denote a price vector whose
elements are ratios of prices in period 1 to prices in period o,
and let » and s denote vectors of unknown constants. Then,
on the assumptions made
(IV. 50,
1] Stone - pag. 6;
10
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
say, where A*=pAp1. If we have, for period 1, an intermedlate
output vector, u say, an intermediate input vector, v say,
and a vector of total outputs, q say, then
(IV. 51)
Ag=u
and
(IV. 52)
GA i=v
We can make an initial estimate, u, say, of » by premultiplying
q by A*. Thus
(IV. 53)
A*q =
=u,
In general uy#u, but we can force an equality by an appropriate
multiplication of the rows of A*. Thus
(IV. 54) (un, -A*)g =u
[f we regard the term in brackets as an estimate of A, it can
be seen that it satisfies the row conditions but not the column
conditions. These can be satisfied by a substitution for A from
(IV. 54) into (IV. 52), followed by an appropriate multiplication
of the columns of A*. Thus
(IV. 55)
A $7 AN 1
gA* wu," li=v,
and
‘IV. 56)
A An 1 , AN I.
g(vvy~'A* uu, i =v
[f we regard the term in brackets as a revised estimate of A,
we can see that it now satisfies the column conditions, but not,
in general, the row conditions. We can, however, repeat the
cycle of operations until we obtain convergence with both the
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SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
row and column conditions satisfied. Thus, after » + 1 iterations.
we shall obtain
Sadly —1 451 A xR+L, =1 991
(IV. 57) ("Huet out Axo tly = oe =u,
For a sufficiently large #, the term in brackets in (IV. 57)
can be taken as an estimate of A. This we call the RAS method.
The Belgian tests described in [9] show that the RAS
method works well, provided that it is possible to estimate
the controlling totals # and v, accurately and that certain coefficients,
whose determination is different from that expressed
by the theory, can be detected and estimated directly. For
example, there has been a general tendency for coal input-coefficients
to fall as a result of the competition of electricity and
oil; but this tendency is at work only where coal is used as a
fuel and not where it is a raw material, as in coke ovens. Since
the theory is incapable of handling such exceptional cases and
since coke ovens use a lot of coal, it is important to estimate the
input of coal into coke ovens independently, remove this
amount of coal from the transaction table and the controlling
totals, and add it back after the remaining items in the matrix
have been calculated.
An up to date matrix obtained in this way can only be
approximate, and the next thing to do is to discuss the entries
with experts in the different industries. In many cases significant
improvements can be made in this way, but there will
always remain a number of industries for which little or no
up to date information can be obtained and which will still
have to be handled in a theoretical way.
For purposes of projection, the theory can only offer the
simple method of extrapolating the coefficients along exponential
trends. Thus if A, denotes the estimated coefficient matrix
for year 1, and if A,, denotes the coefficient matrix for a future
year, 2, expressed at the prices of year 1, then
(IV. 50,
1] Stone - pag. 69
=
4
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
[f the intervals between 0 and 1 and between 1 and 2 are the
same, then ® =1.
The effect of this method of projection will be to change
each coefficient in the way that it has been changing in the
past. This can be only approximately correct, and so it is of
particular importance to get as much direct information from
outside as possible. For example, in Britain between 1954 and
1960 the oil component of the input of fuel into electricity generation
rose from a very small figure to nearly 20%. A conlinuation
of this trend would make oil the dominant fuel input
by 1970. But we know that this particular trend will not
continue as in the past because of the kind of fuel used in
generators that have been built very recently and are planned
for the immediate future. The direct information can come
either from current statistics of input-output coefficients which
are available in a limited number of cases, for example coal
used in coke ovens, or from outside knowledge, as in the case
of oil used in the generation of electricity.
We have used such direct information wherever possible
and then applied (IV. 58) to the remaining elements of the
input-output matrix. We have then discussed the results of
this exercise cell by cell and changed the projections subjectively
if this seemed desirable. The results of such a survey are
illuminating. For example, we were surprised to find that in
our projections the coefficient for machine parts and repairs
into agriculture rose, indicating a greater use of machinery,
while the coefficient for petroleum products fell. We were told
that this was not really surprising because of the substitution
of the less processed and cheaper diesel oil for petrol and that
our results were a reflection of this particular type of fuel
economy. We therefore did not change our provisional result
in this case, although we did in manv others.
d) Output levels. If we bring together the foregoing calculations,
we can estimate output levels in 1970. In the preli-11
Stone - pag. 70
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
73
minary calculations described in [5] we found that the increases
of output required of our thirty-one industries during
the decade of the 1960’s varied from zero to over 100%.
Apart from the allowance, already mentioned, for a trend
projection in the prices of the main groups of private consumption,
the estimates were all made at 1960 prices. In other
words, only one cycle of calculations was made. We did not
calculate shadow prices for 1970 and allow these to modify
the composition of consumption or the technical coefficients
of production. We plan to do this as soon as our production
functions are readv for use.
e) The industrial distribution of labour. These calculations
were based on (IV. 10). We first estimated the labour
force available to our thirty-one industries in 1970, starting
from official projections of the total labour force [49]. We
allowed for 1.5%, of unemployment and for government demands
in line with our estimates of government expenditure
on different purposes. The result was a 5% increase in labour
available over the decade compared with a 50%, increase in
output.
In applying (IV. 19) we took the increases in output at
1960 prices, already calculated, to represent pAy. We based
our estimates of the initial rates of return, r*, on the experience
of the period 1948-1960, subject to a minimum rate of return
of 5%. We estimated investment in fixed assets over the
decade, v*, as five times the sum of the 1960 level and the
1970 level, the latter being obtained by adding together our
estimates of industrial replacements and extensions. We estimated
the industrial real-wage rates over the decade, w, by
increasing the 1960 wage rates in the different industries by
one half of the increase in labour productivity required over
the decade in industry as a whole. On this basis, the total
labour demanded was 49%, in excess of the estimated supply
.] Stone - pag.
71
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
This suggests either that more capital would be needed in 1970,
or that investment through the 1960’s would have to average
more than the arithmetic mean of the initial and terminal
years, or that initial rates of return in the 1960’s would have to
be a little higher than in the 1950’s. We did not attempt to
resolve these questions in [5] but simply increased by a
constant proportion the initial rate of return in each industry,
thus changing the allocations of labour until, in total, they
were equal to the supply.
From these calculations we obtained preliminary estimates
of the changes required in the distribution and productivity of
labour during the 1960’s. We found that virtually the whole
of the increase in labour would be absorbed by the distributive
and service trades, leaving a stationary labour force to be
shared by the rest of industry. The calculated increases in
productivity varied widely from industry to industry. On the
whole they were higher than in the 1950’s but often not sensationally
so; in some cases they were lower.
We propose to discuss these estimates with outside experts
as opportunity arises, but we shall not go out of our way to
do this until our work on production functions is complete.
f) Changes in the spectrum of skills. The work we have
done so far on this subject is described in [5] [6]. We have
divided the labour force into three main functions, managerial,
clerical and technical, and have subdivided the last of these
into five categories, qualified manpower, technicians, craftsmen,
operatives and unskilled. From the census of population
of 1951 we can estimate the number of men and women in
each of these seven classes for each of our thirty-one industries
and for government services. For 1961 the census results are
not yet available and so we have made provisional estimates
based on statistics of unemployment and vacancies. In making
these estimates, we have tried to allow for the economic fluct]
Stone - pag. 72
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
tuations, technological changes and sociological pressures which
affect the labour market.
The picture which emerges from unemployment statistics
we take to represent a past state of technology, and the picture
which emerges from the statistics of vacancies we take to represent
a future state. Our estimates for 1961 are obtained
by giving each picture equal weight. We found that some
skills, namely managers, qualified manpower, technicians and
operatives, appear to have moved in line with the demands of
technology over the 1950's. We then worked out the weights
that would best reproduce the 1951 position for these four skills
and found that they were 0.9 for unemployment and o.1 for
vacancies. We then reversed these weights and applied them
to all skills to obtain a first approximation to the spectrum of
skills for 1970. Obviously this is a very crude method of estimation,
and we have only used it as a first device to feel our way
into the subject.
We now have for 1970 a provisional vector showing the
distribution of labour by skills and, from the calculations
described under e) above, a provisional vector showing the
distribution of labour by industries. If we consider an employment
matrix with skills in the rows and industries in the columns,
we see that these two vectors provide its marginal
totals for 1970. We can now try to fill in this matrix for 1970
by the RAS method, using the corresponding matrix for 1961
as a starting point and the 1970 vectors as controlling totals.
We have carried out this exercise and found that convergence
is almost immediate. Since the marginal totals were
estimated independently, this result suggests that it may be
possible to derive a simple relationships between changes in
productivity and tbe skill distribution of the labour force in
each industry. But at present this is a surmise whose verification
must await further analysis.
1] Stone - pag. 734
hy
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
5. THE COMPUTING SEQUENCE
The model has been programmed for EDSAC 2 at the University
Mathematical Laboratory and, more recently, for FEr-RANTI'S
Atlas. The programme is written in stages, each of
which can be modified as the need arises without disturbing
‘he others. The present computing sequence, which does not
include either the educational or the financial circuit, consists
of seven stages. These are summarised in table 2 on p. 75,
taken from [44].
So far, only the first five stages, o through 4, have been
fully programmed. We are now engaged in programming
stage 5, which had previsiously been calculated separately,
and this will enable us to introduce shadow prices into our
results. Stage 6, the final compilation, has still to be carried
out by hand.
Some idea of the size of the programme as it now stands
can be gained from the following figures. The numerical inputs,
‘hat is the parameters and conditions, needed for a computerrun
number between five and six thousand. A run involves
about thirty million multiplications: on a desk calculator this
is equivalent to some sixty man-years of work; on Atlas it
takes twanty-two seconds.
i Stone - pag. 74
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
+ 3
TABLE
THE COMPUTING SEQUEN(
Operations perforr
Prepare estimates of private and public consumption,
expenditure on social capital and foreign demands
for exports in 1970, and their trends through that
vear
Convert the elements in stage o into demands for
the products of the thirty-one industries, complementary
imports, etc., and the rates of change in
these demands
Project past input-output matrix and combine with
outside information to give 1070 matrix.
Combine stages 1 and 2 to give estimates of the
outputs of industrial products based on 1) demands
from stage 1, 2) capital expenditure to provide for
growth in demand and 3) indirect requirements for
intermediate products, allowance being made for import
requirements and the need to achieve a trade
balance
Convert the outputs of stage 3 into production levels
.n the thirtv-one industries.
Allow for changes in the productivity of labour and
capital, and estimate requirements for them in each
3f the industries
Combine the foregoing results with certain other intormation
and print out a provisional social account-‘ng
matrix for 1970
{] Stone - pag. 75
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
THE TRANSIENT MODEL
I. INTRODUCTION
This model is still at the preparatory stage. The variables
are the same as in the steady-state model. The relationships,
already described in [40], are outlined below. The estimation
problems are similar to those in the steady-state model. The
computations are carried out by dynamic programming. We
are assembling the material for a trial run with a four industry
model and have written a programme in which the calculations
proceed year by year. In this way the full model is kept within
‘he capacity of the computer.
Since we have not yet made any calculations with this model,
I shall describe here only the, relationships we propose to
use, limiting myself to the case of a closed economy with
constant parameters. In [40] it is shown that these simplifications
can be dispensed with.
2. THE RELATIONSHIPS
The relationships of the model are set out below. Unless
otherwise stated, the notation follows that developed in the
preceding chapter.
First, in the base year at the outset of the transitional period,
the vector, s, of the economy’s stock of capital goods is
equal to a given value s say. That is,
V. 1
‘1] Stone - pag. 76
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
74
The elements of s and s relate to the total stock of capital goods,
grouped according to the producing industry, not according
to the using industry.
Second, at the end of the transitional period, which runs
from ®=1 to ®=1- 1, the terminal stock vector, E,s, must
have a certain minimum value, s say, to make possible the
production levels in the first year of the state of steady growth
as determined by the long-run model. That is,
(V. 2)
Third, at all times there must be adequate capacity to make
possible the level of production decided on. That is,
(V. 3)
pa
- KE’ ¢q
KX (I-A)! E' (e+v)
FE? (e+ Es
where F = K (I - Ay-!. Equation (V. 3) can be written in the
form
(V. 4) FE!e< (1+ F)E’s - FE*+!
Fourth, at all times the labour force must be large enough
to make possible the level of production decided on. If À
denotes the total labour force and f* denotes the vector of labour
requirements per unit of output in the different industries, then
(V. 5)
ES
EM
Fifth, if the demand functions ot (lV. 20) are premultiplied
by Fp, it follows
(V. 6)
a
Sy
EE —
F
g
+
FRE
Cs
1] Stone - pag. 77
Te
je
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
where g=(I - hp’)c and h=p-"b. By combining (V. 4) and
(V. 6), we see that
(V. 7) FhE'p<(L+F)E®s - FR*+15s - Fe
Sixth, we should probably wish to ensure a minimum level
of consumption in each year of the transitional period. This
can be expressed as
(V. 8)
Ew
>u*
where p* denotes a preassigned minimum level. Alternatively,
we might prefer to ensure that consumption did not fall during
the transitional period, in which case p* in (V.8) would have
to be replaced by E®! p.
Finally (V. 1), (V. 2), (V. 5), (V. 7) and (V. 8) form constraints
which limit any short-run policy; the outstanding
question is what policy to pursue. Since the terminal conditions
ensure that the long-run policy can be realised at the end ot
the transitional period, the obvious course is to maximise the
utility of consumption during the transitional period subject to
the above constraints. If we denote the utility of consumption
in year ® by Ev, then
V. 9)
ESu = & [IN (Ef es — cg)'t]
where D denotes an arbitrary monotonic function and § denotes
the typical commodity. Thus we should have to maximise
(V. 10)
+1
2
—-0)
T-1
Edu=S ® [ll (ES e; — c's]
8-0 =
1] Stone - pag. 78
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
In order to make use of this expression we should have to
assign a form to ®. The simple thing to do is to put ®=log; in
this case we maximise a weighted sum of the logarithms of the
excesses of the consumption of the various commodities in the
different years over the quantities that enter into the basic
standard of living. This form is only possible if at all times
each element in round brackets in (V. 10) is positive. As even
simpler practical alternative would be to replace v by , that
is to maximise not utility but consumption itself.
As I have stated them, these relationships apply to an economy
which is not only closed but stationary, that is has a
fixed technology and fixed preferences. The way to remove
these limitations is described in [40].
The maximisation of (V. 10) subject to (V. 1), (V. 2),
(V. 5), (V. 7) and (V. 8) is, for practical purposes, a problem
in dynamic programming. If the terminal stock requirements
are set too high there will be no solution: we cannot meet these
requirements and have a consumption in excess of * throughout
the transitional period. If we insist on p* as a minimum,
then we must reduce our terminal stock requirements. If we
insist on the rate of growth originally intended for the steady
state, we must reduce the level of consumption in the first year
of the steady state. By experimenting with different initial
consumption levels for the steady state, and perhaps also with
different forms of the maximand, we may hope to obtain a
complete path for consumption that is acceptable as an object
of policv
2] Stone - pag. 79
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
V1
CONCLUSIONS
I shall bring this paper to an end by summarising the philosophy
of economic model-building which we are trying to
follow in our work.
First, it is useful at the outset to picture a model against
the general background of knowledge, objectives and controls
which would have to be taken into account if the model were
ised for practical purposes.
Second, the main use of a model is to help us in exploring
possible worlds; in examining, that is to say, not only how a
particular economic system works but how it might work, and
n relating the performance of the component parts to the
needs of the system as a whole.
Third, a model must have coherence and realism. Coherence
can be achieved by giving the model a suitable structure;
realism is quite a different matter. To achieve realism we must
use relationships which recognise the changing character of
preferences and technology. Our initial efforts to do this may
not give very accurate results, partly because of the difficulty
of formulating relationships which are both realistic and manageable
and partly because the factual information relating to
‘he past is incomplete. But this does not mean that we should
stay content with first approximations.
Fourth, in choosing the relationships that are needed, such
as consumption functions, production functions and so on,
accepted economic theory provides a valuable guide. In formulating
these relationships, however, economic theory is much
less useful because most of its elaborated parts are based on
a narrow, static view of the world. Within this narrow view,
1] Stone - pag. 80
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
SI
great generality is usually sought and frequently achieved. In
practice it is often necessary to adopt a formulation at the
same time less general than theorists would like within the
range of phenomena they consider, and less restricted in its
range. For example, the consumption functions we are using
cannot handle complementary or inferior groups of commodities
but they can handle systematic changes in preferences and also
adaptive behaviour.
Fifth, is assembling observations on which to base our
projections, we can achieve a great deal by the careful processing
of existing data, but however thoroughly we do this our
knowledge remains incomplete. We must therefore try to gain
the cooperation of outside experts in practical walks of life,
who may be less well placed than we are to attempt a synoptic
view of the whole economic system, but whose specific knowledge
is always greater than ours. The realism of our projections
can only increase as we succeed in getting more reliable information
into the model.
Sixth, to be a useful tool for policy-making a model must
enable us to make not just one but many alternative projections
based on different assumptions. When it has reached this stage
the model becomes in its turn a source of information in the
light of which a policy can be drawn up. If this policy is
carried out, the model can then be used to make predictions.
Seventh, no policy can be carried out without a control
system which keeps the plan in touch with events. This control
system consists of a mixture of centralised and decentralised
administrative machinery, including all private arrangements
for the management of businesses, cooperatives, labour unions
and so on. The model can be used to review the control system
and show how administrative methods might be modified so as
to improve the economy’s inherent tendency to stability. I have
not tried to formulate this range of problems because as yet we
have done very little work on them.
Eighth, it is impossible to plan unless one knows what one
‘11 Stone - pag. 81
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
is planning for. With this in mind, we have divided our model
into two parts, a long-run model and a short-run model, and
a solution comes from iterating between the two. The purpose
of the long-run model is to help in choosing a direction and
a rate of steady expansion to be achieved from 1970 onwards;
and the purpose of the short-run model is to choose a path to
this goal. If the model were used for policy-making, it would
of course be necessary to take a new view of long-run objectives
at regular intervals.
Ninth, an economic model should cover all aspects of economic
activity. As it stands at present, our model is restricted
mainly to the real side of the national economy, but we are
now beginning to extend it into the financial sphere and hope
eventually to develop it in other directions as well.
Finally, we have started with a purely economic model,
conceived on traditional lines, because there we felt on reasonably
firm ground. But we believe that the main motive forces
of economic growth are to be found in human abilities and
attitudes: organising capacity, acceptance of education and
training, response to innovation, labour mobility, and so on.
However, we could hardly have begun with these indefinite and
on the whole badly documented areas of interest; and in any
case it would have been useless to do so until we could embody
them in a coherent picture of the socio-economic system. So,
naturally enough, we decided to build out from the familiar
and to use our working experience as the starting point for
our work.
Pythagoras’ remark, dgyn 8e tot fjpov mavtog, ‘the beginaing
is half the whole’, applies to social and economic modelbuilding
as it does to all complicated human endeavours. The
only practical course is to build a prototype and then improve
on it in the light of experience and needs. It is fairly safe to
say that the modern aeroplane would never have come into
being if no aeroplanes had been flown until they were as good
as they are today.
'1] Stone - pag. 82
SEMAINE D’ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
~
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U.K., Boarp oF TRADE and CENTRAL STATISTICAL OFFICE: Input-Outpu-Tables
for the United Kingdom, 1954. Studies in Official Statistics
No. 8. H.M.S.0., London, 1961.
"481 U.K., CENTRAL STATISTICAL OFFICE: National Income and Expenditur.
H.M.S.O., London, annually.
49] U.K., MiNisTRY OF LABOUR: Ministry of Labour Gazette. .l.i
London, monthly (September, 1962).
‘50] U.K., NATIONAL Economic DEVELOPMENT COUNCIL: Growth
United Kingdom Economy to 1966. H.M.S.O., London, 1963
D.
1] Stone - pag. 85
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CONTENTS
FOREWORD.
MODELLING ECONOMIC SYSTEMS. 1. The background. - 2. Models and
their environment.
MODELS, POLICIES AND PLANS. I. Models: a) Variables, b) Relationships.
c) Estimation, d) Computability. - 2. Policies. - 3. Plans.
II. A DUAL MODEL OF ECONOMIC GROWTH. I. A new development. - 2. Steady
states and transient states. - 3. Why a dual model?
THE STEADY-STATE MODEL, 1. Introduction. - 2. The variables. - 3. The
relationships: a) The circuit of real flows, b) The price circuit, ¢) The
foreign trade circuit, d) The circuit od education and training, e) The
financial circuit, f) The circuit of research and development. - 4. The
methods of estimation: a) Exogenous final demand, b) Endogenous
final demand, ¢) Intermediate demand, d) Output levels, e) The industrial
distribution of labour, f) Changes in the spectrum of skills. -
5. The computing sequence.
«THE TRANSIENT MODEL. 1. Introduction. - 2. The relationships.
VI. CONCLUSIONS.
À LIST OF WORKS CITED.
DIAGRAMS
1, À model in its environment. - 2. A corrective device for economic
models. - 3. A dual model of economic growth. - 4. The steady-state model
TABLES
1. A provisional social accounting matrix for Britain, 1962. - 2. The
computing sequence.
Stone - pag. 86
DISCUSSION
MAHALANOBI:
The introduction to what I may perhaps call the genera’ -' o
sophy of development of Professor Stone, I feel, has © >= in
extremely appropriate, and a most significant contribution the
Study Week. At a later stage I should like to make some observa.
tions on the logical and philosophical aspects of Professor STONE’s
paper rather than on the details of the mode’
FRISCH
If Prof. MAHALANOBIS feels that he is at this moment prepared to
put questions or discuss the philosophical aspect., * think “+ II
means that he should be allowed to do so now
MAHALANOBI:
My first point is that any domain of interest in which observations
have an operational meaning, that is, have a meaning in the sense
of science, may be considered to be a part of (what I shall call) the
world of reality. This world of reality must be distinguished from
the world of mathematics, of logic, of purely formal relations. My
next point is that in this world of reality we are always concerned
with a bounded part of it, that is, limited in both space and time. So.
1] Stone - pag. 37
"= oy
du
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
we have to start with some bounded part of the world of reality
which I may call « R» (in a diagram on the blackboard). This
hounded part of reality consists of certain elements or elementary
units which also, in my view, are finite in number. It is possible
to make observation(s), one or more, on each or some of these
elementary units. In this way we get a system of observations which
[ may call « O » (shown in the blackboard) which has to be distinguished
from « R », the reality, but is based in some sense on this
reality. With any given set of observations « O », it is possible to
make a model, which I may call «M». Now, there are two gaps,
which are important: the system of observations « O » may not be
valid, that is, may not be adequately representative of « R » the
world of reality. Secondly, a model « M » (or models) may not be
relevant or adequate. At this stage, objectives or aims must be
taken into consideration to examine whether any particular model
« M » is relevant or adequate. For the Study Week the objective
or aim is economic development; I am including the question of
fluctuations within development.
Model making, in the sense of Professor STONE, would belong to
the world of reality, and not to the world of abstraction — if I have
1nderstood him right, and must be, therefore, limited in time.
Now, I shall pass on to a second point. In this Study Week we
are interested in models in the world of reality. But models of what
types? Coming from the under-developed areas, I suggest, ultimately
not necessarily during this Study Week) the aim of economic development
must be that of the world as a whole.
There is the system of the world as a whole, bounded, of course,
in space and also in time, that is, time up to what is of interest to
as, 1970 or 1980 or 2000 but perhaps not 5,000 A.D. If I take the
system of the world as a whole, we have three broad areas: briefly:
WEST and EAST (in the political sense) and DEVELOPING countries.
I think we have to think of the world system as acting and
interacting between these three areas, rather than as a single unit.
I shall use the symbols W, E and D (on the blackboard) without
trying to define where is the exact geographical boundary of the
-q
Stone - pag. 88
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_
West, or of the East, or of the Developing countries; these areas are
defined in fact in terms of political tensions and interrelations which
also are shifting over time. This band (points out something on
blackboard) is the relation between East and West, and there is
the Developing countries; so, the world system would consist of W,
E and D and their interrelations. To some extent, Professor LEONTIEF
has taken this into account in his paper in calculating what should
go from either W or E to D.
We thus have a kind of world system of W, E, and D. Then we
have to consider also the national system of each particular country.
In model making, there will continually be the need to think of the
different national systems as constituent elements of the world
system. However, there would be need of delimiting the sphere of
‘nterest according to the purpose in view. We may have to consider
a particular national system or a particular sector or groups of sectors
within a national system. A nation or a country is, however, not
the only basis of delimiting the sphere of interest. There will be
many other ways of delimiting the system, for example, in external
trade we have bilateral or multilateral systems. It seems to me that
it would be a good step forward to have an agreed terminology in
this whole business of model making. The two concepts « macro »
and « micro » require to be arranged in some kind of a logical system
of hierarchy involving dimensions in space, in terms of economic
sectors, and also of time. It would be then possible to view the
micro-models at different levels fitting into a macro-model at a
higher level, and then these macro-models fitting into wider macromodels
at still higher levels until the global world system is reached.
Going back to my first point, if I have understood Professor
STONE correctly, there would be need of taking into consideration
experience or results of experiments to assess the validity of the
model « M » in respect of the system of observations « O », and
finally in relation to the world of reality « R ». When we speak of
experience it would, I think, be of great help to keep in mind that
a model developed in one country, on the basis of experience of that
country, it may be verified or experimented upon, or modified in
11 Stone - pag. So
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 20
some other country. It is possible to make models which are realistic
in the sense of Professor STONE but which cannot be verified
in the U.S.A., because there is not much of central planning there,
but which are capable of being verified in the U.S.S.R., or which
may find great opportunities for experimentation in a centrally planned
country like U.S.S.R.
To sum up, I suggest that it would be useful to go into the
question of terminology at one stage, to identify the hierarchical
levels of multi-dimensional analysis. Professor FRISCH may be able
to think of suitable words. An appropriate terminology would be of
help in fitting « macro » or « micro » models at different levels into
a global frame of the world as a whole, and to delimit different
domains of interest (also of a multi-dimensional character, and covering,
as necessary, both social and political aspects) to suit the purpose
in view.
WoLD
It is most appropriate that Prof. MAHALANOBIS has given us some
broad indications towards a general frame of reference for our Study
Week. Prof. STONE’s excellent paper gave a very good start in this
direction, and Prof. MAHALANOBIS has now emphasized the regional
view of the globe. I would like to take up another basic aspect,
also with a view to clarify fundamental ideas, namely the general
philosophy of model building. The need for a broad consensus about
views and terminology is here so much the more pressing, as the literature
of professional philosophy leaves much to be desired in this
respect. The situation is somewhat paradoxical, for in almost any
creatise of philosophy of science we find a laudable introductory
statement to the effect that the area of the treatise is the procedures
and methods in current use in the many branches of science, the
purpose of philosophy of science being to study and assess the general
principles of scientific inference; on the other hand it is clear that
although the approach of model building by and large is all pervading
'1] Stone - pag. 90
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
03
in the entire domain of natural and human sciences, the treatises of
philosophy of science have very little if anything to say about the
general principles of model building.
It is not my intention to give here a general review of the prin
ciples of model building. My main point is to emphasize a fundamental
distinction between three aspiration levels in scientific model
building. The first is fact finding, the second is understanding, the
third is prediction. Correspondingly, we may talk about
(1) descriptive moc
(2) explanator. a:
(3) forecasting moc.
Fact-finding answers questions of the type: « What has happened?
» The answers are given by observed actual facts, and the
observations are organized in a more or less systematic fashion into
a descriptive model. Explanatory models answer questions of the
‘ype « Why has it happened? » Speaking broadly, the answers involve
an element of causal inference, and the model makes a joint
construct of theoretical analysis and empirical observation. Forecasting
models answer questions of the type « What will happen? »
Explanatory models are based on past experience, and when such
a model is used for prediction it constitutes a forecasting model.
The three types of models thus represent stages of scientific evolution
towards higher aspiration levels. If we examine the various
branches of scientific endeavour from this point of view, we find that
there are great differences. Some sciences stride along happily at the
level of fact-finding and description. Others have succeeded to
assess a stable causal pattern in the observed facts, thereby ascending
to the level of reliable explanatory models. Still more advanced are
the sciences where the explanatory models are reliable enough to
provide valid forecasts of future events.
Economics and econometrics, I believe, have reached this third
stage rather recently. The situation is far from uniform. Certain
areas of economics are well covered by reliable explanatory and fore-[1]
Stone - pag. 91
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 23
casting models, whereas other areas are more or less white spots on
the map of successful model building. This point of view goes to
the core of our proceedings: The very theme of our Study Week
makes a challenge to econometrics to give evidence how far our
science has reached towards the goal of valid explanatory and forecasting
models in two areas of paramount importance: economic
growth and business cycles.
I should like to develop this point a bit more. Let us take very
briefly one example: meteorology. Fact-finding includes of course
everything about the weather. The theory of meteorology has developed
gradually, but it is rather recently that explanatory models
were built that could be exploited for reliable forecasts. The breakthrough
was the thermodynamic theory of cyclones, founded by the
Bergen school around 1918, with J. BJERKNES and T. BERGERON as
leading names.
Cold r
——,
Warm
With reference to the graph, the theory says that cyclones
tend to develop on the border between cold polar air and
warm equatorial air; on the northern hemisphere the cyclones move
eastwards, go toward culmination in some few days, and then dissolve
in an occlusion of the warm front and the cold front. On the basis
of the thermodynamic theory the forecasting techniques have gradually
improved and become more reliable. Here is a clearcut case
where we have seen an important development in model building in
the short span of some twenty years, during which all three levels
of model building have come into the picture. Two points of general
‘1] Stone - pag. 092
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
scope will be noted. One is the feed-back on fact-finding of an
explanatory model: the construction of the model makes use of
empirical observations, and once the model has been constructed it
directs the fact-finding to new types of observations that are of relevance
for the improvement of the model. The second point is that
the problems of short-range and long-range forecasting often are distinctly
different, and require quite different types of explanatory
models. This is so in meteorology: The thermodynamic theory provides
fairly realistic forecasts over the next 24 or 48 hours; the
cyclones usually die out in a matter of days, so if we want weather
prediction over weeks instead of days we must find another theoretical
basis for the forecasting model.
At the other extreme, I should like to refer to the science of
history, and then place economics and econometrics as intermediate
between meteorology and history. History, of course, is in a sense
very near to economics, and reference is here made to the brilliant
review in Prof. STONE’s paper of the interdependence between the
economic developments on the one hand, and political objectives
and policy making on the other. Here we are on the border between
economics and history, and if we adopt Prof. MAHALANOBIS’ global
point of view this is perhaps more history and politics than economics.
Anyway, it is interesting to examine this area with regard to
the three types (1)-(3) of models. The prevalent view among professional
historians is that forecasting lies outside the realm of their
science. In a way, of course, this is true, or rather a truism; history
looks back into the past to explore « wie es eigentlich gewesen sei, »
to quote a famous dictum. The point I wish to make is, however,
that history does not show an entirely blank record when it comes
to forecasting. For one thing, the study of history is one of the lines
of university education that by long tradition qualify for a career in
the diplomatic corps and other strata of civil service where judgements
and counsels about future developments are important elements of
the professional activity. Of special interest in this respect is the
great work of ARNOLD TOYNBEE. It is a question what is most
admirable, his encyclopedian approach toward historical fact-finding.
1
Stone - pag. 93
36
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
or his paving of new ways in historical analysis by the use of general
explanatory categories, typical examples being the notions of
enormity and mimesis. But the main inspiration of ARNOLD TOYNBEE
is, I believe, the area of forecasting: he is deeply worried about the
present trends of the world, and he tries to tell what will happen
if we do not understand things better and take action accordingly.
Well, to repeat, my point is not to defend or criticize historical
views; what I want to emphasize is that the distinction between
>xplanation and forecasting is of crucial relevance in history as in
any other science.
How about economics? Here the distinction between fact-finding
and forecasting is well recognized, and in some areas of economics
the explanatory models are reliable enough to be successfully exploited
for purposes of forecasting. Areas where the techniques of
forecasting have been successful include demand analysis, production
analysis, cost analysis, and on the whole the key areas of microeconomics,
In macroeconomics, on the other hand, we are still in
the beginnings in the transition from explanatory models to forecasting
models, and, to return to the starting point, the theme of our
Study Week is a challenge to assess the present status about valid
model building in the areas of economic growth and business cycle
analysis.
From the general point of view of scientific method, model
building is a pluralistic endeavour. Science can be regarded as a
collection of models: meteorological models, economic models, historical
models, etc. And in each science there is a plurality of models:
short range models, long range models; we have purely theoretical
models, integrated theoretical-empirical models, all sorts of models.
Science is not a unified system which embraces everything; it is a
myriad of different approaches, partly consistent with each other,
partly inconsistent, and each approach takes the form of a model.
The pluralism is often to distinct advantage. À case in point is the
Eastern and Western approaches towards economic growth. It is
a question whether it would at all be meaningful to construct an inteerated
model to cover both the Eastern and Western types of long
1] Stone - pag. 04
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
range economic models; the differences would be too deepgoing. I.
would be more instructive to have one clearcut model of each type,
and compare the models to explore where and what the relevant
differences are.
My own report to the Study Week, when it comes up, will deal
with three specific aspects of model building. I had prepared some
notes about the general principles of model building as an introduction
to the oral presentation of my report. Then when I heard
Prof. MAHALANOBIS’ plea for a general consensus about basic terminology
and notions, I felt it was in line with his plea to cut out these
notes and present them right now, as a contribution towards unifying
our basic notions about model building
LEONTIEF
I would like to comment on the relationship between theoretical
construction and factual observation within the framework of Pro
fessor STONE’s presentation. The relationship is essentially an ite
rative one,
The theoretical model in its first experimental version is formulated
so as to be capable of being implemented with the available, or at
least, obtainable factual information. At the same time it should
serve as a guide in determining the most promising direction of
further empirical inquiry.
The bulk of official statistics is still being gathered to serve various
administrative needs or to supply up-to-date information tc
general users. However, an ever larger volume of data is being collected
with the specific purpose of being fitted into explicitly formu.
lated analytical models.
In the past the theorist was too often inclined to leave the respon
sibility for the factual implimentation of his conceptual schemes to
the statistician. Now the model builders have come to realize that
they have to take active part in the preparation and planning — if
not the actual performance — of the data gathering task. As a result
‘1] Stone - pag. 95
08
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ;
of this they find themselves forced to replace the essentially symbolic
concept of traditional economic theory by concrete operational concepts
referring to directly observable and measurable facts with which
a business man, an engineer or a practical economic planner has to
reckon in the couse of his every day activities.
In some instances we find ourselves seeking collaboration with
other disciplines. In describing and analysing processes of production,
an economic model builder must be prepared to speak the
language of a production manager and of an engineer; in dealing
with the structure and behavior of households, he must be prepared
to speak the language and use the concepts of a demographer and
a social-psychologist.
The time when the economist or the econometrician could limit his
efforts to construction of mathematical models — or devising more
sophisticated methods of statistical inference — is passing fast. He
has to take increasingly active part, not only in the actual use of these
models and application of these methods, but in the organization and
the direction of the fact gathering activities without which all theory
and methodology will continue to be no more than intellectual
exercise.
VIAHALANOBIS
There is one point in Professor WoLD’s observation about which
[ am not clear. When he said « forecasting » did he include
i targets »? It is possible to take suitable action to attain certain
targets which are considered desirable; this is the system, for example,
in central planning in U.S.S.R. It is not forecasting based essentially
on historical experience or time-series analysis but setting up
certain targets which it is desired to achieve over a certain period of
time. Would Professor WoLp include setting up of such targets
within forecasting?
1] Stone - pag. 96
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39
WoLD
Prof. MAHALANOBIS’ question is important, and my answer is in
line with Prof. STONE’s views on the integration between forecasting
and objectives. The notion of targets is closely related to the notion
of objectives. Reference is also made to the distinction between
instruments and targets emphasized in the works of Prof. TINBERGEN.
In my brief comments on descriptive, explanatory and forecasting
models I was limiting myself to the purely scientific aspects of model
building. In the analysis of policy problems we come to a fourth
type of model, policy models. A frequent type of policy model
specifies different alternatives of policy in terms of objectives and
targets to be achieved on the one hand, and instruments to be used
on the other. Speaking generally, a nonscientific element enters in
the policy model when it comes to the actual choice between the
different alternatives of pnlicv
MAHALANOBIS
Another supplementary question: in meteorology, some experi
ments have been made to find out whether rain can be influencea by
artificial means. That would bring in —- I take Professor
STONE’s point of experimentation. Would such experimentation be
included in forecasting? I am trying to get Professor WoLD’s views
clear in mv mind.
WoLD
This question points to the very important distinction between
two ways of gathering knowledge: by controlled experiments and by
nonexperimental observations. These two ways of getting access to
knowledge cut through all levels of scientific model building - - descriptive,
explanatory and forecasting models. My brief outline was
not intended to be complete, and I choose the three areas meteoro.
‘1] Stone - pag. g7
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
logy, history and economics so as to be mainly of a nonexperimeatal
kind, It is very true that experiments also belong in the picture,
but that was just to simplify.
To comment more in detail on the question about meteorology,
[ would say that while most of the fundamental law of physics can
be demonstrated by controlled experiments, the empirical evidence
on the thermodynamic theory of cyclones is essentially nonexperimental
— you do not experiment with cyclones on a world global
scale, and the same holds true of many other meteorological applications
of physics. At the same time there are elements of genuine
controlled experiments in meteorology — rainmaking devices are a
case in point. The experiments of rainmaking provide material for
model building at the descriptive and explanatory as well as the
forecasting level, showing that experimental and nonexperimental
evidence combine and merge at all levels of model building, including
the level of forecasting.
FRISCH
I want to congratulate both Prof. STONE and Prof. MAHALANOBIS
for their presentations, I am very much in agreement about what
they say about the philosophy of models and as a matter of fact
they have relieved me of a good portion of my own task when I am
going to speak in a few days from now. Of course I also agree
completely to the three points which Prof. Worp put before us —
the three levels of aspiration so to speak, but — and this is something
which I took down while he was talking — there is a fourth
point which must absolutely be added (and here of course I am
talking in the same vein as Prof. MAHALANOBIS) namely the analysis
of the decision on action. What are we going to do? That of course
depends on what we would like to see happen sometime in the future
and as a matter of fact this will really be a very essential point in
my presentation, This is the essence of the distinction between a
forecasting model and a decision model. I understand that Pro-"1]
Stone - pag. 98
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fessor WoLD agrees with me on this, We really have three types of
models: the explanatory models, the forecasting models and the decision
models. My primary concern is with decision models.
HAAVELMO
I have a brief comment on Prof. STONE’s paper at the point
where he mentioned the role of estimation, If I understood him
right he suggested that estimation of certain parameters in a model
might be made, so to speak, in advance, by the scientists, but the
question is whether this can be done in a way that is independent of
the use of the model. I think the answer is NO because the method
of estimation will depend essentially on what kind of statements you
want to make. Ordinarily you will not have point estimates, you
will have confidence intervals, and depending on the « gambling
attitude » of the policy makers, the kind of confidence intervals
which you will want for your estimates will depend on the purpose
of the model. As an illustration, consider two Ministers of Finance,
one being afraid of losing his job if there is unemployment, the other
being afraid of losing it because of inflation: the estimates, in the
way of confidence intervals for the multiplier, which you would give
to these two gentlemen, could be quite different.
KOOPMANS
I have two different comments, one to Prof. STONE’s paper.
think the model on which he and his associates are working is
highly interesting because of its scope and the detail of disaggregation
and because it gives an opportunity to test the return on disag
gregation; one can experiment with the model, re-introduce aggregation
and in that way determine how much of the information that
is obtained by the extreme disaggregation is lost owing to going over
to higher degrees of aggregation; I would like to ask whether Mr.
STONE has made plans to exploit this possibility and urge him to do
‘11 Stone - pag. 99
102 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
so, because it might be that the return on disaggregation after a
certain point ceases to be worthwhile.
My second comment has to do with the proposal towards agreeing
on a terminology in which to discuss model construction. I have
misgivings about that proposal. Scientific terminology generally
comes about by a process very much like consumer’s choice in the
market place; some authors use new terms that are picked up; the
same authors may have used other terms that were not taken up.
There is a social process of accumulation of terminology that is an
extremely valuable screening process, not only of terms, but also of
the ideas associated with these terms. If we make a concerted effort
‘0 set up a terminology, we may find ourselves unwittingly and
unwillingly setting up ideas rather than terms, because some terms
are most suitable as vehicles for certain ideas. I would therefore put
in a plea for allowing the natural selection of terminology to take
its course also in this area.
VIAHALANOBIS
I should apologise for having failed to convey what I meant by
:erminology. Sometimes we speak of « macro » and « micro » models.
My point was that it would be useful to have, for example,
an agreed terminology of using the two concepts « macro » and
« micro » in relation to the particular domain of interest of which
we are making a model. Now if we simply wait for general agreement,
such agreement may never be reached. I may give a specific
instance; in social accounts, in national income, it became necessary
continually to have a standard terminology; for example, what should
be meant by « net national income ». I suggested that model making
has now reached a stage where an agreed terminology, in a purely
scientific sense, would be extremely useful for purposes of description.
[ think it is possible up to a point to have a terminology without
Injecting any views in it. ... I may give a specific example. In
my own country, we have distinguished between « projection » and
« target ». « Projection » is what we get from historical experience,
11 Stone - pag. 100
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103
that is, what is likely to happen as long as the same historical regime
continues. On the other hand, we can use the word « target » in
the sense of purposive selection of something which is desired to be
achieved. The two words « target » and « projection », before a
clear distinction was made, led to much confusion in India. This
is the type of terminology I had in mind; I do not know whether
Professor KooPMANS will have any objection to that.
KOOPMANS
[ think, as Prof. MAHALANOBIS notes, that « macro » and « micro
» economics, is a fine example of the natural selection of terms
that I have referred to. This terminology, originally coined, |
believe, by Prof. FriscH in the ’30s, caught on and is now part
of the language. I think the example of « East » and « West » as
a terminology in model construction of the world economy is an
example of the thing I am afraid of; the content of « East » is changing
before our eyes; the content of « West » will as well be changing
sooner or later. If we set up standard terms for parts of the world
which we wish to distinguish, terms which in some way get a stamp
of approval from a terminology creating committee, we may actually
inhibit thought and analvsis.
PASINETTI
I should like to make simply a short remark. There is a distinction
which I thought emerged quite clearly both from Prof. FriscH’s
paper and from Prof. STONE’s paper, but which has been left into
the shadow in the discussion so far. The distinction is between those
relations which in an economic system are so fundamental as to be
independent of the institutional set-up that society has chosen to
adopt and those relations which are specific to a particular institutional
set-up. For example Prof. LEONTIEF’s input-output inter
industry system is independent of institutions: it is a kind of ana
‘11 Stone - pag. 101
104
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - Z(
lysis which can be carried out for a socialist country as well as for
a capitalist country. On the other hand, for example, the processes
through which prices are actually reached are specific to particular
institutional set-ups: ‘they are different according to whether we
consider a socialist economy, a capitalist economy, or any mixed
type of economy. It seems to me that this distinction is preliminary
to, and should be put behind the classification which have been put
on the blackboard by Prof. WoLp completed by Prof. Friscx.
ALLAIS
I would simply like to make a few remarks on the points which
have been raised during the discussion.
In the first place, the thinkers of earlier times do not appear to
have been as preoccupied with method as in our day. I think that
this difference results from the unequal development of our science.
Three centuries ago, at a time when mathematics and physics were
still only stuttering, DESCARTES felt it necessary to study method. If
today we economists speak of method, it is simply because our
science has not yet reached a sufficiently high degree of attainment.
A second remark: I am struck by the fact that several speakers
have paid great attention to the question of aims in the construction
of models. Personally, I feel that models ought to be neutral, and
constructed independently of objectives. I would willingly associate
myself with Professor FRISCH’s suggestion that explanatory, forecasting,
and decisional models should be distinguished. It is not
possible to bring science back to a single type of model. Personally
I consider the most significant type of model to be the explanatory
one, and I believe that to subordinate the construction of explanatory
models to the pursuit of certain objectives is potentially very dangerous.
A third point is that Professor Koopmans has just said that there
is a social process of selection, which he described as being very
useful, and fruitfull both in ideas and in terminology. On the con-‘11
Stone - pag. 102
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105
trary, it is my view that this is a process which must be regarded
with the greatest suspicion. Past experience with theories in physics
shows that what was fashionable at one time or in a certain period
was subsequently completely invalidated. We must therefore be very
critical about the choices made by current opinion, and personally
I would abstain from taking up any position on the question of
whether the choices which are presently those of the majority are or
are not useful or fruitful. TI only think that it is necessarv simply
to be very careful.
A fourth point is that there are three essential stages in the construction
of a model. The first, the working out of the basic assuptions
follows a process of successive approximations similar to that
mentioned by Professor STONE. The second stage is purely logical,
a deductive stage in which essentially mathematical techniques are
applied. Finally, the third and probably the most important stage
is the confrontation of the theory and the facts. Further, I believe
that these are Professor STONE’s ideas, but it seems to me that his
conception is more neutral, and it appeals to me personally more than
some of the points of view which have been expressed. I do not
think that we ought to have a priori or normative ideas about what
we are going to do. It is first necessary to understand well and to
describe properly, and the basic aim of those who construct
models ought to be above all to give as complete information as
possible, and I would add, as neutral as possible. It is also desirable
that there exist decisional models in parallel with this, but these are
completely separate fields. There is the description of facts, their
explanation, and action, which implies normative choices, but each
of these stages must be kept separated carefully. Thank vou.
SCHNEIDER
What does it mean: « Models must be neutia
; Stone - pag. 103
106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ALLAIS
This is the same difference as that existing in pure and in applied
science. The model is neutral if it constructed by a scientist who has
a non-emotional attitude to it, has no ideological aims, and who does
not include views on what ought to be at the base of his construction.
The model is neutral if it aims to describe and explain the facts. It
is no longer neutral once it is intended to act on the facts, and this
is the reason for my complete acceptance of Prof. FriscH’s terminology,
distinguishing between explanatory, forecasting and decisional
models. Decision must not be confused with explanation,
and neither decision nor explanation should be confused with
forecasting; these are different things. During the discussion I was
struck by the fact that several speakers appeared to have been
mainly guided by decisional considerations when constructing their
models.
[LEONTIEF
Our discussion seems to reveal the existence of two different, not
to say, opposite approaches to the choice and formulation of economic
models. Some favor a fully integrated approach associated usually
with the concept of a decision model. The so-called objective
function, the description of all structural and behavioral relationships
and even the methods of statistical estimation of relevant parameters
are viewed in this case as interdependent parts of a single tightly
‘ntegrated system. A change in any one of them requires, accordingly,
modification of all the others, In another alternative approach,
the description of basic structural relationships, the estimation
of relevant parameters and the choice of relevant objective
functions are approached as three interrelated, but separable
problems. A change or a modification in the solution of one of them
does not necessarily require in this case a corresponding modification
of the solution of the two others.
Although the first approach appears to be more elegant, the second
might prove to be more useful since it permits replacement or
1] Stone - pag. 104
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modification of its individual elements without requiring each time
a complete reconstruction of the entire system.
FRISCH
Time will not permit now to go into a detailed discussion ot
what is an explanatory model, what is a forecasting model and what
is a decision model, but there is one point which I must mention
regarding a decision model, to avoid misunderstanding. When we
start out consciously to build a decision model we begin by an attitude
which in a sense is neutral, At this stage we are not at all deciding
anything about what the outcome ought to be. We are to begin with
perfectly neutral in this respect. We as scientific analysts are perfectly
neutral even in another respect, namely regarding what the
preference function ought to be. This is a matter to be decided by
the politician not by the scientific analyst. True enough the scientific
expert will have to help the politician with respect to the
form, i.e. the language in which the function is expressed, but
certainly the scientist has not to decide the substance matter ex
pressed by the preference function, therefore, in these very fundamental
aspects we are still neutral when we decide to work consciously
on the construction of decision models. There is no political
attitude involved.
DORFMAN
Some time ago, in discussing this methodological problem, Professor
LEONTIEF called to mind a very fruitful standpoint which 1
always associate with the name of KARL PopPPER, though he probably
did not originate it. That is: scientific advance is an iterative process.
The models we build, the terms we use to express them, the objectives
for which we build the models, the measurements which they dictate
and on the basis of which we verify them — all these are in constant
interplay and as we learn from each step we revise all the others
1] Stone - pag. 105
108
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2:
I therefore subscribe to Professor KooPMANS’ thesis that we should
not try to freeze the meanings of the terms used in econometrics at
this stage. I think it will be our experience here this week that as
we discuss econometrics we shall learn, and as we learn the words
we use will come to have different and somewhat sharper meanings
‘han they have now. I fear that we can bog ourselves down by
attempting to clarify points of terminology. Science is groping, and
as we grope we shall wish to change the meanings of the technical
words we use.
VIAHALANOBIS
I should like briefly to make two points. First I shall remove the
apprehensions of Dr. Koopmans; I agree that to try to define what
's East of West would be absolutely futile; but we may say a
« macro-national » or « micro-national » or use such neutral terms.
[ agree also with the points made by Professor LEONTIEF. It seems to
me that what is a decision model or what is a forecasting model also
nvolve the question of terminology. ;
However, terminology is in one sense a procedural point. I do
‘eel that we should have some discussions regarding the objectives of
model making. Whether the question of terminology is pursued or
not I have no strong views; but even the present discussion indicates
that some clarification of terminology would be useful. It is purely
my own personal ignorance. I should like to understand clearly what
is meant by such term as « objective », « neutrality », « forecasting
model » or « decision model » and such things. What are the different
‘ypes of models in relation to different spheres of interest? I am not
suggesting that we should be interested in only one type.
I agree generally with the observations made by Professor FrISCH
that even if we have decision in view by politicians or others, the
role of the scientist is to keep a « neutral » mind in advising how
those « objectives » may be attained. I do not see that « neutrality »
is in any way destroyed by keeping certain « objectives » in mind.
On that point I am in complete agreement.
:] Stone - pag. 106
SEMAINE D’ÉTUDF SUR LE ROLE DE I ANALYSE ECONOMETRIQUE ETC.
NC
As far as I have understood Professor DORFMAN, I am in broad
agreement with him, but I think that our discussion today shows
the need for some clarification of ideas which can only be done
through words and therefore through the use of agreed meanings
of words.
AILAIS
I must stress that there is a very good reason for my not comp
letely agreeing with Prof. FriscH. There is a very great difference
between three types of model: explanatory models, forecasting
models and decisional models. This difference is the following. As
far as the explanatory model is concerned there is a judge, you can
verify your explanation when vou observe the facts. The same is
true when you have a forecasting model. Your forecast may be
wrong, but you can see if it is right or wrong. But when you
develop a decisional model, what is the criterion of truth? I cannot
see that there is one. You may think « I am neutral ». You may
think this is always true, but you can be wrong; and if I think
« you are not neutral » and if you think « I am neutral », and if we
are in disagreement, who is to decide? You see here a very great
difference between the first two types of model and the third. For
the first two models there is a judge: nature. Nature can answer
« You are right » or « You are wrong », but with a decisional model
nobody, nothing can answer.
MAHALANOBIS
Neutrality is a word which we should not press too far because
even in gathering facts it is necessary to have a conceptu-! framework;
in one sense, you can collect only such facts a: vou are
looking for. Also, all observed facts would be affecte“ * -rrors
of observation arising from personal bias. One has ‘, even
farther; ultimately, according to the HEISENBERG principle .: un-[1]
Stone - pag. 107
{10 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
certainty it is not even possible to make an observation without
disturbing the system itself. We should not press the question of
neutrality in an extreme way, but take a broad view that for certain
purposes certain models would be more appropriate than others.
The point stressed by Professor LEONTIEF is extremely important
the validity and also the precision and the effectiveness of a
model to serve certain purposes which may be purely explanatory
or may be decision making or may be forecasting or may be of other
types. I think this raises questions of substance which we should
discuss; and we should, among ourselves at least, provisionally agree
on what kind of words we should use.
WoLD
I am not sure whether there is really any disagreement between
Prof. Arrars and Prof. FriscH, but their debate does confirm Prof.
MAHALANOBIS” view that it is a good thing to clarify our terminology
— not necessarily to establish it for the indefinite future, but at least
for the purpose of our discussion during the Study Week. The
question has been posed: What is a decision model, and how does
it differ from a forecasting model? It is my undestanding that
decision model and policy model are essentially the same notion. If
so, the question can be answered along the lines of a famous argument
by the distinguished Swedish economist GUNNAR MyrpAL. Political
actions, including actions of economic policy, are based on value
judgements, and it is typical that the judgements are radically
different for members of different political parties. The analysis of
economic policy and other decision systems takes the form of a
policy model, where the value judgements underlying alternative
political actions are included as specified hypotheses, hypotheses
which in themselves are politically neutral. In this way the policy
model becomes, in principle, an instrument for strictly scientific
analysis of alternative lines of political action.
‘1] Stone - pag. 108
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STONF
I should like to thank you all for the very interesting discussion
on my paper and I am sure that you will forgive me if I do not
attempt to reply individually but merely try to summarise the position.
In 1927 the biologist J.B.S. HALDANE published a book called
Possible Worlds in which he tried to examine how far certain things
were possible from the standpoint of physical, chemical or biological
laws. Could one imagine, for example, an elephant ten times larger
than the elephants that are actually known? And, if not, what was
it that prevented us from imagining that such animals could exist?
My type of model building, and, I think, many other people’s model
building too, is concerned with doing exactly the same sort of thing
for an economic system: we should like to obtain certain results;
could we imagine that they could come about? Now one may ask the
question: how is one to decide such things? And this seems to me
to get to the heart of the difference between predictive models and
models which aim at projections based on hypotheses about changes
that might be introduced into the world. If we assume that the
world will continue to work as it has in the past, and if we do not
like some of the consequences of this, we may ask: could we change
things for the better? The answer is not necessarily « yes » but,
equally, it is not necessarily « no », because the way in which the
world in fact works is not by definition the best possible way. It
seems to me that in trying to decide questions of this kind one must
follow the principle that upholders of laissez-faire claim that laissezfaire
maintains, namely the maximising principle: that wherever you
see economic action taking place it is always successfully directed to
maximising something that one wants to have maximised. Now if
this were actually the case there would be very little need to build
models because we should already have a perfect system which could
not be improved. It can only be in the belief that this is not actually
the case that we have interested ourselves in the rather exacting anc
energetic pursuit of economic model building.
Another question that came up and which I think is wotiu
| Stone - pag. 109
112 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
discussing can be summarised briefly under the term « disaggregatlon
». It certainly is not my belief that the way to get better models
and a more realistic representation of the economy is to start with
a general model of the kind I described and then simply make it
larger and larger and larger. The reason why this is not a good idea
is, firstly, that it is impossible to get, in any group, sufficient information
to build so large a model and, secondly, that it is quite
unnecessary to do so. If you are interested in the operation of the
chemical industry, even if you are only interested in a certain group
of activities brought together in a single firm, you can set up a model,
as large or larger than mine, to analyse the operations of that
ndustry or firm. But it will be quite impossible for any single
group of scientists to integrate into their own work models of this
degree of detail. But why should they; if the industries concerned
are willing to do the job themselves and are obviously very much
setter at doing it? And this goes not only for the industrial side of
‘he model. The same can be said about many of the activities of
government. It is too much to expect that a group of economists
who have the problem of a general model on their hands will also
be able to build models of, say, the health service, the educational
system and the defence system. The right way, I think, to get this
sort of disaggregation is to have a series of sub-models (which,
however, must be linked to the main one), to decentralize the building
of these models and to put this work in the hands of people with the
necessary specialized knowledge.
finally, I should like to say something about another recurring
:heme in this morning’s discussion: the question of iteration. It seems
:o me that this is the fundamental principle on which all learning
and all model-building is based. One has to start somewhere. One
knows perfectly well that one’s prototype model will not be a very
perfect tool, but the really important thing is that one should set it
up, see how the parts of the system interact, and check how the
relationships of the model work out in practice. We are bound to
start with relatively simple ideas, we are bound to start with relatively
inaccurate facts. We can try to find out how far the facts
1"
Stone - pag. 110
Biblioth~k des Instituts
fiir Weltwirtschaft Kiel
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 13
need to be more accurate than those we have at present and we
can try to find out what relationships in the model are really important
for the main purposes for which we want to use it. But if
we build models, put them in journals and allow them to be forgotten,
if we do not keep them up to date and if nobody builds on them,
I do not think that we shall ever get satisfactory applied economics
1] Stone - pag. 111
TOWARD A VERDICT ON MACROECONOMIC
SIMULTANEOUS EQUATIONS
HERMAN O.A. WOLD
Universitetets Statistiska Institution - Uppsala - Sverige
INTRODUCTORY
[t is now 25 years since JAN TINBERGEN launched his
pioneering macroeconomic models for the NETHERLANDS (1937),
US (1939) and UK (1951). His approach marks a bold raise
of aspirations levels, in economic thinking as well as in the
statistical techniques of fact-finding and analysis, and above
all in the systematic coordination of theoretical and empirical
approaches. If we think of the allembracing sectors of economic
life and the huge complex of static or dynamic interrelations
that the model builder sets out to master, the dimensions
of his task bring to mind LAPLACE’s fathom who could
forecast in detail the future course of events in the entire universe
by solving an enormous system of differential equations.
It adds to the greatness of TINBERGEN’s work that large masses
of economic-statistical data had gradually accumulated, that
the young discipline of econometrics was in the starting holes
for vigorous advances, and that it did not tarry long until his
approach, now known as causal chain systems, was followed
by other keen innovations, in the first line input-output analysis
(LEONTIEF, 1041) and interdependent systems (HAAVELMO,
2] Wold - pag.
(16
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
1943). The literature of the area is large and rapidly expanding.
The high goals of macroeconomic model building, that
much is now clear, are an important incentive for the development
of new techniques in quantitative economic analysis.
Thus in the short period of 25 years it has become manifest
that macroeconomic model building is a mighty challenge to
econometric method. At the same time it has emerged as a
significant factor in the long range scientific evolution.
It is appropriate to evoke the long range perspective if we
call to account in the challenge, asking for the results thus far
obtained in macroeconomic model building. Immediate benefits
of an indirect nature have often and rightly been emphasized,
including the build-up of knowledge about macroeconomic
facts, the improvement of statistical data-collection, and the
training of new cadres of econometricians in theoretical and
applied work, all of which are tangible boons that cannot be
overrated. As to direct results in the form of forecasts and
other types of operational inference from the models, the
horizon is eagerly watched for signals of progress. On this
score the outlook is more undecided, and leading authorities
have voiced scepticism and disappointment about the reported
achievements. A symptomatic feature is the recent symposium
in Econometrica (!) with the motto « Simultaneous equation
systems: Any verdict yet? » But here patience is in order,
for, to repeat, 25 years is a short time to master the tremendous
tasks at issue.
The Study Week, a most felicitous initiative of the Pontifical
Academy of Sciences, provides a forum for unprejudiced appraisal
of ends and means, aspirations and actual achievements
in macroeconomic model building. The timing of the Study
Week could not have been better. There is plenty of progress
in the many avenues of scientific research and development
that run together in a fullfledged macroeconomic model. On
(1) Econometrica, October 1960, Refs. 6 and 7. Cf. also Refs. 8 and o.
21
Wold - pag. 2
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117
the empirical side, the assessment of quarterly data sets new
standards of fact finding, standards that are highly influenced
if not called forth by the subject-matter considerations embodied
in the model building. For another thing, causal chains and
interdependent systems have turned out to be genuine innovations
in dynamic model building, not only in econometrics
but in the wide realm of nonexperimental model construction.
At the level of the stochastic foundations of model building the
innovation has posed new types of problem; the debate of the
1950’s on the rationale of simultaneous equation systems shows
that it has taken a long time to sort out and come to grips with
these key issues. As to the design and actual construction of
macroeconomic models, several projects are in the picture, of
different types, of different size, and at different stages of completion.
It is an important task to subject the accumulated
material to systematic scrutiny and comparative studies, with a
view to assess the theoretical and practical value of the achievements,
and, what is perhaps even more important in the present
stage of development, to obtain guidance for further work.
The present report focusses on three specific aspects of
macroeconomic models:
1) The three types of simultaneous equation systems
known as vector regression (VR-), causal chain (CC-) and
interdependent (ID-) systems are presented as three levels of
generality in the mathematical design of dynamic models. The
step from CC- to ID-systems is crucial from the point of view
of the operative use of (a) the behavioural relations, and (5) the
reduced form. Thus in CC-systems all relations (a)-(b) can be
specified as eo ipso predictors, that is, as conditional expectations
subject to random disturbance, whereas ID-systems
allow such specification for relations (b) but in general not for
relations (a). This parting of the ways is the stochastic aspect
of the much-discussed feature that the behavioural relations
of CC- but in general not of ID-systems are designed for a
[2] Wold - pag. 3
34
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
causal interpretation in the sense of stimulus-response relationships.
2) The parting of the ways between CC- and ID-systems
is examined from the point of view of the transition from
disturbance-free to stochastic relationships in the formal design
of simultaneous equation systems. While the dualism at issue
does not exist in disturbance-free systems, it arises in the
stochastic specification of the models because the rules for
operating with disturbance-free relations sometimes but not
always extend to eo ipso predictors. Thus in CC-systems the
transformation to the reduced form is a matter of iterated substitutions
such that eo ipso predictors are carried into eo ipso
predictors, whereas in ID-systems this transformation in general
is an operation that does not preserve eo ipso predictors. The
dualism can be removed by a respecification of the behavioural
relations of ID-systems, namely, by letting those explanatory
variables that are current endogenous variables be replaced by
their conditional expectations as given by the reduced form.
In the resulting systems, calling BEID- (bi-expectational ID-)
systems, both the behavioural relations and the reduced form
make eo ipso predictors, and accordingly allow a corresponding
causal interpretation.
3) In experimental situations the empirical testing of a
model is essentially a matter of replications under controlled
conditions. In nonexperimental situations, and in particular
in macroeconomic model building, no routine techniques are
available for testing the model with regard to its practical value.
Here predictive tests are of key importance, that is, follow-up
studies in which the forecasts obtained from the model are
confronted with the actual course of the time series that are
subject to analysis and forecasting. Thus far it is only for
very few macroeconomic models that such follow-up studies
have been reported, and a plea is made for the systematic use
of predictive tests. Reference is made to the Janus quotient,
a predictive test which is designed as a criterion whether the
2] Wold - pag. 4
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Etc. 116
time series under analysis have the same structure of interrelations
in the observation range and the prediction range. As
applied to CC- and ID-systems the Janus quotient can be
adapted in various ways so as to take into account the distinction
between genuine forecasts from the model and ancillary
forecasts of the exogenous variables. A specific use of the
Janus quotient is as a danger signal against overfiting, that
is, the pitfall of estimation techniques where illusively small
residuals are obtained because the parameters to be estimated
are numerous relative to the available observations. This aspect
of the Janus quotient comes to the fore when ID- and BEIDsystems
are estimated by the two-stage method of least squares.
The predetermined variables then pile up with unknown coefficients
in each relation of the reduced form; hence unless
special precautions are taken the reduced form will be overfitted,
with risk to run into the pitfall to conclude that the
current endogenous variables coincide with their conditional
expectations, and that the ID-system coincides with the corresponding
BEID-system.
The topic of the report is an area of active interest to my
research seminar. I am greatly indebted to Professor ENDERS
A. ROBINSON, with whom I collaborate at present in conducting
the seminar, and to Messrs. A. Gapp, E. LYTTKENS, S. MARTI-NELLE
and G. STojkovic for allowing me to incorporate
unpublished research results in the report, as indicated bv
specific references to their work.
. THREE LEVELS OF GENERALIZATION IN DYNAMIC MODEL
BUILDING (2)
We shall consider three types of approach which (a) have
the form of simultaneous equation svstems: (b) are designed to
(?) The reader is assumed to have some orientation in the literature of
dvnamic econometric models. For textbook treatments of CC- and IDsystem,
see Refs. 10 and 11. The present exposition leans heavilv on Refs.
t2 and 12.
[2] Wold - pag.
5
{20
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
include forecasting as-a main purpose in their applied use;
(c) make use of the device called forecasting by the chain principle.
We say that a forecast that spans % periods ¢+1,
+2, ..., t+k is obtained by the chain principle if the calculations
proceed step by step so that when the forecast for
the period #+i has been obtained (t=1, 2, ..., k- I) it is used
as an actual observation when calculating the forecast for the
period £+{+I.
t-4h t-3h t-2h t-h
Observations: O O e
Forecasts
t t+h t+2h
Q O O
“ENG
FiG. 1 — Forecasting by the chain brincible
All through sections 1-2 we shall be concerned with purely
theoretical aspects of the three types of model. For the specification
of the models we shall make use of the notion of
go 1pso predictor, that is: If a random variable y allows the
representation
I)
y=f(x)+v
where f(x) is the conditional expectation of y for given x,
2)
E(vlx)= f(x)
then f(x) is called an eo ipso predictor of y. The notion extends
to vector variables y, x, v. As to the empirical treatment of
20 1pso predictors, we note that they can under very general
conditions be consistently estimated by least squares regression
(3).
(*) For a detailed proof. see Ref. 14
“21
Wold - pag. 6
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 121
It will suffice for our purpose to consider relationships and
eo ipso predictors that are linear. The following terms and
notations will be used throughout:
(3)
>= (Vie Vopp
A
“I
is the vector of current endogenous variables:
(4)
<= (X1p Xap --Xomt
is the vector of exogenous variables: and
(5)
2,=(V, 1s Vas ee) 5
is the vector of predetermined variables.
(.X1. Vector repression (VR-) systems
The general formula for linear vector regression is given .
'6) y, = Li(V,_15 Vio ori Xu Xo qs ...) FV;
where 1=1 ..., n; the functions L; are linear; and we assume
that L. is an eo ipso predictor of v., for all : and £,
(7)
E(ly._1 Yi
L;y,_..
In matrix representation we write the vector regression
system (6)-(7) as follows,
‘8
Mg
+
4 4
‘21 Wold - pag. 7
72 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
with
0
E(y,lz,
RP -
We note the following general features of vector regression
systems:
1) The system contains one (and only one) explanatory relation
for each of the current endogenous variables;
2) In each relation, all of the explanatory variables are predetermined;
3) Each relation is an eo ipso predictor subject to random
disturbance.
[.2. Causal chain (CC-) systems.
The general formula for causal chain (also known as recursive)
systems is in the linear case given by
(10)
Vir=Li(y14 Vars +, Vici Nieto Vio» ++"5 Xp Xi_ 1» …)+U;,
where ¢=1, ..., n; the functions L; are linear; and L; is assumed
to be an eo ipso predictor of y., for all 7 and ¢,
(IT) Elyse Yatr wor Victts Verts Vr-ns +05 Xp Xp_45 ++)
Lyi Var oes Vi-1,e5 Veste Ve-2 +03 Xt oon)
To give the system (10)-(11) in matrix form we write
à" +B 3
12)
with
E(yly", 2)=A y, -B Z,
2] Wold - pag. 8
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 123
where the prime in y’, indicates that when the vector y, serves
as an expectational condition for the variable v,,, this variable
is deleted in the vector, and where
(14)
a.;
=
1
=I. ....0n: k=t.1+71, .... n).
To state (14) in words: the matrix A is subdiagonal in the
sense that all elements in and above the main diagonal are zero.
System (10) is called the primary (or structural) form of the
model (4). Thanks to the subdiagonal design of the matrix A
the current endogenous variables y,, can be eliminated from
the right-hand members of the primary form (10) by a sequence
of iterated substitutions. This operation leads to the reduced
form of the model,
'15)
where the matrix R is given
(16a-b) R=(I- A)! B
and the following relation can be established,
(17)
E(v,lz,)=R 2.
showing that the component elements of R z, are eo ipso predictors
for the corresponding variables y,,, or briefly stated, that
the reduced form relations make eo ipso predictors for the
current endogenous variables.
7 feature in the generalization from VR-systems (6)
(*) With regard to its theoretical content, a model consists of a set
vf assumptions and a group of theorems deduced from the assumptions. The
term primary form brings in relief that this form contains the basic assumptions;
the term structural form accentuates that the specification of this
same form determines the theoretical structure of the entire model, including
the reduced form and other relationships deduced from the basic assumptions.
2] Wold - pag. 9
124 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
to CC-systems (10) is that in CC-systems the reduced form (15)
differs from the primary form (10), whereas in VR-systems the
primary form and the reduced form are one and the same
system (6).
The following general features of CC-systems will be noted:
1) The primary form (10) contains one (and only one)
explanatory relation for each of the current endogenous va-~iables:
2) In the transformation that leads from the primary
form (10) to the reduced form (15) the current endogenous
variables can be eliminated from the right-hand members by
a sequence of iterated substitutions.
The causal aspect of the substitutional design has given the
model its name « causal chain system ». To paraphraze, the
current endogenous variables form a causal chain of the same
type as in the nursery fad: « The cat on the mouse, the mouse
on the rope, the rope on the hanged man’s neck. »
3) Each relation in the primary form and the reduced form
is an eo ipso predictor subject to raridom disturbance.
In view of the potential use of the primary form and the
reduced form for predictive and other inferential purposes,
3° is a key feature of CC-systems. A model which has property
3° will be called bi-expectational. The bi-expectational
property of CC-systems is closely tied up with the substitutional
design 2°, inasmuch as the substitutions at issue are an operative
procedure that carries eo ipso predictors into eo ipso
predictors.
[.3. Interdependent (ID-) svstems.
A broad class of interdependent systems is covered by the
following model:
(18)
v,= À y,+B z,+v,
2] Wold - pag. 10
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125
where the vector
(19)
y it Ji
is formed by current endogenous variables in such manner that
for any specified : the variable y, may or may not occur in y,,
and mav occur twice or more: and where
20)
a, 4
— i
that is, matrix A is such that the relation for y,, does no
involve this variable in the right-hand member.
System (18) is the primary (or structural) form of the model.
Regarding the primary form (18) as a system of implicit relations
for determining the current endogenous variables, and
solving for these variables in terms of predetermined variables,
we obtain the model in reduced form,
(21)
R z,-wm,
where
(22a-b)
RK=—l
A
y IB: w="
and where we assume that the relations make eo ipso predictors
for the current endogenous variables.
(23)
Ew, IR z)=R
We note the following general features of interdependent
systems (18):
1) The primary form (18) contains as many relations as endogenous
variables;
2) The matrix (I - A) 1s nonsingular.
21 Wold - pag. 11
26
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
3) Each relation in the reduced form is an eo ipso predictor
subject to random disturbance.
1.4. Incentives for the generalization from VR- to CC- and
ID-svstems.
1. Aggregation over time. Vector regression (6) is of old
standing in dynamic model building, and especially in the
natural sciences a good many dynamic theories can be quoted
that make use of this type of approach. An important feature
is that if the data so permit, the time unit can be chosen very
small; hence, in principle, VR-systems (6) cover also the approch
of differential equation systems of any order. Model (6)
thus lies near at hand in situations where the data are registered
continuously (recording barometres, seismographs, etc.), or,
more generally, are registered periodically with time intervals
that are short relative to the changes in the variables between
the recordings.
CC-systems (10) and ID-systems* (18) are recent innovations,
both emerging in econometrics in the decade 1935-1945.
It is not by chance that this generalization was initiated in
econometrics, an area where theoretical model building had
been well developed since long ago, and where by long tradition
a large part of the available time series data had the form
of annual aggregates. Hence there was — and is — a twofold
incentive for the generalization from (6) in the direction of (10)
and (18). One was that annual data were known to involve
lots of information in the form of interrelations between economic
factors observed in one and the same year, a source of
information that cannot be exploited in the VR- approach (6).
And the very existence of large masses of annual data reinforced
the incentive for the generalization at issue.
2. The chain principle of explanation and forecasting. More
recently, another incentive for the generalization has come to
2°
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121
the fore, inasmuch as VR-, CC- and ID-systems make use of
the chain principle in extracting inference from the model. If
a forecast is to span 18 months, say, and weekly data are
available, a model of type (6) with weekly data would require as
much as #8 links, a design that involves the danger that the
forecasting errors will aggravate by accumulation; it would
then be attractive to build a model of type (10) or (18) on the
basis of quarterly data, say, a model that exploits the interrelations
between variables as observed during one and ‘the
same period; such an approch would require only 6 links in
the forecasting procedure. À specific point in this connection
is that in economic statistics the observational errors are in
practice relatively more important if the aggregation period is
short (°); hence there will be a downward tendency in the
magnitude of regression coefficients, and the accuracy of the
forecast will not be optimal if the aggregation period is too
short. To put it otherwise, if the disaggregation goes too far it
works against the law of large numbers, and thereby attenuates
the inference from the model. This is just one aspect of the
problem what period of aggregation is optimal in the design
of the model, a highly important, many-faceted and difficult
question that falls outside the scope of this brief review.
The chain principle is a unifving feature of VR- CC- and
[D-systems. More specifically, the forecasts are generated by
the chain principle as applied to the reduced form, and this
has the same mathematical structure in all three models, as
seen from (8)-(g), (15)-(17) and (21)-(23). At the same time
the chain principle brings in relief that the models work at
three different levels of generalization. Thus in VR-systems (8)
the primary form coincides with the reduced form; in CCsystems
the primary form (12) is transformed to the reduced
form (15) by a sequence of iterated substitutions: in ID-svstems
(°) Ref. 15, Appendix B, discusses this point with reference to correlation
coefficients, and the same argument applies to regression coefficients.
2] Wold - pag. 13
28
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
the reduced form (21) cannot be obtained from the primary
form (18) by iterated substitutions. This last feature is closely
related to the fact that ID-systems are not bi-expectational :
the predictor specification (13) of the primary form (12) of
CC-systems has no parallel in ID-systems (18). The lack of
parallel to (13) in (18) is clearly a stochastic feature of the
models, for it would not appear if the ID-systems (18) were
deterministic in the sense of disturbance-free relations; in fact,
the left-hand members of (23) would then be nothing else than
the component variables y, and (23) would be precisely the
same system of relations as (21).
[n the much-discussed dualism between CC- versus IDsystems
it has been a veritable stumbling block that ID-systems
are not bi-expectational (%). As briefly noted above, this key
feature results from the merging of two lines of generalization,
namely from VR- to CC- and ID-systems on the one hand,
and from deterministic to stochastic specification of the models
on the other. We shall return to this matter in section 2 for
a more detailed review.
3. Accounting identities vs. equilibrium relations. To summarize
the argument, accounting identities make no incentive
in the generalization from VR- or CC-systems to’ ID-systems,
whereas the incorporation of equilibrium relations into the
model is one of the main incentives in the generalization from
CC- to ID-systems (7).
The argument will be illustrated by simple cases in point.
A typical accounting identity is given by
a
Y,=C, +8,
(°) See, also for further references, Refs. 16 to 18 and Ref. 30. ;
() The exposition makes systematic use of eo ipso predictors; otherwise
‘he argument of this subsection is well known.
2,
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126
where Y, is gross national product, C, total consumption, and
S, total savings. As to equilibrium relations, a typical case is
the assumntion
(25)
where S, is saving and I, investment.
Speaking generally, if the primary form of a VR- CC- or
[D-system involves an accounting identity, an equivalent model
can be constructed by interpreting the identity as a behavioural
relation for one of the variables. For example, if the model
involves the identity (24) and the two behavioural relations
(26a-b)
(27a-b)
À
IY
>
you
x
Wal
A
—
05174
“1%;
ir
cer Yo z,)=L,
1»
we obtain
(28a-b) S,=Y,
C,=L,(z,) - Ly(Y,, z,)+v" witn
and under general conditions the relations (24) and (26)-i.7
further imn!-(202-0
—~
gis
LAP
PE
+
EE
3
Liz, -
Ly(Lylé,,,
“5
-t
Turning to equilibrium relations, for example (25), the situation
is fundamentally different. The model involves one
behavioural relation for each of the variables that are subject
to the equilibrium assumption, and the equilibrium is regarded
as the result of corresponding changes in an equilibrating va-2]
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130 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
viable, in this case the interest rate, say g. In symbols, let the
two behavioural relations be
30a-b) S,=L(q,, z)+ v, With E(S,lg, z,)= L(g, 2,)
(3ra-b) I, = L,(q,, Z;) +U ; » E (Llg, 2) — L,(q, Z;)
where for simplicity we have assumed that g, is the only current
endogenous variable that influences S, and I,. Further let
M, denote the common total of savings and investment,
(32)
M,=S,=1,
Then under general conditions of regularity we may substitute
(30a) and (31a) into (32) and solve for the equilibrating
variable, say
33)
g,=L,(z,)+ v,
Thus we may regard (32) as an impliéit and (33) as an explicit
behavioural relation for the equilibrium variable g,. Now with
regard to the rationale of the generalization from CC- to IDsystems
the following points will be noted.
The assumptions (30)-(31) make two behavioural relations
for the endogenous variable M,, and no explicit behavioural
relation for the endogenous variable g, and this situation is
incompatible with the general design (10) of CC-systems. This
is so even if the eo ipso predictor specifications (30b) and (31b)
are abandoned. In this connection it is important to note that
if specifications (30b) and (31b) are adopted, relations (30)-(31)
imply
‘34,
E(q,|2,)Æ L3(2,)
showing that relation (33) cannot be specified so as to make
an eo ipso predictor.
2]
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In an ID-system (18), on the other hand, it is perfectly legitimate
to incorporate relations (30a) and (31a), and the ensuing
relation (32) may perfectly well be specified so as to make an
eo 1pso predictor,
(35)
E(g,lz,) —1
à,
but in the specification of the ID-system we must in general
abandon the assumptions (30b) and (31b). This last point is
entirely in line with the lack of counterpart to (13) in (18).
The upshot of the argument is that accounting identities (24)
can be incorporated into any VR-, CC- or ID-system, whereas
an equilibrium (25) makes a parting of the ways between VRand
CC-systems on the one hand, and ID-systems on the other.
The salient point is that (24) is an exact identity, whereas (25)
is an approximation, inasmuch as the deviations from equilibrium
are ignored. This comment also gives a clue to how
the situation may be dealt with in CC-systems, namely by
taking the difference between the two members (30a)-(31a) of
the « equilibrium » into explicit account, and exploiting it,
possibly with a suitable lagging, as an explanatory factor for
the equilibrating variable ¢,. A simple example is given by
the following model for the balance between demand and supply
in a market under free competition (8).
Demand relation:
3ba)
u
:.580 - 0.390 p,+ 0.520 f,+ u.
(*) Model (36) refers to the US market for pork 1939-1956. Refs. 19 and
20 give similar models for several agricultural products, based on US data,
Ref. 21. The data having been revised, Ref. 22, I am indebted to Mr. Stoj-KOVIC
for recalculating his pork model, Ref. 20, on the basis of the revised
data for the purpose of the present report.
To list the exogenous variables, f, is a price index for farm products;
¢, is corn price; g, is a dummy variable representing war effects 1942-46;
w, is the farm wage rate. Quantities are per capita values; prices and wages
are in real terms.
2] Wold - pag. 17
132
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
supply relation:
36D) $;=1.825+0.055 p,_; — 0.097 C,_, + 0.123 g, + 4,
price mechanism:
(36c) P,= 0.493 (d,_, ~ 5) +0.229 w, + 0.414 c, + u,”
with all three relations specified as eo ipso predictors, giving
E(d|p, f)=1.580-0.390p,+ 0.520 f,
and similarly for (36b) and (36¢c). Model (36) has the formal
design of a CC-system, inasmuch as s, is influenced only by
oredetermined variables, p, by s, and predetermined variables,
and d, by p, and predetermined variables.
These simple illustrations bring in relief the distinction between
the notion of instantaneous equilibrium and other modes
of equilibria. In an ID-system that includes (30a), (31a) and
f32) the instantaneous equilibrium enters among the basic assumptions
of the model. In CC-systems, on the other hand,
instantaneous equilibria have no place; instead, equilibrium
tendencies may enter the picture by way of theoretical deductions
from the model. Thus in model (36) demand and supply
will under general conditions of stochastic regularity be in
stationary balance, a state of never-ceasing random fluctuations
around a limiting equilibrium level.
4. The general scope of vector regression and causal chain
systems. Brief reference is made to two groups of theorems
that establish the general scope of VR- and CC-systems. The
theorems refer to a set of observed time series (3)-(4) that
are conceived of as extending into the indefinite past
(¢{-1, t-2, ...), and they have the nature of representation
theorems that yield predictive inference under the assumption
that the structure of the interrelations between the time series
27
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SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
135
be the same in the future as in the past. Thus it is shown that
the observed time series can be represented in the form of a
VR-system, a representation that holds under very general
conditions and to any prescribed accuracy in the stochastic
specification (*2). A similar theorem holds for CC-systems.
Both theorems exist in two versions, one where the observed
set of time series is regarded as a (multidimensional) realization
of a stationary process, and the expectational properties of the
VR- and CC-systems are specified in terms of cross section
averages of the various possible realizations. The other version
refers to no other realization than the observed time series,
and specifies the expectational properties of the system as
averages over time based on the single realization. The theorems
are closely related to the general representation theorem
known as predictive decomposition of stationary stochastic
processes; Refs. 15, 24 and 46. The predictive decomposition
is parametric, and the parameters are uniquely determined. An
important feature of the predictive decomposition and of
CC-systems is that representations of this type yield predictions
that are optimal in the sense of minimum-delay of information
(°). Thus far we have referred to the given time series as
stationary, but the representation theorems extend to the case
of nonstationary processes; Ref. 25.
Mathematical generalization is not an unmixed blessing.
When a theoretical model is generalized so as to cover wider
areas, the basic assumptions are relaxed to some extent, and
the relaxation brings on that the inference from the model is
attenuated in some respect or other. The ensuing balance between
generalization and attenuation of inference is a most
important aspect of the three models under review. To summarize,
any set of observed time series can be cast in the form
(#) For this and the following theorem, see (also for further references)
Ref. 23.
(°) Announced in Ref. 46. the full proof of the CC-representation is as
vet unpublished
2] Wold - pag. 19
134
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
of a VR-system, And CC-systems, too, are of general scope
in the same sense. Here the substitutional design referred to
in 1.2 (2), is the salient point. Thanks to this design it is possible
to carry through the specifications (13) of the primary
form and (17) of the reduced form in terms of conditional
expectations, although this bi-expectational specification of
CC-systems might seem highly restrictive at first sight. At the
same time the substitutional design marks the limit beyond
which the generalization from VR-systems in the direction of
CC- and ID-systems cannot be pushed without losing the
bi-expectational property. Hence, as noted in 1.4 (2), IDsystems
in general are not bi-expectational. Another important
property that goes lost in the generalization from CC- to IDsystems
is the optimality with regard to minimum-delay of
information.
2.
ENDS AND MEANS IN THE TRANSITION FROM DETERMINISTIC
TO STOCHASTIC APPROACHES
Whe shall in this section consider a number of situations
that are special instances of the universal mathematical rule
that generalization of a theoretical model is always accompanied
by attenuation of inference from the model. The theorems relevant
to the argument of this section belong to the foundations
of probability theory; thus (50) goes back to the beginnings of
correlation and regression analysis around 1900, and the origin
of (41) is still more remote.
2.1. A review of basic notions.
The transition from deterministic to stochastic specification
is a radical generalization of a theoretical model, and the
ensuing attenuation of inference goes down to the very foundations
of model building. To emphasize the basic arguments
we shall start from scratch and give some few simple illustra-2]
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SEMAINE D’ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 135
tions of the notions of univariate distribution and bivariate relationship.
With reference to Figs. 2a and 3a, let x be the (unspecified)
molecular weight of sugar. Physical chemistry tells us that
the sugar molecules are crystals, all of which have the same
weight, w=3.01 x 107% grams, that is 180 times the atom
weight of hydrogen. If the weight pn could be measured exactly,
the situation would be as shown in Fig. 2a. In practice, the
weighting is subject to observational error, and if the errors
follow the normal distribution the measurements will be distributed
as shown in Fig. 3a. The observed average x of this
distribution provides a point estimate of the unknown molecular
weight i. Next let x be the molecular weight of a polymere,
say a specific make of nylon. The nylon molecules are bands
of different length; that is, x is not a specific number, but a
variable subject to a specific distribution, say as shown in
Fig. 4a. Here jv denotes the mathematical expectation of the
distribution,
(37)
—
rt,
‘
‘
Distinguishing between the theoretical and the observed distribution,
as illustrated in Fig. sa, the observed mean x gives a
point estimate of the theoretical mean p. Conceptually, the
dotted curve represents the distribution of a variable x =x* +¢
which is composed of a variable x* with the same distribution
as in Fig. 4a, and an observation error € which for fixed x* has
a distribution of the same type as in Fig. 3a. In the present
illustration it so happens that the molecular distribution can
only be observed indirectly, since the individual molecules are
too small for direct observation. Conceptually, we may think
of the observed distribution as referring to the individual molecular
weights subject to observational error.
Comparing the situation in Figs. 2a and 3a with the more
general situation in Figs. 4a and 5a we note two simple instances
of attenuated inference:
21 Wold - pag. 21
136
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
1) In Fig. 2a the weight pn refers to each of the sugar molecules,
and similarly for the estimate x in Fig. 3a. In Fig. 4a
the weight p holds only as an average for all of the nylon molecules
subject to observation, and similarly for its estimate x
in Fig. sa.
2) Let M be the (average) second order momentum of the
molecules. For the sugar molecules the theoretical model illustrated
in Fig. 2a gives
38)
M =?
and in the model illustrated by Fig. 3a a point estimate of M
is given by
(39)
Mer
For the nylon molecules the theoretical distribution gives, as
Ulustrated in Fig. 4a,
FAW
)
M =u
+
Nonstochastic variables subject to exact observation. a) Molecular
weight of sugar. b) Boys law PV —=c
where ¢ is the standard deviation of the distribution. The
simple point we wish to illustrate is that the inference (38) for
2] Wold - pag. 22
SEMAINE D’ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC
r —
Si
Nonstochastic variables subject to observational erro:
a)-b) Same variables as in Fig. 2 a-b
the deterministic situation does not extend to the inference (40,
for the stochastic situation. or more generally,
(AI)
HEl[a "FE '
that is, the expectation of a nonlinear function f(x) will only ...
exceptional cases equal the function of the expectation.
Turning now to bivariate relationships, Figs. 2b and
refer to BOYLE’s law
2u
(42,
FIG. 4 - Stochastic variables subject to exact observation. a) Moleculas
weight of nylon. b) Consumer demand v as function of market price
x.
21 Wold - pag. 23
138 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2‘
or in words: For an ideal gas kept at constant temperature in
a closed container with alterable volume, pressure P times
volume V is constant. In the accuracy of ordinary scale readings,
BOYLE’s law holds as a deterministic relation. Typical
inferences from BoYLE’s law are that if the volume of the container
is known, say V, the gas pressure is given by
PLY,
“IG
- Stochastic variables subject to observational error. a)-b) Same
variables as in Fig. 4 a-b.
and if the pressure is known, say P,, the volume is
(44)
V=c/P,
If an experiment is performed to demonstrate BoyLE’s law
there will be small deviations owing to observational errors in
P and V, as shown in Fig. 3b. Taking the constant c to be
unknown it can be estimated from the data, for example by
the method of least squares as applied to the logarithmic
relation
45,
log P+log V =log c
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29
The resulting estimate can then be used for the inferences (43
and (44).
Next we come to Fig. 4b, which illustrates a demand rela
tion specified bv wav € an 2 “her predictor, sav
(46)
wilh
(47
A
where a 1s the demand elasticity with respect to price. A typical
inference from the model (46)-(47) is that if price is known,
say p,, the expected value of consumer demand is given by
(48)
Gi
Comparing with (43) and (44) we note that the present inference
is attenuated in two respects:
1) Whereas the inference (43) is deterministic, exact, the inference
(48) about consumer demand is designed to be true
only as an expected or average value. This is so because
demand is influenced by many other factors than price,
influences that are summed up in the residual variable vu.
Whereas the deterministic relation (42) allows the twofold
inference (43)-(44), the stochastic model (46)-(47) allows
only the prediction (48) of d for known p. In fact, if we
solve (48) for p, and drop the subscript the ensuing relation
>
JU,
is not an eo ipso predictor
E(pld) Zc «
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
More generally, considering mode! (1) and letting f= (+) denote
the inverse function of (+), we have
50)
E(x|y)=f-1(v)
The inference (49) from d to p would hold good if the model
(46) were disturbance-free (v having probability one of being
equal to zero), or if our model were not (46) but instead
(51)
{4
p=ced a
+
with
52)
E[pldi=c «
In such case the ratio 1/« would be an operationally meaningful
quantity, namely, the price flexibility with respect to demand
('%). Again, of course, model, (51)-(52) does not allow
the reverse inference (47)-(48).
Coming finally to Fig. 5b, the measurements are here subject
to observation error. Thus for fixed p, the dotted curve
represents the distribution of d =d* +¢, where d* has the same
distribution as in Fig. 4b, and e is an observation error with
the same type of distribution as in Fig. 3a.
2.2. Operational aspects of deterministic and stochastic models.
The simple illustrations in 2.1 have been selected with a
view to elucidate three aspects of the transition from deterministic
to stochastic approaches.
(') See Ref. 26, which is a basic reference for deterministic approaches
in econometrics.
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SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 14}
(1) Determunistic and stochastic models do not obey the
same operational rules. Since stochastic models cover deterministic
models as a special case, the general rules for operating
with eo ipso predictors are valid also for deterministic models,
but the converse is not always true. Operations that in a general
way extend from deterministic relationships to eo ipso
predictors include addition and substitution. Procedures that
never extend to eo ipso predictors include squaring and inversion.
The explicit solving of a system of implicit relationships
extends to eo ipso predictors only in the special case when the
solving can be performed by iterated substitutions (1). It is
this last restriction that lies behind the fact, noted in 1.2 (3)
and 1.4 (3), that the predictor specification (13) of the primary
form of CC-systems has no counterpart in ID-systems.
Causal relations ('?). If we compare the operative aspects
of cause-effect relationships and eo ipso predictors we note a
far-going isomorphism, and specifically so with regard to the
basic operations of inversion and substitution. Thus if y is influenced
by a causal factor x, this does not imply that x is influenced
by y; isomorphically, if f(x) is an eo ipso predictor
of y this does not imply that f~!(y) is an eo ipso predictor of x.
As regards substitution, if y is influenced by a causal factor x,
and x is influenced by a causal factor z, we say — and in
principle this is a piece of causal inference — that y is influenced
by z via x. For linear eo ipso predictors we have
the corresponding theorem that if the variables x, - a”
interrelated by
E(ylx, 2Y=1(x, 2) and E(x|z)=e(2)
then (13)
(") Ref. 27; cf. also Refs. 28 and 29.
(') For a more detailed discussion of the causal aspects of model build
ing, see Refs. 16-19 and 30.
(7) See Ref. 12 for a detailed treatment of the linear case. The substitutional
theorem is in (53)-(54) quoted for three one-dimensional variables
x, y, z. It extends to the case when z is a vector variable, the kev feature
being that functions f and g involve the same vector »
Wold - pag. 27
42
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(54)
E(vlz)=f(e(2), 2)
Thanks to the isomorphism here briefly touched upon, eo ipso
predictors are a most convenient tool for the analysis of causeeffect
relationships. Since behavioural relations are cause-effect
relationships, this last comment leads us to the second point,
namely:
(2) The rationale of making use of eo ipso predictors in
the specification of behavioural relations. The main argument
ts, of course, that unless a behavioural relation makes an eo
1pso predictor it cannot provide forecasts that are unbiased in
the sense of expected or average values. This point is brought
in relief by (47) and (50), and the reader will have no difficulty
to supply any number of similar illustrations.
(3) Eo ipso predictors in multipurpose model building.
Speaking broadly, the transition from deterministic to stochastic
models makes no trouble, in principle, if the model involves
just one relation of potential use for forecasting; all that is
needed is to design the relation so as to make an eo ipso predictor.
It is quite another matter that the relation can be a
bad forecasting device because of specification errors, but in
this respect there is no difference between deterministic relations
and eo ipso predictors. The trouble begins when the
model involves two or more predictive relations, inasmuch as
the corresponding eo ipso predictors may be incompatible. It
is important to note that the ensuing questions of compatibility
or noncompatibility belong to the pure probability theory; no
empirical or substance-matter considerations enter into these
matters. Such is the situation in (48)-(49), where the inference
from p to d and from d to p cannot be obtained by way of
two eo ipso predictors that form a pair of inverse functions.
This nonexistence theorem in probability theory was one of
the cornerstones when KARL Pearson laid the foundations of
correlation and regression analysis, but its implications for
causal analysis by regression methods remained obscure for
2] Wold - pag. 28
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a long time, as witnessed by the debate on « the choice of
regression » in the 1920’s and 1930’s. Such is also the situation
in multirelation systems with regard to predictive inference
from the primary form and the reduced form. In VR-systems
this dualism does not arise since the two forms coincide; in
CC-systems both forms can be specified in terms of eo ipso
predictors thanks to the substitutional design of the primary
form; but the design of ID-systems is too general to allow this
bi-expectational specification. The situation makes a genuine
dilemma for the ID-approach, for if the primary form with
its behavioural relations cannot be specified in terms of eo ipso
predictors and thereby as cause-effect relations subject to random
disturbance, the operational meaning of the entire mode!
comes in doubt.
The dilemma of ID-systems is reflected in the debate on
the rationale of « simultaneous equation systems » in the 1940’s
and 1950’s. In hindsight, what has made the controversial
and partly confused debate on « the choice of regression » and
« simultaneous equation systems » so persistent is the old and
strong tradition of deterministic model building in economics,
combined with the fact that empirical treatment of the models
and thereby the need for their stochastization came into the
picture at a relatively late stage. In the research literature the
need for stochastic models was fully recognized in the early
1930’s, but in economic textbooks the deterministic models still
dominate the scene. Specific reference is made to the deterministic
cobweb model of 1930, in the simplest case given by (4)
55a-c)
supply relation
demand relation
instantaneous equilibrium
(*) See M. EzEKIEL (1938) for an excellent review. Cf also H. SCHULTZ
1938), pp. 77-80
21 Wold - pag. 20
44
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2.
in which model we further note the ensuing relation for current
price,
(56) p, = D-'(q,)=D-'(S(p,_,))
This famous model that has entered a great many textbooks
is perhaps the single feature that has contributed most to the
obscurity in the debate on « simultaneous equation systems. »
We see that the model paves the way into two main pitfalls
that hamper the stochastization of multipurpose deterministic
models, one being the inversion of single relationships, the
other being the explicit solving for the current endogenous
variables in nonrecursive multirelation models. It should be
clear from the above that this comment is not written in a
critical vein, but rather to emphasize the innovating features
of CC- and ID-systems. If an appraisal is in place, it is to
pay homage to JAN TINBERGEN, one of the three initiators of
the cobweb approach, whose superb intuition led him around
these pitfalls later on when he constructed the first CC-systems.
A way out of the dilemma referred to is provided by a
recent theorem that makes ID-systems bi-expectational by
means of a respecification of the primary form. We proceed
to a brief presentation of this new twist of the ID-approach.
2.3. On bi-expectational interdependent (BEID-) systems (1).
Given an ID-system (18)-(23), the corresponding BEIDsystem
is obtained as follows: The primary form (18) is respecified
by the definition
y,=A v'+B z,
(°) For equivalent results in less elaborate form see Ref. 13, Theorem 10
and Ref. 12, Remark 3.2.2b
2] Wold - pag. 30
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where A. B and z, are the same as in (18), while (152)
58)
y" =E(v,IR z)=R
is the vector of conditional expectations of the current endogenous
variables as given by the reduced form, which as before
ts assumed to be given by (21)-(22) with (23), to repeat
(50)
B -
Then the following relation can be established,
(60)
E(y ly, z)=A vy. +B ¢
showing in conjunction with (58) that the BEID-system is biexpectational
in the sense of 1.2 (3).
As to the proof of (60), we note that the model becomes
deterministic if we respecify the primary form (57) and the
reduced form (59) by deleting all residuals and in the left-hand
members substitute y; and y, for y, and y,. This follows as
an immediate corollary from the substitution theorem (53)-(54),
allowing z to be a vector variable.
In the debate on « simultaneous equation systems » it has
been a key point what causal interpretation, if any, can be
given to the parameters a; of the behavioural equations in the
primary form (18) of ID-systems (again, see footnote 6). The
parameters a;, being numerically the same as in the corresponding
BEID-system, relations (57) and (60) give the answer that
the parameters allow the same cause-effect interpretation as
in CC-systems. except that whenever a current endogenous
(59) In the manuscript as presented at the Study Week, the matrix R
was missing in E(y, | R z,) in formulas (23) and (58). Cf. the paper (b)
referred to in the subseauent discussion. page 6. footnote (1
2] Wold - pag. 31
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2.
variable y,, occurs as causal (explanatory) variable it must
be replaced by its expected value y,, as given by the reduced
form. Or to paraphraze in terms of MARSHALL elasticities, if
all variables y, z; in (18) are logarithmic, and y i,t 1S current
demand and y, current price, then a,; is the elasticity of
demand with respect not to observed price y, but to expected
price y 7.
It will be noted that the expected value yj, of a current
endogenous variable is in (58) introduced as a purely stochastic
concept. It is an entirely different issue whether this expected
value can be given a subject-matter interpretation as an expectation
in the psychological sense. Thus if y; is observed
market price, and the consumers’ anticipations of market price
could be assessed, say y;;, for example by interviews on a
sampling basis, the definition (58) involves no implicit conjec-‘ure
as to whether y;, and y;; will be approximately equal.
The parameters of a BEID-system are numerically the same
as for the corresponding ID-system. Hence the problem of
parameter estimation is precisely the same for BEID- as for
[D-systems. Among the estimation techniques developed for
[D-systems, specific reference is made to H. THEIL’s two-stage
method of least squares, Ref. 33, which conforms operationally
to an extension to BEID-systems. Briefly stated, the procedure
is to estimate the reduced form by least squares regression,
substitute the resulting estimates for the left-hand members into
the right-hand members of the primary form, and then estimate
the primary form by least squares regression.
In the following illustration we shall consider three types
of model, all with the same patterns of nonzero coefficients A
and B in (18), but in general with different numerical values
for the nonzero coefficients.
(x) ID-systems, or RFUE- (reduced form uni-expectational)
systems. This is an arbitrary system of type (18).
(2) PFUE- (primary form uni-expectational) systems.
This model is obtained from (18) by respecifying the nonzero
2] Wold - pag. 32
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147
parameters so that all relations in the primary form make
eo 1pso predictors.
The notation RFUE-system serves to emphasize that the
reduced form but in general not the primary form makes a set
of eo ipso predictors. In PFUE-svstems it is the other way
around (19).
(3) BEID- (bi-expectational interdependent) systems.
Here, to repeat, both the primary and the reduced form are
specified in terms of eo ipso predictors.
Illustrations ('7). Whereas an ID-system and the corresponding
PFUE- and BEID-systems in general generate
three different stochastic processes, the following three models
have been designed so as to generate one and the same stochastic
process. Hence if a realization has been generated from
one of the models, the realization by itself cannot indicate from
which one of the three models it has been generated. The
process involves two endogenous variables p,, g, and no exogenous
variable, and it is stationary and Gauss-MARKOVIAN with
the following nine parameters,
‘61)
E,
(62)
- x
4
1°
A
(16) PFUE-systems are what I have earlier, Refs. 12, 28 and 30, called
implicit or conditional causal chain (CCC-) svstems. covering as special cases
circular and bicausal chain systems.
(7) Models (65)-(67) and (68)-(70) are quoted from Ref. 30. I am indebted
to Dr. LYTTKENS for pointing out an erratum in Ref. 30, p. 394, where
the relation that corresponds to (69 ¢) is wrongly stated as E(v;, vi. =) =O.
The erratum does not affect the statement that the three models there
considered define one and the same stochastic process, but it does destroy
the Markov character of model (68)-(70). For example in (66b) we have
E(qlp.1) = Elg.lPr-1, Pi2G. 2 Dr as
but in general not so in (7ob)
Wold - pag. 33
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
where p is a constant that can be fixed arbitrarily in the interval
(63)
0<p<0,06 .
The basic ID-system is a highly simplified demand-supply model
where instantaneous equilibrium is assumed,
64) demand d, = supply s,=q,
To bring out the characteristic differences as clearly as possible
the primary forms of the models have heen constructed so as
to differ only in one parameter, namely the coefficient of price p,
in the demand relation.
An ID- (or RFUE-) system. The primary form:
d,=q = -3 p +U
(6sa-b) | se t 11
SMF ÉP-st Ou With E(g lp.) =~
>
Diy
The reduced form:
(66a-b)
pe —P Paty,» E@Ip_)y=-p pi;
q:= = P_1 +97, » E(glp_))= = Pr_1
ind
(67a-c)
Ei 0120) = E(Uy Vg000) = E(u, V2) =0; k=1, 2, ..
The corresponding PFUE-system. The primary form:
(68a-b)
[7 —-i
4 pet vy: with E(q;lh;)= - =
Elglb,_1)= Zp
£
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14C
and
(6ga-c)
Evi, vi; 2) = E(Va, Vogsr
E(v
H
-' 9
The reduced form:
(70a-b) ) ,
|
'
4
FTV.
,+v, with E(q,l;,.
The corresponding BEID-system. The primary for...
(71a-b)
ye
/
WY
Ly
- 4%
and
(72a-b)
* * * * * *
E(vy, Viger) =E(vy Va ak) = E(vy, Vor) <0; R=TI, 4, …
Reduced form:
(73a-b)
{vo
WIL. E(p;lPe_1,
Elo ¢
giving
(74)
{1 Wold - pag. 35
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - %¢
The first point we wish to illustrate is that all four relations
(71) and (73) make eo ipso predictors. It will be noted
that in any BEID-system the expectational variables y ;; are
linear expressions in the predetermined variables z,. Owing
to the very simple structure of the model (71)-(74) the expectational
variable p; is in the present case nothing else than
-P P,_1- As a consequence, the demand relation (71a) coincides
with the supply relation (71b). This last feature illustrates
how the reduced form may in the BEID-approach contain
more information than the primary form.
Another point for which the three models provide clearcut
illustration is that once the stochastic structure of the model
is specified the parameter estimation is technical matter and
therefore, in principle, a noncontroversial problem. For eo ipso
predictors least squares regression provides consistent estimates;
hence, for example, when applied to time series data generated
from the stochastic process specified by (61)-(63) the
regression of p, on d, will provide a consistent estimate for the
coefficient — 0.8 in the demand relation (68a) of the PFUEsystem,
but in general not for the coefficient - 0.6/p in the
demand relation (65a) of the ID-system. We see that if the
least squares regression is applied to (65a) the bias may be
quite substantial, depending on the numerical value of p, and
that the least squares estimate will be unbiased only in the
special case when p=0.75.
3. PREDICTIVE TESTING OF NONEXPERIMENTAL MODELS (18)
In the big arsenal of statistical methods, the techniques for
the design and analysis of experiments are on the whole much
more developed and refined than the techniques available for
nonexperimental data. This is in particular so for the statistical
*) The general argument of this section borrows from Ref. 34.
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SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 151
procedures of hypothesis testing. In experimental situations
the empirical testing of a model can be based on replications
under controlled conditions. It is here a recognized principle
to treat the experiment as self-contained, not allowing the test
procedure to exploit any information outside the observed
sample (!?). For the testing of a variety of models a great
many routine techniques are available that are of maximum
power under the specified experimental conditions. In nonexperimental
situations the principle of the self-contained experiment
1s an unattainable ideal. Instead, approaches come to
the fore that in a more or less systematic manner exploit other
sources of information than the theoretical model and the empirical
observations used for the estimation of its parameters.
Such sources include comparisons with the results of similar
or related models constructed for other regions or time periods;
testing the validity of the model by ancillary theoretical arguments
ad hoc; and, first and last, predictive tests where forecasts
from the model are followed up by observation ex post
and comparison with the actual course of events. Confrontation
with fresh evidence is, clearly, the real touchstone for the
scientific validity of the model as well as for its practical use.
Specific reference is made to KLEIN-BALL-HAZLEWOOD-V AN-DOME’s
macroeconomic quarterly forecasting system for UK,
Ref. 37, and its predictive testing, Refs. 38 and 39. Here is
a keen follow-up study that without hesitation sets forth how
the predictions conform or fail to conform to the actual developments,
and the ensuing lucid and instructive comments sort
out the weak and strong points of the model. Such follow-up
studies are highly important, and indeed an indispensable supplement
to the published models, both for improving the model
subject to scrutiny, and as a guide for other related forecasting
projects. Some twenty models have been constructed in seven
different countries for predicting the boom-recession pulsations
(**) Stated by R.A. FIsHER (1935), the principle has been brought te
full significance by 1. TUKEY (1954)
2 Wold - pag. 27
152
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 4.
and the number of models is rapidly increasing (¥). The models
are huge and complex systems, with scores of relationships,
and up to hundred or more variables, and often with shifts of
emphasis in the design that render it difficult or impossible to
make comparative studies on a theoretical basis. In this situation
predictive tests are at a premium because they bring the
models on the same line in comparing the forecast performance.
In the present stage of development the need for predictive
testing is especially urgent. Once a model is in permanent use
the ensuing forecasts will in due course automatically give material
for predictive tests. As yet, however, very few, if any,
of the published models are going concerns, and therefore it
takes a nonautomatic decision — not to speak of the courage —
to plan and carry through predictive tests.
In view of their key importance I wish to make a plea
for the systematic use of predictive tests in the construction
of dynamic macroeconomic models. This last section of my
report will have fulfilled its main purpose if it can stimulate
to a joint move in this direction by the participants of the
Study Week.
In making this plea I wish to emphasize that the entire area
of dynamic model building is as yet in an early stage of development.
It is perhaps too early as yet to expect forecasts
that look neat in the sharp light of a predictive test. So much
the more pressing, however, is the need for predictive tests
for the guidance of research in the many branching complexities
of macroeconomic model building.
3.1. The pluralism in model building for different purposes.
The rest of this report takes up some few specific aspects
of the techniques of predictive testing. The various questions
(*) The many-faceted and rapid developments are well pictured in M. NEr-LOVE’s
survey, Ref. 40
[21]
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are dealt with from the point of view of model building in nonexperimental
situations in general, not only in econometrics.
The first point refers to the basic pluralism when it comes to
model building for different purposes. Typical in this respect
is the difference between short range and long range forecasting
and the ensuing differences in the model construction. To
bring the general perspective in relief, reference is made to
similarities in this respect in economics and meteorology.
Meteorology . . 24 or
Economics 6 or
Short range
forecasts
hou
ve R
months
Long range
forecasts
4 or 6 weeks
5 or 10 years
In meteorology the short term forecasts are fairly reliable
over 24 to 48 hours. The principal basis of the short term
forecasting techniques is the thermodynamic theory of cyclons
of the BERGEN school (?!). The importance of this theoretical
innovation can be read off in the gradual increase in the reliability
of the forecasts from 1920 or thereabout.
Meteorological forecasts over the « long range » of 4 or 6
weeks is a more recent development. Here the forecasting has
to be based on other phenomena than the cyclons and their
individual paths. The technique is still in its beginnings, and
the reliability is much lower than for the short range forecasting.
In economic forecasting, « short term » means ranges from
3 or 6 months up to 6 or 8 quarters, and the model building
here focusses on the boom-recession pulsations. « Long range »
usually means something like 5 or 10 years, and the allimportant
purpose of the model is to analyze and forecast economic
growth. Hence, just as in meteorology, the subject
matter content of the economic model is radically different in
short range and long range forecasting. On the other hand
(*') J. BJERKNES (1919) is a basic reference.
of the co-founders. see T. BERGERON (105g).
For a recent review bv one
.,» Wold - pag. 39
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - .
there is a notable disparity, for in meteorology the emphasis
on forecasting is the same in both approaches, whereas in
econometrics this emphasis is more pronounced in short range
than in long range forecasting. As is well known, econometric
models of economic growth often are a hybrid between strict
forecasting and economic programming and policy making.
3.2. The Janus quotient: A predictive test criterion.
The Janus quotient, an adaptation of the FISHERIAN F-ratio,
has recently been proposed for purposes of predictive testing
(7). Its formula is
(75)
LHI ;
_ (n—ey
ns
: (a, — 7)?
or | = + VJ?
depending on whether we prefer to compare variances or standard
deviations. The test refers to a specified model, say M.
The notations are a,, a,, ... for the actual observations, and
Yi» V2» for the corresponding theoretical values as obtained
from the model. Conceptually, the JANUS quotient refers to
the instant between the past and the future, and as indicated
by its metaphoric name, the JANUS quotient looks in two
directions, backward in time over the observation range
/=I, ..., n to form the denominator, forward in time over the
prediction range t=#n+1, ..., #+m to form the numerator.
The design of the JANUS quotient can be generalized and
varied so as to adapt to different types of application. Let
(3) See Ref. 43, also for further details. Cf. the U-criterion earlier proposed
by H. Tue, Ref. 33, from which the J-criterion differs in being
invariant to linear scale transformations.
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155
us consider a situation often encountered in applied work, namely,
when the forecasts y,.,, ..., y,,, from model M are
formed by means of ancillary forecasts of one or more exogenous
variables (?); let y;,, ...,y;, be the quasi-forecasts
obtained when the exogenous variables are known at the end
of the forecast period and substituted for the ancillarv forecasts
in M: then
(76)
is a measure of the accuracy of the forecast model M when
those forecasting errors are removed which arise from imperfect
ancillary forecasting.
Since the numerator and denominator of the JANUS quotient
measure the deviations between theoretical and observed values
in the observation range and the forecasting range, respectively,
the JANUS quotient may be regarded as a criterion of stable
model structure in the two ranges. To elaborate this point we
shall consider two types of forecast
(1) Forecastin
i
extrapolation (*
This approach includes forecasting by deterministic extra
polation,
(77 a) vy,=f()+v. with E(y.\=f(#
where f(¢) is a specified function, usually with parameters
estimated from the ohservation range For example. f(#) max
(3) Cf. HazLEwoop-VANDOME (1961), where forecasts of type y*,., are
referred to as being obtained by extrapolation. For applications of the same
device in unirelation models, see R. BENTZEL (1959).
(*) See Ref. 43 for a more elaborate treatment. including application.
M (75)-(76)\ th unirelation models
Wold - pag. +
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2S
be a linear or curvilinear trend, a sinusoid or a sum of periodic
components. Now assuming that the model (76) is valid both
in the observation range and the forecasting range we have,
as a first approximation,
(77 b)
E(J)m1
valid if the forecast range is short relative to the observation
range.
The present approach further includes forecasting by
exogenous variables. Considering the linear case, model M is
(78)
Ve=B80+B x; +... +B, x, +,
with
(79)
E(y,lx,, > Hing) =03, +8, Xgut... +3, XL}
where the coefficients 3; of the exogenous variables x; usually
are estimated from the observations. Assuming stable model
structure we have in this case
(80)
E(J%)~1
provided the forecast range is short.
The criteria (75) and (76) can be developed in the direction
of significance tests. The following result is due to S. MARTI-NELLE
(*). As applied to a model (77 a) with k linear compo-(*)
Ref. 45. I am indebted to Mr. MARTINELLE for kindly placing his
unpublished results at disposal for the present report.
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nents f;(#) and with residuals v, independently and normally
distributed, the TANUS quotient (75) is distributed as
m—r
(81)
Here the ¥2-variates are independent w
indicated within paranthesis: x.
squation
.u degrees of freedorr
are the roots ol
det (C. 2C,—U
where C is the product sum matrix
«
Wliüil
and C, is the corresponding product sum matrix for the prediction
range. It will be noted from (81) that the JANUS quotient
tends to increase with the prediction span and with the
number of free parameters.
2) Forecasting by the chain brincible
We shall consider the case of a stationary unirelation model
M which we specify bv the representation
(82a-b)
*
5 - Br V1 +Ba V2 +
«Ly U t 1, +X, v, , =
[2] Wold - pag. 43
58
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
with
(83)
EWdyi—1 Viras )=Bo+B1vi_1 +85 vy, 0+...
which is a special case of the predictive decomposition of y,
referred to earlier in this paper [section 1.4 (4)]. The forecasts
Yn+1> Yn+2 --- are obtained by the chain principle, making iterated
use of (83). Thus when y,,; has been obtained, v,,;.,
is calculated from (83) in the basis of y,.;, v,.; 1, ... The
variance of the resulting forecasts is given by
84)
E(yrumT Ans) = (1 + a? + + a? _,) ok
showing that the accuracy of the forecast will decrease as the
forecast span me increases. This last feature is reflected also
in the JANUS quotient, inasmuch as (84) gives
I
(85) E(J)=1+(1- 2) o24 (1-5) a+ + — ab,
The predictive decomposition (82a-b) has the property that
the variance (84) is the smallest possible of all representations
of type (82b). This is the fundamental property of minimumdelay,
established by E. RoBINsoN, Ref. 24, and referred to
earlier in this paper.
The approach (82)-(85) extends to the general stationarv
case when y, allows the predictive decomposition
‘86a-b)
Vi=Y +B ya th
=W+v,+%, v, +a,
‘À
where V, is the deterministic (also called singular) component
of y,. The procedure of forecasting first settles the prediction
of the deterministic component over the entire forecast range,
>]
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15¢
and then applies the chain principle in the same way as in
(82)-(83) to forecast y,~W¥,. We see that from the point of
view of forecasting, the deterministic component ¥, of the model
is, in principle, equivalent to an exogenous component.
Furthermore, the approaches (82) and (86) extend to mul
tivariate models; Ref. 46. The corresponding predictive decomposition
(82a-b) will then yield a representation in the
form of a CC-system. The minimum-delay property extends
to multivariate predictive decomposition. Hence the ensuing
forecast variance (84) is smaller than in other linear forecasting
models, such as ID- or BEID-systems.
The application of the JANUS quotient extends to multivariate
systems, and in particular to VR- CC- ID- and BEIDsystems.
By suitable adaptations the JANUS quotient can test
the entire system or a specific behavioural relation. Ii can
also, as illustrated below, be adapted so as to focus on the
extrapolation aspect of specific relations by removing those
forecasting errors which arise from imperfect forecasting of the
explanatorv variables.
ILLUSTRATIONS
1) Market model for pork. US 1932-1056.
Ref. 20).
(G. STOTKOVIC.
We shall consider the recalculated model as given by
(36a-c). For the demand relation the fit in the observation
range is moderatelv close.
S(VY=0.60 sid)
The Janus quotient (76) as calculated for a forecast span of
three years, treating the explanatory variables as known ex
post. gives the vearlv valnes
<1 Wold - pag. +45
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(87)
T.990 =1.08
J,4958 =0.94 J. 199=0.68
and the aggregate value
Je — 0.02
The price mechanism gives a similar fit in the observation
range,
s(v”)=0.58 s(p)
Here the extrapolatory JaNus quotient (76) gives
(88) J.495)=0.69 J 19 =0.01
T1999) = 0.02
and the aggregate quotient
Ts =0.66
Using the three relations of the system (36) to generate forecasts
by the chain principle, and applying the JANUS quotient
(75) to the ensuing forecasts for price p,, the material gives
the following quotients
89)
JUD = 36 JU =1.56 JU) =0.04
and the aggregate
J=o0.02
According to (80) and (85) the Janus quotients (89) could be
expected to be higher than in (88), but the forecast refers to
only one sample series and here it so happened that the tendency
did not materialize very clearly.
2] Wold - pag. 46
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 16.
(2) Macroeconomic quarterly model of UK, 1946-1956.
(KLEIN-BALL-HAZLEWO0OOD-VANDOME, Refs. 347-39).
As applied to a model like (36) for a sector of an economy,
a predictive test of type (75) or (76) is of course of limited relevance
because the sector is liable to exogenous influences that
may disturb or upset the model and thereby the forecasts. So
much the more relevant is a predictive test of economic models
that comprise the economy of an entire nation, although here
too there are external disturbances on the international plane.
The following figures report briefly an attempt to apply the
JANUS quotient to the abovementioned model for UK 1946-1956.
The model has not been published in such form as to
give the theoretical values obtained from the model in the
observation range; hence the quotient has only been calculated
for three variables, and is partly based on reading off the
graphs of the residuals. Since the tests refer only to a small
fraction of the model it need not be emphasized that the figures
are only given to illustrate the technical procedure of the predictive
test (26).
Year
1957
1958
1959
Quart
Janus
nuoti-Industrial
productior
Price index o
‘inal output
tere
abe
)
The various generalizations and adaptations of the jaNUS
quotient focus on iust one aspect of forecasting accuracy. na
(**) My thanks are due to Miss INGRID AGERHOIM and Mr. K. NAzIMUn.
DIN for assistance in the computations
[2] Wold - pag. 47
(62
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 4
mely the relative size of deviations between theoretical and
observed values. It goes without saying that this is a serious
limitation, and that the JANUS quotient therefore by no means
is a panacea in the testing of forecasting accuracy. Specific
reference is made to the importance of paying special attention
to turning points in the phenomena under analysis. To quote
a wellkown example from meteorology, Ref. 47, the use of
digital computers in short range forecasting was tried out on a
cyclone which on Thanksgiving Day 1950 swept the US continent
in a wide and softly curved swing, and when coming to
‘he Atlantic made a sharp turn northwards through New England.
It was no difficulty to simulate and forecast the wide
swing on the computer, but for a considerable time all of the
trial models led the forecast path of the Thanksgiving Day
cyclone right out into the Atlantic, and it took a qualified combination
of meteorological thinking and data compilation to
construct a model that reproduced the sharp turn.
3-3. Overfitting.
In the nonsense department of statistical method everybody
has seen the pitfall of overfitting — the situation when a model
gives illusively close fit to the given data because the available
observations are outnumbered by the parameters. A case in
point that is actually on record is the time series analysis of
a sea level, in which study 122 annual data were graduated by
a sum of 40 sinusoids with different periods, phases and
amplitudes. The resulting fit in the observation range was
very very close, and the forecast for the next year was included
in the report. Just as the report was published the next observation
emerged, showing an ample deviation from the forecast.
The comment of the author was that unfortunately he had
forgot to include the 41st component.
The parameter estimation of ID- and BEID-systems by
‘21
Wold - pag. 48
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 163
the two-stage method of least squares sets the trap of overfitting
in a new disguise. The transformation (22a) tends to carry
all parameters of the entire primary form into each relation
of the reduced form (*7). Thus if there are some 3 or 4 parameters
in each behaviour relation of the primary form, and the
primary form involves some 50 behaviour relations, each relation
of the reduced form will involve some 150 parameters, far
more than the number of observations usually available. Well
to note, I am not saying that the model builders walk straight
into the pitfall of overfitting. On the contrary, from the beginnings
of the theory of ID-systems it has been a rule to specify
each relation of the reduced form as involving all predetermined
variables of the entire system, and the ensuing dangers
of collinearities in the empirical parameter estimation have
been recognized for a long time (*). The specific point I wish
to make is that the risk of overfitting is tangible already with
a modest number of variables in the reduced form, owing to
autocorrelation and inertia effects in most or all variables of
the system. Such overfitting will blur the distinction between
an ID-system and the corresponding BEID-system, with risk
that the resulting parameter estimates will be biased. To assess
and evaluate the bias by an aprioristic analysis is extremely difficult
or — at least in practice — impossible, because it leads
into overwhelming complexities even for forecasting systems
of moderate size. The approach of a predictive test will however
reveal the overfitting, and thereby the test will also reveal
the difference between the ID- and BEID-systems and the
ensuing bias in the parameter estimation.
(7) Explicit illustrations are given in Ref. 13, p. 484, and with mors
detail in Ref. 27.
(*) The difficulties at issue were amply emphasized by Professor D W
JORGENSON in the oral presentation of his report. Ref. 48. to the Copenhagen
meeting of Feonometric Society. Tulv 1062
ar
old - pag. 40
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
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— Ends and means in econometric model building. Basic considerations
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An outline in the history of ideas and hints of a program. In « The
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,7 CHARNEY J.G. and N.À. PHrLL1Ps: Numerical integration of the quasigeostrophic
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ey
Wold - pag. 52
YL
+ >SION
FISHER
As always, Professor WoLb has given us an interesting paper
My remarks are in the nature of supplements.
In the first place, I am interested in the case of bi-expectational
interdependent systems. As Professor Worp has shown, in
this case the elasticity of demand for example becomes elasticity
in terms of expected rather than actual price, where expected is
interpreted as the expected value of price given by the reduced
form equations. Now there is of course another sense of expectation
in economics; that is the sense in which a variable is expected by
people who make a decision based on it. An interesting question, it
seems to me, is under what circumstances and in what types of
models this will in fact be the same as the expected price given
from the reduced form. Only in such circumstances will it be the
case that elasticity with respect to expected price in fact is a meaningful
parameter which describes interesting behavior. I suspect
that the two coincide in a rather general framework. There is in
the literature a hypothesis known as the rational expectations hypothesis
due largely to JouN MurtH, according to which it is assumed
that the decision makers who are being studied expect the values
of the relevant variables to be those on the average which will be
predicted by the model. MuTH has shown that this is not logically
circular, Further, in this context it sounds like the sort of behavior
which would lead decision makers to expect that price predicted by
the reduced form
+
Wold - pag. 53
.68
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ©
Secondly, it is of course possible that Professor WoLD’s J? can
be less than 1. This can happen if structural change takes the form
of only a small change in the parameters but a large downward
change in the variance of the disturbance terms.
My next point is that Professor WoLp has been careful to avoid
an error which is occasionally made in the literature, namely that
least squares provides an unbiased forecast of the dependent variable.
That is false unless the least squares parameters are themselves unbiased
and this is not the case if, for example, a lagged dependent
variable appears on the right-hand side of the equation. Professor
Worp, however, has not said this although he has said something
which sounds like it. What he has said is that least squares is a
consistent estimator of an unbiased predictor and this, as he has
shown, is true.
Finally, it seems to me that Professor Wozp is unduly worried
about what he calls the danger of over-fitting in the reduced form.
The circumstance in which the reduced form cannot be estimated
by ordinary least squares because there are too few observations
relative to the number of exogenous variables in the model, is quite
a common one in dealing with large econometric models. This is not
of great consequence as a fundamental matter, however, because
one then uses intrumental variables methods which drop some of
the exogenous variables for purposes of estimation. No difficulty
of principle arises, although there is then a problem of how one
ought to choose the instrumental variables to be retained. This
is a question which I cover in my paper.
[HEIL
I. Regarding the difficulty of over-fitting in the reduced form,
Messrs. T. KLoEK and L.B.M. MENNES formulated a procedure (in
a recent issue of « Econometrica ») which is designed to handle
this problem, which is indeed serious when the number of predetermined
variables is not small compared with the number
"2] Wold - pag. 54
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 166
of observations. Their procedure amounts to replacing predeter
mined variables by a certain number of their principal components
The specification of that number remains an arbitrarv choice
2. I like the idea of the JaNUs quotient. Its application requires
that the forecasts are generated by some kind of probabilistic model,
e.g., a regression model. It is therefore not applicable when
there is no such model, e.g., when we wish to determine the accu
racy of entrepreneurial investment forecasts derived from an invest
ment survev.
3. As to your bi-expectational procedure, I would like to sug
gest that you subtract the reduced-form disturbances, not only from
your right-hand dependent variables, but also from your left-hand
dependent variable . Doing so, one finds that there is no disturbance
left in the equation at all, because all random parts are removed.
Wolbp
The comments by Professors FISHER and THEIL reflect that
simultaneous equations as an area of research cover a wide range
of theoretical and applied problems. The discussion is mainly oriented
towards the general foundations of the approach. Specifically
the following aspects are referred to :
1) The rationale of the approach from the point of view of economic
theory, probability theory, statistics, and the theory of knowledge.
Hereunder, much of the discussion is concerned with :
a) Predictive aspects of simultaneous equation svstems:
b) Causal aspects of the systems.
2) The statistical estimation of the parameters of simultaneous equa
Hons
(21 Wold - pag. 55
170 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
The discussion further reflects that the theory and application
of simultaneous equations is in rapid progress along several lines
of development. Hence the research situation was not quite the
same during the Study Week as now ten months later when the
replies are edited. In my replies I shall stick to the notes and
tape record from the round table discussion; when reference is made
to later developments, they will be made by way of footnotes (1).
With gratitude and satisfaction I note that the discussants of my
paper have to a large extent been concerned with my approach of
defining interdependent systems in terms of conditional expectations;
Refs. 12, 13, 30. The ensuing approach of bi-expectational interdependent
systems is in an early stage of development, and so
much the more I welcome a thorough scrutiny of its foundations and
implications. For easy reference in my replies, let an interdependent
system be written :
y=0By+T 2
when defined in accordance with the classic assumptions of the
approach, and
(B,)
y =
Ey*+T z+¢
hal val
+B (y—y#)=(i—8
() Reference will be made to the following two papers:
(a) L. R. KLEIN, Problems in the estimation of interdependent systems.
Forthcoming in the « Transactions des Entretiens de Monaco 1964 »;
Centre International d’Etudes des Problemes Humains, Monaco.
‘b) H. Worp, À fix-point theorem with econometric background. Forthcoming
in « Arkiv f. Matematik ».
2]
Wold - pag. 56
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 171
anc
(B'
when defined as a bi-expectational system. We see that an inter
dependent system when written in the classic form (A) has the same
numerical parameters as the corresponding bi-expectational system
(B). Hence the problem of parameter estimation is precisely
the same for the two versions of the model,
Professor FISHER’s reference to the pioneering work of J. F
MUTH, « Econometrica » 29 (April 1961), is greatly appreciated.
MuTH’s hypothesis of rational expectations opens up vistas towards
highly fruitful syntheses by assuming that expectations in the economic-psychological
sense are in the first proxy equal to expectations
in the sense of probability theory. In the context of my paper,
the conceptual distinction between the two notions of expectation
is referred to on page 32. To simplify matters in a first approach.
MuUTH considers market models of the cobweb type and assumes
that they are deterministic except for the supply relation, and he
makes a most interesting comparison with other theories of economic
expectation in the realm of cobweb models. The illustration in terms
of cobweb models makes a point of contact with my own studies;
see especially Ref. 30. The contact is tangential, and there is little
or no overlapping between the problems under analysis. MuTH
has explored the models with regard to economic-psychological expectation,
and economic-psychological vs. probabilistic expectation;
my own interest has focused entirely on the rationale of probabilistic
expectation, and in particular on the general rules for operating with
probabilistic expectations.
Professor FISHER comments that the coefficient of an expectational
variable is a meaningful parameter « only » in case there is
no difference between the economic-psychological and the probabilistic
expectation. I have here put « only » within quotation marks
hecause I think the statement is somewhat too strong. Specifically
oe
Wold - pag. 57
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
200
another case I am thinking of is when we use current GNP as an
endogenous explanatory variable y; but are aware that the statistical
assessment of GNP is not quite adequate in the explanatory context
of the model; it may then be meaningful to use the expectation y}
as explanatory variable, in the hope that it gives a better proxy to
the nonobserved explanatory variable than the observed value of
GNP. Another potential application that comes to mind is that
if y, is individual consumer income, y;* might serve as a proxy for
permanent income in the sense of M. FRIEDMAN’s well known theory.
In specifying the subject matter content of his models J. F. MUTH
makes use of causal notions, and the expectational variables enter
both as causal factors and as effect variables, His use of causal
notions makes for a general affinity with my own work, which to
à large extent has been concerned with the much debated questions
that arise if we wish to provide a causal interpretation for the relations
and individual parameters of interdependent systems. When
an interdependent system (A) is respecified by way of (B,), this
transition makes for a clearcut interpretation of the expectational
variables y, * as causal factors, and from ;(B,) we see that the variables
y * will also play the part of effect variables. It will be noted
that J. F. MUTH’s model is not quite in accordance with the biexpectational
framework (B,)—(B,), for he specifies the -demand relation
as deterministic by not including an error term, and in the
customary manner of cobweb models he treats the demand relation
as causally reversible by taking current demand to determine current
price. It would seem however that MuTH’s line of argument
only requires some slight qualification to be in accordance with the
bi-expectational form (B,)—(B,) of interdependent systems.
Professor FISHER’s comment on J2 is certainly to the point, and
it brings in relief that the simple proxy J2 cor primarily refers to
‘he case of stationary deviations from the theoretical model.
I appreciate very much that Professor FISHER emphasizes a pitfall
about least squares: If a theoretical relation is an eo ipso predictor,
it can be consistently estimated by least squares regression,
but it does not follow that the reverse is true (that is, if least squares
[2] Wold - pag. 58
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 172
regression is chosen for the estimation of a specified theoretical relationship,
the choice of estimation method will not make this relationship
an eo ipso predictor). I take the opportunity to emphasize
the truth of a related negative statement: If least squares regression
is not a consistent estimate of a specified theoretical relationship
then this relationship is not an eo ipso predictor.
Coming to the last paragraph of Professor FISHER’s comments,
[ am afraid it reveals rather deepgoing differences between our views.
As regards the dangers of overfitting, they are certainly a real headache
(see e.g. the comments by H. THEIL, Ref. 33, section 6 D), and
the trouble does not become less real because it is « quite a common
one in dealing with large econometric models. » (2)
As to the approach of instrumental variables, this is a surrogate
of an ad hoc nature, inasmuch as the instrumental! variables are not
specified a priori in the model. It remains to be seen whether the
results of the approach are as a rule good enough to pass the test ot
confronting the ensuing forecasts with actual evidence by way of
predictive tests. Hoping for the best, I have no desire to disencourage;
all I want to say is that this is one of the open questions
in the present stage of development.
In reply to the first point made by Professor THEIL, I am confident
that KLOEK-MENNES’ adaptation of the principal components
approach goes a long way to overcome the difficulty of overfitting
and related headaches in the statistical estimation of interdependent
systems. The approach has the nature of a shortcut, however, and
obviously it runs the risk that in sieving forth the principal components
of the predetermined variables z, it may throw away one
or more z's that contribute relatively little to the total variability
of the predetermined varnables. and vet are highly important as
(?) Professor L. KLEIN in the paper (a) referred to in footnote (') strongly
emphasizes how troublesome the current techniques for parameter estimation
of multi-relation models are in the present stage of development. To
quote from the Monaco discussion of Professor Klein's paper, the results
he reports from the estimation of a 12 relation model give clear evidence
of overfitting, inasmuch as most of his OLS (ordinary least squares) and
TSLS (two stage least sauares: estimates coincide up ta the third figure
‘21 Wold - pag. s0
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
sources of variability of the endogenous variables, the variables
which the system has for purpose to explain.
As to the second point, I feel sure we see the J? quotient in the
same light, inasmuch as it is only a slight modification of Professor
THEIL’s coefficient of inequality, J? being more similar to the
Fisherian F ratio. The J? is designed to exploit the information
obtained when a model is confronted with past observations on the
one hand, and future comparisons between forecasts and actual
developments on the other, and it is clear that if there is no model
such information is not available.
In his third and last point, Professor THEIL expresses relation
B,) in words. We note the sharp contrast relative to model (A),
which in general implies
(C}
E (y|y, 3) # 8 y + T7
The respecification (B,) - (B,) gives a clearcut answer to those
questions which I have seen as obscure issues in my studies into
the rationale of multi-relation models. I would fain to repeat that
the respecification (B,) - (B,) dates only from a few years ago;
see Ref. 13, Theorem 10, also Ref. 12, remark 3.2.26. In a first
phase of my studies, Refs. 23, 34, a main theme was the comparison
between recursive (also known as causal chain) systems and interdependent
systems (A), a key point being that while recursive
systems have a form that is directly amenable to a causal interpretation
of behaviour relations and individual parameters, this is
not so for interdependent systems (A). This first phase includes the
joint attempt with R. Strotz, Ref. 18, to provide a causal interpretation
for the individual parameters of interdependent systems
(A). In a second phase, Refs. 27-30, my approach was to specify
the models in terms of conditional expectations, called eo ipso predictors,
or briefly predictors. One type of model considered, called
CCC (conditional causal chain) systems, was a straightforward
2] Wold - pag. 60
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIOUE ETC.
,
EU
respecification of interdependent systems (A) into predictor relations
giving
‘n°
*
-
Unlike (B), the respecification (D) in general involves a change
of the numerical parameters 3, T of the system. It would seem
that the respecification (B) is more fruitful than (D). Respecification
(B) is however so recent that only some of its implications have
been explored “
AI LAIS
I wonder if the distinction between the three cases you hav.
denoted as « vector regression, causal chain, interdependent system »
corresponds to a real difference from the point of view of the facts.
What we observe in nature is continuous. Thus if we consider
discrete series instead of continuous series, we introduce something
which is handy for the calculation, but something which does not
correspond to reality and which may result in the artificial creation
nf a certain number of difficultie-HAAVELMO
I enjoyed listening to Prof. WoLD’s paper. I think it was extremely
clear in its presentation. I only have one comment — about
something that bothers me a little bit. When he says to use this
nr that method of approach. does he mean to use this or that
(?) Respecification (B) provides a new approach for the statistical estimation
of parameters in interdeneudent svsteme: see reference (b) in
footnote (1)
or
Wold - pag. 61
16
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
statistical method to handle a certain economic theory? Or does
he mean that he is talking about methods of constructing a certain
kind of theory? I think many of us consider the construction of an
economic theory, including the specification of stochastic elements
in the theory, as one thing, and to confront it with the facts as
something different. If we have constructed a stochastic economic
‘heory based on certain principles of economic behaviour, we have
a model which can be interpreted as an indirect specification of the
joint probability law of the observable economic variables. From
such a model we may derive various kinds of « relations », expected
values, and other kinds of statements concerning the properties of
‘his joint probability law. It is perhaps unfortunate that we talk
about relations between economic variables, when actually we should
regard our economic theories as just indirect ways of saying something
about the probabilistic aspect of the variables we are talking
about. What do we actually mean when we say that a model or
theory fits the facts? Do we mean that it fits the facts as these will
be if the economy is not disturbed by changes in economic policy?
Or do we mean that the model would fit the facts as these would
be after a new kind of policy? The meaning may be one or the
other, depending on what we are after. Moreover, a model that fits
the facts in the first sense may often be used as a basis for extracting
information concerning a model that fits the facts in the second
sense, Some of the parameters to be estimated may be the same in
ooth cases. But ultimately, we may not be so much interested in
‘he joint probability law of the economic variables as it has been in
a past period, our final interest may be related to a future joint
probability law which has been modified by certain economic-political
actions. From this I think we may draw two conclusions,
first that it is extremely important to specify the economic meaning
of a theory or relation and, secondly, that closeness of fit for forecasting
purposes is by no means a decisive feature of a « good » economic
theory.
2] Wold - pag. 62
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 177
WoLD
Professors ALLAIS and HAAVELMO comment upon econometric
models from the point of view of epistemology, the genera] theory
of knowledge. The main point I wish to make in my reply is that
this is a highly important aspect of econometric model building.
Several of the much debated issues about econometric models are not
specific to econometrics, they are fundamental issues in the
social sciences by and large; and many of the econometric techniques
are pioneering in the still wide realm of nonexperimental model
building.
As to the first part of Professor ALLAIS’ comment, I would like
to emphasize the sharp distinction between on the one hand the
hypotheses that constitute the theoretical part of a model, on the
other hand the actual facts that the model serves to explain. There
is always a pluralism of models, and there is never perfect agreement
between a model and the facts as actually observed. The choice
between different types of model is to a large extent a matter of
economy of thought, to quote ERNsT MacH. More specifically, it is
often a matter of choosing the model that uses the smallest number
of parameters when representing the facts. I do not think that at
this point there is really any difference between our views. And
similarly with the last part of Professor ALLAIS’ comment. Experience
has shown that it is sometimes (and in economics quite often)
convenient and useful to approach the reality by way of models
where time is a discrete variable, but if it turns out that if discrete
time gives rise to difficulties, then of course we. should respecify
time as a continuous variable.
Professor HAAVELMO’s comments and five question marks have
direct bearing on the epistemological foundations of econometrics:
more specifically a bearing upon econometrics as nonexperimental
model building. This is a most important topic for our round table
discussion, and I welcome his remarks so much the more as they
give me an opportunity to state agreement all through, except
perhaps on one point which I think requires some qualification
"1
Wold - pag. 63
= à
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
I gather that the two first question marks are largely rhetoric,
and in any case I agree with the answer he gives in the next sentence.
To paraphrase, the construction of economic theories is one
thing (namely, the theoretical part of model building), and to confront
it with the facts is something different (namely, the empirical
part of model building). Thus at this point I would only emphasize
more clearly that although the theoretical and empirical aspects of
model building should be kept distinct, at least in principle, we
should always keep in mind that a fullfledged scientific model is
a synthesis of theoretical and empirical knowledge.
As to the second group of Professor HAAVELMO’s question marks,
it seems to me that they will get clearcut answers if the model
builder has taken care to specify in not too vague terms for what
broad array of facts, applications, his model is designed to be valid,
and if the applications include policy making he should specify
what changes in policy, if any, it is the purpose of the model to
cover. If not for anything else, such specification is essential when
it comes to the verification and testing of the model. Furthermore,
the spectrum of potential changes of policy, is extremely wide, and
the substance of a model would in many cases become too diluted
if the model tried to cover more than a relatively small sector of
potential changes. For example, a relation of consumer demand
may remain the same under very different regimes of economic
policy, whereas many other parts of economic life are quite susceptible
even to small changes of policy.
A more specific reply to the last sentence of Professor
HAAVELMO’s comments is that in case an economic forecasting model
‘nfluences government policy, this creates a feedback problem which
n principle belongs under the construction of a more comprehensive
model that includes the interaction between forecasting and policy.
Feedback phenomena may be more or less difficult to handle, but
even if they are difficult they do not make model building impossible.
For example, feedback models are commonplace in the theory
of servomechanisms.
Since the argument about a change in policy has been in fre-2]
Wold - pag. 64
SEMAINE D’ETUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIOUE ETC.
176
quent use in the debate on the rationale of interdependent systems,
[ should like to comment a little more on this type of application.
Nonexperimental model building cannot be based on the results of
controlled experiments; speaking generally, the empirical basis of
the model is instead some kind of regularity in the observed phenomena,
regularities that the model builder tries to explore and explain
by his model. In presenting his model, he should broadly specify
these regularities as the intended domain of validity of his model.
And if a scientific model is to be used for forecasting the results
of a change in economic policy, the observed regularities should
include some evidence from earlier changes in policy. There is here
a fluid border between science and politics. Several aspects of
science and politics have come to the fore in other sessions of out
Study Week. The only point I wish to make in the present context
is that politics has other social functions than science, and therefore
political activity can never be completely rationalized as an application
of scientific model building.
Coming finally to my point of disagreement with Professor Haa-VELMO’s
comments, it lies in his broad statement that a stochastic
economic model is nothing else than a joint probability law, and
he even goes as far as to put between quotation marks the « relations
» that can be derived as properties of the joint probability
law that constitutes the model. True, the stochastization of deterministic
models is a key development in modern econometrics,
and in this connection I was nearly to say that the part played by
joint probability laws in the specification of nonexperimental models
cannot be exaggerated — but the point I wish to make is just that
Professor HAAVELMO’s statement is such an exaggeration. Joint probability
laws can express much, but they cannot express everything.
They are symmetric in the variables involved, and as such they
cannot express asymmetric features of the model, and in particular
they cannot express causal relations that enter as part of the model,
for causal relationships are directed (from cause to effect) and thereby
asymmetric. Causal relations in general are directed and asym
metric both in deterministic and stochastic models, and the stochasti
x
Wold - pag. 63
80 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
zation of a deterministic mode] brings the asymmetry in further relief;
in fact, as illustrated by (B;), in a stochastic specification of
the model the causal relations are expressed.in terms of predictors,
that is, conditional expectations, and conditional expectations always
are irreversible, and thereby asymmetric.
ALLAIS
To express myself more clearly may I comment briefly using
an example? I recently studied hyperinflations. My formulation
was a continuous one but for simplicity I used only monthly data.
It was therefore impossible to represent the last months of the hyperinflations
correctly, and with monthly data the conclusion would
have been that the assumptions made were incorrect. But when
weekly data were considered the verification of the model was very
good and the hypothesis confirmed.
Thus my conclusion is that some models can introduce artificial
difficulties which could otherwise have been avoided.
WOoLD
Yes, this is surely an illuminating example to show that the
choice of time period is an important element in the specification
of hypotheses in a model. But it does not really bring home the
previous point about continuous time, for monthly data are discrete
in time, and so are weekly data. From the theory of stochastic
processes it is easy to give examples of problems that are easier to
handle in discrete time than in continuous time.
HAAVELMO
Let P(x, y, 8) be the joint probability law of the variables x
and y in the past, or under one kind of economic regime. fis a
"21
Wold - pag. 66
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 181
parameter. Let P*(x, y, GB) be the corresponding probability law
for the future, under a new economic regime, B being the same parameter
in both cases. Then the link, the essential element of invariance,
as between the past and the future may be the value of
3 rather than e.g. the expected value of y for given x, the latter
relation depending on the form of the probability laws P and P*
KOOPMANS
In the discussions comparing inter-dependent systems and other
systems, one element has been important which I have not heard
Professor WoLD mention in this summary. This is the idea of the
autonomy of individual equations of the interdependent systems.
My question is whether the concern with autonomy can be
conserved when we go from the ID to the BEID and alternatively
when we go from the ID to the CC svstem.
WoLDp
Since Professor KooPMAN’s question is closely related to Profes.
sor HAAVELMO’s comment, I shall reply to them jointly.
Professor HAAVELMO’s clarifying example has the advantage that
it is so simple that there could not possibly arise any misunderstanding
about the mathematical aspects. Yet there are at least three possible
interpretations of the example to consider.
I) Parameter §3 is symmetric with respect to the variables x, y;
for example, B is the correlation coefficient of x and y. This is the
case I thought of in the first place when finishing my previous reply
to Professor HAAVELMO by a critical remark. Parameters that are
symmetric in this sense make a conceptual category that is far
too narrow to represent all meaningful parameters. For example, a
demand elasticity with respect to price, or an interest rate, do not
possess this kind of symmetry
<
Wold - pag. 67
32
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -~
2) Parameter 3 is a demand elasticity, an interest rate, or some
other parameter that typically enters as a coefficient for one of
the explanatory variables in an economic relationship. In the simplest
case we have
‘E)
y =x.
It is unclear to me what Professor HAAVELMO means when he says
that B can be an essential element of invariance rather than the expected
value of y for given x, for what could the right hand member
of (FE) be assumed to give if not just the expected value of y for
given x?
And in the last two lines of his comment I am afraid Professor
HAAVELMO is not only unclear, but actually mistaken. It is easy
to give examples where (E) gives the expected value of y for given
x, and the relation is an invariant that does,not depend on the form
of the probability laws P and P*. It is even so that this kind of
invariance has been exploited for assessing the direction of a causal
relationship (a first approach of this type was initiated by-H. Work-ING
In 1034; see Ref. 12, section 6).
3) In the theory of interdependent systems (A) it is a characteristic
feature that the behavioural relations are dealt with as being
reversible with regard to the current endogenous variables y. This
feature being in sharp contrast to the irreversibility of ordinary regression
relations, I have many times voiced scepticism about interdependent
systems on this basis (see e.g. Refs. 23, 34). Now the
respecification (B,) goes some way to clarify the situation. The
reversibility at issue requires that if we rearrange the current endogenous
variables by shifting two or more of them from the one side
of the relations to the other, then the rearranged system should
satisfy the corresponding relations of type (B,). On the classic as-F217
Wold - pag. 68
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 183
sumptions of interdependent systems, this requirement will actually
be fulfilled (1).
I welcome very much Professor KooPMANS’ questions about the
notion of autonomy. Prof. FRISCH’s original concept of autonomy
refers to an economic feature, for example a parameter B, that remains
invariant when other things change; this concept is closely
related to the notion of invariance as formulated in Professor Haa-VELMO's
previous comment, Professor KooPMANS refers to autonomy
in a related sense that emphasizes-the model aspects, a relation
being called autonomous if it can be broken out of a model and
inserted in some other specified model. In such autonomous relations
the parameters could be called autonomous in the sense of
the above points (2) and (3). As regards the argument about a
change in policy, the autonomy refers to the case when the model
builder uses different models before and after the change.
In reply to Professor Koopmans, it is my understanding that
the notion of autonomous relations is highly relevant for the theory
of multirelation models in general, and in particular so for ID, CC
and BEID systems. The difference in approach may perhaps call
for some slight modification, mutatis mutandis, depending upon what
type of model we are considering. Thus for ID systems, the autonomy
of a behavioural relation would require that it remains the
same if it is broken out and inserted in another system which includes
those current endogenous variables that enter as explanatory
in the autonomous relation. For CC and BEID systems the concept
of autonomy might well be generalized somewhat so as to require
only that the residual-free part of the relation remains the same when
it is inserted in some other model. This Jast remark emphasizes the
point I wish to make in (2), namely that the invariants of primary
importance in non-experimental model building are directed predictors
such as (E) rather than joint probability laws.
('} This is however not the whole story, for it turns out that the
classic assumptions constitute a special case that covers only a subspace
of lower dimension than the entire parameter space: see reference (b) in
footnote
Wold
pag. 69
84
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
LEONTIEF
The structure of an analytical system which one decides to use in
any particular instance must to a very large extent depend on the
type, that is, amount and accuracy of factual information actually
available for its implementation. Refinements of statistical methodology
are often wasted on all too crude raw material of primary data.
FRISCH
I have been listening with interest to the discussions in a field in
which I took great interest some yars ago and which I think I was
at that time qualified to speak. Today I do not think I am qualified,
quite to the same extent, but since I have been called upon
let me just give one example which is connected with what Professor
LEONTIEF said right now. Suppose for a moment that you
have a curve whose mathematical form you actually know. . It is
for instance a second degree parabola. For a moment you forget
this mathematical knowledge and you look for some numerical data
to determine the tangent at a specific point A. You have observation
in the vicinity of this point. Let each of these observations
be affected by an error of measurement. In order to have actually
the tangent (not the secant) in the point considered, you want to
use observations that are as close as possible to the point A. If
you really want the tangent (not the secant) at this point you would
have to creep up to the point A as close as you can. But in so doing
you expose yourself more and more to the inaccurancies involved
in the numerical observations and if you get two points that are
very close to A you get something absolutely absurd. So you have
to make a compromise between. giving up a little of the ideal you
are looking for, i.e. the tangent instead of the secant, you have to
do it in order to get a method which is more robust. You must
make a compromise, and a practical man will understand more or
less intuitively how far he should deviate from the mathematical
ideal in order to have a result which is useful for his purposes.
»
Wold - pag. 70
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 185
ALLAIS
In fact, I agree completely with what Professor FriscH said; but
my point was different. It was that for the discussion and for the
analysis of the difficulties we meet we must distinguish between real
difficulties and artificial difficulties which arise only from the consideration
of discrete series. That is quite different.
WOLD
This last group of comments refer to shortcomings of model building
that arise because the empirical observations for some reason or
other do not match the theoretical model.
Professor LEONTIEF very rightly -emphasizes that there must be
a sound balance between the accuracy of the statistical observations
and the degree of refinement of the statistical methods applied. A
caution in the same vein is that the application of refined statistical
methods should not become an end in itself, and thereby become
futile. Or « sieving moscitoes, but swallowing camels », as the
Swedish proverb goes, the moscitoes being sampling errors that are
reduced by refined techniques but tend to zero anyway in large
samples, whereas the camels of specification errors are ignored although
they are finite entities that do not tend to zero with increasing
sample.
Professor FRISCH very instructively points out a crucial feature
in the transition from deterministic to stochastic models, namely
that when it comes to differentials versus finite differences, it is often
easier to work with differentials if the approach is deterministic.
and with differences if it is stochastic
Wold - pag. 7.
ECONOMETRIC ANALYSIS FOR ASSESSING
THE EFFICACY OF PUBLIC INVESTMENT (*)
R. DORFMAN
Harvard University - Cambridge, Mass. - U.S.A.
In my country and in most countries of the world the role
of government activity in the economic sphere has become
increasingly important during the last few decades. We no
longer grant even lip service to the doctrine that « That government
is best which governs least ». Governments are
expected nowadays to intervene vigorously in economic affairs
in the interests of general prosperity and economic advancement.
A major part of this enhanced, or at least more candid,
concern with economic development on the part of govern.
ments has been an increase in the importance of governmentoperated
enterprises and, consequently, of government invest.
ment. Today, socialist countries apart, we are all mixed
economies.
For this reason, and for a number of others about which 1
claim no particular competence, the process of deciding on
government investments has become increasingly self-conscious.
In the good old days when some local or special need or
opportunity made its appearance the legislators debated the
'*) This paper was written in conjunction with research sponsored bv
the Corps of Engineers, U.S. Army and Resources for the Future. They
are not, however, responsible for any of the opinions expressed in ©
Dorfman - pag.
188
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
possibilities and might appropriate the funds for ‘the requisite
investments. Beginning in the 1930’s in my country, and at
other dates elsewhere this process has become increasingly
formalized and bureaucratized. Now a project proposal must
be accompanied by an elaborate dossier setting forth estimates
of its various economic and non-economic consequences and
purporting to justify it by appeal to objective criteria. If the
American Congress or other local legislature doesn’t insist on
such a formal justification the World Bank will. We have even
extended the range of activities counted as government investments.
Expenditures on education, for example, are now frequently
considered under that heading.
It is therefore appropriate for us to consider the application
of quantitative methods of economic analysis to the appraisal
of proposed public investments. We shall assume throughout,
as a background, a mixed economy, one in which the public
and private sectors are both significant. The issue of public
investment cannot arise, by definition, in a pure private economy,
if theré is one anywhere in the world. It takes an ‘entirely
different form and meaning in a pure public economy.
Thus we shall assume a mixed economy, and though we shall
concentrate on the public sector the private sector will always
be visible in the background.
The broad subject of government investment has already
accumulated an imposing literature, which I shall not survey.
Rather I shall advance a proposal concerning one central issue
that the literature, as I know it, treats unduly lightly.
The analysis of public investments is strongly analogous
to the problem of capital budgeting in the context of a private
firm. Both are motivated by the need for wise allocation of
scarce resources, and similar conceptual issues arise in the two
cases. In both cases the benefits promised by the project must
be weighed against the costs it entails and competing projects
laying claim to the same resources must be compared. But in
the case of government undertakings both the benefits and the
3] Dorfman - pag.
2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 189
costs, but particularly the benefits, are far more difficult to
ascertain and measure than in the case of private investments.
This follows from the very natures of the firm and the.government
and from the kinds of enterprise that a government is
likely to engage in in a mixed economy.
When a firm contemplates an investment it will be concerned
very predominantly with the increase in the salable
product or the decrease in the operating costs that will result
from it. Both of these are readily expressible in monetary
terms and the conversion from physical units to economic value
is relatively easy because prices for the commodities at issue
can be obtained, with some degree of haziness and extrapolation
to be sure, from the free markets on which those commodities
are traded.
A government’s interests and the results in expects from
its investments are quite different. Indeed, the greater the
extent to which the kind of appraisal that a firm uses is adequate,
the more likely is the government of a mixed economy
to leave the undertaking to the private sector. The kinds of
results in which a government is interested can be outlined. as
follows :
1. Frequently government enterprises do produce salable
products, comparable to those of private enterprises. This is
likely to be the case where the product is a « natural monopoly
» or is particularly important to the health and safety of
the community or to the operation of the government itself.
2. In recent years, especially, governments have entered
fields characterized by significant economies of scale, to gain
which requires larger commitments of capital than private individuals
are able or willing to mobilize.
3. The government is likely to undertake the production
of goods for which there is a marked discrepancy between market
price and social value, or for which“a private enterprise
51 Dorfman - pag. 3
00 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
would find it difficult to collect adequate recompense because:
of either technological or institutional peculiarities. So-called
. collective goods » fall within this category.
4. The gap between social value and market price mentioned
above may be due to external economies of many different
kinds, which deserve to be mentioned explicitly. One
important category consists of external economies of consumption:
types of consumption that are deemed to confer benefits
on the community over and above those that are perceived
by the individual consumer. Education is the leading example,
but there are many other instances. Without government
action, undesirably small quantities of such goods would be
produced and consumed. The appropriate government action
may take many forms: subsidization of production or consumption,
or direct government provision, with or without
charge. .
, 5. Another type of external benefit is the development of
economic skills. A government may undertake specific forms.
of enterprise or enterprise in specific localities in order, to
promote the growth of technical and managerial skills, i.e. to
introduce modern industries into regions that lack them.
6. In emerging economies, the government. frequently invests
in « social overhead », roads, port facilities, urban housing,
and the like. The external economies sought in such:
projects are reductions in operating costs in the private enter:
prises that are hoped to follow.
7. -Turning, now, from external econgmies, a frequent motive
for public investment is improvement in the distribution
of income, either by reducing costs of inputs to impoverished
sectors of the nation or by providing consumer goods at low
prices to various low-income groups. Such subsidies in kind
are often more feasible. administratively than direct transfer
"3] Dorfman - pag.
4
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 101
payments and are likely to contribute more to general economic
advance.
8. Traditionally, of course, governments have always invested
largely in the facilities required by their own minimal
functions of national defense, maintenance of law and order
administration of justice, revenue collection, and the like.
o. Finally, and again this is a recent development, governments
invest in enterprises intended to enhance the prestige
of the nation. Atomic power plants and steel mills are typical
examples.
This is a list of motives, undoubtedly not complete, that
induce governments to undertake investments. It is also a list
of the considerations that must be applied to any government
investment, because it is a rare project indeed that contributes
to only one of these objectives.
Thus the appraiser of a government project, as contrasted
with the appraiser of a private one, must be concerned with
many kinds of consequence, not all measurable in monetary
units and not all comparable among themselves in any natural
unit. Even in dealing with the consequences that are
measurable in monetary units, in principle, he is not likely
to find that market prices are an ‘adequate guide, partly because
of the prevalence of unpriced external effects and partly
because consumers’ surplus (a treacherous concept that cannot
be avoided here), though not reflected in the markets, is im
portant to governments. -
There are similar complexities in the consideration of costs.
though not as severe. To be sure, a government must recog:
nize, while a private.firm can ignore, such external diseconomies
as congestion and smoke contamination, but these are
usually not of the essence. The government is more likely to
be concerned with discrepancies between the market prices
and social values of certain factors of production: labor, when
there is substantial unemployment, is a famous instance. It is
oJ
Dorfman - pag.
5
92 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
also constrained to take account of the effects. of investments
on the- foreign balance of the nation, and, indeed, the general
budgetary position. If financing the project will lead to a more
inflationary (or less deflationary) budget, these are external
consequences that must be taken into account. Furthermore,
there is the problem of the « social rate of discount », an issue
[ intend to avoid as much as possible. If the social rate of
discount differs from the market rate, then the social value of
investments that will be displaced by the government investment
will be different from the market value. In this circumstance,
the appraiser of a government investment is confronted
by the problems of estimating how much private investment
the project will displace, if any, and of evaluating the
social value of this displaced investment. In short, even when
it comes to costs, the usual accountants’ and engineers’ estimates
are likely to be unsuitable for economic analysis. Nevertheless,
we shall concentrate on the problems posed by benefit
evaluation, though some of. the more important problems on
the cost side will force themselves upon ,us.
The essential problem on the benefit side is that the benefits
expected to flow from a public investment tend to be diverse,
non-monetary, ‘ incommensurable, and difficult to measure in
any units. Several expedients for meeting this problem are
available or conceivable. The only one much used in practice
has come to be called « benefit-cost analysis », though, as we
shall see, it hardly deserves this proud name. The first step
in this procedure is to have engineers prepare preliminary
designs for the physical facilities and to estimate the costs of
construction and the outputs and other physical characteristics
of the system. Frequently the engineers will submit two or
three alternative designs. The economic analysis, which follows,
concentrates in the first instance on the monetary results
of those designs. It consists essentially in placing values upon
those physical outputs of the system that either have market
prices or to which monetary values can be imputed readily.
3] Dorfman - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
197
The estimated benefits of the project are then sum of the values
of the physical outputs produced during each year of its life.
discounted to the date of inception at the social rate of discount.
The costs are the construction cost plus estimated annual operating
costs similarly discounted. It will be seen that this procedure
is, in essence, very similar to the one followed in capital
budgeting by private firms. The only divergence comes now:
note is taken of the various nonmonetary effects of the project,
frequently referred to as « intangibles ». No attempt is made
to incorporate these « intangibles », which may be very tang
ible indeed, into the analysis: that task is left for higher autho:
rity and, in consequence, frequently is never performed. It is
a fact of government sociology that as a result of this procedure
the monetary effects of a project receive more emphasis in
decision making than they should in comparison with the nonmonetary
effects, which tend to be slighted because they are
difficult to measure and express. Nevertheless, that is how
things are with the current state of the art.
The econometrician has two or three suggestions to make
for coping with this inadequacy, and the main object of this
paper is to consider them. One expedient that I shall not consider
is that we attempt to ascertain a full-fledged social utility
function in which the various objectives I have listed above
enter as arguments. I am interested only in devices that might
conceivably be implemented.
The first suggestion that comes to mind, however, comes
pretty close to that. It is that the analyst should attempt to
establish « shadow prices » for the various objectives, thereby
becoming able to compute a value-sum that can be compared
unidimensionally with a value-sum of costs or value-sums of
benefits from alternative projects. These shadow prices would
reflect the willingness of the community to trade off one type
of benefit against another. For example, redistribution effects
could be incorporated by ascertaining that the community was
willing to sacrifice a dollar’s worth of national income in order
[3]
Dorfman - pag.
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
to increase the income of some impoverished group by $.75.
In that case, income to the group to be favored would receive a
shadow price of $1.33.
This proposal may seem visionary, but it contains the essence
of all the proposals I am about to make, and, in fact,
expresses a kind of comparison that is ineluctable and is made
every day. For this reason it may not be as impracticable as
it seems at first blush. Since these decisions are made frequently,
a study of administrative or legislative records should
disclose bounds, at least, on these shadow prices by showing
the rates of trade-off between different objectives on projects.
that are accepted and rejected. Such a study would have to
assume that the shadow prices remain stable for reasonable
periods. Perhaps they do, but perhaps also they change with
changes in the political and economic climate. There is another
difficulty, too, for this line of empirical research. Though in
every project plan, choices of the sort at issue have to be made
somewhere along the line, the form of benefit-cost analysis
rather obscures them. As I mentioned,’two or three variants.
of a project at best are submitted for decision by responsible
authorities. These tend to differ in many dimensions, and the
amount of trade-off between different objectives is not likely
to be brought out very clearly. If variant A of a flood-control
project provides more protection to property than variant B,
it may also cost more, provide less protection to life, and
provide better by-products in the form of recreational facilities
but less hydro-electric power. Knowing that the legislature
has preferred variant A tells little about the shadow prices;
though more may be disclosed by inspecting the record of the
debate. I emphasize this difficulty because it suggests that the
entire process of generating a benefit-cost analysis which TI
sketched above — the sharp distinction between the spheres
of the engineer and the economist, the small number of variants
produced — may be unsuitable for analyses of public invest-'31
Dorfman - pag.
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195
ment projects even though it serves adequately for private investment
decisions.
On the positive side, one should not underestimate how
sophisticated government officials and legislators can become
in economic matters. It won’t do to say they will never understand
a technical concept like shadow prices. They already
understand very well indeed the parallel concept of a social
rate of discount and, though I can’t vouch that they use that
very phrase, debate it very intelligently. If the concept of
shadow prices for different objectives were introduced in any
government it would, in its early years, lead a stormy existence.
But pretty soon the responsible officials would learn
what 1s at stake and start to debate about what the shadow
prices should be rather than what features particular projects
should incorporate. This would be a constructive improvement
in the decision-making process.
Another proposal that comes naturally to an econometrician
is that projects might be designed from the very beginning
to meet certain specified target value for the various objectives
that they are intended to serve. In particular the designers
might be instructed to meet the specified targets at minimum
possible capital cost. This would require a revolutionary
change in the whole design procedure because, as inspection
of the list of objectives suggests, they are not all of the sort
that engineers feel at home with. Formally speaking, the task
to be imposed is this: An engineering design is a choice of
a vector of specifications. Given this vector, estimating the
cost of building a structure that meets them is fairly straightforward,
though it should be mentioned that engineers do not
have a proud record as cost predictors. The prediction of
physical results is less straightforward because economic and
other nonengineering factors have to be taken into account.
If the project produces electric power, its annual output will
depend on the load-factor which, in turn, depends on the economic
composition of its market. If the project produces irri
Dorfman - pag.
106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
gation water its usable output will depend on local hydrology
and on the relation of the timing of the local hydrologic cycle
to the timing of irrigation demand in its market area. And
so on. The design typically determines capacity, but economic
and social factors determine output.
Be that as it may, the ancillary factors are usually not
subject to decision (or if they are they should be regarded as
part of the design), so that the economic performance of the
project is determined (leaving uncertainty aside) once the design
specifications have been settled. We can denote the rela-‘ionship
between design and output by writing fi(x,, ..., x,)
for the extent of performance with respect to the ith objective
or target, where x, ..., x, is the design vector. Similarly we
can write c(y, ..., x,) as the cost of meeting the design specifications,
x4, ..., x,. In this notation the designers are
charged with the task of choosing x, ..., x,, so as to minimize
c(x;, ..., x,) while satisfying fi(x,, ..., x,)>T, i=1, ..., k,
T; being the target level of the ith objective, and % the number
of objectives considered.
You will leap at once to several objections. One is: where
can we find these ambitious production functions, fi(x,, ..., x,,)?
On this, I hope that you will be willing to suspend -your disbelief;
I want to discuss that topic after we have seen what
we can do with these functions if we have them. A second
objection is: the design specifications do not determine the
outputs. By varying the way in which a given structure is
operated, one form of output can be substituted for another.
A third is: how are the various target levels, T,, to be established?
I shall deal with the second and third obiections immediately,
and more or less together.
Let us reformulate slightly our model of a design problem
to create room for inserting some more complicated considerations.
Let Fi(xy, ..., x,, uy, ..., u,) denote the production or
performance function with respect to the ith objective, where,
in addition to the previous notation, #,, ..., #, describe a par-"3]
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ticular operating policy. For example, if the structure is a
school, the #’s would specify the average size of class, the
number of sessions per day, and the like; if it is a dam, the
u's would specify the release rules as a function of reservoir
content, etc. The target values, T, need not denote deterministic
results if the process is stochastic; they should denote
parameters of the probability distribution of outputs. For a
hydroelectric project, T, might be the expected level of power
output in June and T, might be its variance. The problem is
then to minimize c¢(x,, ..., x,) subject to Fi(x,, ..., x,
up, ul) >T, 1=1, ..., k.
This 1s a standard constrained minimization problem to be
solved by any of the usual methods. In general it will be a
very difficult problem to solve, but when projects are planned
with costs expressed in eight or nine digits, the expense of solving
a minimization problem of any imaginable difficulty is as
dust in the balance. Indeed, the expense of computation is
likely to be insignificant in comparison with the cost of gather
ing the data.
You will note that though project selection and design are
at issue, the determination of operating policy has intruded
itself into the problem. This is inevitable, as has long been
recognized. « Operations studies » are a standard component
of project design work.
Assume this minimization problem to be solved. A by-product
of the solution is a set of shadow prices associated with
the assigned targets. In the early stages of the work these
by-products are the main product of the analysis. They inform
us how much costs could be reduced by a one-unit relaxation
in each of the targets. Ratios between them are the trade-off
ratios between different objectives. If several projects are being
considered simultaneously, discrepancies between their shadow
prices for the same objective indicate misallocations and inconsistences
in the overall investment plan
Dorfman - pag. 11
08
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus the shadow prices are an instrument for appraising
the wisdom of any specification of targets. They disclose one
of the major implications of such a specification: the marginal
cost of achieving each target. The appropriateness of the
assignment of targets can be debated in the light of this information,
which is a fruitful improvement over current practice
which often requires responsible officials to make policy decisions
at the level of design specification (the x; in our notation).
Judgements about the magnitudes of incommensurables,
like the diverse objectives of a public investment under-‘aking,
can be made in a more than off-hand way only when
responsible officials confront the trade-offs implicit in their
decision.
The process here proposed begins with any a priori plausible
selection of target levels, which are revised and refined as the
marginal costs of achieving them become clearer. This procedure
envisages that the design and its objectives will evolve
together: the design following pretty mechanically from the
objectives; the objectives following from a critical appraisal
of the design.
This same model and approach can be formulated in a
somewhat different, and instructive, way. The construction
costs, which are to be minimized, are a function of the design
specifications, but the outputs depend on the operating policy.
The role of the design, as far as outputs are concerned, is
largely to make desirable operating policies feasible. For
instance, one cannot have a policy that calls for dispatching
100,000 kw. of electric power from a plant whose installed
capacity is much below that figure. Therefore, it generally (not
quite always) fits the structure of the problem best to regard
output in each dimension as a function of operating policy
alone, and the design as setting limits to the choice of operating
policy. The problem then takes this form: Choose design specifications
to minimize c(x,, ..., x,) subject to the constraints
"37 Dorfman - pag. 12
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y.
1
=)
This form, which emphasizes the lack of parallelism between
design choices and operating policy choices, has empirical
advantages. The major one is that it poses the problem in a
manner that technicians find manageable. It asks, « If you had
a structure with specification x, ... x,, how would you operate
it and what would the resultant outputs be? ».
This basic formulation has to be modified clearly to fit
the particular circumstances of particular projects. For roads,
for example, achievement of objectives depends directly on
structural characteristics (lane width, maximum grade, etc.)
as well as on operating policy (speed and weight limits, level
of maintenance, etc.). For reservoirs, structural characteristics
will set limits to simple functions of the operating parameters
as well as to the parameters themselves. In all these variants,
however, the logical structure of the model will remain the
same.
The final approach that econometrics suggests to the
problem of handling non-comparable benefits is closely related
to the second. Instead of meeting specified targets at minimum
cost, one can pose the problem of maximizing performance
with respect to some one objective, subject to meeting targets
with respect to the other dimensions of performance. In this
approach the most likely objective to choose for maximization
is the discounted present value of the net benefits that have
convenient monetary equivalents. Construction cost is likely
to enter as one of the constraining targets.
The sacrifice of symmetry in the treatment of objectives is
probably more apparent than real, and this approach has some
compensating advantages. One advantage is that it reduces by
one the number of target outputs that have to be specified in
advance of serious analysis. The need to specify a maximum
Dorfman - pag. 13
200 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -
construction cost is only a partial offset; it is much easier
generally to make a plausible guess at an allowable construction
cost than to guess at an efficient output level. Another advantage
is that there is some obscurity about the proper costs to
minimize in the cost minimization approach. The most plausible
choice is total cost — the cost used in the denominator
of a conventional benefit-cost ratio — but this kind of cost
does not constitute a drain on any definable scarce resource,
it is an amalgam of construction costs in the near future and
operating costs extending through the life of the project. The
present value of the monetary net benefit stream has much
more appeal as the dimension of performance to be singled
out for special treatment.
The net benefit maximization approach also leads to shadow
prices, which may be even more usefully interpretable than
the shadow prices yielded by cost minimization. One of the
shadow prices will pertain to the construction cost constraint.
This price should surely be approximately the same for all
projects that are to be initiated in any brief time period, thus
facilitating inter-project comparisons and allocations.
My entire discussion has concentrated on the problem of
incommensurable benefits. There are also often incommensurable
costs, and these can be handled in precisely the same way:
by establishing target levels and revising them in the light of
trial results. A few special cases of costs not measured adequately
in dollar terms deserve explicit mention. These are
all cases in which, perversely, a dollar is not worth a dollar.
Foreign exchange drain is an obvious instance: a country
may well be willing to forego more than a dollar in discounted
net benefits to save a dollar in foreign exchange. Inflationary
budgetary impact is another case. Most public investments
require heavy expenditures in some pattern extending over a
number of years. Anticipated budgetary tightness or inflationary
pressure may well be substantially different in some years
in the near future than in others. In such a case. total con-3]
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20]
struction cost is not the only relevant cost figure (even if
discounted); separate attention should be given to fiscal drains
in individual years. As a final instance, the impact of the
financing of a project on the private sector gives rise to a similar
distinction. To the extent that the financing induces retrenchment
in consumption expenditures there is one kind of cost (one
might even want to distinguish among the socio-economic strata
which retrench). To the extent that the financing induces a
reduction in private investment, there is another kind of cost,
especially if the social and private rates of interest diverge.
The two cannot be added meaningfully dollar for dollar, nor
Is it easy to assign an exchange rate between them a priori.
In short then, what I as an econometrician contemplate is
that the assessment of a public investment be based on a model
of that investment that recognizes its consequences in many
dimensions and that exhibits the full range of choice and substitutability
among these dimensions. Final adoption or rejection
of a project can be decided only when the best design it is
feasible to produce is at hand, one that takes account of al!
the significant dimensions.
This approach requires a new kind of cooperation among
engineers, economists, fiscal analysts, and senior policy officials.
The engineer is not asked to submit a design as a kind
of fait accompli to be analyzed and perhaps accepted by the
other officials. He is rather made more integral to the ap.
praisal and decision process. He is to collaborate in the formulation
of the model of the investment and to present estimates
of the requisite functions, where possible. Where that is
not possible, a frequent situation, the engineer will have to
contribute and analyze some more detailed technological relationships
that I shall discuss more fully below. The economist
contributes his guesses of plausible targets and estimates of
money value for those outputs for which monetary values are
appropriate. All contribute to the construction, the testing,
the appraisal, and the revision of the model of the contemplated
ov Dorfman - pag. 1+
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
investment. The result of the analysis will be, not a single
benefit-cost ratio with some marginal comments, but model of
the investment whose performance can be ascertained in respect
to any targets that higher policy may dictate.
Is such an ambitious mode of analysis really practicable?
[ believe it is, and should like to submit some technical suggestions
for its implementation.
The critical sticking point is the various performance functions
which are very complicated, unknown, and hard to
ascertain. In most cases there is a great deal of pertinent
technical knowledge, but it is not in the proper form. Consider,
for example, one of the more favorable cases: the relationship
between the height of a dam and the amount of irrigation
water it can provide. In the first instance the height of the
dam controls the amount of water that can be impounded in
the reservoir behind it. Given any height, a hydrologist can,
by studying the contours of the land to be flooded, estimate the
the volume of water that can be retained. By making such
studies for a number of dam heights he can generate a functional
relationship between height and contents. Because of irregularities
in contours this is likely to be a very complicated function
but, generally, reasonable smooth and simple approximations
to it can be found.
But this is only a half-way step because the relationship
between reservoir capacity and usable irrigation water supply
— «yield » for short — is even more intricate. There are at
least two complicating features. One is that the reservoir may
not fill annually. As a general rule, the larger the reservoir,
the lower the probability that it will fill in any year. Therefore,
although usable water supply is an increasing function of capacity
up to a point, it increases at a diminshing rate. The
other complicating feature has to do with timing. If the annual
inflow to the reservoir is concentrated in a rainy season that
does not coincide with the time of year at which irrigation
water is demanded then, clearly, the contents of the reservoir
"31 Dorfman - pag. 16
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203
and the amount of usable irrigation water are the same. That
is one extreme. The other extreme occurs when the natural
inflow happens to coincide with the period of irrigation demand;
then the reservoir may do no good at all, the water
would be available when needed even without it. In that
case the yield is zero. Of course, zero-yield reservoirs are never
built; they are simply a conceptual possibility. Many genuine
cases lie between these two extremes, and there are patterns
of inflow and demand in which the yield is greater than the
capacity. All this is complicated enough, but the case here
described — of an isolated reservoir operated for irrigation
supply only — is excessively simple. These considerations
make it clear that the relationship between height of dam (which
costs resources) and usable output is by no means an easv
one to ascertain.
Current practice avoids determining this functional relationship.
Instead it determines one or two points on it by postulating
a specific height for the dam and then estimating the
usable yield provided by that height by means of quite elaborate
calculations.
The instance of the dam height-usable output relationship
is cited to illustrate the complexities hidden in every one
of the performance functions required for the analysis of an
investment project. There seem to be two ways to proceed and
it appears that in practical analysis both should be followed.
One is to impose drastically simplified functional relationships.
For example, the relationship between dam height and
average reservoir content might be approximated by a quadratic
function, and the other relationships would similarly be
held to the simplest expressions that state the problem in a
meaningful way. This procedure would both facilitate empirical
estimation of the parameters required and would simplify the
formal problem of finding the optimal values of the design
specifications. It would also, of course, be extremely un
reliable.
Dorfman - pag. 17
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
But the purpose of the simplified model is not to produce
a decision or a design, but rather to establish plausible ranges
of values for the design and operating parameters. When this
has been done the second approach can be invoked. It depends
on the fact that the elaborate calculations required to
obtain a single point on the performance functions can be
mechanized and thereby made quick and cheap. Thus although
it may not be possible to write out these complicated functions
explicitly, or even to manipulate them if written out, it is
feasible to ascertain a finite number of points on any of the
functions. Even these mechanical calculations may be quite
elaborate. In particular, if there is a large stochastic element it
may be necessary to simulate the operation of the investment
over a substantial period of time in order to estimate a single
point on the function. But such simulations are now feasible in
the great majority of practical instances and are rapidly becoming
cheaper to carry out.
Since it is possible to estimate points it is possible to estimate
differences between points, i.e. first derivatives. And therefore
it is possible to apply any one of a number of steepest ascent
procedures to determine iteratively the solution to any of the
problems I discussed earlier. I shall forebear to go into details
since the procedures I have in mind are fairly well standardized
and improved ones are appearing constantly. All the procedures
have the property of producing the shadow prices required
for testing preassigned target level along with the solution
for each target assignment.
I do not wish to give the impression that the approaches
[ have suggested solve all the problems of deciding on public
investment. Far from it. This is a very difficult field with
many, many open questions. To mention only two of the
pressing controversial issues, there is the problem of chosing
the social rate of discount, and the problem of incorporating
uncertainty into decision criteria. Besides, many problems.
remain to be solved with respect to the kinds of benefits that
31 Dorfman - pag. 18
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can be assigned monetary values. What this procedure does is
provide a format into which the best current understanding
of the conceptual issues can be inserted.
You will have noted that I have dealt at least as much with
problems of design as with problems of assessment of efficacy,
my assigned topic. This is because these two problems are
inseparable: a fair apprisal of a project requires a good design,
a good design must be based on the standards to be applied
in making the appraisal. In essence, the designer must also
be an appraiser; a separate appraiser if there is one, has onlv
an auditing function
Dorfman - pag.
1
1
501 0N
MAHALANOB!
I have a small question. i believe, at the beginning of your
paper you point out that we are concerned not so much with €
choice of public investment in a completely planned economy but
in an economy in which there is both a private sector, and a public
sector, and these features raise many difficulties; about this T completely
agree. Have the implications of this complicating factor
been taken into account in vour n---DORFMAN
The presence of the private sector is a helpful factor as well as
a complicating one. It helps in the preparation of public investment
programs and their evaluation by providing a set of market prices
for the resources and factors of production needed to construct the
public investment projects and also in many cases for the goods and
services produced by the public investment. The market prices of
the factors required to produce a public investment are, of course,
measures of the worths of the goods and services that those factors
could produce if they were not devoted to the public investment.
Therefore, they measure real cost of the project. The market prices
of the goods and services produced by the project measure the
Dorfman - pag.
pa
z
238
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
worthwhileness of the results. Such valuations are clearly quite
essential to a sound appraisal of the worthwhileness of a proposed
public investment project, and I think that all project appraisers
should be grateful to the private sector for providing them with such
values. I know of no other reliable way of obtaining them. I should
consider the problem of appraising public investments much more
difficult if there were not a private sector that established reliable
market prices for the goods and services with which they are concerned.
VAHALANOBIS
Well, your recollection of the paper must be better than mine.
If I may make a point here, if you are in a mixed economy, when
you are planning investments you would have to take into account
the behaviour of the private sector, which you cannot control directly
but only indirectly, before making your decisions about public
investment. From that point of view, I think the programming of
public investment becomes much more difficult than in a fully planned
economy in which everything would be decided more or less
on mostly endogenous variables.
DORFMAN
Of course the presence of an unpredictable private sector presents
the project appraiser with certain difficulties. My impression
is that the same difficulties arise in a fully collectivized economy.
In a collectivized economy there would not be a private sector to
confound and confuse a project designer or appraiser, but there
would, of course, be other bureaus and other administrations within
‘he public sector. The project appraiser and planner dealing with
andertakings in one branch of a centralized economy, for example,
‘he branch that has to do with irrigation projects, would have
31 Dorfman - pag. 22
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209
somehow or other to foresee plans and responses of the administra
tors of other sectors of the same economy. Any genuine economy
is too large and intricate to be globally planned and globally directed
from one center. I think in fact that it may be easier for a project
planner to predict the consequences of his plans in a competitive
context, such as is afforded by a mixed economy, than to do sc
where the sectors over which he has no control are other depart
ments of a centralized economy. The problems of economic coor
dination are always severe, collectivization does not obliterate or
solve these problems rather it places them candidly on the doorstep
of the planners and deprives them of the assistance of decentralized.
on-the-spot, decision makers.
MAHALANOBIS
On the last comment of Professor DORFMAN I should like to say
that in a centrally-planned economy you could approximate the
competitive equilibrium probably better than you can in a capitalistic
economy with many monopolies. In saying this, I am not expressing
any political views. I just want to stress my first point about
the difficulty which you may face in a private or free economy
because it is really not competitive. I agree that if a private or
free economy is fully competitive then it has its advantages. I also
agree that there are some difficulties for a centrally-planned economy
to realize something which would be similar to a good state of competitive
equilibrium. It is the question of imperfection of competi
tion which is of concern to me
DORFMAN
I think that we now understand each other fully The effectiveness
of economic coordination in a mixed economy certainly does
depend on the extent of the distortions introduced by monopolies
Dorfman - pag. 23
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
and other deviations from competition in the private sector. If
the private sector did not exist, those problems would certainly
disappear, but they would be replaced by other serious problems,
namely the problems of placing rational valuations on the goods
and services dealt with in the economy. In effect, there is no competitive
sector in a centralized economy; every sector is monopolized
by one government bureau or another. Whether a centralized economy
can, in some manner, simulate the operation of a private
competitive economy is still open to grave doubt — it has not yet
seen done.
Whether the difficulties presented by an independent market
sector or those resulting from the lack of free market prices are the
more serious is an empirical problem that I am not in a position to
resolve. I am, however, more impressed than Professor MAHALANO-BIS
with the inefficiencies that have been experienced when attempts
are made to plan an economy without having a frame of reference
such as is provided by a system of market prices.
MAHALANOBIS
I was not referring to a centrally-planned economy of the type
in which there would not be any market, I was referring to a
centrally-planned economy in which everything which a government
would like to plan, it could, and everything it would like to
leave to a market, it could again do so. I was also referring to the
difficulties in an economy in which some particular aspects could
not be planned, because of institutional factors.
JOHNSON
There are two comments about the procedure as outlined by
Prof. DORFMAN in his ingenious paper. It seems to me that you
indicate partly in the manuscript and partly in vour discussion
3] Dorfman - pag. 24
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21
that the procedures you outline relate essentially to all types of
government investments. By that I mean those government investments
where there is essentially a monopoly, on the one hand, and
no private individual is engaged in such activities — in most economies
— building roads or schools. The questions that I wish to
bring up with respect to technique relate to those cases in which
the government output is to a degree comparable to private output —
public or private electric power and relative to other forms of energy
such as natural gas.
And the two points are these: first, though it is true that in
making its investments a private firm includes only those returns
which it can capture, many of the kinds of returns that come from
government investments of the external or social sort where the
social and private return diverge, will also exist in the case of private
investment. That is, for example, when General Motors builds
an assembly plant, there are substantial social gains to the particular
community involved — higher property values, various increases
for certain other services and so on. And it seems to me that
the procedure that you outlined tends to ignore these retums in
the case of the private investment and where there is competition
between the public and the private sector not really take them
into account in comparing private and government investments
[f this is true, there is a bias in favour of government investments.
The second point is really along the same lines and this is the
question, of how one again creates a real competition between the
public and private output, whether it is direct, as in electric power
or somewhat indirect, as between electric power and natural gas
How are the taxes paid by the private sector included in the ana
lysis? In some industries, such taxes may amount to a significant
fraction of the total value of the output, and here again it seems
to me that this is not taken into account, and you again bias the
decisions in favour of the government’s investments
Dorfman - pag. 25
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
DORFMAN
I think that you have raised some very pertinent points which
are not adequately dealt with in the paper. What I contemplate
's that private investments that are displaced by public investments,
as in the case of electric power generation in the United States where
both public and private owners are of substantial importance, should
be valued in precisely the same way as public investments even
to the extent that one ought not to value private investments in the
same way that the prospective investors will. For example, one
should value private investments, using the rate of discount that
you would use in valuing the public investments, normally a lower
rate of discount than private investors employ.
I would answer your question about taxes in the same spirit.
Taxes are purely a transfer payment and should be taken account
of in dealing with competitive private investments just as they are
in dealing with public investments that do not generate taxes.
JOHNSON
Another point relates to the paper, but was not referred to specifically
in your presentation this afternoon One of your reasons
for government investments was to improve the distribution of
income by providing goods at low prices (below marginal cost) to
various low income groups. « Such subsidies in kind are often
more feasible administratively than direct transfer payments, and
are likely to contribute more to general economic advance ». I don’t
know of many cases in which a government in mixed economy can
produce goods which it can make available generally only to certain
lower income groups; housing is perhaps one example, but this is
about the only one. Governments can make direct transfer payments
to low income groups on a very broad scale and for all purposes
thal you want to have such payments. Do we want to rely
upon the judgements of the recipients on the ways of expending the
income given to them or do we want them to rely on the decisions
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of committees or of local administrators of public aid on what you
have and how vou should spend the income:
DORFMAN
The two cases that I had in mind in introducing that motivation
were public housing and education. In both cases it is generally
felt that there are very important external economies of
consumption so that not only does the public want to provide those
services to the lower income groups, but also it is the public will
that the beneficiaries of such subsidies spend them on the goods
that they are intended for and on no others. This is candidly a
form of sumptuary legislation in which the community is consciously
and explicitly substituting its judgment for the preferences of some
of its consumers because it is felt that the community as a whole
will benefit from the types of consumption that are being subsidized
KOOPMANS
[ should first confess that I have to apologize for having read
only one half of the paper. However from the beautiful exposition
that Mr. DorFMAN gave, I understood that his discussion was concerned
with evaluating specific projects, one by one, so to say,
or three at a time, and he described TINBERGEN’S proposal of an
iterative procedure by which to get the optimizing quantities and
their shadow prices in line. My question is, how does the rate of
return itself, or if a different rate is applied to different parts of the
future, how does the sequence of rates of return applicable to individual
one-year periods in the future get determined? Is this also
part of the iterative process of dialogue between the economic planner
and the political decision maker? If so, is that problem not of
a higher order of complexity in the data requirements in that really
all relevant projects, including those likely to be forthcoming from
the private sector of the economy, enter also?
A
Dorfman - pag. 2;
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DORFMAN
Professor KooPMANS has asked a very searching question dealing
with a very central issue. I should like to try to respond to it
in two stages. First, it seems to be essential that there should be
some rate of discount to be used in comparing results that mature at
different times, If there is none, inconsistencies will certainly creep
into the governments’ investment program. Different and conflicting
choices will be made in designing different projects and projects
mn different sectors of the economy. In some cases substantial
resources will be expended to secure early accrual of benefits, and
in others, the opposite choice will be made. The wastes resulting
from such discrepancies can be avoided only by applying a uniform
rate of time preference to all projects. I believe that Professor
Koopmans would subscribe to that, but then he asks where are the
necessary rates of time preference to come from. The second part
of my response deals with that issue,
In some project planning, in the United States, for example, an
explicitly formulated rate of time preference is incorporated. More
usually there is no explicit rate, but if one reviews the choices
made in the project design, often an implicit rate can be discovered.
The annals of project design and of legislative debate of projects
submitted to legislatures should, I am conjecturing, disclose what
these implicit rates of time discount are. I do not believe that there
is any economic market from whose behavior the social rates of
:ime preference can be ascertained. As a substitute for such a market,
I suggest that we review from this point of view the behavior
of governments, and particularly of their legislative branches.
VIAHALANOBIS
I should agree with Professor Koopmans on technical points
which I may briefly mention; but I have got some points of a very
different nature. On p. 4, education has been given as a leading
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example.
I presume education does include science and scientific
research.
DORFMAN
[ am afraid that I think that it would strain the conventional
meaning of « education » to use that word to include science and
scientific research. The promotion of science and scientific research
is, of course, an important function of government. These activities
do have important external economies. The discoverer of an important
scientific truth cannot usually retain the benefits of his discovery
even by means of a patent system. This kind of external
economy, however, is sufficiently different from an external economy
of consumption, as exemplified by the benefits to society of
having its members better housed and better educated, so that it
is best to deal with them separately and not to combine science
and scientific research with education for analytic purposes
MAHALANOBIS
I meant when education is treated as an item of external economies,
I simply enquired whether science would be included as
an essential part.
Another point on p. 5 is a practical one. In para g I agree that
governments may invest in uneconomic enterprises with the inten:
tion of enhancing the prestige of the nation, which would be undesir
able. Steel plants are sometimes mentioned as prestigeous but uneconomic
enterprises. Although I have been trying for a long time,
I have not found any adequate discussion of the circumstances in
which the setting up of a steel plant would be economic and proper.
I could well imagine that a very small country like Panama should
not do so. But it would be extremely useful to find out in specific
cases, for example, for Ceylon which has rather a small population
3
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216 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
or for countries with even a smaller population of two millions or
so, whether it would be advisable to set up a steel mill and this
also in an extreme case when domestic resources of iron or coal are
not available. I am giving this particular example to widen the
scope of the discussion.
[ am in agreement with the use of shadow price in principle.
There is however great difficulty in practice in isolating one project
from another. It seems to me that it is often necessary in making
practical decisions in a country like India, to focus attention not
only on the project but also to consider the external economies at
a national level. The time factor also seems to me to be extremely
important, that is, the span of time over which the benefits would
accrue. An interesting case which I analysed several years ago was
the choice in India between importing foodgrains, or importing fertilisers,
or manufacturing fertilisers with imported machinery, or
finally, building machinery in India for the manufacture of fertilisers.
The answer would depend essentially on the time horizon.
The really important point which I should have stated earlier,
's where do you want to go and when? We, have to set up certain
targets at the national level which have to be achieved over a certain
time period, in I0 years or in I5 years or in 20 years. Without
such targets and the time period over which these have to be
realised, I doubt whether shadow prices would be useful. Firstly,
a set of target at the national level and secondly a given time period
over which these have to be achieved have to be supplied for optimization.
DORFMAN
I think Professor MAHALANOBIS is raising two distinguishable
questions here. The question of the steel plant is not a particularly
difficult one. A steel plant may have some external economies, in
particular the educational consequences of helping to train a labor
force which is skilled in working with ferrous metals. A steel plant
3]
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also involves certain external diseconomies because of its impact
on air and water pollution. But these are all so conjectural and so
minor in comparison with the directly measurable consequences of
an investment in a steel plant that it is best to ignore them and to
analyze a steel plant by a straightforward benefit-cost analysis or,
much the same thing, a capital budgeting analysis. In this way we
could ascertain whether the prospective output was valued highly
enough to justify the required expenditures.
The question about time horizons and time periods seems to be
very closely related to Professor KooPMANS’ question. The answer
lies in the choice of an appropriate rate of time preference which
will allow us to evaluate consequences that accrue at different times
whether they be expenditures, which normally occur early in the
life of a project, or beneficial outputs which normally accrue over
a very long period of time. The shadow price idea is simply a
device for placing comparative values on consequences which occur
on different dates. They express the importance of targets as well
as the time periods over which they are realized and are not separable
in concept from social targets to be attained by public investment
undertakings.
MAHALANOBIS
[ agree entirely with what Professor DORFMAN had said; certair.
value judgments have to be made in some way or other by policy
makers. In using econometric models sub-optimization is also necessary.
The advantage of a complete aggregate model is that we
may be able to reach some of the special decisions when these are
not conflicting. But in making calculations for optimization even
in the case of a particular project, it seems to me that there will
be much gain in objectivity in choosing the shadow price if we
set up certain targets which belong to the whole of the national
economv over a given period of time
Dorfman - pag.
3
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
DORFMAN
T think we are in very complete agreement whenever there is a
sound way to approximate the consequences of a particular project.
When this is possible those consequences should be inserted in the
calculations and should be evaluated by means of the shadow prices.
It has been my experience that in practice the only consequences
that you can estimate reliably are those that are very proximate to
the project itself. The more remote consequences, which may be
very important in some instances, are connected with the direct results
in extremely obscure ways that must be left in the realm of
judgment,
MAHALANOBIS
We are not in disagreement, but I am suggesting, on the basis
of some of our own experience in India, it would be useful to attach
a time period for public investments, even in sub-optimization in
the case of a particular project. Without building in an assigned
time period, I do not see how we can get a shadow price. If we
build in a time period, the solution may differ for 5 years or 10
years or 15 years, then it will be a question at a national level what
should be the proper period of time to be taken into consideration.
Whether the decisions are to be implemented by central planning
or by wise decision in a free economy is a separate question; I am
simply considering what would be the proper model.
DORFMAN
Professor MAHALANOBIS and I have both asserted repeatedly
that we are in agreement, but now I begin to feel that also we are
not understanding each other very well. Professor MAHALANOBIS
37
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insists that «it would be useful to attach a time period for public
investments ». Of course it would, and I fail to see why he feels
it so necessary to insist on this. Every consequence of a public
investment occurs at some date. The importance of that consequence
naturally depends on the date or dates at which it occurs.
The time period and time sequence is incorporated in the analysis
by means of the shadow prices which are different for every date at
which something can happen. An event which occurs 10 years
hence will receive less weight in an analysis than one that occurs
5 years hence if the shadow prices for events 10 years in the future
are lower than those for events 5 years in the future, and this wil
be the case whenever there is a positive rate of time discount.
SCHNEIDER
When we consider an economic system which is mainly based
on the principles of a market economy with a public sector, then
I don’t see that the problem of shadow prices has so much ‘mportance
as it has for example in developing countrie- « ‘- -
centrallv planned economic swvstem
DORFMAN
It is very helpful of you to remind us that where we have marke:
prices to guide us we have much less need for shadow prices. In
fact, we may have no need at all. But when public investments
are being considered, we do not have market prices for all the goods
and services that concern us. In particular we do not have market
prices for events that will occur in the future or for the external
economies and diseconomies that will result from the contemplated
investment.
Shadow prices are required for all the consequences of an invest
ment undertaking that are not valued in the ordinary markets
y
Dorfman - pag. 3;
220 PONTIFITIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
~ 8
For example, in designing a road, we may need shadow prices for
the saving of lives or for the saving of time that would result from
designing and building the road to higher standards.
SCHNEIDER
If you would call this additional cost the shadow price, then
[ am in agreement with you; I have understood shadow prices as
prices that are not paid on the market but which are calculated for
internal deliveries from one firm to another firm.
KOOPMANS
The term and concept of shadow price has caught on very much,
and like Prof. SCHNEIDER, I am all for its use. I would only point
out that the theory on which the concept is based does not cover as
many conditions and circumstances as our tendency to use the term
would suggest. In particular there is the tase of choices in which
important indivisibilities are present. I would surmise that the
concept of shadow prices can be precisely and carefully extended
to such cases, but I have not seen it worked out in the literature.
There is one article by BaumorL and Gomory (« Econometrica »,
July 1960) that goes in this direction, but it does not go all the way.
Prof. MAHALANOBIS asked: is there a specific study of a steel
mill from this point of view. I may mention a study not of a steel
mill, but of a fertilizer plant — the question of whether in the Latin
American area there should be one, two, three or up to five or six
plants, and if so, for each number what are the best places to put
them. There is a study by MaNNE and VIETORISZ devoted to this
question in a volume called « Studies in Process Analysis », edited
by Manne and Markowitz (Wiley, 1963). That study used combina-‘orial
methods because of the indivisibilities of the plants concerned.
Again the shadow price concept has not been fully elaborated, and
‘hat is how the reference occurred to me in this context.
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29]
FRISCH
I am glad that Prof. DorRFMAN raised the question of what is a
« project », of what is to be understood by the description of a
project. What kind of thing should be included in the project description?
In my mind there are certain things that can, without
difficulty be included in a project description, and which can be
specified without relating this « project » to other projects.
The quantity of various types of inputs needed, in order to
carry the project out, the distribution of these inputs over time; the
capacity increase which the project will cause if accepted, etc., all
these things can be specified perfectly well in the project description
of this individual project. These data are more or less of the
engineering kind. But there are other effects of the decision to
accept a project, which can only be described by considering this
project as part of the whole economy. Here is where the economist
come into the picture.
Some of these effects — particularly the indirect ones — are very
difficult to trace. This is just the raison d’étre of the complete decision
model.
There are good effects and bad effects. You must define the
preferences of the political authority before you can say what is
«good » and what is «bad ». If you really want to go to the
bottom of this description of effects, you will by necessity be led
more and more in the direction of the global model which I presented
LEONTIFF
The operational significance of Professor DORFMAN’s scheme can
possibly be best judged in terms of the amount and nature of factual
information that would be required for its practical implementation.
He proposes to derive the production function comprising all possible
input combinations that could be used for the manufacture of
51 Dorfman - pag. 13
222 PONTIFICTAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
the finished output in terms of which he defines a particular project.
Should these inputs be described in terms of intermediate or only
primary factors? What source of information can be tapped to determine
prices considered as given in the proposed computations?
In other words, how should the cut off point or points be determined
beyond which he does not intend to push the application of maximizing
procedures involving explicit consideration of several well-defined
alternatives.
ALLAIS
I want to underline only one point. It is in fact very difficult
to choose the right indicative prices without very close connection
with the market, without the pressure of the market. And I can
give two very striking examples. The first one concerns the calculations
of the French National Electricity Board. A few years
ago the engineers were calculating projects using a very low rate
of interest. When these calculations had been made they had the
choice between different projects. The rate of interest chosen for
the calculations was something about 3 per cent, I don’t remember
exactly. Later in order to make a final choice, they used a new
rate of interest of 8 per cent, as a result of general instructions issued
by the Commissariat au Plan. Correspondingly, the result that each
project effectively retained represented an over-investment. The
final outcome of this error of principle in the calculations was a
waste of resources for the French economy. This waste would have
been avoided if the decisions of the Electricity Board had followed
the rules of the market economy.
The second example is the indicative price for coal which the
Three Wise Men designated by the European Coal and Steel Community
in 1957 advised should be used for the energy policy of the
Europe of the Six. This shadow price was 22 dollars per ton of
American coal c.i.f. European port. But four years later the price
was only 12 dollars. If there had been a real and reliable market,
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such an error could not have arisen. This error was a very regrettable
one, for in the meantime there was an incentive to invest much
money in projects which in fact implied a waste of scarce resources
Thank vou.
DORFMAN
I greatly appreciate the comments. All of them are valid and
some deal, of course, with problems that I, myself, encountered in
my thinking, but due to defects in my exposition, were not set forth
clearly in my discussion. I share with Professor ALLAIS and also
with some of the others the feeling that it is extraordinarily difficult
to evaluate projects without heavy reliance on prices established
in markets. I judge that market prices are much more reliable
indicators of social worth than any shadow prices that a group of
experts or politicians may establish, and I think that the proposals
I have made would not be practical without a substantial substratum
of market prices, expecially those dealing with costs and the values
of certain salable outputs. But there cannot be reliable free market
prices in principle for some of the aspects of the kinds of projects
we are here discussing. In particular, this is true of the external
consequences of projects, as in the case of education, and of some
convenience aspects like those relating to speed and safety, which
come up in the design of roads. The shadow prices that I am thinking
of are used to fill in the gaps, so to speak, in the system of
market prices. Take the market prices where you have them and
be grateful. Where market prices do not exist, we need a device
for estimating social values of types of factors and of consequences
that are not priced on the markets.
I think that this response deals also with the issue that Professor
LEONTIEF has raised. I should recommend a very businesslike approach
to implementing my proposal, in effect drawing up a pro
forma profit and loss account for the project being examined. All
inputs used should be included on the cost side. be thev interme-CY
|
Dorfman - pag. 37
224 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
diate or primary factors, and similarly all outputs produced should
be assigned values; market prices when available. shadow prices
when necessary.
And, finally, if I have another 30 seconds, I should like to say
something about the relationship of my thinking with Professor
FriscH's, He asserts that one should begin with a social objective
function or some measure of social welfare. My own contribution,
if it be a contribution, is merely to set forth a way to discover
certain aspects of the social welfare function. I do not feel that we
can begin with it literally because at the beginning we do not know
what it is. We must discover it by trying out proposals and seeing
how the public and political organs react to them. In this way social
preferences will be revealed, and a social welfare function will be
discovered which can be used in designing and appraising investment
projects.
51 Dorfman - pag. 38
ON THE CONCEPT OF OPTIMAL ECONOMIC
GROWTH (*)
TJALLING C. KOOPMANS (**)
Cowles Foundation for Research in Economics at Yale Universita
New Haven. Conn - USA
APPROACHES 11
TiIE LITERATURE
The search for a principle from which an « optimal » rate
of economic growth can be deduced holds great fascination to
economists. A variety of attitudes or approaches to this problem
can be discerned in the literature.
One school of thought, represented among others by Professor
BAUER [1957], favors that balance between the welfare
of present and future generations that is implied in the spontaneous
and individual savings decisions of the present generation.
A policy implementing this preference would merely
seek to arrange for tax collection and other government actions
affecting the economy in such a way as to distort or amend the
individual savings preferences as little as possible.
(*) This research was undertaken by the Cowles Foundation for Research
in Economics, under Task NR 047-006 with the Office of Naval Research,
and completed under a grant from the National Science Foundation. An
abbreviated version of this paper was presented at a joint session of the
American Econcmic Association and the Econometric Society on « Intertemporal
Economic Theory » in the Boston Meetings, December 1963.
(**) I am indebted to S CHakRavarTy. E S. Purrps and H. F. SCARF
for valuable comments
4 1
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226 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 98
Contrasting with this view is the position, expressed among
others quite explicitly by Professor ALLAIS [1947, Ch. VI, X],
that the balancing of the interests of different generations is an
sthical or political problem, in which the competitive market
solution has no valid claim to moral superiority over other
solutions that depend for their realization on action by the
state. À more specific optimality concept is implied in the
strictures of Professor HARROD [1048, p. 40] and of FRANK
RAMSEY [1928, p. 543] against any discounting of future utilites.
These authors leave little doubt that they regard only
equal weights for the welfare of present and future generations
as ethically defensible.
The purpose of the present paper is to do some « logical
experiments », in which various mathematical forms of the
optimality criterion are confronted with a very simple model
of technology and of population growth, to see what their
maximization leads to. Our study is similar in purpose to
RAMSEY’s classical paper, and to TINBERGEN’s recent exploration
[1960] of the same problem. The underlying idea of
this exploratory approach is that the problem of optimal
growth is too complicated, or at least too unfamiliar, for one
to feel comfortable in making an entirely a priori choice of
an optimality criterion before one knows the implications of
alternative choices. One may wish to choose between principles
on the basis of the results of their application. In order
‘0 do so, one first needs to know what these results are. This
is an economic question logically prior to the ethical or political
choice of a criterion.
What is a suitable mathematical formalization of the idea
of an optimality criterion? The most basic notion is that of a
preference ordering of growth paths. Such an ordering states
for each pair of alternative growth paths whether they are
equally good, and if not, which is preferred. Indifference, preference
and preference-or-indifference are usually required to
be transitive.
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pra
An important class of preference orderings is that represent
able (!) by a continuous preference function (utility function,
indicator, etc.). A particular function which has been frequently
used has the form
1°
for consumption paths (x,, x,, ...) of infinite duration with
discrete time #=1, 2, … . This form can be interpreted as
a discounted sum of future one-period utilities u(x,) with a
discount factor of x per period. This form has been derived
by the present author (?) from postulates expressing, among
other requirements.
(a) noncomplementarity of consumption in an three subp
y p y
periods into which the future mav be partitioned:
(D) stationarity in the sense that the ordering of any two paths
is not altered if both consumption sequences are postponed
by one time unit and identical consumptions are inserted in
the gaps so created in each path.
The utility function so obtained is « cardinal » only in the
limited sense that the simple form of a discounted sum is conserved
only by linear transformations of the utility scale. If
below we occasionally use the expressions « utility difference »,
« marginal utility », these must be interpreted as elliptic
() Conditions of continuity under which a given preference ordering per
mits such a representation have been studied bv Worp [19431 and bv Dr
BREU [1954].
2) KooPMANC
"1060
especiallv Section
4] Koopmans - pag. 3
228
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
phrases referring to a preference indicator of that particularly
simple form. There is no intent to claim that, even in the
absence of risk or uncertainty, there is some physical or
psychological significance to the comparison of utility differences
in such a scale.
There still remains a logical gap between the derivation
of the above utility function from the postulates referred to
and its use in the present study: For present purposes a continuous
time concept is more appropriate.
2. PLAN OF THE PRESENT PAPER
We shall freely borrow from PHELPS [1961] and others
mentioned below the assumptions of the main model considered
in Section 4, from RAMSEY [1928] a device for maximizing
utility over an infinite horizon without discounting, together
with methods for applying the device, frorh SRINIVASAN [1962]
and from Uzawa [1963] information about the results of
maximizing a discounted sum of future consumption, and from
INAGAKI [1063] results about the generalization of the present
problem to the case of predictable technological progress (1). If
this particular brew has not been served before, it is not put
together here for any novelty of the combination. Rather, our
eclectic model appears to have in it the minimum collection of
elements needed to serve the two main aims of the present
paper.
The first aim is to illustrate the usefulness of the tools and
concepts of mathematical programming in relation to the
problem of optimal economic growth.
The second aim is to argue against the complete separation
(') Note added in proof: A study by PucacHEv [1963], which has
several similarities with the present study, has been brought to mv attention
by ].M. MONTIAS
"41 Koopmans - pag. 4
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of the ethical or political choice of an objective function from
the investigation of the set of technologically feasible paths.
Our main conclusion will be that such a separation is not workable.
Ignoring realities in adopting « principles » may lead
one to search for a nonexistent optimum, or to adopt an
« optimum » that is open to unanticipated objections.
In connection with the first aim, Section 3 recalls a few
results of the theory of linear and convex programming in a
Anite number of variables, that bear on the problem of optimum
growth. The reading of this section is believed to be
helpful rather than essential for what follows. Indeed, in most
of its formulations, the problem of optimal growth is a special
problem in mathematical programming. The main new element
arises from the open-endedness of the future. If one
adopts a finite time horizon, the choice of the terminal capital
stock is as much a part of the problem to be solved as the
choice of the path. Terminal capital, after all, represents the
collection of paths beyond the horizon that it makes possible.
An infinite horizon is therefore perhaps a more natural specification
m many formulations of the problem of optimal growth.
The mathematical complications so created are the price for
the greater explicitness of long run considerations thus made
possible.
Sections 4-6 analyze a model with a single producible good
serving both as capital in the form of a stock, and as a consumption
good in the form of a flow. It is produced under a
constant technologv bv a labor force growing exogenously at
a given exponential rate. Proofs for many of the propositions
labeled (A), (B), ... in Section 4 are given under the same
label in an Appendix (1).
In Section 7 the findings of the logical experiments of Sections
5. 6 are examined. The main conclusion is that some
(") Approximate equivalents of propositions (E), (F), (H), (I), (J) were
obtained independently by Davip Cass [1963]. The connection between the
limiting case of a zero discount rate and the « golden rule of acenmnlation »
{see Section 5) is also observed and discussed in Cass’s paper
Konbmans - pag.
230 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
22
utility functions that on a priori grounds appear quite plausible
and reasonable do not permit determination of an optimal
growth path even in a constant technology. Tentative and
intuitive explanations for this finding are offered.
Section 8 discusses in a tentative way, and without proofs,
possible extensions of the analysis to a changing technology
and/or a variable rate of population growth, with none, one,
or both of these regarded as policy variables.
3. PERTINENT ASPECTS OF LINEAR AND OF CONVEX PROGRAM-T
MING
Let linear programming be applied to an allocation problem
in terms of the quantities x,, j=1, ..., » of a finite number #
of commodities. Then the feasible set D is given by a finite
number of linear inequalities
n
2 aj; XL; = b; ,
fo
2 1, .….. M
The objective function, or maximand, is a linear form in the x,
<
i
U =
2 i x;
The feasible set D is always closed, and may be bounded
(as in Figure 1) or unbounded (Figure 2).
The range R of the objective function on the feasible set
{the set of values assumed by the maximand on the points of
the set D) is an interval. If D is bounded (contained in some
hypercube), then R is necessarily also bounded. If D is un-"41
Koopmans - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
Pa
LJ
4
bounded, then R may still be bounded, but may also be
unbounded from below, from above, or both. If R is bounded
from above, an optimal point exists (Figure 2). If R is
unbounded from above, no optimum exists (Figure 3). Both
cases can arise on the same feasible set ID through different
choices of the maximand.
A highly special form of linear programming has been used
by KANTOROVICH [1959]. In this case the objective is defined
by prescribing the ratios of the quantities of all desired goods
Koopmans - pag. 7
232
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
2?
i.e., goods entering into the objective, and by maximizing a
common scalar factor applied to these quantities (Figure 4).
This problem can also be formulated in linear programming
terms: One adds to the constraints (1) linear equalities expressing
the prescribed ratios, and chooses as a maximand (2) the
quantity of any one desired good, say.
In convex programming the feasible set is defined by
g(x, ..., x,)=20, 1=1, ..., m,
where the g; are concave (1) functions, and the maximand
U Ux, .... x,
A
$
concave function g(#y, ..., #,) is represented by a hypersurface
£.%, ..., %,) in the space {y, x, ..., #,} that is never « below » any
its chords (if the ++ direction is « up »)
Koopmans - pag.
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233
is another concave function (Figure 5). The term convex programmung
derives from the fact that the feasible set, and each
set of points on which the maximand attains or exceeds a given
value, are convex (!). Linear programming is a special case
of convex programming.
With any optimal point in a convex programming problem
one can associate a hyperplane H through that point, which
separates the feasible set from the set of points in which the
maximand exceeds its value in the optimal point (H is a line
in Figure 5). The direction coefficients of such a hyperplane
define a vector of relative prices implicit in the optimal point.
One interpretation of the implicit prices is that the opening up
of an opportunity to barter unlimited amounts of commodities
at those relative prices does not allow the attainment of a higher
value of the maximand. Moreover, if the maximand is a differentiable
utility function, one may be able, by treating utility
as an additional « commodity » and choosing its « price » to
be unity, to interpret the implicit prices of the other goods as
their marginal utilities either directly in consumption, or indirectly
through the extra consumption made possible by the
availability of one more unit of that commodity as a factor of
production.
|
A ONE-SECTOR MODEL WITH CONSTANT
STEADILY INCREASING I ARBOR Fore
TECHNOLOGY AND
We assume that output of the single producible commodity
is a twice differentiable and concave function F(Z, L), homogeneons
of degree one. of the capital stock Z and the size o
1 : . __. .
(') A convex set is a set of points containing everv line segment con
necting two of its points
., Koopmans - pag. «
234
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
2K
the labor force L. These assumptions imply full employment
of labor and capital, constant returns to scale, and nonincreasing
returns to an increase in only one factor of production.
Since capital is treated as a stock of the single producible commodity,
output is at any time # to be allocated to a positive
rate of consumption X,, and to a positive, zero, or even negative
rate of net investment Y,. Hence, if we use a continuous
time concept, and denote derivatives with respect to time by
dots, we have
5)
(6)
X, Y= F(Z, L,) (
Y.=7. .
F(Z, L) is defined for all Z>o0, L=0. We assume further
that both labor and capital are essential to production, that
either factor has a positive marginal productivity, and that
returns to increases in only one factor are strictly decreasing,
“ (7a, 8)
' (7e, d
tye, *
21s,
Vt
OF
tt
).
~
Sd, 0)—=«
OF
> 0 .
Lu
ô?F
=> < O
SL?
Finally, we assume that the labor force increases at a
constant positive exponential rate À, from a given initial magnitude
L, ,
ro
(u a, ob
L,
L Nn © :
J
"41 Koopmans - pag. 10
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 255
The homogeneity of the production function enables us tu
go over to per-unit-of-labor-force concepts. Calling the unit
of labor force briefly a « worker », let x denote consumption
per worker, y ditto net investment, z ditto capital stock, and
(9)
output per worker. Since we then have
the feasible set in the space of per-worker variables +
romes
(ro) (10a
) (IOd.€
Ver
The term Jz represents the (net) investment needed if one
wants merely to supply the growing labor force with capital
at the existing ratio of capital per worker.
To be specific we shall call a path (x, z,) satisfying (10)
attainable (for the given z,), and use the term feasible path ir
the wider sense of a path attainable for some z,>o0.
It is implied in (7 e) that f(z) is strictly concave
{!) A strictlv concave function is one that is strictly « above » all its
rhorde
41 Koopmans - pag. 1)
236 ‘PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
The per-worker production function f(z) is therefore represented
by a curve such as is shown in Figure 6. The curve rises from
the value f(0)=0 with a decreasing slope. In particular, any
line Az through the origin and of slope A such that O<A<f(0)
will ultimately intersect the curve and continue above it 1)
oy oan.
x > o such that o
> such that
-
: f' (0) there is
-
3
(') To obtain (11) suppose that, for some such À, f(z)=)\z for arbitrarily
large values of z. Then, by f(o)=o0 and the concavity of f, f(z)=\z for
all z>o0. Bat then, for any Z>o, (7b) and the continuity of F(Z, L) imply
7 .
‘he contradiction o=F(Z, o)=lim F(Z, L)=Ilim LF(_, 1)=Z lim Haz
>Z)>o. ts 1 —0 r=» Z
4] Koopmans - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE LETC. 237
It 2 represents the rate of growth of the labor force, Z represents
a capital stock per worker so large that the investment
required to keep it at the same level absorbs all output, leaving
nothing for consumption. If z,>2z, it will therefore be necessary
to allow gz, to decrease at least to some level below Z. To
avoid the uninteresting complication arising if z,>Z we shall
from here on simply define « feasibility » so as to imply
O< 2,52.
Although we have not vet defined a maximand, it may be
observed that the attainable set is now defined in a space
where the « point » is a pair of positive functions x,, z, of time,
defined for o=¢<oco. This is an infinite-dimensional space for
the double reason that we use a continuous time concept and
an infinite horizon. It remains infinite-dimensional if we limit
ourselves (!) to twice differentiable functions z, and once differentiable
functions x.
5. THE PATH OF THE GOLDEN RULE OF ACCUMULATION
To answer an important preliminary question, we first consider
a KANTOROVICH type restriction of the problem to a onedimensional
one. The latter problem has been formulated and
solved in the last few years, independently and in one form
or another, by (*) ArLaIs [1962], DESROUSSEAUX [1961],
PHELPS [1961], JoAN ROBINSON [1062], SwAN [1060], VON
WEIZSACKER [1962].
Remove from the definition of the attainable set the restric.
(") Due to twice differentiability of the data functions f(z) above and
u(x) below we will not be excluding any optimal paths by that requirement.
However, a slightly weaker requirement will be found useful in the Appendix.
(°) Dates are bibliographical only and refer to the list of references below.
Some of these authors used somewhat more general models involving ar
exponential technological improvement factor in the production funcHon
; € :
Koopmans - pag. 13
238 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
tion that z, is given, thus making initial capital a free good.
Restrict the attainable set instead by an arbitrary stipulation
that consumption per worker and capital per worker are to
be held constant over time.
x. =x, 2,=z for all =o
[he new « attainable » set then is given by
{12 a, b, ¢) x=f(2)- Az, x>o, z>o0.
Finally, choose z so as to maximize x, the permanent level of
consumption per worker. This leads to the choice of that value
Zz of z for which.
(13 a, b) f(8)=), so =f) 2%
where f(z) denotes the derivative of f(z).
Figure 7 shows the construction. Because, of the essentiality
of labor to production, i.e., assumption (#7 b) as reflected
in (11), there is for any given slope A such that o<<A<f'(0) a
point 2 for which the tangent to the production function per
worker has that slope. To interpret the condition (13 a) note
that, if we hold L fixed, then by the homogeneity of F,
(2,
SF(Z/L, 0 SF Z,
3 ZL) 7
"41 Koopmans - pag. 14
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 23C
Hence (13) expresses equality, at all times #, of the marginal
productivity of capital (in producing capital, say) to the growth
rate A — a prescription known as the golden rule of accumulation
(©
1G
6. EXISTENCE AND CHARACTERISTICS OF OPTIMAL PATHS
We now return to the original problem that allows x, and z,
to vary in time and recognizes the restriction (10 d) of a historically
given initial capital stock, and look about for a suitable
maximand. We admit to an ethical preference for neutrality
as between the welfare of different generations. After some
hesitation, we tentatively and arbitrarily resolve another ethical
conundrum by interpreting this « timing neutrality » in a percapita
sense. That is, we assume first of all that labor force
and population grow in proportion. Furthermore we thus imply
(") PHELPS [1061
+1 Koopmans - pag. 15
240 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
that, starting from the golden rule path #, £ of the preceding
section as a base line, we welcome equally a unit increase in
consumption per worker in any one future decade, say. Mere
numbers do not give one generation an edge over another in
this scheme of values.
The next difficulty we face is a technical one. A previous
investigation by KooPmANs [1960], continued by Koopmans,
DIAMOND and WILLIAMSON [1962], has shown that there does
not exist a utility function of all consumption paths, which at
the same time exhibits timing neutrality and satisfies other
reasonable postulates which all utility functions used so far
have agreed with. A way out of this dilemma was shown by
RAMSEY [1928]. One can define an eligible set of consumption
paths on which a neutral utility function can be defined.
Moreover, the eligible set is a subset of the feasible set such
that the remaining, ineligible, paths are clearly inferior to the
eligible ones, in a sense still to be defined. In RAMSEY’S case,
in which population was assumed stationary, the criterion of
eligibility was a sufficiently rapid approach over time to what
he called a state of bliss. This state was defined as either a
saturation of consumers with consumption goods, or a saturation
of the productive system with capital to the point where
its marginal productivity has vanished — whichever state would
be encountered first. We shall find that in the present case of
a steady population growth the golden rule path can take the
place of RAMSEY’s state of bliss in defining eligibility. Thus
RAMSEY’s device can be applied to our case with what seems
a lesser strain on the imagination in regard to situations outside
the range of experience.
We have one more technical choice to make. For reasons
of mathematical simplicity, and at some cost in « realism »,
we shall model our utility function after the finite-horizon
example of
/T
u(x,, dt
4] Koopmans - pag. 16
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LA
As explained already in Section 1, this simple integration
of an instantaneous utility flow u(x,) implies noncomplementarity
between consumption in anv two or more parts of the
future.
We shall assume that the instantaneous utility flow is a
strictly concave, increasing and twice differentiable function
u(x) of the instantaneous consumption flow x. This function
does not change with time, and is defined for all x>0. Strict
concavity implies that we attribute greater weight to the marginal
unit of per capita consumption of a poor generation as
compared with a rich one. To assume u(x) increasing rules
out saturation. Finally, instead of introducing a subsistence
minimum, we shall require that
(15)
line ae.
a strong incentive to avoid periods of very low consumptio..
as much as is feasible.
Let #=u(£) denote the instantaneous utility flow derivec
from the consumption flow per worker of the path x, =x, ., .,
of the golden rule. We shall now work with the difference between
the integral (14) for any given feasible path and its value
for the golden rule path, and study the behavior of this difference
as T goes to infinity. The following propositions can be
proved (for proofs see Appendix).
(16)
(A) There 1s a number U such
(ce ve
ha
u
dl ~~
for all feasible paths (x,, 2, and for all horizons T
‘al Koopmans - pag. 1°
242
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
26
Thus, if utility is measured in conformity with (14), no path
is « infinitely better » than the golden rule path. In particular,
no feasible path x, can indefinitely maintain or exceed a level «
of utility flow that exceeds #. Thus the golden rule path continually
attains the highest indefinitely maintainable utility
flow.
(B) For every feasible path, either lim U, exists (is a
T+00
finite number), or Ur diverges to — oo as T tends to oo.
In the first case, we call the path eligible, in the second
ineligible. Then (B) establishes a clear superiority of each
eligible path over each ineligible one. On the eligible set we
choose as the utility function
ry
U= Je {x;) — @) dt
In propositions (C), (D), an optimal path is defined as a
path maximizing U on the set of eligible and attainable paths.
It is not hard to find eligible and attainable paths for
every admissible initial capital stock z,. If z,>2, one only
needs to refrain from net investment until the capital stock
Z,=3L eM of the golden rule path has caught up with the
given initial stock Z,=z,L,» and to continue along the golden
rule path thereafter. If o<z,<É, one can through a finite
period of tightening the belt arrive on the same path.
(C) For any initial capital stock z, with o<z, SZ there
exists a unique optimal path (X,, Z,) in the set of eligible and
attainable paths. For z,#2, both &, and Z, exhibit a strictly
monotonic approach to & and Zz, respectively, from below if
0<2,<2, from above if 2<z,ZZ. For z,=2, the optimal path
is X,=X, Z,=Z for all t, the golden rule path.
Fal Koopmans - pag. 18
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245
(DY) The optimal bath satisfies the condition
(18)
wx) 2.=14 — ulx.
that at any time the net increase in capital per worker multiplied
by the marginal utility of consumption per worker equals
the net excess of the maximum sustainable utility level over
the current utility level.
This condition is similar to the KEyNES-RAMSEY condition
[RAMSEY, 1028, equation (5)] formulated in terms of absolute
amounts of consumption, and reverts to it for A=0. KEYNES
intuitive reasoning in support of this condition carries over
with only slight reinterpretation.
A number of analogous results can be obtained if the utility
of a consumption path is defined as an integral over the
instantaneous utility flow discounted at a positive instantaneous
rate p.
(EY The utility function
(19) Va) =
20
[ oot ulx,) dt. where 5 > o
is defined for all feasible paths for which x,>>x for all t, whenever
X >0
RAMSEY’s device is therefore unnecessary in this case.
We shall however obtain an economy of notation if instead
of V(p) we use the utility function
20)
“lo.
U.a
~~ 0
.| Koopmans - pag. 19
«44 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
which differs from V(p) by a constant. As before, we shall
write Ur(p) if the integral in (20) extends from o to T<{oo.
The stipulation in (E) that keeps consumption from becoming
altogether too small is necessitated by (15), merely to
prevent Uz(p) from diverging to - oo as T—oco. However, we
shall for p>o define as the eligible-and-attainable set the set
of all paths with the prescribed z, for which V(p) exists. (E) assures
us that no paths worth consideration are excluded from
the eligible set. If z, were to be very small, we could still
allow for growth by taking x correspondingly smaller.
In the following propositions (F) through (J) optimality
is defined by maximization of (20) on the appropriate eligibleattainable
set. It is assumed in propositions (F), (G), that an
eligible-attainable path (£, Z,) is given, which is under scrutiny
for its possible optimality. The propositions associate with
such a path tentative implicit prices of the consumption good
and of the use of the (identical) capital good. Once optimality
of the path (£,, 2,) is confirmed, these prices are no longer tentative,
and generalize to an infinite-dimerisional space the idea
of a hyperplane separating attainable from better-than-optimally-attainable
programs, illustrated in Figure 5. The (dated)
price of the consumption good is defined from (20) by
>
fy
{,=e-" X
; u(x),
the present value of the marginal instantaneous utility of consumption
at time ¢ if the given path (%,, 2,) is followed. The
price of the use of the capital good is similarly defined by
2.
i
eo Ky
4] Koopmans - pag. 20
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
245
as the present value of the marginal productivity of capital at
time ¢ multiplied by the marginal utility of consumption at
that time. Finally, we denote by
(23)
Oto) = Üx(s) , where Ûnte, _
et {n(x
3 ~ Wdtfor T < oo
the utility of the path (£, 2,) under scrutiny.
Propositions (F) and (G) hold for any eligible-attainable
path (£, Z,) and its corresponding prices P;» q, regardless of
whether that path passes the test for optimality. Propositions
(H), (I) together express that test.
(F) If (x, 2,) is an eligible and attainable patn, and 1]
(X,, z,) is any bath, feasible or not, then
(24)
Ug.
l
A N
py - Xs) at
for all finite T. and for
T
x whenever the integral con»,
Since both members vanish if x.=4, for all sz, this means
that both
(1) the utility Ur(p) of the path (x, z,) 1s maximized, subject
onlv to the « budget constraint »
if (a,
.
A
+2
1
<,,, and
- xy Ue
—
+1 Koopmans - pag. 2.
A
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(il) « consumption expenditure at implicit prices »
I
| 2 x, dt
.
reaches its minimum, on the set of paths with utility
Uz(p) equal to or exceeding that of the path (£, 2), if
(x,, 2,)= (%X,, Z,).
(G) If (x, 5), (x, z,) ave two eligible and attainable paths,
5)
r I
Joue — dt = [late — 3) — bé, — 21) dt =
9 5
= lea, +5) (2,— 2,) ar — plan — 2,
for all finite T, and for T=oo if the integrals converge and
the last term has a limit.
Again, all three members vanish if (x,, 2,) is itself the path
(#,, 2,). The inequality in (25), rewritten as
I
[puta + 2, — Ly — 2. di
1
[ae —2ndt=o ;
4] Koopmans - pag. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
says that, at prices implicit in the path (%,, 2,), « revenue
from total output minus « rental cost of use of capital
maximized in that path.
(24) and (25) together give rise to Propositions (H), (7
(26)
Then 1j
for ali
(H) L.+ *
«À
qi
a
z(p) be defined as the solution =,
A
AZ=Xx, where ©
oT
nique optimal bath 1s
La
The determination of z(2) is shown in Figure ecaus
of the strict concavity of f(z, « * end £tu, evist ana sie um
que, and 0<_z(g)<2(¢*)<2 fu, 9. and hence
since f(z) — Az increases for
(27) Z(o)=flz\,
r2lo) < aye) =< x for f'(o)—\ > p.
wu
4
Koobmans - pag. 27
48
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 98
The constant optimal path made possible by the initial capital
stock z,=2(p) is found to be an asymptote for the optimal
paths associated with other values- of z,.
(I) For all z, with 0<z,=Z, the unique optimal path
(X,, Z,) is uniquely characterized by the two conditions
(2) lim 2:=2(p),
T=00
(B) the prices (21), (22) implicit in the path (X,, Z,)
satisfy the differential equation
a,
7
JJ
a
g,+p,=0 for all =o.
To interpret condition (B), let (x, z,) be a path which differs
from the optimal path only slightly and only on a short
open interval IJ, on which 2,>2%, (see Figure 9 a). Then x,
will differ from #, first because the slightly higher capital stock
on J allows a slightly higher product, and secondly because
acceleration of investment during the first part of J and deceleration
during the second part leads to some postponement
of consumption within J. In the light of (21), (22), the condition
says that, for an arbitrarily small difference z,- 2, of
arbitrarily short duration, the utility effects of these two components
of x,- x, must cancel if the path (£, 2,) is to be
optimal.
If condition (8) of Proposition (I) is satisfied, the inequality
between the first and third members of (25) becomes
— ~
x, — pl
Pe t ay) dt — PrlEr -- 21, ÉO .
r.] Koopmans - pag. 24
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. Qu-4]
Konbmans - pag. 25
250 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The interpretation of this inequality is that, in an economy
that keeps its own capital accounts (rather than renting the
capital it employs), the revenue from deliveries to consumption
plus the value of the capital stock at the end T of the planning
horizon is maximized in the optimal path (%,, 2,), in comparison
with all eligible-attainable paths. This is so for all finite T,
and in the limit for T—oo if such limit exists.
Finally, (24) and (29) together yield
Tr
30) Ux(p) Uap) = [pia x) dt = NE TT 25)
[If condition (x) of Proposition (I) is also satisfied, then (21)
and Proposition (J) below imply that the price p, associated
with the optimal path approaches zero as #+oœ. In that case
im pr(2zr — 8r)=o0, and capital disappears from the accounts
for an infinite horizon. The inequality between the first and
last members of (30) carried to the limit for T—oo then confirms
the optimality of the path (£,, 2,). In the Appendix we show
that the middle member of (30) also converges. We can therefore
supplement statements (i), (ii), made in interpreting (F)
above (with T—oco) by the statement
(iii) « revenue from deliveries to consumption
2
J xy dt
2
is maximized, on the set of eligible-attainable paths, if
(x, 2,4) = “(Ep 2)
Proposition (J) describes the path characterized by propositions
(H), (1).
4] Koopmans - pag. 26
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
~
ZI:
(J) In the unique optimal path (X,, Z,) for any initial
capital stock z, with 0<z,<z, z,#7(p), both =, an’ ~~ exhibit
a monotonic and asymptotic approach to x» a - mt
tively, from above if z_> 7z(p), from belo:
For later discussion, we note from (27) that the asymptotic
level £(p) of consumption per worker, while independent of
the initial capital z,, is reduced as the discount rate is increased.
The maximum of #(p) for p=o0 is attained at p=0 We shal
not examine the cases where p=>f'(0) - A.
Finally, a word about the case where one tries to apply
a negative discount factor p<o. Writing - g=0, this means
looking for a utility function extending the finite-horizon
example
et u(x,) dt
to an infinite horizon. This is not as far-fetched as it may seem.
After all, we have so far given no weight at all to mere numbers
in comparing generations. If we were to weight each generation
in proportion to its number, and otherwise seek neutrality
with regard to timing, the population growth parameter
would take the place of ¢ above.
In order to apply RaMSEY’s device in the present case, one
would have to find a feasible path (x,, z,) such that
(31)
V"
n
, €”
VU,
— u(x) dt
[4] Koopmans - pag. 27
252 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
is uniformly bounded from above for all feasible paths (x}, =)
and all values of T. The following statement says that no
such path exists.
(K) For each 0>>o0, for each attainable path (x,, z,), where
O<Z, SZ, and for each number N>o, there exist another attainable
path (Xi, z;) and a number T* such that
2)
W*(-0)>N for all T>T*
This says, essentially, that there is no upper bound to the
range, on the attainable set, of a utility function of the type
we are seeking to define. The case p<o is therefore analogous
to the case in ordinary linear programming illustrated by Figure
4. The same difficulty was noticed and discussed by
TINBERGEN [1960] and by CHAKRAVARTY [1962] in connection
with the case p=o0 for a model with constant returns to increases
in the amount of capital alone.
In the present case, the reasons for the absence of an
optimal path for p<o can be illustrated in terms of the path
(Xp 2;)=(%, 2), optimal if p=0 and z,=2. From (21) we see
that the implicit price of the unit of consumption good per
worker, associated with this path would have to be a constant,
p,=u'(à) for all
t
This means that a sacrifice of one unit in per capita consumption,
now made for a short period as a slight departure
from this path, can be taken out by any future generation in
the form of an equal augmentation of per capita consumption
beyond that provided by the path, for a period of the same
short duration. Now if either the discount rate p<<o, or if o=0
4] Koopmans - pag. 28
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 255
but some weight is explicitly given to population size, it will
always increase utility to delay still further the time at which
the fruit of the initial sacrifice is reaped.
In the proofs of Propositions (A) - (K), given in the Appendix,
one common characteristic of the problems considered
is repeatedly used without explicit mention. At any time in
an optimal path (%,, £,), the capital stock #, is the only link
between the past and the future. This is due, on the one hand,
to the utility function being an integral over time of instantaneous
utilities (discounted or not). On the other hand, it
arises from the fact that the feasibility constraint (10 a) restricts
z, but not %,. Hence the function x, is in principle free to vary
discontinuously (even though it is found optimal for it not to
do so). However, Z, is bounded by (10 a, b), hence z, can only
vary continuously. The resulting property can be expressed.
formally as follows: If (£,, #,) is an optimal path for given
then, for any T, the path (-"* ” defined +-is
optimal for <-7.
ADIUSTING PREFERENCES TO OPPORTUNITIES
What have we learned from our « logical experiments »?
We have confronted a simple model of production with a utility
function representing a sum of future per-capita utilities, discounted
by a positive, zero, or negative instantaneous rate of
discount p. We have found that g=o0 is the smallest rate for
which an optimal path exists
41 Koopmans - pag. 29
254 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Let us assume for the sake of argument that the present
model is representative enough to be looked on as a tentative
test of the applicability of the ethical principles under consideration.
Then we have just managed to avoid discriminating
against future generations on the basis of remoteness of the
time at which they live. However, this close escape for virtue
was possible only by making welfare comparisons on a per
capita basis. If instead we should want to weight per capita
welfare by population size, then we are forced to discriminate
on the basis of historical time by positive discounting. There
seems to be no way, in an indefinitely growing population,
to give equal weight to all individuals living at all times in the
future.
This dilemma suggests that the open-endedness of the future
imposes mathematical limits on the autonomy of ethical thought.
The suggestion may come as a shock to welfare economists,
because no such logical obstacles have been encountered in
the more fully explored problems of allocation and distribution
for a finite population. It is true that the mere fact that we
are considering an infinite number of people does not fully
explain the dilemma. For Ramsey was able, albeit by artificial
assumptions, to indicate a fair solution to the problem
for the infinite future of a population of constant size. Our difficulty
is therefore connected with the assumption of an indefinite
growth in the population.
The following reasoning may further illuminate the reasons
for the nonexistence of an optimal path with negative p. Assume
that 0>>p>f (2) - A. (Of course, p= - À would correspond
to equal weights given to the utilities of all individuals. However,
f(z) -A>-X, and our illustration is simpler if we do
keep 2(p)<<z by taking p>f(z) - ». Consider now an optimal
path for the finite time period 0<¢<T, defined by initial and
terminal per-worker capital stock levels z,=2r=2 both equal
to that level £ which, if maintained at all times, would secure
the maximum maintainable consumption per head. The analy-F4]
Koopmans - pag. 30
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 255
sis associated with the proofs of (H), (I), (J) in the appendix
now indicates that, if the level 2 is prescribed only for t=c
and £=T, the optimal path bulges out toward the level of Z(p),
as indicated in Figure 10. The interpretation is roughly a-:
FIG. TQ
follows. The negative discount rate gives the greatest weight
to the per capita utility of the last generation living within the
planning period [o, T]. In response to this weighting system,
the optimal path provides for a reduction in capital per worker
(a « disinvestment » in a per capita sense) during a terminal
segment of the planning period, in order to allow for high
consumption at that time. To make this possible, all preceding
generations make a sacrifice. For the first generation, this
takes the form of heavy investment needed to increase the
capital stock more than in proportion to population growth.
For the intermediate generations, it consists in approximately
maintaining the capital stock — by continued proportional
growth — at a per capita level in excess of that which would
maximize per capita consumption.
141 Koopmans - pag. 31
256 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Now if T is increased, the benefitted generation becomes
a more and more distant one. If T=o0, there is no benefitted
generation, and the limiting position of the curve in Figure 10,
while mathematically well-defined, merely describes a path of
indefinite and fruitless sacrifice.
The problem appears in even sharper light if technological
progress is also recognized. A study by INAGAKI [1963] uses
a CoBB-DOUGLAS production function
F(Z, L, t) = const. ef Z*1.1-¢
subject to exogenous technological progress at the constant proportional
rate 3, an instantaneous utility function
(53)
wr) ~ logs -- log x
as
exhibiting suitable behavior for large values of x, and a labor
force growing exponentially at the rate A. Among other results,
INAGAKI finds that, for the integral V(p) as defined in (19)
to converge on the counterpart of our path (x,, z,) =(£(p), #(e)),
it is necessary that
-— Œ-“4]
Koopmans - pag. 32
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ZI
Let us assume that RAMSEY’s device can be used also in this
case, and that it would again merelv result in addine the borderline
value ç= te the set of discount rates defining a
utility function for which an optimal path exists. Then a predictable
positive lower bound to the rate of technical progress,
valid for an indefinite future period, precludes application of
the ethical principle of timing neutrality in terms even of per
capita utility — not to speak at all ot weighting generations by
their numbers.
Thus, if in the face of technological progress we want tc
hold on to the idea of maximizing a utility integral such as (35,
over time, we must invent a discount rate © satisfying (34), or
its equivalent for another production function. Such a discount
rate might just have to be a pragmatic one having no basis
in a priori ethical thought. While it might well be a result,
conscious or unconscious, of political processes or decisions, it
would have to be revised upward if it is estimated that technological
progress will accelerate to such an extent as to « overtake
it », and could be revised downward if it is expected that
progress will slow down.
One might instead conclude that the whole idea of maximizing
a utility integral is not flexible enough to fit the inequality
of opportunity between generations inherent in modern technology.
Two alternative notions have been partially explored
by the present author, using a discrete concept of time. In
one of these [ KOOPMANS, 1960, see also KooPMANS, DIAMOND
and WILLIAMSON, 1964], the utility function of a consumption
path x, t—1, 4, .…, can be defined bv a recursive relation
J(r,. x,
== Viu(x,), U(x,, 3. y
.1 Koopmans - pag. 3-
258
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
in terms of a one-period utility function #(x) and an aggregator
function V(x, U). This formulation allows the (scale-invariant)
discount factor
(Fe 7)
OU u=u(x), U=U(x, x ..
associated with a constant path to increase or decrease with
the level x at which the path proceeds. The second alternative
"KoorPMmANS, 1962] is an attempt to express formally the idea
of a present preference for flexibility in future preferences between
different commodity bundles of the same timing, or
between physically the same bundles spread out differently
over time, or between bundles differing in both respects.
Further analysis will be needed to determine whether the first
idea is sufficiently flexible to enable us to avoid the difficulties
we have encountered, or, if not, whether the second idea can
be made workable.
8. TECHNICAL PROGRESS AND POPULATION GROWTH AS Pos-SIBLE
Poricy VARIABLES
So far we have treated both technical progress and population
growth as exogenously given. It should now be recognized
that both variables can be, and are in many countries, influenced
by public and private policies and attitudes. Technical
change is furthered by government conduct or support of research
and of education, by the tax treatment of depreciation
and obsolescence, and by business policies with regard to
research and development. Population growth is influenced
4] Koobmans - pag. 34
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256
by expenditures for public health, by family allowances, by
government policies toward family planning, and by general
cultural and religious attitudes toward the idea of population
control. In addition, both variables are in part endogenously
affected by the level of income.
Both possibilities of partial control raise new conceptual
problems in formalizing the idea of optimal economic growth.
In the middle of the scientific explosion, it is hard to assess
whether technological progress can go on forever, so that also
its rate can be raised or lowered forever. It is conceivable that
a higher rate of discovery and invention in the present will
entail a lower rate of progress at some later time when the fund
of knowledge usable in production nears completion. Another
consideration is that technological progress raises transition and
dislocation difficulties that affect the relative welfare of different
individuals within the same generation.
The possibility of influencing population size raises the
question of the value of population size in itself — as distinct
from the question of the weight given to numbers in aggregating
utility over generations, discussed above. It should be noted
that all utility functions discussed in this paper imply neutrality
with regard to population size as such. The question is of some
importance because a different attitude might lead to a different
balance between the « value of numbers » and the loss of per
capita income that may result from an increase in the ratio of
population to land and/or other resources. This problem did
not come up in the more formal analysis of the preceding section
because the assumption of constant returns to proportional
increases in both labor and capital precluded the recognition
of resource limitations.
Koobmans - pag. 35
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
APPENDIX
A 1. NOTATIONS
Instantaneous discount rate . . . .
Exponential growth rate of labor force
Pà
At time t Integrated
absolute per worker over time
(per worker)
Consumption flow
Capital stock .
Labor force .
Production function . . .
Utility . . .
A
7
L,
F(Z, L) f(2)
u(x) U,V, W
dz
Derivatives with respect to time are denoted by dots, #,= =
. . d , d
other derivatives by dashes, f'(2)= , uw (x)= =
a
generally denotes optimal paths and their asymptotic levels.
denotes equality by definition.
"41 Koopmans - pag. 36
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36.
5SUMPTIONS
(a)=(8
(6) =(
(ce; ’
(d)=(
II
/
€
for all £=0, where o<,
F(Z, Ly=L f(T)=L f(z) for all L_-u, ZZ.
f(z) >o0, f’(2)<o for ox..,
(0) =o,
for each À >0o such that o<)<f/(0) there is a z; -c
such that f(Z))=2X2, (the subscript À of Z, is
omitted in what follows),
ux, >0, U"(x}\<o for o<x<o, lim u(x,
VY ed
A
SOME IMPLICATIONS OF FEASIRITITS
Given the initial stock z, of capital per worker, the attain.
able set of growth paths (x,, z,) is now given, in terms of pei
worker variables, bv the requirements that. for all
(35)
(26)
(37)
350) @; > 0, .4 7 0
.35b) 2; is continuous.
(35¢) 2; æ, are differentiable to the right,
(35d) 2, and x, are continuous to the right,
}
x,+2,=f(2,) - à z2,=g(z,
sav
2,
is prescribed. where
(y ~
.
The feasible set is the union of all attainable sets with o.
41
Koopmans - pag. 3-
262 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
We note that, by Assumption (c), both the feasible set and
the attainable sets are convex, and that the function g(z) defined
in (36) 1s strictly concave. Since g(z) vanishes for z=o0 and for
z=2, it reaches its maximum # in a unique point #, so that
(38) (38a) æ = g(2) > 9(2) for all 22, where o<z<2,
3 A A
(38b) 9(2)>9(2)= 0 > g'(z*) whenever 0 <Cz7<2<2*<Z.
From (35 a), (36), we have
‘
a
3:
2<x,+2 = g(z,
and hence for all feasible paths, using (35 a), (37), and the
fact that g(2)>0 only for 0<z<Z ,
40)
0<2,<z for all t=o0
Here o0=2z, has been ruled out because it would not allow the
positive consumption x, for #=¢ required by (35 a).
À 4.
A BASIC INEQUALITY AND ONE APPLICATION
The concavity Assumption (e) of u(x) implies that
41) u(x)- u(x*)<u (x*)e(x-x*) for all x, x*
4]
Koopmans - pag. 38
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. "260:
and the concavity of g(z) implied in Assumption (c) and (+0
that
(42) £(z) 2a
<7 (2%) (z-2%) for all
orn
We shall make many comparisons of utility intezs.: .
feasible growth paths (x, z,) and (#}, 2), based or … ‘ an
on either (38) or (42). To avoid repetition we stat- * r~
parison here in its most general form, where -
and © is as vet unspecified
“ot (yar y—ul:
et u'(ay) gz) — gle ¢)
, —
F
(43 a
u(x) gz) —g(2 A) dt —
t
7 Ux
PEN
(43
l<¢—&¢t ) dt _-u
(we) a
Lt
4] Koobmars - pag. 30
204 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
If T*=o0 the validity of (43) depends on convergence of the
integrals involved.
One application of (43) will be used repeatedly. We define
a bulge in a growth path (x,, z,) as an interval [T, T*] such that
‘44,
r
(44a) o=T <T* <0, 252 25 = 2% say, and
(446) either * < #() and 2, <z* for T<t < T*
> 7(s) and 2,>=z* for T<t< T*
where the definition (26) of 2(p) is extended to all values of o,
( (154) A) =7 for psg)
“ (450) g(2(e) = for g(3) <p < glo)
(45¢) Zp)=o for g'(o)-4+|
Koopmans - pag. 40
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
26.
Figure 11 shows z, for a path with two bulges, both denoted
[T, T*]. The effect on the utility integra! «= strai<htening
out» a bulge is found from (43) by taking Giz#,
and satisfies
(46)
ve
|
ee!
(tx
— Wl
if z*#%(p), because in that case g'(z*)-¢ and z,-z* are
opposite in sign. If 2*=2(p), and if for instance fo.
for T<¢<T* as in the second bulge in Figure 11, we can
by suitable choice of a number z*- <{3(g) write the left hand
member of (46) as the negative sum of two such integrals,
one comparing (x,’, z; ) defined by z;=max {z**, z} with
(x*, 2*)=(£(¢), #(e)) on [T, T*], the other comparing (x, z,)
with (x**, 2**), where x**=g(z**), on an interval [T**, T*#+]
such that TT**<{T***<T*. Since of these integrals the
former is nonpositive, the latter negative, (46) is valid also if
z*=2(c). We thus have
LEMMA 1: For any ç, a path (x,, z,) optimal on any finite
or infinite time interval cannot contain a bulee.
This conclusion, and the inequality (46) on which it is
based, remain valid for T° æ and çZo if the definition of a
bulge is extended to read « (44 b) and either (44 a) or (44 a”, ».
(44 a) o< 1
-
Tc
, and ui ; =o then! . ,
Foy A
as illustrated in Figures 12 (¢=o0) and 1, vu
4] Koopmans - pag. 4
4
nn
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
38
><
Fic. 17
Fic. 13
A 5. INFERIORITY OF INDEFINITELY FLUCTUATING PATHS
IF ¢=<0
We define the asymptotic range of the path (x,, z,) as the
nonempty closed interval
(47) [5 3], {=lim inf z,, {= lim sup z,
— Too t>T Too t=>7T
A positive length T - ¢ of the asymptotic range implies that z,
continues to fluctuate between any neighborhood of ¢ and any
neighborhood of €, infinitely often, and for arbitrarily large ¢.
LEMMA 2: If p<o and if §<Z for the attainable path (x,, z,),
then there exists for each N>o an attainable path (x*, z*) and
a Tx>o0 such that
1 7-73
[
Urle)= | ee (u(x,) — u(æt)) dt < — N for all T>T, .
4] Koopmans - pag. 42
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267
For the proof of Lemma 2 we must strengthen (46) to
obtain a positive lower bound on the gain Us (gp) associated
with the « straightening out » of a bulge [T, T*]. For this
purpose we choose an interva. "2. 7 such that
(49) (a) &
~ Ba e
C and either (b) 2(p)<z, or (c) 2* 2
which is always possible. If for definiteness we assume (49 €,
we have from Assumption (c) of Section A 2
(50) g'(2
— v
0]
for
Now 2, has infinitely manv bulges
ww"
[T, T*' with the
pre
perties
(SI)
iN
for some ?.
Because of the c --tinuitv of z, we can for each ot ‘hes
an interval “we write
1
hoc.
(52)
7/0
Koopmans - pag. 43
268 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
The construction is illustrated in Figure 14. Since the las
inequality in (46) holds also for all subintervals of [T, T*],
we have from (52), (50), if x*=g(z*) and p<o,
T*
2Une(p) = — jeu (u(æ;) — u(æ*)) dt > — u'(x*) Y [te 2*) dt >
ve (z¥F— 1) .
> w'(x*) ve
since e>0 and #’(x*) y>o0. On the other hand, we have from
(52), (39) with x,>0, (36) and (38 a) that
dt< fg) dt < (1*—=) æ
whence 1* —- t™>z/%#>0 and
33)
— Uz (p)> o/(x*) y e2/2Za*>o.
Finally, we define the feasible path (x3, 2}) by (36) and
2, =max §2., z*} for 0=<t<Tkx, 2 =z, for t>Ty ,
subject to a later choice of Ty in such a way that 24. <z*,
Then
Ur (o)=— le <t (u(æ;) — u(æË)) dt > ny, ** , where 2*>o,
4) Koopmans - pag. 44
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 26s
it ny denotes the number of completed bulges in [0, T'? FE i
lim #np.=oc because there are infinitelv manv bul se =
T’>00
[0, oc]. Hence the choice of Ty such that ny >N/«- estaplishes
Lemma 2 in case (49 c) holds. The proof from (4G 5) i.
similar.
A 6. PROOFS FOR A ZERO DISCOUNT RATE (.
Proof of (A). In (43 a) take
Then if we write u(x)Za, u’.
(54)
Uu
di =U + |.
y wu
. À
Aw
by (38), (40), regardless of T, T*, hence also for .
Proof of (B). We distinguish three cases regarding
asymptotic range [§, ©] of the given path (x, z,).
Case (1), {<Ç. In this case we have from Lemma 2 and
from (54) applied to (7%, 27), for anv N c
for an
r—
T> 1
waa — uly) dr -
ULX + » -
In this case. therefore. .
u. a
N +.
+ diverges to - \ a
a
. Koopmans - pag. 45
270
a
Zr
$<
55)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
HS
Case (2), {=8#2. For definiteness assume E<£ and let
{=2e. Then £>0 and there exists T<oc such that
2,=<C+e=%-¢ forall £&T
If now in (43 a) we take o=o0, z}=2, x'=#=g(8) for all #>o.
then,
T+
(56) Up = fa) — û) dt < u
rors
[, a
J (9(z0—2) dt + 2p — 24x | =-_T
—o (T* MT +8
where by (38), (40),
=u Nz —gz—2) > o
©
>
A _
— UU + 2
Hence Ur, = Ur + Urs diverges to - oo as T*—oo in this case,
and by similar reasoning in the case £<¢, hence in the entire
Case (2).
Case (3), {=C=2. In this case clearly
37
. A
lim z, = 2
Teo
[t follows from the third member of (54) that
G-EU- +4 ° ZT
4] Koopmans - pag. 46
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 27.
is a nonincreasing function of T. Hence G: either possesses a
limit for T—oo or diverges to — oc. ln view of (57), the same
must then be true for Ur.
This completes the proof of statement (B). In addition.
we have found
LEMMA 3: If o=o0, a necessary condition for eligibility o:
the bath (x,, z,) is that (57) is satisfied.
Proof of (C). An optimal path (x,, 2,) is now defined as
one that maximizes
(58)
u(æ,;) — u) dt
on the attainable-and-eligible set. A beautifully simple procedure
used by RAMSEY in his slightly different problem can be
adapted to the present problem as long as g=o.
From Lemmas 1 and 3 we conclude that, in any optimal
path, 2, exhibits a nondecreasing, constant, or nonincreasing
approach to lim 2,=2 according as z,<2, =2 or >2 is
t=>00
establishes the second and third sentences of statemen.
with the term « weakly monotonic » substituted for « stric’
monotonic ». Now consider an attainable-eligible path (a
for which
(59)
tor
; NVST*, where
Then. along the lines of (56),
4} Koobmans - pag. 47
212
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
Ty) — u —T) gz" — x) < o
‘u(r, à) dt <u'(T*—T) :9(2"— x) <
7;
by (38 a). It follows that the path
(xt, 2) = (Xs 2p) for ot <T ,
(4 prop ; Zean*_g) for T = t
is likewise attainable, and indeed eligible and preferable to
(x,, 2,), because it achieves a utility
(=U +. U*=Ur+ UUs; + +Urs +. U=U .
Therefore (59) cannot occur in an optimal path.
It follows that, if z,#%, an optimal path shows a strictly
monotonic approach of 2, to the value zr =% for 0<¢<'T, where
I'=co. We shall call any eligible path with that property a
superior path. To complete the proof of the second and third
sentences of (C) we only need to show that for an optimal path
T=oo. This is best obtained as a corollary of the proof of (D).
The proof of the first sentence of (C) will also be combined
with that of (D).
Proof of (D). For all superior paths we can now make a
useful change of the variable of integration in (58) from ¢ to z.
Since, by (36), z,=2% for £=T implies x,=X, u(x,) =14, we have
for all superior paths, using (36),
mn.
) u(æ(z)) —u a
12) — X(2)
Koopmans - pag. 4.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 275
where #(z) denotes the inverse of z, on [o, T], and x(2)=x,,.
The unknown function is now x(z), defined, like #(z), on the
interval 2,<z<{2, or on 2<z=z, as the case may be. The
advantage from the change of variables lies in the fact that
only x(z) itself, and no derivative thereof, occur in the integrand
in (60). Hence (60) is maximized on the set of superior
paths if and only if x(z) is given a value £*(2) such that the
integrand is maximized for almost every value of z in its domain.
This requires £*(z) for almost every z to equal the solution
x =x(z) of
‘61,
u X) J
}
“,
21H
- u(r
IG
Koopmans - pag. 49
21%
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
18
Figure 15 shows the determination of x = £(z) for the two cases
z<%Z and z>%. It is easily seen from the diagram or analytically,
using Assumptions (c), (d), (e), that a function #(z) can
be uniquely determined from (61) for all values of z on 0<z<Z,
so as to be independent of z, continuous and increasing for
all z, and differentiable for 2-24. In particular,
(>
9
lim 22) =o, af =2=gF)
Moreover, since any feasible x, is by (35 c) continuous to the
right, and since for any superior path #(z) is continuous and
monotonic, #*(z) must be continuous to the right if z,<2, to
the left if z,>2. Hence £*(z)=£(z) for every value of z in its
domain, and the asterisk can now be omitted from X*(2).
Once %(z) has been determined in the manner indicated,
one reintroduces the time variable #=?(z) by
HE pt
37
Way =i)
= gy) — x(y)
The function 2(z) and its inverse #, are monotonic and differentiable
with the proper range and domain in each case because,
by (61) and the monotonicity of £(2),
g(2) — 2(2) =.
u
-ux(z
°
<
k
A
&
4] Koopmans - pag. 50
3EMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
OTF
Hence #, is monotonic and differentiable. In order to see that
T=œ whenever z 72, one readily computes from TAYLOR
expansions of g(z) — g(£) with respect to z - 2, and of #27. Nwith
respect to Rs
|;
/
|.
1
u
a negative real number.
A
{2 <2
AN
i i)
_1 .ollows that
- ~ .
t zy, >2 lm
He
—
Therefore T=oc. The proofs of (C) and (D) are thereby complete.
In addition. we note that £(z) is differentiable also for
=
A 7. PROOFS FOR A POSITIVE DISCOUNT RATE (0< g<_f (0)
Proof of (E). Let (x,, 2z,, be a feasible path with : =
for all &. In (4°) we insert}: ..< — z <2 such that gi.
Then, if ..f uw YY x we have #<u(x, and hence
o<T
hence lim 1»
T Tar
+
rf
—-u whenever
1: }
yes
+1 Koopmans - pag. 5.
276 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
>
Proofs of (F), (G). These propositions express, and provide
economic interpretation for, the inequalities (43) if we
take T=o0, T*—oo, and if the « candidate-optimal » path
(&, 2,) is substituted for (x% 27). This is seen by reference to
the definitions (21), (22) of the implicit prices p,, q, of the consumption
good and of the use of the same good as capital good,
respectively. Proposition F represents the first inequality in
(43 a), which does not require feasibility of (x,, z,). The inequality
in Proposition G is obtained from the fourth member of
43 a) by using (42), the equality through integration by parts.
Proofs of (H), (I), (J). Proposition (I) states two conditions
(«), (8), as necessary and sufficient for the optimality of
a path (fe, &). We shall first look at the implications of condition
(B) in isolation. Called the Euler condition in the « calculus
of variations », this condition is, for a path denoted
just (x, z,),
64)
gs + pp = w(x) (9°(2;) — e)+ w"(x;) - æ,=0 forall t>o
Together with the identity (36) this condition leads to the system
of differential equations
ES,
|
(650) 2,=g(2) — =;
, wey),
656) æ,=— w(x) (9 (20) — €
i
|
| t>o.
for the solution of which we have a prescribed initial value
z, Of z, but as yet no given value of x,. Figure 16 partitions
"41 Koopmans - pag. 52
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21
au
x /0) -
41 Koopmans - pag. 5,
278 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the quadrant x>0, zZ0 according to the signs of à, 2, that
‘follow from (65). Figure 17 sketches the trajectories of the
point (x, z,) starting from arbitrary initial values (x,, z,).
Each trajectory is defined as the solution x(z) with x(z,)=x,,
or z(x) with z(x,)=z,, of the corresponding differential equation
dz _ u(x) g@)—x dr w@) g'(&—e
( regs I esis ET me, JT ts mele, Bore
66) de u(x) g@@—o" RE æ) g(2)—æ
respectively, obtained from (65) by elimination of #. Any segment
of any trajectory defines a path optimal on a suitable
time interval with prescribed initial and terminal values z,, zr
of z,. If we prescribe only z, and examine the trajectories for
various x, We find that there is one unique value £, of x,
which together with z, identifies a trajectory (shown in the
diagram as a heavier line) that meets condition (x) of Proposition
(I) of an asymptotic approach, for f—oo, to (£(p), 2(p))
as defined in (60), Proposition (H).
We now denote the path resulting from that particular
choice à, of x, by £,, 2). If z,<2(p), the initial consumption
low £, leaves room for growth in the capital stock per worker,
and both #, and 2, increase with ¢ to approach their asymptotic
values £(p), 2(p), respectively, as t—oo. If z,>%, both £,, £,
decrease, and approach the same asymptots from above. Finally,
if z,=2(p) we must have £, = £(p), #,=2(p) for all #=o.
Since #, approaches the positive number £(p) as t—oo, p,
is by its definition (21) asymptotic to e=°* u(£(p)). Hence, in
30), lim pr(zr- 25)=0 by (40), and (£,, %,) is optimal.
Too
Moreover, if (x, z,) differs from (£, #,), we must have x, #$,
for some à, because in the contrary case (36) and z,=2, would
imply z,=#, for all £. But then, by the attainability condi-Hon
(35 ¢), we have a strict inequality in (24) and, by (25),
a strict inequality in (30), hence (x,, z,) is not optimal. There-"41
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27C
fore (£,, £,) is uniquely optimal for the given 2,. This completes
the proof of Propositions (H), (I), (J).
For completeness we consider two additional questions. LI
(x,, z,) is eligible and attainable, we have from (2)
F.
mn
*F}
r
9,
q(z4)-“
dl 2
ar.
Pp, G(2) + psy) dt — py 2,
Examining the behavior of p, and br for t> one finds from
this formula that all the integrals occurring in statements (i
(ii), (iii) interpreting (F) and (I) converge for T—oc,
Finally, what rules out the trajectories in Figure 17 for
which x,#2%,? Those with x,>% reach the boundary z=o0 at
some finite time, making it impossible to satisfy both (65) and
(35 a) for all £Zo0. For each x; with o<x' <£y there is a
unique attainable path (x7, Z}) satisfying (65) for all #Zo, but
in such a way that lim x;=0. This must entail either the
t>o
ineligibility of (x} z;), or the unboundedness of p* associated
with that path by (21), because otherwise (30) with (xt. «+ #,
substituted for (£, 2,, p,) would imply the optimalitv of :
path (x3, 23) already proved nonoptimal.
A 8.
PROOFS FOR A NEGATIVE DISCOUNT RATE 1
We shall need the following lemma.
LEMMA 4. If ¢(x) ts a positive and nonincreasing function
of x defined for all x>o, and if x, is a positive integrable function
of t on the interval [T'. T2Y TY<T2. such that
+1 Koopmans - pag. 55
280
(67)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
o,
E>
here
EL ow
T,)E,
(T, —
ts
xpd
then
=
58)
Je
| (æ,) dt > —
> (T, TT
T,) 02%,
=)
Proof: We define
pr) = 7 = - measure of } {| T, <t<T, and a; <x
Then p(o)=o0, p(co)=1, and, from (67) and x,>o ,
0 oo 2E oo
69) E> TT | = NE dote) = [x dpa) “fe dp. (x) >
r, 1 0 2
> 0 + 2% (1 — p(2£))
Likewise, from the nonincreasing property of (x),
(70)
a a ( ) 2
ax x) du(x + (æ) du. x
| “ 2k oO
x T J o( 2) . 2,
Since, from (69), (28)= — , (70) implies (68).
"41 Koopmans - pag. 56
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Proof of (K). We again distinguish the three cases with
regard to the asymptotic range of z,, used in the proof of (B).
Case (1), £<&. In this case Proposition (K) is equivalent
to Lemma 2.
Case (2), {=8=ty#2. For definiteness assume Ç<2 and
let 2-¢=3e. Since now lim z,=G we can choose T such that
ta
(71)
10r
A
and at the same time large enough for there to exist an attainable
path (x}, 2;) on [o, T] such that z°=zr+e. For {=T we
choose (x3, 2) according to
(72)
x, + g(27) —2(z) for all t=T
Then, (x},2}) is attainable throughout, and from (42), (71),
for t=T,
(73) x" -
- g(a
ow LV ay a \~,
Ho 1878) * €
Hence for
(74) «Was (=,
p-WU...)
-U VUE)
rd 8
i (arty
(Xr —
£1
u(y) dt
On the other hand, by (36), (40 a), (71),
[4] Koopmans - pag. ,
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
73)
T4 T+
tai dt = ER 23 <(T*—1)2
provided T* - T=1 and §=%+2¢. It follows from (75) and
Assumption (e) that x7 and «(x*) when substituted for x, and
p(x) in Lemma 4 satisfy the premises of that lemma on the
interval [T, T*]. Hence, from (74), (68),
I A
Ws (~0) 2, n-u(22 + 4e) - (T* — T) eoT
from which (K) follows directly. The proof for $< ¢ is similar.
Case (3), &={=2. For any e>o, subject still to later
choice, there now exists an integer T such that
76)
5
e<z2,<g2+e¢ for t=T
It will be useful to write tWrs(- 0) as the difference of
two integrals
-
4 y;
r* T*
Ways (-0) = | eo ur) =) dt — fee ut) =) dt =
; ;
= Uns (—7) — i. § J (—0)
Taking first the second term we have, from (43 a) with x; = À.
(36) and (38 a).
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283
à a
plore —
Jy dt
(79)
For the firsy t
{ai =} which ©
(79a
| (790)
1
vw
choose an attainable path
Lr =((~ ,
| (79¢) T
where 23€ and the number T* T+1 are still subject to late.
choice. In addition, T should be sufficiently large thai, besides
(76), there exists an attainable path (x5, 2%) on [c. T1 sucw tha
2 =%+7.
To obtain a lower bound on the first term in (7) we note
that, in view of Assumption (c) and (38 9%, there cast numbers
1,>0 and y>o such that. whenever 71,
[8a
implies
—— AK
ae) 4
Hence, it 0< v=7, and «= wg(2+7,),, we have from (79 a
and (38 b), in analogy to (78),
4
Koopmans - pag. 5c
284
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
T*-1
(81) LUss_,(~0) = est (g(2+n) —g9(2))dt >— u,y n° 57* (es(T*-D_ goT) =
— 94! 2 ——4 96T*
> u yn Ge
For T* - 1=<¢<T*, x varies and must be boxed in.
0 ee, OCNÉN,, We have from (70 b), (38),
If
x*=min {g(t -¢), g3+n)-e x, <t+e +n = a
where €, 7, are chosen small enough to make #*>0. Then,
because # (x) decreases with x,
“=max uw @*), uo J=u' (x) > #(@*)=u">0 for T* - 1=<¢t<T.
We therefore have from (79 b), (80),
352)
+ Us (—5) = fe u (x?) (g(et) — 9) — 2#) dt >
T+_1
> —U' y n° e0T* Lu” - (a —3) eo TT
Pulling together these inequalities we have, for any T**=T*
from (79 c), (77), (78), (81), (82), since w'=u’ >i,
Wop (<6) > |—u' (yn? (1 +671) + 2:) +u'(n—e) e-s|eo1* = A esT* , say.
[4] Koopmans - pag. 60
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285
It is now possible, within the restrictions already imposed, to
choose first n and then € small enough to make A>o0, next to
choose T to correspond to € according to (76), and final:
given N >o, to choose T* large enough to make
Ws
y
! ‘
a.
-q-’
“>
.4| Koopmans - pag. 61
236 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
REFERENCES
ArLLais M., Economie et Intérét. Imprimerie Nationale, Paris, 1947.
The Influence of Capital-Output Ratio on Real National Income. « Econometrica
», October 1962, pp. 700-728.
Bauer P. T., Economic Analysis and Policy in Underdeveloped Countries,
Duke University Press, 1957, especially pp. 112-126.
Cass D., Optimum savings in an Aggregative Model of Capital Accumulation.
« Technical Report » No 5, Institute for Mathematical Studies in
the Social Sciences, Stanford University, Nov. 27, 1963.
CHAKRAVARTY S., The Existence of an Optimum Savings Program. « Econometrica
», January 1962, pp. 178-187.
DEBREU G., Representation of a Preference Ordering by a Numerical Function,
in Thrall, Coombs and Davis, editors, Decision Processes, Wiley,
New York, 1954, Chapter XI, pp. 159-165.
DESROUSSEAUX J., Expansion stable et taux d'intérêt obtimal. « Annales des
Mines », November 1961, pp. 31-46.
Harrop R.F., Towards a Dynamic Economics. Macmillan, 1948.
INaGAKT M., The Golden Utility Path. « Memorandum dated November 13,
1963, Netherlands Economics Institute, Rotterdam, 31 pp., mimeographed.
KanTorovicH L. V., Ekonomicheski raschet nailuchshego ispol’zovania resursov
(Economic Calculation of the Best Utilization of Resources), Academy
of Sciences of the U.S.S.R., Moscow, 1959 (French Translation
published by Dunod, Paris, 1963).
Koopmans T. C., Stationary Ordinal Utility and Impatience. « Econometrica
», April 1960, pp. 287-309. This issue was also published separately
ander the title Econometrica Essays in Honor of Ragnar Frisch.
On Flexibility of Future Preferences. In Bryan and Shelly, editors,
Human Judgments and Optimality, Wiley, New York, 1964.
Diamond P. A., and WiLLiamsoN R. E., Stationary Utility and Time
Perspective. « Econometrica », January-April 1964, pp. 82-100.
Pueres E. S., The Golden Rule of Accumulation. « American Economic
Review », September 1961, pp. 638-642.
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SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
Pucacuev V.F., O kriterii oplimal’nosti ekonomiki (Criteria of Optimality
for the Economy), in FEkonomiko-matematicheskie metody. Narodno
khoziastvennye modeli: Teoreticheskie voprosy potrebleniia, À … Vainsn
tein, ed., Moscow, 1963.
Ramsey F. P., A Mathematical Theory of Saving. « Economic
December 1928, pp. 543-559.
RoBiNsoN JoaN, 4 Neo-Classical Theorem. « Review of Economic
June 1962, pp. 219-226.
SrinivasaN T. N., On a Two-Sector Model of Growth. « Econometrica
July, 1964, pp. 358-373.
SwaN T., On Golden Ages and Production Functions. Memorandum for the
Round Table on Economic Development held in April 1960 in Gamagozi
Japan, under auspices of the International Economic Association” mimeo
graphed, 18 pp.
TINBERGEN J., Optimum Savings and Utility Maximization over Time. « Econometrica
», April 1960, pp. 481-489, issue also published separately as
Econometrica Essays in Honor of Ragnar Frisch.
Uzawa H., Optimal Growth in a Two-Sector Model of Capital Accumulation
« Review of Economic Studies », XXXI (1), 85, January, 1964, pp. I-24
VON WEIZSACKER C. C., Wachstum, Zins und Optimale Investitionsauot
Kyklos Verlag, Basel 1962, pp. 96
Worp H., À Synthesis of Pure Demand Analysis, Part
Aktuaritidskrift », Vol. 26, 1943, pp. 220-2623
Koopmans - pag. 63
alu
DORFMA}
{ feel very strongly moved to express my admiration ‘mn
Prof. KCOPMANS’ paper. As Dr. JoHNSON might have said, its pic
fundity is equalled only by its ingenuity. And it reaches some re
markable and exceedingly significant conclusions.
[t is important to point to two quite strong assumptions on
which the conclusions rest. One of them is that Prof. KooPMANS has
replaced the three factors of classical economics — land, labour and
capital — by a two factor model. He omits land. The technical
effect of introducing land into Prof. KooPMANS’ model would be to
change one of the essential mathematical functions. Since land is
in virtually fixed supply, it would no longer be permissible to think
of constant returns to scale to the two variable factors. I can only
conjecture the consequences of that change, but perhaps it would
strengthen the moral case for saving on the part of the present generation
because they will be better endowed per capita with the
third factor than future generations if population continues to grow.
The other strong assumption made by Prof. KooPMANS is that
capital does not depreciate. I think it was RICARDO who originally
defined land to be the original and indestructible powers of the
soil. We have since generalized this concept to mean the original and
indestructible power of anything that we inherit. The findings
of capital theory indicate that if either of those two adjectives is
4+] Koopmans - pag. 65
290 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the more important, it is « indestructible ». Whether a good possesses
original powers or not is a matter of the past, which is now
dead. From the forward looking point of view, the essential difference
between capital and land or natural resources is that land
does not require maintenance, but capital does, so that what
Prof. KooPMANS has been calling capital here might realiy come
closer to being a kind of land or what would be from the point
of view of future generations an increase in their supply of natural
resources that they inherit from both their forebears and from
nature itself.
Depreciation can be allowed for in this model by reinterpreting
equation (10 a), the basic feasibility equation on page II. We need
only think of Prof. KooPMANS’ À as being composed of A, +),
where À, is the rate of population growth and À, is the rate of
capital deterioration. With this amendment, would not this samo
analysis apply to an economy in which population grew and capital
deteriorated?
However this may be, the fact that capital depreciates does go
against the strong conclusions that Prof. Koopmans arrived at because
it means that an increment to the capital stock cannot be
infinitely productive because it will decay, and it may carry with it
a responsibility for permanent maintenance in the face of diminishing
returns and perhaps also of diminishing marginal utility in the
U functions. In that case, an economy can reach a state of capital
saturation in the sense that, although the utility per capita resulting
from an increase in the stock of capital does not fall to zero, it
falls so low that it does not exceed the social cost of maintaining it.
This consideration may lead to some changes in your conclusions:
and help avoid some of the problems created by the infinite horizon.
KO0PMANS
I agree to the comments made by Prof. DORFMAN that this exercise
simplifies matters a great deal by ignoring land and deprecia--tion
of capital. Land would undoubtedly become a p-oblem if
4] Koopmans - pag. 66
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20.
population were to increase to such an extent that just space on
which to exist would become very scarce, and it might actually become
a problem well before that. It seems however that technology
has so many possibilities for food production that are less landintensive
than the ones that are now being used, chat there might
be at least a temporary offset in technological progress.
As to the assumption that capital does not depreciate, I think this
is a difference in degree but not in kind. As Pro:. DORFMAN indicates
you can indeed reinterpret A as the sum A=), +X, where
)., refers to population growth and }, to capital depreciation
MALINVAUD
I should like to argue that Prof. Koopmans has been quite wise
in these kinds of simplifications. We are certainly not living in a
one commodity world in which capital would not depreciate. But
some of the difficulties of intertemporal choices will appear with
full clarity in very simple models. At the present stage of our
research we are therefore justified to study carefully and exhaustively
such simple models.
In particular it seems to me that difficulties occurring in more
complex models have not been fully understood in their natures,
because several sources of complications mixed their effects, and
the origin of each new result was not clear. From the point of view
that interests Prof. Koopmans, I do not think we should reach different
qualitative conclusions if we introduce many commodities, if
we assume that capital depreciates, or if we take into account the
fact that land is not reproducible
PASINETTI
I have very much enjoyed Prof. KcopMaNs' skilful, elegant anc
perspicuous analysis. Yet, I find it hard to accept his conclusions,
which seem to me best summarized by the title of his section
4 | Koopmans - pag. 67
292
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
{(« adjusting preferences to opportunities »). Prof. KoopMANs has
found that the traditional approach to optimal growth (which consists
of maximizing utility over time, by accepting a certain rate of
discount of utility, called p, as given by individual preferences) cannot
always be applied. More precisely, he has found that it can be
applied only when the time horizon considered is finite. When time
is allowed to run from o to oo, then p<70 becomes impossible
calthough p>o still remains possible) because there simply would not
exist a utility function to be maximized,
Thus — Prof. Koopmans concludes — the open-endedness of the
future imposes limits on individual preferences. He seems to be
so surprised and even so afraid of this result as to prefer, at this
point, to begin to speculate on the meaning of all this.
I would suggest that the mathematical exercise should be completed,
by allowing time to run from —oo to +oo (and not only
from 0 to +00). I may add perhaps that to consider time as running
from —oo to +oo does not mean allowing time to run in reverse.
It simply means putting ourselves in a slightly different position with
respect to the one Prof. KooPMANS has chosen. Instead of saying,
as he does: suppose we begin our process of maximization at time
zero, whatever happened before; we say: suppose that optimization
has been taking place since the beginning of time. (This, by the
way, appears to me a more logical approach to take in the context
of Prof. KooPMANS’ stationary society). Now, if we allow time to
run from —oo to +00, it is easy to see that, in a stationary economic
system, also p>>0 becomes impossible. The only value of p that
makes any process of utility maximization over infinity possible is
0=0.
Prof. KooPMANS might be even more surprised. For, by following
his arguments, we should conclude that individuals have not even
a limited inter-temporal preference choice: they have no choice at all.
But is it so? This conclusion — it seems to me — is fallacious,
although of course the mathematical results are correct. And the
fallacy stems from not bringing out explicitly the implications of the
following theorem: on the optimum growth path (by which I mean
4| Koopmans - pag. 68
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205
Prof. KooPMANS’ « golden rule path »), the rate of interest of the
economic system is determined independently of consumers’ utility
functions. This theorem follows from Prof. KooPMANS’ own analysis.
For, on the optimum path, the rate of interest must be equal to the
natural rate of growth (which, in the particular case of a stationary
economic system, is equal to zero).
But this simply means that individuals’ preferences are not a
determinant of the optimum rate of interest. It does not mean (as
Prof. KooPMANS seems to fear) that any restriction comes to be imposed
on individuals’ preferences. To argue otherwise would sound
to me rather similar to saying, in the usual case of a one-period problem
of utility maximization, that the fact that market prices are
the same for all consumers imposes restrictions on individual preferences.
Traditional economic theorists have solved this problem a
long time ago, by referring their analysis only to what happens at
the margin. Individual preferences are accepted for what they are,
however different from one individual to another they may be. Yet,
given these preferences, each individual will push the consumption of
each commodity to the point at which the ratios of marginal utilities
are equal to relative market prices. This means that we can make
definite statements about ratios of marginal utilities, without imposing
any restriction on utility functions.
Our case is similar. Consumers’ preferences may be quite different
at different levels of consumption, at different times, and for
different individuals. Any social preference function expressing all
these preferences may be equally different; and it must be accepted
for what it is, without any restriction. Yet, if behaviour is to be
rational, the consumption of each commodity will be distributed over
time so as to equate marginal intertemporal rates of substitution in
consumption to the externally given rate of interest. This is all we
can say.
Professor KooPMANS’ results have been obtained because he has
added something else. He has imposed on utility functions at all
levels (and not only at the margin!) the restriction that utility always
differs by ¢ at anv two adjacent points of time. This restriction is
4, Koopmans - pag. 69
204 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
28
arbitrary, and has no justification; and I would interpret the results
of his mathematical analysis as simply showing the impossibility of
such an arbitrary restriction. I should conclude, therefore, that the
misunderstanding has arisen from having introduced a p into the
analysis at all. For, this has meant introducing exactly what Prof.
KooPMANS has been afraid of, namely restrictions on consumers’
preferences.
KOOPMANS
I do not understand the operational meaning of Prof. PASINETTT’s
suggestion to maximize utility over a period from —oo to +oo. The
following comments apply therefore to maximization from o to oo,
although I may thereby fail to do justice to PASINETTI’S thought.
In the sentence in which Prof. PASINETTI refers to the « golden
rule path », he uses the term optimum in a sense different from mine.
[f an optimal path is defined as one that maximizes a utility function
of the type I have discussed, the golden rule path is optimal only if
both (a) the initial ratio of capital stock to labor force happens to
coincide with that characteristic of the golden rule path, and (b) the
chosen utility function has no discounting (p=0). If both these
conditions are satisfied, the golden rule path is optimal in my sense
as well, and as PASINETTI observes the interest rate p+ A equals the
exogenously given growth rate \ of the labor force. However, if even
only one of these conditions fails to hold, the optimal path, if one
exists, differs from the golden rule path, and the interest rate differs
from À most or all of the time, and is determined by the interplay
of preferences and production possibilities I have analyzed.
Finally, Prof. PASINETTT’s analogy with the one-period problem
of utility maximization misses the main point of my paper. In the
one-period problem with a finite number of commodities, an optimal
consumption choice is bound to exist if the utility function is continuous
(a slight restriction on preferences!) and the opportunity set
closed and bounded In the infinite-horizon case. there is a new
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mathematical situation, and the existence of an optimal program is
found to depend on a stronger restriction on the utility function used.
To facilitate analysis, I have studied this restriction only within the
class of stationary and additive per capita utility functions, expressible
as a sum of future per capita utilities derived from a constant
one-period utility function #(x) and discounted at a constant rate ç.
Within that arbitrarily chosen class, an optimal path is found to
exist if and only if g>0. The question of existence of an optimal
path is so far unresolved within the wider class of continuous but
not necessarily stationary or additive utility functions. It is plausible
to assume that within that class there will again be a subclass for
which, in a given technology, no optimal path exists
MORISHIMA
Your argument is based on the assumption that the rate ot
growth of population is constant. This, together with others, implies
the uniqueness of the Golden Rule path. Suppose, instead, that the
rate of growth of population is an increasing function of the consumption
per capita until it reaches a certain level, after which the
population growth-rate will decrease. Then there are possibilities
of multi-Golden Rule paths. You shall have to be concerned with
the comparison between Golden Rule paths (the best Golden Rule
path, the second best Golden Rule path and so on) and also with the
locality of the stability of the best Golden Rule path.
You treat capital and labour in an asymmetric way: capital may
be unused if too much capital is available, while labour is fully
employed throughout the whole process. May I say that a certain
degree of optimality has already been presupposed in your assumption
of automatic maintenance of the full employment of labour?
Is the full employment of labour maintained even if labour
is treated in the same manner as capital, i.e. if there is a possibility
of unemplovment of labour?
Koopmans - pag. 71
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KooPMANS
I want to thank Prof. MORISHIMA for the information he gave
us for a model in which the rate of population growth is a function
of consumption per head, initially increasing and thereafter decreasing.
This is, 1 think, a realistic generalization that it is important
:o have.
On his second point, the asymmetry in treating capital and labour
with regard to disposal, I would agree that that is something not very
consistent in the paper as presented. I do not think it has affected
any of the results in any of the optimal paths being considered —
paths in which at all times the capital stock is fully used as much as
the labour force (1).
MAHALANOBIS
I am very deeply impressed by the approach and the broader
results of this paper. It seems to me that although a very simple
model has been used on, if you like, somewhat intuitional grounds,
the conclusion seems to be extremely important from the point of
view of the underdeveloped countries. The conclusion to which I
am referring is that neutrality regarding timing between generations
is not possible, the very nature of the process of development or of
industrialization discriminates in favour of future generations. Thus
I am taking to be a basic point, on page 28, the argument that accumulation
of capital, permitting a higher output of consumption goods
in later years must discriminate in favour of future generations, The
broad conclusions may have important educative effects regarding
programmes in underdeveloped countries.
I am not going into details of technical arguments, but suppose
nstead ofthe one commodity model we have three types of capital
() Note added after the conference: In the corrected version of the
paper printed in the volume, disposal of capital use has been excluded.
11 Koopmans - pag. 72
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goods producing three types of consumer goods; the basket changes
but there need not be any change in the total aggregation of capital
goods which goes on increasing. The one-commodity model seems to
me to be illustrative, and the conclusions may remain valid even
if we have many commodities. This question, of course, cannot be
settled without going through the full exercise of rigorous analysis.
There are three points which have been summarised in sections
5 and 6. The point about open end I think has been clearly brought
out and is logically valid.
[ am repeating the question again: whether one commodity or
many commodities would affect that result. Theoretically some preference
functions may change. The order with which Prof. Koop-MANS
started, that pattern of ordering, may also change. Bu: I
think Prof. KooPMANS took care to point out that some kind of
good ordering is all that he was keeping in mind.
A further point which is discussed on page 32, that it is likely
that technological advance would continue; even if it slows down,
even then, from the point of view of the present generation, the
discrimination would be in favour of future generations. The question
of basic decision is not whether the hundredth generation from
now would be in a somewhat less favourable position; the decreasing
return in terms of generations may be there even if the technological
advance continued at the same rate. I do not see this as a very
important consideration for present decisions.
This particular paper seems to me to be of value because of
the wider implications of this Study Week. It is not a Study Week
arranged by a specialist econometric society but by the Pontifical
Academy of Sciences. My interest, as I have continually stressed,
's in the broader implications. And from that point of view I welcome
this paper as likely to have a very valuable educative effect.
It is more difficult to speak on the question of population size.
[t may differ from country to country. Prof. KooPMaNs has pointed
out that where the density of population is extremely small, there
may be some advantage in increasing their number, but where the
density of population is high the position may be different
4+) Koopmans - pag. 73
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KOOPMANS
I share the beliefs expressed by both speakers, Prof. MAHALANOBIS
and Prof. MALINVAUD, that the one-commodity study, one-commodity
model does a service here in isolating just those problems that come
from the open-endedness of the future. Considering those apart from
or complications which arise in two- or more-commodity models,
I used the term « belief » because I do not feel that a stronger term
can be used at this point. One would want to go on to two- and
more-commodity models to examine whether that belief is valid but
this is my present state of espectation.
On the question that Prof. MAHALANOBIS raised about flexibility
in the ordering, that idea is merely mentioned at the end of my
study. I have given some thought to the possibility of working from
an ordering which has built into it the possibility of its own revision
at a later time, and more particularly an ordering in which the
decision maker is willing to make a present sacrifice of immediate
satisfaction in order to leave open more doors in the future. He may,
for instance, prefer to build a capital stock of such a nature that it
can be applied to a wider range of different kinds of production,
even though that kind of capital stock would not be the most economical
one if the composition of production in the future were already
fully specified. For somewhat longer-range planning this seems to
me a matter of practical importance. My speculations on this are in
a reference at the end of my paper, identified by the year 1962, and
‘hey did not carry very far. I believe that from the formal point
of view it is a difficult matter to formalize this idea. I do find that
the need for such an idea is strong enough to justify further formalizing
effort in that direction.
FISHER
Prof. KooPMANS has as always presented us with an illuminating
and beautiful piece of work. I am, however, disturbed a bit at the
conclusions that he wants to draw from it. He has shown that there
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may be severe conflicts between our ethical notions and the analysis
of certain kinds of growth models. His conclusion is that we must
change our ethical notions. My preference would be to say that there
is something wrong with the model. In either case a good deal of
further thought seems called for. More particularly, one frequently
in the theory of optimization over time encounters peculiar difficulties
when one uses an infinite time horizon. Now, in fact, the device of
the infinite horizon was probably originally introduced because the
choice of a finite horizon is an arbitrary one and because with a finite
horizon one has difficulty in deciding what to do about terminal
capital stock. Infinite horizons were, however, introduced primarily
as a convenience. They have in several contexts now been shown
to lead to difficulties all associated with divergence of the improper
integral obtained in the problem. Now this suggests to me that
infinite horizons are not in fact the convenience they appear. The
obvious conclusion from KooPMANS’ paper, therefore, seems to me
to be that one ought to abandon the use of infinite horizons — not
that one ought to abandon certain ethical notions.
Now, of course, this may be wrong. It may turn out — and
Prof. KooPMANS assures me that it does — that even with a finite
horizon one has similar problems which are not so severe. In that
case, it may not be worth dropping infinite horizons. Still, the role
of this sort of analysis is surely to tell us how one can best achieve
one’s ethical and social ends. In the course of analyzing that, it
may turn out that such ends are unachievable. In such a case, one
has to moderate one’s ends. The usual circumstance, however, is
that one’s ends are not achievable in the sense of being contradictory
whereas Prof. KooPMANS has shown that our ends may be unachievable
in the sense that no solution to the problem exists — that the
whole analysis breaks down if one insists on certain kinds of ethical
goals. This sort of circumstance does not persuade me to give up
my ethical goals, but rather to refine the mode of analysis. I can
understand that the end result may be that I will have to give up
certain goals as unachievable, but the demonstration of that ought
i Koopmans - pag. 75
300 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 78
to be that they are unachievable and not that the problem becomes
ill-defined.
Despite the fact that I am unable to agree completely with Prof.
KooPMANS’ conclusions, however, I should like to state that his paper
like that of Prof. MALINVAUD is a pleasure to read and to listen to.
The two papers set a standard that one wishes all the other papers.
and discussion in this Study Week had met.
4] Koopmans - pag. 76
CROISSANCES OPTIMALES DANS UN MODELE
MACROECONOMIOUE
E. (MALINVAUD
Institut National de la Statistique et des Etudes Economique:
Paris - France
ORAL PRESENTATION (*
I may be brief on the motivations for this paper after what
was sald by Professor Koopmans. Like him, I wanted to
understand better the logical problems raised by choices between
intertemporal program and, in particular, to deal, as
rigorously as I could, with the case when the utility is defined
over an indefinite future. Moreover, I wanted to explore the
relations between various particular models, proposed by RAM-SEY,
TINBERGEN, RADNER and SRINIVASAN, all dealing with the
choice of optimal programs. but starting from different hypotheses.
In this oral presentation, I shall briefly survey the various
sections of mv paper.
After a short introduction in section 1, section 2 defines the
model. Like Professor Koopmans. I am dealing with a world
*) Le texte de la présentation orale est reproduit à titre de résumé en
langue anglaise du mémoire nresenté a nartir de la page aq.
5
Malinvaud - pag.
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
=
in which there is just one good that can be used either for consumption
or as capital in production. Notwithstanding differences
in notation, my model is very similar to his. I am, however,
considering a discrete time represented by the index
t=o0, 1, 2 ... ad infinitum.
C, is aggregate consumption during period ¢, i.e. from time
t to time £+1. Similarly N, measures the labor services provided
in period {. Equation (1), page 11, defines per capita
consumption ¢, and per capita input of labor #,. Capital at
time ¢ is denoted as K,, and the stock of good available at
time ¢ before consumption as S,=K,+C,. One constraint of
‘he model specifies that the initial stock is equal to a given
number S, Another constraint results from the production :
function, namely equation (6).
Among all programs which are feasible, I am using the
same kind of utility function as Professor Koopmans does,
except that, in some parts but not everywhere in my paper,
[ am taking the per capita labor input as an argument of the
utility function, thus allowing the amount of labor services
provided to be determined simultaneously with consumption
by the choice of the optimal program.
In order to make possible a choice among infinite programs
[ am not using the technique presented by Professor KooPMANS,
out relying on the criterion given by my definition 1, page 17,
namelv :
A feasible program @! is optimal if there is no value of T
and no other feasible program Æ such that the inequalities (14)
and (15) be simultaneously fulfiled.
Taking this as a definition, I am avoiding considering infinite
sums that might not converge.
Section 3 is devoted to the determination of sufficient conditions
for a program to be optimal. At this stage the model
remains general except for some assumptions on the production
and utility functions, notably that they be concave and possess:
partial derivatives.
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In the derivation of these conditions, inequality (21),
page 20, plays an essential role. It permits a comparison between
a feasible program # and another feasible program
fo+¢ : it might be written as:
SU.
= Apr at LS ;
Lon trang veo Ngo - 72, 2S,
T_
%, 8; and Ar depending on the values of the variables in M.
The coefficient Jr being non-negative, / is optimal if the
2's and B,’s are equal to zero and if & S,=o for any T and
any feasible program #+¢ Æ such that t>T would imply
> U,zo.
This is essentially what proposition 1 amounts 2. I first
define as « regular » a feasible program for which the &,s
and $s are all zero; this is equivalent to requiring that the
equations (22) be fulfiled. I then specify a condition 1 that
automatically implies the condition quoted at the end of the
preceding paragraph. Condition 1 requires that, in Æ, the
marginal utility of consumption be positive at all times and
that there exist a number % larger than 1 such that, at least for
large ¢:
2
>
fx being the marginal productivity of capital at time ¢ Froposition
1 states that a regular program that satisfies condi
tion 1 is optimal.
Let me point out here that any regular program would appear
as optimal if time were restricted by a finite horizon T and
if the values of both S, and S- were taken as boundarv cons
Yi
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37
304 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
traints. Condition 1 is specific to the consideration of unlimited
programs. We shall have occasions to see in the following
examples that it is not vacuous.
We may also observe that the equations «,=o0, §,=0 imply
together with the equilibrium condition at time # a set of difference
equations which may usually be solved recursively for
all C,, K,, N, starting from any given allocation of S, between
Co and K,. Thus, the regular programs generate a one parameter
family of programs, the parameter being C, (or equivalently
Ko, S, being given). Typically, one and only one
program of this family will meet condition 1 and therefore be
optimal. A possible computational procedure would be first to
determine the regular program corresponding to any fixed
initial allocation, and then to find by trials and errors the initial
allocation that leads to fulfilment of condition 1.
Before turning to specific examples, we may look at the
economic interpretations of equations (22), stating that the «,’s
and §3,’s are all zero, and of condition 1.
The first equation (22) may be written as:
34) I+fæ = (1+e) (1+T,) (I+U,)
f= being the marginal productivity of capital, or the equilibrium
rate of interest, e the social rate of interest used for the
discount factor of the utility function, =, the rate of increase
of the population, and u, the rate of decrease of the marginal
ut'ity of consumption. This equation exhibits in a suggestive
way the three components which explain the equilibrium rate
of interest. It is similar to relations recently put forwards by
Sir Roy HARROD and Professor RAGNAR FRISCH.
Condition 1 implies that the equilibrium rate of interest be
larger than the rate of increase in the stock of good S,, and therefore
that the present value of S, decreases to zero with £.
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In section 4, I present an application of the above analysis
to an aggregate version of the model recently built bv Roy
RADNER.
Population and labor input increase at a constant rate.
The production function is such that S, appears as a CoBs-DoucLas
function of N, and K, with neutral technical progress
expanding at a constant rate. The period utility function is
proportional to the logarithm of c,.
In this model, the recurrence equations (22) can be solved
explicitly to give equation (43). The regular programs are
thus determined. For instance, the diagram here shows the
evolution of K, for three programs satisfying equation (43)
but starting from different initial allocations, the consumption
C, Increasing when we pass from program r to programs ?
and -
Programs 1 and 2 are feasible and therefore regular.
Program 3 is not feasible since K, becomes negative after
some time. Program 2 is definitely prefered to program 1 be-17
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306 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - oR
cause it gives a higher value for C, at all times. However, if
the total period considered were restricted to a horizon T as
shown in the diagram, programs 1, 2 and 3 would appear as
optimal, each one with respect to an appropriate terminal condition
on Ki.
If the social interest rate €, used in the definition of the
utility function, is positive, then condition 1 is satisfied by
the regular program @” that gives the highest value to Co. This
program #7 is therefore optimal. Its optimality may still be
proved directly for the case e =0. But, when the social rate of
interest € is negative, no program is optimal.
For instance, program #2 is preferred to the program #!
exhibited on the diagram. However a better program can be
found as follows when e<o : up to time T, select a regular
program close to # but allowing a little higher value of K,,
and therefore a little smaller value of C,; at time T take an
extra consumption by reducing Kr to its value in #2, thereafter
continue with program #. The program thus defined is
not optimal either, because one would prefer to postpone ever
farther in the future the time T at which one switches back
to 4?
In section 5, I consider the case in which the production
function would simply imply a fixed capital-output ratio. The
determination of regular programs then boils down to the solution
of a recurrence equation on c,.
Choosing a type of utility function proposed by R. Frise
and more recently used by J. TINBERGEN, I find that an optimal
program exists only when the social rate of interest ¢ exceeds
a positive minimum that may be of some 10% per annum.
This suggests that, in the programming of future development,
one can hardly avoid taking into account the decrease in the
marginal productivity of capital, unless one discount heavily
against the future.
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Section 6 is devoted to the case of a linear utility function,
a case that is often considered, for instance when one chooses
to maximize a discounted sum of the future consumption
stream.
The formulas found in section 3 are no longer applicable
as such, because the solution of the recurrence equations leads
to non-feasible programs. However, the general approach can
be maintained if one introduces upper and lower bounds on
consumption per head c, and labor input per head #,.
Depending on the values of the parameters, the shape of
the optimal program varies a great deal. I consider precisely
a few cases which may be of some interest. For instance, if
the initial endowment of capital is small and if leisure has some
value even when consumption is at its minimum, the optimal
program may imply that the labor input be not pushed at its
maximum in the first periods, but be increased progressively
as capital accumulates, consumption being nevertheless kept at
its minimum until a sufficiently high capital stock has been
reached (see figure 5 in the text).
Notwithstanding the fact that it permits interesting insights.
the assumption of a linear utility function is not quite satisfactory.
In all cases, the optimal program exhibits some discontinuities
in its time shape, discontinuities that go against common
sense. For instance, consumption may switch in one
period from its minimum to its maximum value.
This unsatisfactory feature is partly due to the simplifications
made in the model, notably to the assumption of a onecommodity,
one-sector world, and to the assumption of independence
among the utilities for different periods. However,
optimal programs in more elaborate models would present similar,
even though less extreme, discountinuities as long as
linear utilities would be assumed. The unescapable conclusion
seems to be that. for the problems considered here, one must
Malinvaud - pag.
7
308 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
take into account the decrease in the marginal utility of consumption.
[ end up, in section 7, with the model first studied by
RAMsEY. I there show that my treatment with discrete time is
really quite similar to the one proposed by RAMSEY for a continuous
time model, and to the one exposed by Professor Koop-MANS
a moment ago.
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NTRODUCTION
La programmation économique nous amène souvent à considérer
des problèmes d’optimisation portant sur plusieurs périodes.
En particulier, tout choix d’investissements suppose la
prise en compte des conséquences futures de nos décisions présentes.
Les modèles économétriques qui servent à éclairer ce
choix font donc intervenir un nombre plus ou moins élevé de
périodes.
Or il subsiste d’importantes lacunes dans l’étude théorique
de ces problèmes d’optimisation intertemporelle. Par manque
de principes bien établis, le praticien qui doit choisir et résoudre
un modèle hésite souvent sur les formalisations et sur les hypothèses
simplificatrices à retenir. Il connaît mal, notamment, le
caractère plus ou moins limitatif des conditions terminales qu’il
est amené à poser.
Le présent mémoire a pour objectif d’aider à l’édification
d’une théorie qui puisse servir de guide aux travaux de programmation.
Afin d’éliminer à ce stade des complications qui
ne sont pas inhérentes à l’optimisation intertemporelle, je me
contenterai d’une formalisation purement globale dans laquelle
existera un seul bien produit. En revanche, j'introduirai une
suite illimitée de périodes de manière à n’imposer a priori
aucune condition terminale et à éviter ainsi un certain arbitraire
(!) Les recherches qui ont abouti à ce texte ont fait l'objet de conférences
à l’Ecole Nationale de la Statistique et de l'Administration Economique
ainsi au’à l’Ecale Pratinue des Hautes Etudes en mai et juin 1063
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310 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Je rechercherai d’abord une définition et une caractérisation
des croissances optimales. Puis je présenterai une méthode qui,
au moins en principe, peut suffire à la détermination de la
croissance optimale. J'examinerai ensuite quelques cas particuliers
qui ont été envisagés dans la littérature économétrique.
Je montrerai alors qu’on ne saurait se contenter de formulations
dans lesquelles soit la fonction d’utilité, soit la fonction de
production seraient linéaires. Je-traiterai enfin de deux modèles
proposés respectivement (!) par R. RADNER (1062) et F.P. Ram-SEY
(1028).
2. LE MODELE
Soit une collectivité n’effectuant aucun échange avec l’extérieur.
Son développement futur est décrit pour une suite de
périodes séparées par les instants #=o, I, 2 … ad infinitum.
L'instant #=0 correspond à l’époque actuelle, ou plus précisément
à l’époque initiale d’un plan de développement. Par convention,
la période # est celle qui s’écoule entre les instants
Let /+T.
L’évolution future de la population est considérée comme
une donnée exogène, résultant de perspectives démographiques
indépendantes de la croissance économique. Soit P, Ja population
prévue pour l’époque ¢.
Désignons par N, la quantité de travail à fournir pendant
la période ¢, et par C, la consommation qui sera effectuée du-)
Les références sont rassemblées à la fin du mémoire p. 78.
CD
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rant cette période. Le travail d’une part, le bien produit et
consommable d’autre part constituent les deux seuls biens introduits
dans le modèle. Ils sont l’un et l’autre parfaitement homogènes.
Dénotons par n ' -. le travail et la consommation par
personne, sc’
Nous admettrons l’existence d’un maximum nm à 1a quar
tité moyenne de travail par personne, et celle d’un mi:'mum
C, à la consommation par personne. En d’autres termes, nous
admettrons que la croissance future doit satisfaire aux inégalités
suivantes
J
+
J
nu et c,, étant deux nombres non négatifs donnés.
Soit K, le capital à l’instant #, c’est-à-dire la quantité du
bien produit qui est maintenue dans le système productif pour
participer à la production durant la période £#. Soit Q, la production
nette de cette période; O, est disponible à l'instant
L+I. À cet instant, l'équilibre entre les ressources et les emplois
pour le bien produit et consommable s’écrit
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312 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
À l’instant o, le stock disponible du bien consommable est
donné. Désignons le par S, et écrivons la condition d’équilibre:
3.
Pour représenter les contraintes techniques sur la production
de la période #, nous admettrons l’existence d’une fonction
de production (1):
‘6)
Q,=f, (N, , K,)
f, étant une fonction donnée qui a priori peut varier d’une période
a une autre. Nous supposerons que f, est définie pour
toutes valeurs non négatives de N, et de K,. Dans la suite,
nous n’aurons pas besoin d’introduire Q, explicitement. Nous
remplacerons les égalités (4) et (6) par la suivante:
7)
C,.,=K.-K, +f (N,, K).
Nous appellerons « programme » ¢ une spécification des
valeurs données aux grandeurs N,, C,, #,, c, et K, pour les
instants successifs ({=o, I, 2 .….). Nous repèrerons différents
programmes par des indices supérieurs. Ainsi #! sera le programme
correspondant aux valeurs N!, Cl, n!, ¢l et K!,
(") Cette représentation des contraintes techniques est évidemment quel-Jue
peu restrictive. Avec un modèle dans lequel le nombre de biens serait
quelconque, on pourrait représenter les produits intermédiaires et les produits
en cours de fabrication par des biens particuliers. Définir les contraintes
techniques à l’intérieur de chaque période n’introduirait alors aucune
restriction réelle. Il en va différemment avec un modèle purement global. La
formulation retenue ici suppose que la production de la période ¢ est disponible
en totalité à la fin de la période et pourrait éventuellement être consommé.
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Nous dirons qu’un programme est « possible ~ . :l satisfait aux
contraintes définies par les conditions oY 15)
et par la suivante:
‘g
Le choix d’un plan de développement se traduit dans ce
modèle par la détermination du programme qui peut être considéré,
en un certain sens, comme « le meilleur de tous les
programmes possibles ».
Pour donner une signification à cette expression, définissons
tout d’abord « l’utilité » du programme pour la période £.
Nous admettrons que cette utilité dépend seulement de la consommation
et du travail par personne durant la période en
question (!); ce sera une fonction U,(c, n,) qui, en principe.
pourra varier d’une période à une autre.
Sous une réserve qui sera examinée ci-dessous, l’utilité du
programme entier sera définie par l’expression :
,
~
“A
dans laquelle y est un nombre positif, constituant un facteur
d’escompte grâce auquel sont combinées les utilités U, relatives
aux différentes époques. Nous l’appellerons « facteur d’escompte
normatif » et nous lui ferons correspondre un « taux
d'intérêt normatif » = défini nar le relation
1
(') Les préférences relatives à la période à seront ainsi indépendantes des
valeurs prises par la consommation et la quantité de travail pendant les
périodes autres que #. La nature de cette hvonthèse a été précisée par G Dr
RRFIT
Malinvaud - pag. 13.
314
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
Avec la formulation générale présentée ici, l’introduction
du facteur d’escompte y est évidemment inutile puisqu’une
modification dans la définition des U, ramènerait l’expression
(9) à une autre semblable dans laquelle y‘ aurait disparu.
Mais la forme (9) facilitera l’application des résultats généraux
au cas dans lequel la même fonction U(c,, n,) sera retenue pour
représenter les utilités relatives aux diverses périodes. Seul ce
cas sera examiné dans les applications qui seront faites du modele
général (1).
L'introduction d’un facteur d’escompte différent de 1 peut
d'ailleurs sembler contestable pour des raisons plus fondamentales
qui prennent tout leur sens quand la même fonction
U(c,, n,) s'applique à toutes les périodes. Sans doute convient-il
d'ailleurs de distinguer deux situations différentes.
S'il s’agit de décrire la croissance d’une société libérale, la
fonction d'utilité doit représenter à l’échelle macroéconomique
le résultat des choix individuels (2). Depuis BôOHM-BAWERK il
est admis, comme une loi psychologique, que les individus ont
une préférence naturelle pour le présent et qu’ils attribuent
dans leur choix une pondération d’autant plus faible aux époques
futures qu’elles sont plus éloignées dans l’avenir. Si nous
admettons cette thèse, qui n’a d’ailleurs jamais été établie de
façon parfaitement convaincante, nous devons naturellement
introduire dans la fonction d’utilité un facteur d’escompte Y
plus petit que un.
Mais il s’agit ici de décrire les choix collectifs d’une économie
planifiée et on peut trouver moins de raison à la présence
d’un facteur d’escompte. On a fait valoir que l’utilité intervient
() Une utilité de la forme
©
SY Ule, un’
:=0
a été dite « stationnaire » par T. KooPMANS (1960) qui a discuté la nature
des préférences représentables par des fonctions de ce type.
(? On connaît la gravité des problèmes d’agrégation qu’une telle représentation
pose.
[5] Malinvaud - pag. 14
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 313
seulement pour les décisions de l’époque initiale; d’autres décisions
seront prises dans l’avenir qui se traduiront par une
révision continuelle du programme choisi à l’origine. Il peut
dès lors sembler naturel que l’on attribue dans les choix présents
la pondération la plus élevée aux conséquences immédiates,
qui ne pourront guère être évitées. Cette argumentation
ne semble cependant pas très forte. On peut préférer une éthique
collective selon laquelle toutes les époques et toutes les générations
recevraient la même attention dans les choix actuels.
Notre formulation ne l’interdit évidemment pas. Il suffit de
prendre yY=1 dans l’expression de IU.
Certains économistes estiment que, pour combiner les utilités
relatives aux diverses périodes, on doit tenir compte des
effectifs des populations qui en profiteront. Ces économistes
préfèrent à l’expression (09) la suivante:
1
(Comme ci-dessus, la distinction n’est réelle que si la même
fonction U(c,, n,) est retenue pour les diverses périodes). Le
choix entre (9) et (II) pose une question d’éthique sociale sur
laquelle il est difficile de se prononcer a priori. Le lecteur qui
a une préférence pour l’expression (II) pourra adapter sans
peine les résultats présentés ci-dessous. Nous nous en tiendrons
dans la suite à des fonctions d’utilité définies conformément à
la formule (9).
Toutefois cette formule ne serait satisfaisante que si était
établie la convergence de la somme infinie qu’elle fait intervenir.
La convergence est assurée dans certains cas présentant
de l’intérêt: par exemple si y est plus petit que 1 et si les fonctions
U, sont uniformément bornées, inférieurement par un
nombre U_ et supérieurement par un nombre Uy. En revan
9
Mahnvaud - pag. 15
316 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
>?
che, cette convergence n’est pas réalisée pour divers cas considérés
dans la littérature.
Aussi devons-nous éviter d’introduire l’expression (9) telle
quelle dans la définition des programmes optimaux. Pour tourner
la difficulté, la seule solution consiste sans doute à limiter
la comparaison à une période finie, celle s’écoulant de l’instant
0 a un instant horizon T. On considèrera alors l’indicateur
d’utilité :
(12)
T—1
U = > y" U, (cs ’ n,)
t=0
et on s'imposera de ne comparer un programme donné qu’à
certains programmes qui lui sont au moins équivalents pour
chacune des périodes à partir de l’instant T.
La méthode la plus simple, et la plus courante, consiste à
ne comparer que des programmes absolument identiques à
partir de l’instant T. C’est ce que l’on fait en particulier quand
on fixe a priori la valeur terminale du stock de bien consommable,
soit:
13)
S,=K,+C, .
Mais le caractére peu satisfaisant de cette méthode est bien
connu. Elle ne donne aucune garantie que ce stock Sy soit
approprié pour le développement économique après l’instant T.
Aussi retiendrons-nous ici un principe plus souple. Nous
comparerons un programme donné à tous les programmes qui
donnent les mêmes valeurs aux utilités U,(c, 7,) des périodes
postérieures à T. D'ailleurs il n’y a aucune raison a priori pour
se fixer un horizon T particulier. C’est pourquoi nous adopterons
la définition suivante pour les programmes optimaux.
51 Malinvaud - pag. 16
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Définition 1. Le programme #* est optimal s’il est possible,
et s’il n'existe aucune valeur T et aucun programme possible #
tels que d’une part:
14)
U
4!
co
a
+, rc, LE, » 71,
a
lu
“gt
l,
et d’autre part:
1-U,
(c,, n,)
- 1 1
> U, cu, > M,
pour tout :
En somme, un programme possible est dit optimal si on
ne peut l’améliorer sur toute période finie sans en réduire l’utilité
à un moment au moins après cette période.
Il est clair que, si la somme infinie de l’expression (9) converge
pour tous les programmes possibles, et si un programme
possible particulier maxime Ü, alors ce programme est optimal
au sens de la définition qui vient d’être donnée: À priori, nous
n'avons pas la certitude qu’un programme optimal maxime 7
quand la somme infinie converge. Si cette propriété paraît
souhaitable, il faudra l’établir dans les cas particuliers considérés.
Pour déterminer les programmes optimaux, nous allons encore
poser une hypothèse générale sur les fonctions de production
et d’utilité.
Hypothese 1. Les fonctions f, (N,, K,) et U, (c,, n,) sont
des fonctions concaves (1). Elles ont des dérivées premières
(') Une fonction g(x, y) est dite concave si quels que soient les nombres
x', 2%, y' et y? et quel que soit le nombre x compris entre zéro et un. l’inégalité
suivante est satisfaite
gla + (i + 25 (1a) YI > aga, y- x) \
Si elle existe, la matrice des dérivées secondes d'une fonction concave
négative semi-définie
Malinvaud - pag. 17
318
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
8
que nous désignerons respectivement f, , f,., U,. et U,.
La dérivée U’,. n’est jamais négative.
Supposer l'existence de dérivées dans le modèle macroéconomique
considéré ici n’introduit aucune restriction sévère. La
concavité des fonctions de production et d’utilité est une hypothèse
habituelle. On sait qu’elle exclut les cas dans lesquels.
les fonctions de production impliqueraient des rendements
croissants.
Le modèle qui vient d’être défini est extrêmement simple.
[I] comporte cependant deux biens: le travail et le bien consommable,
biens dont les quantités ne sont pas données a priori
mais doivent être déterminées quand le programme optimal est
recherché. En un certain sens, ce modèle n’est donc pas purement
global. Son étude peut déjà faire apparaître les problèmes
que soulèverait la solution de modèles moins agrégés.
Par ailleurs, il peut sembler assez adéquat pour la première
phase des études dans la recherche d’un plan de développement.
Le problème consiste bien alors à déterminer, dans leurs grandes
lignes, les évolutions futures du capital, de la production,
de la consommation et de la durée du travail.
Néanmoins, de nombreux modèles considérés dans la littérature
sont encore plus simples en ce sens qu’ils considèrent
comme une donnée exogène l’évolution de la quantité de travail
N, ou même qu’ils ne la font pas intervenir du tout. La
solution de notre problème s’en trouve grandement facilitée.
Pour le montrer, nous examinerons aussi un modèle général
simplifié dans lequel le taux d’activité n, sera une donnée exogène.
La fonction de production pourra être écrite simplement
f; (K,) et la fonction d’utilité U, (c,). Nous remplacerons alors:
l'hypothèse I par la suivante:
Hypothèse 2. Les fonctions f, (K,) et U, (c,) ont des dérivées
premières f', et U’, jamais croissantes. La dérivée U’, n’est
jamais négative.
5, Malinvaud - pag. 18
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3. CARACTERISATION ET DETERMINATION DES CROISSANCES OPTTI-MALES
En vue de dégager une méthode pour la détermination des
croissances optimales, nous allons rechercher certaines conditions
suffisantes pour qu’un programme soit optimal. Nous
dégagerons de la sorte des propriétés de certains programmes
optimaux, propriétés auxquelles nous pourrons donner une in.
terprétation économique.
Etudions tout d’abord quelles relations existent entre deux
programmes possibles. Désignons par N,, C, ... etc. .. les
valeurs des différentes grandeurs dans le premier programme
et par N,+2 N, C,+& C, ... etc. ... leurs valeurs dans 1- -
cond programme. Tenons compte de l’hypothèse 1.
En vertu de la concavité (!) des fonctions U -‘ ¢ nous
pouvons écrire les inégalités suivante--IH)
+s
w
17)
‘1 Pour tonte fonction
Dans l’'inégalité on:
effet x!1= y +5 -
CONCavr
~
sivable, on
concavité de g(x, y) prenons ei
L’inégalité ‘écrit alors:
311 encore *
Tr
>
La relation annoncée s’obtient comme limi.
nombre ~ tend vers 7érn
récédente auand
fF
Malinvaud - pag. 1a.
320
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Reportées dans 1’équation (12) définissant Uy et dans la condition
d’équilibre (#7), ces inégalités impliquent:
(18)
T-1 at
3U, = 3 SU, 5C+ U, 3N|]
~
(19)
8C,., =f,.8N,+ (1+f,) CK -¢K, ,
De plus, la donnée du stock initial S, implique:
(20)
à C,= -à K, .
En reportant les relations (19) et (20) dans l’inégalité (18) et
en tenant compte de ce que U”, ne peut pas être négatif, nous
obtenons:
+4)
5U =3 |Tv, Ti
T= “tl P, te p_ Ute (t+/x) | ôK,
y ,Ç yet , ; | T ;
- Iz u, +20, Tad sNy— JU, 35,
formule dans laquelle Sy est égal 4 Kr+ Cr et représente le
stock de bien consommable 4 1’horizon T.
L’inégalité (21) s’applique quelles que soient les valeurs des
à K,, à N, et de 5 S,. Pour que le programme # soit optimal,
51 Malinvaud - pag. 20
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391]
il suffit que le second membre de (21) ne puisse pas être positif*
c’est-à-dire gue. pour tout T:
I) les coefficients des à K, et des à N, dans (21) soient nuls:
2) et qu’il n’existe aucun programme possible donnant, a partir
de T, des valeurs aux utilités U, au moins égales a celles
que donne #. mais comportant un stock S; plus faible.
Notons ici que.la condition (1) seule suffirait si nous avions
fixé le stock terminal Sy. La condition (2) est propre à l’étude
des programmes à l’horizon illimité. Négligeons la pour le moment;
et étudions ce qu’implique la nullité des coefficients des
> K, et des & N,.
La condition (1) s’écrit:
!
I+
t+l,e ~ -
K
“1 +
r
rT ory
— J
te
22)
P,.. ’
| n= - (x + e) P U,
égalités dans lesquelles intervient le taux d’intérêt normatif €
défini par la formule (10).
La condition d’équilibre (7) et les deux égalités (22) peuvent
être considérées comme constituant un système d’équations
de récurrence sur les grandeurs C, N, et K,. Plus précisément,
les trois égalités en question peuvent généralement être
résolues pour donner N,, C,, et K,, , en fonction de C, et de K,.
Pour chaque valeur donnée C,, et pour la valeur correspondante
So - Cy de K,, il existe donc en général des suites de valeurs
C, N,, K, satisfaisant les équations de récurrence.
Nous pouvons conjecturer qu’un programme optimal est défini
par celui qui, de tous les programmes possibles satisfaisant
les équations de récurrence, comporte la plus grande valeur de
"s] Malinvaud - pag. 21
322 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 23
la consommation initiale C,. Nous allons en effet obtenir une
propriété assez analogue. Pour en préciser le sens, posons
d’abord la définition suivante -
Définition 2. Un programme est dit « régulier » s’il est possible
et s’il satisfait les équations de récurrence définies par la
condition d’équilibre (7) et les égalités (22). Un programme est
dit « régulier maximal » s’il est régulier et si aucun programme
régulier ne donne une valeur plus forte à Co
Posons également les deux conditions suivantes:
Condition r. Le programme @ satisfait la condition 1 si
U’;c.>>0 pour tout # et s’il existe un nombre k plus grand que I
tel que, au moins à partir d’une certaine valeur de £-(23)
+ —— I
: > hk >
(1 fix) S,.1 =
Condition 2. Etant donné deux programmes réguliers quelconques
#! et #?, l’inégalité C! > C2? implique S! < S2 pour
tout ?.
Etablissons le résultat suivant:
Proposition 1. Un programme régulier Æ qui satisfait la
condition 1 est optimal.
Supposons en effet qu’il n’en soit pas ainsi. Il doit exister
une valeur T et un programme possible +5 Æ tel que 5,
soit positif et que à U, ne soit négatif pour aucun ¢# > T. En
vertu des égalités (22), de l’inégalité (ar) et du fait que U's
ne peut être négatif, &'Ur ne peut être positif que si à Sy est
négatif.
Par ailleurs, l’inégalité (16) implique:
(24)
U. à C,+U/,, à N, > o pour tout ¢ > 1
"51 Malinvaud - pag. 22
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323
et la condition d’équilibre (10) peut être écrite:
(23
à Sur EF N+ (1+f,) OK,
Comme les U”, sont positifs, les équations (22) impliquent :
26)
“4
= _—
AN
rs
’
|
1.
La première équation du système (22) implique aussi que I +
soit positif, de sorte que l’inégalité (24) peut aussi s’écrire:
foON, <(x+f,) 8C, pour tout :
En combinant cette inégalité avec l’inégalité (25), nous obtenons
277) 6S, 2 (x+f,) 8S, pour tout # -
T
Comme à S, est négatif, il en résulte que tous les à S, sont
négatif pour £ = T. Soit 6 la valeur de # à partir de laquelle
l’inégalité (23) s’applique, si cette valeur excède T, ou 6=T
dans le cas contraire T.es inégalités (23) et (27) impliquent:
pout tout £ > 9
5; Malinvaud - pag. 23
324 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
224
Comme À» est plus grand que 1, = tend vers -oc quand ¢
croît indéfiniment ce qui contredit le fait que S,+8 S, ne soit
jamais négatif. (Les conditions (3) et (8) impliquent bien S, > o
pour tout programme possible). Cette contradiction complète la
démonstration de la proposition r.
Pour déterminer un programme optimal, il suffirait en principe
de trouver un programme régulier qui satisfasse la condition
r. Mais cette recherche serait assez laborieuse si on
n'avait aucun fil directeur. En pratique, il peut être plus simple
de rechercher d’abord un programme régulier maximal, et de
vérifier ensuite qu’il satisfait la condition 1. La proposition
suivante suggère que cette manière de faire sera généralement
efficace quand la condition 2 est satisfaite.
Proposition 2. Si la condition 2 est satisfaite, un programme
régulier maximal est optimal dans l’ensemble de tous les programmes
réguliers (*).
La démonstration est immédiate. Supposons en, effet que
soit un programme régulier maximal et Æ+5 Æ un programme
régulier qui lui soit préférable. Par hypothèse à C, € o. De
plus on établit, comme au début de la démonstration de la proposition
1, que à S,<o. Mais ceci est contradictoire avec la
condition 2.
La condition 2 peut sembler un peu difficile à vérifier. Il
est donc intéressant de connaître des hypothèses sous lesquelles
slle est bien satisfaite. Posons :
Hypothése 3. La production s’effectue a rendements constants,
c’est-à-dire qu’il existe des fonctions ¢, (x) possédant
des dérivées ©’, décroissantes, et telles que:
28)
A(N,, K,) = N, Es
() La définition 1 introduit, entre les programmes possibles, un ordre
partiel: Il se pourrait donc a priori qu'un programme régulier non maximal
soit optimal. Mais cela semble peu vraisemblable.
‘5] Malinvaud - pag. 24
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Les utilités de la consommation et du travail sont additives,
c’est-à-dire qu’il existe des fonctions V, (¢,) et W, (n,) telles
que:
\
“20
CU, (c, n)=V, (c)-W, (xn).
Ces fonctions sont dérivables; V”, est positive et décroissante ;
W', est non négative et non décroissante.
Nous pouvons démontrer le résultat suivant:
Proposition 3. Si l'hypothèse 3 est satisfaite, la condition
l’est aussi.
Considérons en effet deux programmes réguliers Æ et #- cA,
et démontrons que à C, = 0€“ F 7 o impliquent < ,
3 C,,1 Z 0; 8 K,, 0e. .: 0. La condition . sera
bien alors établie puisque « ‘’mnliaue à K, - o en vertu
de l’égalité (5).
Nous avons ici:
Les égalités
LUN
122,
peuvent alors être écrites:
*9
3)
Malinvaud - pag. 25
326 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
1) Pour obtenir à N, < 0 à partir de $C, > o et à K, <o,
K
supposons & N, > o. Alors & (=) < o. Dans l'égalité (31),
#
le membre de gauche aura avec # +3 # une valeur plus
faible (*) qu’avec #. Mais dans le membre de droite de (31),
à W°, 2 0 et à V", < o. Cette contradiction implique bien
5N, < o.
2) Comme à N, < o et à K, <o, alors 8§S,,, < 0 en vertu de
l’inégalité (25). (Les égalités (30) et (31) impliquent
fa >oetf > o).
3) Pour établir à C,,, Z o et à K,,, < o, il suffit de montrer
que à C,,, ne peut pas être négatif, car 5 K,,, = O résulte
alors directement de à S,,, < 0. Supposons donc 8 C,,,<o
et considérons deux cas:
i K
a) Si 8 = =o, alors le membre de gauche de l’éga-£
lité (30) ne peut avoir avec Æ+5 Æ une valeur plus faible
qu'avec Æ. Mais dans le membre de droite 5V",<o et eV’, >o.
Nous obtenons bien une contradiction.
'K
b) Si à ‘ = > 0, considérons l’égalité:
¢
132)
fu =(14 6) Fer We
P, Vis
(”) La productivité marginale du travail # sy Ëst en effet une fonction
croissante de = . K
| Si nous désignons par x le rapport = » et si x <0, nous pouvons vérifier
ies inégalités suivantes: i
àSyp—p'5x <o en vertu de la concavité de la fonction ©.
Donc: p—xe" 2 p+do—0'(x+È7) .
Mais aussi:
P+Èp—p' (+5 %)>p+5p—(a+5 7%) (5 80°)
puisque: 5 o>o et x+B8 x>o.
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Elle est contradictoire avec les inégalités à f,.>o, à V°,,,>0
et à W, < o.
Ainsi. nous avons démontré la proposition
Terminons cette étude générale par quelques remarques.
tr. Nous avons prêté attention à des conditions suffisantes
pour qu’un programme soit optimal. Nous pourrions chercher
à généraliser quelque peu ces conditions. Ainsi, supposons que
dans le programme possible @ considéré N, soit égal 4 ny P,
pour une certaine période. On ne peut alors envisager que des
variations @ N, négatives. Dans les conditions suffisantes pour
que @Æ@ soit optimal, l’inégalité
, ‘
J f_- t4l.c-
€N
>
Pr =f
— (1 te) 5 LU’
doit remplacer l’une des égalités (22) pour cette valeur de à.
D'une manière générale, on peut toujours se reporter à l’inégalité
(21) pour appliquer le type d’analyse présenté ci-dessus à
des cas qui ne respectent pas exactement les conditions que
nous avons retenues.
De même, si les dérivées des fonctions U, et f, sont continues,
la réalisation des conditions marginales (22) est nécessaire
pour l’optimalité de tout programme possible @ qui soit
tel que l’on puisse encore définir un programme possible en
modifiant isolément chaque N, et chaque K,, et ceci à la fois
dans le sense de la baisse ou dans celui de la hausse. En effet,
la différence entre les deux membres de l'inégalité (21) est un
infiniment petit du second ordre par rapport aux & N, et aux
5K.
2. Nous pouvons donner une interprétation économique des
conditions mareinales introduites ci-dessus.
[5] Malinvaud - pag. 27
328 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Supposons U’, ,; , positif et considérons le taux 7, de croissance
de la population et le taux u, de décroissance de I’ utilité
marginale de la consommation:
Ls
35,
Pi - P,
TT, = —p
! ’
Uy IB Une
E=———————
U...
La première des égalités (22) s’écrit:
(34)
I+fu =(1+¢) (1+7,) (1+u,).
La productivité marginale du capital, pour la période #, c’est-àdire
le taux d’intérét durant cette période, doit satisfaire une
relation particulièrement simple contenant trois termes dépendant,
le premier du taux d’intérêt normatif, le second du taux
de croissance de la population, le troisième du taux de décroissance
de l’utilité marginale (*).
Les deux relations (22) impliquent aussi:
33)
’
oN _ Un
’ TU 117
1 +/ x U,
c'est-à-dire l’égalité de deux taux marginaux de substitution
entre le travail et le bien consommable, substitutions envisagées
(*) Des relations analogues ont été obtenues récemment par divers auteurs.
Voir par exemple R. FriscH (1062)
_5] Malinvaud - pag. 28
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du point de vue de la production d’une part, du point de vue
de la consommation d’autre part.
Enfin l’inégalité (23) de la condition 1 implique que la
valeur actualisée à l’instant o du stock de bien consommable
disponible à l'instant # tende vers zéro quant # augmente indéfiniment.
Soit en effet s, cette valeur actualisée. Par définition :
La condition 1 énonce que, au moins à partir d’une certaine
valeur 6 de #, le membre de droite soit inférieur ou au plus
’ \ I * 2’
égal à un nombre > plus petit que 1. Il en résulte que
(36)
O < S,
Par suite s, tend bien vers zéro avec t.
3. Nous avons considéré ici le modèle dans lequel les quantités
de travail sont déterminées de manière endogène. Nous
pouvons aisément adapter les résultats au cas dans lequel ces
quantités de travail seraient données de manière exogène.
Dans l'inégalité fondamentale (21), il n’y a plus lieu de
faire intervenir le terme en & N,. La seconde des égalités (22)
disparaît donc. Les équations de récurrence qui définissent les
programmes réguliers comprennent l’égalité (7) et la première
des égalités (22). Elles peuvent généralement être résolues pour
donner C,,, et K,,, en fonction de C, et de K,. Avec ces mcdifications,
les propositions 1 et 2 restent valables.
L’hypothése 3 peut être notablement réduite. T1 suffit de
5
Malinvaud - pag. 29
330
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
supposer que les fonctions f, (K,) et U, (c,) ont des dérivées
décroissantes et que U’, est positive. La proportion 3 reste alors
valable elle aussi.
4. UN MODELE DE R. RADNER
La méthode proposée ci-dessus est en général trop laborieuse
pour être appliquée telle quelle sur des modèles abstraits ;
car on ne peut pas obtenir explicitement la solution des équations
de récurrence. Toutefois, R. RADNER (1962) a proposé
pour les fonctions de production et d’utilité des formes analytiques
qui facilitent beaucoup les calculs. Nous allons examiner
brièvement la version la plus agrégée de son modèle. Ceci nous
permettra de vérifier comment s’applique la méthode en
question.
RADNER considère le cas dans lequel la quantité de travail
serait donnée de manière exogène. La population croîtrait à un
rythme constant, de même que la quantité de travail. La même
fonction d’utilité s’appliquerait à toutes les périodes; elle serait
linéaire par rapport au logarithme de la consommation. Le minimum
vital c,, serait pris égal à zéro. Enfin, l’output total
K,+Q, obéirait à une fonction de CoBB-DouGLAS à progrès
technique neutre et de rythme constant. Ces hypothèses s’expriment
par les égalités suivantes:
P,=(1 +x) P, N,=(1+v} N,
57)
IO,
U, (c,)=a log c,+b
f, (N,, K)+ K,=q, N* K* (1+0)
T, v, à, b, qu, a, B et 8 étant des nombres fixes.
5] Malinvaud - pag. 30
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 331
(Le modèle n’est réaliste que si v = =, sans quoi la quantité
de travail par personne croîtrait indéfiniment).
Si B est plus petit que 1, ce que nous supposerons, ce modèle
satisfait l’hypothèse 2, et même l’hypothèse 3 modifiée
comme il a été dit à la fin de la section précédente.
Les expressions analytiques retenues sont évidemment restrictives.
Avec la fonction (37), l’utilité croît indéfiniment avec
la consommation. Toutefois, cette forme peut sans doute convenir
pour des collectivités qui auraient un niveau de vie bien
supérieur au minimum vital et encore éloigné d’un niveau de
satiété.
La fonction de production (38) a une forme inhabituelle.
Elle implique pratiquement que la production nette ait à tout
moment un maximum qui serait atteint pour une valeur du
capital assez peu supérieure à celle observée effectivement.
Pour fixer les idées on pourrait retenir par exemple les valeurs
suivantes des constantes œ = 2 ( 5 . La productivité marginale
du capital serait égale alors
, pour un coefficient de
i
., + K . . :
capital 5 égal à 3. Le maximum de la production nette serait
= 0
: K I
atteint avec un coefficient de capital > égal à = 4,3. Ces
=U
ordres de grandeur ne semblent pas invraisemblables a priori.
L’expression (38) peut encore être écrite:
f, (N,, K,) + K, — ha nw! K;
avec les constantes h, et :L définies par:
184
(L+V)
(1 + 8)
M5
Malinvaud - pag. 31
332
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
On en déduit:
20
[+f =0hy pt KB 1
Par ailleurs, la forme retenue pour la fonction U implique:
LEA
U,, - Pi C,
Ainsi, les conditions marginales (22) deviennent:
(49)
y Sea
Eg) —
I+ fe =O + Cc
ou encore:
(+01
Cran
© = YA p KP
De même, la condition d’équilibre (7) devient :
44,
C1 + K,.1 =h wt K?
5] Malinvaud - pag. 32
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Erc. 333
La récurrence définie par les égalités (41) et (42) prend une
forme particulièrement simple. du fait que ces deux égalités
impliquent :
\
{
Ou encore:
Posons
et écrivons
Kia
Cr
rN
LA
dont la solution est immédiate:
A
»
>}
Malinvaud - pag. 33
334 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 238
Les programmes réguliers sont les programmes possibles qui
satisfont les égalités (43). Pour qu’une valeur de C, corresponde
à un programme régulier, il faut donc que le membre de droite
de ces égalités ne soit négatif pour aucune valeur de £, c’est-àdire
que:
a.
C= n
pour tout /
Or n est supérieur à I, du moins si B est plus petit que 1 +¢,
ce que nous supposerons. La solution des équations de récurrence
(41) et (42) définira donc un programme régulier si:
K
C.
La valeur C, qui correspond au programme régulier maximal
est alors définie par l’égalité
S,—C,
Cc
A
En reportant cette valeur dans l’égalité (43), nous trouvons
que, dans le programme régulier maximal le rapport K,/C, est
constant:
++,
“
“5] Malinvaud - pag. 34
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC, 335
Il en résulte que S,/C, est aussi constant. En vertu de l’égalité
(40), nous pouvons écrire:
+}
-
La condition 1 est donc satisfaite, et le programme réguliei
maximal est optimal, si le taux d’intérét normatif « € -nsitf
(1).
Pour déterminer complètement le programme régulier maximal,
reportons (44) dans (41). Nous obtenons une équation Z:
récurrence sur K, seul, soit:
(45)
K ; vBh KS}
Définissons k, et k par les égalités:
(46)
ot
—_p
"| Dans un programme régulier non maximal, on peu:
avec un nombre ) positif. 1, a!
st 1’égalité (40) implicue
ay
— -
S.….
I +
+
Comme n est plus grand que 1, cette quantité tend vers 8 qui est plus
petit que 1 par hypothèse. La condition 1 ne peur donc être satisfaite nar
aucun programme régulier non maximal
Malinvaud - pag. 35
336 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
7)
1 -1
F = (18 A) T7P ui?
Nous vérifions tout de suite que l’équation (43) implique:
43)
k,, =[k]}-8 RP
la grandeur k, obéit donc à une loi de récurrence très simple.
Sa valeur à l’époque # est une moyenne géométrique de sa valeur
à l’époque #- 1 et d’une valeur fixe k. Elle tend progressivement
vers k. La rapidité de cette tendance dépend du coefficient
B puisque l’on déduit de (48):
+9)
k
log =f = B'log #e
kb h
. I
A titre d’exemple (1), retenons a= 2, B= >, v=1% et
1
0=0,5%. Le coefficient p 7-8 est alors approximativement
égal à 1,037. La croissance limite du capital et de la consommation
s’effectue au rythme de 3,7% par période.
Si le taux d’intérêt normatif est nul et si 0 est égal a 3,
() Comme le montre la formule (38), le coefficient de progrès technique
s'applique à l’output total et non à la production nette. Il convient donc
de retenir pour lui une valeur plus faible que celles adoptées habituellement
pour caractériser le développement de la production nette.
51 Malinvaud - pag. 36
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
337
k CL
le rapport —— est approximativement égal à 1.26. Il aurait
suffi que le capital initial fut supérieur de 26% pour que la
croissance s’effectue de façon strictement proportionnelle au
taux de 3,7% par période. Même avec un coefficient de capital
initial de 3, l’écart entre k, et k n’est plus que de 3% après 10
périodes, comme le montre l’application de la formule (49).
Enfin, le taux d’épargne nette limite est de 13,4%-.
LEA y . K, ,
Si le taux d’intérét normatif est de 10%, et si 5. est éga.
k CL
a 3, le rapport est approximativement égal à 0,7€ Le
capital initial excède celui qui conduirait à une croissance optimale
strictement proportionnelle. Le taux d’épargne nette li
mite n’est plus are - ©
La figure 1 représente l’évolution de la consommation
dans les deux programmes réguliers maximaux correspondant
aux valeur o et 10% du taux d'’intérêt normatif. La
seconde permet une consommation supérieure de 40% dans
la première période; mais elle a pour effet une consommation
inférieure de 8%, dans le régime asymptotique.
5] Malinvaud - pag. 37
338
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
>
Retournant aux formules générales, nous observons qu’à
la limite le capital et la consommation globale croissent à un
taux constant égal à
2 0
(I +v)1-6 (1+60)1-8B — 1
La consommation par unité de travail tend de même à augmenter
au taux
a+f—1 LL
(1+v) 8B (r40)1-B_ 1
Si la fonction de production satisfait l’hypothèse des rende-1
ments constants, ce dernier taux est simplement de (1 +0)1-6 - 1;
4 gr [ 6
il diffère peu en pratique de r—B°
Soit p, la productivité marginale du capital, ou le taux
d’intérét, de la période ¢.
La formule (40) montre que r+p, est égal au produit de
I+£ par le facteur de croissance de la consommation globale.
À la limite 1+p, tend vers
1+p5=(1+e)p!-B
Enfin, nous pouvons vérifier que la croissance asymptotique
pour laquelle la consommation se situe au niveau le plus élevé
est justement celle dans laquelle le taux d’intérét normatif est
.51 Malinvaud - pag. 38
SEMAINE D'ÉTUDE SUR LE ROLE DE I’ANALYSE ECONOMETRIQUE ETC. 330
nul. (Le taux d’intérét p, est alors égal au taux de croissance
de la consommation globale). En effet, comme 1 ne dépend
pas de y et que k, tend vers k, le niveau de la croissance asymptotiaue
de la consommation varie comme
(+
-1) (vB 5) TB
(voir formules (44), (46) et (47) ci-dessus). Il varie donc aussi
comme
B
v2) y1-8
La dérivée par rapport à y du logarithme de cette expression
est égale à:
Elle s’annule justement quand y
L’étude qui précède a permis de caractériser les programmes
réguliers maximaux. Dans quels cas ceux-ci constituent-ils
des programmes optimaux? Nous avons déjà observé que la
condition 1 était satisfaite quand le taux d’intérêt normatif était
positif. Nous avons donc déterminé un programme optimal
pour toute situation dans laquelle il en est ainsi. Afin de mieux
comprendre les complications qui apparaissent quand le taux
d’intérêt normatif est négatif ou nul, essayons d’examiner di-[>]
Malinvaud - pag. 39
340 PONTIFICTAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA -
2%
rectement l’optimalité des programmes réguliers maximaux
dans le cadre du modèle de RADNER.
Nous devons donc comparer le programme régulier maximal,
que nous désignerons par #*, avec un programme possible
qui, à partir d’un certain horizon T, donne aux U,, c’est-à-dire
aux consommations C,, des valeurs au moins égales à celles
que leur donne Æ*. Nous avons déjà vu que, pour que cet
autre programme donne une valeur supérieure à ÜU,, il fallait
qu'il donne une valeur plus faible à Sr. Le problème consiste
donc à trouver la réponse à la question suivante: Est-il possible
d'obtenir, au moins à partir d’une certaine époque T, la
suite des consommations Cr, Cr;, … ad infinitum, à partir
d'un stock de bien S, plus faible que S;? Si la réponse est
négative, nous savons que Æ* est optimal. Si elle est affirmative,
nous savons que Æ* n’est pas optimal, car on améliorerait
ce programme en augmentant la consommation de Sy - Sr pendant
la période T, sans la modifier dans aucune autre période.
Pour répondre à cette question, considérons la relation qui
existe entre les variations à K, et à K,,, du capital lorsque l’on
passe du programme Æ* à un autre programme possible
#* +8 M qui assure la même consommation C;,, dans la période
#+1. Plaçons-nous dans le cas où à K, est négatif.
Partons de l'égalité suivante qui exprime la condition d’équilibre
à l’instant +1:
50)
8,1 =h KP
Comme la dérivée seconde de KP est croissante et que & K, est
négatif, nous pouvons écrire:
(51) OK ,41 = 98,41 =< BA, uv KB-1 3K, + ' Ce D h, vu K8-*(3K,)?
[5] Malinvaud - pag. 40
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
341
Nous savons aussi que la différence entre les deux membres de
l’inégalité peut être rendue aussi petite que l’on veut si 7 K,
est pris suffisamment petit.
Par ailleurs, les équations (50) et (44) impliquent que,
dans le programme Æ* :
(1+¢) K,, ,=8 h, nf KB
Nous pouvons donc réécrire (51) sous la forme:
-
(I +
,,
A
D
K
1 + 2,1
ON
a
K
Cette inégalité implique que à K,,, soit négatif, car 3 est plus
petit que 1. Elle implique aussi:
52)
| = à partir de
l’instant T. Si cette suite peut être choisie de manière à ne
jamais dépasser la valeur I, la réponse à la question posée est
affirmative, et Æ* n’est pas optimal. Si non, la réponse est négative,
et Æ* est bien ontimal.
1) Si le taux d’intérêt normatif n’est "as névaui (€
quel que soit à K+-<o, la suite ue
> 0), alors
augmente au delà
Malinvaud - pag. 41
342
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
25
de toute limite en vertu de l’inégalité (52). Cette suite finit
donc toujours par dépasser la valeur 1. Le programme régulier
maximal étudié ici est optimal.
2) Si le taux d’intérêt normatif est négatif (e<o), alors on peut
prendre à K- suffisamment petit pour que
RL
K* |
a
et, a fortiori, que
SK,
= +2
T +492
=.
A
]
et ainsi de suite. Le programme Æ* n’est donc pas optimal.
Que se passe-t-il quand le taux d’intérét normatif est négatif?
Nous pourrions alors vérifier sans trop de peine qu’aucun
programme possible n’est optimal.
Les raisons de ce fait peuvent paraitre plus claires si nous
considérons le programme possible #* +8 Æ qui est apparu
préférable au programme @Æ* dans la démonstration précédente.
Ce programme permet les consommations C; pendant
toutes les périodes sauf la T-ème, et permet C}+|à Kr| pendant
celle-ci. Mais, supposons qu’au lieu de consommer
8 Kr| dans la période T, on envisage de l’utiliser dans la production
et de consommer dans la période T + 1 le produit supplémentaire
qu’il permettra d’obtenir. L’inégalité (52), jointe
5] Malinvaud - pag. 4
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
343
au fait qu’il existe une proportion constante entre Cj et KJ.
montrent aue:
3
a
Ceci implique que l’opération envisagée est avantageuse du
point de vue de l’utilité U,,;-Ainsi,
bien que à chaque moment le stock de capital existant
dans le programme régulier maximal soit surabondant, à aucun
moment on a intérêt à en consommer une partie, car il est toujours
préférable de reporter à plus tard cette consommation.
En somme, l’étude du modèle de RADNER comporte deux
conclusions qui pourraient avoir une portée générale.
t. Si on n’y prend pas garde, la formulation du modèle peut
conduire à des situations dans lesquelles aucun programme
optimal n’existe. Il semble que cette particularité se présente
surtout quand la fonction ‘d’utilité attribue une grande importance
au futur, c’est-à-dire quand le taux d’intérêt normatif est
faible (négatif dans le cas présent).
2. En revanche, on peut parfois définir des programmes
optimaux dans des cas dans lesquels la fonction d’utilité UU,
ne converge vers aucune valeur finie quand T croît indéfiniment.
Nous l’avons vérifié ici puisque nous avons trouvé un
programme optimal pour le modèle comportant un taux d’intérêt
normatif e ="
Malinvaud - pag. 43
344
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
5. CROISSANCES OPTIMALES A COEFFICIENT DE CAPITAL FIXE
Les équations de récurrence qui interviennent dans la définition
des programmes réguliers prennent une forme plus simple
dans deux cas que nous allons considérer successivement: celui
dans lequel la fonction de production f, est linéaire, et celui
dans lequel l’utilité U, est linéaire.
Si la fonction de production s’écrit:
33
f: (N,, K,) =a, N,+ b, K,
avec des coefficients numériques fixes a, et b,, les équations
de récurrence (22) peuvent étre écrites sous la forme:
34)
2 , P ,
\ Ui, (1 + 6) — (I + )—p- U,
. a, ,
| V.. = —777 Vu
Elles ne font plus alors intervenir la grandeur K,. On peut donc
en général les résoudre par rapport aux C, et N, à partir d’une
valeur donnée de C, En principe, la deuxième équation de (54)
détermine N, en fonction de C,. Si une solution explicite de
cette équation peut être obtenue et reportée dans la première
équation, alors celle-ci définit une récurrence sur la seule grandeur
C,. La détermination des programmes réguliers est donc
orandement facilitée.
51 Malinvaud - pag. 44
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 345
Malheuresement, une fonction de production du type (53)
ne peut offrir qu’une représentation très médiocre des contraintes
techniques. Elle suppose la constance des productivités marginales,
constance contraire aux résultats de nombreuses études
économétriques.
Toutefois, on admet assez souvent que la main-d'oeuvre
est surabondante dans les pays peu développés, et que la production
y est proportionnelle à la quantité de capital disponible.
Nous allons nous placer ici dans cette perspective et retenir pour
la fonction de production l’expression (53) avec une valeur
nulle de a,. Nous allons donc rechercher des croissances optimales
pour une économie peu développée dans laquelle le coef
ficient de capital peut être considéré comme fixe.
La deuxième équation du système (54) montre qu’alors
l’utilité marginale du travail doit être nulle dans tout programme
régulier. On ne perd dès lors guère en réalisme à éliminer
complètement du modèle la quantité de travail et à retenir
une fonction d’utilité U, qui ne dépende que de c,.
Pour déterminer les programmes réguliers, il suffit de con
sidérer la condition d’équilibre
55
gE
1
1+b)K,
vr
Ie
et l’équation de récurrence:
56)
P
(1 +6)\U" = (1 + 8) —
La résolution de (56) donne la suite des C, en fonction à
La suite de K, est ensuite facile 4 obtenir grace a 1'équation 33,
Pour que la solution ainsi trouvée convienne, il faut ence
5
Malinvaud - pag. 15
346
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
vérifier qu’elle satisfait les contraintes, et notamment qu’elle
ne donne pas à certains K, des valeurs négatives.
Nous allons appliquer cette méthode à une spécification particulière
du modèle: celle dans laquelle la population croîtrait
de manière exponentielle, le coefficient de capital serait le
même pour toutes les périodes, et l’utilité marginale U’, serait
de la forme:
U’, (c,)=(c,- c,)7"
avec un coefficient æ positif et au plus égal à 1 (la grandeur Cm
définit comme ci-dessus le niveau minimum de la consommation
par personne). Nous remplacerons donc b, par la constante b
et écrirons:
P,=P, (1+)
avec un nombre T fixe.
Le modèle ainsi défini a été déjà considéré dans la littérature
économique (!). La forme retenue pour la fonction d’utilité
est due à R. FriscH. Elle semble devoir bien convenir, au
moins en première approximation, pour les économies qui sont
encore très éloignées d’un quelconque niveau de saturation.
Avec cette spécification, l’équation (56) s’écrit:
(Cra = Cm) = (c, = Cm)
() Voir notamment J. TINBERGEN (1960), ainsi qu’une note de M. BorrEUX,
Taux d'épargne obtimal dans une économie en développement (3
novembre 1061).
5] Malinvaud - pag. 46
SEMAINE BD ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 347
~ étant la constante définie par:
(57°
. + 0)
T) (I +
En conséquence l’évolution de la consommation obéit :
8
(c, - Cpr) =o! (co - Com) +
Désignons par k, le capital par personne (K,,
tion (55) implique:
4
, +.
1» L,1
>
17
… équa
qui s’écrit encore:
(R, 4 - k,) = E(R, - En) T (c,,
mp,
3 et k,, étant les deux constantes définies par:
Malinvaud - pag. .7
348 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
En tenant compte de l’égalité (58), on trouve aisément la solution
de la récurrence sur k, - k,,, soit:
59) Ke — ka) = PE [Ce — Ka) — 1. te + ‘ty — Cp)
pv désignant le rapport «/f. La formule (59) s’applique pour
toute valeur positive de # si 4 diffère de 1. Nous étudierons
ci-dessous le cas dans lequel (4 serait égal à 1. Pour #=o, la
formule (59) doit être remplacée par:
(60)
k _
Sn
Cc,
Il nous reste a vérifier que ces formules ne conduisent jamais
à une valeur négative pour k,.
Etant donné les ordres de grandeurs que peuvent prendre
en pratique b, 7 et e, nous admettrons aue «x et B sont plus
grands que I.
Comme c, ne peut pas être inférieur à c,, il faut que k
soit au moins égal à k,,; sans quoi k, - k,, tendrait vers - oo
et k, deviendrait tôt ou tard négatif. Nous supposerons donc
que le stock initial permet au moins une consommation par
personne égale à c,, et un capital par personne égal à k,,, soit:
(OI)
So = P, (Cm + Ron)
15] Malinvaud - pag. 48
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346
nu encore:
On peut facilement voir qu’il n’existerait aucun programme possible
si cette condition n’était pas satisfaite.
Nous considérerons encore trois cas suivant que 4 est plus
grand, plus petit ave 1 ’
Premier cas:
La formule (59) montre alors que «=! (k,- R,,) tend > r:
(cy - ¢,,,) quand ¢ augmente indéfiniment. La grandeur a Z,
tend vers la même limite. Pour que k, ne devienne jamais négatif,
il faut que c, soit égal à c,
Le seul programme régulier est donc celui dans lequel:
v
,--
k,)
8° (Ra
+
k,
p=La
consommation est maintenue continuellement à son niveau
minimum. L’économie accumule du capital indéfiniment au
rythme le plus élevé possible.
Ce programme régulier est le seul programme possible si
ko=k,,; C’est-à-dire si S, est juste égal à P, (c,, + k,,). C’est
bien alors aussi le programme optimal.
En revanche, si S, excède P, (c, + R,.)» Ry excède k, Le
programme régulier n’est évidemment pas optimal. De plus,
aucun programme n’est optimal. La situation est tout a fait
Malinvaud - pag. 49
350 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
comparable à celle que nous avons rencontrée dans l’étude du
modèle de RADNER pour le cas dans lequel le taux d’intérét
normatif e est négatif.
Second cas: n=1.
Si œ est égal à B (c’est-à-dire si u=1), la formule (59) doit
être remplacée par:
(R, = k,.) = a'[(k, = k,) = t(c, = Cm)
Il ne résulte que (k,-k,,)/t a’ tend vers - (c,-c,) quand €
augmente indéfiniment. Pour que k, ne devienne jamais négatif,
il faut que c, soit égal à c,, Il existe donc un seul programme
régulier. Si S,=P, (ce, + k,,) c’est le programme optimal.
Sinon, il n’existe aucun programme optimal.
Troisième cas: w<1.
Dans la formule (59), le second terme de l’expression entre
crochets décroît à partir de la valeur -p (cp — c,,) pour £=1,
et tend vers - = (c, — c,,).- Pour que k, ne devienne jamais
négatif, il faut et il suffit que l’expression entre crochets ne soit
jamais négative, donc que:
JZ
(kK, To Km)
1
La à
0 — Cm,
“ wr
5] Malinvaud - pag. 50
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIOUE ETC. 351]
En remplaçant k, et k, par leurs expressions et en tenant
compte de la définition de 4, nous pouvons transformer cette
inéealité en la suivante:
( \
62)
f
: 1
%
J
Le programme régulier maximal donne évidemment
co — C,, ‘a valeur du second membre de cette inégalité,
“ |
ko — k, la valeur —|i (co-€,)- La formule (59) montre
qu’alors (» k,) croît en progression géométrique suivant:
(64)
(k, - k_) =o! (Ra - R,)
La consommation et le capital suivent deux évolutions analogues
comme le montrent les égalités (58) et (64).
Pour ce programme régulier maximal, on a:
[AR S, a P,
1 fie 3 >6+9)5
Coy
C +
Quand # augmente indéfiniment, cette expression tend vers
Malinvaud - pag. 351
352 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
La condition 1 est satisfaite, de sorte que le programme en question
est optimal.
A titre d’exemple, retenons les valeurs suivantes des di-I
verses constantes: b= + correspondant à un coefficient de
capital de 4 et à une productivité marginale du capital de 25%,
I
T= 50 Correspondant à un rythme de croissance démographique
I » 2 A
de 2% par an et y= = correspondant à un taux d’intérêt normatif
de 10%. Admettons encore deux valeurs pour le coefficient
#: la valeur 0,6 qui a parfois été proposée, et la valeur 1
qui correspond à une fonction d’utilité logarithmique. L’application
de la formule (57) conduit à des valeurs de œ égales respectivement
à 1,19 et à I,r1. Le rythme de croissance de Ch - Cm
serait de 19% par an si w valait 0,6, et de 11% par an si u
était égal à I.
Il nous faut examiner maintenant quelles valeurs des paramètres
conduisent à une valeur de 1 plus petite que 1. D’après
la formule (57), x est plus petit que 8 si (*)
(65)
faze (EST
I +
Le taux d’intérét normatif doit être positif si l’utilité U(c,)
est proportionnelle au logarithme de c,. Il doit être supérieur
2 : I I
a une valeur voisine de 8,5% si b= 4° T= 50 et u=0,6.
Quand la formulation du modèle suppose que l’accumulation
du capital n’entraîne aucune décroissance de sa productivité
marginale, le taux d’intérêt normatif doit être suffisamment
(!) On peut observer que c’est aussi la condition pour la convergence
de l’utilité LU. d'un programme dans lequel (c.—c.\ croît au taux ».
51 Malinvaud - pag. 52
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 353
élevé pour que l’on soit en mesure de trouver un programme
optimal. Ceci n’est évidemment pas une raison pour rejeter une
éthique sociale qui impliquerait un taux d’intérêt normatif faible,
et par exemple un taux nul. Les résultats obtenus suggèrent
au contraire que l’hypothèse de constance de la productivité
marginale du capital ne peut être maintenue que si le taux
d’intérêt normatif est notablement supérieur à la borne inférieure
de l'inégalité (65). Quand les principes éthiques qui président
aux choix collectifs ne satisfont pas cette condition, une
détermination valable du programme de croissance suppose la
prise en compte du fait qu’une accumulation accélérée du capital
se traduit nécessairement par une certaine diminution de
sa productivité marginale.
Cette conclusion pourrait être évitée grâce à une modificaton
de la fonction d’utilité. Il nous suffirait d’introduire un
niveau de satiété pour retrouver un programme optimal (!).
Mais les résultats obtenus alors n’auraient sans doute pas grande
signification, car ce programme dépendrait dès les premières
périodes du niveau auquel serait fixé la satiété. Il semble bien
irréaliste de faire dépendre la croissance de pays peu développés
d’hypothèses sur l’importance de la consommation qui leu:
apporterait la satiété.
(Y) J. TinBeErGEN et H.C. Bos (1962) ont étudié le cas dans lequel il
existerait une consommation de satiété ~ =t l’utilité marginale serait de
la forme -
Ils ont montré que, dans le programme optimal, c,—c, croissait suivant une
courbe logistique. Ils ont noté toutefois que. si € était nul. le taux d’épargne
devait nrendre des valenre tràs AlavéAes
WMalinvaud - pag. 53
5,4
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
0. FONCTIONS D’UTILITÉ LINÉAIRES
Les équations de récurrence (22) prennent une forme particulièrement
simple dans le cas où l’utilité U, (c, , n,) est une
fonction linéaire de la consommation et du travail par personne.
Nous allons examiner ce cas qui peut être intéressant en luimême
et qui présente un avantage du point de vue méthodologique.
Il nous conduira en effet à étudier comment adapter
les procédés décrits ci-dessus au cas dans lequel les contraintes
sur la consommation et sur le travail deviennent effectives.
Nous poserons donc:
66)
Uc, , n,) =, C,- v, nu,
les y, et v, étant des constantes données non négatives.
À vrai dire une fonction d’utilité de ce type est assez particulière.
Elle suppose que le taux marginal de substitution entre
consommation et travail est indépendant de la consommation
obtenue et du travail fourni. Elle suppose de même que le taux
marginal de substitution entre les consommations de deux périodes
différentes est indépendant des niveaux auxquels s’établissent
ces consommations. On peut douter a priori qu’un critère
de choix présentant ces caractéristiques soit vraiment adéquat.
Nous allons voir en effet qu’il conduit à des programmes
optimaux peu satisfaisants.
Cependant, dans la littérature sur le développement économique,
on prend souvent comme critère une valeur actualisée
de la suite des consommations futures par personne, ou des
consommations futures globales. On se réfère bien alors à une
5] Malinvaud - pag. 54
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355
fonction du type (66) dans laquelle les v, sont nuls et les y,
sont tous égaux ou proportionnels aux populations P,.
Avec la fonction d’utilité retenue ici, les équations de ré-~urrence
(22) deviennent:
67)
7
Elles ne font plus intervenir que les variables N, et K, relatives
à une même période et n’établissent plus une récurrence véritable.
Pour déterminer un programme régulier, il nous suffit en
principe de résoudre, dans chaque période, le système (67) par
rapport à N, et K, et de déduire C, de la condition d’équilibre
relative à l’instant #. A partir de la seconde période, le programme
régulier ne dépend donc plus de la valeur du stock
initial Sa.
Ce paradoxe tient au fait que, le plus souvent, les opérations
décrites ci-dessus ne définissent pas un véritable programme
régulier. Les valeurs trouvées pour certains des C, sont
négatives, ce qui est contraire à la contrainte c, = c,-Dans
ces conditions, on ne peut espérer obtenir un programme
optimal en utilisant seulement les équations de récurrence
qui supposent la réalisation d’égalités marginales. Il faut
tenir compte plus directement des contraintes sur c, et #,.
Afin d’aboutir a des résultats suffisamment précis, nous allons
restreindre quelque peu la généralité du modèle. Nous
supposerons que la population croît à un rvthme constant.
3
Malinvaud - pag. 55
356
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
qu’il n’y a pas de progrès technique et que la production s’effectue
à rendements constants. Nous poserons :
et nous écrirons:
f: WN, K,) =N, ¢ ix,
¢ étant une fonction donnée. L’hypothése 1 implique que ¢ ait
une dérivée ¢’ qui soit une fonction jamais croissante de X,.
Afin d’éviter des complications supplémentaires, nous supposerons
que ¢” est une fonction décroissante de x,. Nous admettrons
que l’utilité U, est la même fonction pour toutes les périodes.
Comme nous pouvons la multiplier par une constante
positive quelconque, nous l’écrirons :
“68)
U, (c, R,)= c,-vn,
v étant un nombre donné non négatif. Enfin, nous préciserons
les contraintes sur c, et n, en fixant un niveau de satiété cm à
la consommation et un minimum à la quantité de travail:
(69)
(70)
Cm = C, = Cn
n, =n, _~ th
Cm» Cum, #,, et ny étant quatre nombres donnés.
5] Malinvaud - pag. 56
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
357
Dans ces conditions. nous pouvons écrire-Nous
poserons encore:
(771)
A=(14¢) (1 +i
L’inégalité de base (21), qui nous a servi pour comparer les
itilités Ur de deux programmes possibles, devient maintenant:
(72) P, dU, =
T.
T_O
me,
-, Ad
%, et 3, étant définis par:
73
‘74,
rT
Les égalités (67) impliqueraient que «, et 3, soient nuls.
Mais on observe que ces égalités sont incompatibles en général.
car il n’v a aucune raison pour que les deux équations en x
A
WA) -
)
| Malinvaud - pag. 57
358
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
aient une solution commune. Chacune d’elles a une solution
au plus; en effet 4’ (x) est continuellement décroissante par
hypothèse et p (x)- x 4’ (x) est continuellement croissante puisque
sa variation dans tout intervalle infiniment petit (x, x + dx)
est égale à la quantité positive - x dy’ (x). Désignons par xx
la solution de la première équation, si elle existe. C’est la valeur
du rapport N Pour laquelle la productivité marginale du
capital est égale à À} - 1. Désignons de même par xy la solution
de la deuxième équation, si elle existe. C’est la valeur du rap-K
242 . :
port y Pour laquelle la productivité marginale du travail est
égale à Av.
Notons immédiatement que «, est positif quand x, est inférieur
à xx, négatif dans le cas contraire. De même B, est négatif
quand x; est inférieur 4 xy, positif dans le cas contraire.
La condition d’équilibre à l’instant # s’écrit ici:
(75)
C1 =K,- K,,1 + N, © (x,)
ou encore:
Con 2,41
76) (1+7) == 0 (x)+ x, — (1 + ®) = Tye
Nous devons repérer la valeur æ de x qui correspond à un programme
dans lequel l’utilité serait continuellement égale à son
minimum (c,=c,, et #,=Nn). Nous pouvons de même repérer la
valeur % correspondant à un programme dans lequel l’utilité
serait continuellement égale à son maximum (c,= cm et n,=n,,);
5] Malinvaud - pag. 58
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35G
la valeur xy correspondant à c,= Cm et #,=Nm; et la valeur x,
correspondant à c,=c, et n,=n, Il résulte de (76) aue ces
valeurs, si elles existent, sont solutions respectivement de:
(77)
|
De même, nous pouvons définir x, et %, comme les valeurs
que prend x dans la première périod: si c, et #, sont fixés d’une
part à c,, et ny, d'autre part à c- "
78)
x, est évidemment plus grand que %,-Pour
une étude complète du problème qui nous intéresse
maintenant nous devrions examiner de nombreux cas suivant
les positions respectives des valeurs: xx, Xn, Ÿ, Xm> Xms Lo Et Xv.
Nous nous en tiendrons à trois qui suffiront sans doute pour
illustrer la détermination et les caractéristiques des programmes
optimaux.
Si x, était inférieur à r, il n’y aurait aucun programme
possible. Si, était supérieur 4 &, un programme optimal serait
aisément défini, programme dans lequel la consommation serait
constamment égale à son maximum et le travail à son minimum.
Ces deux cas ne sont guère intéressants. Nous supposerons
donc 7 = T
Malinvaud - pag. 59
360
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
I) Premier cas: xy<®,<xx<xy (ou, plus généralement, si Xn
et xy n’existaient pas: x,<xg;
P (20) = X P'(X0)>kv: et gag) - 7 Xx<(T+T) cpr) nr).
Si e>0, un programme optimal Æ* est défini comme suit:
— La quantité de travail est continuellement maintenue à
son maximum (#,=mn, pour tout ?).
— La consommation est maintenue à son minimum (c,=c,,)
tant que
(79)
L=35 ’ < X,
> Cm
7 na
“ie y
simultanément x,=æ, croît.
S, est défini par:
(80)
S,= 7m Pt [x ;_1 +@ (%,_1,
— À partir de l’instant # où 2, > xk, x, prend la valeur
Xx; la consommation par personne prend dans la période
S . .
la valeur P — ny Xx et dans les périodes suivantes la valeur c*
t
définie par (*):
à
-
(x +7) c* = [9 (xx) — 7 Xx] Ny
() On peut vérifier que la valeur de la consommation par personne dans
la période t, est inférieure à ce qu’elle est dans les périodes suivantes.
“51 Malinvaud - pag. 60
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQOUE ETC.
361
Les évolutions de c, et x, sont schématisées sur la figure 2.
Pour démontrer que ce programme est possible, il nous
suffit de vérifier que x, croît avec / et finit nécessairement par
atteindre xx. (Les égalités (79), (80) et (81) établissent que la
condition d’équilibre est bien satisfaite à chaque instant).
Des égalités (77), (79) et (80), on déduit aisément:
82) (1+7) (x—x,_)=[3(2,_)- m2, ]-[¢(®)-7x
Or ¢(x)- =x a la dérivée décroissante ¢’(x)- =. Pour toute
valeur de x inférieure à xx, cette dérivée est plus grande que
(xg) 7" 7). Comme x, , - æ est positif:
R
3)
Malinvaud - pag. 61
362
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
La différence æ, - æ,_, est bornée inférieurement par un nombre
positif tant que w,<xx. Ainsi, æ, croît et finit certainement par
atteindre xx.
Le programme Æ* est aussi optimal. Soit en effet un autre
programme possible #*+à Æ. Nécessairement, & N<o pour
tout # et à K,£<o pour tout #<t,. Or B,>0 pour tout t; x,>o
pour tout £<Z, et a,=o0 pour tout £=¢,. L’inégalité (72) implique
donc Py & Ur =< - 17 3 S;.
Admettons maintenant que le programme /#* +8 # donne
à l’utilité au moins la même valeur que Æ* à partir d’un certain
instant T que nous pouvons supposer postérieur à Zo
Pour que à Us, soit positif, il faudrait que & Sy soit négatif. Mais
nous allons montrer que ceci n’est pas possible, ce qui établira
l’optimalité de Æ*.
En effet, en vertu de la concavité de la fonction de production,
nous pouvons écrire, pour tout t=T:
8S, = [T+ ¢"(xx)] 8 K, + [p(xx) - *x 9'(xx)] à N, .
Par ailleurs, pour tout #>T.
àC,=> vôàN,.
Or à N, est nécessairement négatif, et à K,=à S, - à C,. Par
suite de la définition de xx et du fait que xx<xr, nous pouvons
écrire:
5S, =18S,- A3C,+Av8N, =13S,.
5] Malinvauu - pag. 62
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 363
Cette inégalité montre que à S, sera effectivement négatif pour
tout à —
Enfin, comme S*,
ia
Comme e est positif, cette inégalité montre que la valeur abso-5S
- . ;
lue de = croît au delà de toute limite, et finit nécessairet
ment par dépasser I. Ainsi S}+à S, devient négatif. Le programme
Æ* +8 Æ n’est pas possible.
Nous n’examinerons pas ici ce qui peut se produire quand
le taux d’intérêt normatif est nul ou négatif. Ceci nous entrainerait
trop loin. Nous retrouverions des situations comparables
à celles décrites à propos des deux modèles précédents.
2) Second cas: x,
a
La détermination du programme optimal devient plus dé
licate si le rapport = atteint avant xx la valeur xy qui permet
une consommation maximale avec un travail par personne égal
a nu. Nous ne procéderons pas ici à une vérification précise de
l’optimalité du programme que nous allons définir. Quelques
indications suffiront.
Il est clair qu’une fois xy atteint l’accumulation du capital
doit permettre une réduction de la quantité de travail par personne.
Suivant quelle évolution cette diminution va-t-elle avoir
lieu?
Pour découvrir la réponse à cette question, nous pouvons
~omparer le programme optimal Æ* à un autre programme pos-“5
Malinvaud - pag. 63
364
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
sible #* +8 / qui attribue les mêmes valeurs à la consommation
(à C,=0o). La variation de l’utilité de la période # sera:
(84)
JU =—-3N,
La condition d’équilibre à l’instant # et la concavité de la fonction
f impliquent :
à K,,, < (I+f, ) À K,+f.n Ô N,
D’
où :
a —v 9 ’
YU, < 57, R Kip — (1 +/%) SE
et pour l’utilité U,:
23)
T-1 AFT
POU, < — vA tp, 0K, — — 3K,
t=1 T-1.N
les coefficients 4, étant définis par:
30)
15] Malinvaud - pag. 64
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIOQUE ETC. 365
Le programme Æ@* que nous cherchons sera vraisemblablement
tel que tous les i, soient nuls à partir du moment où c,
atteint son niveau maximal cm. Examinons donc ce qu’impli
que la nullité de ces coefficients.
Les productivités marginales f,y Et fx dépendent seulement
de x,; de sorte que les égalités u,—o0 établissent une récurrence
sur les x,. Une méthode graphique peut servir à la
résolution de cette récurrence. Sur une figure portant les x en
abscisses, soit F, et T, les courbes représentatives des fonctions
g=log (1+fx)- log À et h=log f'w. Pour que , soit nul, il faut
que h(x,) - h(x,_1) = g(x,); c’est-à-dire que l’accroissement de
h entre t- 1 et ¢ soit égal à la valeur de g en ¢{. La figure 3
ci-dessous illustre comment les valeurs successives des x, peuvent
être déterminées
Comme la courbe TI', coupe ’axe des x au point d’abscisse
xx et que I’, est continuellement croissante, la grandeur
croît continuellement et tend vers xx.
Les x, une fois déterminés de la sorte, on peut utiliser la
condition d’équilibre (76) comme une récurrence sur les gran-"51
Malinvaud - pag. 65
366
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
deurs n,. Le travail par personne #, tend vers la quantité »*
définie par:
Cu
nik
)
I +
(I Te
x TX ==)
K
La grandeur n, doit décroître de ny vers n*,
Enfin, l’évolution de K, peut être aisément déterminée puisque
K,=P, x, n,. Le capital par personne x, n, doit croître à
partir de la quantité xy #, pour tendre vers xx n*. (On peut
effectivement vérifier que xm #v est plus petit que xx n*; mais
je n’ai réussi à démontrer ni que n, décroît continuellement de
Nm à n*, ni que x, n, croît continuellement de x, #, 3 xy ny).
La figure 4 ci-dessous représente l’évolution de la consomry
mation et du travail par personne, ainsi que celle de x,, dans le
programme qui vient d’être déterminé.
Il resterait à vérifier que ce programme est bien optimal
quand le taux d’intérêt normatif est positif. Nous ne le ferons
pas ici; car ce serait un peu fastidieux.
‘51 Malinvaud - pag. 66
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 367
3) Troisième cas: Ty
Revenons-en a une situation analogue a celle du premier
cas, xx étant inférieur a xy. Mais supposons que la productivité
marginale du travail serait inférieure à Xv, si, dans la
première période, la consommation était maintenue à son minimum
et le travail à son maximum. Le programme qui a fait
l’objet de la figure 2 n’est plus nécessairement optimal. Il se
peut que l’utilité soit accrue si l’on réduit le travail fourni dans
la première période et que l’on retarde corrélativement le moment
auquel la consommation par personne passera de cç,
à g*
Si une telle éventualité se produit, il semble clair que, dans
le programme optimal, le travail fourni sera inférieur à ny P,
depuis l'instant #/—=o jusqu’à une certaine époque 6, que la
consommation par personne restera fixée à son minimum jusqu'à
une époque #, postérieure à 0 et qu’elle passera à la valeur
c* a partir de la période £,+ 2.
Afin de faire apparaître des conditions suffisantes pour
l’optimalité d’un programme #*, nous allons comparer ce programme
à un programme possible #æ* +8 Æ choisi de telle manière
que:
En opérant comme pour la dérivation des formules (+
(85), on obtient:
87)
Malinvaud - pag.
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
les coefficients a, et |, étant définis par les formules (73) et
(86). De plus. on sait que:
o-1 +
SK, S TT (1 +72) 3 Kon,
° saut
Des conditions suffisantes pour l’optimalité de Ææ* semblent
donc être définies comme suit:
I) les 4, sont nuls pour t=<o
2) les x, sont nuls pour t >t,
3) l’égalité suivante est satisfaite:
(38,
Foe TT
0 4
N (x + fix) — yAlo—0+1
fm
J.
Il serait fastidieux de vérifier que, si le taux d’intérét normatif
est positif, un programme possible qui satisfait les conditions
1), 2) et 3) est effectivement optimal. Contentons-nous
de quelques remarques.
La condition 1) est une égalité marginale traduisant le fait
qu’il n’est pas avantageux de réduire le travail dans la période
{ — 1 pour l’augmenter dans la période # de telle façon que le
capital K,,, reste inchangé. L'égalité «,=o0 implique qu’il n’est
pas avantageux de réduire la consommation de la période ?
pour augmenter celle de la période #+1 sans modifier K,,;.
Enfin, l’égalité (88) implique qu’il n’est pas avantageux d’augmenter
le travail durant la période 9 pour augmenter la consommation
dans la période #,+ 1 sans diminuer le capital à
l’instant &, +1.
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Il n’est pas très aisé de déterminer un programme possible
qui satisfasse les conditions 1), 2) et 3). Néamnoins, on peut
observer que ces conditions pertant uniquement sur les variables
x, et aue l’égalité (88) peut s’écrire:
0
g et h étant les deux fonctions introduites ci-dessus et repré
sentées sur la figure 3. De plus, la condition d’équilibre
plique:
(go)
(1
=) La — Ql
x
j
DO
On pourra dès lors opérer de la manière suivante
Utiliser la récurrence (go) en sens inverse pour calculer les
valeurs des x, à partir de x, 41 = *x, by étant supposé connu
(Faire comme si cette récurrence s’appliquait pour t—8
et t=1.).
Déterminer la valeur de #, - 0 à partir de laquelle le membre
de gauche de l’égalité (89) excède la quantité log à v.
(Pour ce faire, on peut utiliser les courbes re .sentatives
de la figure 3).
En retenant provisoirement la valeur obtenue pour x,, ulilicer
en sens inverse la récurrence
f “
Ou
+
re
hix, .
a
~~
pour déterminer les valeurs antérieures ue
FT
Malinvaud - pag. 60
370
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
A l’aide de la condition d’équilibre (76), déterminer par
récurrence en sens inverse les valeurs de n, à partir de
Mp,1 7 "m (en retenant c, ,=c,,).
- Utiliser la définition des x, et les valeurs obtenues des n,
pour en déduire les valeurs correspondantes de
K,
P,
- Choisir la valeur de #, et réviser en hausse par approximations
successives les valeurs de x,, et de m de telle manière
K S [ ‘( pe
que P = P — C Que l’égalité (89) soit satisfaite exactement,
que x, soit inférieur à xx et m inférieur à ny et que
h(x,,,) — h(x,) soit inférieur à g(*,,1)- (On profite alors de
ce que l'égalité (90) ne s’applique ni pour t=t, ni pour
t=B).
L'évolution du programme optimal est schématisée sur la
figure 5. Le cas étudié ici correspond à une situation dans laquelle
le capital initial est peu important. Néanmoins l’accu-IG.
5] Malinvaud - pag. 70
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIQUE ETC.
371
mulation du capital n’est pas poussée dès le début au rythme
le plus élevé possible. Au début la productivité marginale du
travail serait trop faible. Bien que pauvre, la collectivité préfère
ne pas travailler au maximum possible.
On peut observer que la valeur x,,, est inférieure à la valeur
xy pour laquelle la productivité marginale du travail est
égale à Àv. En effet, h(xy)=log Av. L'égalité (89) implique
alors :
Ivan
nxn,
Or, par construction, g(x,,,) excède h(x,,,) - h(xe). Par suite
h(xw) excède h(x,,,); et xx excède x, Il se peut donc très
bien que le travail fourni par personne soit égal à #m dès l’instant
o, bien que x, est plus faible que xy.
Quelqu’incomplète qu’elle soit, cette étude du cas dans lequel
la fonction d’utilité est linéaire doit suffire à illustrer une
méthode qui convient pour la déterm‘nation d'un programme
optimal.
Elle montre aussi le caractère peu satisfaisant d’une formulation
dans laquelle les utilités marginales ne dépendent ni
de la consommation ni du travail fourni par personne. Dans le
programme optimal, la consommation subit des sauts brusques
qui s’accordent mal avec la notion intuitive d’une croissance
harmonieuse.
Il semble que l’on devrait se méfier, même pour les applications,
de modèles dynamiques dans lesquels la fonction à
maximer est purement linéaire. Les solutions obtenues avec de
tels modèles dépendent fortement des contraintes imposées aux
variables sur lesquelles porte la maximation (ici Cm» CM» ”m
et nm). Or il y a toujours un certain arbitraire dans le cnoix
de ses contraintes.
Bien entendu. si les conclusions auxquelles nous sommes
3 |
Malinvaud - pag. 71
372
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
parvenus sont aussi tranchées, c’est que nous nous sommes
contentés d’une représentation très sommaire des contraintes
techniques. Considérant un modèle avec deux secteurs,
T.N. SRINIVASAN (I964) a toutefois obtenu des résultats qui
présentent, quoique à un degré moindre, le même caractère
paradoxal. La difficulté semble bien être générale pour les modèles
contenant une fonction d’utilité purement linéaire.
7. LE MODELE DE RAMSEY
Le modèle qui a été employé dans cette étude peut être
rapproché de celui introduit par F.P. RAMSEY dans son article
rélèbre (1928).
RAMSEY utilise une représentation continue du temps. Les
symboles C, N, c, n désignent alors le flux de consommation
par unité de temps, le flux de travail par unité de temps … etc.
Ce sont évidemment des fonctions de l’instant # considéré. Les
symboles K et P désignent le capital et la population à l’instant
£.
RAMSEY suppose aussi que la production est instantanée.
La production a chaque instant dépend uniquement du flux de
travail et du stock de capital au méme instant. Le flux de production
a I'instant ¢ est une fonction f(N, K, ?).
Cette production est affectée immédiatement soit à la consommation,
,soit à l’accumulation du capital. Il n’y a aucun
délai dans la mise en oeuvre de ce capital. La condition (7)
d'équilibre à l’instant # devient ici:
dK
Les simplifications qu’implique une telle représentation sont
5] Malinvaud - pag. 72
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC,
345
assez sévères, peut-être même plus que celles résultant du mo
dèle introduit au début de ce mémoire.
Avec une représentation continue du temps, la fonction
d’utilité 7, doit être écrite sous la forme:
(12°)
7
Con,
: désignant le taux d’intérêt normatif instantané. Moyennant
ces modifications, nous pouvons maintenir la même définition
pour les programmes optimaux. Le programme @Æ% sera dit
optimal s’il est possible et s’il n’existe aucune valeur de T
et aucun programme possible Æ*+5 M tel que &U,>0 et
? UZo pour tout /=T.
De même, la définition des programmes réguliers peut être
aisément adaptée. Les égalités marginales qui remplacent les
équations (22) pourraient être obtenues directement par application
des règles du calcul des variations pour la maximation
de l'intégrale (12’). C’est ainsi que RAMSEY les avait déterminées.
Afin de montrer la similitude avec l’approche emplovée
ici, nous allons les déduire des équations (22).
Dans le modèle comportant une représentation discontinue
du temps, admettons que la période de production ait la durée
dt, et non plus la durée 1. Admettons que la quantité de travail
utilisée pendant cette période soit égale à N,dt, la production
à f,(N, K,)dt et la consommation en à à C, df Acanettons
enfin que l’utilité pour la période soit égale à U,(c, n,)di
et le taux d’intérêt normatif par période à edt. Les équations
(22) deviennent :
J
Malinvaud - pag. 75
374
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Supposons alors que la période dt tende vers zéro, la fonction
f, (N,, K,) tendant vers la limite f(N,, K,, #) et la fonction
U, (c, n,) vers la limite U (c, mn, #). Les équations cidessus
impliquent alors:
(TE 3}
{
d T7” 1#? ,*
7 (Uo) + uv K
uv’
a
I dP\
77 LV
Un programme régulier sera par définition un programme
possible satisfaisant le système des équations (7’) et (227), équations
différentielles sur les fonctions N, K et C de £.
L’inégalité (23), qui figure dans la conditions 1, deviendrait
pour une période de production égale à dt:
K+ Coat,
(1 tx at) Kio, + Co pdl =
Lorsque dt tend vers zéro, cette inégalité implique:
(23)
K
aR
dK _
7 ">
m étant le nombre 4 - I.
Moyennant cette modification, on pourrait encore démontrer
l’ontimalité de tout programme régulier qui satisfait la
5] Malinvaud - pag. 74
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC, 375
condition 1, dans un modèle où les fonctions f et U satisfont
l'hypothèse I.
RAMSEY considérait en fait un cas particulier de ce modèle,
cas dans lequel la population était constante, le taux d'intérêt
normatif était nul et les fonctions f et U ne dépendaient pas
de # (ce qui excluait tout nrogrès technique). Les cond:
tions (22) devenaient alors:
(227)
RAMSEY avait observé que l’on pouvait procéd«T uis“m
à une première intégration du v:teme défini par °
et le remplacer par le suive
(na,
u, étant une constante d’intégration. En effet, toute solution
de (92) est bien solution de (7) et de (227). Il suffit de vérifier
que la première équation de (22”) est satisfaite. En dérivant
par rapport à # la seconde équation du système ( 92), on obtient :
Talinvaud pag.
5
376
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Apres dérivation, la premiére équation devient:
dC ,dN dK dK
oN tw
qui, reporté dans l’équation précédente, implique:
rer, 4 ]dK , AN _
Ul fat 5 UDG + (WU, fe +0) di — 0
En tenant compte de la dernière équation de (92), on retrouve
bien les équations (22”).
La détermination et l’étude des programmes réguliers sont
évidemment facilitées par la substitution de (92) au système
défini par (7°) et (22”). Malheureusement la première intégration
a laquelle RAMSEY a pu procéder ne semble pas se généraliser
aux cas dans lesquels l’une quelconque de ses hypothèses
particulières n’est pas vérifiée.
RAMSEY se limite encore à l’étude de deux types particuliers
de programmes réguliers :
1) Dans le programme régulier, C,, N, et K, tendent quand ¢
croît indéfiniment, vers trois valeurs Cy, N,, et K_ telles
que, pour ces valeurs, U',=U’,=o0 et fx >0. La condition 1
est alors satisfaite puisque, au moins à partir d’une certaine
valeur de # le membre de gauche de (23°) excèdera tout
nombre positif choisi à l’avance et plus petit que la valeur
limite de fx. Quand il existe, un tel programme régulier est
bien optimal.
5] Malinvaud - pag. 76
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
vf |
2) Dans le programme régulier, C,, N, et K, tendent, quand
croit indéfiniment, vers des valeurs c, N et K telles que,
pour ces valeurs, fx=0, U’,>0 et U',<o. Alors, la condition
I n’est pas satisfaite. Il semble que l’optimalité du
programme considéré devra être étudiée dans chaque cas
particulier que l’on pourra rencontrer.
Bien qu’elle ait été limitée & 1’examen d’un modèle tu
simple, cette étude laisse encore sans réponse un certain nom
bre de questions. Elle montre sans doute l'ampleur des problè
mes que pose la détermination des croissances optimales
Malinvaud - pag. ,,
378
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
REFERENCES
FriscH R. (1962), Dynamic Utility. « Econometrica », juillet 1964.
Koopmans T. (1960), Stationary Ordinal Utility and Impatience. « Econometrica
», avril 1060.
RADNER R. (1962), Optimal Growth in a Linear Logarithmic Economy.
Working paper N. 51, Center for Research in Management Science,
University of California, Berkelev. novembre 1067.
Ramsey F.P. (1928), A Mathematical Theory of Saving. « Economic Journal
», décembre 1928.
SRINIVASAN T.N. (1964), Optimal Savings in a Two Sector Model of Growth.
« Econometrica », juillet 1964.
TINBERGEN ]. (1960), Optimum Saving and Utility Maximization over Time.
« Econometrica », avril 1960.
TINBERGEN J. and H.C. Boos (1962), Mathematical Models of Economic
Growth. § 2, 4, Mc Graw Hill, 1062.
5] Malinvaud - pag. 78
SSH, N
KOOPMANS
About the very interesting and remarkable condition 1, 1. Wa.
indicated that it was a sufficient condition — is it also a necessery
condition for the one member family of paths that satisfies the
three recursive conditions to be an optimal path? Or if it isn’.
is there any example of an optimal path that does not meet that
rondition?
MALINVAUD
Yes, there is an example of an optimal path that would not meet
that condition: in the case of the linear logarithmic model with
epsilon equal to zero, condition I is not satisfied because the lefthand
member of (23) is just equal
KOOPMANS
I have the impression, though, that in most cases this condition
is just picking out that one path for which the recursive equations
can hold for all times. Is it so that any path that does not meet
condition 1 but satisfies the recursive requirements necessarily violates
at some finite time, the sign restrictions on capital or consump:
tion? Or is that not correct?
Malinvaud - pag. 79
380 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
MALINVAUD
No, I do not think that is correct. For instance in the linear
logarithmic case with epsilon greater than 1, there is a whole family
of paths which meets the recursive requirements and the sign restrictions.
Just one path satisfies condition 1 and is therefore optimal.
HAAVELMO
I have a somewhat strange question which may or may not be
relevant to what Professor Koopmans said. It is this: it seems that
political groups of powers often see it as their task to try to impose
upon people, or to convince people that they should accept a smaller
discount rate for evaluating the future gains from development
projects than would appear to be the individual discount rate, and
as time passes, people often say that the politicians were right.
Now I have a feeling that this may have some connection with
KoopMAN’s thesis, but I am not quite sure.
ALLAIS
I will begin with a few remarks relating to both papers. First
point: both papers use a single preference function. Perhaps this
may be useful, but it can be quite dangerous for a very drastic
and very strong hypothesis is introduced into the models and some
of the conclusions derived using this hypothesis can be questioned.
Even if a single preference function could be assumed, the form of
this preference function would be open to discussion. In the
(*) From here on the discussion concerns both the paper presented by
MALINVAUD and the earlier one presented bv KcopMANs
-5] Malinvaud - pag. 80
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
38
Koopmans paper the preference function is a function of one varia
ble x, only; in the MALINVAUD paper it is a function of two varia
bles n, and c, work and consumption. The way in which the utility
functions are discounted is also open to discussion. From the point
of view of the theory of optimum allocation of resources I do not
see any economic reason for such discounting. It is a pure hypothesis,
and in the general theory of optimum allocation of resources
no justification of this procedure is to be found.
In my second point I join Prof. FriscH’s position, but only
from a theoretical point of view. I think that from the standpoint
of theory, it is very interesting to separate the problem of optimal
economic growth into two problems. The first is the study of what
happens if we limit ourselves to the production function; the second
is the introduction of preference functions. I think that in economic
programming, this procedure is not valid, but from the point of
view of theory, it is very useful indeed, since the difficulties resulting
from the consideration of the utility function can be avoided.
Namely, is it possible to consider only one utility function? and if
so what utility function must we consider? etc. In fact, we can
obtain very general results even if some strong hypothesis such as
convexity in the ordinary sense is not taken into account.
Third point: for this reason, I think my point is bound up
with the second one. Professor Koopmans has said that there is no
optimal path with a negative value of the rate ¢ of his paper. Certainly
this conclusion is absolutely true for the model considered by
Professor Koopmans. All these models are logically consistent. But
in my opinion it is very interesting to study independently the path
which can be considered as optimum if no attempt is made to take
the psychological point of view into account. And in this case very
precise conclusions can be derived even if the rate o has a negative
value.
Fourth point: so far as MALINVAUD’s paper is concerned, I apologize
for repeating what I said in Cambridge last July. Under the
hypothesis considered by MALINVAUD, the rate of interest must be
greater than the rate of growth of primary income. This result is
5] Malinvaud - pag. 81
382
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
not open to doubt if, but only if, we are limiting ourselves to the
physical point of view. But if human psychology is taken into
account, this result does not remain valid. There are two reasons
for this; I can give two examples. The first case is where the shape
of the time preference curve implies a high preference for the future.
In my book « Economie et intérét » in 1947 I studied a model in
which the difference i-p between the rate of interest and the rate
of growth of the primary income is negative, but nevertheless there
is optimality in the paretian sense with an infinite horizon. My
second example relates to the case where the utility functions are
functions not only of consumption, but also of capital goods. If
people want very much to possess capital goods, then there can be
an optimal path with a negative difference i-p.
My fifth point is that my Econometrica paper is only one study
carrying forward things described in many preceding papers and I
believe that I gave very precise consideration to the problem of the
optimal path as long ago as 1947 in my book « Economie et intérét
», that is fifteen years before the DEsrOUSSEAUX, PHELPS, JoAN
ROBINSON, SWAN and Von WEIZSACKER studies which Professor
KoorMANS mentioned.
MALINVAUD
In order to avoid the conclusion that no optimal program would
exist, one has suggested that we drop the assumption of an infinite
horizon. I cannot accept this point of view. Considering an infinite
horizon often leads to interesting results concerning the non-optimality
of programs which would appear as optimal if time were
limited to some specific date, however far in the future this date
may be. In such non-optimal programs, the economy is accumulating
too much capital all the time and never take for consumption
the full benefit of its high capital endowment. I see no way of
discarding these programs if a finite horizon is adopted and if the
terminal capital stock is taken as a constraint.
51 Malinvaud - pag. 82
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 383
With regard to one remark made by Professor ALLAIS, I should
make clear I did not prove, in the paper presented here, that the
interest rate had to be larger than the growth rate. The inequality
between the two rates was introduced as a sufficient condition for
a general result concerning optimal programs. However, in most
particular cases I considered here, the rate of interest is larger than
the rate of growth all the time.
ALLAIS
The point is, if this proposition cannot be proved in a genera.
way, there cannot be an optimal path with the condition : smaller
than g. I therefore cannot see the meaning of the preceding proposition.
AAT INVATIT
In this paper, I introduced the condition only because I was
unable to find a result without it. But I may remark incidentally
that a finite horizon was present in the cases where optimal programs
were found with an interest rate smaller than their
orowth rate
KOOPMANS
Supplementing Professor MALINVAUD’s remarks I do not think
that the response to the difficulties I have pointed out should be to
drop the infinite horizon. I think it you make a very large horizon,
the same difficulty that shows itself starkly with an infinite horizon
will also show itself somewhat less starkly but in an equally disturbing
manner with a verv large finite horizon. Thus the infinite
Malinvaud - pag. 83
354
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
horizon is a mathematically explicit way of bringing the presence
of a mathematical limitation to ethical thought to our attention.
Professor HAAvELMO pointed out that sacrifices enforced at one
time may later be endorsed by public opinion, perhaps when the
oenefits from that earlier sacrifice become apparent. This would
re-enforce an idea I have expressed in my paper but not in my
presentation. Perhaps the discount rate p itself should be a function
of the level of consumption reached. I would expect that if we get
close to affluence p diminishes, It is conceivable that one could find
ways of making p depend on the consumption level in such a way
as to avoid the difficulties that I have encountered.
Several speakers have asked whether these strange results are
due to the assumption of a double commodity that can both be
eaten and used as capital. So far I do not know of more detailed
or elaborate studies directed to the same question; so I can only
state my hunches. 1 would think that essentially the same results
would be found with other forms of indefinitely continuing population
growth as long as the percentage rate of growth stays above
some positive percentage. I think one would find the same difficulty
even more strongly if one introduces technological progress in addition.
It is possible, however, that resource limitations not ultimately
compensated by technological progress could work in the opposite
direction and would do away with the conclusion. If the single
social preference function is replaced by individual preference functions
and a market mechanism is introduced of the type that
Professor Arrais has stressed several times in the discussion, I
would not want to venture a guess as to whether the difficulty I have
encountered would remain or disappear.
As to Professor ALLAIS’ statement that there is no reason for
discounting, I started out with that idea myself. But I found that
for there to exist an optimum path, I had to either discount or
discriminate against people on the basis of how many there are in
a given generation.
51 Malinvaud - pag. 84
DYNAMIC STRUCTURE AND ESTIMATION
IN ECONOMY-WIDE ECONOMETRIC MODELS
FRANKLIN M. FISHER (*)
Massachusetts Institute of Technology - Cambridge, Mass. - U.S.A.
.. INTRODUCTION
ND CLASSIVICATION OF ESTIMATORS
1.1. General Introduction
This paper is concerned with the techniques of and the
problems in the structural estimation of economy-wide econometric
models. Briefly stated, the essential general features
of such models which raise special problems for estimation are
as follows. They tend to involve a large number of equations
and variables; they are nearly closed in the sense that most of
the variables of the model are endogenously determined; they
are dynamic and essentially interconnected in the sense that,
considered as dynamic systems, they are indecomposable; finally,
the disturbances from different equations tend to be correlated
with each other and with their own past values. All of
these features will be discussed at greater length below, and all
(*) This paper was largely written during my tenure of a National
Science Foundation Postdoctoral Fellowship at the Econometric Institute
of the Netherlands School of Economics. I am indebted to T. J. ROTHENBERG
for helpful conversations and to L. R. Krein and E. Kun for criticism of
an earlier draft but remain responsible for errors. The paper forms part
of mv contribution to the Brookings-SSRC econometric model project.
|
Fisher - pag.
386
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
25
of them raise problems of varying magnitude for structural
astimation.
As is well known, there are now a fairly large number of
alternative estimation techniques available for such estimation.
Such methods fall into classes which differ in the assumptions
made or amount of information taken into account. They also
have different properties. In general, a great deal is known
concerning the asymptotic properties under ideal conditions of
most of these estimators; rather less is known of small sample
properties; and a very few results are available on their relative
robustness — the relative degree to which they stand up to
such things as multicollinearity, specification error, and serial
correlation in the disturbances of the model.
This paper begins by reviewing the known properties of
the principal estimators in the context of economy-wide models.
We observe that the features of such models mentioned above
make the use of even the best of such estimators rather suspect
in its original form, while the size of such models makes them
literally unavailable when time series of lengths usually
encountered are the data involved. This leads naturally to
estimation using instrumental variables in some form, and the
second half of the paper is devoted in one way or another to
exploring the question of how appropriate instrumental variables
should be chosen. It is argued that this is best done
through continual application of the a priori structural information
which governs the formulation of the entire model in
the first place, rather than through relatively arbitrary statistical
devices.
1.2. Classification of Estimators
For our purposes, the estimators which have been proposed
for structural estimation may be divided into three classes. The
first of these consists of ordinary least squares and its generali-‘6]
Fisher - pag. 2
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
387
zations. The second includes: two-stage least squares; limitedinformation
maximum likelihood; the other members of
Theil’s k-class; Theil’s h-class; and Nagar’s double k-class (2).
All the estimators in this group have the common property that
whereas (unlike ordinary least squares) they take account of
the simultaneous nature (if any) of the equations in the model
to be estimated, they use only a priori restrictions on one
equation at a time. Accordingly, we shall call such estimators
« limited information » methods. The last class of estimators
consists of those methods which do use information on all equations
at once, what we shall term « full information » methods
Among these, of course, is full-information maximum likelihood,
but the class also contains A. ZELLNER and H. THEIL’s
three-stage least squares, an estimator recently proposed by
T.J. ROTHENBERG and C.T. LEENDERS called « linearized
maximum likelihood », and the simultaneous least squares
estimator of T.M. Brown (3).
In principle, all of the above estimators make use of all
exogenous and lagged endogenous variables in the model as
predetermined instruments. As indicated above, for reasons to
be discussed below, this cannot always be done or is not always
desirable, and in such cases other methods which so employ
only some of the exogenous or lagged endogenous variables
must be used. We shall discuss the problems raised in such
situations below, observing here only that, given the choice of
variables to be treated as predetermined, most of the estimators
just classified have exact counterparts in such circumstances.
(?) See THEIL [32, pp. 353-354] and Nacar [23].
‘\ See ZELLNER and THEIL [37]. ROTHENBERG and LEENDERS [26j, and
[M Brown (TF
bi
Fisher - pag. 3
388
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
2. ORDINARY LEAST SQUARES
2.1. Assumbtions and Proberties
Ordinary least squares has a number of desirable properties
when appropriate assumptions are satisfied. Briefly, if the
explanatory variables in the equation to be estimated are either
non-stochastic or distributed independently of all past, present,
and future values of the disturbance term in that equation, if
the disturbance term is serially uncorrelated and homoscedastic,
and if there are no a priori restrictions on the parameters to be
estimated, then ordinary least squares is the best linear unbiased
estimator. In addition, if the disturbances are normally distributed,
then ordinary least squares is the maximum likelihood
estimator.
These assumptions can be weakened in several ways. First,
if the explanatory variables are not independent of the disturbance
term but are uncorrelated with it in the probability limit,
then ordinary least squares ceases to be unbiased but is consistent.
If the disturbances are serially correlated, ordinary
least squares loses efficiency but retains consistency provided
that such serial correlation does not affect the validity of assumptions
concerning the correlation of the current disturbance
term and the explanatory variables (a matter to which we shall
return) (*). Finally, ordinary least squares presents no particlar
difficulties of computation.
As is well known, however, the minimum assumption for
the consistency of ordinary least squares — that the explanatory
variables are uncorrelated with the disturbance term — cannot
be maintained if the equation to be estimated is one of a system
of simultaneous structural equations. In this case, ordinary
least squares loses even consistency when used as an estimator
(*) See THEIL [32, pp. 219-225] or JOHNSTON [15, pp. 192-195] for a
discussion of this case.
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of a structural equation, although not when used to estimate
the equations of the reduced form.
This argument is not sufficient, however, to dismiss ordinary
least squares from consideration as an appropriate estimator
in large econometric models. In the first place, there is the
question emphasized by H. WoLb (°) as to whether such models
really should be simultaneous given the nature of causation.
Second, the issue is not of the yes-or-no variety as it is often
made to appear; rather, if the model is such that correlation
between the disturbance term and the explanatory variables
in the given equation can be appropriately assumed to be small
(rather than zero) or if the variance of the disturbance is known
to be small, then least squares will be almost consistent (°).
One may then be willing to accept the small inconsistencies
involved for the sake of the other properties of the estimator,
principally its relatively small variance around its probability
limit. We must therefore go on to ask when this is likely to
happen and when the assumptions of WoLD’s recursive model
are likely to be approximately satisfied.
2.2. Recursive Systems and Necessary Assumptions
Suppose. that the model to be estimated is:
(2.1)
Ay,+ By, ,+Cz,+u,
where u, is an m-component column vector of disturbances;
y, is an m-component column vector of current endogenous
variables; z, is an #-component column vector of exogenous
variables (known at least to be uncorrelated in the probability
limit with all current and past disturbances); A, B, and C are
constant matrices to be estimated; and (I - A) is nonsingular,
{ WoLD and JURÉEN [34, 50-51] and other writings
(*) Worn and FaxÈr lag"
wo
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while A has zeros everywhere on its principal diagonal. The
assumption that there are no terms in y,_, for 8>1 involves
no loss of generality in the present discussion, since it can
always be accomplished by redefinition of y, and expansion of
the equation system and will be used only for convenience in
dealing with the solution of (2.1) regarded as a system of stochastic
difference equations.
If:
(R.1) A is triangular;
(R.2) The variance-covariance matrix of the current disturbances
is diagonal;
(R.3) No current disturbance is correlated with any past
disturbance;
then the model is recursive and does not violate the assumption
that in each equation the disturbance term is uncorrelated with
the variables which appear therein other than the one to be
explained by that equation. Ordinary least squares is then
a consistent estimator and is the maximum likelihood estimator
if each element of , is normally distributed and homoscedastic.
To see that the no-correlation assumption is not violated,
we solve the system for y,, obtaining:
(2.2) y,=(1-A)'By, ;+(I1-A)Cz,+ (I-A) lu.
fo wv bl
Denote (I- A)~! by D and note that it is triangular by (R.1).
We may take the zero elements to lie above the principal diagonal.
Assuming that DB is stable, we have: (7)
(2.3)
æ 6
y, = 2 (DB) (DC z, * Du,
(7) We shall not discuss the assumption of the stability of DB in any
detail at this point. If it is not stable, then it suffices to assume that the
model begins with non-stochastic initial conditions. Obviously, if stability
fails the assumption of no serial correlation becomes of even greater importance
than if stability holds. We shall return to this and shall discuss
the question of stability in general in a later section.
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Denoting the covariance matrix of u, and y,_, by W(6) with
columns corresponding to elements of u, and rows corresponding
to elements of y,_, » and that of w, and u,_, by V(8) (which
is assumed to be independent of ¢), with columns corresponding
to «#, and rows to u,_, :
(2.4)
W{o)=E (DB) (DV(0i,
8=0
Since, by (R.3), V(8)=o0 for 6>o, this becomes:
(2.5)
W(o) = DV(o).
By (R.2), V(o) is diagonal, hence W(o) is triangular with
zero elements above the principal diagonal. Thus any element
of y, is uncorrelated with all higher-numbered elements of =,
Similarly,
(2.6)
W(1)=S (DB) (DV(0.
Hence all variables which appear in any given equation in (2.1)
save that variable which is to be explained by that equation
are uncorrelated with the disturbance from that equation, as
stated.
We have gone through this demonstration in detail partly
for later purposes and partly to exhibit the way in which each
of the assumptions (R.1)-(R.3) enter. We must now ask
whether those assumptions can be weakened.
In the first place, it is clear that the triangularity of A is
crucial. From (2.5), if A and therefore D is not triangular, then
W (0) will not be triangular either in general, and the elements
of y, cannot be taken as uncorrelated with higher-numbered
disturbances. This is well known, as in this case the system
(2.1) is truly simultaneous. In such a case, ordinary least
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squares will be inconsistent for at least one equation in the
model.
What is less often realized in practice is the role played by
the assumptions on the disturbances. Because of the simplicity
and other advantages of ordinary least squares, there is a natural
tendency to settle for a triangular A and to overlook the
fact that such triangularity does not suffice to make ordinary
least squares consistent (%).
To see that such assumptions are generally required, consider
first the assumption that V(o) is diagonal. If this fails,
then (2.5) shows that W(0) cannot generally be taken to be
triangular, whence ordinary least squares will be inconsistent.
This corresponds to the intuitive idea that if a high-numbered
and a low-numbered disturbance are correlated, the endogenous
variable corresponding to the low-numbered disturbance cannot
be taken to be uncorrelated with the high-numbered disturbance
even if thre is no direct influence through the explicit
equations of the model. Indeed, not only is the diagonality of
V(o) required for the consistency of ordinary least squares,
but also, if nothing more is known of the coefficients of the
model save that A is triangular, such an assumption is necessary
for the very identifiability of the equations (°).
It is possible, however, to alter the assumption of no serial
correlation. Clearly, this enters in both (2.5) and (2.6) because
y,_ appears in the model. If this were not the case, the assumption
in question would not be needed for consistency.
In most econometric models, however, and certainly in economy-wide
ones, lagged values of the endogenous variables do
in fact appear. We are nevertheless able to weaken the noserial-correlation
assumption (R.3) to:
(R.3*) B (as well as A) is triangular with zeros above
the diagonal, and for all 0>o, V(0) is triangular with the same
arrangement of zeros so that high-numbered disturbances are
() In fairness, it should be pointed out that Worp’s theoretical writings
are entirely clear on this point. See Worp 36, pp. 358-359], for example.
(®) See F1sHER [10]
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ancorrelated with lagged values of low-numbered disturbances.
Further, either B or all V(6) (A >o) have zeros evervwhere
on the principal diagonal.
If (R.1), (R.2), and (R.3*) hold, every term in (2.4) will
be triangular, so that W(o) will likewise be triangular as required.
Further, W(1) will also be triangular rather than zero
and will have zeros on its principal diagonal, but this will be
all that is needed, since if B is triangular no lagged endogenous
variable appears in an equation of (2.1) explaining a lowernumbered
current endogenous variable.
Intuitively, the general necessity of no serial correlation for
the consistency of least squares is that an element of y,_, is
influenced by an element of «, ;. If that element of u,_, is
itself correlated with a lower-numbered element of #,, then the
corresponding element of y,_, cannot be assumed to be uncorrelated
with that element of u,. Even if V(8) is triangular for
8>0 (or even diagonal) and B is not triangular, the dynamics
of the system will carry serial correlation into relations between
any current disturbance and any current endogenous variable.
If both V(6) and B are triangular, however, such effects are
only carried toward higher-numbered equations.
That triangularity of both V(6) for all 6>>0 and B are
generally necessary in the presence of serial correlation may be
seen from the fact that since D is triangular, the terms in (2.4)
will generally not otherwise be triangular and the fact that if B
is not triangular, even triangularity of W(1) will not suffice.
(The condition as to the principal diagonals can be easily seen
to be required by considering a single-equation model).
Of course, as is also the case for the assumption of the
diagonality of V(0), even if such assumptions fail generally,
similar weaker assumptions concerning certain off-diagonal elements
may hold and yield the consistency of ordinary least
squares for certain equations. The indicated assumptions are
necessary for such consistency in all equations, however. (Such
weaker conditions are fairly readily obtained from the genera
lization of the current discussion given in a later section’
Lo
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2.3. Recursive Systems in Economy-wide Models
Are the assumptions of the recursive model just discussed
likely to be valid for an economy-wide econometric model? In
general, the answer appears to be in the negative.
In the first place, the argument for the triangularity of A
that causation takes place sequentially in time (which, incidentally,
would imply diagonality) is not conclusive if the data
are collected as averages over a much longer period than the
causal interval involved. It may be true that simultaneous
structures are but approximations to underlying recursive ones
with very short time lags; this does not make the matrices involved
triangular, however, whatever it implies about appropriate
estimators (19).
Even if triangularity of the A matrix is satisfied in an
economy-wide model, however, the other conditions discussed
are unlikely to be fulfilled. Even the best specified econometric
models inevitably omit variables the effects of which then enter
the disturbance terms. If the model is well specified, these
effects will not be large and systematic, rather they will be
small and random. Even so, the omitted variables appearing
in the disturbances cannot all generally be expected to be different
ones for different equations. Indeed, one expects there
to be some events which act as shocks on many or all the equations
in an economy-wide model. Such action may indeed be
of different magnitudes for different equations, but it is surely
extremely restrictive to assume zero correlation among the
(1) StroTz [30] considers a model in which the variables are observed
at discrete intervals longer than a short causal period which is allowed to
approach zero — a problem not quite the same as that considered in the
text. He argues that the usual estimators are not approached in the limit
by the maximum likelihood estimator of his model. The status of the
argument is presently in some doubt as Gorman [11] has suggested that
« natural » assumptions on the continuity of the stochastic process generating
the disturbances do lead to the usual estimators in the limiting case
nf simultaneitv.
6
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different disturbance terms. Thus it is very unlikely that V(o)
will be diagonal.
Similarly, it is rather unrealistic to assume no serial correlation
in the disturbances. Disturbances from econometric
models do in fact tend to be serially correlated and while we
shall later argue that correlation between a given element of u,
and a different element of «, , may be small, even a diagonal
V(0) for 0>o will not help. This is especially the case if the
time lag involved in the model is small (the very situation in
which triangularity of A is relatively likely), as in such a case
the effects of a random shock due to an omitted variable are
likely to persist for more than one time period. To put it
another way, it is natural to suppose that as the time period
involved goes to zero, V(1) approaches V(o) which is certainly
not zero (1).
Moreover, there seems little direct comfort in the points
made above that it is sufficient to have B=o0 or to have both
B and all V(0) 6>0 triangular and either B or all such V(6)
with zero principal diagonals. Economy-wide models are generally
dynamic ones so that lagged endogenous variables do
appear. Further, while we shall argue below that a diagonal
V(8) for 6 >o is not quite so unreasonable as it may seem, a
triangular B matrix is wholly unlikely, since this would be a
case in which there were no feedbacks (simultaneous or lagged)
from one variable to another and economy-wide models simply
do not have such a hierarchic structure in view of the interconnectedness
of economic activity.
It is thus evident that even if one is willing to assume a
triangular A matrix, the assumptions of the recursive model
cannot generally be taken as valid in an economy-wide econometric
model. This is especially true if triangularity has been
achieved by the introduction of relatively short time lags. At
the risk of over-emphasis, we repeat. Ordinary least squares
CY See GORMAR
U|
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
does not become consistent when one changes a current endogenous
variable to a recent past value of the same variable even
if triangularity of the A matrix is achieved in this way. The
assumptions which lead to the consistency of least squares require
more than this and all the same difficulties will still be
encountered even if they go unrecognized.
2.4. The Proximity Theorem and Near-Consistency
As already remarked, however, the issue of the use of
ordinary least squares (or indeed of any particular estimator)
is not whether the assumptions thereof are precisely satisfied.
Rather the crucial question is that of how closely they are
satisfied, of how those assumptions stand up as approximations
rather than as exact statements. The problem is not a discrete
one; rather it is continuous. Moreover, the question of goodness
of approximation is itself dependent on the sensitivity of
the properties of the estimator to variation in the assumptions
thereof. In general, the less sensitive is an estimator, the greater
the tolerable deviation from the strict conditions under which it
has desirable properties.
In the present instance, our discussion has largely run in
terms of consistency. Consistency, however, is a rather weak,
although desirable property. Since ordinary least squares has
several other attractive features, we might plausibly be willing
to tolerate small inconsistencies to gain, for example, computational
ease, small variance around probability limits, and so
forth. Tt is thus not sufficient to ask whether the assumptions
under which ordinary least squares is consistent are satisfied;
we must ask whether the fact that they are not generally satisfied
in economy-wide econometric models is likely to be of
much Importance.
This question is formally answered by the Proximity Theo-‘61
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rem of Worp (!?). That theorem states that the inconsistency
of least squares will be small the smaller are the correlations
between the explanatory variables in the equation to be estimated
and the disturbance from that equation and also the smaller
is the variance of that disturbance. A perhaps more illuminating
way of looking at the same thing is to consider the disturbance
as made up of a linear combination of omitted variables.
The inconsistencies in the parameter estimates can then be
shown to be equal to the coefficients of the multiple regression
of the disturbance term on the explanatory variables (!3).
For our purposes, the Proximity Theorem shows that if
(R.1), (R.2), and (R.3) or (R.3*) hold approximately, the
inconsistency of ordinary least squares will be small. Indeed,
that inconsistency will be small in a given equation if the appropriate
columns of
(2.7
premultiplied by the inverse of the variance-covariance mau
of the variables appearing on the right of that equation is smal.
Since that inverse enters the ordinary least squares parameter
estimates in precisely the same way, we may say that (roughly)
relative inconsistencies will be small provided that W(o) and
W(x) are small. Thus, if all terms above the diagonal in A
are nearly zero; if cross-equation covariance between contemporary
disturbances is small; and if there is little serial cor
relation, ordinary least squares will not do too badly.
Unfortunately, there is reason to believe that this will not
generallv be the case. The arguments given above for the
(1) Worp and JURÉEN [34, p. 189 and pp. 37-38]. The Proximity
Theorem as stated by WoLD is one concerning bias; we discuss inconsistency
since unbiasedness is not in any case a property of least squares in models
with lagged endogenous variables. See Hurwicz [14]
(13) See FISHER [8] and THEIL [31]
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failure of (R.2) and (R.3), in particular, are arguments that
the excluded effects are likely to be substantial in practice.
While one may be willing to assume that they are not so in
particular cases, depending on the structure of the model to
be estimated, this seems a dangerous procedure in most economy-wide
models given the high degree of approximation
which such models inevitably involve. The Proximity Theorem
in more general form will be of considerable help to us below
and is of substantial value in other contexts; for structural
estimation in economy-wide models, it seems a weak reed on
which to rest estimation by ordinary least squares.
2.5. Reduced Form Estimation
Our discussion thus far has run in terms of the estimation
of the parameters of structural equations. The simultaneous
model context in which ordinary least squares is most often
thought to be appropriate, however, is not this at all, but
rather in the estimation of the equations of the reduced form.
Here the difficulties in the use of ordinary least squares which
arise from simultaneity apparently disappear as all variables
on the right-hand side of reduced form equations are either
exogenous or lagged.
In this connection, the argument against the use of ordinary
least squares has generally run in terms of lack of asymptotic
efficiency when compared with estimates of the reduced form
which are derived from structural estimates using overidentifying
a priori information. Such lack of asymptotic efficiency
may be particularly important in the event of a structural break
or in the prediction of turning points (14). The argument in
favor of ordinary least squares estimates of reduced form equations
has been the desirability of having forecasts of the endo-(%)
LesNoy [18]
‘61 Fisher - pag. 14
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genous variables which are unbiased conditional on the values
of the predetermined variables (*°). It has also been suggested
that the added asymptotic efficiency in the use of other estimators
stemming from the employment of a priori information
may in fact frequently be quite illusory as such information
may be incorrect (1%).
There is substantial merit in all of these arguments in
various contexts. Fortunately, the issue is rather easy to
decide in the context of estimation of the reduced form of a
dynamic economy-wide econometric model. In the first place,
such a model generally involves lagged endogenous variables.
To estimate even the reduced form by ordinary least squares
when such variables appear on the right-hand side does not
yleld consistent estimates in the presence of serial correlation,
substantially as seen above. Moreover, even if the assumption
of no serial correlation is made, ordinary least squares still
does not give an unbiased estimate of the parameters nor a
conditionally unbiased forecast of the dependent variable.
Nevertheless, one might plausibly be willing to accept such
defects in ordinary least squares for the sake of greater efficiency.
Such efficiency fails, asymptotically, however, if the
overidentifying information on which structural estimation by
other means is based is approximately correct as the issue is
again one of good approximation rather than of correctness (7).
Since restrictions on coefficients are more likely to be good
approximations than are restrictions on disturbances concern
ing which economic theory provides relatively little informa.
tion, ordinary least squares is unlikely to be asymptotically
efficient.
On the other hand, such information as is available on the
small sample properties of the limited-information estimators
‘discussed below) does suggest that asymptotic efficiency may
VV.
“rT
J
-
SH
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
not be remarkably important as the small sample variances
of such estimators are infinite in some cases. Ordinary least
squares certainly does have the property of finite small sample
variances under ordinary conditions, however defective it may
be for other reasons. Ordinary least squares estimates of the
reduced form equations may therefore be appropriate ones to
consider if one is willing to assume that serial correlation is
animportant.
Note that this is not quite the same as the situation as
regards structural estimation already discussed. In that context
several strong assumptions have to be nearly satisfied in
order to justify the use of ordinary least squares. In the present
context, only the assumption of no serial correlation must be
approximately satisfied; if it is, the remaining argument against
ordinary least squares is the one of lack of asymptotic efficiency
and this may be by no means decisive in a world of
relatively small samples (18).
In practice, however, ordinary least squares estimation of
the reduced form of a large economy-wide model is simply
incapable of accomplishment. If all lagged endogenous variables
are treated as predetermined, the number of exogenous
and predetermined variables in any but the most aggregative
economy-wide model is simply too large to permit this type of
estimation in the presence of the relatively low number of observations
ordinarily available.
2. FULL-INFORMATION ESTIMATORS
We now discuss the class of full-information estimators out
of what is perhaps the natural order. because it is relatively
(5) All of our discussion of the effects of serial correlation has overlooked
the existence of estimation techniques designed precisely to deal
with that problem. See for example JoHNsTON [15, pp. 192-195] and
THEIL (32, pp. 219-225]. All of these techniques, however, assume that
there are no lagged endogenous variables in the model, and we have principally
been concerned with the problems raised bv serial correlation when
there are such lagged variables
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easy to dispose of it. We shall then be free to turn our attention
to the class of estimators ordinarily used in these problems
the limited-information class.
It is customary in these discussions to pay lip-service to
full-information maximum likelihood as the optimal estimator
using all information available and then to dismiss it in practice
as too difficult of computation. While it is still true that
such computational difficulties are still prohibitive in practice
for even moderately large systems (1%), such dismissal no longer
suffices. This is the case because there are now two full
information estimators which are known to have the same
asymptotic distribution as full-information maximum likelihood
and which are not particularly difficult to compute. These
are the three-stage least squares estimator proposed by ZELLNER
and THEIL and the linearized maximum likelihood method of
ROTHENBERG and LEENDERS (!°). Since the known virtues of
full-information maximum likelihood are all asymptotic, com:
putational difficulty can no longer be considered a valid reasor
for not using some such method.
As it happens, however, there are more cogent reasons thar
computational difficulty for the abandonment of full-information
methods in practice. However desirable the properties of
full-information methods may be in principle when all assumptions
are met, such estimators suffer relatively heavily from
a lack of robustness in the presence of common practical difficulties.
Thus KLEIN and NAKAMURA have suggested that
full-information maximum likelihood is more sensitive to
multicollinearity than are limited-information estimators (%)
(18) The difficulties are being overcome, however. See EISENPRESS [7%]
(19) ZELLNER and THEIL [37]; ROTHENBERG and LEENDERS [26]. Ro-THENBERG
and LEENDERs give the proof that these estimators have the same
asymptotic distribution as full-information maximum likelihood. See also
SARGAN [27] and MapaNsky [20]. BrowN’s simultaneous least squares {6?]
[which is a member of the full-information class] is known to be consistent
but is not known to have the same asvmptotic distribution as the other
members.
(2°) KLEIN and NAKAMUR/
‘vi Fisher - pag.
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Further, it is evident that all full-information methods are
rather sensitive to specification errors of the types that are
unavoidable in the foreseeable state of econometric models
which are only approximate. In particular, such estimators
have the defects of their merits in that by using information
on the entire system to estimate any single equation, they carry
the effects of specification error in any part of the system to
the estimate of any other part. Since it is clear that some
equations may be thought to be better specified than others as
the quality of economic information is by no means constant
in an economy-wide model, this is a highly undesirable feature.
It seems clear that specification error should be quarantined and
hence that limited-information estimators which are known
to accomplish this are preferable to full-information ones in
large models (*). While it may be desirable to use intermediate
estimators which apply full-information type methods to sectors.
rather than to the system as a whole, -the theory of how this
should be done remains to be worked out.
4. LIMITED-INFORMATION ESTIMATORS
4.X. Availability in Practice
While a great many limited-information estimators have
been suggested, in practice, none of them are available for use
in their original form in estimating an economy-wide econometric
model. This is the case because all such estimators begin
in one way or another with the ordinary least squares estimates
of the reduced form equations. As we have already seen, such
estimates are likely to be difficult or impossible to secure in all
but the most aggregate models because of the low number of
(1) Fisuer [8, p. 155]
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observations generally available relative to the number of predetermined
variables, if all lagged endogenous variables are
treated as predetermined.
In addition, if lagged endogenous variables are so treated,
then we have already seen that such treatment raises considerable
difficulties in the likely presence of serial correlation of
the disturbances. On the other hand, economy-wide models
are generally sufficiently closed to make their equations unidentifiable
if only truly exogenous variables are treated as instruments.
We shall discuss the problems raised by serial correlation
in the next section which will be concerned with the question
of how instrumental variables should be chosen in practice to
avoid inconsistency. In the present section, we shall discuss
the properties of the limited-information estimators ignoring
these problems. Such a discussion is not rendered irrelevant
by the practical difficulty of using all variables that are not
current endogenous ones as instruments when the number of
observations is relatively limited. This is so because given the
variables which are to be treated as predetermined, treatment
of all remaining variables as endogenous results in a situation
in which every known limited-information estimator has its precise
counterpart (#2). Thus, for example, if only certain lagged
endogenous and exogenous variables are to be used, replacing
every other (save the normalized one) in a given equation by
its value as computed from a multiple regression on the instruments
and then regressing the normalized variable on the resulting
variables provides the exact analogue of two-stage least
squares.
Of course. in such a situation, and especially where serial
(3) It must be admitted, however, that if different predetermined variables
are used in replacing each included endogenous variable (as suggested
below), them it is not clear how limited-information maximum likelihood
carries over to such cases. Fortunately, this will not matter for our
purposes
61
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correlation in the disturbances cannot be presumed absent, the
choice of instrumental variables is likely to be of considerably
greater importance than the choice of the particular limitedinformation
method in which such instruments are to be applied.
The latter choice is clearly worth discussing, however,
although the major portion of our discussion will be reserved
for the former one which will be taken up in the two following
sections.
4.2. Classification and Large Sample Properties under Ideal
Conditions
The limited-information estimators in common use are those
of THEIL’s k-class. Chief among these are two-stage least
squares, limited-information maximum likelihood, and an
estimator due to NAGAR (¥). Another class of estimators, the
h-class, has also been suggested by THEIL, and NAGAR has
recently proposed still a third class, the double k-class (2).
For our purposes such subdivisions will not be particularly
important. What will be important is the fundamental distinction
between limited-information maximum likelihood and all
other proposed limited-information estimators. Alone among
suggested members of the k-, h-, and double X-classes, the
fundamental distinguishing parameter (k in this case) is
stochastic in limited-information maximum likelihood, being
determined as a root of a stochastic determinantal equation.
As we shall see below, this distinction aside from making
limited-information maximum likelihood somewhat cumbersome
to compute leads to a lack of robustness in that estimator in
the presence of multicollinearity. Such a lack is not shared
by the other estimators of the limited-information class.
(¥) THEIL [32, pp. 231-232]; NAGAR zz].
(3 THEIL [32. DD. 353-254]: NAGAR [23]
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Indeed, when one looks only at the properties of limitedinformation
estimators under ideal conditions, there are relatively
few grounds for choice among them. In the next subsection
we shall consider the little which is known of their small
sample properties, here we merely observe that they all have
essentially the same large sample properties. It can be argued
that limited-information maximum likelihood has the desirable
property of treating all included endogenous variables in an
equation symmetrically; indeed, CHow has shown that it is a
natural generalization of ordinary least squares in the absence
of a theoretically given normalization rule (?).
On the other hand, such an argument seems rather weak
since normalization rules are in fact generally present in practice,
each equation of the model being naturally associated with
that particular endogenous variable which is determined by the
decision-makers whose behavior is represented by the equation.
The normalization rules are in a real sense part of the
specification of the model, and the model is not completely
specified unless every endogenous variable appears (at least
implicitly) in exactly one equation in normalized form. For
example, it is not enough to have price equating supply and
demand, equations should also be present which explain price
quotations by sellers and buyers and which describe the equi
librating process. (For most purposes, of course, such addi
tional equation can remain in the back of the model builders’
mind, although the rules for choosing instrumental variables
given below may sometimes require that they be made explicit.)
Thus, symmetry may be positively undesirable in a wellspecified
model where one feels relatively certain as to appropriate
normalization, although it may be desirable if one wishes
to remain agnostic as to appropriate normalization. So far
as arguments of this type or from large sample properties under
=,
*.-Fisher
- pag. 2.
106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
vel
ideal conditions are concerned, then, there seems little or no
reason for preferring one limited-information estimator to
another.
4.3. Small Sample Properties
The situation is not very different at the present time when
one considers small sample properties. To date, relatively little
is known about these and work has proceeded largely by means
of Monte Carlo experiments. Moreover, all such experiments
and such analytic work as is available have been exclusively
concerned with the case in which lagged endogenous variables
do not appear (or at least are not used as predetermined instruments)
and the analytic work has dealt only with those members
of the k-class with non-stochastic &. In the present context,
the former limitation is a severe one. Nevertheless, it seems
worth briefly discussing what is known about small sample
properties in such cases as the situation when lagged endogenous
variables are present is probably no more hopeful.
The principal point that has emerged on small sample properties
of limited-information estimators is that the sampling
variances involved are infinite in some cases. Such a
conclusion is borne out both from the analytic work that
has been accomplished to date and by the results of the Monte
Carlo experiments that have been performed (%). It seems
idle to hope that this circumstance does not occur when lagged
endogenous variables are present in the model.
In practice, this unhappy circumstance has a number of
consequences. First, it is clearly the case that relatively little
reliance can be placed on judgments of goodness of fit derived
from consideration of asymptotic standard errors. Such asvmp-(?°)
See BASMANN [4], [5], BERGSTROM [6], NaGAR [22], and SARGAN
[28] for the analysis. JOHNSTON [I5, pp. 275-295] summarizes most of
the Monte Carlo experiments: see also OUANDT [2:5]. [26]
[6] Fisher - pag. 22
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totic standard errors are obviously inappropriate when directly
taken as approximations to an infinite sample variance and may
or may not be reliable when used to derive approximations to
the probability that an estimate diverges from the true parameter
by more than a given amount. In general, the latter
approximation is probably better for small divergences than
for large ones as the normal approximation to the small sample
distribution is almost certainly worst in the tails (¥).
Second, as already indicated, the absence of this property
in ordinary least squares makes the latter estimator rather more
attractive than would be the case if limited-information estimators
always had finite variance. Certainly, there is a certain
amount of justification for using ordinary least squares as an
approximation while building the model provided that assumptions
(R.1)-(R.3) are not too badly violated (which we have
argued cannot be assumed in economy-wide models). Further,
QUANDT has recently suggested combining ordinary least squares
and limited-information estimators to take advantage of the
fact that the latter are consistent while the former has a finite
variance (%).
Furthermore, the infinite small sample variance of limitedinformation
estimators casts doubt on the convergence in some
cases of the expansions used by NAGAR to demonstrate the
unbiasedness of his suggested estimator to order 1/T, where T
is the sample size (¥). When such expansions do converge,
such unbiasedness is about the only known sample property
in which one limited-information estimator is demonstrably
superior to the others. As it happens, however, NAGAR’s demonstration
assumes that there are no lagged endogenous variables
in the model so that, even aside from the convergence
problem iust mentioned, his results are not applicable in the
present . ~
(77) See BASMANN [5]. Sarg:
for the probabilities just described
(3) QUANDT [25].
(2%) NAGAR [22]. See SarG'
derives approximate expressions
Fisher - pag. 2:
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4.4. Robustness
Thus neither large nor small sample properties under ideal
conditions provide much guide for the choice of an estimator
from the limited-information class in the present state of knowledge.
This is not entirely the case as regards the robustness
of such estimators, however. KLEIN and NAKAMURA have
shown that as a consequence of the stochastic nature of % in
limited-information maximum likelihood, that estimator is more
sensitive to multicollinearity than are the other members of the
k-class (*). In the absence of other criteria, these seem grounds
for abandoning limited-information maximum likelihood in
practice in favor of some other limited-information estimator.
There seem to be no very strong reasons, however, for
choosing among the limited-information estimators other than
maximum likelihood. The paper on robustness just mentioned
indicates that these do not differ among themselves as regards
this property (*!). Since two-stage least squares is the easiest
of these estimators to compute and since it does provide a natural
generalization of ordinary least squares in the presence
of theoretically given normalization rules (2), it seems natural
to choose it in the present state of our knowledge.
5. NEAR-CONSISTENCY, BLOCK-RECURSIVE SYSTEMS, AND THE
CHOICE OF ELIGIBLE INSTRUMENTAL VARIABLES
5.1. Introduction
In this section we begin the discussion of the choices of
predetermined instrumental variables which are available and
(°°) KLEIN and NAKAMURA [16].
(*') See also Fisuer [8] for proof that the same is true as regards sensitivity
to specification error.
(32) See Cuow [7].
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the circumstances under which such choices are likely to be
appropriate. Most of our discussion will be in terms of the
assumptions that must be approximately satisfied if a given
variable is to be eligible for inclusion as an instrument. We
thus postpone to the next section the important question of
how the eligible candidates ought in fact to be used. Until
further notice, then, we discuss only whether and under what
circumstances a given single variable ought to be treated as
predetermined.
In general, we desire two things of a variable which is to
be treated as predetermined in the estimation of a given equation.
First, it should be uncorrelated in the probability limit
with the disturbance from that equation; second, it should closely
causally influence the variables which appear in that
equation and should do so independently of the other predetermined
variables (®¥). If the first criterion is not satisfied,
treating the variable as predetermined results in inconsistency;
if the second fails, such treatment does not aid much in estimation
— it does not reduce variances. In practice, these requirements
may frequently not be consistent and one has to compromise
between them. The closer the causal connection the
higher may be the forbidden correlation. Thus, in one limit,
the use of ordinary least squares which treats all variables on
the right-hand side of the equation as predetermined perfectly
satisfies the second but not the first criterion. In the other limit,
the use of instrumental variables which do not directly or indirectly
causally influence any variable in the model perfectly
meets the first requirement but not the second (3%). In general,
one is frequently faced with the necessity of weakening the first
requirement to one of low rather than of zero correlation and
{(*) It should therefore be relatively uncorrelated with the other variables
used as instruments so that lack of collinearity is not really a separate
criterion. We shall return to this in the next section.
(*) If only such variables are available for use (or if an insufficient
number of more interesting ones are) then the equation in question is
underidentified and even asymptotic variances are infinite.
Fisher - pag. 25
410 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
accepting indirect rather than direct causal relations between
instruments and included variables. (Such a compromise may
result in different instruments for different equations when a
limited-information estimator is used; this will be the case
below).
In the present section, we discuss the circumstances under
which zero or low inconsistencies can be expected, leaving explicit
use of the causal criterion to the next section.
Now, two sets of candidates for treatment as instrumental
variables are obviously present. The first of these consists of
those variables which one is willing to assume truly exogenous
to the entire system and the lagged values thereof; the second
consists of the lagged endogenous variables. The dynamic and
causal structure of the system may well provide a third set,
however, and may cast light on the appropriateness of the
use of lagged endogenous variables; to a discussion of this
we now turn.
5.2. The Theory of Block-Recursive Systems
A generalization of the recursive systems already discussed
is provided by what I have elsewhere termed « block-recursive
systems » (*). In general, such systems have similar properties
to those of recursive systems when the model is thought
of as subdivided into sets of current endogenous variables and
corresponding equations (which we shall call sectors) rather
than into single endogenous variables and their corresponding
equations.
Formally, we ask whether it is possible to partition the
vectors of variables and of disturbances and the corresponding
matrices (renumbering variables and equations, if necessary)
to secure a system with certain properties. In such partition-(5)
See Fisuer [8]
61 Fisher - pag. 26
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ings, the Ith subvector of a given vector x will be denoted
as x'. Similarly, the submatrix of a given matrix M which
occurs in the Ith row and Jth column ..f submatrices of that
matrix will be denoted br MT
5-1)
We shall always assume the diagonal blocks, M" to be square
If when written in this way, the matrix M has the propert,
that MY =o for all I=1, ..., N and J>I, the matrix will be
called block-triangular. If MV =o for all I=1, ..., N anc
J#I1, the matrix will be called block-diagonal (*).
Now consider the system (2.1). Suppose that there exists a
partition of that system (with N>1) such that:
(BR.1) A is block-triangular;
(BR. ’ V(o) is block-diagonal;
(BR., Y(3)=0 for all 8>o.
(Note that these are generalizations of (R.1)- (R.3)). In this
case, it is easy to show that the current endogenous variables
of any given sector are uncorrelated in probability with the
current disturbances of any higher-numbered sector. Such
variables may thus be consistently treated as predetermined
instruments in the estimation of the equations of such higher
numbered sectors.
To establish the proposition in question, observe that (2.2)-fa
* "wer hold (3). By (BR.3), (2.5) holds also, so that:
W(o)= DV(o0'
‘>
(8) Block-triangularity and block-diagonality are the respective canonica.
forms of decomposabilitv and complete decomposability.
37\ Continuine to assume that DR is stable
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
By (BR.1), however, D=(I- A)-! is block-triangular, while
V(o) is block diagonal by (BR.2). It follows that their product
is block triangular with the same partitioning. Thus:
(5.3) W(0o)" =o for all I, J=1, …, N and J>I,
but this is equivalent to the proposition in question.
As in the special case of recursive systems, assumption
(BR.3) can be replaced by a somewhat different assumption:
(BR.3*) B is block-triangular with the same partitioning
as A, as is V(6) for all 6>>0. Further, either all B" or all
V(0) (6>0) are zero (I=1, ..., N).
To see that this suffices, observe that in this case every
term in (2.4) will be block-triangular.
Note, however, that whereas (BR.1)-(BR.3) patently suffice
to give W(1)=0 and thus to show that lagged endogenous
variables are uncorrelated with current disturbances, this is
not the case when (BR.3) is replaced by (BR.3*). As in the
similar case for recursive systems, what is implied by (BR.1),
(BR.2), and (BR.3*) in this regard is that W(1) is also blocktriangular
with zero matrices on the principal diagonal so that
lagged endogenous variables are uncorrelated with the current
disturbances of the same or higher-numbered blocks, but not
necessarily with those of Jower-numbered ones.
If A and B are both block-triangular with the same partitioning,
then the matrix DB is also block-triangular and the
system of difference equations given by (2.2) is decomposable.
In this case, what occurs in higher-numbered sectors never
influences what occurs in lower-numbered ones, so that there is
in any case no point in using current or lagged endogenous
variables as instruments in lower-numbered sectors. This is
an unlikely circumstance to encounter in an economy-wide
model in any essential way, but it may occur for partitionings
which split off a small group of equations from the rest of the
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model. If it does not occur, then (BR.3) is generally necessary
for the block-triangularity of W(o). Indeed unless either
(BR.3) or the first statement of (BR.3*) holds, no W(0)" can
generally be expected to be zero if Bo.
To see this, observe that (2.4) implies that W(o) cannot
generally be expected to have any zero submatrices unless every
term in the sum which is not wholly zero has a zero submatrix
in the same place. This cannot happen unless every matrix
involved is either block-diagonal or block-triangular. Hence,
if V(8)=0o for all 8>>0, all such V(8) must at least be blocktriangular
as must B (3%
5.3. Block-Recursive Assumbtions in Economv-wide Models
Unfortunately, while block-triangularity of A is not an unreasonable
circumstance to expect to encounter in practice (*)
the assumptions on the disturbances involved in (BR.2) and
(BR.3) or (BR.3*) seem rather unrealistic in economy-wide
models for much the same reasons as did the parallel assumptions
of the recursive model. Thus, it does not seem reasonable
to assume that the omitted effects which form the disturbances
in two different sectors have no common elements; nor, as
already discussed, does it seem plausible to assume either that
there is no serial correlation of disturbances or that the dynamic
system involved is decomposable.
Note, however, that these assumptions may be better ap
proximations than in the case of recursive systems. Thus one
(38) Of course, this does not show that (BR.3) or (BR.3*) is necessary,
since counter-examples may easily be produced in which different non-zero
terms in (2.4) just cancel out. The point is that this cannot be assumed
to occur in practice. To put it another way, since such cancellation cannot
be known to occur, it clearly occurs only on a set of measure zero in the
parameter space. Thus (BR.3) or (BR.3*) is necessary with probability 1.
(**) It is encountered in preliminary versions of the Brookings-SSRC
model. C. Hort and D. Stewarp have developed a computer program for
organizing a model in block-toaneular form
ar
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414 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
may be more willing to assume no correlation between contemporaneous
disturbances in two different aggregate sectors
than between disturbances in any two single equations. A similar
assumption may be even more attractive when the
disturbances in question are from different time periods as will
be seen below. Thus also, the dynamic system may be thought
close to decomposability when broad sectors are in view and
feedbacks within sectors explicitly allowed. If such assumptions
are approximately satisfied, then the inconsistencies involved
in the use of current and lagged endogenous variables as predetermined
in higher-numbered sectors will be small (*0).
Nevertheless, the assumption of no correlation between contemporaneous
disturbances from different sectors, the assumption
of no serial correlation in the disturbances, and the assumption
of decomposability of the dynamic system all seem
rather strong ones to make. If none of these assumptions is in
fact even approximately made, then the use of current endogenous
variables as instruments in higher-numbered sectors
leads to non-negligible inconsistencies. We shall show, however,
that this need not be true of the use of lagged endogenous
variables in higher-numbered sectors under fairly plausible
assumptions as to the process generating the disturbances. We
thus turn to the question of the use of lagged endogenous variables,
assuming that A is known to be at least nearly blocktriangular.
5.4. Reasonable Properties of the Disturbances
The problems which we have been discussing largely turn
on the presence of common omitted variables in different equations
and on the serial correlation properties of the disturbances.
It seems appropriate to proceed by setting up an explicit model
((#) See Frsurr [8]. The theorems involved are generalizations of the:
Proximity Theorem for recursive systems.
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of the process generating the disturbances in terms of such
omitted variables and such serial correlation.
We shall assume that the disturbances to any equation are
made up of three sets of effects. The first of these will consist
of the effects of elements common to more than one sector — in
general, common to all sectors. The second will consist of the
effects of elements common to more than one equation in the
sector in which the given equation occurs. The third will consist
of effects specific to the given equation.
Thus, let the number of equations in the Ith sector be n,
We write:
(5.4)
where
(5.5)
is a vector of implicit disturbances whose effects are common
(in principle) to all equations in the model and ¢' is an #; x K
constant matrix:
(5.6)
is a vector of implicit disturbances whose effects are common
(in principle) to all equations in the Ith sector but not to equations
in other sectors; W' is an » - H, constant matrix: and
(5.7)
31
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
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is a vector of implicit disturbances the effect of each of which
is specific to a given equation in the Ith sector.
Define
Oy
wt
wrt
>}
(5.10)
*
and
(5.11)
Then (5.4) may be rewritten more compactly as:
(5.12)
u, =e. + Vou, + w,
We shall refer to the elements of e,, v, and w, as economywide,
sector, and equation implicit disturbances, respectively,
noting that whether an economy-wide or sector implicit disturbance
actually affects a given equation depends on the relevant
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rows of ¢ and ¥, respectively. (The unqualified term « disturbance
» will be reserved for the elements of u,.)
All elements of e, v, and w, are composites of unobservables;
it is hardly restrictive to assume:
(A.1) Every element of e, v, or w, is uncorrelated in
probability with all present or past values of any other element
of any of these vectors. The vectors can always be redefined
to accomplish this.
We shall assume that each element of each of these implicit
disturbance vectors obeys a (different) first-order auto-regressive
scheme (*!). Thus:
5.13)
(5.14)
(5.15)
where A,, A,, and A, are diagonal matrices of appropriate dimension
and e,*, v,*, and w,* are vectors of non-auto-correlated
random variables. Assuming that the variance of each element
of e,, v,, and w, is constant through time, the diagonal elements
of A, A,, and A, are first-order auto-correlation coefficients
and are thus each less than one in absolute value.
Now let A,, A, and A, be the diagonal variance-covariance
matrices of the elements of ¢,, v,, and w,, respectively.
In view of (A.1) and (5.13)-(5.15) it is easy to show that (5 =
implies:
(5.16)
V °
-
Yi -
-
!
o
+
(*") Auto-regressive relations of higher orders could be considered in
principle, but this would rather complicate the analysis. We shall thus
assume that first-order relationships are sufficiently good approximations.
If higher-order relationships are involved there is no essential change ir
the qualitative results
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418 PONTIFICTAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
Evidently, V(8) will be non-zero unless a number of other
assumptions are imposed. Consider, however, the question of
whether V(0) will be block-diagonal. Since all the A and A
matrices are diagonal, and since ¥ is itself block-diagonal by
(5.10), we have:
G17) Vie) =o'A A (80: I Ter... N:]=1
Thus the off-diagonal blocks of V(8) depend only on the properties
of the economy-wide disturbances.
This result is perhaps worth emphasizing. When applied
to 8 =o, it merely states formally what we have said previously,
that contemporaneous disturbances from the equations of the
model which occur in different sectors cannot be assumed uncorrelated
if there are common elements in each of them, that
is, implicit disturbance elements affecting both sectors. When
applied to 6>o0, however, the result is at least slightly less
obvious. Here it states that despite the fact that contemporaneous
disturbances from different sectors may be highly correlated,
and despite the fact that every disturbance may be
highly auto-correlated, a given disturbance will not be correlated
with a lagged disturbance from another sector unless.
the economy-wide implicit disturbances are themselves autocorrelated.
To put it another way, the presence of economywide
implicit disturbances and the presence of substantial serial
correlation do not prevent us from taking V(6) as block-diagonal
for 6>0 provided that the serial correlation is entirely
confined to the sector and equation implicit disturbances.
Is it then reasonable to assume that the serial correlation
is so confined? I think it is reasonable in the context of a
carefully constructed economy-wide model. We argued above
that any such model inevitably omits variables the effects of
which are not confined within sectors. Effects which are highly
auto-correlated, however, are effects which are relatively
systematic over time. In an inevitably aggregate and approxim-6]
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ate economy-wide model, there are likely to be such systematic
effects influencing individual equations and even whole sectors.
Systematic effects which spread over more than one sector,
however, seem substantially less likely to occur, especially when
we recall that the limits of a sector in our sense are likely to
be rather wide (*?). Variables which give rise to such effects
are not likely to be omitted variables whose influence lies in
the disturbance terms. Rather they are likely to be explicitly
included in the model, if at all possible. If not, if they relate
to the occurrence of a war, for example, and are thus hard
to specify explicitly, the time periods in which they ate most
important are likely to be omitted from the analysis. In short,
systematic behavior of the disturbances is an indication of
incomplete specification. Such incompleteness is much less
likely to occur as regards effects which are widespread than
as regards effects which are relatively narrowly confined, especially
since the former are less likely to be made up of many
small effects (**). (Recall that an economy-wide implicit
disturbance is one which affects more than one sector directly,
not simply one whose effects are transmitted through the dynamic
causal structure of the explicit model.) It thus does not
seem unreasonable to assume that:
(5.18)
and therefore
(5.19) V(0)Y =c
…, N,
as good approximations.
(*¥) As they are in the Brookings-SSRC model.
(“) A similar argument obviously implies that sector implicit disturbances
are less likely to be serially correlated than are equation implicit disturbances.
The analysis of the effects of this on V(@) and the subsequent
discussion is left to the reader. The assumption of no serial correlation in
the sector implicit disturbances seems considerably more dangerous than
that being discussed in the text.
6
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5.5. Implications for the Use of Lagged Endogenous Variables
Of course, assuming (5.19) to hold is not sufficient to yield
consistency when lagged endogenous variables are treated as
predetermined. We have already seen that unless V(6)=o,
the decomposability of the dynamic system must be assumed
in addition to (5.19) to secure such consistency. We argued
above, however, that such decomposability was rather unlikely
in an interconnected economy, although the fact that (5.19) is
likely to hold approximately makes it important to look for
near-decomposability and thus secure near-consistency (**).
Even if such near-decomposability of the dynamic system
does not occur, however, (5.19) has interesting consequences for
the treatment of lagged endogenous variables as predetermined.
To these we now turn.
Consider the expression for W(1) given in equation (2.6).
Writing out the first few terms of the sum, we obtain:
(5.20) W(1)=DV(1)+ DBDV(2)+(DB/DV(3) … .
Since D is block-triangular and V(1) block-diagonal by (5.19),
the first term in this expansion is also block-triangular. Hence
even if the dynamic system is not decomposable, endogenous
variables lagged one period are approximately uncorrelated
in the probability limit with disturbances in higher-numbered
sectors (but not in the same or lower-numbered sectors), to
the extent that the right-hand terms in (5.20) other than the
first can be ignored.
In what sense is it legitimate, then, to assume that such
terms can in fact be ignored? Assume that the matrix DB is
(*) Near-decomposability of a dynamic system has a number of interesting
consequences in addition to this. Se ANDo. FTSHER, and SrMoN [3]
especially Anpo and Frsuer 2]
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similar to a diagonal matrix, so that there exists a non-singular
matrix P such that (®):
(5.21)
DB =PHP
where H is diagonal and has for diagonal elements the latent
roots of DB. Let:
(5.22)
(5-23)
(5.24)
Then every such term can bè Writctai €
(5.25) (DB) DVI(o:-PH
DO.
30
We know that every diagonal element of the diagonal
matrix A is less than unity in absolute value (indeed, we are
assuming that some of the diagonal elements are zero). More
over, if we are prepared to maintain the stability assumption
on DB which was slipped in some time ago, every diagonal
element of the diagonal matrix H will also be less than unity
in absolute value. It follows that every element of every term
in the expansion of W(1) other than the first is composed of
a sum of terms each of which involves at least the product of
(*) The assumption involved is, of course. very weak and is made for
pase of exposition
6 |
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22 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
a factor less than unity and the square of another such factor.
There is clearly a reasonable sense in which one may be prepared
to take such terms as negligible at least when compared
with the non-zero elements of the first term in the expansion
for W(1) which involve only the diagonal elements of A to
the first power. If one is willing to do this, then one is saying
that the use of endogenous variables lagged one period as
instruments in higher-numbered sectors involves only negligible
inconsistency at least as compared with the use of the same
variables as instruments in their own or lower-numbered
sectors.
There may be considerable difficulties in accepting such a
judgment, however. In the first place, it is well to be aware
that there are two different statements involved. It is one
thing to say that the effects in question are negligible compared
to others and quite another to say that they are negligible in
a more absolute sense. If one accepts the stability assumption,
then there certainly is a value of 6 beyond which further terms
in the expansion of W(1) are negligible on any given standard.
These may not be all terms after the first, however: we shall
discuss the case in which there are non-negligible terms after
the first below.
Second (a minor point but one worth observing) even our
conclusion about relative importance need not hold although
other assumptions are granted. While it is true that as 0 becomes
large the right-hand side of (5.25) approaches zero, such
approach need not be monotonic. To put it another way, every
element of the matrix involved is a sum of terms. Each such
term involves a diagonal element of A to the 8 and a diagonal
element of H to the 6 - 1. If all such diagonal elements are
less than unity in absolute value, then the absolute value of
each separate term approaches zero monotonically as 6 increases;
this need not be true of the sum of those terms, however,
and it is easy to construct counter-examples. Nevertheless,
there is a sense in which it seems appropriate to assume
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the terms in the expansion for W(1) to be negligible for :
greater than some value, perhaps for 6>1.
All this, however, has leaned a bilt heavily on the stability
of DB. If that matrix has a latent root greater than unity in
absolute value, then part of the reason for assuming that the
right-hand side of (5.25) is negligible even for high values of 9
has disappeared. Of course, it is the case that the diagonal
elements of A are known to be less than unity in absolute
value, so that the infinite sum involved in W(1) may still converge.
However, such convergence is likely to be slow in an
unstable case and may not occur at all, so that the effects of
serial correlation are even more serious than in the stable
case. Clearly, the stability assumption requires additional
discussion at this point.
The usual reason for assuming stability of the dynamic
model being estimated is one of convenience or of lack of
knowledge of other cases. Since the unstable case tends to lead
to unbounded moment matrices, the usual proofs of consistency
of the limited-information estimators tend to break down in
that circumstance. Indeed, maximum-likelihood estimators are
presently known to be consistent only in the stable case and
in rather special unstable cases (*). It is therefore customary
to assume stability in discussions of this sort. For present purposes,
even if limited-information estimators are consistent in
unstable cases and even if the Generalized Proximity Theorems
which guarantee small inconsistencies for sufficiently good
approximations also hold (¥), the approximations which we
are now discussing are relatively unlikely to be good ones in
such cases. Even if the existence of W(1) is secured by assuming
that the dynamic process (2.1) begins with non-stochastic
initial conditions at some finite time in the past (and even
(*) For example, if all latent roots are greater than unity in absolute
value. See ANDERSON [1]. J. D. SarGaN has privately informed me that
he has constructed a proof of consistency for the general case. The classic
paper in this area is that of MANN and Warp [21]
(47) See FrisHeEr [8
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this does not suffice for the existence of the probability limit),
the effects of serial correlation will not die out (or will die out
only slowly) as we consider longer and longer lags. The conclusion
seems inescapable that if the model is thought to be
unstable (and the more so, the more unstable it is), the use
of lagged endogenous variables as instruments anywhere in an
indecomposable dynamic system with serially correlated
disturbances is likely to lead to large inconsistencies at least
for all but very high lags. The lower is serial correlation and
the closer the model to stability, the less dangerous is such use.
Is the stability assumption a realistic one for economy-wide
models, then? I think it is. Remember that what is at issue
is not the ability of the economy to grow, but its ability to
grow (or to have explosive cycles) with no help from the exogenous
variables and no impulses from the random disturbances.
Since the exogenous variables generally include population
growth and since technological change is generally either treated
as a disturbance or as an effect which is exogenous in some
way, this is by no means a hard assumption to accept. While
there are growth and cycle models in economic theory which
involve explosive systems, such models generally bound the
explosive oscillations or growth by ceilings or floors which
would be constant if the exogenous sources of growth were
constant (*). The system as a whole in such models is not
unstable in the presence of constant exogenous variables and
the absence of random shocks (#). We shall thus continue to
make the stability assumption.
Even when the stability assumption is made, however, it
may not be the case, as we have seen, that one is willing to
take the expression in (5.25) as negligible for all 8>>1. (In
particular, this will be the case if serial correlation is thought
to be very high so that the diagonal elements of A are close
(*) See, for example, Hicks [13] and Harrop [12].
(*) Whether a linear model is a good approximation if such models are
realistic is another matter
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to unity in absolute value). In such cases, one will not be wil
ling to assume that the use of endogenous variables lagged one
period as instruments in higher-numbered sectors leads to only
negligible inconsistencies. Accordingly, we must generalize
our discussion.
Fortunately, this is easy to do. There clearly does exist a
smallest 6*>o such that for all 6°>6* even the diagonal blocks
of V(0*) are negligible on any given standard. Consider
W(6*), the covariance matrix of the elements of #, and those
of y,_,» with the columns corresponding to elements of w, and
the rows to elements of Clearly,
(5.26)
Tr
+
+
JV
_
e= 0% 1
; 0-6"
DB) DV(e)
PH" P DONO
Since D is block-triangular and V(6*) block-diagonal by (5.19),
the product, DV(6*), is also block-triangular. Considering
W(6*)# for ]>1, it is apparent that the covariances of endogenous
variables lagged 6* periods and current disturbances
from higher-numbered sectors are made up of only negligible
terms. Not only is V(8) negligible by assumption for 6>>6%
but also every such term involves at least one power of H,
which, by assumption, is diagonal and has diagonal elements
less than unity in absolute value.
Note, however, that a similar statement is clearly false as
regards the covariances of endogenous variables lagged 6*
periods and current disturbances from the same or lower-numbered
sectors. Such covariances involve the non-zero diagonal
blocks of V(0*) in an essential way. It follows that the order
of inconsistency, so to speak, involved in using endogenous
variables lagged a given number of periods as instruments is
less if such variables are used in higher-numbered sectors than
if they are used in the same or lower-numbered sectors. To put
it another way, the minimum lag with which it is reasonably
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safe to use endogenous variables as instruments is at least one
less for use in higher-numbered than for use in the same or
lower-numbered sectors.
As a matter of fact, our result is a bit stronger than this.
It is apparent from (5.26) that the use of endogenous variables
lagged 0% periods as instruments in higher-numbered sectors
involves covariances of the order of A+!H. Even the use of
endogenous variables lagged 6*+1 periods as instruments in
the same or lower-numbered sectors, however, involves covariances
of the order of only A® +1. No positive power of H is
involved in the first term of the expansion for the latter covariances.
Since H is diagonal with diagonal elements less
than unity in absolute value, the difference between the minimum
lag with which it is safe to use endogenous variables as
instruments in the same or lower-numbered sectors and the
corresponding lag for use in higher-numbered ones may be
even greater than one. This point will be stronger the more
stable one believes the dynamic system to be. It arises because
the effects of serial correlation in sector and equation implicit
disturbances are direct in the case of lagged endogenous variables
used in the same or lower-numbered sectors and are
passed through a damped dynamic system in the case of lagged
endogenous variables used in higher-numbered sectors (°°).
To sum up: so far as inconsistency is concerned, it is likely
to be safer to use endogenous variables with a given lag as
instruments in higher-numbered sectors than to use them in the
same or lower-numbered sectors. For the latter use, the endogenous
variables should be lagged by at least one more period
to achieve the same level of consistency (31).
(°) All this is subject to the minor reservation discussed above concerning
sums each term of which approaches zero monotonically. In practice,
one tends to ignore such reservations in the absence of specific information
as to which way they point.
(*!) The reader should be aware of the parallel between this result and
the similar result for the use of current endogenous variables which emerges
when (BR.1)-(BR.3) are assumed. Essentially. we have replaced (BR.2)
with (5.10) and have dropped (BR.3).
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Now, it may be thought that this result is a rather poor
return for all the effort we have put into securing it. While
one can certainly conceive of stronger results, the usefulness
of the present one should not be underestimated. We remarked
at the beginning of this section that one important desideratum
of an instrumental variable was a close causal connection with
the variables appearing in the equation to be estimated. In
general, economy-wide (and most other) econometric models
have the property that variables with low lags are often (but
not always) more closely related to variables to be explained
than are variables with high ones. There may therefore be
a considerable gain in efficiency in the use of recent rather than
relatively remote endogenous variables as instruments, and it is
important to know that in certain reasonable contexts this may
be done without increasing the likely level of resulting inconsistency.
To the discussion of the causal criterion for instrumental
variables we now turn.
6. CAUSALITY AND RULES FOR THE USE OF ELIGIBLE INoTX.
MENTAL VARIABLES
6.1. The Causal Criterion for Instrumental Variables
We stated above that a good instrumental variable should
directly or indirectly causally influence the variables in the
equation to be estimated in a way independent of the other
instrumental variables and that the more direct is such influence,
the better. This statement requires some discussion.
So far as the limiting example of an instrument completely
unrelated to the variables of the model is concerned, the lesson
to be drawn might equally well be that instrumental variables
must be correlated in the probability limit with at least one
of the included variables. While it is easy to see that some
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causal connection must therefore exist, the question naturally
arises of why it must be one in which the instrumental variables.
cause the included ones. If correlation is all that matters,
surely the causal link might be reversed or both variables influnced
by a common third one.
This is not the case. Consider first the situation in which
the proposed instrumental variable is caused in part by variables
included in the model. To the extent that this is the
case, no advantage is obtained by using the proposed instrumental
variable over using the included variables themselves.
Obviously, the included variables are more highly correlated
with themselves than with the proposed instrument. Further,
correlation with the disturbances will be maintained if the proposed
instrument is used. To the extent that the proposed
instrument is caused by variables unrelated to the included
variables, correlation with the disturbances will go down, but
so also will correlation with the included variables.
The situation is similar if the proposed instrumental variable
and one or more of the included ones are caused in part by
a third variable. In this case, it is obviously more efficient to
use that third variable itself as an instrument, and, if this is
done, no further advantage attaches to the use of the proposed
instrumental variable in addition. (The only exception to this
occurs if data on the jointly causing variable are not available.
In such a case, the proposed instrument could be used to
advantage.)
In general, then, an instrumental variable should be known
to cause the included variables in the equation, at least indirectly.
The closer is such a causal connection the better. As
can easily be seen from our discussion of block-recursive
systems, however, in many cases the closer is that connection
the greater the danger of inconsistency through high correlations
with the relevant disturbances. In such systems, for
example, current endogenous variables in low-numbered sectors
directlv cause current endogenous variables in high-numbered
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sectors (*?) while the same endogenous variables lagged are
likely to be safer in terms of inconsistency but are also likely
to be more remote causes. The value of the result derived at
the end of the last section is that it provides a case in which
one set of instrumental variables is likely to dominate another
set on both criteria.
6.2. Available Instruments and Multicollinearu-There
is obviously one set of variables which has optimal
properties on several counts. These are the exogenous variables
explicity included in the model. Such variables are (by assumption)
uncorrelated in the probability limit with the disturbances,
they also are in close causal connection to the current
variables in any equation; indeed, they are some of those
variables in some cases (*)). In the happy event that such
exogenous variables are adequate in number and in the nonsingularity
of their variance-covariance matrix, and that no
lagged endogenous variables appear, there is no need to seek
further for instrumental variables to use.
Unfortunately, this is unlikely to be the case in an economy
wide econometric model. Such models tend to be almost selicontained
with relatively few truly exogenous variables entering
at relatively few places. This is especially the case if government
policies obey regular rules, follow signals from the
economy, and are therefore partly endogenous for purposes of
estimation ‘). In estimating anv equation, all variables not
(°**) On causation in general and in decomposable systems (or our block:
recursive systems) in particular, see SIMON [29].
(°°) They may not cause all such variables even indirectly if the dynamic
system is decomposable. Such cases are automatically treated in the rules
given below.
(**) This is to be sharply distinguished from the question of whether governmentally
controlled variables can be used as policy as opposed to
estimation instruments.
(61 Fisher - pag. 45
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used as instruments (except the variable explained by the equation)
must be replaced by a linear combination of instruments
and the dependent variable regressed on such linear combinations.
If the second stage of this procedure is not to involve
mversion of a singular matrix, then (counting instrumental
variables appearing in the equation) there must be at least as
many instruments used as there are parameters to be estimated.
Further, the linear combinations employed must not be perfectly
correlated. Current exogenous variables are simply not
generally sufficient to meet this requirement in economy-wide
models. Moreover, they do not cause lagged endogenous
variables which are likely to be present in a dynamic system.
Clearly, however, if the system is dynamic it will be possible
to use lagged exogenous variables as well as current ones.
Such use may be especially helpful if lagged endogenous variables
are to be treated as endogenous and replaced by linear
combinations of instruments which can be taken as causing
them in part. Indeed, if lagged endogenous variables are to
be taken as endogenous, then exclusive use of current exogenous
variables as instruments will not satisfy the causal criterion
for instrumental variables already discussed. Since we have
already seen that lagged endogenous variables should be used
as instruments only with caution, it follows that lagged
exogenous variables may well provide a welcome addition to
the collection of available instruments.
Unfortunately, this also is unlikely to suffice. While it is
true that one can always secure a sufficient number of instruments
by using exogenous variables with larger and larger lags,
such a procedure runs into several difficulties. In the first
place, since rather long lags may be required, there may be
a serious curtailment of available observations at the beginning
of the time period to be used. Second, exogenous variables in
the relatively distant past will be relatively indirect causes of
even the lagged endogenous variables appearing in the equation
to be estimated: it follows that their use will fail the causal
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criteria given and that it may be better to accept some inconsistency
by using endogenous variables with lower lags. Fi
nally, after going only a few periods back, the chances are
high in practice that adding an exogenous variable with a still
higher lag adds a variable which is very highly correlated with
the instruments already included and therefore adds little independent
causal information (*). While the use of lagged
exogenous variables is therefore highly desirable, it may not
be of sufficient practical help to allow the search for instru
mental variables to end.
Whatever collection of current exogenous, lagged exogenous
and (none, some, or all) lagged endogenous variables are used
however, the multicollinearity difficulty just encountered tends
to arise. Some method must be found for dealing with it.
One set of interesting suggestions in this area has been
provided by KLOEK and MENNES (°°). Essentially, they propose
using principal component analysis in various ways on
the set of eligible instruments in order to secure orthogonal
linear combinations. The endogenous variables are then
replaced by their regressions on these linear combinations
(possibly together with the eligible instruments actually appearing
in the equation to be estimated), and the dependent
variable regressed on these surrogates and the instruments
appearing in the equation. Variants of this proposal are alsc
examined.
This suggestion has the clear merit of avoiding multicollinearity,
as it is designed to do. However, it may eliminate
such multicollinearity in an undesirable way. If multicollinearity
is present in a regression equation, at least one of the
variables therein is adding little causal information to that
already contained in the other variables. In replacing a given
endogenous variable with its regression on a set of instruments,
Fa.
(*) This is especially likely if the exogenous variables are principally
ones such as population which are mainly trends.
(*) Kroek and MEexNEs [17]
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therefore, the prime reason for avoiding multicollinearity is
that the addition of an instrument which is collinear with the
included ones adds little causal information while using up a
degree of freedom. The elimination of such multicollinearity
ought thus to proceed in such a way as to conserve causal
information. The KLOEK-MENNES proposals may result in
orthogonal combinations of instruments which are not particularly
closely causally related to the included endogenous
variables. Thus such proposals may well be inferior to a
procedure which eliminates multicollinearity by eliminating
instruments which contribute relatively little to the causal
explanation of the endogenous variable to be replaced (2).
Clearly, this may involve using different sets of instruments in
the replacement of different endogenous variables. Proposals
along these lines are given below.
It may be objected, however, that such a procedure may
eliminate multicollinearity in the regression of the included
endogenous variables (other than the left-hand one of the equation)
on the chosen instruments only to encounter it again when
the dependent variable is regressed on the replaced variables
and the instruments appearing in the equation. This is clearly
true (*'); it is unavoidable, however. The fact that the variables
to be replaced by combinations of instruments are all part of
the system to be estimated guarantees that they themselves
must be reasonably highly collinear and related to the included
instruments. It is impossible to reduce that kind of multicollinearity
without introducing as instruments noise elements
which are unrelated to the included variables, and such introduction
clearly gains nothing. If we can secure instrumental
variables which are closely causally related to the included
variables but relativelv uncorrelated with the disturbance of a
(**) This seems to have been one of the outcomes of experimentation
with different forms of principal component analysis in practice. See Tav-LOR
[30a].
(7) It is also true of the KILOEK-MENNES Drocedures.
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given equation, we have gone as far as we can. The avoidance
of multicollinearity is a necessary part of such a procedure
because multicollinearity is one sort of failure of the causal
criterion discussed above. To attempt to eliminate that multicollinearity
which inevitably results from just those causal relations
which are to be estimated, however, is self-defeating (%)
6.3. Rules for the Use of Eligible Instrumental Variables
We have several times pointed out that the causal criterion
and that of no correlation with the given disturbance may be
inconsistent and that one may only be able to satisfy one more
closely by sacrificing the other to a greater extent. In principle,
a fully satisfactory treatment of the use of instrumental variables
in economy-wide models would involve a full-scale Bayesian
analysis of the losses and gains from any particular action.
Such an analysis is clearly beyond the scope of the present
paper, although any recommended procedure clearly has some
judgment of probable losses behind it, however vague such
judgment may be.
We shall proceed by assuming that the no-correlation criterion
has been used to secure a set of eligible instrumental
variables whose use is judged to involve only tolerable inconsistencies
in the estimation of a given equation. Note that the
set may be different for different equations. Within that set
are current and lagged exogenous variables and lagged endo-(*!)
I want to make it clear that I am not accusing KLoEk and MENNES
of attempting to do this. Their proposals are designed to eliminate multicollinearity
in the first stage of the procedure where it is desirable to do so.
Their « Method 2 » [17, pp. 51-52] does eliminate collinearity in the second
stage between the replaced endogenous variables and the included
predetermined ones, but this is not necessarily the same as the desirable and
irreducible collinearity among the variables in the equation discussed in
the text. It may well occur in practice that the KLoEK-MENNES proposals
lead to desirable results, although an approach using more structural information
than does theirs seems preferable.
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genous variables sufficiently far in the past that the effects of
serial correlation are judged to be negligible over the time
period involved. As shown in the preceding section, that time
period will generally be shorter for endogenous variables in
sectors lower-numbered than that in which the equation to be
estimated appears than for endogenous variables in the same
or higher-numbered sectors (¥). Clearly, other things being
equal, the use of current and lagged exogenous variables is
preferable to the use of lagged endogenous variables and the
use of lagged endogenous variables from lower-numbered sectors
is preferable to the use of endogenous variables with the
same (or possibly even a slightly greater) lag from the same
or higher-numbered sectors than that in which the equation
to be estimated occurs. We shall suggest ways of modifying
the use of the causal criterion to take account of this. For convenience,
we shall refer to all the eligible instrumental variables
as predetermined and to all other variables as endogenous.
Consider any particular endogenous variable in the equation
to be estimated, other than the one explained by that equation.
That right-hand endogenous variable will be termed of
zero causal order. Consider the structural equation (either in
its original form or with all variables lagged) that explains that
variable (). The variables other than the explained one
appearing therein will be called of first causal order. Next,
consider the structural equations explaining the first causal
order endogenous variables (). All variables appearing in
those equations will be called of second causal order with the
(*) It will not have escaped the reader’s notice that very little guidance
has been given as to the determination of the absolute magnitude of that
time period.
(°°) There must exist such an equation if the variable in question is
normalized. As stated above, lack of such normalization is a form of incomplete
specification.
(*) Observe that endogenous variables appearing in the equation to be
estimated other than the particular one with which we begin may be of
positive causal order. This includes the endogenous variable to be explained
by the equation to be estimated.
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exception of the zero causal order variable and those endogenous
variables of first causal order the equations for which
have already been considered. Note that a given predetermined
variable may be of more than one causal order. Take now
those structural equations explaining endogenous variables of
second causal order. All variables appearing in such equations
will be called of third causal order except for the endogenous
ones of lower causal order, and so forth. (Any predetermined
variables never reached in this procedure are dropped from
the eligible set while dealing with the given zero causal order
variable.)
The result of this procedure is to use the a priori structural
information available to subdivide the set of predetermined
variables according to closeness of causal relation to a given
endogenous variable in the equation to be estimated. Thus,
predetermined variables of first causal order are known to
cause that endogenous variable directly; predetermined variables
of second causal order are known directly to cause
other variables which directly cause the given endogenous
variable, and so forth. Note again that a given predetermined
variable can be of more than one causal order, so that the
subdivision need not result in disjunct sets of predetermined
variables.
We now provide a complete ordering of the predetermined
variables relative to the given endogenous variable of zero
causal order (%!). Let p be the largest number of different
causal orders to which any predetermined variable belongs. To
each predetermined variable we assign a p-component vector.
The first component of that vector is the lowest-numbered causal
order to which the given predetermined variable belongs; the
second component is the next lowest causal order to which it
belongs, and so forth. Vectors corresponding to variables be-(¢)
I am indebted to J. C. G. Boor for aid in the construction of the
following formal description.
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longing to less than p different causal orders have infinity in
the unused places. Thus, for example, if p=35, a predetermined
variable of first, second and eighth causal order will be assigned
the vector: (1, 2, 8, 00, 00). The vectors are now ordered lexicographically.
That is, any vector, say f, is assigned a number,
3(f), such that, for any two vectors, say f and kh:
(6.1) B(f)>B(k) if and only if either f,>h, or for some
j(I<7<P)
fi=h; G=1, …, j-I) and f.>h,.
The predetermined variables are then ordered in ascending
order of their corresponding B-numbers. This will be called the
3-ordering.
Thus predetermined variables of first causal order are assigned
lower numbers than predetermined variables of only higher
causal orders; predetermined variables of first and second
causal order are assigned lower numbers than predetermined
variables of first and only causal orders higher than second
(or of no higher causal order), and so forth (2).
The procedure just described gives an a priori preference
ordering on the set of instrumental variables relative to a
given zero causal order endogenous variable. This ordering
ts in terms of closeness of causal relation. Alternatively, one
may wish to modify that ordering to take further account of
the danger of inconsistency. This may be done by deciding
that current and lagged exogenous variables of a given causal
order are always to be preferred to lagged endogenous variables
of no lower causal order and that lagged endogenous variables
from sectors with lower numbers than that of the equation to
(#2) This is only one way of constructing such an ordering. If there is
specific a priori reason to believe that a given instrument is important in
influencing the variable to be replaced (for example, if it is known to
enter in several different ways with big coefficients) then it should be given
a low number. In the absence of such specific information, the ordering
given in the text seems a natural wav of organizing the structural information.
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be estimated are always to be preferred to endogenous variables
with the same lag and causal order from the same or higher
numbered sectors. One might even go further and decide that
all current and lagged exogenous variables of finite causal order
are to be preferred to any lagged endogenous variable.
However the preference ordering is decided upon, its exist:
ence allows us to use a posteriori information to choose a sel
of instruments for the zero causal order endogenous variable
in the way about to be described. Once that set has been chosen,
that endogenous variable is replaced by its regression on
the instruments in the set and the equation in question estimated
by least squares regression of the left-hand endogenous
variable on the resulting right-hand variables (2).
We use a posteriori information in combination with the
a prior: preference ordering in the following manner. Suppose
that there are T observations in the sample. Regress the zero
causal order endogenous variable on the first T-2 instruments
in the preference ordering (a regression with one degree of freedom).
Now drop the least preferred of these instruments from
the regression. Observe whether the multiple correlation of
the regression drops significantly as a result. (The standard
here may be the significance level of R? or simply its value
corrected for degrees of freedom.) If correlation does drop
significantly, then the T-2nd instrument contributes significantly
to the causation of the zero order endogenous variable even in
the presence of all instruments which are a priori more closely
related to that variable than it is. It should therefore be retained.
If correlation does not drop significantly, then the variable
in question adds nothing and should be omitted.
Now proceed to the T-3rd instrument. If the T-2nd instrument
was retained at the previous step, reintroduce it; if not,
leave it out. Observe whether omitting the T-3rd instrument
reduces the multiple correlation significantly. If so, retain it,
(2) An important modification of this procedure is described below
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if not, omit it and proceed to the next lower-numbered instrument.
Continue in this way. At every step, a given instrument is
tested to see whether it contributes significantly to multiple
correlation in the presence of all instruments which are a priori
preferred to it and all other instruments which have already
passed the test. When all instruments have been so tested,
the ones remaining are the ones to be used.
6.4. Discussion of the Rules
The point of this procedure (or the variants described below)
is to replace the right-hand endogenous variables in the
equation to be estimated by their regression-calculated values
using instruments which satisfy the causal criterion as well
as possible while keeping inconsistency at a tolerable level.
Certain features require discussion.
In the first place, multicollinearity at this stage of the proceedings
is automatically taken care of in a way consistent
with the causal criterion. If some set of instruments is highly
collinear, then that member of the set which is least preferred
on a priori grounds will fail to reduce correlation significantly
when it is tested as just described. It will then be omitted
and the procedure guarantees that it will be the least preferred
member of the set which is so treated. If the B-ordering is used,
this will be the one most distantly structurally related to the
endogenous variable which is to be replaced. Multicollinearity
will be tolerated where it should be, namely, where despite its
presence each instrument in the collinear set adds significant
causal information.
Second, it is evident that the procedure described has the
property that no variable will be omitted simply because it is
highly correlated with other variables already dropped. If two
variables add significantly to correlation when both are present
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but fail to add anything when introduced separately, then the
first one to be tested will not be dropped from the regression,
as omitting it in the presence of the other instrument will significantly
reduce correlation (622). While it is true that variables
may be dropped because of correlation with variables less preferred
than the T-znd, which are never tested, the exclusion
of the latter variables seems to be a relatively weak reliance on
a priori information.
This brings us to the next point. Clearly, it is possible in
principle that instruments less preferred than the T-2nd would
in fact pass the correlation test described if that test were performed
after some lower-numbered instruments were tested and
dropped. Similarly, an instrument dropped at an early stage
might pass the test in the absence of variables later dropped
because of the increased number of degrees of freedom. One
could, of course, repeat the entire procedure in order to test
every previously dropped variable after each decision to omit;
it seems preferable, however, to rely on the a priori preference
ordering in practice and to insist that instruments which come
late in the 3-ordering pass a more stringent empirical test than
those which come early. The rationale behind the 3-ordering
is the belief that it is the earlier instruments in that ordering
which contribute most of the causal information, so that it
seems quite appropriate to calculate the degrees of freedom
for testing a given instrument by subtracting the number of
its place in the ordering from the total number of observations
(and allowing for the constant term) (2b).
Turning to another issue, it may be obje~ted that there is
no guarantee that the suggested procedures will result in a nonsingular
moment matrix to be inverted at the last stace. That is
(#) This property was missing in the procedure suggested in the first
draft of this paper in which variables were added in ascending order of
preference and retained if they added significantly to correlation. I am
indebted to ALBERT ANDO for helpful discussions on this point.
(8) Admittedly, this argument loses some of its force when applied to
the modifications of the R-ordering given above
6 |
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there may be some set of » endogenous variables to be replaced
whose regressions together involve less than 7 predetermined
variables. Alternatively, counting the instruments included
in the equation to be estimated there may not be as many
instruments used in the final stage as there are parameters to
be estimated. This can happen, of course, although it is
perhaps relatively unlikely. If it does occur, then it is a sign
that the equation in question is unidentifiable from the sample
available, that the causal information contained in the sample
is insufficient to allow estimation of the equation without relaxing
the inconsistency requirements. To put it another way,
it can be argued that to rectify this situation by the introduction
in the first-stage regressions of variables failing the causal
test as described is an ad hoc device which adds no causal
information. While such variables may in fact appear in such
regressions with non-zero coefficients in the probability limit,
their use in the sample adds nothing to the quality of the
estimates save the ability to secure numbers and disguise the
problem.
Of course, such an argument is a bit too strong. Whether
a variable adds significantly to correlation is a function of
what one means by significance. The problem is thus a continuous
rather than a discrete one and should be treated as such.
For the criterion of significance used, in some sense, the equation
in question cannot be estimated from the sample in the
circumstance described; it may be estimatable with a less stringent
significance criterion. In practice, if the significance
requirements are relaxed, the moment matrix to be inverted
Will pass from singularity to near-singularity and estimated
asymptotic standard errors will be large rather than infinite.
The general point is that if multicollinearity cannot be sufficiently
eliminated using causal information, little is to be
gained by eliminating it by introducing more or less irrelevant
variables.
A somewhat related point is that the use of different vari-6]
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ables as instruments in the regressions for different endogenous
variables in the same equation may result in a situation in
which the longest lag involved in one such regression is greater
than that involved in others. If data are only available from
an initial date, this means that using the regressions as estimated
involves eliminating some observations at the beginning
of the period that would be retained if the longest-lagged instrument
were dropped. In this case some balance must be struck
between the gain in efficiency from extra observations and the
loss from disregarding causal information if the lagged instrument
in question is dropped. It is hard to give a precise guide
as to how this should be done. (My personal preference would
be for retaining the instrument in most cases.) Such circum
stances will fortunately be relatively infrequent as the periods
of data collection generally begin further back than those of
estimation, at least in models of developed economies. Further,
the reduction in available observations attendant on the use
of an instrument with a large lag renders it unlikely that the
introduction of that instrument adds significantly to correlation.
Finally, the use of different instruments in the regressions
replacing different endogenous variables in the equation to be
estimated reintroduces the problem of inconsistency. When the
equation to be estimated is rewritten with calculated values
replacing some or all of the variables, the residual term includes
not only the original structural disturbance but also a linear
combination of the residuals from the regression equations used
in such replacement. When the equation is then estimated by
regressing the left-hand variable on the calculated right-hand
ones and the instruments explicitly appearing, consistency requires
not only zero correlation in the probability limit between
the original disturbance and all the variables used in the final
regression but also zero correlation in the probability limit
between the residuals from the earlier-stage regression equations
and all such variables. If the same set of instruments
is used when replacing every right-hand endogenous variable
6]
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and if that set includes the instruments explicitly in the equation,
the latter requirement presents no problem since the normal
equations of ordinary least squares imply that such correlations
are zero even in the sample (). When different
instruments are used in the replacement of different variables,
however, or when the instruments so used do not include those
explicitly in the equation, the danger of inconsistency from
this source does arise.
There are several ways of handling this without sacrificing
the major benefits of our procedures. One way is simply to
argue that those procedures are designed to include in the
regression for any right-hand endogenous variable any instrument
which is correlated with the residuals from that regression
computed without that instrument. The excluded instruments
are either those which are known a priori not to be direct or
indirect causes of the variable to be replaced or those which
fail to add significantly to the correlation of the regression in
question. The former instruments are known a priori not to
appear in equations explaining the variable to be replaced and
hence cannot be correlated in the probability limit with the
residual from the regression unless both they and the replaced
variable are affected by some third variable not included in
that regression (#). Such a third variable cannot be endogenous,
however, since in that case the excluded instruments in
question would also be endogenous; moreover, our procedure
is designed to include explicitly any instrument significantly
affecting the variable to be replaced. Any such third variable
must therefore be one omitted from the model and it may not
be stretching things too far to disregard correlations between
residuals and excluded instruments stemming from such a
source.
(9) This is the case when the reduced form equations are used, for
example, as in the classic version of two-stage least squares.
(*) If they were non-negligibly caused by the replaced variable itself
they would be endogenous. contrarv to assnmption.
‘61 Fisher - pag. 58
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As for instruments which are indirectly causally related tc
the endogenous variable involved but which fail to add significantly
to the correlation of the regression in question, these cannot
be significantly correlated with the sample residual from
that regression. One can therefore argue that the evidence is
against their being significantly correlated with that residual in
the probability limit.
Such an argument can clearly be pushed too far, however.
[f there are strong a priori reasons to believe that the excluded
instruments should be included in view of the causal structure
of the model, one may not want to reject correlation in the
probability limit because multicollinearity (for the long continuance
of which there may be no structural reason) leads to
insignificant correlation in the sample. A modified course of
action, then, is to include in the regression for any replaced
variable any instrument which one believes a priori to be important
in that regression and which appears either in the equation
to be estimated or in the regression for any other replaced
variable as computed by the procedures described above (%).
Clearly, not much is lost by doing this since the added variables
will not contribute much to the equation in the sample.
Alternatively, one may go the whole way towards guarding
against inconsistency from the source under discussion and
include in the regression for any replaced variable all instruments
which appear in the equation to be estimated or in the
regression for any other replaced variable as computed by the
described procedures whether or not such instrument is thought
a priori to be important in explaining the replaced variable.
This alternative clearly eliminates the danger under discussion.
It may, however, reintroduce multicollinearity and may involve
a serious departure from the causal criterion if a prior:
non-causal instruments are thus included. Nevertheless. it does
f%) Omitting instruments which do mot so appear does not cause in
~onsistencv
>
Fisher - pag. sc
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retain the merit that every instrumental variable used is either
explicitly included in the equation or contributes significantly
to the causal explanation of at least one variable so included.
In practice, there may not be a great deal of difference between
these alternatives and the last one described may then be
optimal (unless it is unavailable because of the degrees of
freedom required).
Whatever variant of our procedures is thought best in practice,
they all have the merit of using information on the dynamic
and causal structure of the model in securing estimates.
The use of such information in some way is vital in the estimation
of economy-wide econometric models where the ideal
conditions for which most estimators are designed are unlikely
to be encountered in practice (%).
(*) The use of the causal structure of the model itself to choose instrumental
variables as described in the text is closely akin to the methods used
by BARGER and KLEIN to estimate a system with a triangular matrix of coefficients
of current endogennus variables See [23]
61 Fisher - pag. 60
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
REFERENCES
[1] AnpeErsoN T.W., On Asymptotic Distributions of Estimates of Para
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and the Relevance of Stability Discussions. « International
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3a] BarGer H. and L. R. KLEIN, 4 Quarterly Model for the United States
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BASMANN R. L., On the Exact Finite Sample Distributions of Generalized
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51 BASMANN R. L., À Note on the Exact Finite Sample Frequency Functions
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°7a] ÉIsENPRESs H., Note on the Computation of Full-Information Maxi
mum Likelihood Estimates of Coefficients of a Simultaneous System.
« Econometrica », 30 (1962), pp. 343-348.
‘81 FIsHER F. M., On the Cost of Approximate Specification in Simultaneous
Equation Estimation. « Econometrica », 29 (1962), pp. 139-170;
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« International Economic Review », Vol. 4, No. z (May, 10963).
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of Birmingham, Faculty of Commerce and Social Science, Discussion
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[12] Harrop R. F., Towards a Dynamic Economics. London, 1948.
[13] Hicks J. R., À Contribution to the Theory of the Trade Cycle. London,
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Hurwicz L., Least-Squaves Bias in Time Series. Chapter 15 of « Statistical
Inference in Dynamic Economic Models » (T. C. Koopmans,
ed.), New York, 1950 (Cowles Commission Monograph 10).
[15] JouNsTON J., Econometric Methods. New York, 1962.
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of Econometrics: Some Aspects of the Problem of Multicollinearity.
« International Economic Review », 3 (1962), pp. 274-299.
Kroexk T. and L. B. M. MENNES, Simultaneous Equations Estimation
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», 28 (1960), pp. 45-6I.
Lesnoy S., Limited Information versus Reduced Form Least Squares
in Prediction. Unpublished (presented at Cambridge meeting of Econometric
Society, 1958).
“19] Liv T. C., Underidentification, Structural Estimation, and Forecasting.
« Econometrica », 28 (1960), pp. 855-865.
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RAND Corporation Memorandum RM-3557-PR, 1963 (to be published
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17 Mann H. B. and A. Warp, On the Statistical Treatment of Linear
Stochastic Difference Equations. « Econometrica ». 11 (1943), pp. 173-220.
[22]
Nagar A. L., The Bias and Moment Matrix of the General k-Class
Estimators of the Parameters in Simultaneous Equation. « Econometrica
», 27 (1959), PP. 575-595.
Nagar A. L., Double k-Class Estimators of Pavameters in Simultaneous
Equations and Their Small Sample Properties. « International Economic
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QUANDT R. E., Some Small Sample Properties of Certain Structural
Equation Estimators. Princeton University, Econometric Research Program,
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Mimeographed and unpublished. 1963.
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OU
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hood Estimates (to be published in « Econometrica »).
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Two-Stage Least Squares. Mimeographed and unpublished, 1563.
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dies in Econometric Method » (W. C. Hood and T. C. Koopmans, eds.)
New York (Cowles Commission Monograph 14); reprinted as Chapter
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of [3].
(30] StrOoTZ R. H., Interdependence as a Specification Error. « Econometrica
», 28 (1960), pp. 428-442.
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toed and unpublished paper, 1963, and Ph. D. Thesis by same title
Harvard University, Cambridge, 1962.
THEIL H., Specification Errors and the Estimation of Economic Ru
lationships. « Review of the International Statistical Institute ».
(1957), pp. 41-51.
Tuer. H., Economic Forecasts and Policy. Amsterdam, 2nd Revise
Edition, 1961.
[33] Waugn F., The Place of Least Squares in Econometrics. « Econome
trica », 29 (1961), pp. 386-396.
[34] WoLp H., in association with L. JUREEN, Demand Analysis. New Yor}
1953.
[35] WorLp H. and P. FAXER, On the Specification Error in Regression Ana
Iysis. « Annals of Mathematical Statistics », 28 (1957), pp. 265-267
136] WoLp H., Ends and Means in Econometric Model Building. « Probability
and Statistics » (The Harald Cramer Volume) (U. Grenander, ed.)
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ZELLNER À. and H. TuriL, Three-Stage Least Squares Simultaneous
Estimation of Simultaneous Equations. « Econometrica », 30 (1962),
pp. =4-78
o| Fisher - pag. 6;
~T
3C"-SSION
WoLD
Prof. FISHER has given us an excellent survey of structura
properties and estimation techniques for the dynamic econometric
models known as interdependent (ID) systems and causal chain
(CC) systems. Having over many years emphasized the clearcut
rationale of CC systems, also known as recursive systems, I appre
ciate verv much the fair treatment he has given this approach. His
review of the ID approach is a very clarifying exposition of the many
estimation techniques that have been proposed for ID-systems, a
multitude of techniques that are known to work and give reconcilable
results when applied to small systems, whereas the situation is
far from clear when it comes to large systems with many unknown
parameters. The very pluralism of the methods is an indication that
the problem of parameter estimation in large ID-systems has not
yet found a satisfactory solution.
The questions at issue are technical matter, Hence the first thing
is to summarize what the two types models have and have not in
common, so as to be able to locate and if possible clear up any
divergencies that may exist. Three points will be mentioned.
(1) In the beginning of section 2, Prof. FISHER states three as:
sumptions R 1-3 that are to be satisfied by a CC system, al. in
accordance with current theory of the probability structure and
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statistical estimation of multirelation models. Assumptions R 2-3
at first sight look very stringent, and in section 2.3 Prof. FISHER
argues that they are not likely to be valid for economy-wide econometric
models. The first point I wish to make is that the narrow
stringency of Assumptions R 2-3 is only apparent. They come in
a better light if the relations of the systems are specified in terms
of eo ipso predictors (an ea ipso predictor is the conditional expectation
that constitutes the residual-free part of the relation). This
type of assumption has the fundamental advantage that it corresponds
directly to the operational use for which the relations are
intended. Furthermore, R 2-3 become automatically satisfied as an
implication, not as an assumption, and this implication in its turn
implies that the relations can be consistently estimated by least
squares regression,
(#) The theory of eo ipso predictors has shed new light on the
much-discussed question about the operational significance of the
structural relations of ID-systems. With reference to my report to
the Study Week for details of the argument, it can be shown that
the clearcut cause-effect interpretation of the behavioural relations
of CC-systems does extend to ID-systems, but only at the price of
a respecification of the system. For example, if the ID-system
involves a behavioural relation which specifies the elasticity of investments
with respect to profits as 0.4, the respecified assumption
will be that 0.4 is the elasticity of savings with respect to expected
profits. Actual profits and expected profits are different notions,
conceptually and observationally; the actual profits are given by the
statistical data, whereas the expected profits in the present context
are given by the reduced form of the ID-system. The snag is that
expected profits as derived from the reduced form stand in no obvious
connection with expected profits in the sense of psychological
anticipation. The respecification of the ID-system thus involves an
element of arbitrariness.
The point just mentioned has a bearing upon the doubts that
Prof. FISHER expresses in section 2.3 about the matrix triangularity
in Assumption R 1. When Prof. FISHER states that it is somewhat
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unlikely that economy-wide models have entirely triangular A matrices,
nobody can disagree, and what he says is even an understatement.
On the other hand, if matrix A is made nontriangular the
system becomes interdependent, and then we are confronted with
the snag of a possible discrepancy between the two notions of expected
values for the current endogenous variables. In other words,
one kind of approximation has been replaced by another, and the
arguments at issue do not indicate which approximation is prefer
able. This question brings me to the third point.
(ii) Prof. FISHER’s considerations for or against CC- and IDsystems
are on the whole of a theoretical nature. For my part 1
would instead make a plea for comparisons of an empirical nature.
Above all, my plea is for studies where one and the same observational
material is used for the testing of fullfledged CC- and IDmodels.
Such comparative studies are not as yet in the picture, so it
is as yet an open question which type of model performs best in
actual applications. In this connection there is the important question
what criteria should be used when comparing and evaluating
the performances. Predictive tests here come to the fore as the real
touchstone for the validity of a model; that is, tests where the model
as constructed on the basis of past experience is exploited to make
forecasts about the unobserved future, and where in due course the
forecasts are confronted with the actual developments
FISHER
Professor WoLD has long been interested in the question of what
are desirable properties of econometric models. This is an interesting
question and his work and his present comments cast considerable
light on it. As an economist, however, I think one ought to be
interested not so much in the question: « What are good properties
of econometric models? » but rather in the question: « What are
the properties which good econometric models have? » In other
words, one does not design a model so that it will have desirable
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properties: one does not design a model in order that it should have
minimum delay; one does not design a model so that it will be an
eo 1pso predictor. One designs a model so that it represents something
that one believes to be true about the real world. After the
model has been designed, and not before, one asks: What properties
does it have? And what properties can I give the estimates of the
model by using different estimating techniques?
Now, indeed, there are always several ways in which a given
real phenomenon can be represented in a model. Given two of these
which we believe to be equally valid in terms of economic theory,
we would of course choose the one which had better properties in
terms of forecasting or in terms of the properties which the estimation
techniques appropriate to the model will have. Frequently, however,
that is not in fact the primary question. In general, we are
not indifferent between two representations and the choice between
models does not come on the question: « Will one be recursive and
the other simultaneous? » The crucial question is rather that of
which model represents what we believe to be true about the real
world and which model and estimation technique is appropriate to
the use which we intend to make of the results.
THEIL
A question of clarification. Do you intend to split up a large
equation system into subsystems and construct principal components
of the predetermined variables in each subsystem?
FISHER
Professor THEIL wishes to know whether I recommend splitting
up a large equation system into subsystems and constructing principal
components of the predetermined variables in each subsystem.
This is not what I recommend, although similar procedures have in
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fact been adopted in economy-wide models in the past. My proposa.
is to take each equation (and indeed each endogenous variable
within each equation) and to get a group of instrumental variables
which are particularly appropriate for it. One does this by consid
ering the causal structure of the system. The block-recursive system
— in which the full system is divided into sectors and subsystems
— is used in my paper to point out that it is safer to use some
endogenous variables in some places than in others so that the
eligible list of instruments is different for different equations.
HAAVELMO
I wonder whether FISHER would agree to a rough summing u
of the situation that might help people who are not so much interested
in technical details and who wonder what we get out of all
this. The summing up could be something like this: We have
small samples in practice, so that large sample properties of esti:
mates are probably not too interesting. Then, if we had a clear
situation as far as identification is concerned, that is to say, if we
need no subtle tricks in order to ensure identification; and secondly,
if by means of mere inspection we can find that there are exogenous
variables which vary a great deal as compared with the disturbances
u, and the exogenous variables are really uncorrelated with the u’s
in various directions, both simultaneously and recurrently; then we
get reasonably good estimates even without refined methods. Now
it is in the in-between cases that we may perhaps gain the most by
being very particular about the estimation methods chosen. This
would intuitively be the conclusion that one who works in practice
with these things would draw. This does not of course in any
way reduce the importance of stringent work on estimation methods,
but I think it might — if it is reasonably correct interest those
who want to know what the practical conclusion .-69
ag.
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FISHER
I would generally concur with Professor HAAVELMO’s summary of
the situation. If, in fact, there are a lot of exogenous variables
which vary a great deal as compared with the disturbances, then
reasonably good estimates can be obtained — although I would not
agree that refined methods are not called for. In my paper I pointed
out that such situations are likely to be very rare in the estimation
of economy-wide econometric models. In such cases, I tried to
show that one could improve the results substantially by paying
attention to the causal structure of the model rather than by incautiously
applying techniques designed for the ideal situation.
FRISCH
I am afraid that what I am going to say will be considered
perhaps as black heresy, and if I am saying it, I hope you will
understand that, when I voice this opinion, if I say it in a very
outspoken way, it is not because of a feeling of unfriendliness towards
members of our fraternity, it is merely to save time that I
am putting it in an extreme way.
Looking upon the problem of estimation in economy-wide models,
I like to adopt the view-point of an economist who is called
upon to give his advice to politicians. And if I look upon the matter
in this way, I have an uncomfortable feeling that, in statistics, we
are too frequently carried away by the terminology we use. You
may say that terminology doesn’t mean very much, you can call
a thing a straw hat, and use the concept in a precise way in your
logical deductions. It is not so dangerons to call it a straw hat,
because then everybody will understand that this is just a sort of
numbering of your concepts. But if you use other words which in
every day life have had a certain connotation, you may be
obsessed by this connotation and, as I say, be carried awav through
your terminology.
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Looking upon the matter from the view-point of an economist
giving advice to a politician, I would suggest that, when we speak
of an estimator and the properties of estimators — we accept to put
as the main classification: 1) Purposefully useful; 2) Purposefully
irrelevant and 3) Purposefully detrimental.
I will explain what I mean by « purposefully useful ». Every
estimator must be thought of in terms of the purposes for which
we are going to use to coefficient or magnitude estimated. In this
particular case the purpose is perhaps a political one. You must say
that this is vague. Yes, I am sorry to say that these words are
vague and they cannot be defined precisely because the purpose
will vary from one application to another, so therefore the specifi
cation of what is meant by these properties must vary and can only
be made more precise if you are explaining very explicitly for what
purpose you are going to use your analytical results. Many of these
properties of estimators which are fundamental from the view-point
of application are very difficult to handle. Of course we like dif
ficult problems, provided they are not beyond the limit of our
ability. Then it would be very interesting to handle the problems.
But when they pass beyond this stage it is very tempting to pick
out certain properties which are not in true sense purposefully useful
— they may even be purposefully detrimental but they have the
property that I am able to handle them. 1 may decide then to
work with these concepts instead of the really important ones which
I am not able to handle. Let me take an example. I am going
to multiply 13 by 27. I scratch my head and I think. « Oh, multiplication
is such a terribly difficult operation, but I am very good
at adding fiures, so why don’t I add instead the two numbers 13
and 27». You wouldn't think that that was a very useful proredure.
I will illustrate what I have said by taking the two words
« unbiased » and « consistent », I am afraid that we have, to some
extent, been carried away by the common meaning of these words.
What is « biased » and « unbiased n? I would rather have pre.
ferred to use the term « not immoral » instead of « unbiased », be:
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+}
cause if you say that, everybody would have understood that you
are Just using the term in a technical sense. But if you say « unbiased
» people can not become aware that you are fooling with
words. It is probably hopeless to suggest a change of terminology,
but I feel it would have been better if you had used these words
«not immoral » instead of « unbiased ». Next with regard to « consistency
». Take a person who is not able to carry on a « consistent »
discussion and a consistent way of using his logic. You would not
respect him very much. About his conclusions you would probably
say « This is a fallacy » I would rather have preferred that you use
the terminology « not fallacious » instead of « consistent » because
then again people would have understood that you are really just
playing with words. I would seriously suggest that we change terminology
to something which is neutral and just say precisely what
we mean. Instead of « unbiased » I would prefer to say « expectationally
hitting » because what is involved, is simply that the
expectation of the estimate is equal to the thing which we estimate.
Instead of « consistent », I would say « targetly converging » in the
stochastic sense, because that’s what we mean. It may be « converging
asymptotically » i.e. it may be « targetly converging » in
the limit when the number of observations becomes great.
« Unbiasedness » or, as I would like to call it, « expectationally
hittingness » may not really be the property in which we are interested.
Take a firm that is selling shoes: women’s shoes and men’s
shoes. The owner of the firm will want very much that a random
customer can be satisfied. Now, there are two types of shoes: men’s
shoes and women’s shoes. The « univers » has probably a bimodal
distribution. If the owner were to make a guess about what shoe the
next customer would ask for, he would be off the mark if he focussed
his attention on the mathematical expectation as derived from that
bimodal distribution.
I would say that in this case « unbiassedness » is purposefully
irrelevant. I don’t think that this discussion about terminology is
aseless because many people are not able to protect themselves
against risk of being dragged into false understanding and false va-16]
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luation, which will lead away from what is useful, If you will
allow me, I will pick out a few expressions in Prof. FISHER's presentation,
which exemplifies how one may be carried away by the
words, I put it down while he was talking. He said that Wozn
has shown that this particular estimate — it was ordinary least
square if I remember correctly — « retains desirable properties » —
well, how do you know what is desirable? And, even worse, later
on in his presentation, Prof. FISHER said — and I also took this down
— « retain all the desirable properties » — I think he spoke about
a full information maximum likelyhood or something that approxi
mated it. It was in this connection that he said « retain all the
desirable properties n. The immediate question is: « desirable for
what purpose? » — May I finally give a third example. Professor
FISHER spoke about using lagged variables as instruments and advised
against this use for statistical reasons. Now, suppose I am in an
underdeveloped country. And suppose I am giving advice to a politician
who is up against the problem whether he should go in for
education on birth control. In an underdeveloped country with a
very heavy population increase and with all the implications which
this means from the economic and social viewpoint, the problem is
important and the politician is pondering very hard the foreseable
effect; this case pertains, of course, essentially to lagged variables.
Suppose I come along and say « Well, I must advise against
such a policy because it would upset certain calculations of mine
regarding some specific properties of my estimates ». As I see it,
these properties of my estimates are purposefully irrelevant, In this
case it is precisely the lag values one would need to explain demographic
development.
Finally, if I may just have one minute, Approaching all these
things from the viewpoint of society at large, extending over time.
we are accustomed to fall back on time series to a large extent
Sometimes we may use cross-section studies, but many economists
and statisticians rely on time series that come from observation of
what has happened in the past. Such series are, however, irrelevant
for a great number of the estimates of the equations that enter
[6] Fisher - pag. 72
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into the equational system we have to use in development planning.
We have for instance investment project specifications. We will
have tens of thousands of such projects. Each of them will be
described in engineering terms; for instance the sequence in
time in which certain input elements are to be made: labour,
products from domestic sectors, imports, etc., the burden which
this entails on the balance of payment is extremely important,
and so on. In most cases these data can be derived fairly correctly
from an engineering analysis. There may, of course, be some uncertainties
and you may — according to the price situation — shift
a little in the use of input elements, but let us disregard that point
for the moment. The engineering analysis will — with a fair degree
of approximation — give the essential information. Similarly with
regard to the time and the volume in which the capacity effect of
the investment emerges. If you are building a hydroelectric power
station, then you can say, from an engineering viewpoint that
«next year I'll have one machine coming along; the year after, I'll
have two more machines », and so on. All these things are given
from the engineering viewpoint with a fair degree of accuracy and
there is no question of trying to estimate the possible consequences
of adopting a section of these 1,000 or 10,000 projects by looking
back in our time series and discussing whether a certain time series
estimate will be « unbiased » or « immoral » or have some other
specific property which you are able to handle mathematically, but
which have little relevance for the actual problem of development
planning.
FISHER
The last point made in my reply to Professor WoLD is of course
in agreement with part of Professor FrRisCH’s remarks. There is a
growing literature on Bayesian estimation in which the decision
problem to be answered by the model determines the estimation
technique. This is a highly interesting development and it is one
with which I am in nearly complete sympathy. There are several
[6] Fisher - pag. 74
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places in my paper in which this appears rather close to the surface.
In general, I have said that the choice is between « consistency >
and « efficiency » and I usually take the direction of opting for consistency.
However, I tried to point out in the paper that this choice
rather depends on what one expects to do with the results after one
has obtained them. On the other hand, detailed analysis of the
dependency of estimators in economy-wide econometric models on
the use that is to be made of the results is difficult and so far only
in its infancy. It does seem likely that properties of estimators
such as consistency, unbiasedness, efficiency, and so forth, are properties
which one believes are likely to be relevant in a rather wide
class of decision problems.
Now for that part of Professor FRISCH’s comments which deals
with the use of persuasive definitions. While I am not out of sympathy
with the argument that use of such terms as « unbiased »,
« consistency », and « instrument », may be misleading in that such
words carry connotations from general application, I must say that
I think Professor FRISCH’s remarks on this point are irrelevant as
regards my paper. However unfortunate the use of such terms may
be, there is a long and distinguished history of their use in this way
and while it may be too bad that such usage has come about, if
is rather late for me to do anything about it.
I must, however, comment specifically on Professor FRISCH's
discussion of my use of the word « instrument » since I am unable
to tell whether he actually misunderstands me or whether this is
just another example of the unfortunate use of a word with several
connotations. « Instrument » in the theory of economic policy,
where we owe much to Professor FRISCH, means a variable which
can be controlled by a policy maker and which can be used to move
toward those goals which the policy maker finds desirable. « In
strument » as I have used it is short for « instrumental variable »
as the context of my paper makes abundantly clear. In this sense,
it means a variable which can be taken to be uncorrelated with
the disturbances but whose movements influence the endogenous
variables of the system. Such a variable is used as an instrument
(6] Fisher - pag. 7°
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in the sense that advantage is taken of its properties to secure parameter
estimates. In Professor FriscH’s example, the birth rate
may perfectly well be an instrument in the policy sense and fail
to be an instrument in the statistical sense in that its movements
when not influenced by the policy maker may be functions of the
disturbances and the other endogenous variables of the system,
Once again, my usage has quite a long tradition in econometrics
behind it.
ALLAIS
This is not at all a criticism of FISHERS paper, but as an economist,
I must confess that I feel myself in sympathy with some of
Prof. FriscH’s words. I did not intend to say anything, but in
the introduction to this Study Week, we read « The econometric
method represents a big improvement over non-mathematical methods
of studying phenomena connected with economic operations ».
I think we must be very careful about the impression which can
be given to outsiders by the methods we use. My feeling is that
there is a terrible gap between the power of statistical methods and
the limitations of economic models and the data. The methods we
have are in advance, and very largely in advance, if we compare
them to the economic models which we use. We may have wonderful
methods to analyse the models which are built but the value
of the conclusions will then depend primarily on the value of the
models.
Thus the danger is that public opinion or politicians may greatly
overestimate what we can do. The precision of the methods is one
thine. Their value is another
FISHER
I am in basic agreement with Professor ALLAIS’ remarks. Anyone
who has ever done a good deal of empirical econometric work knows
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perfectly well that the apparent precision which the theory of esti:
mation gives to the results is not in fact present. Whether this ic
due to the properties of the model or to the character of the data
is a matter of some dispute, but it is a common phenomenon.
What my paper is about is those situations so common in practice
in which statistical techniques which have very nice properties
in the secure situations for which they are designed do not in fact
have such nice properties when applied to models of the type one
actually has to estimate. In such cases we have to make all sorts
of compromises to produce techniques which have properties whick
one deems desirable. For example, in almost all the original lite
rature on simultaneous equation estimation, everyone simply as
sumed that there was no serial correlation in the disturbances. In
economy-wide models, there is such serial correlation and the usual
simultaneous equation estimators lose a good deal of their appeal
if used incautiouslv
[LEONTIEEF
In the context of previous discussion I would like to raise agair
the question of relative advantages and disadvantages of simulta
neous as against independent estimation of the empirical parameters
entering into different parts of an integrated analytical model. Al
though from a fundamental philosophical point of view indirect
inference and direct observation have much in common, in daily
practice of scientific investigation they differ from each other greatly.
So long as we operate with highly aggregative models, indirect
inference must dominate the field. Since aggregative variables and
parameters, in terms of which relationships are usually described.
cannot be observed directly, they necessarily must be estimated in:
directly. As soon, however, as disaggregation reaches the critical
level at which the individual bits of data used in the theoretica
model are identical or nearly identical to those familiar to the
producers and consumers of individual goods and services in their
61
Fisher - pag. 77
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daily practice, direct observation becomes possible and indirect inference
dispensable, The empirical implementation of an analytical
model comprising large sets of simultaneous, or at least interrelated
equations, does not require any more indirect estimation of large sets
of parameters imbedded in corresponding large sets of simultaneous
statistical equations. The observations can be made one by one and
essentially independently of each other. If and when this happens
controversies concerning the choice among alternative methods of
statistical estimation might lose the dominant position which they
now occupy in discussion of econometric problems.
FISHER
Professor LEONTIEF in his interesting remarks looks forward to
the day in which the use of highly disaggregate data will enable
us to concentrate less on simultaneous equation problems. While
I think that such a day is probably very far off, I do agree that
such problems become of less importance as one goes to more and
more disaggregated data. On the other hand, there are at least
some forms of disaggregation which do not lead to this desirable
result. For example, as I argue in my paper, disaggregation in
the form of securing data for smaller and smaller time periods does
not avoid the simultaneity problem since the disturbances for such
time periods are likely to be serially correlated. Whether other forms
of disaggregation aid in avoiding simultaneity seems to me likely
to depend on the model and the type of disaggregation. I think
that so long as one is interested in models of the entire economy,
simultaneity will remain an interesting question, even if one deals
with data in terms of individual units. This is so, because it remains
true, for example, that the income identity holds summing over all
units.
While simultaneity introduces numerous problems, I think one
should beware of the attitude (which I am aware is not Professor
LEONTIEF’s) that simultaneity problems should be avoided at the
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expense of doing violence to the model. There are now several
techniques for dealing with such problems, and the fact that they
may not be fully satisfactory does not justify the use of a wholly
inappropriate technique in a context which is really simultaneous
KOOPMANS
If I understand Prof. FISHER correctly, use was made of the dat:
two or three times in order to select the set of predetermined va
riables that is drawn on for help in estimating a particular equation.
Should one worry about the effect of repeated use of the same data
and if so what kind of errors or lack of efficiency could have beer
produced thereby?
FISHER
Professor KoopMaNs wishes to know whether my recommenda:
tions for the choice of instrumental variables does not involve the
double use of the same set of data. I think that it only does so
superficially. One does indeed use the data in applying a stopping
rule in the regression of endogenous variables on instruments, but
the regressions obtained by applying the stopping rule are just the
ones which are used in estimating the equation of interest. I do
not see that this involves using the data twice any more than does
two-stage least squares, choice of instrumental variables by principal
components, or any simultaneous equation technique in which
reduced form coefficients have first to be calculated.
MAHALANOBIS
I have a purely technical question. Can you break up into two
or more random partitions? And secondly, it would be of interest
61 Fisher - pag. 79
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28
to know what would be the degrees of freedom, that is, in the sense
of independent observations, which might be available in concrete
examples,
FISHER
Professor MAHALANOBIS wants to know if it is possible to break
up the data into two random partitions. Presumably, one would
want to do this to avoid the double use of data which I have just
suggested is not present. I take it that what one would do would
be to use one set of data to choose the instruments and the other
set of data to estimate the parameters.
As indicated, I do not believe that this is necessary. In addition,
it would be impossible in practice. The Brookings-SSRC model in
its first form has something like one hundred equations; in its second,
more disaggregate form, there will be many, many more.
Each equation involves at least a few parameters. The data will
run from the war until about 1962 by quarters, so that rough calculation
gives about 64 observations. There are, however, an enormous
number of eligible candidates for instruments, far greater
than 64. I do not believe that one would want to use only 32 observations
either for the estimation of the parameters, or for the choice
of instruments from a list which will be far, far longer than 32.
However, as stated, since I believe the answer to Professor Koop-MANS’
question to be in the negative, the issue need not arise.
16] Fisher - pag. 80
DECISION RULES AND SIMULATION
TECHNIQUES IN DEVELOPMENT
PROGRAMMING
HENRI THEIL (*)
Nederlandsche Economische Hoogeschool - Econometyisch Instituut
Rotterdam - Nederland
NTRODUCTIG
Many underdeveloped countries, all Communist countries,
and some developed Western countries have their Economic
Plans nowadays. Such Plans may differ considerably as to
character and scope, but it is quite generally true that they are
based on certain considerations as to the goals to be pursued,
as to the relationships between certain noncontrolled key variables
on the one hand and the variables controlled by the
decision-making authority on the other hand, as to the impact
of outside forces, and as to the likely development of such outside
forces over time. Given a number of assumptions one will
arrive at a certain Plan, which specifies that this is to be done
in year 1, that in year 2, and so on. In principle the Plan is
optimal in the sense that it pursues the goals as well as possible
subject to the constraints under which the economy operates.
But there is the problem of uncertainty. The development
of outside forces over time has to be predicted, which cannot
(*) The author is indebted to Mr. J. Boas of the Econometric Institute
Rotterdam, who carried out the simulation experiment described in Sec
tion 6
"71 Theil - pag.
166 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
be done without forecast errors. The relationships describing
the impact of these forces and of the variables controlled by
the decision maker are not known with certainty either. The
decision maker must therefore make his decisions (prepare his
Plan) under conditions of uncertainty. Now there is the important
feature that this uncertainty diminishes in the course
of time. For example, how the outside forces behave in year I
is unknown at the beginning of the year, at least not known
with certainty, so that the decision to be made at the beginning
of that year must be based on a forecast which is generally
imperfect. At the beginning of year 2 one should know more
about how the outside forces have behaved in year I; also, one
may have better ideas about how they will behave in year 2,
because this is now « nearer future » than it was a year before.
Clearly, the decision maker is in a position to take this new
information into account when formulating his decision for
year 2. But the Plan does not! The Plan was made at the
beginning of year 1 and what it has to say about things to be
done in year 2 is therefore necessarily based on the smaller
amount of information which was available at the beginning of
vear I.
This is of course generally recognized. It is the reason why
a Plan is almost never taken completely seriously in the sense
that it is executed literally until the very end. Plans are revised
regularly in the light of new evidence. But this happens
after the events, not before as is done by a « strategy » or
« decision rule ». The main argument of this paper is that it
is worthwhile to consider the possibility of development strategies
instead of development plans. That is, rather than fixing
once and for all what is to be done in year 1, in year 2, in
year 3, ..., we consider a procedure of the following kind:
now, at the beginning of year 1, a decision for that year is
formulated and at the same time also a rule specifying what
is to be done in years 2, 3, ... depending on the information
that will be available by that time. The advantage of this pro-71
Theil - pag.
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cedure is not only that it is more systematic but also (and
particularly) that, in general, taking account of the possibility
that future information may be reacted to contributes to better
results from the standpoint of the goal pursued (}).
The primary objective of this paper is to illustrate the
method of linear decision rules in the context of development
programming. The paper has no pretensions with respect to
the development of model building in this field. Models describing
the constraints under which the economy operates are
undoubtedly important and in fact indispensable for the derivation
of decision rules; but it is felt that the introduction of
innovations in that area would shift the attention away from the
primary objective and, therefore, a very simple capital-income
ratio model is introduced in Section 2. In Section 3 a quadratic
social preference function is formulated, after which the theory
of linear decision rules follows in Sections 4 and 5.
Section 6 deals with the second objective of this paper: the
use of simulation techniques in development programming.
Since many of the structural relations and also the long-term
development of many crucial outside forces are subject to a
considerable degree of uncertainty, particularly in the field
of growth and development, it seems plausible that simulation
techniques can be valuable to indicate the range of variability
of the outcomes which result from different policy procedures.
In order to concentrate on the main idea, the application is
confined to the same simple case as that of the earlier sections.
2. THE MODEL
Let K, be the stock of capital goods at the end of year ¢
and Y, national income of that vear, both in real terms. One
(') An additional advantage is that the strategy approach enables the
decision maker to formulate predictions (no perfect predictions!) of his own
future actions and of their consequences. But this aspect will not be pursued
here
(7] Theil - pag.
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of the equations of the model is
2.1)
K,_, = oY, 1
where p is the capital-income ratio, which is assumed to be
constant over time. Let C, be consumption and x, the savings
ratio in vear ?:
‘2
2)
C,=(1-x)Y,
Furthermore, we use two identities:
t
‘2.
>)
(2.4)
C,+1=Y,
K.=K, ,+1,,
where I, is net investment in year ¢£. Finally, we need an equation
describing the development of the population :
(2.5)
N.=(1+v,)N,_, ,
where N, is the mid-year population size and v, its rate of increase.
In what follows we shall be interested particularly in the
rate of increase of per capita consumption, C,/N,. Using (2.2),
(2.1) and (2.5) we find:
Ci 1—a, Ke
N, 2 (1+v)N,_,
and hence:
a C/N
(2.6)... - CoN
K,
— 9
K, 1
7 4_1 +
“,
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Now K, ,/K, ,=1+1, /K, ,=1+x%x,_; Y,_/K,_,from (2.2,
and (2.3). Since Y,_,/K, ,=1/p from (2.1), we conclude that
the ratio of K,_; to K,_, is equal to 1+x,_,/¢. Taking logarithms
in (2.6), we then find the following expression for the
first difference of the logarithm of per capita consumption:
(2.7) — A [log (C;/N,)] = €
log (1 + x,_1/2) —
log (1 + v,) .
This equation is the only aspect of the model that will be
used in the sequel. We shall consider a decision maker who is
interested in two things: the savings ratio x,, which he controls,
and the logarithmic rate of change of per capita consumption.
which he does not control. The approach to be followed requires
that the latter (uncontrolled) variable be expressed linearly
in the former. Eq. (2.4) is nonlinear and it will therefore be
linearized. Taking all logarithms as natural logarithms, we
shall approximate the log of 1+x,_,/¢ by x,_,/p. We shall
also approximate the log of 1+v, by v, (although this is not
strictly necessary, since the expression does not involve the
savings ratio). For the first term on the rirht we use-(2.8)
But this is still nonlinear in x,_; and we will therefore apply
the following (crude) approximation (3). We should expect
that the savings ratio will be of the order of 15 to 25%, so that
1/(1-x,_,) is then of the order of 1.2 or 1.3. This range of
ancertainty is not very sizable; moreover, we multiply
1/(x-x,_,) by x,- x,_,, which is generally close to zero, particularly
since we shall put a penalty on large values of
‘?) But see footnote 5 below for a more accurate approximation
71 Theil - pag.
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470 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
|*; = *,_,| in Section 3. Therefore, we shall approximate the
expression (2.8) by -b(x,-x,_,), where b is regarded as a
fixed coefficient (about equal to 1%).
Equation (2.7) is now written as
(2.9) À [log (C,/N,)] = - bx, + (b+1/e)x,_, - v,,
which is the form with which we shall work in the remainder
of this paper.
3. THE PREFERENCE FUNCTION
Our decision maker is supposed to formulate a quadratic
preference function which he wishes to maximize. We shall
consider a very simple quadratic function, which amounts to
a sum of squares as far as the successive log-changes in per
capita consumption is concerned. This should not be regarded
in the sense that we really believe that the decision maker’s
desires are quadratic; it means only that we try to approximate
the decision maker’s preferences by a quadratic
function in the relevant range (in the same way as the constraints
of the economy are approximated linearly in the relevant
range).
Specifically, let d, be the « desired rate » of increase of the
logarithm of per capita consumption in year £. For example,
d,=0.1 (10%); then, given the quadratic character of the
preference function, an actual rate of increase of 5% will have
a disutility of (5 — 10)”=25, a 4% increase will have a disutility
of (4 - 10)?=36, and so on. Let us write y, for the discrepancy
between the actual and the desired rate of increase in
year t:
(3.1)
y,=ATlog (C,/N,)1 - d, 3
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then it follows from (2.0) that y, is determined as follows:
(3.2) V; -
- bx,+ (b+1/6)x, 1 -v. a
We consider the sum of squares of the y,’s over a period of
T years, where T represents the horizon which the decision
maker takes into account. It will prove useful to introduce a
matrix notation to handle all years simultaneously:
3.3)
then (3.2) for t=, ...,
(3.4)
! can be written in the form
Ex+s.
where
3-5)
7
Theil - pag.
172
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
5)
Vas
where x, is the savings ratio in the year preceding the first
(which is taken as given from the past). Note that all R-elements
above the diagonal should necessarily vanish, because
they represent the effect of controlled variables (x,) on earlier
noncontrolled variables (y, with #<#). The diagonal elements
specify the effectiveness of controlled variables on noncontrolled
variables in the same year, and the elements below
the diagonal represent lagged effects.
It was stated above that we are interested in minimizing
the sum of squares of the discrepancy between actual and
desired rates of increase of the logarithm of per capita consumption.
This amounts to minimizing y’y. But this will be
amended to the effect that we shall also be interested in moderate
changes of the savings ratio, the argument being that
a savings ratio of 209%, last year followed by one of 15%, this
year and then of 259, next year is difficult to realize. To
handle this, we introduce the sum of squares of the successive
differences of the savings ratio:
4
Uh
J
-
+,
D
0
D
I
x — 200% + X2 —
— mn _ ' 2
— x Ar 2x ex + x;
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where À is the T x T matrix of the second-difference transfor:
mation and e, the first unit vector of order T (i.e., the first
column of the T x T unit matrix). Note that x; is a constant
(it is given from the past), so that it can be omitted from the
preference function. This preference function, which the decision
maker wishes to maximize, is then assumed to be of the
following form:
3-7)
wo
,
T
4
+ YY)
Thus the decision maker is supposed to minimize a weight
ed sum of squares of two sets of differences. One set deals
with the successive differences of the savings ratio, the other
with the differences between actual and desired log-changes
in per capita consumption. Of course, g should be a positive
number; it measures the seriousness of a given change in the
savings ratio relative to that of a discrepancy between the
actual and the desired log-change in consumption of the same
numerical size
. EXPECTED UTILITY AND CERTAINTY EQUIVALENCE
Our problem in mathematical terms can in the first instance
be described as that of maximizing the quadratic preference
function (3.7) subject to the linear constraint (3.4). But it is
readily seen that the real problem is more complicated. For
carrying out this conditional maximization requires that the
determining factors, such as the rates of increase of the population,
are known before. This is evidently not the case,
so that we must conclude that the decisions have to be made
{71 Theil - pag. 9
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under conditions of uncertainty. (The v’s are the only uncertain
factors which enter into the problem as it is posed
here, which is of course highly restrictive, but it is easy enough
to extend the list of such factors). We shall attack this
problem by assuming that the uncertainty is of the probabilistic
type and that the decision maker is interested in
maximizing expected utility. That is, he is supposed to
maximize the expectation of the preference function (3.4)
subject to the constraint (3.4), the latter being interpreted
stochastically (with random v’s).
It is worthwhile to consider the implications of this procedure
in somewhat more detail. Take the first year, at the
beginning of which the decision x, has to be made. The rates
of increase of the population, Vi, Vo, ..., Vr, are then unknown
and are supposed to be subject to a T-dimensional joint distribution.
One year later the decision x, has to be made and
the decision maker will then know more, particularly about v,
but perhaps also about later v’s, because the development of
the population during the first year may have shed some
light on the probable development during later years. Clearly,
the decision maker should be able to use this information
gained during the first year when he formulates his decision x,
at the beginning of the second year. In the same way, at the
beginning of the third year he knows still more (particularly
about v,) and he can use this additional information for his
decision x;. And so on.
It follows that it is the decision maker's task to formulate,
for each year ¢=1, ..., T, the decision x,.as a function of the
information that will be available at that time. Such a series
of decisions x, ..., Xp written as functions of relevant information
is a strategy or decision rule. More specifically,
the decision maker’s task is to find the maximizing strategy,
i.e., the strategy which maximizes the expectation of the preference
function subject to the contraints. In general it is
71 Theil - pag. To
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47°F
not easy to find this maximizing strategy, but it is very
simple when the preference function is quadratic and when
the constraints are linear. The solution is found by applying
the theorem of « first-period certainty equivalence », which
in our case amounts to the following: the first-period decision
of the maximizing strategy is identical with the first-period de:
cision that would be made if the uncertainty aspect would be
disregarded by replacing all random v’s by their expectations.
This means that the problem is reduced, for the decision of
the first year (x,) at least, to an ordinary conditional maximization
problem: maximize the quadratic preference function
(not its expectation) subject to the constraint (3.4) on the
understanding that the vector s of this constraint is replaced by
its expectation; i.e., in (3.6) we should replace the v’s by the
expectations of the v’s. Clearly, this solves the uncertainty
problem in an almost trivial way. For the second-period de
cision (x,) one can proceed in precisely the same way one period
later, because by that time the second period will have
become the first. And so on (3).
5. THE MAXIMIZING STRATEGY
The results mentioned in the preceding section can be regarded
as a separation of the decision problem in two successive
steps: first maximize as if there is no uncertainty,
then replace certain random variables in the result by their
expectations. We shall start with the first step under the
assumption that the horizon (T) is so large that it can be
(*) The first-period certainty equivalence theorem is due to H.A. St
MON [2] and was subsequently generalized by the present author [3, 4]
For applications to a microeconomic (paint factory) case reference is made
to C.C. Hort et alii [1]. For a more extensive discussion including several
other applications. see the author's recent monograph [5].
"71 Theil - pag. 11
476 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
effectively regarded as infinite. The derivations are given in
the Appendix and the result for the optimal first-period decision
(x9) is as follows:
5.1)
2b
x? — he, — bE+ be +9 0 + d,) +
(1 —X) à +I/p eu
"bg à (ve +d),
where
5.2)
, — [I +42°(0*+b/p+g)li—1
[t + 40° (D? + b/o +9)l# + 1
(0 <n <1)
The result (5.1) shows that the optimal savings ratio in
the first year (x{) consists of. two parts, one of which deals
with the savings ratio in the preceding year (x,) and the other
with population increases and desired consumption increases
in the first year and later (v,, d;, v,, d,, ...). We find that
the effect of v, and d, on the optimal savings ratio is negative
but that the effect of later v’s and d’s is positive. This is the
natural consequence of the fact that society has to spend for
immediate welfare but to save for future welfare.
As an example we take the special case v,=v, d,=d for
all #; then (5.1) is simplified to
| . DE ,
(5-3) x) = My TBE Bar +d).
which means that this year’s savings ratio is a certain fraction
of last vear’s savings ratio (Ax,) plus a constant (k, say). For
71 Theil - pag. 12
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477
the following year we then have
+R
22x, + Mk
k=)
=x +
x2=from
which it is evident that the savings ratio in successive
years moves gradually from x, to
(5.4)
h
"ie
-
| a) «+ d)
Take, e.g., p equal to 3 or 4 years, b slightly above I, and ¢
a small positive number [which is tantamount to saying that a
successive change in the savings ratio, x,- x,_;, is considered
much less serious than a discrepancy y, (between actual and
desired consumption increase) of the same size]. Then
(5.5)
b?+
ble+g
sil
so that À according to (5.2; 1s approximately
(5.6)
i.e., about 0.8 when p is about 3 years or so. Then the limit
(5.4) to which the savings ratio approaches is about 3 times
[71
Theil - pag. 13
178 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
» 8
(v+d). Thus, if the rate of increase of the population is
v =0.02 per year and if the desired rate of increase of per capita
income is 0.06 per year, the limit of the savings ratio is of
the order of 25%.
When the v’s and d’s are not constant over time, we have
to substitute their values directly into (5.1). It is seen that
the influence of future v’s and d’s is of the decreasing exponential
type. But it is impossible to compute x? when the future
v's are unknown. We have then to rely on maximizing expected
utility and on the first-period certainty equivalence theorem,
provided of course that the relevant expectations are
known. Note that these expectations are conditional expectations,
given the information available at the moment when the
decision must be made. Suppose, e.g., that the rate of increase
of the population fluctuates around a mean v and that it satisfies
the following stochastic difference equation:
5-7)
+
—_— yy —
I
— (y — WV +e,
where €, is a random variable with zero mean and zero correlations
over time. Then the expectation of v, - v, given the
information available at the beginning of the first year, is
I _— _
3 (Yo-v); that of v,-v (under the same condition) is
[ — . .
i (vo-v); and so on. By substituting these expectations in
the right-hand side of (5.1) we obtain the first-period decision
of the maximizing strategy (under the assumption d,=d):
5-8)
7 : Me (v + d) —
TEN NE Trg)
A (b - 1/p) (vo —v) ;
7 {4 = 2%) (6? + b/a + 9)
71 Theil - pag. 14
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 479
This result shows that when last year’s population increase is
above average (v,>>v), the savings ratio to be applied in this
year is below the level that would be applied if the population
increase would be known to remain constant at the value v.
This is apparently due to the fact that this year’s population
‘ncrease is then also expected to be above average, which has
a negative influence on the savings ratio of this year, see
(5.1). It is true that the population increases of later years,
too, are expected to be above average, which has a positive
influence, but this effect is evidently of less importance.
The desired rate of increase of per capita consumption, d,,
need not to be constant. A simple alternative specification is
(5.09)
”,
, @
A
which implies that the desire of the first year is confined to a,
after which it increases gradually and approaches 4 in the limit.
The first-period decision of the maximizing strategy under con
ditions (5.9) and (5.7 iz:
(5.10)
1
On comparing this result with (5.8), we find that the savings
ratio is equal to that of (5.8) under the assumption that the
desired increase is constant at the level d, except that a certain
"71 Theil - pag. 15
180
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
multiple d - d, has to be added. This multiple is obviously
positive, since the increasing desires in later years require more
saving in the beginning.
6. THE SIMULATION TECHNIQUE
We shall now apply the ideas set forth numerically by
means of a simulation technique. Thus we consider a number
of countries whose economies are taken care of by an equal
number of decision makers. These decision makers control the
savings ratios of their respective countries and do so, year
after year, on the basis of the decision rule (5.10). That is,
each of them maximizes the expectation of the utility function
(3.7) subject to the random constraint (3.4), where it is assumed
that the desired values of the log-changes in per capita consumption
are of the form (5.9) and that the rate of change of
the population satisfies the stochastic difference equation (5.7).
This equation supplies the random element of the process,
which is of course the rationale of the simulation technique.
Fifty countries have been considered, each during a period
of 50 years. All start with an initial savings ratio (x,) of 10%.
The first-year desired increase (d,) of per capita consumption
is 2% %, the long-run desired increase (d) is 74%. The coefficient
7 of (5.9) is put equal to 0.95, which implies that after
about #=15 years there is a desired increase d, half-way between
the first-year and the long-run desire, d,= (4, +d)=5%-The
expected value of the rate of increase of the population,
v, is taken equal to 0.015. The random variables e, of (5.7)
have been generated as normal variates with zero mean and
three alternative standard deviations: 0.005, o.or and 0.02.
These alternatives are chosen to illustrate the importance of
7] Theil-- pag. 16
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48,
different degrees of uncertainty (*). The values of the standard
deviations are of course on the high side when they are considered
as describing a distribution relating to population
changes; but this is hardly relevant in the present connection,
since the random elements of this study should be regarded as
representing uncertainty in development problems in general.
The autoregressive scheme (5.7) is started up with a value for
vo — v, which is a random normal variable with zero mean and
a variance equal to 2 times the variance of the e’s. It is easily
verified that this is the variance of v,-v for any t when we
assumed that the autoregressive scheme has been in operation
since Adam and Eve.
The b-value chosen is 17; in accordance with the remarks
made at the end of Section 2. For the capital-income ratio (g)
we take 3'/; years. The coefficient g of the preference function
(3-7) is put equal to 0.1, which means that a change |x, - x,_,|
in the savings ratio is considered ten times less serious than
a discrepancy y, (between desired and realized log-change in
per capita consumption) of the same size.
It would go too far to mention all results for all countries
in all years separately. One sample case is presented in
Table 1. It shows that the savings ratio is increased from 10
to almost 139, in the first year, to almost 159, in the second
year, and so on, after which the further increases tend to become
smaller and smaller. In some years there are decreases
rather than increases. After 50 years the savings ratio is close
to 30%. In the first two years there is a decrease (between 1
and 2% ) in per capita consumption, witness the negative values
*) Note that the ¢’s generated are identical for each triple (apart from
an adjustment such that their standard deviations are 0.005, 0.01 and 0.02,
respectively). That is, for each country i and each year t (where
t, t=1, ..., 50) one single g is generated. This was done to guarantee a
rserfect ceteris paribus situation for the three different standard deviations
4
Theil - pag.
~
182 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
7 =
of the first two elements of the second column. This is the
natural consequence of the necessity to start the development
by reducing consumption at the beginning. It is of course not
the Law of the Medes and the Persians that the period of successive
decreases should last two years, since the process is
subject to random fluctuations. Table 2 gives a survey of
results and reveals that 2 years is about the average period
during which consumption decreases. When there is little
random variability (0.=0.005) the individual periods of decline
are concentrated closely around this average; but for larger
standard deviations there is also more variability of the decline
periods, as could be expected. The category « other cases »
refers to those in which the period of decline is interrupted
by one or more years during which per capita consumption
increases rather than decreases; e. g., decreases in the first two
years followed by an increase in the third and a decrease in
the fourth year. The signs of changes in the fifth year and later
have been disregarded; they are negative in some isolated
cases for o.=o0.02.
Table 1 shows further that in the first decade the total logarithmic
increase of per capita consumption is 0.1611, which
corresponds to a percentage increase of about 17. In the second
decade it is much larger: 0.6757 corresponding to a percentage
increase of almost 100. These increases, too, are subject to
random variability. A survey of the quartiles of decade growth
rates is presented in Table 3. The results show a gradual
increase of these quartiles over time. Of course, the increase
is not so regular in every individual case! This is illustrated
by Table I, which indicates that in that case the decade growth
rate decreases in the third and fourth decade. For the set of
all data generated it appears that different values of o, do not
lead to much difference as far as the median growth rates are
concerned, but they do lead to important differences in dispersion.
“21 Theil - pag. 18
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G0
TABLE I. — A sample case of 50 years development (0. =0.0="
‘ear
7
J
nN
20
|
y“
22
>
2°
2”
yr
24]
20
3N
3
3:
34
35
36
37
38
39
LA
1
14
Y
t
t9
yO
Savings ratio
(x 1000)
27
49
167
82
96
207
216
224
229
N24
246
254
261
265
265
266
269
272
ITA
a 0
284
284
283
286
287
286
282
2718
277
276
277
277
279
280
282
280
280
280
283
"o£
287
285
288
289
291
293
295
295
Log-change in per capita
consumption (x 10,000)
tai
28
30
195
42
319
268
308
310
357
550
577
155
521
164
503
560
741
799
787
548
669
B86
712
701
437
454
500
488
52.
182
702
541
584
141
558
593
708
341
652
341
133
588
153
169
1
711
8417
Decade log-change
(x 10.000)
SAC
157
701
184
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE 2. — Periods of initial decline of per capita consumption.
Number of cases when the standard deviation of e is
0.005 0.01 0.02
Zero . .
One year
Two years
Three years
Four years
Other cases
n
15
5
N
4
7
13
11
3
‘9
TABLE 3. — Quartiles of decade growth rates of per capita
consumption ( x 10,000).
[Lower quartile .
Median Coe.
Upper quartile
Lower quartile .
Median ..
Upper auartile
Lower quartile . .
Median . . . . .
Upper auartile .
First Second Third
jecade decade decade
Fourth Fifth
decade decade
g,= 0.005
1959 5626 6494 6747 7095
2184 5730 6738 7014 7283
2433 5885 6880 7954 7483
a, =0.01
718 5462 6330 6475 6976
2166 5672 6818 7010 7344
2664 5980 7109 7489 7759
oc, = 0.02
1231 5133 6004 5934 6736
2130 5553 6980 7004 7474
21928 6169 7546 7061 89288
71 Theil - pag. 20
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Go.
The final judgment on the performance of the process must
be based on the criterion function. If we confine ourselves tc
the first fi'ly years, the function can be written in the form:
6.1)
the expectation of which we try to minimize. The values whic
the function (6.1) takes in the 50 x 3 cases are subject to a
distribution and will therefore differ from case to case. Howaver,
the distribution of the performance measures (6.1) is not
very exciting. A more useful measure, particularly in the context
of the present paper, is a relative performance measure
describing how well the policy has done compared with two
alternative approaches: the « perfect » approach, which forecasts
correctly all future values which the chance mechanism
will produce, and the « naive » approach, which does not use
the information which is available to the decision maker at
the moment when he has to make his decision. The former
approach is based on equation (5.1), its v’s being generated
by the ¢’s of (5.7) as these were actually generated by the
normal chance mechanism. [Strictly speaking, (5.1) and (5.7)
require that we use an infinite number of ¢’s, which is somewhat
anpractical, but the 25 which have been generated beyond the
50 years of the experiment are sufficient for our purpose].
The second, naive approach is based on (5.10) except that the
last term (in v, - v) is deleted. That is, the decision maker
just acts as if the rate of increase of the population has been
and will remain constant at the level v. The savings ratio of
the last policy is not affected by any random element and can
therefore be computed for any year ¢ at the beginning of the
Theil - pag. 21
186
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 728
50-year period (°). It is the same for all 50 countries and it
can be regarded as a primitive kind of fixed-plan policy under
disregard of any kind of new information.
The relative performance measure that will be used takes
the form of a ratio:
(6. I )strategy TS (6 I )perfect
(6. I )naive -— (6 . I)perfect
That is, we take the strategy value of the function (6.1) as a
deviation from the value which this function takes in case the
perfect approach is adopted and express this difference as a
fraction of the similar difference for the naive approach. The
ratio is zero when the strategy approach yields as good results
as the perfect approach; it is one if the strategy approach is
as good or as bad as the naive approach. We should obviously
expect that the ratio is generally between o and 1, and this is
indeed the case as we see from Table 4.
TABLE 4. —
Relative performance of the strategy approach.
Standard deviation of the ¢’ s
0.005 0.01 0.02
Lower quartile . . .
Median . . . . . .
Upper quartile
0.816 0.844 0.866
0.896 0.911 0.916
N 995 n 97% 0.976
‘) In eq. (2.8) we approximated the logarithm of the ratio of 1—x, to
t—x,_, by the ratio of —(x,—#,_1) to I—#,_1, which was approximated
further to -b{(x.—x, ,)\ to ohtain a linear expression The latter approxi-71
Theil - pag. 22
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The median turns out to be of the order of 0.9, which
means that in about 50% of all cases the value of the objective
function (6.1) was reduced by the strategy approach by at least
10% (when our yardstick is the distance from naivety to perfection).
Whether this is to be considered as much or as little
is a question of taste; the present author must confess that
he is not impressed by this figure. But a more important
question is whether the strategy approach leads to sufficiently
superior outcomes in general, and this can only be taught by
future research. The results of Table 4 suggest further that the
dispersion of the individual relative performances tends to decrease
when the standard deviation of the €’s becomes smaller.
This is a somewhat surprising result if it would be true; to find
out whether it is true or not requires a larger experiment.
Simulation experiments are useful to show the random variability
of outcomes whenever the model on which they are
based is too complicated to be handled analytically by the
ordinary methods of probability theory. The model used here
cannot claim to be complicated, but, in a more general con-‘ext,
it is not difficult to think of other models in development
programming which do have this feature. Even in this simple
case, however, the method illustrates clearly something which
is obvious to those who think in probability terms but which is
not so obvious to those who think in terms of nonstochastic economics;
viz., that a procedure which is good in general is not
necessarily good (at least not good in a limited time period)
in all individual cases. For example, the relative performance
measure is above I in almost 20%, of all cases. Thus, if we
mation can be improved upon if we replace b by the reciprocal of 1 minus
che x,_, of the fixed-plan policy, because it turns out (in the present
application at least) that the fixed-plan savings ratios do not differ too
much from the corresponding savings ratios of most of the strategy cases.
This alteration would imply that the multiplicative coefficients of eq. (2.9)
“ease to be constant over time, which makes the computations more comp
cated, but probablv not in a hopeless manner
Theil - pag. 2-
188 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
advocate the strategy method to Turkey, Egypt and Iran, it
may turn out after half a century that our advice was good
for Turkey and Egypt, but not for Iran. This should be stated
frankly to the policy makers concerned, which can be done
easily with the aid of a simulation experiment. In fact, this is
another, practical advantage of the simulation method: presenting
its results in tabular and graphical form to laymen is
far simpler than the presentation of many sheets of algebra.
This advantage remains even when it is true, as in the case
considered here, that the model used is not very complicated.
7 Theil - pag. 24
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APPENDIX
Maximizing the quadratic preference function (3.7) subjec:
to the constraint (3.4) can be done conveniently by using iui:
constraint to eliminate the vector of noncontrolled variables
The result is
‘A.1)
where
‘A.2)
a (à, Po +8) —
F
Tr
k, rk oi
Eo
7xge, Rs
‘= - R'R,
x'Ka
Che optimal decision is then
‘A.3) x= -K k= (gd + RR)! (gx,
which shows that our first task is to find the inverse of gA + R'KX
Let us write
A.4)
Theil
pag. -
190
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
then we have
_ ao
2
a \
0
he
de
2
-be
bh? 4 ¢2
—
"eos
ve
L_
and hence
(A5) gd + RR
b +c +29 —(bc+g) oO co]
(be+g) b'+c+29 —(be+9) …
—(be+g) b’+c+2g
\
which is an infinite band matrix in the infinite-horizon case.
The derivations will be somewhat simplified if we write (A.5)
in the following form:
(A.6)
ha 1
gA+RR
=2(bc +9)
thr
he + q)
The matrix on the right [disregarding the factor 2 (bc+ g)]
can be written as a scalar (p) times the product of two simple
71 Theil - pag. 26
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Le
matrices :
so that we can identi?
I+A) = 1-oy
r.
A
This implies 2pA =1. When added to and subtracted from the
equation for p(1+ Xx?) we obtain expressions for p(x +1)? and
p(x — 12:
pI+4)? — 2 |
r)
4)
pa - 22
Taking the ratio and then square roots leads to a linear equa-“on
in À with the following solution:
‘A
{be + @
she + q
f
*
and the solution for p is then 1
with (A.6) we conclude:
—,
On combining these results
A.5,
Dheil - pag.
192
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
It is now easy to determine the inverse of 9A + R’R, since
1d
x
LS
A
+
yr
AS L340
AEE 2
SK
1%
rar
À ae
pv
4024
”
The typical (¢, #)" element is therefore
Min (t,t)
5 NEHV—2i —
I
WE] _ ptt
)
and so we obtain for the typical element of (gA + R’R)-—!:
(A.9) (gA + RR) =
rT —
St] test
A) ergy T= An
771 Theil - pag. 28
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493
Our second task, according to (A.3, is to postmultiply this
‘nverse bv tle <r!
“A. IV} NS
Then, after postmultiplying the inverse by this vector we obtain
the optimal decision ver!ur - " of which the #" component is:
AI
which is eauivale..t
5°
sui the first component
Theil - pag. 2c
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
REFERENCES
‘1] Horr C.C., F. MoOpIGLIANI, J.F. MUTH and H.A. Simon, Planning
Production, Inventories, and Work Force. Prentice Hall, Inc., Englewood
Cliffs, N.J., 1960.
[2] Simon H.A., Dynamic Programming under Uncertainty with a Quadratic
Criterion Function. « Econometrica », Vol. 24 (1956), pp. 74-81.
3] Tuer. H., 4 Note on Certainty Equivalence in Dynamic Planning.
« Econometrica », Vol. 25 (1957), pp. 346-340.
[4] Tue. H., Economic Forecasts and Policy. Second Edition. Vol. XV
of Contributions to Economic Analysis. North-Holland Publishing Company,
Amsterdam. 1961.
5
THrrL H., Optimal Decision Rules for Government and Industry. Vol. 1
of Studies in Mathematical and Managerial Economics. North-Holland
Publishing Company. Amsterdam, 1964
71 Theil - pag. 30
SST
[SARD
First, I want to ask what you would consider to be a significant
value for the median (on page 23), or the criteria which would govern
what a significant value might be. Could you develop this
point further? Second, I was wondering whether, in doing your
research, you had considered any other concept of rational behaviour
besides the one of maximizing expected utility. Suppose you had
raken a more conservative approach toward the problem, say a
max-min approach. What would have happened? Would there
have been much difference? After all, how many political figures
nvolved in economic planning would think in terms of maximizing
expected utility?
FRISCH
Let me start by complimenting Professor THEIL for the simplicity
of the model he used, and I am not saying this in a sarcastic way.
[ really mean that frequently we can exhibit certain fundamental
properties of a problem, by using simplified examples. And of
course you save a lot of time and mental effort if you use simplified
examples.
Theil - pag. 3
196
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€
Now for other aspects at Professor THEIL’s paper I may perhaps
be a little more critical. First of all, regarding the very principles
of using rules of strategy. I really doubt whether that is a practical
proposition in actual economic planning. To me it would seem
much more efficient, much more practical, to use what I call and
what I will speak about on Thursday — the principle of moving
planning. That is to say, each year you are going to work out a
completely new, say, five-year plan, taking account of the new informations
you have got, what ever new slants to your preference function
you would like to give and so on. Let me give you an example,
suppose I am out in the woods and I am lost in the darkness of
the night. I am lost; I do not know where I am. All right, what
to do? Will have to wait till morning and the sunrise to find my
way. Now what would be the use of sitting down and enumerating
to myself the various alternatives that might happen and how I
should then react to them? Suppose in the morning, when the sun
comes up, I see that there is an abyss to the north. I would not go
there. A river to the east; I would not try to pass that. If I discover
a hill to the left I may decide to go up to the top of that hill and
look around. It is little use from the practical point of view, to
enumerate all the possibilities and to decide beforehand about
the action I ought to take if any given alternative materialises. It
is much better to wait till the moming comes and then decide on
the basis of actual alternatives, what to do. That would be an
Alustration of my principle of moving planning. Then there is the
question of this desirable rate of increase, or desired rate of increase
in consumption. If I have understood Professor THEIL’s presentation
correctly this is taken as a datum. That is correct, isn’t it?
(Professor THEIL answers yes) All right. Now I can easily understand
that from the pre-programming viewpoint you can impose a lower
bound on consumption and on the rate of increase in consumption,
perhaps from nutritional data, calorie contents, right vitamins and
so on. But it is very difficult for me to understand that the precise
size of the rate of increase of consumption can be fixed in advance.
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This would be what I call the pre-programming attitude. To me the
rate of increase in consumption is something that will come out of
a programming analysis, which starts from much more basic data.
Now only one more remark which is not as important as the two
ârst I made; that is regarding the procedure of taking the rate of
saving as an instrument in the political sense. I do not see how
you can take the rate of saving as a political instrument because it
's not something on which the policy maker can decide. It is not
one of his parameters of action. The saving rate is rather one of the
consequences that will emerge from his decisions on a lot of other
things. It is a consequence from his decision upon action and
ran only be spoken about in the post-programming sense, not in
a pre-programming ses.”
| EONTIEF
In presenting his method of sequential decision-making, Profes
sor THEIL apparently commits the decision maker to use the same
procedure in each stage of the process. But is this really necessary:
Can one not view each step of the process as involving a decision
over a limited interval of time at the end of which all the available
‘nformation, including the information concerning the correctness
of previous anticipations, provides a new, essentially independent
basis for the next decision? The possibility of modifying the decision
rule itself from one period of time to the next should not
ye excluded.
ISHER
i have a question related to a simulation experiment not
performed by Professor THEIL. One of the results of hi: —--hat,
in terms of the criterion function as a whole, .he
approach does about ro ner cent better than what Professor
Ty
““HEII
7
Theil - pag. 33
108 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
calls the naive approach. I am curious to know how that 10 per
cent is made up. In particular, I would like to know how the approach
does in terms of the rate of growth of consumption taken
separately on the one hand and in terms of the changes in the savings
ratio necessitated also taken separately, on the other hand. Tt
would seem to me to be of some interest to know whether the gain
comes principally from the variable which one is ultimately interested
in or whether it comes from being able to cut down on the number
of changes of policy needed to get the variable of ultimate interest
up as high as possible.
THEIL
Professor LEONTIEF’s question is clearly the most fundamental
one that has been raised in the discussion. I think it is worthwhile
to make a distinction between strategies in general and the specific
case of quadratic preference functions and linear constraints. As to
strategies in general, it can be proved fairly easily by means of an
example that if you take a decision step by step and if you do not
combine your present decisions with those of later decisions (as is
the case when a decision rule or strategy is used), you may lose in
terms of expected utility. In fact, it makes also difference as to
the first period decision. However, in the special situation of quadratic
preference functions and linear constraints, this is different.
The first-period certain equivalence theorem enables us to compute
this maximizing first-period decision in a very simple way by just
neglecting this difficult procedure of comparing all these possible
strategies,
Professor FriscH dislikes the idea of taking the rate of savings
as an instrument. This is the result of the simplicity of the model.
Making it more realistic would have forced me to go into a much
more elaborate model and I am afraid that in that case I would have
lost the main point, which is, as Professor LEONTIEF put it so aptly,
not about the particular model or the particular preference function,
71 Theil - pag. 34
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but just the general idea of this kind of decision making with regard
to problems of development planning. Regarding another remark
made by Professor FRISCH as to the distinction between moving
planning instead of a strategy, I think really that the difference
is not as big as he thinks it is. Particularly this possibility of using
‘he certainty equivalence procedure makes that effectively we are
working with a moving horizon, we could call it moving planning.
This can be particularly easily justified when one works with an
nfinite horizon.
I should also add that this approach of maximizing the expectaon
of a quadratic preference function over time subject to linear
constraints is the only case for which it is possible to handle three
difficulties simultaneously: uncertainty, maximization over time,
and many variables. The method of « dynamic programming »
breaks down when the number of variables is not extremely small.
However, inequality constraints cannot be handled by my approach
‘Professor FriscH suggested the use of lower bounds). On the
other hand, the approach has several interesting advantages, for
which I would like to refer to my Optimal Decision Rules for Go.
vernment and Industry.
Furthermore, Professor FriscH talked about a priori desired
values of the quadratic preference functions. I agree with him that
t would be worthwhile to have a preliminary study as to how we
should really define these desires in any scientific manner. I think
t is rather difficult to do and for the moment I would like to suggest
that this is being solved by a process of experimentation.
Professor Isarp asked two questions. He wondered — it was
page 23 — what I had expected for this relative performance of the
strategy approach and I expressed more or less my dissatisfaction
that 0.9 came out of it. I have no precise ideas about what is
zoing to come out of this in general. This is a matter that will be
‘ound in future research. To take an example, right now we know
rather well when a correlation coefficient is high or when it is low.
But if you go back a sufficient number of decades, when this coefcient
was formulated for the first time, the man who invented ”
= -
2 F
Theil - pag.
35
500 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
certainly had no precise idea about when he should judge his coefficient
high or when low. The only thing he knew was that the
coefficient was bound to lie between —1 and +1; whether .7 was
high or whether .99 was high, that certainly was not a matter that
could be decided at that time. In the same way I hope I am able
to give an appropriate answer after ten years or so.
Regarding expected utility to be replaced by the max-min approach,
I have my doubts. I think the max-min approach is always
a little difficult by the time your random variables have an infinite
range. And also regarding expected utility, this has a rather firm
foundation given by von NEUMAN and MORGENSTERN; they showed
that under a number of rather innocent assumptions a rational man
behaves as if he maximizes expected utility. It is of course quite
another affair whether these preferences can be represented in the
way I do, but I think that given the large number of difficulties
which we are bound to have in formulating a preference function
anyhow, this is a matter of relatively minor concern.
Finally, regarding Professor FISHER, I must say that no computations
have been made on the part of the loss function which
is due to the instruments and the part which is due to the noncontrolled
variables. But the basic data are available, so the computations
can be made.
FRISCH
Professor THEIL excused himself for the non-realism of the choice
of instruments. He excused himself by pointing to the simplicity
of the model. I should say that this is an excellent example of how
a simple model can reveal very pertinent facts and conclusions.
In this particularly simple model, which I complimented Professor
THEIL on, I should say that the capital output ratio is a much
better example of a real instrument, but THEIL took that simply as
a constant that was given. That was mv first remark.
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Second, it seems that Professor THEIL connected the idea of
moving planning with the idea of an infinite horizon. There is absolutely
no connection between the two. When I speak about moving
planning, I simply mean that I am not committed to any predeternined
rule of strategy but simply I am evaluating the whole situation
in a new and entirely free way, and that, of course, I may do
even if my planning period is a short one — three, four, five years
or something. So it has nothing to do with the idea of an infinite
horizon.
Professor THEIL also made a remark on the MORGENSTERN theory
of expected values of utility. I do not think you can justify this
approach by saying that it is an expression of rational behaviour
or something else pertaining to the substance matter of the problem.
The whole thing here is only a formal one, and resides in the fact
that you introduce an assumption which makes utility additive.
And if you do that, if you put that up as an axiom, the you can derive
a lot of consequences regarding sub-optimality. Prof. THEIL’s use
»f the word suboptimal is really an example of what you may call
persuasive definition, because optimality in this particular case is
in the end precisely an expression for the idea of expected utility.
THEIL
This idea of moving planning and moving horizons can also be
applied in the case of a « truncated » horizon of, say, three years.
Then the decision maker is supposed to look three years ahead at
he beginning of every year, and he should evaluate all things all
over again. Hence there is not as much difference with Professor
FRISCH’s ideas as he thinks there is. Regarding sub-optimality, 1
define as the optimal decision the decision which maximizes the prelerence
function, not the expectation of the preference function,
subject to the constraints as they actually are. There is therefore
no reason to speak about a « persuasive terminology »
Theil - pag. 37
SOME OBSERVATIONS ON COUNTERCYCLICAL
FISCAL POLICY AND ITS EFFECTS ON
ECONOMIC GROWTH
TRYGVE JAAVELMO
l'niversitetet à Oslo - Oslo - Norge
{NTRODUCTORY REMARKS
A leading idea in « Post-Keynesian » economic thinking
has been the strong emphasis on fiscal policy as a means of
steering an economy towards full and efficient use of its resources.
But for this purpose, it is held, the policy must be
« radical » and « unorthodox », and not of the old-fashioned,
balanced-budget type. The line of reasoning, stated briefly,
seems to run approximately as follows: « Orthodox » fiscal
nolicy based on a balanced budget and relatively modest economic
activity on the part of the state is responsible for unemployment
and waste of resources during business depressions.
The task of a good fiscal policy should be to keep effective
demand high enough — if necessary through very large budget
deficits — in order to maintain full employment also when
private investment activity is low. Even more far-reaching
is the particular ideology that often seems to accompany this
kind of reasoning, viz. that a government actually has fulfilled
its main obligations as an economic policy-maker if i
adheres to a fiscal policy as described.
Haavelmo - pag.
504
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
This strong emphasis on fiscal measures as an instrument
of economic policy raises several interesting questions. For
one thing, it seems quite curious that, once possible defects
of an automatic market mechanism are admitted, the explanation
should simply be: « wrong fiscal policy ». The choice to
pin all the blame for depressions and idle resources on « oldfashioned
» budget practices alone seems rather arbitrary, in
view of the many other possibilities there are to remold the
economic system by means of policy decisions. Another question
is whether a flexible budget policy can actually solve the
relatively simple problem of maintaining full employment in
a macroeconomic sense. A constraint in the form of demand
for relatively stable prices could be sufficient to make the
problem very difficult, even hopeless (!). Here we shall, however,
not concern ourselves with these and related questions,
although they are interesting enough. What we want to discuss
is the following more straightforward question.
Let us accept the idea that full employment can be maintained
by regulating effective demand and that this regulation
can be carried out by means of a policy of deficit spending.
Let us further assume that such a policy can be practiced
without too serious and unwanted side effects in the form of
inflation, reduced labor efficiency, or the like. And let there
also be no minimizing of the tremendous improvement that
such a policy could mean as against the alternative of a passive
laissez faire principle. But this being recognized, there is
still the important question of whether such a full employment
policy would be « optimal » in any reasonable meaning of that
phrase. In other words, is the goal that such a full employment
policy sets itself sufficiently high to satisfy modern requirements
concerning efficient use of a nation’s resources? That is the
question to which the rest of this paper is devoted.
(') Cf. P.A. SAMUELsON and R. Sorow, Analytical Aspects of Anti-inflation
Policy. « American Economic Review ». Mav 1060. D. 177-104.
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TX ITIONG
AND ASSUMPTIONS
505
We shall carry out our analysis under the assumption of
a closed economy consisting of two sub-sectors, a private sector
and a public, or government, sector. We shall need the following
variables in the models to be considered.
(1) x(t) = total net national product
(2) «xp(t) = part of x consumed by private sector
3) xo(" - part of x consumed by public sector (« collective
consumption »)
(4) ke(t) = part of x representing private net investment
(5) Ro({\ = part of x representing public net investment
(6) RC, Rp(t) + Rolf)
(7 K total stock of productive capital of all kinds, in
the whole economy
disposable income of the private sector
rate of total government deficit spending
(8) w(t
(9) ml
All variables are to be regarded as in constant prices
Except for K, the variables are all flows, i.e. they have tae
dimension of « per unit of time ». More information concerning
the meaning ascribed to the variables is given implicitly
by the following definitional relations (using hereafter the sim
pler notation of x, xp, etc. instead of x(£), xp(: -* °
2.1)
(2.2)
(2.7)
Haavelmo - pag.
306 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
We propose to work with the following simplifying assumptions
concerning the structure of the economv considered.
Assumption 1. The capacity to produce of the economy is
assumed to be proportional to the stock of capital, i.e. equal
to aK, where ‘a’ is a constant over time. We shall assume that
population and total labor force are constants and that the
part of total labor force employed at any time is a function
of the part of total capacity actually in use. The coefficient a,
therefore, includes also the effect of labor input. (More specifically,
if N is the total (constant) labor force, N actual
employment, and x actual total output, we could assume that
N/N is equal to x/aK).
Assumption 2. The propensity to consume of the private
sector is given by
(2.4)
Xp=0V +
3
where a and {3 are constants over time, and such that o<<a<(1
and 8>o.
Assumption 3. Public consumption, xo, is assumed to be
proportional to the size of the economy as measured by its
production capacity. We thus have
(2.5)
Xa=%aK
where y is a constant over time.
Regarding the coefficients a, «, B, Ÿ and the initial value
of K, we shall assume that a(z - v)aK + + yaK<aK, i.e. that
the maximum rate of public and private consumption does
not exceed capacity (cf. Assumption 4 below).
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Assumption 4. The level of public investment activity is
assumed to be proportional to total « desired savings » in the
economy when operating at full capacity. The meaning of this
needs a little further explanation. Available resources at full
capacity, after allowing for public consumption xo, is equal
'o (1-ÿ)aK. If an income corresponding to this flow of
resources were at the disposal of consumers, they would want
to spend «(1 - Y)aK +f of it for xp. The remainder, available
for public and private investment activity, would be equal to
(1—a)(1—7y)aK—_ (which is assumed to be initially positive).
Our « assumption 4 » can then be written as
2 6)
«
Ro -
5 j\a —0)(1—y)aK—5
where 8 is a constant over time. Alternative levels of à (including
&=o or &=1) may be considered.
Under the specifications describing the details of Assumption
4 the maximum flow of production available for private
investment would be 7 gM NAN TU
More special assumptions concerning the nature of fiscal
policy on the one hand, and the nature of private investment
activity on the other hand will be introduced in Section 4.
The simplifying assumptions introduced above are, of
course, very drastic and grossly unrealistic in many aspects. In
one respect, however, they are not disturbingly unrealistic,
namely in their degree of similarity to the kind of models on
which much of the theory of fiscal measures for full employment
is based. If this is correct, it would seem of interest and fair
enough to study the efficiency of fiscal policy under the assumptions
that we have introduced.
Haavelmo - pag. _
508 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
pe
3. SOME PRELIMINARY CONCLUSIONS CONCERNING THE CONSE-QUENCES
OF THE « OLD » VS THE « NEW » FiscAL PoLicy
On the basis of the — as yet incomplete — model above it
is possible to illustrate the differences between the « old » and
the « new » ideas of fiscal policy and to bring out some of their
striking characteristics.
We shall regard private investment, kp, as the independent
and freely variable element of the model and study the consequences
of variations in this part of total activity.
Let us define « old-fashioned » fiscal policy as a policy
where m is kept constantly equal to zero. (It may perhaps
also be reasonable to assume that this policy would be coupled
with a state of affairs where 3, indicating the public part of
investment activity, is rather small.) From (2.2) and (2.4)
we then derive the now rather « threadbare » textbook relation
(3.1)
T —
Y
(m=0o)
provided, of course, that x=xp+x0+kp+ko<aK. This
« classical » relation of a neutral fiscal policy shows one property
that would make good common sense to most people,
viz. that when the economy is in a state of stagnation or slow
progress (i.e. kp small), then people do not feel that they can
spend much on consumption either. The senseless aspect of
the situation is, of course, that both xp and kp may be far below
the level that available capacity permits.
The « new » type of fiscal policy, on the other hand, can
be illustrated by requiring that xp+ xo + ke + ka=aK. which
81 Haavelmo - pag. 6
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30
will determine the necessary level of m for a given kp. Using
the relations of Section ‘His ~Telds the following ecnations
r
3.2,
3.3)
3.4)
These relations of a « fiscal policy for full employment »
show some interesting — not to say disturbing — aspects.
First, we find that the less economic progress the society is
making (i.e. the smaller kp is), the more it spends on current
consumption. Second, and this explains the high rate of consumption,
the less economic progress the society is making,
the more people earn! This follows from relation (3.4). Third,
the less economic progress the society is making, the more
people feel that they are providing for the future in the form
of saving! This follows bv calculating v - x-. which yields
Another way of putting this last statement is to say that the
slower the real economic progress of society the higher the rate
(m — ko) of « unfounded paper claims » added to the weall!
of the private sector.
In spite of the obvious improvement over the « old » poicy
in that capacity is being utilized, it would seem rather di.
ficult to imagine that the people of an econorv would have
Haavelmo - pag.
ir
SIL
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
welfare function under which the strange properties exhibited
above could be an optimal choice if only the « real » constraints
of the system. viz. capacity to produce, were to count.
4. EFFECTS OF CYCLICAL VARIATIONS IN PRIVATE INVESTMENT
ON THE RATE OF ECONOMIC GROWTH
In order to illustrate the possible effects of variable private
investment under a fiscal policy for full employment we shall
now make a strict, but not entirely unrealistic, assumption
about the behavior of private investment. We shall assume
that it is cyclical with variable amplitude around some variable
(but presumably positive) level. More specifically, we shall
assume kp to have the following time shape:
(4.1) ke = A(f) [sin wt+B1,
where A(#) is some non-negative function and B a constant.
We want to propose some fairly reasonable assumptions in
order to determine the factor A(#) and the constant B. We shall
suppose that when the desire of private investors to invest is
at its peak, it is sufficiently high to exhaust whatever capacity
is available for this purpose. When, on the other hand, the
desire to invest is low, we shall assume that its practical lower
level is zero.
The second of these assumptions will be satisfied if we put
(4.2)
R=The
determination of A(#) is a little more complicated. It
depends on, among other things, to what extent it is possible
to vary the government parameter m.
In point of principle the available capacity for ke could
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be made large by operating with a negative and numerically
large value of m. However, experience in recent years from
many countries has shown very clearly that it is, politically,
very difficult to get m very much below zero. In fact, if government
investment ko is of some size, it has proved next to
impossible to make even (m - ko) substantially negative (except,
perhaps, through the camouflage of an inflationary process
yielding « unintended » higher tax returns). As a practical
assumption in the present context we shall, therefore, impose
the constraint that
4.3) Minimum value of (m - &
Consequently, we find that the maximum value of kp is limited
by the government policies for x, and k, as determined by
(2.5) and (1.6), respectively, and by the lowest value of xp
that can be obtained under the constraint (4.3). This implies
that
(4.4)
na
J, max
.
Jai
From (4.4) and (4.2) it then follows that it may be reason
able to determine A(#) in (4.7) by setting
4.5) Alf) =
y
»—0) [(I—x)(1—y)aK— 8].
We then obtain the following differential equation describing
the development of * stal canital under the full-employment fiscal
policy consider.
4.6)
CC Isinw/-«raavelmo
- pag.
512
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The general solution of this equation has the form
4-7)
K = He”) + K
where H is an arbitrary constant to be determined by the initial
conditions and where the function A(#) and the constant K are
given by
(4-8)
h(z) —
»—æ)(1—y)(e -- = cos té +
+ (1—a)(1—Y)a | e-s + 5 É,
(4-9)
IB
I—à
2
It is seen that %(¢) is composed of a pure cyclical component
and a trend element, while h(t) is a linear function of the pure
cycle alone. From the constraints that we have imposed upon
ko and kp, making them uniformly non-negative, it is obvious
that H must be positive and that h(t) can never be negative.
Inspecting our basic model equations we find that x, xo, and
ko are linear functions of K alone, while xp, kp, y, and m are
linear functions of K and K. From these considerations we
can draw the following conclusions.
Conclusion 1
Even if fiscal policy is sufficiently « radical » to maintain
full use of capacity (i.e. x=aK) at all times, the rate of growth
will depend essentially upon the extent, (1-8), to which the
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initiative concerning investment is left in the hands of private
nvestors.
It is interesting to compare the two extreme cases &=0 and
>=1. In the first case there is no investment activity on the
part of the government. The second case corresponds, formally,
to a situation where the government is the sole investor. This
second case need not, however, be interpreted in such a strict
sense. It may well include the case where the government is
not actually an investor but merely acts as planning and coordinating
body for all investments in the community. (Indeed,
various fiscal measures, other than a simple policy of
deficit spending as discussed above, may well be useful in
order to implement such a general investment program.‘ From
‘4.6) we can then draw the following conclusion
Conclusion
If all investment is private (8 =0), the trend rate of growth
will be one half of the trend rate of growth when all investment
is public or publicly directed, (8=1).
In judging these rather strong conclusions it is, of course,
essential that we keep in mind the assumptions on which the
results are based. Thus, to mention only one thing, we have
not even touched upon the relative merits of technical efficiency
of public vs. private investment activity. Nevertheless, the
results obtained show a definite tendency to reduced growth
because of fluctuations in private investment, a tendency which
cannot be waved aside as just a peculiarity of our special model
The model has several other rather interesting aspects.
Thus, for example, the rate of accumulation of « unfounded
wealth » of the private sector will, on the average
account for a part of total accumulation which is the larger
the higher is the share (- ¢&) of private investment. How thi:
Haavelmo - pag. .
5,
4
PONTIFICIAF ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2.
is to be interpreted from a welfare point of view, could be a
subject of considerable speculation.
Another obvious characteristic of the model is that the part
of total product which is consumed will, on the average, be
the higher the larger is the share of private investment. But,
consumption too, will be cyclical and will not be based on a
consideration of the actual resources that the consumers, taken
together, have at their disposal at any time.
5. REMARKS ON POSSIBLE EFFECTS OF INDUCED PRIVATE IN-VESTMENT
It might be argued that the assumptions made about private
investment, kp, in the preceding section are unreasonable because
no account is taken of the possibilities of induced investment.
Such investment could have two main components. The
first of these, relating investment to the level of total production
activity, could be based on the hypothesis that when total
activity is high, profit expectations are high, making some latent
investment projects more attractive. The second main
component could be some kind of acceleration effect.
Without going into details it is, nevertheless, fairly obvious
row such elements of induced investment would work in the
model framework considered. Roughly speaking, the effect as
far as the maximum need for deficit spending () is concerned,
would be similar to that of a combined increase in « and in 8.
As for the rate of growth of K, the effect (under a fiscal policy
for full employment) would be similar to an increase in 3.
Of course, one cannot say, without further investigation,
whether the absolute magnitude of the cyclical variations in
total investment, or the absolute maximum level of m at any
given time, would be larger or smaller than they would have
been without such induced investment. The obvious reason
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for complications in this respect is the fact that the whole time
path of the variables will be changed as a result of a change
in the characteristic parameters of the model.
There are at least two reasons why we do not want to follow
ap this analysis of inducement any further in the present context.
One reason is that the theoretical foundation of the idea
of induced investment is in itself rather dubious. Thus, as 1
have tried to show elsewhere (!), it is not easy to justify a
systematic tendency to induced investment by any « classical »
principle of producers’ behavior in maximizing profits. But
‘he second and more important reason is that the possibility
of large-scale induced investment is in fact irrelevant to the
main argument to which this paper is devoted. This last sta-:ement
may need a little further explanation.
The main line of argument of the « new » and « radical »
principles of fiscal policy for full employment has, as far as
[ have understood it, been a) that the government should not
be afraid to accept very large budget deficits if necessary,
b) that in certain periods very large budget deficits will actually
be required, and c) that if such a « radical » policy is adhered
to, the government has done its main job as an economic policv
maker.
Now, if this is a correct interpretation of the line of thinking,
it would seem to me to be rather strange to argue that
induced investment is so important that only modest variations
in the budget deficit would be required. Such an assumption
is also contradicted by facts. Experience has shown that private
investment may be subject to large variations which cannot
easily be explained by any initial change in effective demand
for consumer goods.
The idea that a « radical » fiscal policy for full employment
would not in fact need to operate with very large budget de-(!)
Cf. my book on A Study in the Theory of Investment. University ol
Chicago Press. 1960
Haavelmo - pag.
13
16
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 5
ficits is, I think, based ofi a confusion between two different
behavior patterns as far as private investment is concerned.
On the one hand we have the dubious effect of increased effective
demand upon investment in the case where investment is
assumed to be based on existing capacity and current earnings.
On the other hand we have the possible stimulating effect of
the mere knowledge that the government would use very large
deficits if necessary. This latter effect would then be a result
not of the fact that total activity is high but of the conviction
that total activity would not be permitted to become low.
The second of these two investment theories has, I think,
considerable strength. But it has the rather interesting implication
that the actual use of very large budget deficits would
not be the proper means of stimulating private investment,
except perhaps for effects in the way of « teaching a lesson for
the next depression ». It is likely, however, that definite and
concrete over-all plans for the various parts of economic activity
feasible within the capacity available would be more convincing
as far as creating « business confidence » is concerned.
Haavelmo - pag. ..
AN
USSIGRK
[LEONTIEF
May I ask Professor HAAVELMO to answer the following question:
His equation 3.3 involves a relationship between the magnitude
a deficit and that of public investment. A deficit is measured in monetary
terms while investment in real. Professor HAAVELMO visual
zes apparently the possibility of a situation in which large or
small deficit might be combined with either large or small public
nvestment. It would be helpful if he could interpret in somewhat
more operational terms the practical meaning and implication of
-ither one of these different possible combinations
HAAVELMO
I refer to the definitions (2.1) and (2.2). I can derive the equa:
tions (3.2) - (3.4) from the definitions plus the consumption function
(2.4) and the relation (2.5). m is simply government outlay net
of taxes. Of this outlay the only real counterparts are public consumption
x, and public investment k, The rest is government
subsidies or expenditures on « worthless things ». m for full employment
will be large if total investment, public and private, is small
Haavelmo - pag.
318
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
[SARD
My comment is general. The paper is excellent, but I miss the
regional variable, HaAAVELMO talks about investment programming
and yet we know in national economic planning that a basic problem
is how to allocate investment among the different regions. I am
very conscious, too, of the different investment incentives required
in different regions. And so forth. Thus, althoug I do not wish
to be critical of HAAVELMO’s excellent statement, I do hope that
he and others might perhaps consider sometime in the future introducing
in a simple wav the regional variable in their model.
ALLAIS
I must confess I have been provoked by the conclusion of the
model according to which the rate of growth is doubled under a
regime of public investment. I have searched for the explanation,
because, in a model which is mathematically coherent, the explanation
of the conclusion must lie in the hypotheses adopted. I therefore
have four remarks to make,
The first is that there is one apparently absolutely inoffensive
hypothesis on page 4. That is, the capacity of the economy to
produce is assumed to be proportional to the stock of capital
(assumption 1). At first sight that is quite natural and in my paper
I intend to stress the fact that the capital output ratio is practically
constant and I intend to propose to vou an explanation of this
constancy.
But, from the practical constancy of the capital output ratio,
it is impossible to conclude to any proportionality of real income
to real capital and specifically I intend to show that, for a given
population and with given technological knowledge, there is a maximum
value for real national income whatever the value of real
capital.
If you make the assumption that population is constant and if
you don’t make it explicit somewhere that there is some technical
8] Haavelmo - pag. 16
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514
progress (and this is the case in HAAVELMO’s paper), it is impossible
to derive the conclusion that there can be an indefinite increase in
real consumed income, as is the case in HAAVELMO’s paper (rela
ion 4.8).
My second observation is the following: in any case is the formula
verified by facts? Do we observe that there is a correlation
between the rate of growth and the volume of public investment?
As far as I know, it is impossible to derive that conclusion from
:he data we have.
A third point: It seems to me it is very difficult to assume
some oscillations of private investment (equation 4.1) without introducing
the monetary aspects explicitly. In fact it is certain that
f there is a depression it can be fought efficiently by making public
nvestment, but under one very important condition. This is, that
he public investment be combined with creation of money, Ii
public investment is not combined with the issue of new money,
‘he global effect for the whole economy will remain the same as it
was in the past. With public investment there can be a positive
multiplier effect but if this public investment is financed by a diminution
of spending elsewhere, there is a negative multiplier effect
2lsewhere and the global effect on the whole economy is o. Thus
to have the advantage of full employment, it is not public investment
which is important but the creation of new money. And if,
nstead of undertaking new public investment, one could imagine
the state spending its money in some other way, for instance by
giving subsidies to people, anything which is not investment but
new expenditure, the same effects will be generated. So, in my
opinion, what is very important for full employment is the overall
expenditure of the economy as a whole and not public investment.
My fourth point is that public investment is also subjected to
cyclical fluctuations and if equation (4.1) were valid for public investment,
the results would be absolutely different.
The HAAVELMO paper is very interesting, and, for me, quite
thought provoking, but in my opinion the conclusion at which i
arrives derives directly from hypotheses which are very questionable
Haavelmo - pag
520 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
2e
First, theoretically, the conclusion concerning the influence of the
public investment on the rate of growth depends largely on a very
strong hypothesis which I personally cannot accept (assumption 1).
Secondly, empirically, there is no correlation showing that the rate
of growth is doubled when all investment is publicly directed. Third,
for full employment, what is important is not investment but spending,
and spending depends essentially on monetary policy. Finally,
the fourth hypothesis (4.1) is a very questionable one to the extent
that it assumes that public investment does not fluctuate at all.
FISHER
I have a question concerning the conclusion of part three of the
paper. I am concerned particularly about the place in which it is
found that the fiscal policy for full employment has some peculiar
properties, In particular, HAAVELMO shows that such policies lower
the rate of private investment and for that matter the rate of public
investment. The more that is spent on current consumption and
the more that is being earned evidently the more it is felt that the
future will take care of itself. HAAvVEUMO states that it seems difficult
to imagine that people would actually have a welfare function
in which a policy with these properties would be an optimal choice.
However, this overlooks a crucial point which is expressed in the
equations in the paper. This is that capital stock does in fact appear
with a positive coefficient. It is thus true that while with a
given capital stock people would invest less under full employment,
this would have the result of lowering the capital stock below
what it would have been otherwise and therefore such reduction
of investment could not continue to happen for a long period
of time. It follows that if people had a welfare function which
extends over a reasonably long period, they might well find that
an optimum choice would be fiscal policy for full employment because
they would not then find it to be true that the less progress
they were making, the more they were spending. Professor Haa-'8]
Haavelmo - pag. 18
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521
VELMO’s conclusion overlooks the role of capital stock in the model
ind therefore overlooks the presence of long-run effects
ALLAIS
May I add a remark? The conclusion that the rate of growth
s greater if we have public investment in the model, is evident
without any calculation since Professor HAAVELMO introduces two
strong hypotheses. The first, as I have already stressed, is that
real national income is proportional to real capital. The second one
s the acceptance of an oscillation in K, (formula 4.1), at the same
ime making the assumption that public investment is increasing
when this oscillation is present. With such an hypothesis, it is
vident without any calculation that in the end there must be a
greater rate of growth when the volume of public investment is
significant.
HAAVELMO
First, some general remarks. Let me be quite emphatic about
my ideas on public investment in this connection. There is no kind
of political preference involved, as to who is to carry out investment.
Those who actually carry out investment activity might well
pe the private sector in all cases. That is, what I call public investment
here may just be the part planned, financed or otherwise supported
by the government — that doesn’t change my formulae. If,
1s has been suggested, a bigger and better model were developed,
his might come out more clearly. I am personally not particularly
ond of this kind of simple models, certainly not for planning purposes.
I have just presented it, as I said, to use it as a base for
criticism of a way of thinking.
Now for the more specific points — ALLAIS made his points in
‘wo rounds, perhaps we could take them jointly. Given my objective
for this paper I don’t think the assumption about production capa
Haavelmo - pag. 1g
322 PONTIFICTAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
city being proportional to capital is very central. Rut this would of
course depend essentially on what alternatives are considered more
realistic. I should also like to add an explanatory remark in connection
with my assumption of constant population. The reason
why I made this assumption here is one of simplification and at least
off hand I don’t think that this particular assumption is central in
explaining my main results. Then there was the remark that actually
in statistical figures there is not very much correlation between the
rate of public investment and growth. Well, I don’t know about
that in detail, but I would dare say that probably we have not
had so much of such policy that it would show up very much in
the figures. My model illustrates more a policy that is being talked
about, rather than a policy that has actually been carried out in
full. Then there was the point about business cycles as an element
in the model. I admit that this is not a business cycle theory and
it was not meant to be. I have made the strong assumption about
cyclical movements in private investment just to see how the system
works if investment runs that way. I could refer to many sources
where assumptions have been made about the autonomy of private
investment.
Then there was the comment from Mr, FisHer. He correctly
pointed out, in connection with the formula he referred to, that if
you want to consider matters over time, of course the amount of
capital changes gradually. Now the point of the formulas on page 7,
which he referred to, is just to make some comparative studies,
assuming that you could instantaneously shift the rate of private
investment around, since this is the autonomous factor in my model.
I just wanted to see what the instantaneous effect is and I
can also, at the same time, answer another comment that comes in
here. I am not saying that it is bad that you get more consumption
and less investment or that it is good that you get high growth and
lower consumption. I'm just saying that I doubt very much
whether cyclical movements in the preferences are realistic. I will
also add that one must be aware of the essential dynamic aspects of
the model. It is by no means certain that you will not in the se-[81
Haavelmo - pag. 20
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Ia
cond case get much greater consumption and rate of growth of
capital in the future. Therefore it is doubtful whether one can say
that one alternative leads to more consumption and the other leads
to more investment generally. This may shift over time. That
brings me to the question of the underlying preference function in
general. I have not made a statement of right or wrong here. I have
just said that if one wants growth, here’s how matters work. I'm
not saying that it is impossible that one should want precisely the
development that we get by the model under the assumption of
nvestment being mainly private. There is one point in that connection
however; as far as I see it, the only way that a social preference
function — if we assume such a thing — shows up in this
kind of economic model, is through the consumption function. And
‘here is nothing there that should lead us to think that people have
strange cyclical variations in their preferences. In fact, people may
not even be aware of the investment cycle in my model. They have
constantly full employment. Their disposable income, it is true, is
higher when the economy is not growing than when it grows. But
consumers, counting in money, don’t see that the country is not
growing and therefore they are consuming. I think that this is a
‘air interpretation of the meaning of the consumption function. If
people have high income, they will naturally think that they can
afford to consume more and still provide for the future while
actually they are only piling up money which the government owes
‘hem. I think now that, even though I have not mentioned the
names of all those who commented. I have answered more or less
111 the questions
AILAIS
This conclusion that the rate of growth is increased if you have
public investment instead of private investment is so important from
the point of view of both economic theory and pclic- “hat " hink
I must insist again on three other points
Haavelmo - pag. 21
524
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2R
First point. If in equation 2.4 œ is equal to I, the rate of growth
is 0. In this case, whether there is public investment or private
investment, the rate is exactly the same. This is quite disturbing
because if public investment is more efficient, one would expect that
this conclusion would be independent of œ. The logic of the model
cannot be attacked in any way; it is absolutely correct. But it is
necessary to stress the hypothesis from which the conclusions are
derived. Thus, my first point is: if ¢ is equal to 1, public investment
has no role at all in increasing the rate of growth.
My second point is: HAAVELMO has said that the hypothesis that
real income is proportional to capital does not play any role in the
final results. But if it were assumed that real income cannot increase
indefinitely as a result of increasing investment, the conclusions
of the model would be absolutely different.
And, on the contrary, if it were assumed that real income is
increasing proportionally to real capital, we would be admitting an
hypothesis which is in contradiction with the facts.
If one assumes an indefinite increase of real national income
resulting from indefinitely accumulating capital, this is equivalent
that the stage of decreasing returns to capital is never reached. And
this is not confirmed by observation.
Thirdly, the hypothesis of a cycle of oscillation for K, is essential
for the conclusions, HAAVELMO is completely right in saying
hat fluctuations of K, have been observed in the past, and I am
ready to admit equation (4.1) at least as a first approximation. But
as far as public investment is concerned it has never been observed
to be able to compensate for the fluctuations of private investment.
In addition, we have never observed an absence of fluctuations in
public investment. Thus, I could propose another model conforming
with information relating to the past and show that public
investment has exactly the same drawbacks as private investment
for the rate of growth.
At all times, we must be very careful about the use politicians
could make of such a model. They might believe that a general
demonstration had been . given of the superiority of public invest-[81
Haavelmo - pag. 22
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525
ment, ignoring the very strong and very questionable hypotheses
from which this demonstration was derived
MALINVAUD
[f I have rightly understood Professor HAAVELMO, it seems tu
me he is comparing situations for which the degree of exogenous
irregularity in the economic system is not the same. For instance,
his result according to which the rate of growth would be smaller
if all the investment were private than if it were public, depends
on a comparison between a situation in which fluctuations in effective
demand would come from private investment with a situation
in which all the investment would be public and steady. If such
s the comparison, is it quite fair? Should not we compare situations
in which the degree of exogenous irregularity would be about the
same? A French economist could well argue that in postwar France,
fluctuations did not come much from private investment but much
more from public expenditures including public investments which
were at times influenced by political changes
KOOPMANS
I have just one question. Is the statement that the fraction .
investment that is private affects the rate of growth dependent on
the presence of fluctuations in private investment. Or is that statement
reached in a part of the paper where the fluctuation had not
been introducec’
HAAVELMO
Professor MALINVAUD had a question about cycles in public 1
vestment. I am studving the effects of a certain kind of ~~lir an
77 Haavelmo - pag. .
526
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
cerning public investment, not the effects of empirically observed
public investment. Incidentally, in my model there will in general
be included cycles in public investment too. Koopmans has asked
whether the conclusions about growth depend on the assumption
of cycles in private investment. The answer is « yes ».
WoLD
When reading Professor HAAVELMO’s paper I was struck by
‘he contrast between the simple and seemingly innocent assumptions
on the one side, and the rather startling implications and conclusions
on the other, and at the same time it puzzled me that he gives
little or no comment whether his results can or cannot be reconciled
with current theories. Somehow I got the feeling that Professor
HAAvELMO has written his paper with tongue in cheek, and this
impression was confirmed as I consulted him about his paper one
of the first days of the Study Week. I should now like to ask
whether I have understood his intentions correctly, namely that the
paper illustrates the danger of mixing together the theories of two
different regimes of economic conditions: on the one hand the serious
depression around 1930, and the Keynesian theory of measures to
get rid of the depression, on the other hand the modern theories
of economic growth? More specifically, is it the point of the paper
that the Keynesian assumptions are appropriate for a regime of
unemployment and unused capacity, whereas these same assumptions
lead to unrealistic and startling conclusions in a regime of full
employment and full utilization of capacity? If so, how are the
hypotheses underlying Professor HAAVELMO’s simple model to be
sorted out between the two regimes? A clarifying answer to these
questions would add greatly to the appropriate understanding of
Professor HAAVELMO’s important model.
(81 Haavelmo - pag. 24
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HAAVELMO
[ am not sure that I understand which parts of my paper Professor
WoLD wants me to explain. My model is similar to other elementary
models frequently used as a basis for recommending certain
short-run countercyclical fiscal policy measures. The purpose of
ny paper is to illustrate possible long-run effects of such policy
measures. My main conclusion is that such countercyclical policies,
though very much better than nothing, may still be quite far from
what could reasonably be called an optimal policv for economic
orowth
Haavelmo - pag.
BALANCED GROWTH AND TECHNICAL
PROGRESS IN A LOG-LINEAR MULTI-SECTORAL
ECONOMY
MICHIO {MORISHIMA
Osaka University - Osaka - Japan
~TRODUCTION
In this paper we are concerned with an economy where
each good may serve the capital requirements as well as the
current production requirements of all the various industries.
We assume that each industry has a production function of the
CoBB-DoucLas type (1). We also assume that the constant
returns to scale prevail in each industry and that the marginal
productivity of any factor equals the price-ratio between the
factor and the product.
As for the consumer’s behaviour we follow Mrs. Joan RosiNSON
and J. voN NEUMANN in assuming that only workers
consume and only capitalists save. We assume that all workers
{') Such a system may be called a WALRAS-MOORE system. See HENRY
L. MooRE, Synthetic Economics (New York, Macmillan, 1929). A similar
model has recently been examined by RADNER. See R. RADNER, Notes on
he Theory of Economic Planning, Center of Economic Research, Training
Seminar Series 2 (Athens) 1963, and Optimal Growth in a Linear-Logavithmic
Economv. unpublished (November 1062).
jo
Morishima - pag.
1
530 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
are identical in the sense that each of them can only offer one
unit of labour and their utility functions are identical. Furthermore,
we assume that the utility function is log-linear in
the quantities of the commodities consumed by the worker.
We obtain a set of output-determining equations and a set
of price-determining equations. In the former, we explicitly
take into account not only the current inter-industrial demand
but also the investment demand due to the multi-sectoral acceleration
principle; and the latter states that prices are determined
so as to cover capital losses as well as the unit cost of production.
As the input coefficients (as well as the consumption of
various goods per worker) depend on prices, the mechanism of
determination of outputs is influenced by the price-valuation
mechanism, though the latter is independent of the former by
virtue of the prevalence of the constant returns to scale. We
can show that the characteristic roots of the whole system
appear in pairs with their reciprocals. This leads to a Turnpike
Theorem which asserts that there is a long-run tendency
for the optimal path of economic growth to approximate to
the path of steady balanced growth at the maximal rate.
In the second section of this paper, we examine various
effects of technological changes, neutral and biased, on the longrun
equilibrium prices and on the long-run output configuration
to which efficient paths converge. We shall also deal with
effects of technological changes on the allocation of labour
among various industries.
.. A TURNPIKE THEOREM
1. Let us consider an economy consisting of n industries,
whose products may serve capital requirements as well as current
production requirements of various industries. Let
(o] Morishima - pag. z
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x;;, (=I, ..., n) be the volume of output of good 7 (by industry
7) in period ¢, and x, (i=1, ..., n; j=0, I, ..., n) the
volume of current input of good j into industry ¢ in period £,
where the o-th good stands for the sole primary factor of
production, ‘labour’. Finally, let s;, (i, j=1, ..., n) be the
volume of capital input of good j into industry ? in period ?.
It is assumed that the production function of each industry
is of the CoBB-DouGLaS type, i.e.
1)
MT a n t
A …
x, =F, I, HI Set +
;
i—=1, .., 7)
where F;, a;, and b;; are all constant and non-negative. It is
also assumed that the constant returns to scale prevail, so that
2)
ri
3)
In each industry, unit cost is to be minimized; furthermore,
it equals the price of output when competitive equilibrium prevails.
It is well-known that the marginal conditions may be
put in the form:
-…., M
y| Morishima - pag. 3
532
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 4
where p;, is the price of good j in period ¢ and g, , the price of
capital-service k in period à. Substituting for x;,/x;, and
Sp: 1/X;, rom (4) and taking logs, we may put (3) in the form
n n
(5) logpii— > 4; log py, — 2 bi; log ¢;1 — ay logp, s+ Gy=o0
i= iwhere
(6)
n n
G; = log F; + 2, a; log a; + 2 6 108 6j;
j= I=Let
us now consider a person who has a given sum of money
M available for expenditure. If he lends that amount to
someone for one period at the prevailing rate of interest », he
will enter the next period with amount (1 +7)M. Alternatively,
he may spend M on goods; if he spends it exclusively on a
capital good k in period £, he obtains M/p, ; units of that good.
By letting them out hire, he receives income by the amount
Jn,1(M/P;;) in period £, which will grow to (1+7)g,,(M/P; 1)
at the beginning of period #+1. Although the capital goods
he owns will be worn at a certain rate (say) d, in the process
of production, he will still own (1 - d;)(M/p,,) units of good k
at the beginning of period #+1, which will be evaluated as
(x — d,)(M/Pr1)Pr141 at the price in period #+1. In equilibrium
neither option can be advantageous over the other, so that
(147) ME (1+7)9ne (M/An 0) + (1 —d) (M/bx) Prt +1
By dividing both sides by M/p, ,, this may be put in a simple
form
7)
(I + 7) pr, = (1 + 7) Gre + (1 — dy) Priv
9] Morishima - pag. 4
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In the state of the long-run equilibrium where all mriceremain
constant over time, we have from (5) and
(8)
lor
+
9)
Substituting for g, from (g), and subtracting (2) multiplied
by log p,, we may rewrite (8) in the form
(10) logy; -s
3 r+ 4d,
a; + by) log v;== 2 b;; log
-G; (i=1,...,n,
where v; is the price of good : in terms of labour, i.e. v,=p,/p,.
Equations (10) allow us to determine the long-run equilibrium
wage price, v,, ..., v,, Once the rate of interest r is given.
Let us now turn to the output-determination side of our
system. At the beginning of period #+ 1, industry ? has good ;
of the amount (1 - d;)s;;,p so that the total amount of good
available in the economy i- - (1 -d;)s,. This, together wit.
the output of good j in period ¢+ 1, is distributed among p:.
ducers and consumers.
As for consumers, we assume that capitalists do not consume
and workers who are identical and can offer only one
anit of labour spend their income upon various commodities
without making any savings. Let y, (¢=1, ..., n) be the consumption
of good i per worker in period t. These amounts will
be determined so as to maximize the utility function
d
ty veey 4
Morishima - pag. 3
534 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA - 28
subject to the budget constraint
n
Pot — 2 Vabu "
The conventional procedure of maximization leads to
Fyn Pu
hy va Pu
(i=1,..,n)
These conditions, together with the budget equation, yield
J)
Pot
Yn= 8
' Du
(i= I,.., n)
where g,=h;/ > h;.
j=As
the industrial demand for good j in period {+1 is
n
> (Xjitrr+Sjiçy1) and workers’ consumption of good j is
i=l ’ ’
Vjt+1 X *oit+1> the supply-demand balance of good 7 is estabt+1
Æ, “oi,
lished when
+4)
K
<
z= 45) Spip + juan
1=Xj
1 + Spier) +
n
+ 7;
j,t+1 2 Koi
= ist+]
‘9] Morishima - pag. 6
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535
Let us now write:
….,
v4
a M.D
Tin Poi! Pre
)
11 Pre/ 91
" Pat 3
where c;=a,g. Taking (4) and (11) into account, we have
from (12)
13) 2B (I—dY+ x, =x. A + x40 Bi + Teri Ci
When long-run equilibrium prices prevail, it is shown that
‘he dynamic input-output system (13) is reduced to-(14)
x, B(I—d) + x, , (I—
C
whose characteristic equation is
3)
BI—ad)+ u 1I-Ly
Morishima - pag. ,
536
PONTTFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
where A, B, and C are matrices A,, B,, and C, evaluated at
pu=p; (i=0, 1, ..., n). In view of (2), (9) and 3 g;=1 we find
i=1
that one of the characteristic numbers {4,, …, |, is I+r. We
can also show that the eigen-vector % associated with 1 +7 is
non-negative. It is clear that we have
(16) x B(1+—d)+(1+n)x([—+A—B—C)=o.
Thus (14) has a particular solution (1 + 7)! which is referred to
as a balanced growth solution. We also refer to a state fulfilling
(8) (9), and (16) as a state of balanced growth.
2. So far we have treated the rate of interest as a given
constant and have shown that to any assigned value of it there
corresponds a state of balanced growth. It is impossible, however,
for the rate of growth of outputs to exceed the rate of
growth of the working population for a long time, because the
scarcity of labour will sooner or later emerge. In the contrary
case where the labour force is increasing at a rate higher than
the rate of growth of outputs, the ratio of the number of unemployed
to the number of employed workers continues to rise.
In the following, therefore, we are concerned with finding a
rate of balanced growth at which the growth of outputs is in
harmony with that of the labour force.
We begin with examining the effects of a change in the
rate of interest on the long-run equilibrium prices. Differentiating
(10) with respect to », we get
a log :
ax — (us — @-— 0)"
ar
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where
rem:
H =
It is seen that a+b is a non-negative matrix whose column
sums are less than unity; hence (I. a °° ‘>o As
d,<1, H>o. Therefore, it is at once seen that
zl.
da
sry YZ
»
Lo
[n words, all the long-run equilibrium prices increase when
‘he interest rises.
When the long-run equilibrium prices prevail, the consumption
of good ¢ (per worker) will be g;/v; (see (11)). As
g,>0 for all ; and g,>o for at least one i, we at once see from
the inequality above that an increase in the interest rate gives
(3) Furthermore, when the capital-labour ratios of all industries are
>qual to each other, i.e. when the Marxian composition of capital prevails,
chen ¢ is proportionate to the column vector a. so that
d log :
1 log
‘hat is, all the prices in terms of labour increase
3» yportionatel.
Morishima - pag. 9
538
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
rise to a decrease in the consumption of at least one good with
no increase in the consumption of any other good: that is,
(17
a
À < 0,9
where
; = [21/91, Z2/V2> <> EnlVn)
Let us now assume that the working population grows at
a rate depending upon the long-run equilibrium consumption y :
The rate of growth of the labour force p is negative for very low
level of vy, o for the subsistence level of y, and then increases
with the rise in y until p reaches a certain maximum, after
which p will decrease gradually but will increase again when ¥
reaches the level of an affluent society (4).
We are now in a position to be able to fix the rate of interest
(°). Measure the rate of growth of the labour force and
the rate of balanced growth of outputs along the vertical axis,
and the rate of interest along the horizontal axis. Considering
(17), we find that our assumption on the labour-forcegrowth-rate
function implies the curve pg’ in Figure 1. On the
other hand, as the rate of balanced growth of outputs equals
the rate of interest, the relation between them is simply expressed
by the 45° line. It is obvious that, to sustain the demandsupply
balance of labour, the condition that the rate of balanced
growth of outputs equal the growth rate of the labour force is
to be fulfilled. It is seen from Figure 14 and 1b that there are
at least one and at most three equilibria. The greatest equilib-(®)
Let X be a vector; X<o (or X>0) means that all components of
X are non-positive (or non-negative) and at least one of them is strictly
negative (or strictly positive).
(*) Professor S. C. TsrANG makes a similar assumption in his analysis
of the Rostovian stages. See S. C. TsianG, 4 Model of Economic Growth
in Rostovian Stages, « Econometrica », XXXII (1964), pp. 619-48.
(>) A similar argument is found in my Equilibrium, Stabilitv and Growth,
Ch. III (Oxford Clarendon Press. 1064).
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rium rate of interest, #°, gives the maximum rate at which
all industry can grow in balance without labour shortages.
We may, therefore, refer to »° as the voN NEUMANN rate of
interest. The long-run equilibrium prices v°; corresponding
to 7° are referred to as the voN NEUMANN normalized prices.
GURE
‘IGURE 1
3. In the following we fix r at the voN NEUMANN rate : .
We assume that wages are paid at the rate such that it enables
‘he workers, if they chose, to buy the same amounts of all
goods as those which they would buy at the long-run equilibrium
prices, v,% ..., v,° (corresponding to #°); that is, the
money-wage rate p,, is adjusted so as to maintain the real-wage
rate at the voN NEUMANN state. We have
IS,
- a
“i 3
#
Furthermore, we assume that the workers grow at the voN NEU-MANN
rate throughout the period during which the real-wages
are kept at the voN NEUMANN level.
9 1
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340
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -~- +
Let us denote the voN NEUMANN non-normalized prices
(i.e. the long-run equilibrium non-normalized prices corresponding
to the voN NEUMANN rate of interest) by p,°, p1°, … b,°,
4:° ..., 4°. Equation (5) may be linearized in the neighbourhood
of the voN NEUMANN prices in the following way.
We get from (7) and (0)
(10)
_ 1 —d,
Agy,= App, — Tir À Pryt+1
where Aq, ,=q;,-q’, and Ap, =p, ,-p,’. Expanding the
left-hand side of (5) in a TAYLOR series and neglecting higherpower
terms, we have by virtue of (19)
20)
_— n _ n I —
Pix — 2 An Pau — (1+ 7 ) >. {riz by) Pi, +
7 [1 —a — —
—_— by) ; — a, o =0
+2 AE | Pied Pot
where pi =Ap. pC. We also have from (18)
(ZI)
et n —
Po HS Li Pis -
Substituting for Por from this and writing (20) in matrix form,
we get
(22)
M—a—(1+P)best—c] p,+ b(I—d)e-!p,,, =0
9] Morishima - pag. 12
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where
Next define diagonal
ri
16
Pp
Similarly, let O°, Q,, and Q, be diagonal matric
lements q%, ¢;,, and q, respectively, wher
In view of (7) and (9), we have
23)
(24)
0°= (1+) "1g Po,
LC
(1+) 1P, - (I - de-'P,
We can at once verify the following relations:
23)
A
4
wv A° + P A° _ A°F,
oB°{PB
- Ce 1
9] Morishima - pag. 1-
542
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
where
(26)
Ao —=Po a(P°)—! , Be =Pe b(Q°)=! , Co = Po c(Po)-1 ;
Let us denote the balanced growth solution to (14) corresponding
to the VON NEUMANN rate of interest by (1 + 7°)'x°;
then x° is the eigen-vector x of (15) evaluated at »=#°, P,=Pe
and Q,=0Q°. Define a vector z, as
‘27
2, =(1+7) "x, — x°
We may now put (13) in the form:
(28)
(x°+2,)B,(1 - d) +
+ (I+°)(X°+2,,1) (1+—A,,,—B,1—C;41) #0.
Substitute for A,, B,, and C, from (25), and neglect higherpower
terms such as z,P,A°, z,B°Q', etc.; in view of (16), and
(24), we may linearize (28) as:
(20) (I- d) (Bz + (1+)[I- (A°+B°+C°'] 2/41 +
+ D°p, + E°p,,; + F°p,,3 0,
where D°, E°, and F° are some n x n matrices whose elements
are independent of z and p, and a prime applied to a vector
(or a matrix) denotes the transposition of that vector (or that
matrix).
Equations (22) and (29) describe movements of prices and
9] Morishima - pag. 14
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54.
outputs in the neighbourhood of the voN NEUMANN equilibrium.
The characteristic equation of the whole system is:
(30)
2(b(I—d)e
X2Fo+)E6 :
« -)be -
0 1- (AC+B°+C°) 1 4 (I d, (Be
[It is clear that X?F° +) E° + D° has no effect on the determination
of the characteristic numbers. It follows that % roots,
Ai» ..., À, of (30) equal # roots, v,, ..., v, of the characteristic
equation of the price system (22),
371) Ivb(I d+ 1-a-(1+7be
and the other n roots, X,,,, ..., ka, of {30) equal the roots,
4, ..., wo, of
wfI- (A°+Be+Co)'1+ (I -d)(B
divided by (1+7°), i.e. 4, ;=t,/(x1+7°). Asoneof p/sis1 +7,
the À corresponding to it, say 2, ,, is unity. Furthermore, by
virtue of (23) and (26), we may write
n(I
[t 1S cles
l1=1,
, H.
Boy:
À
)be~! - ¢c)+ be (I - d)
J
cils, together with (51), yield
uence
{P-Gi
Morishima - pag. 1+
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
AL
(1=1, ..., n).
That is, the characteristic numbers of the whole system appear
in pairs with their reciprocals.
We now make the following assumption which is the joker
allowing us to avoid cyclic exceptions to the Turnpike
Theorem; that is, all characteristic numbers X; other than
\i=2X,.1 have absolute value different from unity. We also
assume, for simplicity, that 1 is the sole characteristic number
which is multiple. It can be shown that the solutions to (22)
and (29) may be written in the following forms:
b, = Ty + XA + +a, A
(32)
z,=[I+D+ … + D] «+o, + vee +a, kA, +
+BEnn1 + BaEn rad + ee FB.E A,
where
D=(1+7) [1 - (A0 + B°+C°)]-"(1 - d) (BOY,
and ©; ({=1,...,n) and §, (¢=1, .., 2n) are n-dimensional
column vectors; in particular, €, ; is the eigen-vector of D associated
with X,,; Scalars, &;, > Xp» By» ---> By Can be determined
bv 2# equations:
zy =k + ob + + ok + BE + Bb t oo +86.
wrt
ZT >
1+D+...+DT Ja,ËE, + ab AT +. +E ATH
+BE,1 + BaEnnad 142 + FBLA
9] Morishima - pag. 16
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Let L stand for the number of workers available in period o.
We have assumed that throughout the whole process the realwages
are maintained at the voN NEUMANN level, so that the
labour force grows at the von NEUMANN rate. The labour
force in period ¢ is, therefore, given by (:+#°)'L. On the other
hand, we at once find from (4) that the demand for labour in
period € is given by 5 a, Pr x. Hence the deficiency of
i=l Pe
labour does not emerge when
34)
T--
Pu x - €
mr CE
Dot
hold for all ?.
It is obvious that x, (+. x, when ¢ =0) is historically given
We have from (27)
Let x* (>o) be the vector of output configuration at the terminus
which is prescribed by the growth programme. A path
%X,, %{, ..., xr) which maximizes 7 such that
35)
nx*=(1+)1;
s said optimal.
[n view of (18), we have from (34,
36)
7s
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546 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
where v;=g;/v,, and the i-summation takes place over all goods
t with g>0 and the j-summation over all goods j with gi =o.
It follows from (35) and (36) that v is bounded and hence
zr=Nx* - x° is also bounded.
When T becomes large with x, and x* remaining unchanged,
it is seen from (33) that those œ, which are associated with
|A;|>I and those B, with |\,,;|>I, as well as «, are shown to
become small, because the absolute values of SE Ar. and
I+D+ ... +D" are shown to become very large, but zr is
bounded. All the other a; and 8; are not very small, but those
A; and A, which are associated with them have absolute values
less than unity. It can now be seen that when the programming
period T is sufficiently long, 2’, well approximates f; £,,; most
of the period T. From the definitions of E,,, and (x°) it can
be shown that they are proportionate to each other, i.e.
Eur1=0 (x°). By virtue of (27) we finally find that in the very
long-run programme outputs of industries almost always grow
like x, 2 (1+7°) (B,0+1)x°. The turnpike property is thus
established.
5. TECHNICAL CHANGE
I. In this section we concentrate our attention on the state
of the voN NEUMANN balanced growth, the convergence to which
nas been discussed in the preceding section. We shall be concerned
with comparative statical analysis of effects on the von
NEUMANN equilibrium of technological changes (or changes in
the parameters of the production functions) which may be
arranged in the following classes: (1) an increase in F;, all
other parameters remaining unchanged, (2) a substitution of b,;
for a, accompanying an increase in F; and (3) a converse
substitution between b,; and a,; associated with an increase
in F,. If there were no changes in prices and the wage rate,
9] Morishima - pag. 18
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a technical change of the class (1) would not give rise to any
effect on the capital-labour ratio of industry ¢ (6), while that
of the class (2) (or (3)) would increase (or decrease) the capitallabour
ratio of industry i. We may, therefore, call technical
changes of these three classes neutral, labour-saving, and cap-‘tal-saving
respectively.
It is true that we may also conceive of a technical change
such that a substitution is made between a,; and a,;. We have
a technical change of the material-using type if the substitution
is in favour of a;;, and of the material-saving type if it is in
favour of a,. To these cases the following analysis of the
biased invention can be applied mutatis mutandis.
2. Let us first be concerned with a technological change of
the neutral type. Suppose F; to vary, other parameters and
he rate of interest remaining unchanged. We have from (10)
37)
; 107
[4
J
where d log v/dF; is a column vector with components
d log v;/dF; (j=1, ..., n) and J; is a column vector whose i-th
component is 1/F;,, while all other components are zero. Since
a+b is a non-negative matrix whose row sums are less than I,
it is shown (7) that (I- a - b)-">o and that the i-th element
of the main diagonal of (I-a- b)-! is greater than any off-(®)
We see from (4) that the capital labour ratio of industry i (at the
constant prices $,°, a,° a. is
-ki
EX bu 9 qx
oO
Aoi pel Par
() MErzrer L. A., 4 Multiple-Country Theory of Income
journal of Political Economy », LIX (February. 1951), D. 2:
Éransfers.,
9]
Morishima - pag. 10
548
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2H
diagonal element of the i-th column of (I-a- 5)". Accordingly,
dlogv; _ dlog y;
38) 7F “dr =°
=~
In words, if industry ¢ makes a technical improvement of the
neutral type, the long-run equilibrium prices of good 7 decrease
proportionately more than the other prices.
Clearly (38) implies dy/dF;>0; therefore, the curve pp’
in Figure 1 shifts upward in the diminishing phases and downward
in the increasing phase (see Figure 2a). The intersection
of the new curve p” p’” after the technical change with the 45°
line results in a new voN NEUMANN rate of interest #!. Usually
we have >r. But as in the case illustrated in Figure 2b a
technological change of the neutral type may perversely bring
forth a decrease in the von NEUMANN rate of growth. We shall
refer to the increasing part ww’ of the curve pp” as the perverse
part and to the other part as the normal part. (See Figures 2a
and 2b). In any case, however, we have dy /dF,>0. so that
“'GURE
IGURE 2-b
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5x:
the technological improvement gives rise to an increase in the
real-wages.
The voN NEUMANN output configurations x° and x! before
and after the technical change are determined bv
x°Bo(I — d) + (1 + °)x°(] Ac
Be (Co) —o.
and
x'B! 1 d)
(T+!
T
respectively. Taking the definitions of A, B, and C into «.
count, we have
. —
(1 +°)x°P° IT
-@a- ie}
Pp
J >
30)
‘1 + 71) x!P!
-*
a
—.
>
À
Hence x°P° and x!'P! are proportionate to each other.
28) implies p*' <= <htat
Since
[In words, if industry ¢ makes a technical improvement ot tne
neutral type, the relative weight of that industry in the von
NEUMANN state becomes heavier than before.
Finally, it is at once seen from (4) that when the von NEU-MANN
equilibrium prevails labour is distributed among industries
in the ratio:
40)
vol
“ov
D.
1 IX pp = Q
wv,
Aon uve
9 | Morishima - pag. 2.
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
As x'P! is proportional to x°P°, we find that the technical
change does not affect the distribution of labour.
3. Let us now examine effects of a technological change of
the labour-saving type. Suppose an increase in F, gives rise
to an increase in b;; and a decrease in a, other parameters
remaining unchanged. We assume that the constant returns
to scale prevail before and after the technical invention.
Taking account of the fact that b,; and a,; are functions of
F, fulfilling (2) and remembering the definition of G,, we obtain,
by differentiating (10) and solving,
(41
dlogv _ 1
dE = = (I —a—0b) M,
where M; is a column vector such that its i-th component ms, is
, i I +7 , d by;
(42) m=+ + log | (4, HT) / (Moi 0.) “ZF.
and all other components are zero. We have db,;/dF,>0 by
the definition of the technical change of the labour-saving type.
I+r
But (by, rd) /(a,v;) many be greater or less than unity. As was
k
shown in Section 2, v, increases when 7 increases. This to-.
. . I+ . ee
gether with 1>d, implies that (b,; SF )/(@ojV;) is à diminishk
ing function of r; it may take a value less than unity when 7
is large, but greater than unity when small.
When wm. is positive, we have from (41)
dcr ov
gu °°
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Furthermore, we havc
d]o
7 ln
because a+ b is a non-negative matrix with row sums less than
one. This leads to dy/dF,>o. It is obvious that in the opposite
case where m; is negative we obtain dy/dF,<o. Taking
into account the fact that m; may be negative only when 7 is
oreater than a certain rate », we have Figures 2° = 77 —-“IGURE
FIGURE
Suppose that the curve gg’ intersects the 45° line on a point
in the normal part. The intersection gives the old voN NEU-MANN
rate of interest »° before the technological change. We
can easily verify that the technical change of the labour-saving
type results in increases in both y and r, if <r, and decreases
in both y and r, if >>». The latter is a situation which would
aot be preferable from workers’ viewpoint as well as from
capitalists viewpoint. We may, therefore, say that a new
method of production of the labour-saving type is not adopted
9]
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552
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
FIGURE 3-c
when 7° exceeds r. (Similarly, a new technological method
of the capital-saving type is not adopted when 7° falls
short of 7). When #° belongs to the perverse part of the curve
pe’, however, we obtain the following result: A technological
improvement of the labour-saving type yields an increase in y
but a decrease in 7 if <7, and vice versa. Thus, either y
or 7 increases, so that the biased technological change may be
adopted, irrespective of the relation of #° to 7.
The following argument is independent of the location of #°
and its relation to ». Let us normalize the voN NEUMANN output
configuration x° so as to make the i-th component of x°P° (i.e.
x°vP) equal 1. It follows from (39) that before the technical
change we have
—)
Ï
”
> L(a;, + D,
= =7)
n
> Les — aj; + b;; + Ci
3=7
Li-1I, 141, ..., 0),
‘
19] Morishima - pag. 24
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2933
where c;;=a,.g; and /; is the j-th component of the normalized
voN NEUMANN output configuration, ° Nif
‘erentiating (/7* with respect to F +
44)
i -4- Tic as
where
1 Aùuoil dF
Lo ,; / dF,+ 2; du,
and a; is an (»-1) x (#-1) matrix obtained by omitting the i-th
row and the :-th column from a; b; and c; are similarly defined.
As a;+b;+c; is a non-negative matrix with row sums less than
unity, any diagonal element of (I—a,——b;—c;)7! is greater
than off-diagonal elements of the corresponding column. On
the other hand, by our definition of the technical change of the
labour-saving type, we have
45,
In view oi
m
+
We assume tha. ,,
positive.
Si
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Hence we have from (45)
46)
a:
dr
> 0.
In words, a biased technical improvement which a substitution
of by; for a, accompanied increases the ratio of the value of
output of industry % to that of industry i. It is also seen
from (40) and (46) that the amount of labour used by industry
k is increased in comparison with the amount employed by
industry 1.
4. A similar argument may be applied to examine effects
of a technological change of the capital-saving type. Suppose
a substitution between a,; and b,; in favour of a,; accompanies
an increase in F;. Differentiating (10), we obtain (41) and (42),
dby; . : — —
where 7F 1s now negative. Let r be the value of # which
makes m;=0. We shall arrive at the following final results:
(47)
(48)
d log v, dlog f;
JF. < 77 SO
a,
— -_-<O0.
a=1f
#C
ld
(47) states that when industry ? makes a technical improvement
which saves the capital good k, and is greater than r, then
the prices of all goods (in terms of labour) will diminish, and
that of 7 will do so by the largest percentage. (48) implies that
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labour is redistributed in favour of industry ¢ in ‘tz relation
to industry k Inequalities “;~° nd “48) yield
that is, a technical change of that type decreases the ratio c.
the quantity of output of industry %& to that of indust—
9] Morishima - pag. 2;
3SIC.
MALINVAUL
May I question Professor MorisHIMA on the implications of his
analysis? The studies made thus far on the turnpike theory have
heen concerned with the optimal pattern of development of an economy
in which there is no final consumption. You have succeeded
in escaping from the latter restriction; but you have tied up consumption
to the general growth of the economy by other rules. I do
not see very clearly what kind of conclusion you intend to draw
from the analysis.
You may be thinking that, by extending the turnpike theorem to
a case allowing for consumption, you are giving indications about
the qualitative features of the programs which would be optimal
for development. You may also be aiming at describing how the
actual process of growth has occurred.
[ should like to know exactly your intentions. Do you explore
ndications for programming? or do you describe what happens in
a capitalist econo-—
MORISHIM.
The aim of this study is to extend the recent results of growth
economics (especially the turnpike theorem) to a model with endoenous
population growth and flexible consumption demands
bv 2
Morishima - pag. 24
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
MALINVAUD
Yes, I understand that it is your immediate intention; but you
probably have farther reaching intentions. You want to find new
sxtensions of the turnpike theorem; but the final purpose of the
exercise is not clear to me,
According to the purpose, I would have to look at the paper
in different ways. If it is purely descriptive, then we should be
careful that the hypotheses provide, at least as a first approximation,
a proper description of what happened during the process of growth.
[f it is oriented towards planning, then we must look at whether
too many constraints have not be imposed; because, if such were
‘he case, the results might have little significance for planning.
Thus, I should appreciate if you could say a little more about
these broader issues. Perhaps this is not the rule of the game.
But Professor MAHALANOBIS induced us to look somewhat beyond
the formal aspects of our theories.
MORISHIMA
Well, in various turnpike theorems so far established as well as
in the original voN NEUMANN model, it is assumed that the supply
of labour can be expanded indefinitely at the subsistence level of
real wages; so that it completely ignores the problem of deficiency
of labour, one of the most serious obstacles to a rapid growth. In
fact, it is a defect of NEUMANN’s theory of growth that no attention
's paid to HARROD’s observation that the natural rate of growth sets
a limit to the maximum average value of the actual rate of growth
over a long period.
In this study I am concerned with an economy where the planning
authorities (or capitalists collectively) make an efficient investment
planning to produce, at the end of the programming period,
various outputs in desired proportions. If wages were fixed at the
subsistence level, capitalists could accumulate stocks of capital goods
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at a maximum rate, but labour would remain stationary or could
grow at a very small positive rate; so that a bottleneck due to a
deficiency of labour would inevitably emerge sooner or later; hence
the rate of real wages should be one of the variables of the programming.
Once wages are greater than the subsistence level, workers can
make choice among consumption goods, and the growth programme
lepends on consumer’s choice unless the consumption goods are
rationed by the planning authorities as will usually be done during
1 war. I am sure that a planning model in which workers are allowed
to choose consumption goods freely and the rate of growth of
population is finite and depends on the per-capita consumption deserves
serious attention and should be granted a citizenship of our
science.
DORFMAN
Professor MorisHiMA’s elegant model illustrates the ethical perplexities
that Professor KooPMANS has just discussed, For look: the
balanced growth rate is given by the intersection of the curve showing
the growth rate of the labor force as a function of the real wage
with the curve showing the equilibrium rate of interest also as a
function of the real wage. Technical change shifts this point of
ntersection, but if it increases the real wage it decreases the rate
of balanced growth, and vice versa. This model therefore portrays
1 trade-off between high real wages and high rates of growth, which,
of course, is to be expected in a full employment model. The novelty
is that this conflict ef interest between present and future results
from the possibility of technological change, whereas previous derivations
of this conflict assumed constant technology. Be that as
t may, this conflict presents just the sort of ethical problem that
concerned Professor KooPMANS.
This problem might disappear from Professor MoRISHIMA’s molel
if the curve portraying the rate of growth of the labor force also
shifted, It probably does shift in real economies. In fact, one of
lq,
Morishima - pag. 31
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the crucial problems of economic development is to make it shift
so that increases in the real wage are not absorbed by increases in
the population. This raises a somewhat different ethical problem.
MORISHIMA
In this study I have pointed out that the conflict of interest
between workers and capitalists (or the conflict between present and
future) will be observed only when the original per-capita consumption
before the technical invention is located in the « perverse » part
of the curve showing the growth rate of the labour force as a function
of the per-capita consumption (i.e. that part of the curve in
which an increase in the per-capita consumption gives rise to a
decrease in the rate of growth of the labour force). In the « normal
» part there is no conflict; the extra output due to the increase
in productivity is shared between the « present » and the « future ».
MALINVAUD
Concerning the model itself, I am not very happy about the
hypothesis that savings come only from capitalists. Not so much
because I would strongly disagree on the assumption that capitalists
are saving the larger part of their income, and workers consuming
the larger part of theirs. But, for already some time in our economies,
governments have interfered in the distribution of income.
Hence, I do not find the hypothesis made by Professor MorisHIMA
very well suited for the practical questions which it is our ultimate
air to answer.
MORISHIMA
My model, although it is oriented toward planning, has no public
sector. It is implicitly assumed that anv plan for a rapid growth
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proposed by the authorities is accepted by capitalists; so that they
may make a plan requiring that all capitalists’ income is automat
ically reinvested.
It is true that our assumption does not well fit any actual economy
unless all industries are nationalized; in fact, in a mixed economy
there are private and public sectors. The planning authorities
will push forward their plan mainly through public sectors, while
private sectors can act against it. It would, however, be beyond
‘he present state of our techniques to establish the turnpike theorem
n a mixed economr
PASINETT:
[ have some doubts on Professor MorISHIMA’s paper, which du
not concern, of course, the mathematics of it (which is admirable)
out its assumptions and thus its bearing on the type of world in
which we live. Professor MorisHIMA’S model is intended to be a
development of that of voN NEUMANN, and in order to explain mv
doubts, it may be useful to compare the two models.
Von NEUMANN, as we know, was concerned with a hypothetical
society in which there is no technical progress and economic growth
:akes place at constant technical coefficients; relative prices as well
1s per-capita incomes remain constant as time goes on. In such
conditions, no assumption is necessary about individual preferences,
which can be accepted as given, whatever they may be. (It is only
necessary to assume that people are, by and large and irrespective
of time, of the same type). Such a scheme thus happens to have
‘he mathematically interesting property that balanced growth (i.e.
expansion of production of each commodity according to the incomeelasticity
of total demand for it) is a proportional economic growth.
I have had the opportunity of arguing myself in the paper I
nave presented to this Study Week, that the economic expansion
which von NEUMANN has considered is a very unrealistic type of
zconomic growth. Yet, I would take voN NEUMANN’S model as a
very important first analytical sten And I should always look
61
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favourably on any development of it, provided that this development
goes in the direction of a more practically relevant type of
analysis, by relaxing some of the unrealistic assumptions.
Now, my question is: does Professor MORISHIMA move in this
direction? If I understand him correctly, I must say he does not.
By considering technical change — and thus increasing per-capita
ncomes — he has been compelled to introduce specific assumptions
on consumers’ preferences. He has assumed that all individual
utility functions, besides being « identical », are « log-linear in the
quantities of the commodities consumed. » This means that, when
per-capita incomes increase, each consumer — if relative prices do
not change — is supposed to increase his demand for each commodity
in exactly the same proportion.
These are further assumptions with respect to vON NEUMANN’S.
And what is unfortunate is that these additional assumptions are
not only unrealistic; they postulate a behaviour which we know to
be impossible, at least among human beings. As ERNST ENGEL
pointed out more than a century ago, when per-capita incomes
increase, the demand for each commodity does not tend to increase
proportionately; which means that utility functions are not loglinear.
To postulate a consumers’ behaviour which goes against one
of the strongest empirical laws of economics (ENGEL’S law) makes
Professor MORISHIMA’S analysis more — instead of making it less
— artificial than that of voN NEUMANN. I have been wondering
why Professor MoriSHIMA has made such assumptions; and the only
reason that occurs to me is that they are the only ones that allow
a model with technical change to keep the mathematically elegant
property of proportional economic growth.
If this is the case, I must confess to be very disturbed. I feel
that this is just the way in which mathematics can do economics
a great disservice, For, in this direction, instead of using mathematics
as an analytical tool for the interpretation of economic phenomena,
we risk developing elegant mathematical models for their
own sake, and then making whatever assumption may be necessary
to give them an economic interpretation.
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MORISHIM/
My interpretation of voN NEUMANN's theory of growth is somewhat
different from Dr. PASINETTI’s. He says that NEUMANN’s
model does not need any assumption about individual preferences,
which can be accepted as given, whatever they may be. But,
according to my interpretation which, I hope, is an orthodox one,
NEUMANN made drastic homogeneity assumptions, not only on productions
but also on the consumption side: He assumed that wages
are held at the subsistence level so that necessities of life consumed
oy a worker (the consumption coefficients) are biologically determined
and independent of prices. On these assumptions he showed
‘hat an economy can expand in constant proportions.
Now, in order to make his model more realistic we must, first
of all, release it from the assumption of the subsistence wages. As
people get richer, the proportions in which they divide their expendture
between various consumption goods, will depend on prices
because they can now choose among goods); and the proportions
will vary when the per-capita income increases (ENGEL’s law). In
such a situation, it will be wiser and more profitable to forget as
a first approximation the effects of the per-capita income on the
consumption coefficients and to take full advantage of the assumpions
for simplification than to confront the actual world directly
and to loose one’s way in its complexity. I believe it would be
aseful in making economic policies as well as in advancing theories
0 have found that a Neumannian growth equilibrium with constant
proportions is still possible even when the consumption coefficients
lepend on prices, unless the income-elasticity of the demand for
some consumption good is different from unity. It is of course
tue, however, that a second step toward the reality should take
account of increasing or decreasing returns to scale on the producion
side and of deviations of the income-elasticities from unity on
‘he consumption side.
9, Morishima - pag. 35
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MAHALANOBIS
I shall not speak on many of the detailed aspects of this
particular model, because I am not competent to do so. The treatment
of an important question by a very simplified model, I believe,
throws some light on policy decisions.
Also, I should like to raise one question, not with reference to
the particular model just now discussed, but of a broader nature,
which is of importance to an underdeveloped country like India,
as distinguished from Japan and advanced countries, regarding the
usefulness of complicated models generally. 1 should like briefly
to mention our experience that when some complicated models are
used, the question of unreliability of data becomes of crucial importance;
Dr. JoHNSON has drawn attention to this point in his
paper, and Professor LEONTIEF has also referred to it earlier. In
another intervention I tried to indicate two gaps between the world
of reality and the model. Firstly, the gap arising from the lack of
availability or the lack of reliability of data; and secondly, another
gap between the data and the model. These are questions, of course,
of a very general nature, which however deserve serious attention
of econometricians.
KooPMANS
I appreciate the support of Prof. MAHALANOBIS for what I tried
to say earlier. At the same time I do not go as far as he does if I
understand what he said to mean that there is something wrong in
complicated-ness itself. I think we are entitled to make our models
as complicated as we can manage as long as by that extra complication
we obtain added insight, and the extra complications do not
prohibit communication of the findings. We are working at the
frontier of our collective understanding of these problems, and while
altimately we hope to end up with models that reflect reality better
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and which therefore have to be more complicated, at the moment
we can’t do much better than stumble along as we are doing.
\[AHALANOBIS
[ should agree entirely. A model may be, should be indeed, as
complicated as is necessary for an explanatory model or a decision
model or to serve some other purpose. I am in complete sympathy
with the study of such models. Even when adequate statistical
data or other types of information are not available, working with
such models in an imaginative way may lead to the collection of
relevant data; this itself is a very useful task. Or, such studies
may lead to advances in theory as often happens in physics. I have
‘herefore nothing against using complicated models in any way.
The questions which I raised had some ambiguity, and I think
Dr. KooPMANS’ observation were justified. One object was that
sometimes simple models are quite adequa?
DORFMAN
Yesterday we had a long discussion of the types of economic
model, which left me very uncomfortable. We then seemed to
conclude that there were three types of model, explanatory models,
forecasting models, and decision models, in Professor FrIscH’s terminology.
But I went away feeling that we had forgotten about
some other kinds, and today we have had three examples of models
that cannot be used for explaining any empirical observations, or
forecasting the future, or making practical decisions. Today's
models are intended to illuminate the logical consequences of some
assumptions or conditions. We might call them a fourth type, logical
or hypothetical models. I suspect that there are many other types
sf model alsa
Morishima - pag. =.
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MALINVAUD
Since we apparently have some time this afternoon, I may
perhaps be permited to come back on the motivations for my own
paper.
Professor KooPMANS has indicated this morning that one of the
reasons for his own research was to explore the consequences of
assuming a particular kind of utility for choices over time. This
was certainly also a part of my own motivations. But I was still
more strongly motivated by the need to see clearly what we should
do when we use models with several periods, as a guide for planning
or programming.
We can certainly do some programming by considering timeless
models, Such models proved useful in various countries, in India
for instance. But multiperiod models are necessary for many questions:
the choice of investments, the discussion of public policies
nfluencing the saving ratio, even the study of consequences that
would follow a progressive reduction of the length of work.
In the multiperiod models now used by programming, terminal
conditions are imposed. For instance, the future up to 1970 is
divided into three periods and the capital structure to be in place
at the end of 1970 is fixed a priori. Since such terminal conditions
are, to some extent, arbitrary, one may have serious doubts on the
adequacy of the procedure.
My research should be considered as an attempt to explore fully
the nature of the solution in a much simplified model in order to
understand better the implications of assuming arbitrary terminal
conditions, and perhaps also to find new ways of dealing with the
difficulty. All this is done for a one commodity world, but should
give us some useful insights on what is likely to occur in more disaggregated
models.
DORFMAN
Are you disagreeing with me:
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VIALINVAUD
No, I am not! I am in full agreement with Professor DORFMAN.
[he preceding comments are merely intended to show what I had
n mind when speaking of the motivations for a research.
VAHALANOBIS
[ do not think there are basic differences of opinion but while
agreeing with many of the observations made here, I am trying tc
Iraw attention to one point — that when certain periods, 10 years
or 15 years, are taken as the time horizon in making government
Jecisions, there are necessarily makeshift arrangements. In an
inderdeveloped country which starts experiments in the way of
making policy decisions which are intended to be implemented (I am
calling these experiments), and also starts exercises in the way of
building models, sometimes it may be possible to feed the models
oy numerical data or sometimes it may not be. There is a more
general point about models which also require information other than
statistical data in a numerical form. I was therefore speaking of
wider experience by which I meant in respect of any model, judgement
as to what are relevant factors, or what are their order of
priority. I have an impression that there is sometimes a good deal
of faith in model making which with the help of very highpowered
“omputors would supply push-button solutions. This is why I refer-‘ed
to the limits of usefulness, depending on whether a model is
“apable of being fed by quantitative data or involving non-quantiative
judgements, or factors of selection, and also to what extent
he solution depends on the degree of accuracy of the information
which has to be supplied in a quantitative form. One great service
io the underdeveloped countries would be to discuss the usefulness
1s well as the limitations of models in application to practical affairs.
|
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WoLD
I have two comments. The first refers to the entire group of
four papers presented to-day (1), all of which deal with purely
theoretical highly simplified models of economic growth. This kind
of logical analysis has a strong appeal, and that it has a wide
appeal is clear from the fact that the three other papers are closely
interrelated both with regard to problems and results.
My second comment refers specifically to Professor MORISHIMA’s
paper, and is a question about the proportionality assumption on
page 7 (3). Am I right to understand that this assumption requires,
for example, that the demand for food remains a constant fraction
of GNP as the economy grows? If yes, the assumption is in radical
contrast to the whole outlook of a later paper to-day, that of Professor
GALE JonnsoN. There the whole emphasis is shifted. There
exists no such proportionality; on the contrary, the lack of proportionality
is a universal handicap for the farming population.
ALLAIS
I have two remarks to make. The first was made by Prof.
Worp, namely that there are some quite strong hypotheses in the
model.
Secondly, I have unfortunately not had the time to study Professor
MORISHIMA’s paper very carefully since I was working in the
other group. But so far as I can judge, the MorisHIMA’s model
is a special case of a general theory I presented this year in my
paper for the Cambridge Round Table of the International Economic
Association. Thank vou.
(!) Koopmans, MALINVAUD, MORISHIMA, PASINETTI.
(3) The discussions having been organized in two separate groups during
a first phase of the Study Week, my question belongs to the second phase
of joint discussion. Now when reading the proofs I see that a similar
question was posed in the separate discussion, but T repeat the question
to mark the importance of the issue
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\[ORISHIMA
With regard to Professor WoLD’s second question, please see my
reply to PASINETTI. [7 am sorry that I cannot replv Professor
Arras, for I have not yet read his Cambridge paper. As I am a
very slow reader, it will unfortunately spend a long time before it
‘inally appears at the front of the queue of my backlogs, although
the time required to bring it from Paris to Osaka has been shortened
to 20 hours. Thank vnu
| Morishima - pag. .
A NEW THEORETICAL APPROACH
TO THE PROBLEMS OF ECONOMIC GROWTH
LUIGI L. {TASINETTI
King's College - Cambridge - Great Britain
atroduction . .
"he process of production in the short run . .
"he simplest case of economic expansion - Poyulation
growth with constant returns to scale
Problems connected with technical change - Set-‘ing
the bases for a general dynamic analysis
CHAPTER V. A general multi-sector dynamic model .
CHAPTER VI. The empirical significance of the model
Appendix to CHAPTER VI
CHAPTER I.
CHAPTER II.
“HAPTER III.
CHAPTER IV.
I consider myself very fortunate to be given the opportunity
of presenting this work of mine to a group of so eminent ecoaomists.
At the same time, I feel I am at some disadvantage
secause I am not inserting myself into the prevailing line nf
contemporary economic thought. I am rather trying a different
approach altogether and I may be faced with some
ack of communication.
It may be useful, therefore, before I come to what I might
call a multi-sector model of economic growth, to make some
efforts to sketch out a few introductory remarks on the general
line of thought within which I propose to move.
(10! Pasinetti - pag.
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CHAPTER 1 (*)
INTRODUCTION
+. The historical background of economic analysis
I shall begin with a few historical remarks which might
sound at first rather general and far-fetched but which will
soon turn out to have a justification.
If we consider the historical context in which economic
analysis has come into being, we may say that this context is
represented by the modern world; namely by the stage of our
history which is known as the age of experiment and science,
because of the dominating idea that man, by using his own
critical intellect, by observing nature and experimenting, can
learn in a systematic way and can pass on his improved knowledge
to the following generations.
In economic terms, the direct consequence has been a process
of unprecedented increase of material wealth. The process
may be distinguished, for analytical purposes, into two relevant
phases, which we may call the phase of trade and the
phase of industry. There is no clear-cut distinction between
the two, as they have a common origin and are intermingled,
but they appear nevertheless with very definite characteristics
on the historical scene.
(*) The present work is a summarized version of a Ph. D. dissertation
submitted by the author at the University of Cambridge in September
1962. Chapters II to VI have appeared already, as a publication for limited
circulation, under the title A Multi-sector Model of Economic Growth,
King’s College, Cambridge, July 1963. Acknoweledgements for criticism and
Comments are gratefully due to: R.M. Goodwin, N. Kaldor, J. Robinson,
R.F. Kahn, J.S. Duesenberry, D.G. Champernowne, [.M.D. Little. Responsibility,
of course, is entirelv mine
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The phase of trade is the first to break through. It can be
perceived even as early as at the turn of the first millennium,
but more clearly later on, after the Renaissance « opening of
the minds » towards the outside world. A few important improvements
in the technique of transportation lead to discoveries
of new lands and extend the horizon of the known
world to include countries with climates and products previously
unknown. New possibilities of trade open up, with
a striking impact on the economic conditions of the whole
world. The trading nations are suddenly better off, not
because of a rise in world production, but because of a better
utilisation of the production which already takes place. Each
nation keeps her own institutions and organisational structure
of production, but now she can advantageously exchange the
products which are proper to her particular climate or localized
resources for products which she could never produce or which
she could produce only at much higher costs. The material
wealth of all peoples is increased just by exchange, by a better
spatial allocation of existing resources and products. This is
‘he merchant era, an era which represents perhaps the most
outstanding example of how all people can gain from trade.
Much slower to manifest itself is the phase of industry,
which requires already, and thus presupposes, trade. Industry
is a process of augmenting wealth through a material increase
in the quantity and number of products, to be reached by
the practical application of the advances of science, division
and specialisation of labour, better organisation, invention and
utilization of new sources of energy and new materials. Unlike
trade, industry requires changes in the organisational structure
of society. Therefore, it comes about slowly; but progressively.
[n fact, it requires long and painful social changes in the relalions
between men and the means of production before it can
fully break out in the English « industrial revolution » of the
sighteenth century. Of course, trade remains the natural and
necessary complement of industry but, as a cause of further
io| Pasinetti - pag.
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increments of wealth, it is bound to subside. Industry, on
the other hand, is bound to remain a permanent cause of
increase of wealth and to become pre-eminent as time goes on,
owing to the very nature of its cumulative process.
These two aspects of the modern world seem to me very
helpful in indicating the directions in which the emergence of
the modern era has stimulated economic analysis.
The concept of trade is, so to speak, a static concept. It is
associated with a situation in which a plurality of economic
systems (or of individuals) are endowed with particular resources
or products and try to gain advantages by exchange. The
interest that such a situation arouses in an economist concerns
the problem of how to reach the best allocation of given resources,
namely of how to make the best use of what one has
already. We may imagine a stationary situation in which a
plurality of economic systems have reached equilibrium internally,
but do not trade among themselves; and then another
stationary situation in which the same economic systems, besides
having reached an internal equilibrium, also trade with
one another. Is is easy to show that the passage from the first
to the second situation — i.e. a once-for-all change from no
trade to a new situation of trade, to be maintained thereafter —
normally brings about a gain for all. The problem involved
is a problem of rationality, which may be expressed by a
mathematical function to be maximised under certain constraints.
The concept of, and the problems entailed by, industry
are quite different. Industry is, so to speak, a dynamic concept.
It means production, i.e. the engagement and the application
of man’s ingenuity to make and shape the products he
wants. But since by doing and experiencing man learns, it
is implied in the very nature of carrying on a production
activity that new and better methods of production will be
discovered. Of course, to find new methods takes time, and
takes time in a persistent way. The economist is faced here
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no longer with a problem of rationality, but with a process of
learning. Any mathematical formulation of it cannot but be
in terms of functions of time, since the process makes short
steps, and may appear quite negligible, in the short run; but,
as it goes on incessantly, it is inevitably bound to become the
more pronounced the longer the period which is considered.
The contrast with the simple concept of trade is now evident.
The passage from a position of no trade to a position of trade
means a jump, which may be quite big but which is temporary,
as it ends when the new equilibrium situation has been
reached. The process of learning associated with industry, on
the other hand, means a persistent movement, not a once-for-all
change, but a rate of change in time — a movement which is
cumulative and indefinite. In this sense, industry comes to
realise most properly the concept of progress which is inherent
in modern society.
Clearly, these are two distinct series of problems. A particularly
important difference between the two, for theoretical
analysis, is that they acquire an opposite practical relevance
in relation to time, the former being relevant (in the short run)
just when the latter is practically irrelevant and the latter becoming
relevant just when (in the long run) the former becomes
irrelevant. Both series of problems have been of course considered,
in the course of development of economic thought. But
according as to whether the economic theorists have been
mainly impressed by the former or by the latter, the attitude
they have taken to the type of hypotheses to choose has been
diametrically opposite.
2. Scarcity versus learning tn economic analysis
There are clear indications — it seems to me — that the
Classical economists, writing under the strong impact of the
English « industrial revolution », were well aware of the two
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376 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
aspects of the modern world mentioned above and had in fact
no hesitation in singling out the industrial aspect as by far
the more important.
At the very beginning of his Wealth of Nations, ADAM
SMITH takes great pain in pointing out that it is « the skill,
dexterity and judgment with which labour is applied » which
is by far the pre-eminent factor accounting for the wealth
of nations, « whatever be the soil, climate or extent of territory
» (!). DAvip Ricarpo in his turn, again on the very
first pages of his Principles, sets out this opposition in terms
of types of commodities. « There are some commodities — he
says — the value of which is determined by scarcity alone. »
These are the goods which are given by nature. We may call
them the commodities of the scarcity type. RicarDo says that
they « form a very small part of the mass of commodities daily
exchanged in the market. » And he continues: « By far the
greatest part of those goods, which are the object of desire,
are produced by labour; and they may be multiplied, not in
one country alone, but in many, almost without any assignable
limit, if we are disposed to bestow the labour necessary to
obtain them. » (?) We may call these, the commodities of the
production type. It is on these commodities that Ricarpo
concentrated his analysis.
The whole economic theory that followed preferred, on the
other hand, to abandon this approach and to go on to taking
exactly the opposite view. The stream of economic thought
which came to dominate the second part of last century (Marginalism)
and which still now-a-days provides the backbone
of most of contemporary economics, appears as a tendency to
concentrate on the other type of commodities: the commodities
(1) ADAM SMITH, An Inquiry into the Nature and Causes of the Wealth
of Nations, ed. by E. Cannan, p. I.
(2) Davip Ricarpo, On the Principles of Political Economy and Taxation,
p. 12. ‘The references are to the edition by PrEro Srarra, with the collaboration
of M.H. Dos, The Works and Correspondence of David Ricardo,
in 10 vls., Cambridge, 105I.
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of the scarcity type. As we know, the typical marginalistic
scheme of general equilibrium is a model of what has been
called a pure exchange economy. (3). The model presupposes
the existence of given natural resources in fixed quantities, and
of a given number of individuals (owning the resources) with
well-defined utility preferences. The economic problem these
‘ndividuals have is one of rational choice. They have to find
‘hose prices (equilibrium prices) which bring about, through
exchange, an optimum allocation of the given resources relalively
to their original ownership-distribution. The problem
can be represented analytically by a mathematical function
which is being maximized, subject to certain constraints fixed
oy nature.
Of course, marginalist economists have then gone on to
investigating all other economic problems as well (production
included). The relevant point is, however, that they have
done so by an extension of the theory they had originally
developed for scarce commodities. Professor SAMUELSON, at
the very beginning of what is one of the most rigorous versions
of marginal economic analysis, claims exactly this. He claims
to have been able to isolate a simple theory which can be
applied to every economic investigation: a mathematical funclion
to be maximized under given constraints (%).
This has been a crucial step. For, in this way, every single
corner of economic theory has come to be permeated by the
character of scarce goods and by the rational problem of
making the best use we can of them, at the expense of the
character of produced goods and of the learning process of
human beings. It has meant that modern economists have
(°) See, for example: J.L. Mosak, General Equilibrium Theory in Interrational
Trade, Bloomington, Indiana, 1944. The same scheme can be found
n the very first chapters of all standard treatises of marginal economics.
See, for example: V. Pareto, Cours d’économie politique. Lausanne, 1895;
J.R. Hicks, Value and Capital, Oxford, 1939; P.A. SaMUELsoN, Foundations
of Economic Analysis, Cambridge Mass., 1947).
) P.A. SAMUELSON, op. cit., pp. 1 and ff
iO |
Pasinetti - pag.
/
578
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARA - 25
been led to look at the whole of economic reality through the
lens of a scarcity scheme, and thus to magnify or shrink the
various aspects of reality according as to whether they do or
do not fit into the pattern of a world of scarce goods.
I have thought it may not be altogether useless — after a
century of marginal economic analysis — to go back to explore
the possibilities, which may have remained unexploited, inherent
into the other approach to economic reality, which has been
left into oblivion since the time of the Classics.
3. A pure production model
It is my purpose to develop, in the following pages,
a theoretical model of economic growth for an industrial economic
system. In the whole theory, a central rôle will be
played by the learning process of human beings, in its twofold
aspects of technical improvements and of consumers’ preference
evolution. Scarce resources will not be considered, although
of course this does not mean that they do not exist. It only
means that the theory will be developed independently of any
rational problem concerning their best utilization. All commodities
considered are produced, and can be made practically
in whatever quantity may be wanted, provided that they are
devoted to the amount of efforts they technically require. Limitations
of course exist, but not in the material world: they
only reside in the knowledge and power of activity of Men.
As will be realised, this means adopting a procedure which
is exactly opposite, though symmetrical, to the one followed
by marginal analysis. The scheme itself might be called a pure
production model, as against the pure exchange model of marginal
economics. It will refer to a certain type of commodities
(this time the commodities of the production type), will centre
around a definite problem (the problem of production), and will
L10] Pasinetti - pag. 8
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3s
be dominated by a general principle (represented by the learning
process of human beings) (%).
There is one important respect, however, on which the
following analysis will not take a path symmetrical to the one
followed by marginal analysis. The model will be kept at a
sufficiently high level of generality, so as to remain neutral
with respect to the institutional organization of society. The
preoccupation is that of singling out the necessary conditions
for efficiency and equilibrium growth — conditions which will
emerge as independent of any particular institutional set-up
that society might choose to adopt.
{°) It may be useful to add a warning to the reader at this point. To
ake human activity, i.e. labour, as the only non-produced factor of proluction
must not be interpreted as meaning that «labour is the only
scarce factor », as — I have noticed — some of my friends, while discussing
the present work with me, have tended to do. Such an interpretation
would be incorrect: scarcity presupposes some aim the attainment of
which is limited by the existing quantity of the factor. But no such aim,
imited by the existing quantity of labour, is present here. As the reader
will see presently, the systems of equations which will be considered vield
solutions for relative prices and relative quantities. To say, in such a
context, that labour is scarce has no sense.
10]
Pasinetti - pag.
9]
180 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CHAPTER II
THE PROCESS OF PRODUCTION IN THE SHORT RUN
.. A very simple case: production by means of labour alone
We may begin our analysis by considering the process of
production at a given point of time or, to be more realistic,
within a short period of time, defined in such a way that,
within it, changes in population, productive capacity, technical
knowledge and consumers’ preferences are negligible.
The economic system we are considering is closed and consists
of a society of individuals whose purpose 1s to produce
goods in order to derive enjoyment (or relief from pain) by
consuming them. Two types of activities are therefore performed
in the system: a production activity and a consumption
activity. At the beginning of the period, the production processes
are programmed in the best way that is technically
known and, at the end of the production processes, the commodities
are consumed. Technical knowledge is supposed to
be quite advanced so that each process of production requires
division of labour and a marked specialisation. Therefore, each
individual consumes only a very small part of the commodities
he produces (or contributes to produce); and obtains all the
others through exchange. Total production (and therefore consumption)
is well diversified: the system produces many types
of commodities, according to the preferences of its individual
members, or consumers.
10] Pasinetti - pag. 10
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58.
In order to keep the analysis in as simple terms as possible
only final commodities will be considered. No intermediate
stage will be explicity represented. After all, it is always
possible, when needed, to re-introduce intermediate stages and
intermediate commodities by a simple linear transformation,
as will be shown and discussed in detail later on, in chapter VI.
For the time being, therefore, all production processes will be
considered as vertically integrated, in the sense that all their
inputs are exclusively represented by services from two types
of factors of production: labour and capital.
To begin with, however, it will be useful, as a purely expository
device, to take a very simple step and to develop first
a theoretical model (sections 2 and 3) where production is
exclusively carried out by labour. Capital, as a factor complementary
to labour, will then be introduced from section 4 on.
2. The flows of commodities and of labour services, in physical
terms and at current prices.
Even in a system as simple as the one now described, there
is a whole series of flows — or rather two different series of
flows — which take place inside the period considered: flows
of labour services from the individuals as labourers to the
production processes, and flows of commodities from the production
processes to the individuals as consumers. Suppose
that the number of final commodities produced is (z - 1). Then,
since we consider no intermediate stage, there is a production
process, behind each final commodity, which goes right back
to the original factors of production: labour in our case. We
have, therefore, (n - I) production processes or sectors, each
>f which consists of one labour input and of one product output.
‘10] Pasinetti - pag. 11
582 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
All the individuals may then be considered as grouped in a
final sector, which may be called household — sector n —
which receives all productions for consumption and provides all
labour services for production.
All these flows may be framed in a usual input-output
table, which in our case becomes very simple. As is well
known, the table can be looked at from two different points of
view, and accordingly represented by two systems of identities.
From a physical point of view, the production of each commodity
is identically equal to the sum of all quantities of that
commodity which are delivered as inputs to the other sectors.
(In our simple case this sum is reduced to one term: the amount
delivered to the household sector). Moreover, the sum of all
labour services is identically equal to labour employed. Similarly,
from a value point of view, the production of each
commodity must be equal in value to the sum of the values of
its total inputs, and the sum of the values of all commodities
must be equal to the total income which is distributed to the
factors of production (labour in our case).
We have therefore:
J
-
“ni -
Ann
QO
10| Pasinetti - pag. 12
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385
INC
11.2)
1n
l1-x2 P,
where:
Le
P.
= physical quantities produced, j-=
prices, 1=1, 2,
quantity of labour used in all productive activities
— physical flow from sector : to sector
+*
Ci;
[t appears now very useful, for reasons of symmetry, tc
Xi . .
put a;=3 . Then, systems (II.1, "I1..} me TNTESseG …
j
matrix notation ‘
11.3)
(}) Here and in all subsequent analysis, I shall be using matrices to
‘epresent the systems of linear equations (with the advantage of showing
mmediately the symmetrical properties). However, any compact notation
vill be avoided and matrices will always be written in full. In this way,
the following analysis will require from the reader only a few elementary
notions of matrix algebra.
Pasinetti - pag. 1;
384
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
~
an,
Ans
|—
Py
ry
”
(lg)
A
de,
a, 1» —
a, 7
0
|
0
|
J
The (a, a, .... a, ,_,) represent technical coefficients
of production and the (a,,, a,,, .... a,_; ,) represent demand
coefficients of consumption. An evident property of both these
systems is that they have the same, but transposed, matrix of
coefficients. As they have been defined, they are nothing but
an algebraic representation of the flows which take place in
the economic system under examination. However, they can
be looked at in a different way. If we consider the a;’s as parameters,
then (II.3) and (II.4) form two systems of equations,
and we can enquire into the nature of their solutions. These
two systems are of a particular kind — they are linear and
homogenous. Therefore, in order that they may have nontrivial
solutions (i.e. solutions which are not all equal to zero),
the coefficient matrix must be singular, namely it must
satisfy the following condition (which is the same for both
systems) (3):
7 …
Ton
An)
gg
A, .
(*) For shortness, the zeros will not be explicitly written from now
on; so that all entries of our matrices which are left blank should be
interpreted as zeros
10]
Pasinetti - pag. 1.
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
585
By developing this determinant, the condition may be stated
more simply as:
(IL.5) a, a, + a, a,, + …. +a, ._1 AQ _1n -I=0.
[f this condition is satisfied, then each of the two linear homogenous
systems gives solutions for (n - I) variables, while the
n'" variable can be fixed arbitrarily. In our case, there evidently
is for system (II.3) a quantity which is fixed, namely X,: the
total quantity of labour available. On the other hand, no similar
quantity can be considered as fixed in (II.4). This exdresses
the well-known property that any real system can give
solutions only for relative prices but not for their absolute level.
Therefore, the choice being arbitrary, we may put, for the time
veing, P, =P Hence
(11.6)
11.7)
which represent the solutions of the system for physical quanities
and relative prices. The meaning of expressions (II.6)
and (II.7) is fairly straightforward. Each of the coefficients
‘ay, Gays .. a, ,,) expresses average per-capita demand for
[10] Pasinetti - pag. 15
586 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
each commodity, so that (II.6) say that production of each
commodity exclusively depends on demand. If there were no
demand, there would be no production. On the other hand,
each of the coefficients (a, a, ... a, ,_;) expresses the labour
imput in each physical unit of output, so that (II.7) say that
the price of each commodity is directly proportional to the
quantity of labour required to produce it. In other words,
prices, in this simple case, are explained by a pure labour
theory of value.
3. À necessary condition for full employment
Condition (II.5) will recur time and again in the subsequent
analysis and we may well investigate its economic meaning
immediately.
From a mathematical point of view, the fulfilment of (II.5)
is a necessary condition for each of the systems (II.3) and
IT.4) to have positive solutions. However, non-fulfilment does
not imply no solution. The coefficient matrix of (II.3)-(II.4)
has a particular form (all its entries are zeros, except on the
last row, on the last column, and along the diagonal), which
means that the solutions of the systems can be derived directly,
without substitution, from the first (7 — I) equations of (II.3)
and from the first (n- I) equations of (II.4) respectively.
Therefore, relative prices and relative quantities are determined
independently of condition (II.5), whose binding restrictions fall
entirely on the last equation of each of the two systems. Let us
n—l
see what this means. Suppose, for example, that > a,; 4, CI.
i=1
This inequality implies two things. In the context of system
(IL.4) it implies (3) that Za, P,<P,, namely that average per-()
From now on, for shortness sake, the limits of the summations will
be omitted. In other words, all summations that will appear in our analysis
have to be interpreted as running from 1 to (n—1), unless otherwise specified.
10]
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587
capita expenditure is less than the income each worker receives:
a situation of under-consumption. In the context of
system (II.3) it implies that Ye, X,<X , namely that total
labour employed is less than total labour available: a situation
of under-employment. Conversely, the fulfilment of (II.5)
implies both full expenditure of total income and full employment.
Looking at condition (II.5) more analytically, we may noice
that it is a sum of products a,4,,, (:=1, ... n- 1), where
each of these products has the property of being composed of
one technical coefficient and of one consumption coefficient
both referring to the same commodity. Therefore, each a,a,,
2xpresses the proportion of full employment national income
which is spent in that sector (sector ¢), or — which is the
same — the proportion of total labour which is employed in
‘hat sector. Evidently, the sum of these proportions must be
equal to one in order to reach full employment. In those cases
in which average per-capita expenditure turns out to be less than
‘he income each worker receives, prices and quantities are de-‘ermined
all the same, but total production turns out to be
:ess than potential production, and there will be unemployment
in the system. The situation is expressed by condition
[I1.5) being under-satisfied, i.e. by the summation of (II.5)
being less than unity. The gap between Ea,;a,, and unity expresses
the proportion of labour that remains unemployed.
Let me point out immediately the macro-economic nature
of condition (II.5). It does not depend on the number of sectors
that exist in the economy; it is just one condition referring to
the economic system as a whole. We may express it by simply
saying that there must be a total expenditure equal to potential
national income if full employment is to be achieved.
When put in these terms, condition (II.5) sounds familiar.
[t expresses a conclusion which is very well known in economic
theory, since J.M. KEYNES brought it out explicitly in his
General Theory The novelty here is that this macro-eco-‘10]|
Pasinetti - pag. 17
588 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
nomic result emerges immediately from a pure production model
of the type we are dealing with; a model, by the way, which
has been developed on a multi-sector basis.
It must be added that, with the particular hypothesis of
the present simple case (where no capital is needed) the condition
that all income must be spent also means something
more specific, namely that all income must be consumed. By
adopting a Keynesian definition of savings, i.e. by defining
savings as that part of personal incomes which is not spent,
macro-economic condition (II.5) may also be expressed by
saying that there can be no net savings in the system, as a
necessary condition for full employment. Single individuals, of
course, may save, but there must be other individuals who dissave
by the same amount, so that savings and dissavings, on
the whole, cancel each other out. The only form of savings
which is possible for the system as a whole is in real terms,
i.e. by carrying over durable (produced) commodities from one
period to another. But within each period, total demand must
be such as to induce the full utilization of the production potential,
if full employment is to be reached. This is another
important property of a production system. There must be
demand — and in our simple case demand for consumption
goods, whether durable or non-durables — for such an amount
as to generate that quantity of production that requires the full
employment of the existing labour force.
1. Production by means of labour and cabital
We may now consider an economic system which has all
the properties of the system analysed in the previous sections
with one difference. The production processes, in order to be
carried on, require another factor of production besides labour:
capital. Each productive process includes, as in the previous
case, a flow-input of labour and a flow-output of final commodity,
but now it also requires a stock of capital, on which the
[10] Pasinetti - pag. 18
SEMAINE D'ÉTUDE SUR LE ROLE DE L'’ANALYSE ECONOMETRIQUE ETC.
58y
labour services have to be performed in order to produce the
final output.
Capital itself is a commodity, or rather a series of commodities,
which have been produced in the economy, so that the
final outputs of the system are now of two types: consumption
goods, which are consumed as in the previous case, and capital
goods, which are invested in the process of production. Capital
goods are durable and therefore, once produced, they go
to increase the existing stock of capital. However, although
durable, they are not eternal. As a result of being used by
:abour they wear out, and must be continually re-integrated
if production is to go on. This means that each productive
process leading to a final commodity needs a further item. Besides
a flow-input of labour and a stock of capital, each process
also needs a flow-input of capital goods to keep the initial capital
stock intact. This means that total production of capital goods,
which represents total gross investment of the system, must be
distinguished in two parts. A first part of it (replacement)
simply goes to replace worn-out capital: in fact it is nothing
but a cost of production, since it is needed in order that the
productive processes may end up with the same amount of
capital stock with which they began. The rest of the total
production of capital goods, namely the excess over replacement,
represents a net addition (net investment) to the existing
capital stock.
But how is capital going to be measured? In our analysis,
all commodities have been measured sa far in two ways — in
physical terms — whatever the physical unit may be — and
in current price terms — physical quantities multiplied by
prices. The same procedure will be used for capital goods,
>xcept that a particular physical unit will be adopted, namely
broductive capacity. Of course, a unit of productive capacity
s, in an ordinary sense, a very composite physical commodity:
not only does it include both what has been called « circulatng
» and what has been called « fixed » capital. It may
10] Pasinett: - pag. 19
590
PONTIFICIAE ACADEMIAE SCIENTIARVM SCR PTA VARIA - 28
actually be made up of a long series of different physical goods
in different proportions and with different durability. The
reason why such a unit of measure is here used is the same that
prompted the use of the concept of a final commodity (*). Both
concepts permit useful simplifications of exposition and will
become especially helpful in the dynamic analysis which will
follow in the next chapters.
5. The physical stocks and flows of the system
Let us consider, first of all, the physical aspect of our system
in a given period of time. We are faced with a series of stocks
and a series of flows.
At the beginning of the period, there exists a series of
stocks of capital which have been inherited from previous periods.
We may represent them by a vector
(11.8) K, Ky ooo. Ky ooo. KT
where each K; stands for the stock of capital, measured in terms
of productive capacity, in sector j (j=I, 2, … #-I). When
our analysis begins, the K's are obviously given. However
they are not given by « nature ». They are the result of
production activity in earlier periods of time, each K; being
the sum of all net investments made in the past in sector 7.
(*) Conceptually, the notion of a unit of productive capacity and the
notion of a final commodity have many similarities. Both a final commodity
and a unit of capacity can, at a given point in time, be broken down
into many distinct components: intermediate goods for the former, capital
goods in an ordinary sense for the latter, These break-down relations, however,
are valid only at a given point of time. When a movement through
time is considered, the relations change - intermediate goods on one side
and final goods on the other, capital goods on one side and productive
capacity goods on the other, follow a path of their own. It is in connection
with these time movements that final goods and productive capacity goods
will become particularly useful in the subsequent dvnamic analysis.
'10] Pasinetti - pag. 20
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We might say that there is also another stock in the system,
at the beginning of the period we are considering, namely population.
This stock, however, for economic purposes, is not
relevant as such, i.e. as a stock (except in a slave society,
which is outside our interests). Its economic relevance is connected
only with the flows it calls into being.
Let us come, therefore, to consider these flows. As in the
previous case, all the members of the community are grouped in
a final sector #. This final sector provides the labour services
to all productive processes and owns the stocks of capital.
Moreover, it exerts the demand for all the final goods, which
are now of two types — consumption goods and investment
goods.
The flows of consumption goods, as before, come from
‘n — 1) consumption goods sectors. In addition to these, we
have now a whole series of flows of investment goods, which
will be denoted with the suffix k. As a first step to the more
general formulation of section 7, we shall assume that capital
is required only for the production of consumption goods, while
capital goods can be produced exclusively from labour. Since
the unit of measure of capital is different for each consumption
good sector, the process for the production of capital goods
must itself be expanded into (# - 1) sectors. Our system thereby
acquires (z - 1) new variables: Xp,o Xi,» oor Xi,» cach
of which represents the production of capital for the corresponding
consumption goods sector. This production of capital goods
s devoted partly to new investment (expressed by demand
coefficients in the final sector: a, ; ns Ayn, @ ,) and partly
‘0 replacement of worn-out capacity (expressed by replacement
coefficients : Ak,1> 4,2 +++» @p,_,n-1)- Hence the whole struccure
of physical flows can be represented by the following
system of equations, which takes the place of (II):
10] Pasinetti - pag. 21
392
(IT.9)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
21,
x |
x,
don
a,
€
Ün—1,n
Ann
Kaa
Xi
a
k,_1 #-—1
Ay) Ang + <= Unn—1 Anky Cyn,
frees -
A 0
0)
0
© |
A few words may be added about the replacement coefficients.
Each Ai (¢t=1, 2, ..., n—1I) represents the amount
of capital, in physical terms, which is required in each period
of time in order to keep productive capacity intact. This is the
general definition of the a; /'s. However, it may be useful to
adopt a simplification, because of its immediately apparent
meaning. We may assume that in each period of time and in
each sector, a constant proportion of the productive capacity
vanishes owing to wear and tear, so that T;=1/a;
‘4=1, 2, ... n— I) represents a quantity which is very near
‘though, depending on the rate of growth of the sector, not
exactly coincidental with) the average life-time of physical
capital goods in sector :.
By now following exactly the same steps as in section 2,
the condition that must be satisfied in order that the system
may have non-trivial solutions emerges as
I
11.10) > Api iy + > Fan; Ain + > App; Crm = 4
[10] Pasinetti - pag. 22
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
Su.
and the solutions for the physical quantities take the form
‘1I.11)
17
ar
\
1,
As can be seen, the first (# - 1) solutions, referring to consumption
goods, are again of the simple type of the previous
case: they actually coincide with the (II.6). But the following
(n - 1) solutions, referring to investment goods, now contain
two elements, expressing the fact that production of these goods
is generated by two distinct types of demand: demand for new
investments, and demand for replacement of worn-out capacity.
6. The structure of brices
From the physical flows which have just been considered,
another system of equations can be derived in terms of prices.
At first, one might think that one should write the same matrix
of (II.g) (but transposed) and multiply it by prices. This procedure,
however, is no longer sufficient because the production
processes now require capital, besides labour, and therefore
‘he total income which flows to sector # has two components:
a remuneration for labour (wages) and a remuneration for ca-‘10]
Pasinetti - pag. 23
504
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 7
pital (profits). In each sector, the value of total production must
be equal to the amount of wages due the labourers plus the
amount of profits due the owners of the capital which is used in
the production processes (in terms of productive capacity:
X1> X3, > X,-1) By calling W the wage rate and = the rate
of profit, system (II.4) becomes:
(11.12)
Un,
{>
ZX, p,
I~
a,
Ap 1n—1%,n—1
Ank
a,
“egy
Xin as, ete on Ay_ 1,0 Ayn Ax n ee LL -1
P,._
P,.
Vv
j en
19 GN P,
Xn)
i)
2
x
Here we are faced with a complication. Comparing (II.12)
with system (II.4) we can see that the variable P, has been
eliminated, but only in order to be replaced by fwo new variables:
'W and x. This means that, if we want to keep definite
solutions for relative prices, we must assume one of the
variables as given, or alternatively we must introduce a new
relation.
“10] Pasinetti - pag. 24
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50%
This is an old dilemma in economic analysis. When the
classical economists (RICARDO in particular) faced it, they
thought they could take the wage rate as given from outside
the sphere of economics. Later on, the marginalist economists
chose the other alternative. As is well known, they introduced
a series of new hypotheses, which amounted to treating capital
goods as if they were natural resources, and built up a whole
theory behind the technical coefficients — the theory of marginal
productivity.
In the present work, I am in a sense going back to the
Classics’ approach, but with a reversal of their hypothesis. At
this stage, I shall take as given the rate of profit (x). It must
be made clear, however, that this step does not have, here, the
same meaning that taking a given wage rate had for the Classics.
The rate of profit is not an exogenous magnitude in the
present theoretical scheme (as technology and consumers’ preferences
are). What this taking the rate of profit as given here
means is a simple assertion that the rate of profit is not determined
by the elements of the model so far considered. Of
course, the rate of profit is determined bv elements which come
within the scope of economic investigation and which must
be examined. We shall come back to this problem later on
in chapter V. For the time being, the only anticipation of
that discussion needed here is that the rate of profit is bound
‘0 remain roughly constant, or rather, to show a roughly consant
trend through time.
This result is relevant here because it allows us to keep
the rate of profit among the constants of our analysis. Thereby
we are in a position to express the new vector appearing in
I1.12) in terms of prices, and to incorporate its constant elements
into the coefficient matrix. The rate of profit itself need
not necessarily be exactly the same in all sectors. There may
be particular elements of risk and uncertainty in each sector,
10] Pasinetli - pag. 25
306
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
to be added to a basic rate of profit. Thus system (II.12)
becomes
11.15)
=}
Un.
+ 1
Brat
" T,_,)
Anne 1
Auk,
a,
Ain +++ 00 Anim (an 0 —11 ay,) i (ar, _ n=Tn1 An_1,n) -
| =i
p, |
P. J
P,,
D
“kp
7
al
Û
QO
Here again we have obtained a system which is linear and
homogenous. The coefficient matrix looks different from the
matrix of (II.g). However — interestingly enough — the
determinants of the two matrices are exactly the same, as can
be easilv demonstrated (5°). The condition for non-trivial so-1
(°) The determinant of the matrix of (IL.g), after substituting the T
for the a, ; ‘s comes out as:
€
D2 Wi Tin + 5 Tor Arn + 2 T Anh, Uin —_ 1
and the determinant of the matrix of (II.13) as:
Dani in + > (ann — TiQin) px, + 2 (= + +) in A,
i
This second expression clearly reduces to the first by expansion. We obtain:
>a, a, + 2%, App + > + din Un, — DT: in App, + ST Ai A, — I,
where, as can be seen, the last two summations cancel out.
10] Pasinetti - pag. 26
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lutions of (II.13) is therefore exactly the same as that wn:ch
has been found already for (II.g), namely "I.10) ~n- ‘he
solutions for prices (relative prices) come out as-Ir.
As the reader can see, the last (# - 1) prices are still of the
simple type of the previous section. However, this is only
decause of the simplifying assumption that capital goods require
no capital goods to be produced. The formulation for
the (7 — I) prices of consumption goods are more general and
more interesting. Each price is expressed as a sum of two
:lements: the prime cost (a,, = quantity of labour required
to produce a unit of commodity) and the gross profit mark-up
(=; + a which in turn is composed of the rate of profit
in sector i (7;), and of the depreciation allowance (1/T;), both
of them being proportional to the capital intensity of the productive
process (a,;, = quantity of labour required to produce
one unit of productive capacity).
Already at this stage, a pure labour theory of value is no
longer valid. The only case in which it still stands becomes
a very peculiar one: the case in which either the rate of profit
‘101
Pasinetti - pag. 2;
508 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ~
is zero or the proportion of replacements (i.e. of indirect labour)
to direct labour is exactly the same in all sectors. In
general, therefore, relative prices will depend both on the
sum of direct and indirect labour required, and on the proportion
between the two, i.e. on the capital intensity of the
production processess. However, it is important to notice that
our approach has made it possible — as the (II.14) show —
to express the capital components of prices in terms which are
directly and unambiguously comparable to labour. These capital
components may thus be added to direct and indirect
labour. In the (II.14) they are added, and the total sum is
multiplied by the wage rate. We may, therefore, conclude that
formulae (II.14) express a theory of value which is indeed no
longer in terms of pure labour but is — as we may put it —
in terms of labour equivalents. In the case considered here,
the amount of labour equivalent corresponding to the employment
of capital is a fraction (represented by the rate of profit),
of the amount of labour required to produce capital goods.
7. A more complex case involving capital for the production
of capital
The foregoing analysis has been based on the simplifying
assumption that capital goods are not needed for the production
of capital goods, but the simplification has been made only in
order to keep the formulation as short as possible.
Dropping this assumption does not entail any conceptual
difficulty, it only requires a few more algebraical manipulations.
Assume, for example, in order not to go and infinitum,
that each of the capital goods sectors makes capital goods for
itself and for the corresponding consumption goods sector, and
denote by y; in each sector i, (i=1, 2, ..., n— 1), the ratio
of one unit of capital goods expressed in terms of capacity for
10] Pasinetti - pag. 28
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3GQ
the consumption goods sector to one unit of capital goods
expressed in terms of capacity for the capital goods sector.
Then a new series of (n - I) replacement coefficients for the
capital goods sectors have to be introduced in matrix (II.g), in
addition to those already considered in the consumption goods
sectors. Similarly a new series of (# — 1) profit and depreciation
mark-ups, for the prices of investment goods, has to be introduced
in matrix (II.13) in addition to those already considered
‘or the prices of consumption goods.
In both matrices the new (n- 1) elements fall along the
second half of the main diagonal (°). The solutions, as can
oe easily seen, are again very simple to reach and come out
as follows:
11.15)
(®) The condition under which the matrices of this more complex model
ire singular emerges as follows:
nk; a in
10]
Pasinetli - pag. 29
(I1.16)
600
J
{
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
‘
-
a4
TT.
»
I
1px, + - W
+
74
~
»
n
- _ ) ao, _ tn WwW
“ky m1
TT
n
3
le
OF — lp, WW
“Ry 1 k, 1 n=1
These two new series of solutions are basically similar to
(IL.11) and (II.14), although they are a little more complicated.
Each production of capital goods now covers not only replacement
and new investment for consumption goods sectors, but
also replacement and new investment for capital goods sectors.
Similarly each of the prices now covers not only the profit
and depreciation allowance for consumption goods sectors but
also the profit and depreciation alowance for the corresponding
capital goods sector.
Since, in this case, the productive processes of capital require
as input a part of their own outputs, there is one more
necessary condition for positive solutions — or rather a series
of necessary conditions — explicitly brought out by (II.15) -
(I1.16). Algebraically, the conditions are
(11.1%) T,.>v., 1=1I, 2, ..., 1
=
meaning that the total output from the employment of one
machine, in the whole course of its life, must be more than
to]
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one machine of the same type. If it were not so (i.e. if the
processes to produce capital goods required more as inputs
than what they give as outputs) production would be impossible.
The reader will also notice that the theory of value implied
by (II.16) is analogous to the theory implied by (II.14). AI
components of prices are expressed in terms of labour and
labour equivalents.
These similarities will be very useful in the subsequent
analysis, where it does not make any difference whether the
(II.11), (II.14) or the (Il.x5), (II.16) are used, except in
some particular cases. Normally, therefore, we shall be able,
without lack of generality, to carry on our analysis in terms
of the (II.11), (II.14), which are much simpler. For analogous
reasons of simplicity, the procedure will normally be followed
of using a single rate of profit (x) and the same replacement
coefficient ( =) for all the sectors.
3. The conditions for equilibrium
We have not yet commented on the new form taken by
the full employment condition. Expression (II.10) represents,
in mathematical terms, the condition that must be satisfied in
order that the linear and homogeneous systems (II.g) and
II.13) may have non-trivial solutions. It clearly takes the
place of (II.5) in the previous case and has exactly the same
economic meaning: a necessary condition for reaching full
employment. Unlike (II.5), expression (II.10) now makes a
distinction among three different types of demand: demand
for consumption goods, demand for new investments and demand
for replacing worn-out capital goods. However, the importance
of this distinction is only to be seen in the dynamic
analysis which will follow — a different composition of demand
10] Pasinetti - pag. 3
602
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
in a certain period of time evidently has different consequences
on the stocks of capital goods in the ensuing period. But for the
time being, and as far as a short-run analysis is concerned, the
composition of demand does not matter. To fulfil condition
(I1.10), that is to reach full employment, the only requirement
that is needed is that the sum of all types of demand be
such as to imply a total expenditure equal to total potential
income.
What must be added, however, is that the fulfilment of
(II.10) is no longer enough, because it only refers to the flow
aspect of the system, and thereby implies full employment only
potentially. Some other conditions must be fulfilled with regard
to the stocks. First of all, there must be enough productive
capacity in the system to make it possible to produce what is
required bv potential demand. i.e.
11.18) K;>X. , 1=1,2, .... (m-1).
Yet this could not be defined a satisfactory situation, if the
capacities were far beyond what is required by a full employment
demand. We have to impose also the conditions of full
capacity utilization, namelv:
(11.10) K<X., 1=1, 2, …. (KR -I).
It follows that, for the (II.18) and (II.19) to be simultaneously
satisfied, all the K;’s must be equal to the X;s.
To sum up, two types of conditions are now necessary for
the systems (II.o) and (II.13) to hold, namely
(II.20)
I
Sa, ay + San. Ann + > TF Ank, a, = 1
5
aa
(11.21)
K. — D
wv
—y Ay sees (mn — I,
10] Pasinetti - pag. 32
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603
We may notice that (II.20), like (II.5), is a macro-economic
condition: the whole national income must be spent, if full
employment is to be reached. On the other hand, (II.21) represents
a series of sectoral conditions: there is one condition
for each sector. Each sector : must be endowed with that stock
of productive capacity which is necessary to produce the amount
of commodity ¢ which is demanded.
At this point, one may wonder what would happen if (II.20)-‘IT.21)
were not satisfied. It clearly depends on how the nonfulfilment
comes about. A few interesting cases may perhaps
be usefully considered. Inequalities of type (I1.18) plus the left
hand side of (IT.20) being lower than its right hand side mean
idle capacity and less than full empoyment: a situation which
we may call one of Keynesian under-employment (7). On the
other hand, inequalities of type (II.1g) again plus the lefthand
side of (II.20) lower than its right hand side correspond
‘0 a situation in which the capital structure is smaller than the
one which would make full employment possible: a situation
which may be called one of Marxian under-employment. The
‘wo opposite cases as to condition (IT.20) represent situations
of inflation of different types. due respectively to lack of labour
and to lack of capital.
A further question that may arise at this point is the followng:
if any of these cases, or if any other situation takes place,
in which (IT.20) and (II.21) are not satisfied. how is the svstem
(7) Of course, Keynes had a behavioural theory about this situation,
vith reference to a capitalist economic svstem. In terms of the present
nodel. KevyNEs’ theorv was that, for psychological reasons.
a. a. < i
namely that total demand for consumer goods on the whole tends to be
smaller than full employment income. Of course demand for investment
goods might be such a proportion of total income as to make up for the
difference to unity in the inequality stated above. But KEevNEs pointed
out that there is no reason to necessarily expect this, because the two types
bf demand depend on different factors. As is well known, he thought that
in fact the most likely situation to arise is one in which there is lack nf
~ffective demand and excess of productive capacity.
‘Io] Pasinetti - pag. +3
604
* PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ~
going to react? The answer depends on the particular institutions
that the system has adopted. But, as said already, it
is not the purpose of the present enquiry to introduce any
particular theory corresponding to any specific institutional
set-up. The purpose is simply to find and to specify the conditions
that must be satisfied in any case, if full employment
is to be reached.
To conclude, when both (II.z0) and (II.21) are fulfilled,
then the two systems of equations (II.g) and (II.13) entirely
hold. This means that the economic system they represent is
in a sifuation which we may call one of equilibrium, an expression
simply taken to mean full employment of labour and
full utilization of productive capacity.
9. Towards a dynamic analysis
The foregoing analysis, after introducing capital, has acquired
an important characteristic: although still a short-term
analysis it is not a static analysis. Even if we suppose that the
equilibrium conditions (IT.20) - (II.21) are satisfied, we cannot
say that we are at the end of our enquiry. These conditions
refer to a certain period of time, but just in order to be fulfilled
in that period, they may contain some elements (investments)
whose mere existence means modifying in the following period
those magnitudes (stocks of capital) on which the previous
equilibrium was based. In this theoretical framework, therefore
(unlike what happened in the traditional type of economic
enquiries), the attainment of the situation of equilibrium in a
given period of time does not mean at all that all problems
have been solved.
We are inevitably led, by our own analysis, from the investigation
of equilibrium conditions in a given period of time,
lo the investigation of equilibrium conditions over time.
101 Pasinetti - pag. 34
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CHAPTER III
THE SIMPLEST CASE OF ECONOMIC EXPANSION
POPULATION GROWTH WITH CONSTANT RETURNS TO SCALE
… À simble dvnamic model
Our inquiry will now venture into what happens after the
single period so far considered has elapsed, and into what
happens in general as time goes by. From an analytical point
of view, there are two notions of time which may be adopted.
Time may be conceived of as a succession of finite periods,
with the supposition that changes take place only between one
period and the other. Or, time may be conceived of as bringing
along changes in a continuous way, so that the finite periods
of the previous procedure become so short as to be infinitesimal.
For our purposes, it is irrelevant whether the first or the second
procedure is adopted. But since the second procedure makes
things simpler, from our point of view, it will normally be
followed all through the following dynamic analysis. However,
the arguments will also be re-cast now and then in terms of
the first procedure, when that appears useful and illuminating.
If we look beyond the single period of time, all quantities
considered so far must be dated. The previous short-run model
remains valid within each period, but from one period to
another those quantities which have hitherto been taken as
given (population, technical knowledge, consumption patterns,
capital stocks) may undergo important changes.
10] Pasinetti - pag. 35
606 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
We shall consider in this chapter the simplest of all cases
of dynamic change. We shall assume that, as time goes on,
the only « exogenous » factor which is moving is population.
This simple case has been extensively dealt with already in
current economic literature, and the purpose of the present
chapter is not, therefore, to obtain new results. The purpose
is simply to evince the connections with the known growth models
and at the same time to develop formulations which will
be needed in the following analysis. Our assumptions may be
listed schematically as follows:
a) first of all, the initial conditions are such that, when our
analysis begins, at a time defined as zero, the system is
operating in equilibrium. There is full employment of labour
and full utilization of productive capacity;
b) as time goes on, population increases at a steady percentage
rate g, so that
111.1)
X,(t)=X, (0) eë
where ¢ denotes time, and e is the well-known base of the
natural logarithms;
technical conditions remain fixed over time; expansion takes
place at constant returns to scale. In other words, all
technical coefficients (the a,’s, the a, s and the Ts,
i=1, 2, .... n— I) remain unchanged in time;
1) consumers’ tastes also remain constant, which means that, if
individuals continue to receive the same income, their consumption
— i.e. all the coefficients a;,’s, ¢=1, 2, .... (n-1)
— remain constant through time.
>)
These are all the assumptions that are needed, besides of
course the convention (discussed in section 6 of the previous
chapter) of taking the rate of profit as given. The reader may
‘101 Pasinetti - pag. 36
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607
have noticed that there is a series of coefficients which have
not been mentioned: the Ann’ S, namely the demand coefficients
for new investments. The reason is that these coefficients cannot
be specified in advance. They are themselves unknowns,
in the present model, as they must be determined in such a
way as to be consistent with the growing productive potential
of the system.
Our task is, first of all, to find out the conditions under
which equilibrium may be maintained over time; and then to
nvestigate the time-paths of the variables of the system.
2. The conditions for a dynamic equilibrium
We might begin by saying that, in order to maintain equilibrium
over time, conditions (II.20) and (II.21), which are
satisfied by hypothesis at time zero, must also and constantly
be kept satisfied as time goes on. A statement of this kind,
however, now becomes uninteresting and artificial, because
those two conditions cease to be independent of one another,
when time is allowed to elapse.
If population is growing, both the supply of labour services
and the demand for each product increase, as time goes on.
[n order that the new labour force may find employment and
che growing potential demand may become effective, productive
capacity must also increase. Thus, the equilibrium conditions
for productive capacitv in time become
111.2)
which simply means that there must be an increase in the productive
capacity of each commodity, parallel to the increase
in its demand.
The formulation may be expressed in a more helpful way
‘101 Pasinetti - pag. 37
608
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
by making use of a few definitions. We know from (II.11)
that each production of capital goods (X,) is composed of two
. I
parts: replacement of worn-out capacity ( T %inXn) and new
investment (a;,X,). Denoting these two parts respectively
by X,, and X,,, so that
(111.3)
Xe =X", (6) + Xx, (2) ,
we have
(111.4) x, 0= 2K, @], i=1, 2, .... (n-1).
By substituting now (III.4) into (III.2), we obtain
voy
(IIL. 5) Xp) =o [XO], i=1,2, ... (n-1).
which amounts to expressing the capacity equilibrium conditions
in the form of equilibrium relations between the production
of capital goods in each period of time and the rate of
change in that period of the corresponding consumption goods.
Using the (II.11) and (III.1), the (III.5) become
d :
ay; X,(0)e8 = — [ain X,(0)e#] , i=1,2,....(n-1),
D]
Ann X,(0)e8' = g a,,X,(0)e8
and finally
(111.6)
Ain = E Gin »
ey 2) vee (m=T1).
“10] Pasinetti - pag. 38
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609
This is a formulation of real interest. It states the series
of sectoral productive capacity conditions in terms of flows
‘and no longer in terms of stocks) by specifying a very definite
relation which must hold between each demand coefficient for
new investments and the corresponding demand coefficient for
consumption goods. It states that, as a condition to endow the
system with the equilibrium productive capacity, each sectoral
aet investment, in physical terms, must be equal to the corcesponding
sectoral final demand multiplied by the rate of
growth of population. This determines sectoral equilibrium net
investments in the whole economic system:
TI1.7) X, (t)=g a,,X,(t) 1=1, 2, .... (NA — I).
These conditions may, therefore, be called the capital accumulation
conditions for keeping full employment over time.
[f we like, they may be expressed also in terms of ratios be-‘ween
quantities evaluated at current prices. After multiplying
both sides of """ * hy P. and dividing by PX, we obtain
(111.8)
. 1H
15,
which mean that, in equilibrium, each sectoral ratio of investnent
to production, evaluated at current prices, is equal to the
sercentage rate of population growth multiplied by the corresponding
sectoral capital-output ratio.
When the capital accumulation conditions are kept satisfied,
‘he system — as time goes on — is being constantly endowed
with exactly those stocks of productive capacity which are necessary
in order to provide full employment for all the workers.
This, however, does not yet necessarily imply full employment
and full utilization of productive capacity. In order that equi-101
Pasinelti - pag.
3G
610 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
librium be reached, the other condition — the macro-economic
relation referring to the system as a whole — must also be satisfied.
This condition, after substituting (II1.6) into (II.20),.
becomes
(II.9)
I
5 Uni Ain + g+ T > Ank; a, =I
The economic meaning is again that there must be a total
expenditure equal to total potential gross income if full
employment and full capacity utilization are to be reached.
However, (III.g) now also requires a very definite
division of the total expenditure between new investments,
replacements, and consumption. The effect of substituting into
it the capital accumulation conditions has been to specify the
magnitude of the term (g+ =) 2 a, a;,, which is nothing but
an analytical break-down of the equilibrium ratio of total gross.
investments to gross income in the system as a whole. This
equilibrium ratio, as can be seen, is exclusively determined
by the three exogenous factors of our model: population
growth, technology and consumers’ preferences. Therefore, if
111.9) is to be satisfied, it is the total effective demand for
consumption goods that must absorb the whole remaining proportion
— represented by the first addendum of (III.g) — of
potential gross income. Condition (III.g), since it thereby determines
the size of total effective demand, may be called the
effective demand condition for keeping full employment.
We may notice that this condition, as it stands, presents
no problem through time. Since all coefficients are constant,
once (IIl.g) is satisfied at time zero, it will remain satisfied all
time.
‘Io] Pasinetti - pag. 40
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Ot}
We may also look at it in another and more familiar way,
by re-writing ‘ft =
TI1.10) I- Ya, a.
UMR: Ain
>]
<
2 nk. Ain
The left hand side expresses the proportion of total gross income
which is not spent either on consumption goods or on
capital replacement (the aggregate ratio of net savings to gross
income), and the right hand side expresses the equilibrium
aggregate ratio of net investments to gross income. Now, the
right hand side, by utilizing conditions (111.6), expresses the
equilibrium investment-income ratio as a multiplication of the
rate of population growth by what can be shown to be, after
1 few algebraical re-arrangements (1), the over-all capital-output
() The proof may be given in the following wav.
Call YY = ©. 7 YX": P,. and multiply both sides o.
(III.0)
X.
YW — —
x,
a
nt +
14
Add now to both sides the ratio of equilibrium total profit
qncome *
to nationa
v.
ni
We obtain
>
D TA,h; Ain +
J Gigi, 1 Api .
si din
“'
ne
PVE es
+
10] Pasinetti - pag. qu
612 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - -
ratio. This is a well-known relation. As the reader will realize,
condition (III.ro) is simply a more analytical formulation (referred
to the case of population growth) of what has become
generally known in macro-economic theory as the HArrop-Do-MAR
equation (?).
To conclude, two types of conditions must necessarily be
satisfied in order to keep equilibrium over time. There is, first
of all, a series of capital accumulation condition (IIT.6), ensuring
that each sector be endowed all the time with the additional
productive capacity required by the expanding demand. These
conditions. state that, in each sector, the ratio of new investments
to the level of production must be equal to the technologically
determined capital-output ratio multiplied by the rate
of population growth. Secondly, in order to ensure the full
utilization of the productive capacity that thereby comes into
being and of the available labour force, the macro-economic
effective demand condition (III.g) must also be satisfied. This
condition determines the equilibrium division of total expenditure
between consumption, replacements, and new investments.
It states that, given the total amount of equilibrium
investments by the first series of conditions and by the replacement
requirements, total demand for consumption goods must
be such as to absorb the whole remaining part of potential
gross income.
Lea Pit
4
441,
And since, in equilibrium, X; = K;, we obtain:
=p
0.E.D.
(*) See R. F. Harrop, An Essay in Dynamic Theory in « The Economic
Journal » 1939, and Towards a Dynamic Economics, London 1948; E. Do-MAR,
Capital Expansion, Rate of Growth and Emblovment, in « Econometrica
», 1046.
[10] Pasinetti - pag. 42
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3. A more complete formulation of the effective demand condition
It may be useful to take a further step, at this point, to
complete formulation (III.g). So far, population has been referred
to in general terms, both with respect to demand of commodities
and with respect to supply of labour. This means
assuming implicitly that population and labour force are the
same thing. But in practice, this is not so. All people do contribute
to the demand of commodities, but only a fraction of
them (representing the active population) actually take part
in the process of production. Moreover, a further complication
arises in connection with the fact that our coefficients have a
‘ime dimension. The point may be made clearer by supposing
a finite period, for example a year, as the unit of time. In
‘his case, the consumption coefficients refer to yearly per-capita
consumption; but, as a normal practice, the technical coefficients
are referred to that fraction of the unit of time which
corresponds to actual working time. If this is done, the a,,’s
and the a,;s come to be no longer expressed in terms of the
same number of people and not even to refer to the same unit
of time.
To be consistent, a correction must be made on the technical
coefficients and the simplest way to do it is to divide each of
‘hem by two parameters, the first of which — let us call it à —
representing the proportion of active. to total population, and
the second — let use call it 3 — representing the proportion
of, let us say, working hours to the total number of hours forming
the unit of time considered. Evidently:
10] Pasinetti - pag. 43
614
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
These corrections do not affect the expressions we found for
prices — for example the (II.14) — where the technical coefficients
appear, provided that the wage rate is also referred to
a unit of actual working time. They do affect, on the other
hand, the expression of the effective demand condition for
equilibrium. (This is after all intuitive: the conditions for
reaching full employment are evidently different according to
the ratio of active to total population and to the length of
the working week). Hence, after introducing « and B, (III.o)
becomes
I I I
(III. 11) 3 Sawant (gm) LS an, app =1,
which must be considered as a more complete formulation of
the effective demand condition for a dynamic equilibrium.
+. The dynamic movements of relative prices, physical quanhlies
and other economic variables
To find now how prices and physical quantities move as
time goes by is a very easy task. As to prices, it can immediately
be seen from the (II.14) or from the (II.16) that, under
the present hypotheses, all their components are constant in
time, so that all relative prices remain constant as time goes on.
The expressions found for physical quantities on the other
hand — the (II.11) or the (II.15) — all contain one component,
namely population, which is increasing at a percentage
rate g. Therefore, each physical quantity increases in time at
the percentage rate of growth g.
Besides prices and quantities, there are other magnitudes
in the system which are of economic interest and which are
worth considering. The time-paths of two series of them in
particular — the amounts of employment in each of the sectors
and the production of each commodity at current prices — can
[10] Pasinetti - pag. 44
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615
be derived immediately from the movements of prices and quantities.
In general, denoting respectively by F; and V; the sectoral
employments and the productions at current prices, we
mav write:
(III.12) E.(#) =a,,(t{,X, '
(IT1.13) V,(t)=X,({)P;.(#),
1=1,2,...,(n — I),k,, k,..., BR,
Again, by substituting from (II.11), (IL.14) or from (II.15),
11.16), it can be seen that, under the present assumptions, each
of the E;'s and each of the V;s (taking W as numeraire) increase
in time at the percentage rate of growth of population (g).
There is moreover another series of magnitudes which always
recur in a dynamic analysis and which we may consider
immediately. I am referring to the capital-output ratios in the
various sectors of the economy (which will be denoted by
k;, 1=1, 2, ..., n— I); and to the over-all capital-output ratio
‘which will be denoted by ¥). Using for simplicity the formuations
(II.11), (II.14) referring to a system where capital is
required only for the production of consumption goods, each
sectoral capital-output ratio mav be expressed -
TII.14,
“1;
3) In each sector, the capital output ratio
Remembering that K; and +; coincide, when
using the (IT r= rd M 1) we ht:
ll emplo--me
continued on following
a fw
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and the over-all capital output ratio as (*)
(111.15) xl) =- ‘
Sain(t) anid +( gm ) Dazn(t) Appt
Sa,
n(f) Amp
Âs can be seen, the components of (III.14) and (I11.15)
are nothing but coefficients. And since, under the present assumptions,
all coefficients are constant through time, it follows
that each sectoral capital-output ratio and also the overall
capital-output ratio remain all constant as time goes on.
When the more complex formulations (II.15) and (II.16) are used, the
result js
XY.
a.
î
Ann;
+ [ ns ! Ve
T 21k
(*) The over-all capital output ratio is, in our notations.
Tn
X =< TT
> my ki <x k;
Taking the (II.x1) and (II.14) and again remembering that K, and X,
coincide when full employment is kept, we obtain
—
2, +
B I
Dain Jan + (n+ T
‘ Corn
Vv
ss Ain aun; Ww x,
>a, a
Zip Uni + | T-: — .
When the more complex formulation:
result is
Un a
;Io]
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|: a
r—in Ÿ
11.16) are used, the
Sag, Uk;
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[t goes without saying that, as a straightforward corollary
of the above results, all aggregate quantities — like gross and
net national income, consumption, saving, investment, capital
— increase in real terms, as time goes on, at the same percentage
rate of growth (g) of population.
=. Interesting features of the present case of growth
The dynamic features of the system considered here, as
they emerge from the previous sections, are of an extreme simplicity.
The merits of this simplicity are entirely attributable
to the assumption of a fixed technology and constant returns
to scale, which confers on this case all the elegant properties it
possesses. With an invariant technology and constant returns
to scale, the growth of the system is entirely determined by the
rate of increase of population. This growth does not affect the
position of any single individual: per-capita income remains
constant as time goes on, and economic growth simply means
that the system expands all its sections in the same proportion.
All products grow, with population, at the same percentage
rate, while the structure of the system (its relative composition)
remains constant as time goes on.
This case of growth is well known, of course, in economic
theory. A clear though rudimentary picture of this case can
already be found in CassEL (°). Recently, LEONTIEF and
VON NEUMANN, although in a different way (*), have based
all their well-known dvnamic elaborations exactlv on this case.
(°) Gustav CasseL, Theoretische Sozialékonomie, Leipzig, 1918, pp. 2
and ff. English translation: The Theory of Social Economy, London, 1921
Jp. 34 and fi.
(®) Both von NEUMANN’s and LEONTIEF’s dynamic models will be discus.
sed in more detail in an appendix to chapter VI.
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But, most interesting of all, this is the case to which macroeconomic
models of growth can be correctly applied. Since the
system expands keeping its proportions constant, there is in
fact no loss of generality in framing the analysis in macroeconomic
terms. The usual convention that the variables must
be considered as measured in terms of a composite commodity,
made up of a fixed « basket of goods », is in this case perfectly
legitimate and logically unobjectionable. As all our results
have shown, since all single coefficients remain constant through
time, the composition or structure of the system, once specified
at time zero, remains the same for all time.
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CHAPTER IV
PROBLEMS CONNECTED WITH TECHNICAL CHANGE -
SETTING THE BASES FOR A GENERAL DYNAMIC ANALYSIS
t. Technical progress in macro-economic models
The case of the previous chapter — despite its popularity
among theoretical economists — remains a very particular case
of economic growth. In practice, as soon as we look beyond
a single period of time, there is another series of changes that
ake place besides those of population: the changes in the
‘echnical methods of production. These changes are in fact
much more problematical and much more complex than those
concerning population.
Technical change has been the great neglected factor in
economic analysis. Only in the last fifteen years have economists
begun to deal with it through the elaboration of models
of economic growth. All these models with technical progress,
however, have been developed in macro-economic terms, i.e.
with the implicit assumption that one single commodity (or a
composite commodity of invariable composition) is being produced
in the svstem (!). And technical change has been in-"')
After the path-breaking contribution of Harrop and DoMar, already
mentioned, the macro-economic models of growth which have been elaborated
are so many as to be alreadv difficult to count. On the whole, two main
streams of thought have emerged. The first one has tried to pursue
Harrop-DoMmar’s Keynesian approach in various directions, and the second
has tried to insert HarROD-DOMAR’s ideas into the traditional neoclassical
10] Pasinetti - pag. 40
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
troduced in the form of a « rate of technical progress », which
has been treated exactly like, and svmmetricallv to. the rate
of population growth (?).
Unfortunately, this approach has been accepted rather uncritically
so far. It is my purpose to criticize and abandon it.
But before doing so, I must invite the reader to take a closer
look at all the implications that this approach entails. Our
disaggregated formulations will turn out to be very useful in
this task.
As said above, any macro-economic analysis implies that
all variables considered are measured in terms of a composite
commodity or « basket of goods » of fixed and invariable composition
through time. Therefore, unless the macro-economic
framework is given up altogether, the introduction of a rate
of technical progress in such an analysis necessarily implies
two further and much more specific assumptions: 1) that technical
progress is going on at the same rate in all sectors of the
economy; and 2) that demand for each product is expanding
at the same rate.
Let us carefully consider a hypothetical case of economic
growth in which these two assumptions are satisfied.
theory. The first type of models is perhaps best represented by: J. Ro-BINSON,
The Accumulation of Capital, London, 1956, and N. KALDOR,
A Model of Economic Growth in « The Economic Journal », 1957. The
most representative examples of the neo-classical models are perhaps:
R.M. SoLow, À Contribution to the Theory of Economic Growth in « The
Quarterly Journal of Economics », February, 1956; and J. Meape, 4 Neo-Classical
Theory of Economic Growth, London, 1961.
(*) In the present and following chapters, we shall normally consider
percentage, i.e. relative rates of change. However, for brevity’s sake, and
following what has by now become a custom, in economic literature, the
words percentage or relative will normally be omitted, except in those
cases where their omission may generate misunderstanding.
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2. A dynamic model with uniform technical progress and uni
form expansion of demand
After dealing with the case of population increase and constant
returns to scale, the hypothetical case of economic growth
taking place with uniform technical progress and uniform expansion
of demand can be treated more summarily.
The assumptions are now as follows:
a) the initial conditions, as in the previous case, are such that
at time zero there is full employment and full capacity
utilization:
D) as time goes on, population remains constant;
c) there is the same technical progress in all sectors of the
economy. This means that, as time goes on, all the technical
coefficients of production decrease at a steady percentage
rate ¢, i.e.
IV.1) a, (t)=a,,
— 1, Lis ……0… (n- 1), k,, R-, “ee rR,
d) consumers’ tastes are such that the composition of consumption
is invariant to changes in income. In other words,
when income increases, each individual expands demand of
all the commodities consumed in the same proportion. This
means that, if per-capita income increases at a percentage
rate p through time (as in equilibrium it must do), all coefficients
of consumption will also increase at the rate p, i.e.
IV. 2)
a. (t) = a; U } et
Ly 25 eens (n - Ir
Under these assumptions, the two conditions for a dynamic
2quilibrium come out rather straightforwardly. By following
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
exactly the same procedure as in section 2 of the previous
chapter, the capital accumulation conditions emerse as
(IV.3) a, »(t)=pa,, (1), 1=1, 2, .... (n-1),
which are similar to the (III.6), the only difference being that
now the rate of technical progress has replaced the rate of
population growth. Of course in the (IV.3) all coefficients are
moving, but they are moving at the same rate of change so
that the relation between them remains constant through time.
The effective demand condition also emerges as very similar
to (111.11):
(V4) 1-3 Senda = (e+) 3 5 anid and,
where again the rate of technical progress has taken the place
of the rate of population growth. The interesting property of
this case is that all coefficients of production and all coefficients
of consumption, although moving in time, are moving
in an opposite direction and at exactly the same rate. As a.
result, each single binary product of coefficients under the
two summations remains constant as time goes on — the movements
of the components exactly cancelling each other out.
This means that the contribution to national income of each
single sector remains constant. As in the previous case, condition
(IV.4) does not raise any problem through time. Once
it is satisfied at time zero, it will remain satisfied for ever,
pecause in all sectors productivity and demand are increasing
at the same rate.
The time paths of all economic magnitudes can be found
immediately by substituting (IV.1) and (IV.2) into (II.11),
(IT.14) and (III.12)-(I1I.15). As can easily be checked, if the
rate of profit remains constant, the results emerge as follows:
r) physical production of each commodity increases in time
at the rate p;
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2) all commodity prices — if W is taken as the arbitrarily
fixed price — decrease in time at the rate p. Alternatively
— if the price of any commodity instead of W is taken as
given — all commodity prices remain constant as time
goes on, while the wage rate increases at the rate p. Another
way of stating this result is to say that the commodity price
structure (relative prices of commodities) remains constant
‘through time, while the real wage rate increases at the rate c;
3) employment in each sector remains invariant as time goes
on:
4) production of each commodity evaluated at current prices
remains constant in time (if W is taken as the arbitrarily
fixed price) or increases at the rate p (if any commodity
price, instead of W, is taken as given);
5) all sectoral capital-output ratios, as well as the over-all
capital-output ratio, remain constant.
3. Analytical properties of the two cases of growth considered
so far
The most attractive property of the case of economic growth
just examined is that it still retains the constancy of the proportions
of the system in time. By the device of uniformity both
in technical change and in expansion of demand, all movements
of coefficients cancel out inside each sector and moreover the
structure of prices remains unchanged. Thus, again, the system
expands by multiplying all its sections in the same proportion
so that its relative composition is invariant with respect to
zrowth and time.
From an analytical point of view, there is a remarkable
symmetrical correspondence between the two cases of growth
~onsidered so far. In the case of population growth the wage
101 Pasinetti - pag. 53.
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
rate was constant and the system was growing at the same rate
as population; in the case of uniform technical progress just
examined, population remains constant and the wage rate is
growing at the same rate as technical progress. The two cases
may easily be combined, and the results of such a combination
are so straightforward that it is of no use to spell them out in
detail here. We may however explicitly state at least the conditions
of equilibrium, which emerge as follows:
(IV.5) An; nt) = (g + P)ain(t),
1=1, 2, ..., (n-1),
and
(IV.6)
TS gaan) = 3 om an) alt) =
= (246) 3 — am (0) auld).
The economic meaning is evident. Each single sector of the
system and the system as a whole expand at a rate which is
the sum of the rate of population growth and of the rate of
technical progress, a- sum which is widely known in economic
literature by HARROD’s term of natural rate of growth. This
natural rate appears explicitly both in (IV.5) and in (IV.6),
the latter now giving a complete analytical break-down of
HARROD’s equations. (The aggregate net saving ratio is required
to be equal to the natural rate of growth multiplied by
the over-all capital-output ratio) (3).
At this point, however, after admiring the symmetry and
the analytical beauty of the two cases of growth considered so
far, we must also draw from the results we have obtained at
least two logical and stringent conclusions.
(3) See the proof in footnote (1) of Chapter IIT.
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The first one refers to methods of analysis. If the assumptions
embodied in the two cases of economic growth considered
above were acceptable, then there would not really be much
gain of insight into the working of an economic system by
using a disaggregated model. The merit of such a model would
only be to show the structure of the system at a given point of
time. But since this structure remains the same for ever, the
dynamics of the system is always uniform, which means that
no extra information can be obtained by disaggregating. In
other words, a macro-economic formulation would be by itself
sufficient and satisfactory.
The second conclusion is of a much more practical relevance.
If the assumptions embodied in the two cases of economic
growth considered above were to correspond, even roughly, to
what in the long run is happening in the real world, then any
oreoccupation about problems of economic growth would be
entirely unjustified. The model considered above amounts in
fact to saying that economic growth as such does not present
any problem at all. The only thing that in any economic
system is to be done is the setting up of that particular struc-‘ure
which is most desired — the only constraint being that it
must satisfy relations (IV.5) and (IV.6). This is a once-for-all
problem. Once this structure has been set up, no problem exists
any more. Thereafter, the system will expand for ever, keeping
proportions constant.
Attractive though the first conclusions may be, the second
one is so much in a striking contrast with everyday experience
and with the economic policies of all Governments, that it
should immediately lead one to infer that there must be something
wrong somewhere. And it is my contention that this
something wrong is to be found exactly in the hypothetical
case of economic growth considered in the previous section.
To substantiate this assertion will require all the rest of
the present chapter.
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4. The production aspect of technical change
It will now be useful to consider carefully the meaning
and implications of technical change. When the technical
coefficients of production change as time goes on, there are
two distinctly different series of effects which are called into
being and which may respectively be connected with production
and demand. On the production side, technical change
means a variation in the technical conditions and therefore a
change in the physical quantities of goods which may be produced
out of a given amount of original factors of production.
On the demand side, it means a change in remuneration of the
factors of production, and therefore a change in the amount
of per-capita real income at the disposal of consumers in general.
Let us take the production aspect first. Here the causes of
change may be manifold. For a long time economists have
been impressed by those changes which are connected with
the exhaustion of natural resources. To MALTHUS and RIcARDO,
for example, at the beginning of last century, it appeared
as a matter of logical necessity that the continuation and expansion
of the process of production, on natural resources
which are given, should inevitably lead to decreasing returns,
i.e. to an increasing trend in the technical coefficients of production.
But the economic history of the industrial countries
has by now consistently and persistently brought to the fore
another, more important and widespread process of change,
continuously at work ‘in any modern society: technical progress.
Technical progress is a very complex movement. In the
sense in which it is relevant for economic analysis, it includes
not only, and not so much, the great scientific discoveries,
which by their own nature come about in a discontinuous and
sometimes accidental way, as their practical application on
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an industrial scale, which takes place through a much longer
and continuous process. Moreover, it includes all the innumerable
series of expedients and devices, small if considered
individually, but of great relevance if jointly gauged, which
are the daily upshot of experience, experiments, research and
of re-thinking of the productive organization. This is indeed
a very complicated process, emerging from the learning activity
of men and the application of this learning activity to pro-Juction.
By its nature this process is, therefore, a slow but
persistent one. It consists of long and repeated attempts not
only to reorganize the old methods of production, to utilize
more efficiently the new materials, and to improve the quality
of the products, but also to invent and apply new methods
of production, produce new products, find new resources, and
discover new sources of energy.
It would certainly be out of place to develop a theory of
echnical progress here. If such a theory should ever be
developed, it would pertain to a much wider field than economics,
because it could not avoid some definite conceptions about
the aims and ends of human society. Therefore, as far as the
present work is concerned, the usual procedure will be followed
of taking technology as given from outside economic analysis.
However, what will be taken as given is not a fixed technology
but the movements of the technical coefficients in general
through time.
In this way, the concept of technical change which is
adopted in the following pages will not be restricted by any
particular assumptions (*! But in order to reach practicalr
() It will cover the case of technical progress of any type and also
the case of increasing or decreasing returns to scale. On the other hand,
he case on ‘which neo-classical economists focussed all their attention
‘a change of production methods owing to changes in the rate of profit)
vill automatically not arise because in the present model the rate of profit
remains constant as time goes on.
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628 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2X
relevant conclusions, the analysis will be carried out with reference
to those movements that empirical findings have by now
shown to be everywhere the most typical ones in a modern
society. These movement may be briefly stated in three propositions:
a) in the long run, the effects of technical progress are, on a
(weighted) average, by far more important and more widespread
than the effects of decreasing returns to scale ©).
This means that, as time goes on, the coefficients of production
decrease (i.e. productivity increases) in most sectors,
although in a few sectors the coefficients might increase (i.e.
productivity decrease);
b) as a net effect of decreasing returns and of technical progress,
each production coefficient is slowly but persistently
moving through time. However, each coefficient is moving
at a different speed. In other words, there is a wide dispersion
amongst the rates of change of productivity referring
to the different branches of the economy (5);
c) technical progress consists not only of increases in productivity
but also of continuous additions of new sectors producing
new and better goods for the economic svstem.
y, The demand aspect of technical change
Let us now consider the effects of technical change on demand.
If on the whole technical change is in the direction of
a persistently increasing trend of productivity, it means a
() The simplest empirical confirmation of this proposition is that in all
industrial countries, per-capita income is enormously higher today than it
was when they began to industrialize.
(°) Cf. for example the interesting study by F. L. Hirt, a new Look
at Productivity Growth Rates, in « Survey of Current Business », 1957.
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higher and higher amount of wages and profits, or, more generally,
an increasing trend in per-capita incomes at the disposal
of consumers. It follows that, in each period of time,
‘echnical progress compels the members of the community to
make new decisions; they must decide on which commodities
‘hey are going to spend the increments of their incomes. It is
nere that their preferences as consumers come to play a central
role.
Consumers’ preferences ultimately depend on the character
of human nature, which represents, in the same way as the
technical conditions of production do, a fundamental datum
for any meaningful economic investigation. No commodity,
whatever ingenious technique it may require, can be sucessfully
produced if its utility for the consumers is not sufficient to
justify its cost: it would remain unsold. The relevance itself of
technical progress depends on demand; an increase in productivity,
however large it may be, loses much or even all of
its meaning, if it takes place in the productive process of a
commodity for which demand is small or negligible. This means
that any investigation into technical progress must necessarily
imply some hypotheses (and if not explicitly, it necessarily does
so implicitly) on the character of the evolution of demand as
Income increases. Not to make such hypotheses and to pretend
to discuss technical progress without considering the evolution
of demand would make it impossible to evaluate the very relevance
of technical progress and would render the investigalion
itself meaningless. Increases in productivity and increases
in real income are two facets of the same phenomenon. Since
the first implies the second, and the composition of the second
letermines the relevance of the first, the one cannot be con
sidered if the other is ignored.
Unfortunately, the economic theory which has so far been
developed is hardly able to give us any help on this problem.
[he consumers’ demand theory that we know today is a highly
sophisticated logical framework, built on static premises. Ti
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
25
relies on well-known and consistent preferences defined at a
given level of per-capita income. Such a theory is indeed useful
in showing the consequences of price changes, at a given level
of income, but has nothing to offer us to explain changes following
each successive increment of income.
The regrettable consequence has been, that when economists
— by introducing technical progress in their models — have
been compelled to make definite hypotheses about the expansion
of demand, in the absence of any guiding principle,
have made those assumptions that best suited the mathematical
properties of their models. As we have seen, they have invariably
postulated, either explicitly or implicitly, a uniform and
proportional expansion of per-capita demand (7).
Now, if there is something that we do positively know about
expansion of per-capita demand when income increases, it is
that per-capita demand for each commodity does not expand
proportionally. All the empirical investigators who, in the last
hundred years, have looked into this matter have invariably
and without exception confirmed this tendency.
As is well known, the first empirical generalizations on the
evolving pattern of demand, in response to increases in income,
come from an old discovery in economics which goes back to
ERNST ENGEL (*) in the 1850’s. ENGEL, after studying the
conditions of consumption of workers in the kingdom of
Saxony, stated what has since become known as Engel’s law.
The law says that the proportion of income spent on food declines
as income increases. A more general formulation of this
() The effect of this is that all models of economic growth now-a-days
share a defect which was characteristic of Classical analysis. They have
concentrated their emphasis exclusively on the production side of the’ ecoaomic
process and have entirely forgotten the other half of economic reality,
related to demand.
(®) Ernst ENGEL, Die Productions- und Consumptionsverhiltnisse des
Känigreichs Sachsen, in « Zeitschrift der Statistischen Bureaux des Kôniglich
Sächsischen Ministerium des Inneren », Nos. 8 and go, Nov. 22, 1857; republished
in « Bulletin de l’Institut International de Statistique ». IX (1805).
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empirical law, stating that the proportion of income spent on
any type of goods changes as per-capita incomes increase has
been confirmed ever since, and evinced by all the econometricians
who have been concerned with empirical work on demand
(°).
These results are in fact no surprise. It should not take
long to realise that what they reveal is a basic tendency inherent
in the human nature of the consumers. We shall go into
this matter in the next section. Meanwhile we may safely
draw the conclusion that all the models of economic growth so
far developed have adopted a set of hypotheses which are
‘ncompatible with one of the most fundamental empirical laws
of economics. Since increases in per-capita income necessarily
imply non-proportional expansion of demand, and since technical
progress means increases in per-capita incomes, then the
introduction of technical progress in any dynamic model necessarily
implies a non-proportional expansion of demand. The
assumptions adopted in all macro-economic models of growth
are therefore unaccentable.
6. The evolution of demand in time
The practical importance of ENGEL’s law has always been
recognized by all those who, on any occasion, have been engaged
in empirical work on demand. Yet, in spite of the fact
that it was discovered more than a century ago, very little work
has been done to try to take advantage of the information it
gives and to incorporate it into the theory of consumer’s behaviour.
(Bits of piece-meal theory can be found only in the
works of the econometricians who — faced with facts — have
alwavs been compelled to make additions and to adapt an in-(°)
See, for example: R.G.D. ALLEN and A.L. Bowrey, Family Expenditure,
London, 1935; and also: H.S. HOUTHAKKER, An International Comparison
of Household Expenditure Patterns, Commemorating the Centenary
of Engel’s Law. in « Econometrica ». 1957.
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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - z
sufficient theory to their particular purposes). As a result,
ENGEL’s discovery has remained until our own day at the
state of an empirical law, almost entirely isolated from the
body of demand theory.
It is not my purpose, of course, to engage in a theoretical
investigation on demand here. However, since the tendency
that ENGEL discovered plays a central role in the subsequent
analysis, I shall discuss at least two theoretical aspects of it,
and make to current demand theory (otherwise insufficient
for our purposes) a few essential additions.
The first basic point I should like to make refers to the
nature of human preferences. This point applies even if we
supposed that individual preferences can be represented by perfectly
known utility functions. It would be misleading to
ignore, it seems to me, what we do know on the subject, namely
that the utility that any commodity can give depends on
the previous consumption of other commodities. For, the activity
of consumption is a process in which there is a very definite
order of succesive steps to be taken. For example, the decision
to buy a motor-car presupposes that the consumer has already
bought — and is permanently in the condition to buy — an
adequate quantity of food, clothes, dwelling space, etc. The
motor-car would not have for him the same utility if he were
not well-fed, well-clad, etc. (1). In other words, the utility itself
of the motor car depends on the type and quantity of commodities
which have been consumed already. To talk of these
problems evidently means talking of the shape of consumers’
preferences considered as a whole (i.e. of the whole utility function);
it means talking of absolute levels of utility (and not
only of marginal utilities). Let me point out that there are in
particular some basic human needs (like eating and breathing)
for which the commodities that are necessary may be said to
have an infinite level of utility: without them men would die!
) This interdependence was first pointed out to me by Mr. N. KALDOR.
10]
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Yet, once these needs are satisfied the marginal utility of successive
increments of those commodities may fall dramatically
and very quickly even become negative. Obviously, before this
happens, demand will shift to other commodities. But the property
that those basic needs saturate rapidly in no way alters
the fact that they must be satisfied first of all.
Now, current demand theory (especially — I must say —
after its recent elegant refinements) has focussed our attention
exclusively on what happens at the margin — on marginal
substitution among commodities if a price change, or on equalisation
of marginal utilities. Such an approach may have a
justification in a static world, where everything which may happen
cannot but happen at the margin (which always remains
the same). But in a world where per-capita incomes are
moving, there is very little help we can get from marginal
atilities, unless they are specified over the whole range of the
atility functions, i.e. unless we pass from them to absolute
levels of utilities. It is the absolute level of utility of each commodity
that will tell as which of the various commodities comes
next into the range of consumer’s preferences, even if the corresponding
want will then rapidlv saturate.
To conclude, we may say that, owing to a fundamental
property of human nature, there exists a very definite order
of priority in consumers’ wants. The less basic a want is,
the higher will be the number of wants that must be saturated
before it can be afforded consideration. This order of priority
is especially strong at low levels of income, where satisfaction
of some wants is a conditio sine qua non, even to the appearance
of all the others. But the order persists at a higher level
as well, where the process of decision becomes more complicated
only because the order itself may no longer be so obvious,
and needs first to be discovered. This takes me to my second
point. :
The second addition to demand theory that I should like
to make refers to the nature of the behaviour of human beings.
to] Pasinetti - pag. 63
034
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
Here, once more, traditional theory has taken up only a limited
case to study. It has always assumed that individuals know
their preferences perfectly and behave rationally. There is of
course a justification for such an approach in a static environment,
but no longer in a dynamic world. We do positively
know that human beings are not omniscient, and that the way
in which they come to know new situations is through experience.
They may know reasonably well the problems faced
already in the past; for they have had the opportunity of
trying and experimenting with different solutions. But if a
new situation arises, they have first to learn how to deal with
it, and the decisions they make the first time may not be the
best ones — they are tentative decisions in order to learn.
Now, if real per-capita income is continually rising, each consumer
enjoys an extra amount of income to spend in each
successive period, which indeed puts him in a new situation.
Especially when incomes become high, to pretend the consumer
makes the best decision — according to his preferences —
about the extra income he has just obtained is unreasonable.
He does not know his preferences at that high level of income,
because he never experienced them before — he has to learn
them. This is not all. As time goes on, the quality of old
goods may change, and the price structure may also change;
while old needs may be satisfied with better (superior) goods.
This means that the consumers’ learning activity is a process
required over the whole range of his preferences. As a conclusion,
I should like to propose here to enlarge our views of
consumers’ behaviour and to complete the traditional postulate
« the consumer is a rational being » with the more general one
« the consumer is a learning being ». The latter is more general
because it can be regarded as including the former as a
particular case: the case of a stationary economic system. (In
such a system technology and income have, by hypothesis,
always been constant through time, which means that all learning
activity has been completed already in the past. Since
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consumers had infinite opportunities of trial and error, they
must now know their preferences perfectly. The second postulate
thereby reduces to the first).
This is enough for our purposes. To synthesize these results
for the following analysis, we may simply state them in
three propositions:
a) at each level of per-capita income, the proportion of income
spent on any commodity is generally verv different
from one commodity to another:
h) as per-capita real income increases, each increment of demand
tends to concentrate on a particular group of commodities.
This group of commodities changes from one level
of income to another (it may be mainly food at very low
levels of income, clothes and again food at slightly higher
levels of income, houses, durables and services of various
kinds at further higher levels, etc.). In other words, as
.ncome increases, the tendency of the consumers is not to
.ncrease proportionally the consumption of already-bought
commodities, but rather to buy new goods and services or
also to satisfy old needs with different (better) goods:
~)
AW
there is no commodity for which any individual’s consumption
can be increased indefinitely. An upper saturation level
exists for all types of goods and services, although at different
levels of income: it may be reached sharply — in the
case of goods satisfying physiological needs — or only
through a slow and long process as income increases — in
the case of services yielding very sophisticated types of
satisfaction — but its attainment is eventually inevitable.
Moreover, demand for some particular goods (inferior goods)
may in fact decrease, after reaching saturation, if real income
persistently goes on growing.
If we want to represent these results graphically, by plotdng
the expenditure for each commodity as a function of real
10] Pasinetti - pag. 65
636
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
7
income, we should obtain families of curves of the following
shapes:
Der-capréa
expenditure
Se ration
level
r-capita
eal income
Fic
Curves of type (a) are likely to fit the cases of goods which
are absolutely necessary for physiological reasons (e.g. food),
and curves of type (b) are likely to fit almost all other cases.
Type (c) finally represents the typical behaviour of inferior
goods (1).
Of course, the relation between expenditure on each single:
commodity and real income, represented in fig. 1, is limited
by the two-dimensional character of the diagrams. The actual
time-path of each single expenditure will also depend on the
variation of the structure of prices. It is important to realize,
however, that the shapes of the relations represented in fig. 1
will remain unaffected.
7. The criterion for the choice of the hvpotheses
Before going on to considering the general dynamic model
to which the foregoing discussion has graduallv paved the wav.
(1) J. AITCHINSON and J.A.C. Brown (in The Log-normal Distribution,
Cambridge, 1957, chap. 12) suggest a function based on the log-normal
distribution, as a general function capable of fitting almost all the cases
of ENGEL curves. The shape of the function is of the type (b) represented
in fig. 1.
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it may be useful to face explictly a spontaneous question which
might be raised at this point. Why should the subject of economic
growth be approached in this way? The theoretical
scheme of the previous pages is, after all, full of simplifications
and rough approximations. Now, why should such simplifications
be made in preference to others? And in particular,
why should such simplifications be made in preference to those
that traditional economics has adopted so far?
This is a relevant question to ask because, as pointed out
already, any model does entail a simplification of reality and
it is of paramount importance to be clear about the criterion
according to which assumptions are made. One is bound to
make immediately some simplifications in the choice of the
variables and of the constants of any economic enquiry. For,
In economics, there are — rigorously speaking — no constants.
All quantities one may be dealing with are, in fact, variable.
But since some quantities are more variable than others, it is
commonly said that one should take as constants those quantities
which vary the least. This clearly is a reasonable criterion to
follow, provided that it is applied consistently. But there has
been difficulties, connected with the development of economic
thought.
In the last hundred years, economists have just happened
to be interested in the static characteristics of an economic
system, or at most in its short-term behaviour. And since in
the short run hardly any quantity can vary substantially, they
have tended to make the distinction between variables and
constants coincide with the distinction between unknowns and
data. Those economic magnitudes that were to be explained
funknowns) have also been taken as variables, and those
magnitudes that were to be accepted as given (data) have also
seen taken as constants.
Obviously, such an association did not matter very much
as long as it was restricted to a short-run analysis. But
when, recently, the attention of economists has shifted .uo dy-‘Io]
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638 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
FN
namic problems, some difficulties have began to arise. For,
some data — i.e., factors to be accepted as given from outside
economic analysis — are in fact much more variable in time
than some of the economic magnitudes taken as variables. In
such circumstances, the traditional association has made the
choice between variables and constants slowly undergo a serious
distortion. Some magnitudes have been kept as variables although
their variability is not important, and other magnitudes,
whose variability is an essential feature of the long-run growth
pattern, have been left out of the analysis altogether.
It is precisely to avoid this distortion that the theoretical
framework developed in the present work has been laid down,
straight from the beginning, with reference to the requirements
of a long-run dynamic analysis. The criterion for the choice
of the hypotheses is a consequence of this approach. I have
considered as typical variables those quantities which — independently
of whether they are to be explained or not — undergo
changes of an irreversible character as time goes on,
by incessantly moving on in the same direction. Although the
changes might be quite negligible within a single short period
of time, each period marks for these quantities a step forward,
in a slow but cumulative movement. On the other hand, I
have assumed as constants those quantities which, in the long
run, do not present any tendency (or for which there is no
reason to expect any tendency) to move in any direction.
These quantities, of course, may change quite a lot from
one period to another, but the point is that their changes are
temporary and reversible. Even if they do go for some time
in a particular direction, they cannot go on indefinitely, and
they are bound to come back to where they started.
The difference between these two types of quantities can be
immediately perceived if we consider a long period of time.
Compare, for example, the American economy in 1860 and
in 1960. It is quite possible and easy to claim that, within
this period. magnitudes like the average time-life of the equip-"10]
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ment or the rate of profit have undergone considerable variaons,
as shown, for example, by a coefficient of dispersion.
But the important point is that such a dispersion, if it has
taken place, has done so around a roughly constant trend.
The result is that, after a century of vicissitudes, these magnitudes
are in 1960 practically at the same level as they were
in 1860. The case of population, per-capita income, or composition
of demand, is radically different. These quantities may
have changed very little from one year to another but they
have always changed in the same direction. The cumulative
result, after a century, is enormous. Population has increased
five times, per-capita income has also quintupled, and total
consumption is mostly composed in 1960 of goods and services
that in 1860 did not even exist: in other words, the trends of
these variables have been irreversibly and persistently increasing,
and are going to persistently increase in the future.
To conclude, the distinction between variables and conslants
in the present analysis has been based on variability
chrough time. Therefore, it does not coincide, and must not
be confused with, the distinction between unknowns and data,
namely between quantities which are intended to be explained
and quantities which are accepted as given from outside economic
analysis. Accordingly, there are magnitudes — such
as population and technical progress — which are here taken
as given from outside economics and nevertheless are essential
variables. And there are other magnitudes, like the rate of
profit, which are to be explained by economic investigation,
and which nevertheless have been taken as constant over time.
10] Pasinetti - pag. bo-
b40 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CHAPTER V
A GENERAL MULTI-SECTOR DYNAMIC MODEL
1. The model
We are by now in a position to expound a very general
multi-sector dynamic model. After the foregoing analysis,
the exposition need not be long and we may proceed in a very
schematic way. The hypotheses which are made will be listed
here under two headings, referring to initial conditions and
to movements throuech time.
A. Iwitial conditions. — At the time when our analysis
begins, which we may denote as time zero, the system is in
equilibrium, i.e. there is full employment and full utilization
of existing productive capacity. The system is defined by:
a) a series of (n —- 1) stocks of capital:
K. (0). K,(0). ... K, (0) :
which are the result of past productive activity. For simplicity,
the assumption will be kept that capital goods require no capital
to be produced. Since the system is in equilibrium, the stocks
of productive capacity are exactly of the size required by
demand:
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b) a population X, (0), which is taken as an exogenous
magnitude. The flow of labour services which this population
can provide in each unit of time is equal to X,(¢) divided by
two coefficients — a(¢) and 3(¢) — standing for the proportion
of active to total population and the proportion of actual working
time to the total time composing the time unit respectively.
The subscript ¢ has been added to both coefficients because they
may be undergoing a long-run trend;
c) a series of 2(n - I) technical coefficients:
2,(0), «.. ay, 1(0), ... ay (0), ... a,
(0)
expressing the inputs of labour required in the unit of time
in combination with the appropriate stock of capital -
produce one physical unit of final commodity. There is also
a series of (»n - 1) technical coefficients T,, ..., T,_;, which can
roughly be interpreted as expressing the average life-time of
capital goods in each sector where capital goods are required;
d) a series of (n - I) demand coefficients (-):
a,,(0), ... a,_; qv.
expressing per-capita demand for each consumption good in
the unit of time. There is moreover another series of (n
demand coefficients for new investments:
ul,
ee A+
“po
(') It may perhaps be useful to point out explicitly that although the
technical coefficients and the per-capita demand coefficients occupy a symmetrical
place in the system, they are not of the same nature. Technical
coefficients represent sectoral concepts. Each of them is given by the
state of technology in each particular sector. Per-capita demand coefficients
represent macro-economic concepts. Each of them is an average
taken all over the svstem
10] Pasinetti - pag.
71
5642
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
These coefficients, however, cannot be taken as exogenously
given. How they are determined will be discussed in section 4.
At time zero, since the system is in equilibrium, all demand
coefficients considered together are such as to require the full
utilization both of the labour force and of the existing productive
capacity.
B. Movements through time. — As time goes on, the following
movements take place:
a) population increases at a steady percentage rate g,
so that
(V.I)
X,#)=X,(0) e¥ ;
b) productivity changes at a particular percentage rate of
change in each sector. It will be assumed that these rates of
change (p;) are different from one sector to another but that
they are steady through time in each sector. This means that
(V.2) a(t) =a,;(0) €
at ; (2) =a, (0) eo
1=T. 2, ... (n-1).
Most p;'s and Pr, 8 (j=1, 2, ... n- 1) are positive, but a few
of them (referring to those sectors where the exhaustion of
natural resources is particularly heavy) might be negative:
c) per-capita demand changes at a particular percentage
rate of change for each commodity. We shall denote these rates
of changes as r, ({=1, 2, ... n-1). The r;s are not constant
over time; they change as a result of a very complex process,
as has been explained in the previous chapter. The #,’s as such
are not exogenous magnitudes. in our analvsis. What has been
10] Pasinetti - pag. 72
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v4.3
assumed as an exogenous datum is the set of consumers’ preferences.
Thus, as time goes on, the shape of these consumers’
preferences and the movements of technical coefficients postulated
by (V.2' determine the time movements of the ve
may write
V.3)
cdg eee @
a
dt [an ..
a.
a, IB:
a
Ah
c
where the fs depend on the shape of the consumers’ preferences,
which are defined in such a way as to satisfy propositions
a), b), and c) of chapter IV section 6, graphically represented
in fig. 1. The technical coefficients that appear in (V.3) induence
the 7;’s through the medium of two channels: the level
and the rate of change of real per-capita incomes, and the
variation of the structure of prices.
Here, for purposes of simplicity and symmetry, the assumption
will be made that the movements of per-capita demand
may approximately be broken down (if we represent them on
a logarithmic diagram) into stretches of straight lines. In other
words, we assume that time can be divided into finite stretches
of length s (larger than the unit of time we are adopting)
within which, for each commodity i, the percentage rate of
change of demand 7, remains constant. Then passing from
one stretch of time of length s to the next, r; changes, remainng
then constant again for another stretch of time s And
30 on.
Therefore, bv defining now a new variable v as
101 Pasinetti - pag. 73
+4
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2é&
where pis the greatest integer that multiplied by s and subtracted
from ¢ leaves a positive remainder (8), the movements
through time of the demand coefficients may be written as (?)
‘V.q) a. (D=a. (t-0e®. 1=1,2, ... (n-1),
where each 7; is an f; function of the technical coefficients and
now also of (£-0). In order not to complicate the notation
excessively, this functional dependence is not explicitly written
in (V.4), nor will it be written hereafter, but it must always
be taken as understood.
Of course, the same notation in terms of 8 can be used also
in formulations (V.1) and (V.2), and this will be done, in the
following analysis, any time it is required by reasons of symmetry.
2. A few restrictions
First of all, it may be useful to impose a few restrictions on
the coefficients of our system. These restrictions do not imply
anything new and in fact have always been implicit in our previous
analysis. We only make them explicit now for the sake
of rigour. The reason for these restrictions is that our mathematical
terms are very general and their generality needs to be
limited to the range in which it has economic sense.
(?) As explained in footnote (1), demand coefficients represent average
per-capita consumption. No discussion is carried out here about possible
complications arising from a changing distribution of income among individuals
or from a changing composition of the population as regards sex
and age. Evidently, the simplest way of interpreting our analysis is to
suppose that both these features remain invariant in time. However, even
if they should change, their changes can be neither quick nor big. In any
case, they would not affect the conclusions of the following analysis because
their effect would simply be to anticipate or postpone turning points.
without altering the nature of the trends through time
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To begin with, all technical coefficients, whether referring
io the production of consumption goods or of investment goods,
and all demand coefficients of consumption goods can never
be negative. In mathematical terms:
2...
V.5)
A
which have a quite straightforward meaning. For, a negative
amount of labour or a negative amount of consumption makes
20 economic sense.
On the other hand, the coefficients of demand for net investment
can become negative, although only to a certain extent,
ie. up to that point at which gross investment is still nonnegative.
The restriction applies with absolute certainty to
fotal gross investment. It need not necessarily apply to gross
investment in each single sector, if capital is flexible enough
to allow some transfers of capital from one sector to another
when needed. In other words, taking first the most restrictive
of all cases,
V.6)
T a, (?) + Ax (1) =z 0,
cee (NH —
1
Should a situation arise in which the equilibrium conditions
(which will be discussed in a moment) would require some of
nequalities (V.6) to be reversed, then some more information
about existing capital stocks is needed in order to tell what will
happen. If, in those sectors where inequalities (V.6) are required
to be reversed, capital is very specialized and cannot
oe transferred anywhere else, then idle capacity (i.e. a disconlinuity
in the model) will appear. If, on the other hand
10] Pasinetti - pag. 75
646 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
capital is perfectly adaptable and can be transferred to other
processes of production, then even negative gross investments
can take place in those sectors.
In no case, however, as said above, can total gross investment
become negative. In other words. the inequality
(V.7)
2 au) + + Ain | 2:0
must always and in any case be satisfied.
3. The flows of the svstem
Within each single period of time (finite or infinitesimal as
it may be) there are flows of commodities from the production
processes to the final sector and flows of labour services from
the final sector to the production processes. These flows have
been examined already in the short-run inquiry of chapter II,
and need not be further discussed here. Within each single
period -of time, the structure of the system is represented by
systems (II.g) and (II.13), and we may consider these two
systems as rewritten here. The only difference is that we must
now add a time subscript to each single coefficient and to each
single variable, so that we obtain a pair of those systems for
each unit of time we are considering. And since a very specific
set of hypotheses has been made about how the coefficients
of production and of demand change as time goes on, we
shall now at last be able to look into the whole dynamics of
the system. In other words, after finding the solutions of the
equations in each particular period of time, we shall be in a
position to look at the movements of these solutions through
time.
As explained in chapter II, the rate of profit (nr) which ap-10]
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pears in system (II.13) will be taken as a constant. This does
not mean that the rate of profit is an exogenous magnitude. It
only means that it does not depend on the structure of the
sconomic system, as represented by (II.g) and (II.13), and
must be explained with a separate economic theory. Now,
since I have had the opportunity of dealing with such a theory
already in an independent publication of mine (Rate of Profit
and Income Distribution in relation to the Rate of Economic
Growth in « The Review of Economic Studies », October 1962),
[ shall here simply refer the reader to that publication (*). The
results of that analysis which are relevant for our present purposes
are that in an economic system where the over-all rate
of capital accumulation remains roughly steady over time, the
rate of profit is bound to remain steady as well.
A further point is that the assumptions of section 1 do not
yet provide us with the dynamics of all coefficients that appear
‘n systems (II.g) and (II.13). The movements through time
of one series of coefficients — the demand coefficients for new
nvestments — have not been specified. This series of structural
coefficients is the only one that affects the stocks of the system
(i.e. productive capacity in each sector). It cannot therefore
oe taken as fixed from outside the system. These coefficients
must be such as to be compatible with the process of growth
and will come out themselves to be determined as a part of the
equilibrium conditions
3} To those readers who still find it difficult to follow the logic of the
long-run-equilibrium growth models I would suggest a device. We have
normally been used to think in terms of a free market economy and then
‘0 extend the results to the case of a centrally planned economy. Here,
it turns out ot be much more helpful to reverse the procedure and to
think first in terms of a centrally planned economy. For, in this case, the
relationship (an equilibrium relationship) between the long-run rate of profit
and the natural rate of growth emerges immediately. The corresponding
relationship for a free market economv will then appear much easier to
grasp
10] Pasinetti - pag. 77
648
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
1. The conditions for a dynamic equilibrium
According to our assumptions, when our analysis begins
(time zero), the economic system we are considering is in equilibrium
— there is full employment and full utilization of productive
capacity. We know already that, if this equilibrium
situation is to be kept, two types of conditions must be satisfied.
First of all, since both population and technology are changing,
the system must continually enlarge its productive capacity
so as to keep up both with the increasing demand and with
the increasing labour force. This means that, in each sector,
a very definite relation must be satisfied between new investment
and the rate of change of the corresponding final demand
for consumption goods. The problem has been discussed already
in section 2 of chapter III. By following here exactly the
same procedure. the mentioned relations emerge as
(V.8) A, ,(t)=(g+r;) a,,(t), 1=1. 2. ... (n-1)
which represent the capital accumulation conditions for keeping
full employment over time.
After explicitly inserting the dynamic movements of demand
and of population that have been postulated, the (V.8) mav also
be written as
(V.9) a, (BO =(g+7) a,(t-0 gard
or, if we refer them to total net investment in each sector (instead
of referring them to per-capita investment), they may also
be written as
(V.10) Xpa)=(g+7) X, (t- 0) a,(t-0) 9+"
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These relations define equilibrium investment in each single
sector, and therefore their sum defines total equilibrium investnents
in the system as a whole. As can be clearly seen, equilibrium
investments, in physical terms, are exclusively deter
mined by the expansion of demand.
Yet, as we know, the actual undertaking of all these investments
only provides the required productive capacity, i.e. potential
full employment. For the system to reach full employment,
a further macro-economic condition about total effective
demand must also be satisfied.
In mathematical terms, all this is expressed by the fact
‘hat conditions (V.g) only come to complete the two systems
of equations (II.g) and (II.13), by defining at any point of
ime the series of coefficients (demand coefficients for new investments)
which was still missing. But the two systems of
equations, thus completed, are of the linear and homogeneous
type, and their coefficient matrix must be singular if they are
‘o yield non-trivial solutions. This condition of singularity,
which was expressed by (II.20) with reference to a particular
period of time, must now be kept satisfied over time. After
substituting (V.q) into (II.20) and after explicitly expressing
‘he movements of the coefficients. we obtain
V.11)
1,,(-0) a, (1-9) eri—ed.
+247.) apai-0) a, (1-9) erro
Pl
which finally represents, under our present hypotheses, the
effective demand condition for keeping full emplovment over
time. …
This expression, as pointed out earlier, refers to the ecoaomic
system as a whole. The effect of substituting into it
sxpressions (V.q) is that now (V.11), besides stating the general
[10] Pasinett: - pag. 79
550 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
condition that total expenditure in the system as a whole must
be equal to potential total income, also specifies a very definite
division of this total expenditure between new investments,
replacements and total consumption. Replacements are determined
by the rate of wear and tear, and new investments by the
rate of change of demand for consumption goods. The sum
of these three types of demand — as a proportion of total
potential gross income — is required to be equal to unity.
The right hand side of (V.11) represents gross equilibrium
investment expressed as a proportion of gross potential income
and the left hand side the proportion of gross income which is
not spent on consumption goods. Let me point out explicitly
(since a misunderstanding is possible) that the left hand side
of (V.11) may, therefore, be regarded as expressing the saving
ratio, but nothing more than that. It must not be interpreted,
for example, as a propensity to save. No behavioural relation
about savings has been introduced in the present analysis. We
have simply been looking for the conditions that must be satisfied
to keep the system in equilibrium, independently of how
individuals behave and of how institutions induce them to
behave. And our results have been that — whatever individual
behaviour may be — the equilibrium conditions are two. First
of all, enough productive capacity must be provided, in each
sector, to keep up with the expanding demand — condition
(V.10). And secondly, given total equilibrium investment required
by all the (V.10), the system as a whole must spend on
consumption goods the whole remaining part of gross potential
income — condition (V.1I1).
Total equilibrium investment — i.e. the right hand side
of (V.11) — may of course be given the macro-economic interpretation
we have already discussed. We may say that, in
macro-economic terms, (V.11) expresses HARROD’s equation (4).
(*) See footnote (1) in chapter III.
10] Pasinetti - pag. 80
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However, we may notice that we can no longer stop our analysis
at this point in the present model. The fulfilment of
macro-economic relation (V.11) is now no longer independent
of time and of the structural dynamics which is going on
behind all coefficients.
2. Relevance of a disaggregated formulation
We may now begin to assess the relevance of the disaggregated
type of analysis which has been developed in the previous
sages. Condition (V.11) — as said above — states an important
macro-economic conclusion, which is the same that has emerged
from all macro-dynamic models. But this conclusion now emerges
as expressing only what appears on the surface of the whole
problem of a dynamic equilibrium. Relation (V.11) also implies
that — just in order that it may be satisfied as an overall conition
— a very complex process of structural change must
70 on behind all the macro-economic magnitudes. This structural
process is in fact what technical progress means in a
modern society.
We may note that to have shown the existence, and now
lo make possible the analysis, of this process of structural dynamics
is one of the main innovations of the model developed
in this chapter. The two simpler models, discussed in chapter
III and in the first part of chapter IV, could not deal with
these problems at all. As the reader will remember, both
‘he capital accumulation conditions and the effective demand
condition, in those models, have been assumed to be independent
of time and of the stage of development the system has
reached, so that — once defined at a certain point of time —
they remained the same for all the time. This meant giving
up the possibility of investigating any type of structural dynamics,
and at the same time it meant frustrating the purpose
‘tself of a disaggregated formulation.
10] Pusinetti - pag. 81
652
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
The whole picture becomes radically different in the present
model, where the fulfilment of the equilibrium conditions
at time zero is no longer the end but just the beginning
of the whole story. These conditions cannot remain the same
as time goes on; because technical progress — whether uniform
over the whole economy or not — causes each single component
of the summations in (V.r1) to change. This means that the
way in which each one of conditions (V.8) and (V.11) are fulfilled
must be continually different as time goes on. The size of
the various sectors that may satisfy those conditions in a given
period of time is necessarily different from the size of the same
sectors which ensured their fulfilment in the previous period,
and again is necessarily different from the size of the same sec-‘ors
which can ensure their fulfilment in the following period.
But the discussion of the previous chapter now allows us
to go far beyond these general remarks. By hypothesis, almost
all technical coefficients are decreasing in time. This means
that, unless the demand coefficients increase in the same proportion,
condition (V.r1) is bound to become under-satisfied
as time goes on. But we know already that no demand coefficient
can increase indefinitely, because eventually all demand
coefficients reach saturation level. Therefore we must conclude
that condition (V.11), as it stands, inevitably manifests a tendency
to become under-satisfied, i.e. to generate unemployment,
as time goes on.
Fortunately there are two factors, operating in the long
run, which come to counterbalance the above mentioned tendency.
These two factors must now be introduced into the
model. One of them is the same one which causes the whole
trouble: technical progress. So far in this chapter technical
progress has been considered in the form of increases of productivity.
But it has been pointed out earlier that technical
progress also takes the form of introducing new goods. Our
model must therefore be completed now by opening it to the
possibility of the introduction of new sectors. This can be
[10] Pasinetti - pag. 82
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done by taking the number of sectors as running no longer
from 1 to a fixed (»- 1), but from 1 to let us say x(¢), where
x(t) denotes a number which increases as an effect of techncal
progress. Thus condition (V.11) must be written:
'V.12)
1-3) a, (1-6) e (rie |
on) App (1-0) a; (1-0) é (rise) Bz az, Ble,
All this means that the effects of technical progress on the effective
demand condition are twofold. On the one hand, it
brings about an average decrease through time of the a,a;,’s
referring to the commodities so far produced, and on the other
it keeps on adding new coefficients (referring to new commodities).
The second tendency may succeed in counter-balancing
the first.
However, if this counter-balance still does not come about,
there is a second way in which condition (V.12) may be kept
satisfied in time. That is through a long-run diminishing trend
of the right hand side of the equation; namely of the coefficients
x and 3. This means a decreasing trend of the working time
(i.e. an increasing trend of leisure time), to be achieved by a
decrease either of the proportion of active to total population
or of the length of the working week.
All these conclusions may seem, after all, to boil down to
the common-sense proposition that technical progress gives society
a choice between producing more or new goods, and
enjoying more leisure. Our analytical formulation, however,
just because of its macro-economic implications, reveals something
more than that; it evinces the fixed framework within
which the choice has to be made. It shows that there is a third
slement in the problem — the requirement of keeping full
employment — which restricts the choice to those combinations
‘10] Pasinetti - pag. 83
654 PONTIFICIAE ACADEMIAE SCTENTIARVM SCRIPTA VARIA - 2¢
of goods and leisure which, in terms of time required, add up to
a fixed amount determined by the existing working population
and technical knowledge. To express the same thing in
another way, we might say that the choice between products
and leisure is not merely a possibility, but a necessity, if full
employment is to be maintained. (There does not exist the
alternative of not choosing). Technical progress, which characterises
our societies and which brings with it that choice,
does not come — so to speak — under the form of a gift which,
by being always susceptible of being refused, only could add
to, and never diminish, the pre-existing wealth. It comes
under the form of a flow, which cannot be stopped and has to
be continually channelled in new directions, which themselves
have to be discovered anew because the old ones saturate. This
entails an ever-standing problem of utilization, under a fixed
restriction represented by condition (V.12); a problem for
which the failure to find solutions may cause damage to the
previous situation.
Before closing our comments on condition (V.12), we may
add that there is a third way of keeping equilibrium over time.
This would be io regard — as indeed we have done — the
variations of the r/s as determined by the evolution of consumers’
preferences, but to have among them at least some #,’s
which are not dependent on such preferences and which are
flexible enough and capable of being influenced in such a
way as to keep condition (V.12) satisfied. This is not a fanciful
remark; it is in fact an important one, on practical grounds.
It means that, if there is an external Agent or institutional
Organization that is interested in keeping full employment, it
is possible for this Agent or Organization — when any other
mechanism which might have been put into operation fails — to
act and to attain the aim of full employment simply by inflating
total demand.
10] Pasinetti - pag. 84
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5. The dynamic movements of physical quantities and of relafive
brices
When conditions (V.8), (V.12) are satisfied, each of the
‘wo linear and homogeneous systems of equations representing
the flows of the economy — the (Il.g) and (II.13) — yields
solutions for all its unknowns but one, which can be arbitrarily
fixed. But since, under our present set of hypotheses listed
in section 1, all coefficients and unknowns are dated, the solu-‘ions
no longer take the form of single values but that of movements
through time. It is evidently one of these movements
which, in each of the two systems, can be arbitrarily fixed. In
the case of system (II.g) there is already one of the X’s — namely
X,: population — whose movement has been accepted as
siven from outside economic analysis. On the other hand, in
the case of system (II.13), no one of the prices is given so
that the structural dynamics of the system only determines the
movements of relative prices. As a matter of convenience, we
may take the wage rate as given through time ( W), so that the
dynamic movements of physical quantities and of relative prices
emerge as follows:
QE}
V.13)
&
V.14)
7
3
=Bet ..
LOA
ex
. x(t,
10] Pasinetti - pag. 85
656
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
Ȃ
where A, B, C are constant which stand for the initial conditions,
namely
A=a,(t-0) X,(t- 6),
B = a,;(t = 6) W,
C=a, (t-0) W.
The economic meaning of (V.13), (V.14) at a given point of
time has been discussed already on the occasion of our shortrun
analysis of chapter II (sections 5 and 6), and there is no
need for repetition here. But besides the determination of quantities
and prices at a given point of time, the (V.13), (V.14)
now bring out their movements through time as well.
As can be seen, each physical quantity follows a time-path
of its own in time, expanding at a particular rate (g+7;) which
is the sum of two rates of increase: the rate of growth of population
and the rate of change of per-capita demand. The first
of these two rates is the same for all goods, but the second is
different from one commodity to another. Therefore, unless
per-capita demand remains constant in time (i.e. unless there
is no technical progress) the whole production structure in physical
terms is changing as time goes on. This means that, while
growing, the system is continually changing the proportions in
which it produces the various commodities.
The (V.14) show moreover that each relative price is also
changing in time in its own way. With the convention of adopting
W, the wage rate, as numeraire, each price is decreasing at
a rate resulting from a weighted average of the pace at which
technical progress is going on in the sector to which it refers,
and in the sector which produces capital goods for it. Since
all these rates are also different from one another, the whole
price structure is changing as time goes on
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657
7. The dynamic movements of the other variables
Following the procedure which has been used in chapter III,
‘he movements through time of other magnitudes of economic
nterest (like sectoral production at current prices, sectoral empoyment,
capital-output ratios, etc.) can be derived as simple
~orollaries from (V.13) and (V.14). These other economic
magnitudes depend both on (V.13) and on (IV.14), i.e. both
on technology and on demand. Therefore their dynamic movements
result from a composition of the movements through
ime of physical quantities and of relative prices. Let us con:
sider them in detail.
1. The time-paths of the production of each commodity
evaluated at current prices [the V,(#) and V )] can evidently
be obtained by multiplying each of the © * by each of the
'V.14). The resul’ is
V.I5,
rs
g+tr-e J
I~ 5
+ Uwhere
the constants D and G stand for the initial conditions
namely
a,
vu) a; ke
d; An)
u
VW A, QL
J)
W A.
J
10} Pasinetti - pag. 87
658 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
As can be seen, the V's and the V, ’s change in time for two
reasons: because the physical quantities are changing and because
prices are changing. Having taken the wage rate as
numeraire, the (V.15) show that each rate of change of these
variables comes out as an algebraic sum of the rate of population
growth, the rate of change of per-capita demand and the
rate of increase of productivity (the latter with the negative
sien).
2. Similarly, the dynamic movements of employment in the
various sectors emerge as follows:
(V.16)
Ef) = M elg+riept
Ex?) =N (= +e+r) el@tri-e,,)8 ,
1=1, 2, … x(t)).
where (3):
M=a,;(t co 6) a, (2 - 0) — X(t co 0),
N=a, (t- 0a, (t-0—X,(r 8
In words, employment in each sector ¢ moves through time at
a rate of change equal to the rate of population growth plus the
rate of increase in per-capita demand for commodity i, minus
the rate of increase of productivity in the sector.
(°) The E;(t) and E,;(t) have been expressed here in terms of men per
anit of time. For example, if the unit of time is one vear. thev would
be expressed in terms of man-years; so that:
S[E;(t)- E,. +)1
= o and)
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659
Here we have new problems arising because the E;’s and
E,; 's although representing flow-variables (services from labour
in the unit of time) come from, and are inseparably linked to,
a stock-variable (the labour force) which may not be perfectly
mobile. As appears from (V.16), if population is constant
(g=0) employment in each sector ¢ increases or decreases
through time according to whether r;>p; or r;<p;. This means
that, on the average, half of the sectors are offering jobs and
half of the sectors are dismissing workers, as time goes on.
Clearly, this may be a very serious state of affairs — especially
In a very progressive system, i.e. in a system with very high
2s — if labour is highly specialized and is not susceptible of
being transferred from one sector to another except at the ex
pense of heavy losses in productivity.
Fortunately, the natural movements of population come in
here to help in the right direction and in two senses. First of
all, the natural process of people ageing permits a redistribution
sf employment among sectors by addressing young workers
towards expanding sectors and by not replacing retired people
in the contracting sectors. This may be a slow process, but
it is one which is going on even when population is stationary.
Secondly, when population is growing, its rate of increase is
a net positive addition to the rate of change of demand (and
herefore of employment) in all sectors. It follows that, in
absolute terms, only those sectors will actually lose employment
where the rate of increase in productivity is so high and the
rate of increase in per-capita demand so low that the difference
between the two is not only negative, but negative to such an
extent as to make the sum (g+7;-g;) less than zero, or rather
the sum (g+7;-g;+¢) less than zero, where @ stands for
the rate of people’s retirement from working activities (6)
(®) At this point restrictions (V.6), discussed in section 2, may be rewritten
with reference both to the stocks of capital and to employment of
abour in each single sector. in the following wavy:
V
‘continued on following page)
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560 PONTIFICIAE ACADÉMIAE SCIENTIARVM SCRIPTA VARIA - 28
Agriculture seems to be one of the most typical sectors of this
kind.
To sum up, we may say that, as time goes on, the whole
structure of employment changes, in the sense that the proportions
in which total employment is distributed among the different
sectors of the system are changing. However, actual
dismissal of workers from some sectors will take place only in
those cases where the increase in population is not enough to
counter-balance the effect of a decreasing per-capita demand
or of an increasing productivity, or of the algebraic sum of the
two. This means that the higher the rate of population growth
(when the capital accumulation needed to absorb it can easily
be afforded), the easier it is for an economic system to adapt
itself to a given structural process of change of employment.
3. The movements through time of the sectoral capitaloutput
ratios also follow straightforwardly from the previous
analysis, namely from expressions (III.14) and from the dynamic
movements (V.13) and (V.14). Here they can perhaps
be better examined by considering their reciprocals. which
emerge as follows:
WV. 17)
1 £ An (0) (0, ~0pt
x (NT a, (0)
1=1, 2, ... Xx(f).
(V.2n)
£E+7;—0:+5>0. 1=1, 2, ... x(D)
We may say that, when all (V.1n)-(V.2n) are satisfied, the model retains all
its properties. But if some of these inequalities should be reversed, then in
order to tell what will happen one needs more information about the degree
of flexibility of capital — in the case of V.1n) — and about the degree of
mobility of labour — in the case of (V.an). If no mobility of capital or
Of labour is possible between one sector and the others, then some idle
capacity — in the case of (V.1n) — and some technological unemployment
in the case of (V.2n) — are bound to appear
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The first point to make about the (V.17) is that if T and =
remain constant, the sectoral capital-output ratios exclusively
depend on technology, both at a certain point of time and in
their movements through time. As time goes on, each of them
increases, decreases or remains constant according to whether
the rate of increase of productivity in the sector considered is
higher than, lower than, or equal to, the rate of increase of
productivity in the corresponding capital goods sector. The
reader will recognize in these three possibilities the cases which
in dynamic economics (7) are commonly known as the cases of
labour-saving, capital-saving, and neutral technical progress.
We can only confirm here that, with reference to the sectoral
~apital-output ratios, this classification is perfectly justified, because
the variations can indeed be traced back to a particular
type of change in technical knowledge.
4. But let us now consider the aggregate capital-output ratio.
Again, it becomes convenient to look at its reciprocal (the
output-capital ratio). By substituting (V.13)-(V.14) into
I11.15) and by calling » the over-all average rate of growth
of per-capita demand (i.e. the weighted average of the rates of
increase of total demand in each single sector), the over-all
capital-output ratio emerges as
I i \
'V.18) n= tmred j=Faip(t-0)
a, (1-0) e rime) b
Sa. (-0) a, -0) e (Fe,
Now, the most remarkable difference that ¥(¢) exhibits with
respect to each of the «,(f) is that it depends not only on
technology, but also on demand. As can be seen by a simple
(') These definitions are due to R.F. Harrop who foreshadowed them
already in his review of Joan RoBINSON’s, Essays in the Theory of Employment,
« The Economic Journal », 1937, pp. 328-9, and finally formulated
them in Towards a Dvnamic Economics. London, 1048, Pp. 22-24
ro) Pasinetti - pag. o1
662 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
comparison, expressions (V.17) only contain technical coefficients
and their rates of change over time; while expression
(V.18) contains, besides these coefficients and their rates of
change, also the demand coefficients and their rates of change
over time.
There is one immediate conclusion that may be drawn. It
is not permissible to talk of neutral, capital-saving and laboursaving
technical progress merely on the basis of changes in
the over-all capital-output ratio, because technical progress
is only one of the two factors on which the over-all capitaloutput
ratio depends. It would be quite possible, for example,
to envisage a case in which technical progress is capital-saving
in each single sector of the economy and nevertheless the
aggregate capital-output ratio remains constant or even increases
as time goes on, simply because per-capita demand
is expanding in the direction of highly capital-intensive commodities;
and vice versa. This also means that all recent
discussions on the factors affecting the aggregate capitaloutput
ratio, discussions which have stemmed from macroeconomic
models and have focussed only on technology,
have in fact missed one half of the problem. As emerges from
the foregoing analysis, any explanation of the movement in
time of the aggregate capital-output ratio cannot be correct,
if it does not consider both sides of the problem: the demand
side as well as the technical side.
5. A few final remarks may be added about the dynamics
of the aggregate variables normally used in macro-economic
Investigations (i.e. national income, total capital, consumption,
investment, etc.). From the (V. 13)-(V-14) it follows that all
these aggregate magnitudes move in time in a very composite
way. Each of them results from a sum of physical quantities,
whose proportions are changing, multiplied by prices which
also change as time goes by. Therefore, all of them have a well
defined meaning at any specific point of time. in relation to the
10] Pasinetti - pag. 02
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663
technical and demand conditions prevailing at that point. But
owing to the simultaneous change of the two component structures
(physical quantities and relative prices), their comparisons
through time entail the so-called « index-number problem ».
The difficulties arising from this problem may be negligible
when comparisons refer to short periods of time, but they increase
more and more as the time elapsing between the aggresate
magnitudes to be compared becomes longer and longer.
8. Short-run flexibilitic
Before closing the chapter, it may be useful to hint brief,
at short-run possibilities of adjustments. If displaced from positions
of long-run equilibrium, the system possesses in the
short run, some important possibilities of flexible adiustments,
owing to the parameters «, 3 and T.
The values of x, 3, T which enter the model represent the
long-run normal values of these parameters, which may be
constant or showing mild trends in time. But in the short-run
these parameters can be easily and widely influenced, thereby
affecting the size of the stocks that the svstem can utilize. This
means that productive capacities and labour never represent
absolutely rigid concepts, at least in the same way as the
physical equipment and the population existing at any certain
lime do. Between these latter physical concepts and those parts
of them which are relevant to the productive process, the men-‘ioned
parameters provide a sort of flexible cushion. Tempo:
rary variations of 3 and T mean temporary increases in the
productive capacities of existing physical plants by utilizing
them more intensively or by keeping in operation some machines
which were due to be scrapped. Similarly, variations of {3
and « mean increases of working-hours out of a given population.
obtained by lengthening the working week or by
‘10] Pasinetti - pag. 93
564
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
increasing the proportion of active population (more working
women or later retirement ages).
These flexibilities evidently represent very important factors
in allowing the system the possibility of adjustment and of
keeping stability in the short run, in face of disturbances of
various kinds.
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_11APTER
[HE EMPIRICAL SIGNIFICANCE OF THE MODEL
… The relation of sectoral dynamic analysis to input-output
analvsis
The model which has been developed in the previous pages
has far-reaching theoretical implications and also straightforward
empirical applications. Leaving its theoretical implicalions
aside for the time being, we may concentrate in this chapter
on its empirical significance. At the same time, the opportunity
will be taken of completing the model with reference
to inter-industry relations at a given point of time. It will
be remembered that, in Chapter II, all inter-industry connections
and intermediate commodities at a given point of time
were deliberately left aside, because our aim was to arrive at
a dynamic investigation as soon as possible. We may now go
back for a moment to that stage. Fortunately, there is no need
to develop here any model with intermediate commodities; for
such models have been extensively developed already in the
economic literature, especially during the past twenty years.
Hence our task can be limited to showing how they relate to
the previous analysis.
There are in particular two theoretical schemata, recently
presented, which may be considered as the logical static counterpart
of the previous dynamic analysis. They are WassILY
L_EONTIEF’s input-output model, and PIERO SRAFFA’s produc-10]
Pasinetti - pag. 95
666 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
tion system (!). The latter perhaps corresponds better than
the former to the approach taken in the present work, but the
former has had in practice wider empirical applications. Therefore,
it will be convenient here to take LEONTIEF’s system as
the static term of our comparisons.
First of all, let me point out the similarity of approach,
from an empirical point of view, of the previous dynamic model
and the static input-output model. Both models share the
characteristic of being built on coefficients which represent
actual outcomes and which can, therefore, at least in principle,
be given an empirical content simply by recording the actual
performance of the economic system. Of course, the technical
and the consumption coefficients, by which this actual performance
is represented, come from a choice, made from among a
larger set of possibilities. But all the alternative possibilities
that might have, but have not, been chosen have become irrelevant.
The coefficients that appear both in the input-output
and in the present model must, therefore, be interpreted as
simply representing those real quantities which can actually be
observed.
Let us notice, moreover that LEONTIEF’s system and the
present model also coincide in the way they look at the final
sector of the economy — the last column of the coefficient
matrices are the same in the two systems (with the only difference
that, in the present model, consumption goods and
investment goods are listed separately). However, they differ
profoundly in the way they consider the production processes.
The same production structure of the economy is looked at
from two different points of view — one is very close to it and
to what is immediately observable; the other is placed much
further away, at the final stage of the consumption and investment
goods. The T.LEONTIEF approach can certainly be more
(') Wassiry W. LEONTIEF, The Structure of American Economy, 1919-1939,
New York, 1941 and 1951; PIERO SRAFFA. Production of Commodities
by means of Commodities. Cambridge. 1060
[10] Pasinetti - pag. 06
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667
immediately grasped. One of the things by which one is most
‘mpressed, when looking at the real transactions which take
place in an economic system, is the great number of interrelations
among productive units. One’s first instinct is, therefore,
to inquire into these inter-industry connections and try to reproduce
them analytically. This is the idea which already underlay
QUESNAY’s tableau economique and which has been developed
and given a full empirical content by LEONTIEF.
À different approach is taken in the pr>sent model. Not
« industries », in the input-output sense, but « sectors » are
taken as the basis of the whole investigation. And sectors are
defined in such a wav as to be vertically integrated. All interrelations
which can be observed in the real world are looked
at as parts of a process which has not yet come to an end.
Any process reaches its completion only when the product
which comes out is a final commodity (consumption or investment
goods). A vertically integrated sector is, therefore, from
an inter-industry point of view, a very complex one as it goes
through and through the whole intricate inter-industry connections.
However, from the point of view of the homogeneity
of the inputs, it becomes a very simple one, as it eliminates
all intermediate goods and resolves each final commodity into
its ultimate constituent elements: a (flow) quantity of labour
and a (stock) quantity of capital. It may be interesting to
recall that the procedure has already been used by LEoNn WAL-RAS
in his Elements of Pure Economics. although in a more
rudimentary way (%).
At a given point in time, between the two ways of looking
at the economic system there is really no logical difference.
Both models represent the same thing, looked at in a different
way. The difference, in other words, lies only in the classifica-‘ion,
and we can pass from the one to the other simply bv an
(?) See p. 241 of the English edition of Léon WaLras’s Elements d’économie
politique pure, translated. collated and edited bv W. Jarrè, Homevood
(Ill.), 1053.
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568 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
algebraical re-arrangement, corresponding to a process of solving
a system of linear equations: the production coefficients of
one model turn out to be a linear combination of the production
coefficients of the other.
This can be shown immediately, if goods are expressed in
physical terms. (A further similar algebraical re-arrangement
would then be needed for the investment goods if they are to
be expressed in terms of physical capacities). Thus if we take
an input-output system, we must state consumption goods
industries and investment goods industries separately. We can
then isolate the inter-industry transactions by opening the
system with respect to the final sector. This means that we
take as given the final demands, common to both LEONTIEF’S
and the present model, and drop from the system the last row,
representing the inputs of the original factor (labour) into each
industry.
We obtain:
(VII)
Tg
Corey
“Cay
tu
U;
U,—1
where the c;;’s stand for the inter-industry technical coefficients,
the Z;s for the productions of intermediate commodities, and
the U;s for the final demands (i,i=1, 2, ..., R- 1). Bv solving
(VI.I), we arrive at
(V1.2,
I Comin
-T
TJ.
U
U,—1 |
10] Pasinetti - pag. 08
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where the superscript - 1 indicates the operation of matrix
inversion. Now each column of the inverted matrix represents
the amounts of all intermediate goods which have gone into
one unit of final commodity. This means that by multiplying
each column of the inverted matrix by the row of the inputoutput
labour coefficients which has been excluded from (VI.1),
we arrive at the labour coefficients of the vertically integrated
system.
[n algebraic terms:
1
where the mark ’ denotes the operation of matrix transposition.
Equations (VI.3) now give the algebraic relation, existing in a
ziven period of time, between the labour coefficients of an
input-output model and the labour coefficients of the vertically
ntegrated sectors used in the present dynamic model. The
ones may be obtained from the others — as the (VI.3) now
directly show — by a straight-forward algebraical operation.
>. Fitting empirical data into the moael
The algebraic relation which has just been obtained and
which links the input-output technical coefficients with the techaical
coefficients of our analysis finds an immediate applica:
ro] Pasinetti - pag. 99
670 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
tion in the problem of giving the various symbols so far used
an empirical content. Of course, since each coefficient or variable
appearing in our system refers to actually observable
magnitudes, the problem of fitting empirical data into the
model is in principle very straightforward, in the same way as
it is for an input-output model. But the criterion of classification
is different. In the input-output model, the criterion is the
industry producing a certain commodity, intermediate or final,
and the problem is to reckon where its inputs come from and
where its outputs go to. In the present model, the criterion
is the process of production of a final commodity, and the
problem is to build behind each final commodity a conceptually
integrated sector which, by passing over all the intermediate
~ommodities, goes right back to the original factors.
The procedures for collecting and ordering data for inputoutput
analysis purposes are well known. On the other hand,
collecting data for the purposes of the present model may seem
a rather laborious task, mainly in connection with the production
sectors, if not in connection with the final sector (the
latter being the same as the final sector of an input-output
model). But the difficulties are only apparent. It is true that
to classify production processes in a vertically integrated way
would be almost an impossible task if attempted directly. But
this task need not be attempted directly. The algebraical procedure
which has just been shown allows us to go over to a
vertically integrated type of classification by starting from a
classification of the input-output type.
All this means that, in order to give the present model an
empirical content, one may first of all collect data and fit them
into an input-output table in the usual inter-industry way. Moreover,
data must be collected about capital at current prices
(or about capital-output ratios, as is more usual) in each single
industry of the input-output classification. Then the resulting
system of input-output linear equations can be open with
respect to the final eoods and solved bv computing the inverse
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671
of the coefficient matrix. This inverse matrix, or more precisely
the transpose of this inverse matrix, as shown in the previous
section, provides the link between the input-output type of
classification and the one which is needed in the present model.
This means that to pass from one classification to the other is
simply a matter of computation. The procedure, which has been
shown by (VI.3) with reference to the labour coefficients, remains
exactly the same for the stocks of capital (or for the
series of capital-output ratios). In other words, after multiply-‘ng
the transposed inverse matrix by the vector of the capital
stocks (or of the capital-output ratios) of each input-output
‘ndustry, we obtain the vector of the capital stocks (or of the
capital-output ratios) of each vertically integrated sector. The
‘ransposed inverted matrix appears, therefore, as the linear
operator which may be applied to a classification of labour
and capital according to the inter-industry relations, in order
to reclassify it according to a new type of vertically integrated
sectors.
In this way, each production process is reduced to one flowinput
— labour — and one stock quantity—capital. The coefficients
representing them do not correspond to labour or capital
employed in any particular firm or industry, since the whole
framework of intermediate relations has been consolidated; but
‘hey do represent all the labour and capital which are neces
sary to produce the commodity under consideration, in what:
ever remote corner of the economy they have been applied.
Formally, the new coefficients are, therefore, derived concepts
(derived from the consolidation of the inter-industry coefficients)
out they have a deeper economic meaning and, as will be
stressed in a moment possess more favourable characteristics
‘or a dvnamic analvsis
[10] Pasinetti - pag. 10I
072
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
3. The rationale of framing a dynamic analysis in terms of
verticallv intevrated sectors
At this point, the reader may wonder why, in the previous
dynamic analysis, a classification based on vertically integrated
sectors has been preferred to an input-output type of classification.
This question can be answered simply by considering
the type of analysis for which each of the two classifications is
most suited.
To begin with, I may recall that both the inter-industry and
the vertically integrated way of looking at the production
processes of an economic system are by no means new in economics;
they can be found quite extensively used at different
stages in the history of economic thought. However, it is very
significant that they have normally been used for different purposes
and independently of each other: the inter-relation approach
has mostly been associated with analysis at the microlevel
and of a static nature, while the vertically integrated approach
has mostly been associated with dynamic and macroeconomic
types of investigations. As a result, a kind of gap
has gradually appeared between the two approaches. The
foreigong discussion now puts us in a position to investigate
the nature of this gap.
At a given point of time, the two models which we have
been confronting make the connections between the two approaches
quite obvious and well-defined. LEONTIEF has provided,
for the inter-relation approach, a much more aggregate
framework than the one normally used. The model which has
been discussed here gives, on the other hand, a much more
disaggregated framework for the vertically integrated approach.
Between the two, the inverse matrix mentioned above provides
the analytical bridge. As a matter of fact, once we posses the
inverse matrix. all relations between the two approaches at a
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673
given point of time take the form of one-to-one correspondence.
No gap really exists in this case: the two ways of looking at
the production activities meet half-way, through the above
mentioned inverse matrix, which represents the analytical tool
for re-classifying the same transactions according to two different
criteria and points of view (3). Of course, the inputoutput
model gives us more information. If we were simply
interested in what happens at a specific point of time, the
input-output model would be the obvious one to use because
it provides a more complete picture.
But as time goes on, the input-output coefficients change
and the inter-industry system breaks down. The connections
described above begin to vanish. Then it is only the vertically
integrated model that allows us to follow the vicissitudes of
the system through time. It may perhaps be useful to point
out explicitly how this happens. The process of technical
change, as has been remarked earlier, manifests itself in a continuous
way at the level of the single units of production, espesially
in the form of slow improvements, coming from patiently
‘rying different raw materials, re-thinking the disposition of the
line of production, eliminating bottle-necks in the production
process, trying new similar products, etc.. Even when new
methods of production or new products are invented, their
introduction into the economic system very rarely takes the
form of a sudden change. Most of the time, the new products
or the new methods are operated uneconomically for a period,
antil experience and slow improvements put them on a competitive
footing and prepare the ground for further improve-(3)
Professor LeoNTIEF himself has, on a couple of occasions, adopted the
vertically integrated type of classification. See especially Domestic Proluction
and Foreign Trade: the American Capital Position Re-Examined,
« Proceeding of the American Philosophical Society », vol. 97, No. 4,
Sept. 1953, and Factor Proportions and the Structure of American Trade:
Further Theoretical and Empirical Analysis, « Review of Economics and
Statistics ». Nov. 1956.
10] Pasinetti - pag. 103
674 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ment and development. To an external observer, looking for
the « best-known technique », the methods of production may
sometimes look like being introduced at a certain point and
then remaining stable for long periods of time; but when a
reckoning of the inputs and outputs of an industry is begun,
the most widely warying shifts are found from one moment to
the next. Even at a given point of time, there are many directions
from which the inputs may come, and to which the outputs
may go.
These are well-known problems which have always caused
difficulties to all builders of input-output tables and which sometimes
cast serious doubts on the meaning to be attributed to
the coefficients of very disaggregated input-output systems. Of
course these doubts diminish the more the industries are aggregated,
but in this direction the meaning of an input-output
framework diminishes too. So that input-output experts have
always tried to find — so to speak — a sort of minimax point
at which to stop the process of aggregation-disaggregation, in
such a way as to keep the usefulness of an input-output table
without making its coefficients too unstable.
What has not been sufficiently realized is that the property
of eliminating these shortcomings belongs to the completely
aggregated quantities not because they are aggregated but because
— by being completely aggregated — they are necessarily
vertically integrated. The property extends to all vertically
integrated magnitudes as well. By resolving all products only
into the same constituent original elements — labour and capital
— the vertically integrated approach leads to setting up relations
whose permanence over time does not depend on the different
technical possibilities. For example, two equivalent methods
which, at a given point of time, entail the same cost for the
same output, are represented in an inter-relation system by two
different technical functions. But in a vertically integrated
system, they are expressed by exactly the same function. Their
being equivalent means that they require the same amount of
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inputs, and these are expressed in the same terms independently
of the industry in which they are used. Similarly, a sudden shift
taking place at the level of a particular technical process in
the origin of one of the inputs (for example a shift in the provenance
of a fibre from the textile industry to the chemical
industry) means that less labour than before is required to
produce it through the new channel. A shift of this type changes
an inter-industry relation by causing the disappearance of a
coefficient (and of the correspondent variable) and the appearance
of another, different, one. In the vertically integrated
relation, it only causes a small diminution of the same coefficient.
If we imagine many of these shifts taking place while
ime goes on and, along with them, corresponding changes
'n the consumption coefficients — which is, after all, the normal
path that technical progress takes — the input-output
:able is continuously upset and all functions change from one
moment to the next. On the other hand, the vertically integrated
relations remain unaffected. The only consequence for
‘hem is that their coefficients gradually diminish through time
in a movement which, for analytical purposes, has been aporoximated
in the present model by a smooth trend developing
at a certain rate of change.
Concluding and summarising, we may say that, at any
ziven point of time, there exists between the static inputoutput
model and the model presented in the previous pages
a very definite relation through a fully specifiable matrix of
coefficients. Considered at a given point of time, the inputoutput
model is more analytical — it has much more to say
about the structure of an economic system. However, as time
goes on, and the conditions of production and of consumption
change (owing to technical progress, economies and diseconomies
of scale, etc.) the inter-industry relations break down
and become different from one moment to the next, so that
a particular input-output table is needed for each stage in
the evolution of the economy under consideration. These
10] Pasinetti - pag. 105
676 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
tables can be compared (comparative statics analysis), but
they cannot be analytically linked to one another — no theory
of any generality can be provided for passing from the one
to the other (*). The continuity in time is kept, on the other
aand, at the vertically integrated level, where the relations
which can be set up possess — to use FriscH-HAAVELMO’s
terminology (°) — a higher degree of autonomy. This means
that the permanence of these relations in time is independent
of technical change. In this context, the vertically integrated
technical coefficients acquire a meaning of their own, independent
of the origin of the single parts which compose them.
The movements of these coefficients through time, and the
various consequences thereof, can be investigated and followed
as such. When more information is needed about the
() A note may be added here about how this applies to the work which
is being done at present at the Department of Applied Economics of Cambridge,
where Professor Stone, Mr. BrowN and their colleagues are working
on a model of economic growth for the U.K. from 1960 to 1970. They seem
to be trying to make an estimate of the 1970 input-output table by applying
uniform coefficients of reduction to the rows of the table for 1960. (See:
RICHARD STONE and ALAN BROWN, 4 Computable Model of Economic Growth,
D.A.E. Cambridge 1962, especially pp. 70-71).
The procedure, as such, hardly has any theoretical justification if
technical progress follows the pattern which has been described in the text.
Yet, since the period considered is not too long and since the table adopted
is rather aggregate (of the order of 30 industries), which means that the
industries considered are not very far from being vertically integrated sectors,
the results obtained thereby may turn out to be not too unsatisfactory
after all,
One should realize, however, that whatever degree of satisfaction there
may be in the results, it cannot be attributed the procedure, which is
unacceptable in principle. It is to be attributed to (and will be greater,
the greater the degree of) shortness of the period considered, and aggregation.
i.e. vertical integration, of the industries considered.
(°) TryGgve HaaveLmo, The Probability Approach in Econometrics, Suppl.
to « Econometrica », July 1944. The process of passing from inter-industry
to vertically integrated relations for the purpose of dynamic analysis seems
to be a typical example of what HAAVELMO describes as a way of passing
to more fundamental and autonomous relations. « In scientific research,
our search for explanations consists of digging down to more fundamental
relations than those that appear before us when we merely stand and look.
Each of these fundamental relations we conceive of as invariant with respect
to a much wider class of variations than those particular ones that are displayed
before us in the natural course of events » (ibid.. p. 28)
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industrial structure at a particular point of time, the vertically
integrated coefficients can be split and analysed into inter
industry coefficients particular to that point of time.
In this way the static input-output and the dynamic vertically-integrated
systems appear as mutually complementary
and completing each other. Inter-industry relations, referring
to any particular point of time, represent a cross-section of
the vertically integrated variables, whose movements through
ime express the structural dynamics of the economic system.
10] Pasinetti - pag. 107
678 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
APPENDIX TO CHAPTER V1
A CRITICISM OF THE VON NEUMANN TYPE oF DyNAMIC MODELS
The vertically-integrated model developed in the present
work — as the reader has by now realised — has taken so
marked a departure from all the dynamic extensions of the
inter-industry system, which have become so widespread in
current economic literature, as to require perhaps some justification.
To this purpose, I may append a critical assessment
of the approach adopted in all the dynamic models which are
currently discussed among mathematical economists.
As is well known, the most outstanding of all of them is
voN NEUMANN’s extension over time of the general equilibrium
system (!). Another model of particular interest to us is,
of course, LEONTIEF’s dynamic version of his static inputoutput
analysis (?). AII the other works that have followed (3)
(') Jounn von NEUMANN, A Model of General Equilibrium, « The Review
of Economic Studies », 1945-46.
() W.W. LEonTIEF, Dynamic Analysis, Chapter 3 of « Studies in the
Structure of the American Economy », by W.W. LEONTIEF and others,
New York, 1953. This dynamic analysis of LEONTIEF’s has been anticipated
by Davip Hawkins, Some Conditions of Macroeconomic Stabilitv. « Econometrica
», 1948.
(®) See, for example: KEMENY, MORGENSTERN and THompsoN, 4 Generalization
of the von Neumann Model of Expanding Economy, in « Econometrica
», 1956; D. GALE, The Closed Model of Production, in « Linear Inequalities
and Related Systems », Annals of Mathematical Study, No. 38, ed.
by H.W. Kuhn and A.W. Tucker, Princeton, 1956; R. DorFMAN, P.A. Sa-MUELSON,
R.M. SoLow, Linear Programming and Economic Analysis, New
York, 1958, chapters 11 and 12; M. MorisHIMA, Prices, Interest and Profits
in a Dynamic System, « Econometrica », 1958; Some Properties of a Dynamic
Leontief System with a Spectrum of Techniques, « Econometrica »,
1959: Economic Expansion and the Interest Rate in Generalized von Neumann
Models « Econometrica ». TroÂ6
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have introduced many slight variations in these two models
but have left unchanged their basic features.
Let me begin by considering voN NEUMANN’s celebrated
model, which is built on a set of basic assumptions that may
be listed as follows: 1) there is a wide, well specified, and
invariable set of technical methods for producing separately
or jointly the various commodities; 2) there are constant returns
to scale in the employment of all inputs of production
(labour included, in the sense that labour is imputed a subsistence
wage rate and is treated — like any other commodity —
as an output which requires fixed coefficient inputs of subsistance
consumption); 3) the excess of the output of each commodity
over the input of the same commodity in the production
process is accumulated. Given these assumptions, VoN NEU-MANN
shows that there exists a certain set of techniques and
of corresponding prices, and a certain proportion in which the
various commodities may be produced, at which a uniform
rate of growth of all products is maximum. At this rate of
growth, which is considered to be the optimum one, the system
grows uniformly in all its sectors, i.e., it multiplies all its sections
in the same proportion and therefore keeps constant in
time the structure of prices and the relative composition of
production.
Professor LEONTIEF arrives at results which are very similar
to these, although he does not go into the problem of
choice of techniques and starts instead by immediately assum-‘ng
given production coefficients for each process. LEONTIEF
begins with his input-output flow matrix and adds to it a
matrix of capital coefficients. Then he shows that there is a
well defined proportion among the initial stocks of capital
(determined exclusively by the structural coefficients) which
yields a maximum uniform rate of growth for all sectors.
Even if the system does not start from this particular composition
of the initial stocks of capital, it will all the same tend
10] Pasinetti - pag. 109
680 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
>;
to produce it eventually, although in the meantime it may
run into various difficulties.
These have undoubtedly been important steps in the development
of dynamic economic analysis. There is in fact
nothing in them, which is incompatible with the dynamic model
developed in the present work. We may say that they
represent a concentration of powerful analytical tools on one
particular case: the case of constant returns to scale and no
technical progress. Unfortunately, this particular case, elegant
and exciting though it may be from a purely analytical point
of view, has, on empirical ground, very little practical relevance.
In the foregoing discussion of the relation between the
input-output static model and our vertically-integrated dynamic
model, there is a point to which all the voN NEUMANN
types of models can be traced back. It has been said in
section 3 that when, from an analysis of a system at a given
point of time, we pass over to considering movements through
time, the inverse matrix of the technical coefficients, which
provides the analytical bridge between the two models, breaks
down, because of technical change. All the voN NEUMANN
types of models have been an attempt to resist — for analytical
purposes — this hard fact; and to maintain that analytical
bridge through time by assumption, if nothing else. But such
an assumption, convenient though it is mathematically, has
clearly nothing to do with the real world. It means omitting
deliberately what has been singled out, in our previous analysis,
as the basic force responsible for the dynamism of a
modern society, namely the process of learning which goes
on, both on the technical and the demand side. In point of
fact, nne may even question the type of dynamics these models
have adopted, which has meant introducing time into a static
framework with the careful preoccupation of not affecting the
static framework itself. In a sense, time has no importance
in these models. since the features of the economic svstem
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are described independently of time, once and for all time.
As a result, these models do not say about the structure of
an economic system anything more than what is already said
hy the corresponding static model. They add to it a projection
through time, by taking the structure of the system at a given
point of time and crystallising it — so to speak — for all
sternity. The picture which emerges is that of a hypothetical
economic system growing only in size but with no development.
Each member of the community goes on indefinitely
oroducing the same commodities, quantitatively and qualitatively,
receiving the same per-capita income and consuming
the same consumption goods. We have already discussed
this type of system, and acknowledged its logical consistency
and beauty, as a mathematical exercise, in chapter III, at the
same time pointing out, with reference to any progressive economic
system, its lack of practical relevance.
I should not, therefore, come back to the subject if
it were not for the fact that once concepts are coined, they
tend to be generally used. And although neither von NEU-MANN
nor LEONTIEF ever extended their conclusions outside
the fixed framework they adopted, the economists who are
now-a-days using these concepts do not always appear to be
so strict. It is indeed not infrequent, among economists, especially
to talk of von NEUMANN’s maximum uniform rate of
growth as if it were a concept of general validity; and in
particular as if it were applicable to a growing economic
system in which there is technical progress. Since extensions
of this type are unjustified, and since they distort
che very purposes of VoN NEUMANN’s theoretical scheme, it
may be useful to show explicitely why thev must be firmly
resisted.
Suppose the simplest type of technical progress, from an
analytical standpoint; i.e. suppose that improvements take
he form of increases in productivity uniformly spread over
all sectors. In this case, it is only too natural to abandon
to] Pasinetti - pag. 111
682 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
voN NEUMANN’s assumption of a subsistence wage rate, and
therefore of constant coefficients at which workers can reproduce
themselves, and to replace it with the assumption that the
wage rate increases in time pari passu with productivity. Now
we may ask the question: Is it possible to define, at each point
of time, a maximum technically possible and uniform rate
of growth in the von NEUMANN sense? The answer is yes. But
what is the meaning of this maximum uniform rate of growth?
It means that, of all possible compositions of total production
which follow a uniform rate of growth, there is one at which
this uniform rate of growth is maximum. There is something
here to which we do not seem to have paid enough attention.
The point could have been made earlier with direct reference
to the original voN NEUMANN model but it becomes more
striking when technical progress is considered. The von NEU-MANN
maximum rate of growth entails a very definite composition
of production, a composition which comes to be determined
entirely on technical ground. It means, for example,
that to achieve that maximum rate total production will have
to be composed by a very high proportion of those commodities
which are easier to produce.
Therefore, unless the members of the community are indifferent
to the composition of the basket of goods they consume
(which would be an absurd assumption to make), i.e.,
unless the members of the community do not care about
whether the national product is mainly composed, let us say,
of bread and butter or of juke-boxes, or of nuclear spearhead
rockets, the pattern of growth defined by von NEUMANN’S
maximum can by no means be called an optimum pattern
of growth.
The argument may perhaps be better developed if we
follow three successive steps. First of all, let me make the
obvious assertion that the members of any society are interested
in producing the type of goods thev like best and
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not necessarily the type of goods that are the easiest to produce.
Even if we supposed that consumers’ preferences were
such as to require a uniform expansion of production of all
commodities, there is no reason to expect that the composition
of production preferred by consumers should coincide with
‘he composition of production which technology would require
in order to achieve the maximum rate of uniform growth. If
‘hese two compositions do not coincide — as would always
be the case, except by a fluke — the composition according
to consumers’ preferences might evidently rank much higher,
in terms of individual utilities and welfare, than the composition
according to the voN NEUMANN criterion. The latter
would indeed yield a higher rate of overall growth. but many
commodities might remain unwanted.
Secondly, as has been argued in Chapter IV, the consumers
who enjoy an increasing per-capita income do not want
1 proportional increase of all the commodities they consume.
As soon as their demand for each commodity approaches sacuration,
they are bound to spend the incrasing income on
different goods. This leads us to a stronger conclusion than
‘he one reached above. The voN NEUMANN concept of maximum
rate of growth, if applied to a system with technical
progress, not only may not, but actually can never, correspond
— not even by a fluke — to the optimum pattern of growth,
…e., to the pattern of growth that consumers prefer.
Thirdly and finally, there is also the reverse side of the
coin. The von NEUMANN approach imposes on the system
‘he requirement that all commodities should grow at the same
rate. This is an unjustified restriction. There might well
be just one commodity in the system which is very difficult
to produce, in the sense that its production can only
grow very slowly through time. Now, in the voN NEUMANN
model, which insists on the requirement that all productions
must expand at the same rate, that commodity inevitably
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684 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
slows down the growth of the whole system (4). Empirically
this is absurd. If that commodity, for example, happens to
be one that consumers do not want to increase, there is no
reason why the growth of the whole system should be kept
back simply in order to fulfil the unjustified requirement of a
uniform rate of growth. In practice, the production of that
commodity may quite well be kept constant or even decreased
or eliminated altogether if better and cheaper substitutes can
be invented. All this means that, following consumers’ preferences
both with regard to the composition of consumption
and to the rates of expansion of each single production, it
might quite well be possible to achieve an over-all rate of
growth which might not only be better, on utility or welfare
grounds, but which might also be higher than voN NEUMANN’S
maximum uniform rate.
To conclude: there is no ground whatsoever for an extension
of the voN NEUMANN concept of maximum rate of uniform
growth to an economic system in which there is technical
progress. In no case would such a maximum rate of uniform
growth produce an optimum pattern of growth in terms of
atility and welfare. Moreover it may even turn out to be
numerically lower than the over-all growth rate achievable
by following the sectoral rates of expansion indicated by consumers’
preferences (5).
Now the reader may better understand why the dynamic
analysis of the previous pages has been freed since the begin-(*)
This criticism was already raised by D.G. CHAMPERNOWNE in his
1 Note on J. V. Neumann's Article on « À Model of General Equilibrium »,
« The Review of Economic Studies », 1945-46. This Note is quite indicative
of the preoccupations of economists at that time. Mr. CHAMPERNOWNE
does point out the limitations of von NEUMANN’S assumption of constant
returns to scale but mainlv with referenre to the case of decreasing returns
to scale.
(°) It may be useful to add here that the criticism developed in the text
applies in its entirety to the so-called turn-pike theorem, which was proposed
by DORFMAN-SAMUELSON-SoLOW (op. cit., pp. 320-231) and is now
so widely discussed in the economic literature
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ning from the strait-jacket of a fixed structure of inter-industry
relations, although it has been kept capable of being expanded
in such direction whenever needed, with reference to a specific
point in time. The pattern of development has thereby
appeared in a much simpler and more natural way. The technical
evolution of the system has emerged as determining the
pattern of costs and therefore of long-run prices, and the evotution
of demand, in response to increases in per-capita income,
3s determining the proportions in which the single commodities
must be produced. As a result, the solutions of the
system, in the form of time-paths of relative prices and quantities,
and the over-all rate of growth have emerged as determined
and unique; not from any complicated particular or
ad hoc requirement, but from the simple, common-sense action
of technology on the cost side and of consumers’ preferences
on the side of demand.
10] Pasinetti - pag. 115
386
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CONTENTS
CHAPTER I — Introduction.
1. The historical background of economic analysis
2. Scarcity versus learning in economic analysis
3. A pure production model
CHAPTER II —The process of production in the short run.
I. A very simple case: production by means of labour alone
2. The flows of commodities and of labour services, in physical
terms and at current prices . . . .
3. A necessary condition for full employment . . .
4. Production by means of labour and capital . . .
5. The physical stocks and flows of the svstem
6. The structure of prices . . . . .
7. A more complex case involving capital for the production
of capital . . . . . . . . AL .
8. The conditions for equilibrium . . . . . . 5
3. Towards a dvnamic analvsis
D
à)
»)
))
1)
»
Nn
»)
»
nN
5
Q
IO
TI
16
18
20
23
28
21
34
CHAPTER III — The simplest case of economic expansion - Population
growth with constant veturns to scale.
1. A simple dynamic model . . . . . . .
2. The conditions for a dynamic equilibrium . . . .
3. A more complete formulation of the effective demand condition
s ws 2m ®t 4 3 + mw % mu
The dynamic movements of relative prices, physical quantities
and other economic variables . oo . .
5. Interesting features of the present case of growth .
CHAPTER IV — Problems connected with technical change - Setting
the bases for a general dynamic analysis.
1. Technical progress in macro-economic models . . .
2. A dynamic model with uniform technical progress and uniform
expansion of demand . . . ee ew
3. Analytical properties of the two cases of growth considered
an far
»
»
»
»
3
35
37
43
A Q
10] Pasinetti - pag. 116
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i. The production aspect of technical change
5. The demand aspect of technical change
5. The evolution of demand in time . .
7. The criterion for the choice of the hvpothese:
387
6
CHAPTER V — A general multi-sector dvnamic model
1. The model . . . . .
2. A few restrictions . . .
3. The flows of the system .
:. The conditions for a dynamic equilibrium .
5. Relevance of a disaggregated formulation . . .
6. The dynamic movements of physical quantities and of relative
prices EE
7. The dynamic movements of the other variables .
8. Short-run flexibilities
CHAPTER VI — The empirical significance of the model.
«. The relation of sectoral dvnamic analysis to input-outr
analysis . po® ow ew a om ea
Fitting empirical data into the model . . .
The rationale of framing a dynamic analysis in terms
vertically integrated sectors
\PPENDIZ
criticism of the voN NEUMANN
tvpe of dvnamic models
‘1o] Pasinetti - pag. 117
DISCUSSION
MAHALANOBIS
I am tremendously interested in what I have heard. It is extremely
exciting to me, if I have understood the general outlook of the
paper. I am not an economist and have a very superficial knowledge
pf marginal analysis and other classical approaches. I should like
‘0 check my impressions by asking some questions to see whether I
nave understood the paper correctly. I am not quite clear about
« equilibrium ». The objective is full employment of labour, capital
and, I take it, also of all natural resources. How would natural
resources come into the picture? I am not quite clear about the
mplication of the word « equilibrium » in this connexion. The
anderdeveloped countries have a problem of growth; the equilibrium
approach appears to me to be essentially static, because if any function
of time is introduced which in some way can be calculated
or in some way estimated, that is, of absorbing the time-dimension
50 to say, then the approach would still remain static. That is, a
mere formal inclusion of time does not make the system dynamic;
many so-called dynamic models seem to me to be essentially static.
The implication of the word « equilibrium » in this context is not
lear. In any case, Dr. Pasinetti asked whether any one was inteested
in such questions; I could very clearly and emphatically declare
that there is one person around this table who is.
10] Pasinetti - pag. 119
390 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
PASINETTI
Let me express my thanks, first of all, to Professor MAHALANOBIS
for his kind words of appreciation. I shall try to answer briefly
‘he three main questions he has raised.
The definition of « equilibrium » which I have chosen is not a
static one. It is a definition, used in macro-dynamic analysis, which
simply stands for a situation of full employment and full utilization
of capacity through time. As far as natural resources are concerned,
for the reasons which I tried to explain, I found that introducing
them immediately would have put me into those difficulties which
have kept back for so long marginal analysis from tackling the
problems of economic growth. I have preferred, therefore, to leave
them aside, for the time being, although it is my intention to introduce
them later on. Finally, I must say that I am in full agreement
with Professor MAHALANOBIS on the assertion that simply introducing
time into a static model does not make it necessarily dynamic. This
is a point which I have tried to make myself, when for example I
have criticized the von Neumann-type of growth models, in which
time is introduced simply in order to bring about an expansion of
the scale of the system, without altering the structure. In such
models, the proportions (or structure) of the economic system are
specified at a given point in time, and then kept constant for ever;
in other words the structure of the system is independent of time.
[ have always doubted whether we should call such models dynamic.
They appear to me only half-dvnamic.
MAHALANOBIS
I think I have understood, and I am in complete agreement
about that very important question of structure, because from the
point of view of underdeveloped countries, the main objective — at
least in the initial stages and for a fairly long time — is to change
the structure of the economy. If I have correctly appreciated the
10] Pasinetti - pag. 120
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point mentioned by Dr. Pasinetti, namely, the deliberate intervention
of the human mind in the form of learning and technology,
to which I should also like to add the word « science » and also
the word « research » covering both science and technology. It
will be convenient to define in what way I am using these words,
remembering, that all definitions are partly arbitrary. The object
of « science research » (including not only the natural sciences but
also the social sciences) is to know nature more adequately; the
object of « technological research » is to use scientific knowledge
to do something more efficiently, that is, to solve practical problems,
or to bring about changes in society in an effective way. Using the
words in this sense, « science », « technology » and « research »
are most important factors, perhaps the most important factors in
economic development. All the natural resources have been available
since the beginning of the human civilisation; iron ore and
ther metals and coal, for millions of years; the significant factor
was the intervention of the human mind, learning about nature
which I am calling science and then trying to utilize such knowledge
for useful purposes which I am calling technology. To bring science,
:echnology and research into economics or econometrics seems to
me to be a most exciting way of looking on the modern world.
Natural resources would also come in because the type of technology
‘0 be developed would be determined by the available resources;
so long as we have national boundaries technology would have to
oe oriented towards exploiting given resources. Also, I take it that
‘he object of science, technology, research and learning would be
iimed at continually changing the structure; this would be, I think,
‘he truly dynamic part of the work in-sharp contrast to the static
approach.
DORFMAN
Professor PASINETTI is opening the window on a new and very
significant line of investigation. My main reaction is one of hopeful
anticipation, but I guess I have two warnings to utter.
10] Pasinetti - pag. 121
692 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
In his anxiety to emphasize the dynamic, ever-changing nature
of a production-type economy, Professor PASINETTI does less than
justice to many of his predecessors, particularly the members of the
neoclassical school ranging from WALRAS and JEVONS to SAMUELSON.
They, too, were interested primarily in production rather than
exchange, and it is from the nature of production that they deduced
their theories of distribution and capital. LEONTIEF, by his own
admission, is a lineal descendant of Warras, and LEONTIEF is par
excellence a theorist of production. To be sure he is not a theorist
of the aspect of production that intrigues PASINETTI: the process by
which production engenders technological progress. PASINETTI’s
novelty is not that he is concerned with production, but that he is
concerned with the evolution of productive techniques.
In their effort to understand exchange and production, the older
theorists simplified life by assuming away changes in productive
techniques. In pursuit of his new interest, PASINETTI claims the
same privilege for himself: he wishes to simplify his life by ignoring
factor scarcities. It is no objection to point out that this is a radical
proposal. But it may be more of an objection to remember that
PASINETTI wishes to introduce prices into his system and to explain
them, and it is very hard to say what the social significance of prices
is in the absence of scarcity.
PASINETTI
Professor DorFMAN touches upon a really crucial point when, at
the very end of his comment, he asks: What is the social significance
nf prices in the absence of scarcity?
To answer this question, I think we must consider the conclusions
of the two types of models which have been confronted. In
‘he simplest marginal scheme for scarce goods, quantities are accepted
as given by nature and prices emerge as a sort of indexes of
scarcity with respect to consumers’ preferences. In the theoretical
scheme I am proposing (a theoretical scheme for the lone run) re-10]
Pasinetti - pag. 122
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505
lative prices are determined by technology. Demand (i.e. consumers’
preferences) then determine the relative quantities to be produced.
Prices, therefore, emerge as a sort of indexes of relative
efforts that society is obliged to put into each single unit of the
various commodities. The two interpretations of prices are indeed
-adically different.
The central part of Professor DoRFMAN’s comment may now be
looked at from this point of view. I am, of course, aware that neoclassical
economists did deal with problems of production. But I
am raising objections to them when they try Zo extend to produced
commodities the interpretation of prices and quantities which has
>merged from a model for scarce goods. Of course, I am not raising
any objection — on the contrary I am approving — when they
come near to the interpretation of prices and quantities for produced
commodities which I have hinted at above, as for example MARSHALL
himself does in his analysis of the long run, and as indeed LEONTIEF
does as well
KOOPMANS
[ agree with Dr. PASINETTI that the phenomenon of learning ...
production has been insufficiently recognized in economic the-i:
[he same applies to learning in consumers’ choice. As regards
production, I wish to draw attention to a paper by W. Z. HrrscxH,
Firm Progress Ratios, « Econometrica », April 1956, in which some
attempts to quantifv the learning process in production are reported
on.
PASINETT
[ thank Professor Koopmans for his kind bibliographical sus
zestion. I consider in fact Mr. HirscH’s paper, referring to thc aiiframe
industty, as one of the many empirical studies which confirm
he necessity of the type of analysis which I am trying to propose.
101 Pasinett: - pag. 123
304
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
SCHNEIDER
I can be very brief because much of what I wanted to say has
been anticipated by Prof. DorFMAN or by Prof. Koopmans. I am
convinced that Prof. PASINETTI is on the road to important and
Interesting results; on the other hand I agree with Prof. DorRFMAN
‘hat his attack on earlier theories and his survey of historical nature
is not only unnecessary in this connection but also partially unjust.
But I do not want to go into that because I would need an hour
to explain my argument.
You want to construct your model independent from the allocation
problem, But growth and allocation belong together; they
are inter-dependent things and I have an uneasv feeling that you
overlook this interdependence
PASINETTI
Professor SCHNEIDER is right, of course, in saying that optimum
allocation of resources and economic growth may not be independent
of each other. I do not think, however, that they are necessarily
dependent on each other. For example, I can well conceive of an
economic system where allocation of resources is optimum, and yet
there is absolutely no economic growth whatever. At the same time,
I can well imagine an economic system where economic growth is
very fast and yet there is misallocation of existing resources. For
myself, T have chosen to investigate economic growth first of all.
AILAIS
I have not had the time to study the paper sufficiently carefully
and I have only one question to ask, though there is some discussion
of the point on page 47. When considering a system which
is growing with a growing population. it is necessary to take into
10] Pasinetti - pag. 124
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account the greater scarcity of natural resources to be realistic. My
question is how can the Prof. PASINETTI’s model be modified so
as to take into account the limitation of natural resources when
population is growing? From a theoretical point of view such a
question may not be very important but from a practical point of
view it is very important indeed. Thank vou.
DASINET,
[ am not sure whether I agree with Professor ALLAIS on the
point he has made. My impression is that the problems of scarcity
are theoretically very exciting; and yet in practice have not had
‘hat importance which our theories have tended to give them.
The bulk of contemporary economic theory has started from
‘he investigation of the optimum allocation of scarce resources in
an absolutely stationary world; and has then tried to extend the
same concepts to a growing economic system. I am proposing a
‘heoretical model which starts from the opposite end; namely from
an economic system in which there is no scarcity but there is learnng
and thus economic growth. Later on — I am hoping — it may
well turn out to be easier to introduce scarce resources into a model
for learning and growth than it has been so far to introduce learning
and growth into a model for scarce resources.
VIAHALANOBIS
[t is my conviction that one has to develop a model in which
‘here is no scarcity to begin with and to go on later to introduce
scarcity. In other words, the first type of model has a kind of
conceptual primacy with respect to the second type of a problem —
t is only a conceptual primacy — the second type of problem may
at each point of time be more important than the other. The first
:ype of approach may be extremely valuable to developing countries.
10] Pasinetti - pag. 125
696 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Professor FRISCH’s model is also of great interest to the developing
countries because it takes into consideration, in a very imaginative
way, of problems which have not been discussed or do not
require consideration in connection with the more advanced countries.
The point I am making is that model making and a good deal
of econometrics, a very large part of it, very naturally and very
properly had started on the basis of the experience of the advanced
countries where already a good deal of structural stability had been
reached. It was natural and also proper that such developments
should have taken place. because these were useful for the advanced
-ountries.
On the other hand the approach of Professor Friscx and Dr. Pa-SINETTI
is likely to prove extremely valuable to underdeveloped
countries to supplement the tools already fashioned, and likely to
be fashioned in future, to suit the requirements of advanced
countries.
10] Pasinetti - pag. 126
THE ROLE OF CAPITAL IN ECONOMIC
DEVELOPMENT
MAURICE {ALLAIS
Ecole Nationale Supérieure des Mines
Paris - France
« Science attempts to establish the conceptual identity of
chings which when first perceived appeared to be different ».
EMILE MEYERSON: Du cheminement de la bensée
{On the Paths of Thought).
« The truth of a theory in physics is directly related to
the number of valid relationships which can be put on record
by it »
HENRI PorNcARÉ: La valeur de la science (The
Value of Science)
« The aim of any theory in physics is to represent experimental
laws; the words truth and certainty have only one
meaning in terms of such a theory; they express the agreement
between the conclusions to be drawn from the theory
and the relationships noted by observers ».
PIERRE DUHEM: La théorie bhvsique (Theorv of
Phvsics).
.1| Allais - pag.
598 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ABSTRACT
A general view is given of the results of my work on Capital
theory from 1943, date of publication of my Treatise of Pure
Economic Theory, until 1963.
The first part presents a general formulation of Capital
theory, the second a model illustrating the general theory, the
third confronts the model with the empirical data, and the fourth
presents different applications. In appendix. the model is
studied for a very general case.
The different relations given in Part I between quantities
which are not real quantities express simple accounting identities
valid for any economy. Thus they have a very large range
of applicability. The relations between real quantities derive
simply from the hypothesis that there exists, over a wide range,
a valid index Rg of real consumed national income evaluated
at primary factor cost (services of labor and of natural resources)
when the process considered is optimal in the Paretian
sense. From this hypothesis it follows that
ôR _k 2yèY
R IyY
where Y represents the primary inputs, y their prices and % the
coefficient of homogeneity of the production function. The
different relations in real terms of this paper are derived from
this hypothesis ($ 110)
11] Allais - pag. 2
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The general model rests on the stronger hypothesis that the
slasticity
r
of real consumed national income with respect to the primary
inputs (the services of labor and natural resources) which can
be imputed to it according to Paretian optimality theory can
be regarded as practically constant over a wide range (§ 211}
While rather stronger, this is still quite a weak hypothesis.
Thus, the theory enables an expression of the real consumed
national income to be calculated as a function of the capita!
output ratio y.
It is not possible to summarize this paper in a few pages,
out to ease the task of the reader, the main definitions and
the essential results are presented below
() The numbers of the paragraphs and of the formulae are indicated
n brackets.
11] Allais - pag.
3
700
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE T — Main notations and definitions (§ 110)
Reproducible capital .
National income .
Consumed national income
wages
Primary income
rents
Nominal
values
R
R,
Re
Real
values
C
3
R
Gross national product
IT]
Capital-Output ratios . . ;
Rate of interest
Primary income
Rates of growth{ Real income
Technical progress
Variable
rate
of growth
(§ 120-120)
Quasi
Processes
stationary
processes
Constant
rate
of growth
(§ 130-133)
Stationary Process
(§ 140-142)
Allais - pag.
21
=C
y=C/R Yye=C/Ra
1 dRo
Ra dé
_ die
2. dt
dx
Jd
p=p(t)
i(t)—o(t) = # = constant (120-1)
| w(t, 6)= (6) (120-2)
p= constant
1 = constant
pli, 0)= (6) (120-2)
> (t. A\=o(a) (130-1)
Yu
(0) = (6)
R. .R..
(140-5)
(140-4)
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TABLE IT Characteristic curves
Droductiorn
imortizatior
rod § = primary input supplied at time £- 6 and emerging ii
real national consumed income at time #
vod § =
R. (4) =
primary input supplied at time ¢ and emerging in real
national consumed income at time 7-< H
primary income emerging at instant
R.(#) = primarv income supplied at instant
LO A
c(h)dhrsaid
=
(ITx-5 eng
IT 4
Average period of production
E
dp (£9) dh (112-1,
Average period of amortization © = / 04 (2,6) d0
(112-2)
ir] Allais - pag.
702
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE IIT — Main relations
/ R= RU LE _ R,+:C
¥ dt
Ry=R, + R,
C,=C+C,
R,= Ce
(110-1) (110-2)
(110-3) (110-4)
(110-5)
(116-13)
Production function
By hypothesis, real consumed income is a functional of the
primary inputs
Ro(t)=a(t)F[R, (1) @ (4,017
k = coefficient of homogeneity of the production function
(§ 117)
Case of a quasi-stationary process with p=p (t)
wo (i-p)e
Rel)R (0) ofan
I AR op 1 AC ç(t) (122-3) (123-2)
R, dt C dat
R-R Ÿ for : (123-4) (123-12
Clt)o 2c to ce Or 120 3-4
) 1-7 a (HP)
[° Gi-eis
5R, nv dp(6)e _
R, f“,. (io
© / wlole "de
R, maximum for i(t)-p(¢)=0
— 4
Re oN —- p lize) 29 _[ig=0]
Kom 2 d(i-p)
R.,= maximum value of R
y
All these relations except (117-1) are accounting identities
[relations (110), (112), (123)] or simple consequences of a
Paretian optimality (relations 12%).
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TABLE IV — General Model (Part. 11)
HYPOTHESES
‘a) The process is stationary with a variable rate of growth
(§ 210)
A=1"=constant.
TE, The production function is logarithmically linear (§ 211)
Re
[
/
_X(2) - doy we
3 (8) vt anv forin
-
(217-0)
E113
NOTATIONS (§ 220)
Dit
»10;d8
ajo
93(6,d0
(220-1)
(211-3)
(220-5)
(220-6)
RESULTS
~
“y
sr
(240-11)
4286-14)
Rem =K (2) en
cM
or 2=constant
«33-2)
11] Allais - pag. 7
[04
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE V — Exponential model (§250-253)
HYPOTHESES
(a) and (b) unchanged
(c) constant rate of growth p (§ 230)
(d) exponential decrease of the elasticity B(0) (§ 250)
LI 5
A=(250-5)
and (251-1)
RESULTS
r-€ -6-_6
R I+0,1
C=O,R,
R —0, (i-===
[56 i-p] o ( }
Rom
k .
>= 0: (i-p)?
— k _k@_ 9?
R_—all | Re eo
Le@
(251-6) and (251-12)
(251-0)
(251-15)
(251-18)
(251-17)
©, appears as a quantity whose order of magnitude is
practically constant. ©, is an index of the intellectual difficultv
of conceiving indirect processes
11] Allais - pag. 8
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TABLE V1 — The general model and the embirical data
(8 210-710
|. ESTIMATES OF THE MODEL PARAMETERS
(. The coefficient of homogeneitk
(s 331)
2. The function ¢(u).
The macroeconomic data give one condition only (§ 312)
ÿ_ Cc _ re
R +4
=]
11-3
<r ©
"
if
oO
Ji
312-1), (314-2)
Estimate -
+
4
+
-w R
(§ 313)
Estimate 0:
iole=r,
[t is impossible to estimate & with accuracy.
)—-À-I-Y.-0
314-3)
[I. EMPIRICAL JUSTIFICATION OF THE GENERAL MODEL
.. Justification with respect to the hypotheses
a) Nature of the production process (Hypothesis a
(§ 315).
b) Constancy of the production elasticities over time
(Hypothesis db) (§ 316).
c) Constancy of p (Hypothesis ¢) (§ 317).
Justification with the respect t. the consequences
2) Small variations ot w R at a given time
§
—
18-1;
em \
2)
Small variations of R, with ¥ —C, R in time (§ 319).
11] Allais - pag. 9
706 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE VII — The exponential model and the empirical data
(§ 320 to 325
{. ESTIMATES OF THE MODEL’S PARAMATERS
Estimates of ®_ (§ 321)
©, = - 77
=
(321-1)
United States - Great Britain
1 NCE
QT?
4.41
\vera
of
=
United
States
"Ce
4-34
4.71
3.64
4.12
, era£g
IRSO-IC
1
EMPIRICAL JUSTIFICATION OF THE EXPONENTIAL MODEL
2. Justification with respect to the hypotheses
Hypothesis d) - Exponential decrease of the production
elasticities ($ 322).
2. Justification with respect to the consequences
a) and b) small changes of y at a given time for the
various countries and over time for a given country
($323 and 324)
y=0,/1+0
(323-1)
‘rage 3.46 |
countries): 3.54 |
c) Approximate constanr - of CR, at a given time as
between countries (
34
World (median of 48 estimates
Cle) o
R,(¢)
4) Exponential amortization of primary income ($ 326)
(325-1) and (325-3)
1 _8
tO) =e ©
e) The relation between (6) and the composition of the
labor force ($ 330 to 336).
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TABLE Vail
Applications of the model
[. POSSIBILITY OF INCREASING REAL NATIONAL INCOME BY IN-CREASING
CAPITAL TNTFNSIT cD
US]
0-3) (410-4)
—
N
410-2,
or
2. DIMINUTION OF REAL CONSUMED INCOME AS A CONSEQUENCE
OF THE INCREASE OF PRIMARY "NCOME 320-421)
Decrease « — w= conite
U.S. a
3. COMPARISON
(§ 43¢
ye
ea
Fig
RONDUCTIV ’
a
“WO
1
_
COUNTRIES
Tn
hd
;
D
rt)
‘
ombarison ot American and
Fa
Productivity: ratio . . . .
Equipment per worker: ratio
Output per unit of enninment:
ratic
Capital
Ü =:
IQ
Formula (430-3) shows that it is impossible to explain the greater American
productivity by a greater value of @. characterizing the capitalistic
structure.
L.
CAPITALIST DEVELOPMENT POLICY
(§ 440).
11] Allais - pag. 11
708
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
INTRODUCTION
The purpose of this study is to present a general theory of
the role of capital in economic development.
Although there are numerous suggestive practical applications
of this theory, the limited space at my disposition obliges
me to concentrate mainly on the development of its major
theoretical aspects.
This paper follows a long sequence of earlier publications,
and it tries to give a general and rigorous presentation of the
results obtained earlier.
In chronological order these publications are:
my Treatise of Pure Economic Theory (1943) in which I
generalized the theory of optimum allocation of resources
for the case of an efficient path over time;
my book Economie et Intérét (Economy and Interest)
(1947), in which are set forth the general principles of the
theory of capital which have been the basis of all my subsequent
work;
my study of the influence of the capitalistic structure on
the difference between French and United States productivity,
published in « Le Monde » in autumn 1948;
my paper for the 1955 Congress of French language economists,
in which the exponential model was presented for
the first time:
my paper for the 1960 Congress of the International Institute
of Statistics, held in Tokyo. This study contained an
overall assessment of the exponential model together with
many numerical applications:
11 | Allais - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC
700
the analysis of the influence of the capitalistic structure on
differences in living standards and productivity as between
the French and U.S.A. economies, contained in my
1960 study L’Europe Unie (United Europe);
the BowLEY-WALRAS lecture which I delivered to the Conoress
of the Econometric Society (Dec. 18th. 1961) at New
York, in which I attempted to give an overall view of the
work I had done on the theory of capital between 1940 and
q61 (1H:
the monograph which I presented to the Conference of the
[nternational Economic Association at Cambridge in July
(963, in which a full demonstration was given of certain
propositions which had been stated without proof in the
New York lecture
The present study begins with the presentation of a general
and condensed formulation of the theory of capital as I have
developed it over the past twenty years. This is then illustrated
by the discussion of a model of very general application; up to
now I have only given summarised versions of this model, or
demonstrations of particular aspects of it. I will then show
how this model concords with observed data both in respect of
its hypotheses and its consequences. Finally, some remarks
and certain applications will be given.
[t may be that this study is too long, but the reason for
this is that I have wanted it to be self-sufficient, providing a
complete demonstration of the propositions made. and furnish-(")
This lecture was published in the October 1962 number of « Econometrica
». However, it was necessary to condense the exposition to such
an extent that I fear that this paper is practically unintelligible to those
whose who have not had the occasion to become acauainted with mv earlier
vork.
2} The references to these studies are given in the biblioeranhv
11] Allais - pag. (3
710 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ing as complete as possible an overall view of the present state
of my work on the theory of capital.
To the best of my knowledge, there is no comparable equivalent
in current literature, for the approaches adopted elsewhere
have been very different indeed.
I would like tc take this opportunity to add that the results
of this study, once established, are of great importance from
the standpoint of the orientation of economic development policy.
They show that the capitalistic structure as such has much
less importance than is believed by a large segment of opinion.
These results are the culmination of a long sequence of deductlons,
and it is correspondingly important that each stage in
‘he chain of reasoning be solidly established.
In order to ease the task of the reader, the main notations
and formulations have been grouped together in a summary
section which precedes this study (!). Some further basic noions
which will clarify the approach adopted in the study are
given below.
The object of the theory of the capitalistic optimum which
is given is, on the one hand, to describe the econometric nature
of roundabout methods of production, and also to show that
we cannot expect, from an indefinite increase of available real
capital, an indefinite increase of real national income consumed
per inhabitant. The present study aims to show how the influence
of real capital on real income can be measured on the
basis of a number of fairly weak assumptions.
The general model which is presented in part II, and its
exponential variant in particular, is shown to be very well
borne out by all the empirical data which are at present available
— confirmation which relates as much to the assumptions
as to the resnlts.
() Perhaps the best introduction to this study is my 1962 article in
« Econometrica »: AvLrars. The influence nt the Cabital-Outbut Ratio on
Real National Income
11] Allais - pag. 14
SEMAINE D' ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
{11
THE ROLE OF CAPITA;
aN
ECONOMIC
DEVELG MEN 1
JU MMAIX
GENERAL FORMULATION OF THE THEORY OF CAPi
A
V
A
General considerations.
\ quasi stationary process with variable growth rates
A quasi stationary process with constant growth rates
The case of a stationarv process
J
. MODEL ILLUSTRATING THE GENERAL THEOR:
The assumptions of the model.
Consequences of the assumptions.
The case of a constant rate of growth of primary income.
Limited expansion of the main expressions for small values
the rates p and à.
The case of an exponential decrease of elasticitv
A
3
-
8).
" NFRONTATION OF THE MODEL AND OBSERVED .
The general model and cbserved data.
ne exponential model and observed data
General comments.
APPLICATIONS
The possibility of increasing real per capita national income
»y increasing capitalistic intensity.
Process of maximum growth of production per unit cf primary
ncome for given technical knowledge.
Comparison of productivity for two countries.
Policies for capitalistic develonment.
À PPENDI"
INFLUENCE ON THE RESULTS OF VARIATIONS OF THE FUNC
The case where 8(0) e#0 can be developed as a Taylor series.
Particular cases.
11] Allais - pag. :5
PART 1
GENERAL FORMULATION
OF THE THEORY OF CAPITAL
+
+
GENERAL CONSIDERATIONS
Definitions
IIO. The definitions and economic notations to be used are
siven below (1).
Capital
Income (‘1
TABLE
[
National reproducible capita
Non-reproducible capital (”
Total national capital .
National income
Consumed national income
Primary income (%) .
labour . . .
natural resources
ke
XR.
) Re
rR.
Xe
() See ALLAIS (1954), The accounting basis of macro-economics.
(3) Nominal values divided by the nominal price of an hour’s unskilled
abour.
3) Excluding the capitalised value of natural resources,
The capitalised value of natural resources.
-, Per unit of time.
{®) The value of primary factors of production (wages and rents) (in
trench: «revenu originaire »).
111 Allais - pag. 17
714
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(I10-1)
dC
R=R. +7
R=R,+:C
R.=R, +R,
R.,=:C,
Cr=C+C,
(110-2)
(110-3) Relations
(170-4)
(110-5)
/
(110-6)
-
r
L dR,
"R_z 7 Rate of growth of
” primary income at
time #
(110-7) Growth rates
Vv
. dR,
à 7 = Rate of growth of
Be real consumed national
income
Rate of growth of
technical progress (!)
(110-8)
;
+)
Pure rate of interest
expressed in wage
units
Capital-output ratio
defined in terms of
national income
(110-9) Capitalistic
characteristics
Y —C,R
(110-10)
va=C, Re
Capital-output ratio
defined in terms of
consumed national
Income.
(') See below § 117 and 133, and especially the relation (1232-2).
11] Allais - pag. 18
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[12
The quantity R, represents the value of primary factors of
production, wages and rents; R stands for National Income,
Rc consumed national income, and C the value of reproducible
capital. Measurements are made at factor cost. Real values of
‘hese variables are indicated by a bar over the symbol concerned.
The equations (110-9) and (110-10) define capital-output
as part of reproducible capital and appear in the vear’s total
~onsumption respectively.
For the sake of generality, consumed national income is
defined to include only such consumption as takes place at
‘he time. For example, private motor vehicles are evaluated
as part of reproducible capital and appear in the years total
~onsumption only in respect of the services they render.
Characteristic functions
mr. This study is based on the concept of the characteristic
function, which so far as I know was first described bv JEVONS
in 1871 in the context of a particular model (!). I presented a
systematic analysis of this concept in Economy and Interest
in 1947.
The simplest way of envisaging the curve which is characteristic
of a production process (see Figure I below) would apdear
to be as follows. Assume that in the present period wages
are paid in respect of the construction of a blast furnace, which
will in due course produce pig-iron. The pig-iron will be used
‘o produce steel, which will in turn find its way into different
manufacturing industries, to be used, for example, in making
() JEVONS (1871) in chapter VII. The concept of the characteristic
‘unction was used for the first time for the case in which @(8)z#%0 for
9<6<A and ¢(6)=o0 for 1 <0.
The concept was subsequently applied in penetrating studies by BOHM-BAWERK
(1888) (see especially Exkurs I to VI) and G.H. BousoUEtT (1936),
vol. III. chanter VI
11] Allais - pag. 10
716 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
saucepans which will finally be bought by consumers for their
own use.
A part of the wage bill generated in the construction of the
blast furnace will finally be incorporated in the price of the
saucepan; it will thus appear, with some time lag, in national
income.
It can be seen that consumed national income will at a given
time include wage outlays which were made at an earlier time,
and, at least in theory, it is possible to consider a curve which
specifies the distance from the present of various earlier outlays
(wages and rents) which appear in the consumed national income
of the time # considered. This curve can be called the
« characteristic curve » of the capitalistic process under study.
If there is a Paretian optimum, factor income is imputed
proportionally to the marginal productivities considered in
terms of physical values (1).
In parallel with the characteristic curve which defines the
origin of the different elements of primary income appearing
in the consumed income of time #, an amortisation curve can
be defined which represents the time distribution of the primary
-ncome of time # over the various subsequent periods (>.
The term Ru (2) will be used to represent the global amount
of primary income per unit of time preceding # which is incorporated
in total consumed national income R_(#) of time #. The
global amount of primary income per unit of time at time ?
(') This imputation generates a number of very interesting problems
which considerations of space preclude me from discussing. On this question,
see in particular my Treatise on Pure Economic Theory (1943) and
my Cambridge paper (1963): Some Analytical and Practical Aspects of the
Theory of Capital. Appendix I. See also ALLAIS (1961 - C), The Definition
of Characteristic Fonctions and the Problem of Imputation (La définition
des fonctions caractéristiques et le problème de l’imputation).
() In general, the definition of these two curves results from the way
in which the accounting of imputations is undertaken in practice. If a
Paretian optimum obtains. these imputations satisfv the corresponding
conditions
11] Allaic - pag. 20
SEMA'NE D ETUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIOUE ETC.
717
which will be incorporated into national income in periods
subseauent to # will be denoted bv R. (1).
Rolf) ot, 0) d0 is defined as the input primary income
between #- 60 and t-0+d 6 which appears in the national
consumed income of time à, and R,(¢) ¢(¢, 6) d0 as the input
of primary income of time ¢ incorporated in consumed national
income at time £4 0.
From these definition it follows that:
I1i-I)
11.-2)
-
u
x
“ot (¢ H) d9
<a /
41-51) 40
These definitions are summarised in Table
ot
pelow
[ABLE
Primarv
ncome
Elements
Global
Time
f input
.ppearing in the!
~onsumed income
Corresponding
rrimarv inputs
Aa --u
\
Sve, 0)df
11 | Alluis - pag. 21
718
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Naturally we have
(111-3)
(111-4,
"oo
| (4,9) dH =1
0
© A
/ p(,9) dh =1
0
If primary input at instant (#- 6) but appearing at time ¢
is denoted by Pa db, then
(111-5) 7, dO=Ru(t) ¢ (£, 8) d0=R.(t - 9) ¢ (£- 6, 0)d0
whence
(III-6)
Ro) © (£, 0)=Ru(t-0) © (£- 6, 6)
By definition we have
111-7)
R,(t)=R, (t-0)e Ls ra
11] Allais - pag. 22
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719
The characteristic curve of primary income appearing at
me / is shown in Fieurc
ple, 6
2r18tic curv
roduchon
1.
This curve is obviously asymptotic to the 6 axis, for first
there exists no consumable object to day which does not concain
some input dating back as far as Julius Caesar ('), end
second it is clear that the influence of primary inputs tends tc
zero when the distance in time increases indefinitely.
The characteristic curve for amortisation of primary income
furnished at time # but appearing in consumed income at some
future moment is given in Figure (2).
haracleristic curve
Cr amortizatson
(}) Indeed, certain investments made by the Romans are still yielding
‘evenue. À good example is the Coliseum at Rome.
11] Allais - pag. 23
720 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The functions © (t, 0) and 9 (t, 0) define the capitalistic
structure of the economy.
Average production and amortization periods
112. Taking into account the relations (111-3) and (111-4),
an average amortization period ® and an average production
period ® can be defined by the following formulae:
(112-1)
(112-2)
9() = [ 7 Dep(2, 6) d 9
o() = [ b(2,0) d0
Income and Capital
I13. From the definitions which have been given above, consumed
national income (total consumption) at factor cost is
ojven as
(113-1)
LL,
a.
od 0
I
[
() If the global production function with respect to primary inputs is
homogeneous of order k and if pricing is nronortional to marginal cost
[11] Allais - pag. 24
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
121
whence
(113-2)
R_(¢)-R,
FT
+0
Wei
whence again from (111-6)
(113-3) R,(t)=/ Ro (2-0) © (£-0,0)e
so that it follows from (111-7) that
I13- = - €
ot)
=R 1-6
(
,0
-6
i(u
)
p(
“ja
t ado
The value C(#) of reproducible capital can be considered
as the total capitalized value of the factor inputs furnished prior
to instant # which will result in final output only during a
period (£+1, t+1+ dt) subsequent to: ”
Paretian optimum situation) it can be
valued at market prices will he
shown that total consumption
See below, equation (117-12).
1) ALLAIS (1947), Economy and Intere.
11] Allais - pag. 25
LA
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA - 28
Appearence oF
the primary input
In consumption
es
rot
Input: Ry(t+1- 8) ¢ (¢+1-6,6) db
=Ro(t+1) © (£+1, 6) de .
FiG. 3
The value at instant t of the primary input furnished prior
to ¢ appearing in consumed income Ro(¢+ 1) dt of the period
(+x, t+T+dT) is
So
a [ Ro (#+T-0) 0. (t+-1-0,0)e t+ eae
Thus from (111-7)
(113-5) Clt)=R,, (1) a
re
£
- “than (i-p)du
P(t+T-0,0)e T° Ja
MN
11] Allais - pag. 26
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
723
Alternatively, changing the order in which the integrations are
jone, which may be more convenient in certain cases,
113-6) Clt)=Ry(t}/
“
-
Tr
Clearly, it again follows that
(113-7)
NO
Rll, P(t+1,0)e (es
-(u,-À
Again, the same result could be derived by integration the
lifferential equation
'113-8)
dC) (ect) =Ru(6)-R (1)
-
1d
by the usual methods and taking into account (113-2), (113-4)
and (111-6). In this form, the differential equation is derived
from the relations (110-1) and (110-2).
11] Allais - pag. 27
fe:
PONTIFICIAE ACADEMIAE SCIENITARVM SCRIPTA VARIA - 28
The Value of the Capital which corresponds to the Share of
Factor Input which will appear in Consumed Income between
Years 0, and 9,:
(14. For certain numerical applications, it is useful to evaluate
the share Cg (¢) of capital which corresponds to factor
inputs which are frozen in the form of capital over an interval
of time included between years 6, and 0,.
We have
(114-1)
6, JL. i(u)du
C, ()=/fRo(t+=-0)g (t-1-0,0je did
i
where the surface S is represented schematically as below
11 Allais - pag. 28
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 725
It is then possible to write
(114-2) C,
"x
ay +
which gives
(114-3) (
on
Primary Cabital
r15. The term « primary capital » Ca(#) (!) can be applied
to the global value of the factor inputs incorporated in capital
C(t). From this definition we have
IIS-I)
“w ,
ST
K
lt
+T
-6)
© (ter
-0
6)
d
0
y In french: « Capital originaire ».
.. | Allais - pag. 29
726 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
LIC
(115-2) C(t
/
K, (+7) (¢+1,0) 40
(115-1) and (115-2) can be obtained by putting i=0 in (113-5)
and (113-7).
Clearly, using the same notation as in § 114
(115-3)
% 8 f du
Ce = Ru) | d6 | olor felere 2
8. a
Es
This relation is obtained by putting ¢==0 in equation (114-3).
Macro Economic Magnitudes as Usually Defined and in the
Model
116. It seems to me necessary to emphasise the differences
between the definitions adopted in the present study and those
asually applied to the various economic magnitudes.
To facilitate the exposition, I will denote concepts based
on the usual definitions by adding a prime superscript to the
appropriate symbols. This will distinguish them from the
symbols which refer to magnitudes defined as in the present
study, for which the same notation will be used. but without
primes.
111 Allais - pag. 30
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 727
a) National Cabital €
National capital is equal to the global value C of all reprojucible
goods, whether production goods (factory buildings,
>quipment, semi-finished products, stocks), or durable consumer
soods (dwelling houses, durable household goods, motor vehi
cles, etc.).
This definition is of course the definition which is normally
ased by statisticians in work on the estimation of national
wealth, so that
116-1)
hx
b) Consumed National Income
In the present theory, consumed national income 1s -jual
to the global value of all final services. If, as is or course
necessary, consumer durables are included in capital C, the
value of the services rendered by them, », must be included
in the calculation of Rc. The income arising from a capital
goods is given by
116-2)
vie) —
alt,
Ue,
where a is amortization, v the value of of the good at instant £,
and 7 the instantaneous rate of interest (2). This relation sig
(') Subject to a number of secondary differences which will be discussec
ater (§ 334, table on page (115).
(?) ALLais, Traité d Economie Pure (Treatise on pure economics) (1943)
362. relation 2. and Economy and Interest (10947), p. 70, relation ?
1.1] Alluis - pag. 31
728 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
nifies that the value of the services rendered by capital goods
is the total of the amortization of all capital goods and the
interest on their global value.
Thus, in the present theory, the value of a privately owned
and used motor car is not counted in income Rec at the moment
of sale to its owner. This value appears in the form of
income only in the form of the services rendered by the car
over time. The definitions usually adopted, by contrast, include
the value of an automobile in R’; at the moment of sale.
There is no difficulty in specifying the correction to be made
to Rc to derive Rg, at least as a first approximation.
The adjustment consists of subtracting investment in consumer
durables Ic from consumed income R”c as it is normally
calculated, and adding back the amortisation of consumer
durables Ac and the interest {Cy chargeable against the capital
represented by existing consumer durables Co (1).
Thus
Re=R"- Ic + Ac +ICo ,
Now, from the definition of amortization
d Ce
Ac=le-=1lc-
Pc Ce
!) Relation (116-2) above.
11] Allais - pag. 32
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 729
where
o£
oc denotes the rate of increase of the overall value of holdings
of durable consumption goods Cc .
It thus follows that
116-3) Ro=R’c+ (i - po) Ce
The adjustment (¢ - gc) Cc is rather small in practice.
he United States in 1065, it is possible to derive (?)
or
Hence
Nith
Pc)
ve
Le
R
=0.0053 .
() These formulae should be used in all cases to estimate amortization
rather than direct recourse to the corresponding statistics in yearbooks which
we quite questionable.
1?) See § 333-6 below.
Allais - pag. 33
730
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
c) National Income
From relation (110-1), national income is given in the
model as
(116-4)
RR, 20
dt
and, at least in terms of a first approximation, the concepts
which are normally used satisfv the relation
(116-5)
R'=R'.+ ac
Af
It follows immediately from (116-1), (116-3), (116-4) and
(116-5) that, at least as a first approximation.
(116-6)
R=R"+ (i - po) Co
d) Gross National Product
Whereas the differences between C, Re and R on the one
hand, and C’, R’c and R’ on the other, are relatively insignificant,
this is not true for the concepts of Gross National Product
(G.N.P.) and overall amortization.
Usual Definitions
Under the definitions usually adopted
(116-7)
P—=R’ + A
11] Allais - pag. 34
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 731
where P’ is written for G.N.P., and A’ represents the amortisation
relative to the range of durable goods for the economy
as a whole, per unit of time (,.
Further, from (1.6-5) wc have
116-8)
with
(116-9)
116-10)
where I's and I'y denote gross and net investment respectively
Under the existing convention of national accounting, captal
is for all practical purposes defined as goods whose life
expectancy exceeds one year, and indeed in practise, as goods
whose life expectancy is at least five years, the shortest period
over which amortization is calculated.
As a result of this convention, the lining of a Martin oven,
which normally lasts a few months, is written off as an operat-‘ng
expense, whereas the structure of the oven is considered
1s an investment.
With rules of this type, it is generally found that
116-11)
K Re
n wich K is of the order of 0.1 to 0.z.
(!) ALLAIS (1954), Les Fondements comptables de la macroéconomique
The Accounting Basis of Macroeconomics).
SH
Allais - pag. 35
752
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The Case of the Model
Although it facilitates computation, this convention allocates
a particular vole to a specified interval of time, the year, which
is both arbitrary from a theoretical standpoint, and involves
running a risk of suggesting erroneous ideas.
It is for this reason that in the present study, investment
is defined to include any outlay incorporated in a durable good,
whatever its longevity. Under such a definition any output of
primary factors which does not appear immediately as consumed
income is considered as an investment. It is thus a
simple matter to see the conceptual changes in the notions of
G.N.P., gross investment and amortisation when the framework
of the present theory is adopted.
If we consider the concept of a continuous characteristic
function 9, depreciation Adt is equal to the primarv income
Redt which emerges in the national consumed income Rod? of
instant # and we have
(116-12)
A=R,
whence
(116-13)
P=R+A
~R+R.
using P to correspond to the equivalent in the present theory
(in which any investment outlay, whatever its durability, is
considered as an investment) of the conventional notion of
G.N.P. (in which a minimum durability of one, or even of
several years is taken as the boundary between operating expenses
and investment outlays).
[11] Allais - pag. 36
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 733
Thus. in this study, G.N.P. is equal to the sum of national
income R and of primary income Re.
These results may seem to be somewhat disconcerting at
first sight, but it will become clear after some reflection that
they are more natural than those which follow from the adoption
of the rather arbitrary concepts usually applied in national
accounting.
If, using the usual conventions, amortisation turns out to
he only of the order of 20%, of consumed national income, the
reason is once again that standard systems only allow for the
amortization of goods whose life expectancy exceeds one year,
or even several years, while all the services invested in a shortperiod
process are not brought into the books as such but are
considered as fungible goods.
The example of the Martin furnace has been given already,
and a multitude of others could be added to it. Thus, outlays
corresponding to certain preparatory work in mining underrakings,
those corresponding to a wide variety of agricultural
activities, or again a great deal of the maintenance work done
in various sectors, for example lorry repair, are considered as
operating expenses.
Similarly, the work of the grocer’s boy who stocks sardine
cans on a shelf is an investment, but it is not counted as such.
[n the same way, a window cleaner’s activity represents an
investment, and the only reason it is brought in as an operating
expense is that this facilitates the accounting.
Under the formulation which I have given, all that which
is not immediately consumed is investment.
Thus, the service rendered by a waiter in a restaurant as
he nears the client’s table is incorporated in the capital value
of that semi-finished good, the dish to be served. This incorporated
capital will have been amortized when the customer
has completed his repast; but in the conventional evaluation
of capital it is not considered as an investment, nor is any
account taken of it in the usual estimation of G.N.P.
11] Allais - pag. 37
134 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The value of reproducible capital is several times that of
national income. Errors arising from the conventional estimaling
procedure may not be very considerable relative to capital
(1), but they will be significant in relation to the evaluation
of depreciation and of G.N.P.
The value of the grocer’s boy’s or of the waiter’s services
should be counted as a production of capital; it is an element
of gross investment and therefore of Gross National Product P.
Demonstration of the Equality A =R,
It is a simple matter, using the preceding formulation, to
show the passage from A’ to A. Conventional accounting only
considers as capital those goods whose longevity exceeds a
certain duration. Thus with the notation of Table 2 (§ 111)
the corresponding concept of the general model which is presented
in this paper is
(116-14) A* = [ FR (2,6) db
r J
Le
If the usual convention is modified and 6, made to tend to
zero, then for A. =0 relation (111-1) gives
(116-15) CC A=A*[9,=0]=R, .
(') See below § 334, Table 8 and o.
(*) In fact, under the usual concept the investment in a tool is considered
as made when this tool is put in service. With such a concept
depreciation A’ includes interest.
Nevertheless from an economic point of view, this is only an intermediate
stage and the original investment takes place when the primary
output (as defined in § 111) are supolied.
[11] Allaic - pag. 38
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
735
The Composition of G.N.P.
Denoting ? ‘he income given by capital we have
A
JU
116-16)
iE: Rr. +
~
i.
so that
(116-17)
P=R+A=R+R.=R. +R.
G.N.P. is thus equal to the sum of factor income and ..
ncome of capital.
The results of the preceding discussion are summarise.
‘he following table (1):
r
za
Concepts
Global
Jsual
san.
Defini
{ac.rityn
Defined in Model
Reproducible
capital
Durable consumer
goods
Consumed national
income
National income
Amortization
sross National
Dera viet
UTIOSS
Investment
Net
[nvestment
!\ A detailed discussion of these notions is given in § 333 and 334 below
i1] Allais - tag. 39
736
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Depreciation in the Production Process and Financial Amortization
The quantity A=R, represents the depreciation in the production
process. In a similar way it is possible to define a
financial amortization. In the present theory any primary
outlay is incorporated in capital. Thus we can consider the
consumed national income as the sum of financial amortization
Ar and of capital interest 1C and write
(116-18)
R.=Ar+iC
Then from (110-1) and (110-2) we have
(116-10)
R,.-/C=R__4C
dt
and then
(116-20)
A=R, dC
wT
and taking account of (116-12)
116-21)
A-A__ dC R
d 4 -[R,-R,]
(+
A
(!) The value of R,—R,, is given below for a special case (relation 251-20).
II] Allais - pag. 40
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
737
Case where Consumed Income is equal to Capital Income
In the case where consumed income Rc is equal to capital
income R,. we have from (116-16)
116-22)
A
R,=R_ +1
and from (116-19) we have then
116-23)
R tu K w
Thus in this case the whole increase of capital comes from
primary income. There is no contribution from the capitalisaon
of interest on capital.
From (116-21) and (116-23) we have in this case
116-24)
a — ir
Mlustration by a Simple Example
The significance of the preceding discussion can easily be
understood by examining a simple example.
Consider a stationary process in which the rate of interest
is zero and assume that the lag between use of a factor input
and its appearance in consumed income Rg is the same for all
11] Allais - pag. 41
138
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 23
inputs and equal to ®. Then we have from (116-12), (110-1),
(II0-2) and $ 111.
(116-25) R=Rc=R,=R, = 4
The sequence of events in this case can be represented by
the following figure, which is the same as that demonstrating
the movement of a fluid in a conduit.
Entry into
the capitalrstic
process
Exit From the
rapitalistic
process
Production
persrod
=
Movement of inout-Supply
of
any factor
input
dppearance of
the rfactor
moult
This shows that all factor income R,dt arising between
t-© and #-©+dt appears between # and #+dt and that in
the case of this process it is true that
(116-26) R=R.=R.=R.=A =|.
‘11] Allais - pag. 42
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETc
In this particular example all operating expenses would
clearly be investment outlays, and a clear cold light would be
thrown on the lack of symmetry of the usual definitions.
Real Consumed Income
(17. If it be assumed that real consumed income Rc (t) is a
function of factor inputs 7,d® i.e. a functional
117-1)
R, (t)=a(t)F[R,(¢)& (,0]]
=a (t)F[R,, (8) ¢ (¢-6,8)]
of the function (111-6)
Rul(t) #(8,0)=R,(t-0)e (#-0,0,
and if it is also assumed that this functional is of k-order
homogeneity, then
117-2) R, (t)=0 (e)[R()1* G[#(4,6)
In this discussion, x(#) is used as a coefficient which characterises
the rate of technical progress.
11] Allais - pag. 43
740
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The characteristic function can in fact be used to derive an
expression for R,- As the reasoning is somewhat difficult it
may be helpful to consider two relatively simple cases before
going on to study the general case.
Case 1.
a) Assume in the first place that
(117-3) R=R (A, B, ..., O)=R(X, Y, .... 2)
where R is real national income, A, B,......,C represent different
outputs (including equipment) and X., Y,........Z the
factors of production used.
Assuming that R (A, B,......,C) is of first-order homogeneity,
and that the function R (X, Y, ........ Z) is of E-order
homogeneity, then
(117-4) R (AA, AB, !.., XC)= AR (A, B, …, C)
(117-5) R (uX, uY, … uZ)=uR (X. Y. … 7).
Applying the Euler theorem, this gives
(117-6)
(117-7)
> B R,=R
YV R'v=F R _
[11] Allais - pag. 44
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 74
so that we have
177-8)
R_ZR, 8B _, IR, 3Y
R ER B =R,Y
in which the ËR, &B and 8Y correspond to virtual displacements
compatible with the restraints.
Now, the conditions for Paretian optimum can be written
II7-Q)
Np _
5,
‘Rb a
YB T
where a, ©
AB... C
-, ..., 2 are the prices of the goods
2).
Carrying the values of the derivatives of R given by (117-9)
nto (117-8), we have
JR YbBB _ 5 LycY
SCSAB TN
; À
f
!) ALLAIS, Treatise on Pure Economics (1943), p. 204.
‘111 Allais - pag. 45
742 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The first of these relations is usually used to calculate SR.
In nominal value, the expression for national income is
equal to
°
bh
R
at market prices, and to
> y *
at factor cost. But by (117-9)
(117-11)
4B IBRj
yY YR,
whence it follows from (117-6) and (117-7) that
(117-12)
YbB = 1 SyY
b
If we consider here only the expression for national income
at factor cost. we have
(117-13)
R
= Yay
[Ir] Allais - pag. 46
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. [743
nd
‘I-I17-I4,
4
'
Le
where (à R) is the differential of national income calculated
at constant brices.
ase 2.
bY We now consider a stationary process (+
R, is constant and the functions ¢ and © do not vary over ‘ume.
The relation (111-6) above shows that the functions ¢ and «
are identical (!). Since the process is stationary, national income
and consumed national income are equal. The function
p(0) which characterises the time distribution of factor inputs
is also a function of ¢, and can be written ¢(8,7).
From = © wc hav 7 +
I17-I~
‘\ See also
iso below.
U
‘11] Allais - pag.
/
I
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and, using relation (117-14) we deduce
(117-16)
1 3R
E R
/ 5 (8,i)e'®do
©(8,i) de
TTT
In fact, in relation (117-15), the prices of the different factor
inputs are equal to the exponentials ¢®. The function A(:))
characterises the distribution of factor inputs. But, while ¢(0)
is of course a function of 7 it has not been differentiated with :,
since in the expression (117-16) the Sp play an analogous
role to the 8Y in relation (117-10), and since in this relation
the y are regarded as constant (*).
Thus the relation (114-16) also enables the expression for
the differential of real income to be deduced from the differential
of the characteristic function, and we have in particular
(117-27,
(8,i)e'’de
(7
0.2)d 0
() Of course this does not mean that i is independent of time. In general
7 should be considered as a function of time.
(?) These various formulae were first given in 1947 in mv studv Economy
and Interest (1047), DD. I86-1RK
11] Allais - pag. 48
SEMAINE D’ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 745
The General Case of a Dynamic Process
The generalization of these different results follows immediately.
In the case of a non-stationary process and assuming
k-order homogeneity, we have
Jd.
'117-"
>,
where Rc represents the change in Re when ¢ changes by &¢.
Naturally, taking into account equation (111-6) it is further
possible to write
- 7
117-19)
St1-0) ¢ (t-0)
-0
mp
1-0)¢
Au) a
t-60
Jd ou
This is the fundamental relation from which the different
real values in the present paper are derived.
Allais - pag. 4¢
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The Interest Rate and the Rate of Growth
118. I have shown elsewhere that subject to very general
conditions a dynamic growth process is characterised asymptotically
by the relation
(118-1)
i) =o(t) +1
where ¢ is the rate of interest characterising the stationary
process which can be made correspond to the process under
study (1).
I'he Significance of the Preceding Relations
119. All the foregoing relations which do not incorporate
veal quantities basically express simple accounting identities.
Primary inputs correspond to the accounting imputations which
are made.
The relations between the real quantities are simply derived
from the assumption of the existence of a satisfactory index
of real national income consumed (calculated at factor cost),
when the process considered is optimal in the Paretian sense.
When an optimal allocation of resources (?) is realised, the
primary inputs and their marginal equalities take on physical
significance (3).
(') ALLats (1063), Quelques aspects analytiques et appliqués de la théorie
du capital (Some Analytical and Practical Aspects of the Theory of
Capital).
(?) In French « Maximation du rendement social ».
() See ArILAIS (1963) Cambridge Paper: Some Analytical and Applied
Aspects of the Theory of Cadital. Appendix |
11] Allais - pag. 50
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
fa.
In the neighborhood of a given situation the fundamental
relationship (117-19) is derived from the assumption according
to which the relation
110-1)
SR, _p Ly
Re yy
is derived from (117-10) in the case of a dynamic process and
applies when the use of primary factors (services rendered ™-labour
and natural resources) over time is being studied.
It can thus be seen that the fundamental relationship
'117-19) of the present study results simply from the assumption
of the existence, over a large range, of a valid index R of real
national income at factor cost when conditions of optimum
allocation of resources in the Paretian sense are satisfied
Allais - pag. 5,
748
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
eo
— A QUASI-STATIONARY PROCESS WITH
VARIABLE GROWTH RATE
Definition
120. I will use the expression « a quasi-stationary process
with variable growth » to denote a process characterised by
a) invariance of the difference i(¢) - P(#) over time;
b) invariance of the characteristic function © over time
for a given value of i -p (1).
We then have
(120-1)
(120-2)
(8) - p(é)=i
p(¢, 0) =1¢(0)
where #’ is a constant.
The assumption underlying (120-1) appears to be reasonably
acceptable at least as a first approximation. The hypothesis
(120-2) is equivalent to assuming that the composition
of the labor force is constant over time. This, too, is a quite
natural assumption, at least as a first approximation.
These two relations enable the general relationships which
have already been derived to be considerably simplified.
(') The alternative hypothesis in which ot 0) is assumed invariant over
:ime could also nsefullv be studied
11] Allais - pag. 52
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 749
Amortization Pv"
rar. We have (relation Zi
121-1)
a,
ony 7
174 A)
so that
121-2)
4
Thus the average duration of the amortization period 1.
invariant over time.
Consumed National Income
"22,
122-1)
We deduce from relations (+, 4, (120-1) anu
oo . -
Re(2) = Rolf) [ ce B)as
J20 «4
whence
122-2)
À
Se
4
A
KR.
tas
[
, Allais - pag. 5.
750 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
so that
(122-3)
+. 4Re _p tt)
R. Zi
and finally from (120-1) and (122-1)
(122-4)
Ro(t)=Ra(f) for i(t)=p(t)
Reproducible Capital
123. In the same way from (113-5)
(123-1)
C(t)=R_ (1) a fe 0e (0)d 6
so that
(123-2)
1 dC
Le oft
= p(t)
thus from (122-3) and (123-2)
(123-3)
dy _4 Cl _,
dt © dt R(t)
fIT] Allais - pag. 54
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
Thus the Capital Output ratio defined in terms of consumed
national income is constant over time and depends only on ¢
[t then follows from relations (110-1) and (110-2) that
123-4)
C(t) = R.(¢)-K_. (7
€ (ti-- A
This expression gives C(#) without any need for integration
oy (123-1).
Then from (122-1) we have
123-5)
-,=Ralt
which relation can also be obtained from x.
the order in which the integrations :r ‘«Fon
Again, from (110 = ur
-Cversin,
122-6)
and from (110-2) ana
1
123-7)
Ve
1) ALLAIS (1947), « Economy and Interest », pp. 12-3
.y Allais - pag. 5.
752
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
From (110-10), (122-1) and (123-5) we have
© Li'l
(123-8) | £ = T (0) db = y. [Fer (ng) df
0 Jo
Thus developing (123-8) as a Taylor series
ow [ 2 3 - i ,
(123-9) / | Or due [poas=1. / | Ly
“0 2! 3! o L 21
bz 8
2
+. [a(@y dé
Thus from (112-1)
(123-10)
‘oo ge 2 8
sou t fPTa8 84 Jos
Ce vA 2 31!
and finally
(123-11)
Yeo 0 - [7
T—7 — + =; vo 4
74, #1 (e** — T — 7’6)p(6)d6
Thus necessarily
(123-12) 0 s— To _
L— (1-Pp) Ye
for 1-o0>o0
Whatever the function ¢(8) this inequality is very important
for empirical analysis.
11] Allais - pag. 56
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 75;
Again
(123-13)
n -
Ql NJ
or
Value of the Capital corresponding to the Share of Factor Income
Appearing between Years 6, and 0,
Relation (114-3) can be rewritten
123-8) oN
-Ry,(¢)
¢ (0)d 0
Clearly, in the light of § 115, and putting :=0 in equa
‘on (123-8).
123-0)
co, = 8
8,0 I R(t,
AF
49
Primary Capital
From the definition in § 115, we have here
123-9)
butting 2:=0 and
Col =Rul 7
“
-9(0)du
tb
- —p in relation (125-5, ( ).
- |
Allais - pag.
154
PONTIFICIAE ACADEMIAE SCIENTIARVM. SCRIPTA VARIA - 28
National Income
124.
(124-1)
Since
R=R,,4C
di
it follows from (123-2) and (123-4) that
124-2)
R=R.+pC
“Rp Re Ra
1-0
whence finally
124-3)
R(t)- PO Ro(6)-p() Ra
2 (t)-o (t)
From (123-6) and (123-7) we have
(124-4)
Re () _I—ve _ I
R,() 1—v:
El
— 2 fi
From § 120 and (122-1) this expression depends only of 7’,
which is evident from (110-2) and (123-5) lor from (125-2)
below].
II] Allais - pag. 58
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1
Cabital-Outbut Rat--125.
We have from
(125-1)
ses =
}
K-Ra
whence from (110-9), (110-10) and …
125-2)
p (t,
so that
125-3)
125-4)
anid
125-5)
EP T6) a
= 1
where Ÿc depends only on :’ (relation 123.3,
11
Allais - pag. 5,
756
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Approximate Expressions for the Principal Magnitudes
126. For small values of 7’ it is possible to write from (122-1)
and (123-5)
(126-D
R()=R,(0) [rive 20”
/ —— +...) (0)d 8
œæ 70, 17°8%
(x26-2) C(t)=R,,(t)[ _ Tt q(ojde
9
thus we have from (112-1)
(126-3)
(126-4)
R, (£) + Ro (6) [r+ÿ ©]
| C(t) ~ OR, (¥)
and according to (110-2)
(126-5)
R(t) ~R, (t) [1+i 0]
whence
(126-6)
(126-4)
Ym oN ©
R.
a)
2
ignoring first order terms containing 7 and :.
"111 Allais - nag. 60
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
For small values of ¢ and 7 we can write
‘126)-8)
126-9)
Re(t) © Rolt)e'
R(#) + Ra(f
’
In terms of a first approximation, the process therefore
rorresponds to what would happen if consumed national income
and national income were equal to their values at the
end of a period ©, applying the respective compounded rates
of interest ¢"=7-p and :.
Further from (123-9) and (112-1), for small values vi
126-10)
>
mn OR.()
Real Consumed National Income
127. From (117-19) and for rates of growth p(z) considerza
as given we have
re
127-1"
-B)ù vis.
Le. from …
)
-
Allais - pag. bi
758
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
or again
(127-3)
+
y.
(8)e
3
do
©
de
Now
- 1°g:
e'®=14+70, °
Lee
and from relation (111-3) we have
(1277-4)
(850 (08)d6=n
Thus for small values of i’ and from (112-1)
(127-5)
R if 6S ¢(0)d6
: SR oN -
k R.
whence it follows that
(127-6)
R
oR 1701
— “= — —
1420
(6
Cel©] de
11] Allais - pag. 62
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 759
and finally, from (3z-""
(127-7)
H
~
dR,
a:
When the rates of growth p(#) are considered as given, tne
smaller the :(#) the greater ©. As from (121-2), © is a
function of :”, we have
(127-8)
de
7
~
It can thus bz seen that real consumed income Rc 1s a maximum
for © =o (1). Thus we have
'127-9) Re is maximum for i(t) - p(#)=o
If we write Rom to represent
we have as a first approximation
(127-10
LA V6
the maximum value of Rg,
0 [i-p=0’
d{i-p)
() This result was first stated for the case of a dynamic process by
DESROUSSEAUX (1961 A), using the formulation I had given in « Economy
ind Interest » (1947) for a stationary process (p=o0). However, DESROUS-SEAUX'S
line of approach is completely different from that adopted in the
present study, for he assumes that the function ¢(¢, §) satisfies a « condition
of regularity » over time, but this condition is formulated from a
social, and not an economic. point of view.
Allais - pag. o,
760 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The Functions R(t) and ot, 0)
128. Lastly, it is possible to deduce from the invariance
of ¢ and from equations (111-6) and (111-7)
a + p(u)du
(128-1) Ral6) ©(t,0)=R,(t) p(0)e ”
from which it follows that
(128-2) 1 Ro, 1 0¢ 1 dR,
R, dt § ot TR. at "le(t}-e(t0)|
Thus from (110-6) we have
(128-3)
R A
= Axe ® PS OF (48) = o(t-6)
R, dt @ Ot
Nothing further can be said about the functions R.(#) and
(2, 6).
Changes in i and op over Time
129. Attention is drawn to the fact that the only assumption
implied in the preceding formulas is that the difference i(¢) - p(2)
is constant. They therefore hold where 7 and ¢ are functions
of time.
[11] Allais - pag. 64
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 701
rt
: QUASI-STATIONARY PROCESS WITZ
CONSTANT GROWTH RATES
Hypothesis of Invariance of the Function clé, 0)
130. If in addition to the conditions already discussed,
condition that the function ot, 0) is invariant is postulatea,
can easily be shown that the rates i(t) and c(t) must be constan*
For by hypothesis
130-1)
£(:, 8) = 16,
whence, from (i.
130-2)
But the left hand member is dependent only on i, and ., .
‘ight hand member only on #-6. It follows that 7 muse be
constant and from :20-1) the same is true of the rate of in
‘erest 1.
Thus we hav
Allais - pag. v,
762 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
A process of this type can be labelled a « quasi stationary
process ». It is characterised by the invariance of the rates p
and i and of the characteristic functions © and 9.
The fact that 7 and p are constant and the functions © and 9
variant enable the formulas obtained for the case of a quasistationary
process with variable growth rates to be still further
simplified.
Characteristic Functions
I31. From the relation (112-2) and the fact that 9 is invariant,
we deduce that
(131-1)
d6
— —0O
74
i.e. the production period ® is constant.
Further, from (111-6) we have
(131-2)
©(8)R,({)=R, (#)e(0)6°°
and this enables the two conditions below to be deduced from
relations (111-3) and (111-4)
(131-3)
(I3I-4)
R;(t)=R._(£)) 2°00) do
R (t;=R_1
[Te 8
© (e)de
[11] Allais - pag. 66
SEMAINE D'ETUDE SUR LE ROLE DE L’'ANALYSE ECONOMETRIQUE ETC. 703
30 that
(131-5)
7
-
Equation (131-4) implies that 0) tends to zero when ©
ncreases indefinitely.
Equations (131-3), (131-4), (112-1) and (112-2) imply that
(121.60)
(= 10) Ru(t) < Ru(t) -
1
Roll
-
for
Equations (131-5), (ri2-1) and (112-2) imply that
131-7)
‘2 20) (1+6 (5)
~
~
T
Thus
131-8)
w
National Income
32.
I
Since
Ko +8
- | Âllais - pag. €
764 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and since R and C grow at the rate p while 4 is constant, it
follows that
(132-2)
Ir dR
R A
-C
The relationships derived in part IB naturally remain
anchanged (§ 120 to 120).
Growth of Real Consumed National Income
133. From (117-2) and (131-3) we deduce
133-1) R(t)=a(tR, (t)] [“e"g(0)a0]'G [5 (6)
In this relation, a(f) represents the impact of technological
progress on productivity and we have
(133-2)
a(t). + dælt)
walt) dt
where 7 (Z) is the rate of growth of technological progress.
As the functions © (8) and ® (8) are invariant over time,
we have
1 dR. dot I dR, (¢
(133-3) — —e =—-.— dat) +k (t)
R. dt alt) di R(t) dt
11] Allais - pag. 68
SEMAINE D'ÉTUDE SUR LE ROLE DE IL ANALYSE ECONOMETRIQUE ETC.
705
vhence, from the dz2fnitions in $ TIC
(133-4)
Thus, if the production function is of k-order homogeneity,
the rate of growth v of real consumed national income is given
by adding the rate of growth = of technological progress to the
product kg of the rate of growth of primary income © and the
~oefficient of homogeneity «.
[t follows from the preceding .
133-5)
V
1
[n other words, when the coefficient of homogeneity is equal
lo unity, which is more or less its value in real conditions, the
rate of growth of technical progress is equal to the amount by
which the rate of growth of real consumed income exceeds that
of primary income. The rate of growth of technical progress
‘hen is equal to the rate of growth of productivity per unit of
brimary income.
The growth of real consumed income is independent o. .. _
capitalistic structure (!), since the functions ¢(8) and (0) have
Seen assumed invariant.
ne sense .
, Allais - pag. 55
7166
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
D. — THE CASE OF A STATIONARY PROCESS
General Formulation
140. In the case of a stationary process, we have
(140-1)
~
— nN
and the preceding formulae simplify considerably.
From relation (131-2) we have then
(140-2)
5(0)Ra = p(0)R,
so that
(140-3)
/&(e) R, d6=/"s(6)R. de
whence, from relations (111-3) and (111-4)
(140-4)
R.—R.
C11] Allais - pag. 70
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 767
and therefore
140-5)
5(0) = ¢(6)
The relations (722-1), Z23-5), (124-3) and ..
‘140-6,
8.73
, ,, become
140-7)
140-8)
and we have as first approximations:
140-0)
{
140-10)
140-11)
| Allais - pag. ,.
168
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
These are the various relations for a stationary process given
in my 1947 book « Economy and Interest » (!) (3).
Primary Capital
I4I. In a stationary process, the average amortization period
is equal to the quotient derived by dividing capital by
amortization, whatever the amortization rules. It follows that
for primary capital C, we have
(I4I-I)
C.=0R,.
From the earlier expression for C (3) and for a given funcfon
¢(0), C, can be derived by putting i=0. Thus we have
(I41-2)
Co lim P elr
=m € -Tw(e)do
rr 0¢(6)d6
and this is effectively relation (141-1), (relation 112-I).
Thus ® can be regarded as the average amortization period
of primary capital (4).
(!) « Economy and Interest », pp. 127, 128, 132, 133, 187 and 188.
(3) The hypothesis that k=1 is made implicitly in this study; the function
(6) there considered is the same as Rw ©(A) in the notation of the
present paper.
(3) Relation (140-7).
(*) Of course this equality also holds as a first approximation for small
values of n (relation 126-10)
111 Allais - pag. 72
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
It can further be seen that the interest element incorporat- :
in capital C is
141-3)
or, as a first approximation
141-4)
dd,
(Gross National Produc
142. In the case of a stationary process, the relations (1-.
and (110-2) can be written from (140-4)
P=R+R,=2R-1C
Thus, in a stationary process, gross national produc: .
equal to the sum ~f national income and primary nationa
NCOME.
Since here from
: -
NG (Ac
J
g. 7.
ag
Allais - p
ak]
[70
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
it follows that
(=Thus,
© appears as the average amortization period for primary
capital C, (§ 141), and since the process considered is a stationary
one, this average period © is independent of the amortization
rule.
For positive rates of interest and from (141-3), C/R, is.
always superior to ©.
As far as the ratio y/® is concerned and as will be shown
later, it may be greater than, smaller than or equal to unity (1),
even in the case of a stationary process (2).
227, relation (227-6).
240 (relations 240-5 and 240-11, with p=o).
che same time, in the case of the exponential model, y=0 for p=o
relation 251-171)
11] Allais - pag. 74
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
PART
MODEL ILLUSTRATING THE GENERAL THEORY
200. The general theory whose main elements have been set
forth above can be illustrated by a very simple model. This
model has some very suggestive features and appears to be of
very wide generality.
Among others, the model has the essential advantage of
being easily adapted for numerical applications, and in particular,
of facilitating evaluation of the influence of the volume
nf capital used on real consumed national income.
THE ASSUMPTIONS OF THE MODEL
The model to be examined is based on two hypotheses:
Hypothesis (a): Paretian Optimum over Time and Invariance
of ilt) - p(t)
210. It is assumed that a Paretian optimum over time is
realised (1), and that the difference between the rate of growth
() In French « maximation du rendement social », cr in the terminology
sf English language literature: « optimum allocation of resources » [see
ALLAIS (1943)]
11 | Allais - pag. 753
772 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and the rate of interest is a constant, 7’
(210-1)
W(t) - p(t) =7
Hypothesis (b): Constant Production Elasticities
211. Af anv moment t, the elasticities B(0) of real consumed
income with respect to primary inputs r:d0 can be considered
as being constant over a wide range and independent of the
moment t considered.
As we have
(211-0)
8(0)- ZR
R.
we see that, on this hypothesis, real consumed income Ro(?) can
be represented bv the function
(211-1) LR,(#)=La(t)+/"8(0)LF da
(') These assumptions hold in the case where the process considered
respects hypothesis (c) in my article in « Econometrica » (ALLAIS, 1962 A)
Le. hypotheses 1-7 of the second version of my paper for the Cambridge
Conference (ALLAIS, 1963) Some Analytical and Practical Aspects of the
Theory of Capital.
(3) The operator I. represents the natural logarithm.
11] Allais - pag. 76
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
7,
VITO
211-2)
We now write
211-3)
Thus the production function is assumed to bs logar™
nically linear, and k is assumed to be its degre ect homo“:
reity. (#) is a coefficient which depends on # and allows for
‘echnical progress, which is assumed to operate cn ~ global
hasis.
Relation 11
2y We have
bov.
Using discrete notation and applying a period analysis of period I, the
sroduction function under consideration can be written
1)
=
LA.
Ré.
A Las A
where a, b,, b,, ..., by, ... are constant and R,, Ry, ..., R, ..., are the inputs
»f primary income of moments t, t—1,.., t—@,... contributing to real
income at moment -
» st
LA. +
OQ
«AN
t
+ 4%
Allais - pag. ,,
[T4
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
B.
— CONSEQUENCES OF THE ASSUMPTIONS
Notation
220. It will be useful for what follows to consider the Laplace
transform ¢(u) of the function
(220-1)
b(u)= = / Blo) as
vith
— A
dR _, dR,
—— = 09 —
A
R R,
in continuous notation, relation (2) becomes relation (211-1) of the text.
If we put
b, = T (0)
BR -
we have
LR=La+T B(o) L [T7,] +T B(1) L [T7] +. +T B(6) L [TR] +.
=[La+} T 8(6) LT] + X T B(6) L7e
If we suppose that T tends towards zero and that we have
limit [La+E T B(6) LT]=La(t)
and if we put
we finally derive
T
ade
LR = af?) + | 7 8(0) LA, db
0
which is the relation (211-1) of the text.
(*} The meaning of hypothesis (b) is in particular that, at least over
a large range, there exists a valid index of real national consumed income
R,. This last property corresponds to the general hypothesis I have
made in the first Part (§ 117 and 119) [in particular relations (117-1),
211-1) and 211-2)]
11] Allais - pag. 78
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE EITC.
Tio
Considering small values of u, developing ¢(u) as a Taylor
series, and taking only the first few terms we have
(220-2)
d(u)vI - © 0+ AZ 42
where ©, and À are two constants. We have:
(220-3)
(220-4)
dy(u) _ _e, [1-240,u]
du
d gw oo 240.
du?
and
(220-5)
(220-6)
a3(e)de
"“9’B(0)do
Further
(220-7)
220-8)
(220-0)
p(0)=+
dy lu
A Lu
i
cé
x
‘u=0)-210,
LL
11] Allais - pag. ,
76 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
For a system of points M, of mass m., we have
220-10) Xm, OM. = OC Em,+Em, GM’
where G is the center of gravity. In all cases therefore
(220-11)
2m, OM: >CC Em
(equality arises only when the entire mass is wholly concenrated
in the point G).
We then have by simple transposition
220-12) / 0 (6)de =
glade?
/3(e)de
and from (211-3), (220-5) and (220-6) it follows that of necessity
(220-13)
A
A will be equal to 1/2 only if the distribution of the É(6)do,
whose overall value is k, is concentrated at a given point. and
of course this cannot happen in reality (1).
As will be shown, the value A=1 is a particularly significant
one (2).
(*) Taking account of (112-1), the same analysis applied to the function
0(0) shows that
JE (60) do=er
(?) § 251, relation (251-1)
-1| AÂllais - pag. 80
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
771
It is useful to wri
220-14)
WIi
220-15,
Expression for Consumed Income ani Rear Consums .
at Factor C:
LHCIM.
221.
From relation (113-4) and (zz.
-"
we deduce here
221-1;
R.(t)=R,(¢] 41-00).
id
rrom (211-1) and (211-2), we have
2c &
=“
H
a ANd
""6(6)JL[R, (2-8). 0,6
| Allais - pag. Si
7"
|
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Now,
(221-3)
R, (t-6)=R,(#)e Ls p(u)du
so that
(221-4)
_ . f p(u)du
LR(t)=La(t)+ RLR, (1) B(o)(L[e “° g[t-0,0)]| de
Paretian Optimum Conditions
222. In conditions of Paretian optimum, the cost of production
Ro(#) of real consumed income Rg(f) is minimized.
Subject to the condition of the production function relationship
(221-4) at any time #; and according to (221-1) and (221-4) and
for values of g(#) and R,(#) considered as given, we have
(222-1)
de (t-6,6)d0=o
11] Allais - pag. 82
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
77
for any system of values of the 9 on the condition of restraint
(222-2)
Re _/"p(e) 2¥(t-08) d8=c
R. c{t-8,0)
where the &¢ represent virtual variations of ¢
This implies that
(222-3)
K(
8(0)
ry
from which
(222-4
p.t-6,0)=K(¢)B(0)e
The Condition ls +-225.
(275 …
From
ed
we have
ct
9d
{ Allais - pag. &
780 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
so that from (222-4)
(223-2)
[TK (t+0)8(0)e Can ~1
This condition can hold for all values of t only when K(t + 0)
is independent of t+ 0, in which case we have
(223-3)
1
“0 10
“B(0)e
K/
so that, from (220-1)
(223-4)
KE) amr
whence it follows that
223-5)
o(t-0.0)_ Bl)"
Eos
As the right hand side is independent of #, it can be seen
that ¢ is independent of # and invariant over time. Thus.
222-6)
Tie - 0, 0) = (0)
11] Allais - pag. 84
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
with
(223-7)
v
where 1 and ç are functions of time (§ 210).
It follows that the expression for amortization oc) facror
income 1s invariant over time and depends only on in =
erence i- p.
The process considered is therefore, following the definition
given above, a quasi stationary process with a variable growth
rate, and the relationships derived under this hypothesis are
applicable () (2) (3)
) $ 120.
() Where the function ? depends oa time, we hav
JX)
Tquation (222-3) can then br
we <‘
2)
vhence
3)
ct, €)
and the integral equation (2.
IN K ti
0
must hold for all valur
If we write
i, 0)
rs
Enen
>
ak
6)
foc
ju F&60)e-ÿ%d6 = 1
In this case it does not appear that sufficiently simple results can be
obtained for them to be easily applied.
3) It would also be possible to take the invariance of the function g (6)
as a starting point, deriving the invariance of the function 8(6) as a conclusion.
(See ALLAIS, 1960 A, pp. 11 and 12)
+. Allais - pag. 85
782 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The Expression for Re
224
From (122-1), we derive
(224-1) R,(#)=R, (ef ep (0) do
= Be)
“Ro(t)/ For de
so that
(224-2)
R, (2)
Rolt) = Sra
National Income
225.
From relation (124-3) we have
|
€
225-1)
F
I
11] Allais - pag. 86
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 78:
so that
(225-2)
ny ¢
and from (224 |
(225-3)
R(t) _ i-ed(i-p)
Ro) Tejo[r
Reproducible Capital and Capital-Outpui Ratios
226. From (123-4) and (224-2) we
have
226-1)
Tom which we derive
2260-2,
<r _ 1-¢ (i-p.
a ) pn
11d
220-3
|i
eo.
.… | Allais - pag.
784 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
These relations allow ¢(i - p) to be expressed as a function
of y, and vy, which can be convenient in numerical applications.
Thus
(226-4)
b(E-p)=1~-( Ar
: 1-Y; T(i-p)
_ - = =1 ———
(226-5) ¢(i-p) % =
Eliminating ¢(¢ - p), we derive
(226-6)
I 1
vy =F
Naturally we find again the general relation (125-2) above.
It is further possible to deduce
(226-7)
C(t) %
R(t) 1-[i-p)Y,
Cl) 14
Ra(t) 1
and from (224-2), (225-1) and (226-4)
226-0)
(226-10)
TA
N
R -—
R TT I —vr
This last relation can be deduced from (110-2) and (110-9)
and has general validity (relation 123-7). The same observation
can be made for (226-8) which results from (110-2) and
(110-9).
[IT] Allais - pag. 88
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETt. 785
Primary Capital
From (123-9), (220-1) and (223-7) we nave here
226-11)
Colt) XT = 20)
R,(t) op d(i-p).
Average Amortization Period
227. We have from /2-(2247-1)
-
=
es
-
“iV #4.
)
—i'€
(ee as
Now, from (220
(227-2)
lV
(
03(0)e
d
50 that
227-3,
1d
~~
JV
v
dor
d(i-p)
11, Allais - pag. 89
86
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and taking (226-4) into account.
(227-4)
i-p dY,
I, v cpr
0 Ye d(i-p) Y.
Noting that, according to (226-6) and for e(t) considered as
given, we have
(227-5)
1 day, 1 adv
v2 d(i-p) 1 d{i-p)
we find
~ avy L A
(227-6 Ce + [=e L-p ÿ
2778) (1-Yp) (1-72) TY aio) |
Real Consumed Income
228. From (223-6) and (223-7) we have
228-1)
_f!
e /. p(u)du
L e(t-0,0)!
"p(u)durLe(0)
(2 (u)du-Lk[1')-i°6+LB(8)
i1| Allais - pag. go
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 187
Thus, trom equation (221-4) we obtain
228-2)
LR_=La(t)+kL Rolf) Li kO,-h(t)+
“ko(t)
witi
228-3)
228-4)
ky
9) [ plu)du] de
/8(e)LB(e) de
whence finally
228-58)
R,(t)=æ{tje ot
LR D(i
WHaximization of Real Consumed Income for
We have
220-0,
.| Allais - pag. yi
788 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and
O° [LR]. +, 49li) x de(i)
(228-7) 7 3 |e ir | o(i) di?
Taking relations (220-7), (220-8), (220-9) and (220-13) into
consideration, we have
(228-8)
LR
[LR |=FB
2
228-0) or | LR] = (1-24), < O
for 71=o0
for
1=0
Thus at any instant, Ro is a maximum if
(228-10)
>
7
N
Therefore we have
(228-11) R. is a maximum for 1(t) =¢(t)
Writing Rey for the maximum value of real consumed nalional
income for =0 we have from (228-5)
228-12)
=e [5 “
CM (ip)
11] Allais - pag. 92
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 789
with
(228-13)
R,,~ «(t)
ert
where %(¢) is a functional of g(¢) (relation (228-3)) and where
k’ is given by relation (228-4).
The results presented here are of course particular forms
of the general results derived in $ 127 above.
Further, taking relations (: “+ 2nd (226-5) into consideration,
we have
(223-14)
Real National Income
229. From relations (228-5) and (224-2), it is possible tu
deduce
220-1)
1%
Li’ le
=I
k:"0, Re
It can thus seen that real consumed income is proportional
0 [Re(8)1*
1{1| Allais - pag. 93
790
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Then, in order to calculate real national income R, it is
quite natural to make, at least as a first approximation, the
assumption that
(229-2)
R-[X ‘R
R | Re
C
which, when k=1, gives
(229-3)
R R,
ER
This expression can be considered as valid at least as a first
approximation.
It is then possible to deduce from (124-3) and (229-2)
220-4)
R k
= | R,
R= 0
Whence, bringing in relations (224-2) and (228-12), we
derive
(229-5)
R(t) -| i-p ¢ (i-p) ,~% (ime)!
Rolf) (3-p)b(i-p)
11] Allais - pag. 04
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
Further, from
== relations ‘226-4, and “2° -- we
1AVC
229-6)
9,
ag.
llais - p
.1] À
192 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
-- CASE OF A CONSTANT RATE OF GROWTH
OF PRIMARY INCOME
Hypothesis (c): Constant ç
230. It is impossible to obtain simple expressions for o(t, 0)
and Ro) in the more general case where ç is a function of t,
and it does not appear that invariance of p, or its equivalent,
invariance of the function af, 6), can be deduced solely from
hypotheses (a) and (b) (*) (3.
Thus we are led to pose the constancy of p and therefore
of 1 as a new hypothesis.
The Functions (t, 8) and Ra(t,
231. If the supplementary hypothesis is made that the rate
of growth p(#) is constant, i.e. that the process considered is
a quasi-stationary process with a constant rate of growth of
primary income, the functions l(t, 8) and Ro (2) can be specified.
(Y) § 210 and 211.
(?) Nevertheless it may not be impossible to show that a constant value
of p is a necessary consequence of the general hypotheses made in my
Cambridge paper, from which I deduced the asymptotic property of invariance
of i(t)—p(#), and hypothesis (b). This point still requires further
slucidation. Up to now. it has not been possible +0 make this demonstration.
it] Allais - pag. 96
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 793
From relations (128-1) and (223-7)
231-1)
(6) ae
R,(t)@(40)- BE
Then from relation (111-4), we have
= Rolt) l"B(o)e*
231-2) ko (i-p) el
from which we deduce according to (220-_
231-3)
D /
Re.
LP!
7
——
vhence, from (251-41)
(231-4)
9 (£, 0)=g (6
with
231-5)
P
vi
_ Be
kw
.., Allais - pag. 9,
704 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The characteristic function o(t, 0) is therefore invariant.
Also, we have from (223-7) and (231-5)
(231-6)
#(e) - bli-p) °°
b(i) elo)
Since
b (2-p) > d (1)
for p> o0
the function @ /¢ declines from a value of above 1 to 0 when 0
increases indefinitely.
Average Production Period
232. We have from (112-2) and (231-5)
O05 (0) a0
(232-1)
_[Tog(e) ¢° 4
/ 0) 0
3
Now, from (220-1), we have
(232-2) dé(i) __ 1 [Top(e)e "do
i k y.
[11] Allais - pag. 08
SEMAINE D’ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 795
so that
(232-3
From (227-3) and
(232-4)
{~
164.
pu
a’
}
we derive
A
<cO
So
to
this condition is more restrictive than the condition (131-8)
which is valid in a more general case. It could have been
derived a priori by observing that for invariant values of the
functions (8) and (0) over time the effect of growth of Re
s to reduce © in comparison with ©. It thus follows that ©
seems to be a better indicator of the lengthening of the production
process than 6.
[t can thus be seen that while the various formulae giving
Res (0) and © are dependent on the hypothesis that the rate
of growth ¢ of primary national income R, is constant, the
sreceding formulae are independent of this assumption.
The Expression for kK,
233.
(2:5, +,
Relation (228-3) becomes here from (+.
p68(6)de
» 9,
11| Allais - pag. 9g
796 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus from (228-13) we have
(233-2)
— -ke6
Rom = K(t)e
with
(233-3)
K(t)=e(t)e"| Rell]
where œ(f) represents the effect of technical progress and k
and %’ are constants (1).
It can thus be seen that at any time, the maximum value
of Ron which can be attained will be the smaller the greater
the value of p.
Case of a Stationary Process
234. The formulae corresponding to a stationary process are
derived by putting p=o0 in the preceding relationships. As is
to be expected, the general properties of the process given in
$ 140
(234-1) 9=¢, 60=0, R=R, y=v, R=R..
are again found.
() Relation (211-3) and (228-4).
11] Allais - pag. 100
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 797
D. — LIMITED EXPANSION OF THE MAIN EXPRES-SIONS
FOR SMALL VALUES OF THE RATES e AND :
Limited Expansions
240. For numerical applications it is useful to have the
limited expansions for the most important of the preceding
expressions (!). Results only are given below, as the calcula-Hons
offer no difficulty.
General Model
(240-1) (i-p) 0 1-0, (i-p)+0, (i-p)+ 30, (1-p)
(240-2) R, oo (1-0, p IR,
(2403) p(B) [1+0, [i-p)}B (0) 7"
(240-2) (8) = [1+0,i]8 (0)
(240-5) © Ce ©, [r-0,(;-p)-250,(i-p
(240-6) © o 0,[1-0,i-250,7]
(!) Relations (220-14), (220-2), (231-3), (223-7), (231-.
(224-2), (225-3), (226-1), (226-2), (226-3), (228-12) and (229-2
Il
, Allais - pag. 101
98
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(240-7) Ro/Re » 1+0,(i-p)
(240-8) R/R. 1+0,7¢
(240-9) C/R, ~ 6, [1-3 0, (7-p)]
(240-10) y _C/R, «©, [1-8, (i-p)-30,(i-p)]
(240-11) Y=C/R @, [1-0, 1-30, (i-p)]
R £O. |.
(240-120) TE ~T-— = (i-p)- 30] (i-p)’
R RO: 2 2 2
240- == I+ °_(1- (à
(240-13) —=1 ke®,. —-(i p}-Rè@, {(:-p)
= = 2
(240-14) Rew Re ~ R(1+293) 9, (i-p)*
Rom 2
(!) This is again the general formula (127-10)
Rew — R t-—p)2 dO
LC —2 ( or ee (Z — p = 0)
Roy 2 d(i—p)
since from relation (240-5) we have
ae
dis (ê—9=0 = +23) 0;
This identity of the results is obviously necessary, but it is nonetheless
remarkable in that it demonstrates the underlying consistency of two very
different chains of reasoning; the first starting from (117-19) and the second
from (211-1).
11] Allais - pag. 102
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 700
These results can be used in numerical applications subject
to the assumption that all series expressions converge rapidly
(1).
Expressions for the Main Parameters when i= and
CASE 1=0
241. From the foregoing (*), we have
(241-1)
(241-2)
fo)
¢
ple! of
2°
El
YY 7
1.
241-3)
241-4,
(241-5)
c Or,
Re
241-6)
(241-7) à
€
(') It is possible to check that this is so in practise in a large number
of causes, and, as far as can be judged, in all those cases which are of
practical significance (See Appendix).
In any case, it is possible for specific cases to confront results based on
he development of limit values and those derived from rigorous formulae
‘in this context, also see the Appendix, § 5338).
(*) Relations (223-7), (231-5), (227-3) and (240-,,
240-8). (226-2), (226-3) and (220-2).
1
Allais - pag. 103
300 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CASE 2=0
242. Few of the formulae take on an interesting form when
:=o and p#0.
It is only worth noting that
242-1)
®=0, fori=c
#0
() Relations (232-3) and (220-2).
11] Allais - pag. 104
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Erc. 801
KE.
— CASE OF AN EXPONENTIAL DECREASE
OF ELASTICITY j(6)
Hypothesis (d): The ratios of the elasticities of primary inputs
depends only on their distance in time
250. If the assumption is made that the ratio of the productivities
of two inputs supplied at moments 6, and 6, (f
depends only on 6, - 6,, then considering three moments
and 0,(0,<0,<{0,) we have from (2r
250-1) B(03 - 6,) = B(03 - 05° B(9-From
the above it follows that
(250-2)
3(0) =Ae-"®
[hen from the relations (211-3) and (220-f,
250-3)
250-4,
+] Allais - pag. 103%
302 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
whence
(250-5)
k
ae, ¢
Supplementary hypothesis (d) thus corresponds to an exponential
decrease in the elasticities 3(0).
Characteristics of the Exponential Model
251. We have here from (220-1), (220-6) and (220-14)
¢ (u J
(251-1)
_
+0 u
J
A
Using this expression, the formulae applying to the exponenal
model are derived directly from those already given (1).
They are presented in the table below (2).
1) Relations (223-7), (231-5), (227-3), (232-3), (224-2), (225-3), (226-1),
226-9), (226-2), (226-3), (228-12), (229-5), (233-2), (233-3), (25I-5) and
(251-6).
2) The notation Ry represents the maximum values of RB for i—o
‘117 Allais - pag. 106
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 803
T-honential Mod.
(251-2) B(6,
(251-3) | (8)
©
(251-4) | @(8)-—
6
*
Gt
. {.,
vis
zu
{251-5) jar
251-6) |0=0,/[1+0,
(251-7) R_.=[1+0,(i-p)|R,
251-8) R=[1+0,¢|R,
-(®!
(=251-9)
C_(1+0,7)C,
TE,
251-10) C
(251-11
2251-12
® Rr,
“
-0
-0-e
©
1+6G
tr
.] Allais - pag. 107
304
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(251-13) Teo=Y(i=p)= ©, CC
(251-14) |Y,=Y(i=p)=0 /(1+p 0
Pa
Re _ CA €
(251-15) [2
G,
©,
| i, _— {[x+(i-p)0,]e >"
Ra
5 : 1% 1;
(251-16) — = i e ©
R, L® i
(re i)e %‘1f
R ° |
M
= R, | ko 5 R, ]"
(251-17) R_, =a(t)] == |e R,= a(t) Ze
e® eO,
In these formulae, Ra, Re, R, Rg, R, i and p are functions
of time. The quantities (?- p), Yo, Y, ©, 0, Re/Row, R/Ru
are invariant.
Those formulae in which (0) and Ô appear involve the
supplementary hypothesis that à and o are constants (!). In the
same way the expressions (251-17) given for Rey and Ry are
valid only on this hypothesis (2).
It can thus be seen that in the exponential hypothesis, the
significant expressions depend on only two coefficients: the
coefficient of homogeneity k, and the time constant ©,.
Of all these relations, perhaps the most striking is that
which specifies that the ratio C/R, is a constant and equal to
() $ 230 to 232. _
() Relations (233-2), (233-3), (228-4) and (250-5) for Rom: relations
(229-5), (251-1) and first relation (251-17) for Ray .
‘11] Allais - pag. 108
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 805
the constant ©. This condition signifies that the value of capital
per unit of primary income is independent of the rates
of interest and growth : and © (").
Of course it is again found that R, is maximum for i-p=o.
But it will also be remarked that national income R is maximum
for :=0 whatever p. This latter is not a general finding;
it is true only for the exponential model. The notation Ry
represents the maximum value of R.
It should also be observed that relation (251-12) between y,
vc and p is independent of the model. It is based only on the
constancy of (i - ¢) and the invariance of the function (68) over
ime [§120 and relation (125-2)].
Expansions in series form as functions of à and 7 - o are done
as previously (2) by putting
which value corresponds to the exponential expression for
For small values of (? - ç) we find here
(251-18)
T= es -
oy
T
k 2
© ({-p,
and for the small values of ¢
(251-10
R.,
h
Se,
§ 240.
(!) In my Tokyo monograph (Arrais, 1960 A) I derived and made deailed
comments on the different relations given above for the case n=o0
g. 109
11 | -Allais - pag
306 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Value ot R,- R,
In the case of the exponential model, we have, according
to (251-4)
(251-20) R,-R,=[r_ 2 (i-¢) | R,
1+O,:
oo P 0, R,
for small values of ¢ and p. Now, from (110-6) and (251-9),
we have
(251-21)
at _ e, Ra
di di
=0 OR
Thus we have
(251-22)
dt
(1
Thus, in the exponential case consumed income is practically
equal to capital income (see § 116) and we have from (116-22),
(251-20) and (110-2)
251-23) : R. ~ R.+iC
(!) We have also this property in the general case. In this case. we have
from (231-3), (123-2), (122-4), (224-2) and (220-2)
1
rel vy Ro Ÿ ? ® Ro
I1—4(G—e)]
à GS 5° Re 960 Ro
II] Allais - pag. 110
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
Ji
Here, from (116-13), (110-1), (251-21), (251-7), gross me
ional product as understood in this paper is given as
251-24)
. +R,
aC Ê
+7} a
Van
wr
7
u
-v
6, os
LL 6
‘hus
(251-25)
Re
vv
1° Ko
disregarding terms beyond the second order in à an
interesting to note that:
(251-26)
«
uv
+0: 4
Ur
+
Finally, relations (251-8) and (251-9) show that the share
of interest in national income and in national reproducible
capital is the same.
Allais - pag. 111
308 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Value of the Capital corresponding to the Share of Primary
Income which subsequently appears in National Income
between years 0, and 0,.
252. From relation (123-8) and writing t-p=u we have
8, 1
(252-1) Ce (t) =Rol}—
1
fs 1
5 Jor_ (5%) K 2 |
-——_0le © _e ©
-U
a
whence, in particular
1 _u)® _2
° I | rete ) (re o
~
; q
pk
(252-3)
( —(=-u)e _2
CT re © Le ®
u I-4 ©
or alternatively
( . uo
æ — t É 1
252-4) C, (2) R,( ï 76
UL
‘11] Allais - pag. 112
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
30G
Clearly, in the light of the observations presented above
in § 115, and from relation (123-9), the expressions for C ,
C8, and C$, can be obtained by writing i=o0 in the preceding
relations i.e. by putting u= -p.
For 6=0, we have (relation 25
©
252-5) C(t) = —= Ra(t)=0.Ry
which is relation (251-9) given above.
Case of a Stationary Process
253. The formulae corresponding to a stationary process
can be deduced immediately from those just given by putting
> =0.
In this case, of course, we have
253-1) @=9, ©=0, R=R, y=r,
à
<
as before. Here the Capital Output Ratio 7 is equal tc
average amortisation period ® of primary capital C
11€
(!) Relations (251-5), (251-6), (251-3), (251-4), (251-7), (251-8), (251-11).
251-12), (251-15) and (25I-16).
2 Relations (251-6), (251-10) and (253-1).
“11] Allais - pag. 113
$10 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA V/2IA - 28
PART III
CONFRONTATION OF THE MODEL
AND OSBERVED DATA
A. — THE GENERAL MODEL AND OBSERVED DATA
310. In my Tokyo monograph (1960) and my New York
lecture (1961), I examined the justifications of the hypotheses
made in, and the conclusions flowing from, the model whose
main properties have been described in the preceding sections,
both for the general case and for the case of exponential decline
of the elasticity B(0).
In the space at my disposal, it is not possible to cover
these different points in detail, and I will do no more than
recall the main features. The interested reader can find the
Jetails in my earlier publications. The discussion below also
ireats of several particularly interesting technical points which
[ have not previously had occasion to present.
For clarity’s sake, the general model will be studied first.
. ESTIMATION OF THE PARAMETERS OF THE GENERAL MODEL
() The Coefficient of Homogeneity k
311. I suggested in my Tokyo monograph (!), and in my
Econometrica article that there are three reasons for believing
that
311-1,
(!) Arrars (1960 A) § 30.
11] Allais - pag. 114
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
The evidence for this is derived from study of the RosTAS analysis
of 31 British and American industries in the 1930’s ('),
of the results derived by DoucLAs and subsequent researchers
and finally, the stability of economic equilibrium when suitable
monetary conditions obtain.
2) The function db(u)
312. Unfortunately, in the present state of knowledge, i
appears to be impossible to determine the function ¢(«) of the
general model with precision. Indeed, the only elements ‘or
which accurate numerical data are available on the basis
existine statistics are the ratios
C
45
x
1
A
Now, it has already been shown (?) that for the mode!
(212-1)
312-2)
(312-3)
+
r-¢ (i-p)
-p d (1-p)
4 (i-e)
-pl¢ (i-p)
.Ù
() RosTtas (1948).
?) Relations (226-3), (225-3) and (225-2)
vai Allais - pag. 115
3
)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2R
The expressions for these three ratios contain only the function
¢(i - p), so that for any country at a given moment, there
is in fact available one and one only piece of information about
b(w).
Eliminating ¢(z - p) from these three expressions, we find
(312-4)
(312-5)
R +
R, 1Y
Re - I-Ÿp
R
These two relations are in fact relations (123-6) and (123-7)
which were developed in the examination of quasi-stationary
process with a variable rate of growth of primary income,
and they are absolutely independent of hypothesis (b) (1).
The first has general validity since it results from (110-2)
and (110-9) (§ 226), the second follows exclusively from the
fact that we have simultaneously
(312-6)
(312-7)
rT dC
_ ZZ _p
C dt
1 dR,
| R. ra
1) § 211
11] Allais - pag. 116
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 813
The same is true for all the relations which can be derived
therefrom (1).
Estimation of ©,
313. If the assumption is made that the function ¢(u) converges
rapidly for small values of #, the relation
313-1)
which neglects first order terms which include 7-p can be
used to derive an estimate of the order of magnitude of © .
Estimation of
314. It would be very useful to have an estimate of &, since,
ander the hypothesis of an exponential decrease of the function
3(0), we have &=o (*).
However, even if rapid convergence of the function ¢(u)
for small values of u is assumed, and admitting as a consequence
that we can write (4)
(314-1) ¢(i-p) oo 1-0, (ip) +(1+3) © (i-p)
1y
This is notably the case tor the interesting relation
given above (125-2).
™ Relation (226-3) and
Relation (251-1).
Relation (240-1).
x |
Allais - pag. 11°
314 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the knowledge of ¢(i - p) on its own can be used to derive only
a single relation between ©, and &.
Thus, for example, the statistical knowledge of Ÿ gives the
relation (1)
raze. y €
314-2) ER © ©, [r-O,?-à @(?-p)]
In other words, there is only one empirical relationship available
to determine the two parameters ©, and 8.
The condition (?)
5 2
343) Re [= (i-e}-50;(i-e}]
R_ 2
from which a second expression could be derived, is practically
unusable, for the variation of Ro/Rey With 7 and p is secondorder
and completely negligible by comparison with the influence
of the other factors determining the level of real consumed
income.
Alternatively, we could think of using one of the two relations
(3) (4)
(314-4)
(314-5)
®~ ©, [r-0,(i-p)-28@,(:-p)]
®~ 0,[1-0,i-23 6,4]
Relation (240-11).
Relation (240-13).
y Relation (240-5) and (240-6).
4) We have 0—0 ~ (142 8) 02 pn
11] Allais - pag. 118
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC
but recourse to available statistical material to determine
functions ¢(0) and © (0) directly, and using them to deriv
and Ô would involve such large estimating errors that the «. .-culation
of ©, and & from the two preceding relations we
be completely devoid of significance.
It is easy to see this. For, from relations (240-5) and
'240-10) we have
(314-6)
y © «302%
¥
so that
(214-17)
3 1-0
®: (i-p)
or, putting ©, © ÿc as a first approximation
(314-8)
5,
1.-0
i-p!
It is obvious that, even where the relative error in © is
relatively low, the relative error in the difference Yc - © will
pe high. The same is true of the difference ¢ - ¢ (!). So much
(") An accurate estimate of i poses obvious difficulties; the same is true
for n. We have
>
An estimate of p in which this parameter is taken as being equal to the
rate of population growth can clearly only be a first approximation. It
follows that for small values of i and quite high values of o, the relative
error in the estimate of i—n can be fairlv large.
111 Allais - pag. 1710
316 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
so that in the existing state of knowledge, even if the gratuitous
assumption of rapid series convergence is made, the possibility
of making a serious estimate of the constant & appears to be out
of the question.
At the same time, it should be clearly understood that these
difficulties do mot arise from the model itself but ave inherent
in the nature of things.
2.
JUSTIFICATION OF THE GENERAL MODEL WITH RESPECT TO ITS
HYPOTHESES
General Model - Hypotheses (a), (b) and (c)
The general model is based on three hypotheses:
(a) Paretian optimum and invariance of the difference
(8) - pl),
(b) unchanging production elasticities and
(c) a steady rate of growth p of primary income R, (§ 210-211
and 230).
Hypothesis (a)
315. The assumption of a Paretian optimum can be accepted
as a first approximation, at least insofar as equivalence over
time of primary inputs and the comparison in time of their
relative productivity is concerned.
The hypothesis that the difference i(¢) - p(¢) is invariant is
an asymptotic property of capitalistic processes which can be
demonstrated on the basis of very general assumbtions (1).
(”) Arrats (1962 A) p. 702, ALLAIS (1963) $ 117 and 127 and ArrLals
1964) $ 43.
1] Allais - pag. 120
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 817
Hypothesis {1
316. The assumption of logarithmic linearity is justified by
the results obtained in the research undertaken elsewhere on
production functions (1).
[t is further justified by the fact that a general property of
nature is that the empirically measured regularity of numerous
ohenomena can be expressed a wide range by logarithmically
linear formulae. This explains, for example, the relative
stability of demand elasticities, the approximative constancy
of the rates of growth of population and production of dif-‘erent
kinds, and the general applicability of the lognormal
statistical distribution.
It is worth stressing at this point that this hypothesis can
only be considered as valid in the context of a finite range of
variation of the parameters (?) (3).
The invariance of 3(8) over time is an assumption which
can have no a priori justification. It can only be justified by
its consequences.
Hypothesis (c): Stability of the Rate of Growth ¢ of Primary
Income R,,
317. The hypothesis of a constant value of the rate of growth
o> can be justified by the fact that if economic development is
(") ArLaIs (1960 A), $ 37.
‘?) In discrete notation if the production function (note 2, p. 45, $
R=a R,%» , Rt ,..., Ra’, ...
were considered valid for any value of primary input, the manifestly unacceptable
conclusion would follow that R would necessarily be zero if any
one Rg were zero.
?) Naturally, hypothesis (b) implies the more general hypothesis accordng
to which there exists a valid index R. of the real national consumed
income ($ 115, 119 and 211)
Allais - pag. 121
318 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
considered in terms of trends, the rates of growth appear as
approximately constant over large segments of time.
At all events, it must be stressed that the great majority
of the results are obtained independently of this hypothesis (}).
The hypothesis of constancy of p enters into the culculation of
0, R., ® and Rom (?) only.
3 - JUSTIFICATION OF THE GENERAL MODEL WITH RESPECT TO
ITS CONSEQUENCES
There are at least two points on which the general model
appears to be justified bv the conclusions to which it leads.
a) Weak Relationship between Real Income per Capita and the
Capital-Output Ratio at anv Given Moment
318. We have seen that
(318-1) Ro) _ | Ze gore
R.,(t) Lr-Ye
where Ron(f) is independent of the capitalistic structure (3).
For kev1, © 4, and ¢ and p of the order of a few percent,
this equation shows the relationship between real income
(') § 220 to 220.
(*) § 230 to 233.
(®) Relation (228-14).
by (233-2) and (233-3).
Ron (f) is given bv (228-13) and if o is constant
‘I1] Allais - pag. 122
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
Bis
per head and the capital-output ratio y=C/R to be a very
weak one in different countries over a given time span, and
‘his is in conformity with observed facts.
In practise, no statistical interdependence can be found between
real income per capita and the capital-output ratio y.
While this coefficient has real influence, it is insignificant by
comparison with the influence of other factors.
It can thus be seen that the theory presented leads to the
expectation of, and provides an explanation for, a weak relationship
between real income per capita and the capital output
coefficient.
Such a finding demonstrates only the consistency of the
model with observed facts. It does not prove that the assumption
of logarithmic linearity of the production function is necsssarily
correct. It is evident that there exists an infinite number
of production functions which imply a low interdependence
of real income and the capital-output ratio.
D) Weak Relationship over Time between Real Income per
Cabita and the Cabital-Outbhbut Ratio
319. As the coefficient ©, is assumed to be constant over
time, the preceding formula leads to the expectation of a low
relationship over time between real income per capita and the
capital-output ratio, and again this corresponds to observation.
But here again, no conclusion can be drawn other than that
the model is consistent with real” -
., Allais - pag. 123
320
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
5.
— THE EXPONENTIAL MODEL AND
OBSERVED DATA
ESTIMATION OF THE PARAMETERS OF THE EXPONENTIAL
MODEL
320. The exponential model contains only two parameters,
k and O,.
The estimate of the value of % is derived in the same way
as in the general model.
Estimation of ©,
321. We have found the relation
321-1)
ef
I—7 Ÿ
with y=C/R (}).
Thus ©, depends only in y and 7 and this relation holds
whatever the value of p.
Methods of estimating y and ¢ are given in my Tokyo
monograph and in mv New York lecture (2) (3).
Relation (251-13).
ALLAIS, 1960 A, § 42.
ALLAIS. 1062 Â, DD. 714-715
ir]
Allais - pag. 124
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
XA
Here are the main results
“ABL
\ vera gr”
Average
rXNC
The estimates of ©, are of the same order of magnitude for
he United States from 1880 to 1956, and for the United States,
France and Great Britain : Get
2.
JUSTIFICATION OF THE EXPONENTIAL MODEL WITH RESPECT
TO ITS HYPOTHESIS
Hypothesis (d): The Relative Productivity of Primary Inputs
322. Hypothesis (d) (3) cannot be confirmed directly from
experience, and can only be justified by the concordance of
the conclusions flowing from it with observed data.
However, a priori, it seems intuitively reasonable to assume
at least as a first approximation that production elasticities
decrease exponentially with time.
Further, this assumption is closely related to hypothesis (b,
of invariance of the function 3(8) over time. In fact, postulating
the invariance of the capitalistic production function of
the productive system over time implies not merely the cons-‘ancv
of the function 3(8), but, also, if we think it over, the fact
elow
Allais - pag. 125
322 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
‘hat the relative productivity of two primary inputs depends
only on their distance from each other in time, and this is
exactly what hypothesis (d) says.
3. JUSTIFICATION OF THE MODEL WITH RESPECT TO ITS CON-SEQUENCES
The main points on which conclusions derived from the exponential
model are justified are the following:
a) Small Inter-country Differences in the Value of the Capital-Output
Ratio at the Same Point of Time
323. The relation
(323-1)
yo 9
+0 2
shows that if ©, as a technical constant applies to the different
countries considered, the capital-output ratio y should change
but little, for according to the formula such changes cannot be
very great at observed rates of interest and with a value of ©,
of about 4.
Thus, for 1913, I have calculated ®, as (?):
United States
France . .
Great Britain
Average .
3-44
3.83
3.67
2.64
* Relation 251-6.
?) ATLAIS (1060 A), D. 715; (1962 A), PP. 52-53.
11] Allais - pag. 126
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. wi.
2h
b) Low Variability over Time of the Capital-output Ratio
within the same Country
324. According to the assumption made, the function [(0)
is in variant over time, whence it follows that ©, is a constant,
independent of time (!). This leads to the conclusion that for
any given country, changes in y are the result solely of changes
in ¢, which means that vy should vary but little over time for
any one country (2).
The record shows that this is indeed the case for the U.S.A.
where, considering years in which comparable levels of full
smployment were attained, the coefficient y takes the values
shown below (3).
TABLE =
Year
r880
:890
-000
3.13
3.87
3.69
Period
1880-1900
Period
Average
2.56
Long Period
880-1913:
average: 3.48
1906
[9I0
913
[923
1929
1937
[050
[955
‘056
3.37
3.38
2.44
3-34
3.46
2.88
3.40
3.30
i
1906-1913
1923-1937
1950-1956
2.40
3.56
1880-1956:
average:
3.46
1 § 211, 250 and relation ,23C-5
4) Relation (323-1).
3) ALLAIS. 1962 A, p.
, Allais - pag. L.
324 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The exponential model thus leads to the expectation that
the value of the capital-output ratio will differ little across countries
at different times.
If we use COLIN CLARK’s figures, consisting of 58 estimates
for 21 countries for different dates between 1805 and 1953,
we find that they are lognormally distributed with a median
value of 3.54 (1).
The overall results are therefore as follows:
U.S.A., 1880-1956 (median of 12 observations) . . 3.46
U.S.A., France, U.K., 1913 (average) . . . . 3.72
World - 1805-1953 (median of 58 observations) . . 3.54
The concordance of these different estimates is absolutely
remarkable. It suggests a temporal and spatial regularity
which in any case must be explained.
The explanation given by the model is a simple one.
c) Constancy of the Ratio C/R,
325. The exponential model contains a very remarkable relationship:
325-1)
C()=©, Rolf)
2
|" }
According to this expression, statistical analysis should show
an approximative stability over space and in time of the ratio
of capital to primarv income.
() Arrars (1960 A), pp. 54-57; (1062 A), p. 72:
(?) Relation (251-9). PP. 54-57; (1962 A), p. 743
11] Allais - pag. 128
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
YZ
Since for arv economn
325-2)
.. -R,
t follows that we have
(325-3)
for any economn
In other words, testing that the ratio C/R, remains e--to
a constant ©, is equivalent to testing that the ratio»
remains equal to this constant.
This is precisely what was done above in estimating
wal
$
d) Exponential Form ot Lhe
amortization Function
Ae
*
226.
The exponential amortization relationship
(326-1)
!) Relation (r1o-2’
2) Relation (251-3
~ Allais - pag. 120
326 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
derived from the exponential model appears to concord very
well with the results obtained from analysis of available data
on durable goods in all cases where a sufficiently large competitive
market in these goods exists (1) (2).
An attempt to derive the fonction ¢(6) directly, at least
approximatively, can be made with the help of statistics of the
distribution of the working population. This analysis involves
major difficulties, but at first sight seems to produce results in
conformity with an exponential form of ¢(8) (3).
The length of the production period 6=y is of the order
of 3.5 years for the U.S.A. For the average amortisation period
® =vc, from (251-6)
(326-2)
&
B
— 60
Whence, for Ô=3.5 and p=1.7% (*), © is derived for
the U.S.A. as 3.72, i.e. a value which differs little from that
5
of
(1) Arras (1960 A), p. 22-23.
2) It may be objected that in a rigorous formulation, the composition
ot different exponential amortization functions cannot result in an expoaential
global amortization function over the whole range of variation (0, oo)
of @. Nevertheless, over the useful variation interval (o, 100) of 0, use
of this type of function is certainly possible. as it involves a rather low
-elative error.
() Arrais (1960 A), § 45.
(*) Equal, as a first approximation. to the rate of population growth in
the T7.S A
11] Allais - pag. 130
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
y.
~ _NERAL COMMENTS
Constancy of ®
330. The observed stability over both time and space of the
coefficient ®, may at first sight seem extraordinary. However,
on further reflection, and in the framework of the model, this
stability becomes much less astonishing. It expresses the fact
‘hat whatever the state of knowledge, the decrease in the productivity
of the factors of production as they become more
distant in time remains constant. In a certain sense, this is
an index of the conceptual difficulty involved in thinking about
roundabout brocesses (1.
Estimation of © and the Distribution of the Active Population
331. There is a manifest relation between the value of the
average amortization period ® and the composition of the work-‘ng
population, and it would seem reasonable to envisage estimating
the value of ® directly in order to compare it with its
theoretical value, which in the case of the exponential model ic
(~~ —N
Ou I-2 7
") Relation
‘?) Relation
(211-0) and
251-11)
y
Allais - pag. 15.
328
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
Any hope of doing this is illusory, since it has been shown
hat whatever the function 3(8), for small values of à and © we
have
(331-2)
0 ~ ry,
(1)
In practise, the rates à and p are quite small and it is therefore
practically certain that the order of magnitude of the
value of © should be about the same as yc. But if this concordance
does occur in fact, it only shows that the reasoning
based on the composition of the working population is correct;
in no way does it demonstrate the validity of the model proposed.
This demonstration would only be possible if the determination
of the function ¢(8), and therefore of ©, from the
composition of the working population, could be done accurately
enough for the difference yc- © itself to be determined
sufficiently precisely. As has already been noted, this is an
undertaking which appears to be impossible, at least in the
present state of knowledge (?).
The Influence of the Function [3(0)
332. It is interesting to examine what the results which
correspond to the exponential model become when a non-expoaential
expression is used for B(6).
Appendix I contains a study of the case in which the quite
acceptable hypothesis is postulated that B(8)e*® can be repre-(")
Relation (227-6).
For i=-n=0, we have
mn
3) § 312 to 314.
‘111 Allais - pag. 132
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
sented by a convergent Taylor series for any value of 8. It is
worth remarking that in most of the cases which can reasonatly
be envisaged, the coefficient ¢ remains relatively small, where
apon the properties of the model are quite similar to those of
‘he exponential model.
Amortization of Primary
Tncor
333. The relationship between the amortisation function ¢ (6,
and the usually accepted notion of amortisation raises a number
of problems to which allusion has already been made (!). It is
of interest to examine these problems in the light of the results
which have been obtained.
1) In the first place, the usual definition considers investment
as occurring when an already completed investment good
begins its productive career. By contrast, the act of investment
is considered in this paper as taking place when the
primary inputs are furnished. A distinction must therefore be
made between the amortisation of primary income and the
amortisation of investment. It further follows that under the
usual definition, amortisation contains an element of interest as
well as the depreciation of primary inputs, whereas in the
present paper, amortisation relates only to the primary input
content of the investment.
2) A part of the primary income invested at a given
instant ¢ emerges in the global producion P* of consumption
and investment goods at instant (£40). The primary inputs
which correspond to the latter are thus reintroduced into the
Allais - pag. 15
330 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
oroduction process. Some portion of them will reappear at
‘nstant (£46 +0’) in the global production of that instant. The
primary inputs which correspond to the durable goods element
of this production again enter the production circuit, and so on.
There 1s therefore a close relationship between the amortisation
schedule of primary income (08) and the amortisation
schedule for durable goods from the time they emerge from
‘he production process and are themselves put into operation
to produce further new production of consumption goods.
Clearly, of course, the amortisation schedule for primary income
is not identical to the durable goods schedule.
The supply of reproducible capital at a given moment is
composed of three elements:
a) Inventories (stocks and work in progress);
b) Equipment (machines, lorries, motor cars, washing machines,
etc.);
c) Structural assets (factories. houses. roads. etc.).
No item of primary income arising at any given moment
can emerge in subsequent consumption unless it has been
Incorporated during the interval either in inventories (stocks
and work in progress) or in equipment or structural investment.
There is thus a close relationship between the amortisation
function of primary income ©(0) and the structure of
eproducible capital.
Finally, since inputs of labour constitute a part of primary
Income there is also a close link between the amortisation funclion
of primary income ¢(8) and the composition of the labor
force which corresponds at any given moment to the allocation
of the amounts of labour available as between the different
phases of the production process. In particular, as has been
noted earlier, the invariance of the function (0) over time
11] Allais - pag. 134
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 85
follows from the invariante of the composition of the labor
force (1).
Any assumption concerning the amortisation function <(6)
of primary income can therefore be tested against observed
realitv from three points of view:
2) The composition of global production P* in terms of consumption
goods and investment goods, and the amortisation
schedule for investment goods and durables;
b) The structure of reproducible capital;
cr) The composition of the labor forc..
3) The shape of the amortisation schedule ç,(6) for any
particular economic sector i generally bears little relation to
an exponential curve, at least for sufficiently low values of 6.
7:(0)d0 represents the share of primary income which will
emerge in consumed income during the interval 6 to 6 +de.
But, taking steel manufacture as an example, the industry’s
output can only enter final consumption after a period of at
least some months; it must first pass through a number of
intermediate stages. It follows that primary income absorbed
by the steel industry can only emerge into final consumption
after a certain lapse of time. The function ¢(8) thus starts
with a value of zero, grows, reaches a maximum and then
declines. The same is true of the majority of economic sectors
(3).
Naturally,
333-1)
R.ÿ(0) = 4 R oi 9,"
where R,; 1s the primary income devoted to sector
() § 120.
() The shape of the majority of ¢(0) functions is therefore similar to
that of the function « (0) which corresponds to the para-exponential model
>xamined in annex (Section II A, $ 520 to 524).
- oq
Allais - pag. 135
332
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
In the present state of knowledge, it is difficult, if not impossible,
to assign a particular form to the ¢,(8) functions. At
the same time, the analysis in the present study provides numerous
reasons for believing that the shape of function ¢(6)
should not differ too greatly from an exponential form.
4) Global production P* at any given moment consists
of the total of consumed income Re and investment I:
(333-2)
P*=R~+1
But on the cost side,
(333-3)
P*¥*=R,+ +:C
where primary income R, includes the wage bill and rent from
ownership of property, iC represents the interest charged on
the global value of the existing reproducible capital, and €
represents the effective amortisation of reproducible capital
defined by the relation
(333-4)
dt
{|
Thus, we have
333-5) P*=R~+I=R,+ A +iC
4
(") See ArLLars, Les Fondements Comptables de la Macroéconomique
The Accountine Bases of Macroeconomics)
11] Allais - pag. 136
3EMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
5) Global production P* corresponds exactly to gross national
product P’ as conventionally defined if the two followne
adjustments are made:
a) In the first place, the quantitv
should be added to P’, where C¢ is the overall value of existing
stocks of durable consumption goods, i the rate of interest,
and gc the rate of growth Co. Valuation is in terms of the unit
of primary income, i.e. the basic hourly wage-rate 2.
b) The Gross National Product, P’, as usually defined
covers private consumption, private investment and government
purchases of goods and services. But a part only of
zovernment expenditure should be considered as corresponding
to final collective services rendered directly to the public; the
outlays which correspond to services rendered to the business
sector should be deducted. As a first approximation, this quanity
can be taken as taxes paid by business
Thus the relationship between P anc
333-6)
when Jg represents the taxes 1..d * business.
() 8 116.
2) § 110.
*) On this point, see ALLaIs, Les Fondements comptables de la macroéconomique
(The Accounting Bases of Macroeconomics), op. cit., pp. 48-46
and 54-55, and KUzNETs, Government Product and National Income, In
rome and Wealth, Serie I, Bowes, Cambridge, pp. 178-104.
4llais - pag. 137
334
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
6) The estimates of capital-output ratios y and Yc given
in § 320 and § 326 have been calculated on the basis of available
figures for national income. In point of fact, the compiation
of these figures is subject to many conceptual difficulties,
and the data must be adjusted if they are to be used for even
a midly rigorous application of formulations in the present
study. It may be helpful at this point, to illustrate the problem
by examining data for the U.S.A. economy in 1956.
(n that year, it is possible to take as a first approximation:
i =2.55%
i
7}
?
:=4.7%
=0.85%
() à here is the pure rate of interest in wage units, since the unit of
valuation in the present study is the unit of primary income i.e, the basic
hourly wage rate ($ 110). In my New York lecture, published in « Econometrica
» in 1067. I calculated
- m
os
Z
where i, is the pure rate of interest in nominal terms and ¢ the rate of
growth of hourly wage rate (p. 714).
() § 326.
() Since the composition of reproducible capital is fairlv stable over
ime. as a first approximation, it is possible to take
al
— A
dt
L
11] Allais - pag. 138
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 835
and. in billions:
Then, from (333-6
P* =
=— 410
.2+
0
.00
85 x 163
4 -
35
7=38
4.0
Private investment (excluding consumer durables) amounted
to $64.4 billion (°). To this figure should be added purchases
of consumer durables amounting to $38.5 billion (9),
making a total for private investment outlays of $105.9 bilion.
The value of public investment is not specified, but it
may be assumed that both kinds of investment are in proporion
to their share of existing non-monetary reproducible cavital.
This enables the value of investment to be estimated as
~~ x I05.9 = 121.8
() « Economic Almanac », 1962, p. 121.
(?) Excluding monetary metals (« Economic Almanac », 96.
valued at $26.5 billion:
1226.2 — 26.5 = 1197.7 .
The figure of 1226.2 corresponds to the global value of reproducible capital,
of which 26.5 represents the value of monetary stocks of gold and silver.
There is also a question as to whether military assets ($84.3 billion)
should not be included also. These assets contribute to the supply of the
anal collective service « defense ». On balance, it seemed preferable to omit
them, but it is open to discussion whether they should not have been
counted in.
() Ibid., p. 140.
*} Indirect business Tax, Ibid., p. 121.
() « Economic Almanac », 1962, p. 122
f¢) Ibid., p. 140. In billions of dollars.
Public buildings and equipment =35.6+145.7+ 5=156.3.
The figure of 156.3 corresponds to the overall value of the different classes
of governmental reproducible capital. excluding military assets valued
nt 84.3
Allais - pag. 130
336
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Correspondingly, from (333-5)
Rc=P* -1=384.9- 121.8 =263.1
whence
C 1199.7
rs ——— — 6
Y R- 263.1 45
Again, in billions of dollars:
Value of inventories =122.3
Value of equipment =340.8
Value of structures =736.€
So that the value ) of the share of reproducible capital
accounted for by stocks and goods in course of production
ran be calculated as
X
122.3
— ——— = 0.102
1190.7
Since
1333-7)
1333-8)
(333-9)
333-10)
Ro=[1-Ye(I - p)]Re
dC/dt =pC
dC
a=]- dt =] - eC
2)
3)
4
2)
(5)
{!) « Economic Almanac », 1962, p. 140.
(?) Relations (rro-1), (110-2) and (123-2). It may be recalled that the
relation (123-2) is based on the two assumptions that (i—p) is constant and
that the function (6) is invariant over time, i.e. that the composition of
he labor force does not change (120). Both assumntions are fairly weak.
@) Relations (251-9) and (110-6).
“) Relations (333-4) and (333-8).
3} Relations (110-1) and (333-8)
ac Th
Allaic - pag. 140
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
557
it can therefore be deduced that
Re = (1 - 4.56 x 0.0085,
1C dt=¢C=0.017 x IIG”
Ad =121.8-20.4- 107
263.1 + 20..
C
—U.Uxs +3
<0
or
For the U.S.A. in 1956, relation 533-5, above
P¥*=Rc+I=R, +9 +:C
can finally be written
oO “
*
wy
(') This figure differs greatly from the 34.4 given in the « Economic
Almanac », 1962, p. 12, which related however only to private investment.
On the assumption that the relative values of amortisation of consumer
durables and government investment are the same the 34.4 becomes
34.4 -- — zg
1109.7 — 210.7
1199.7
The 319.7 is derived as the total of 163.4 (consumer durables) and 156.3
public investment). The difference between the two figures 101.4 and 46.9
s a reflection of the inadequacy of the usual methods of estimating amortisation.
Whatever calculating process is adopted. it should respect the
dentity
on which I have based the calculation
4llais - pag. 14
338 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
It follows from this relation that primary income R, is
allocated both to consumption and investment goods, and as
a first approximation it may be assumed that the share of
amortisation and interest in Rc and I is the same (!). On this
basis, the position can be summarised as in the following table:
TABLE 7
J.
~ A Tn-T
Tntal
7,1
8
-~
%
68.4
31.6
100
2t
The table shows that under the assumptions made, the share
of primary inputs in investment at any given moment corresponds
to
fu
R= 0316
only some 329, of primary income, whereas they represent
4
y =0.808
in the region of 909%, of reproducible capital. This quite remarkable
circumstance calls for explanation.
(!) See relations (251-8), (251-9) and $ 251 in fine. See also Tables 8
and o below
11] Allais - pag. 142
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 839
The Case of the Exbonential Model
334. In the case of the exponential model, it is possible
to precise these different indications:
1) In the first place,
(334-1)
(334-2)
334-3)
d=v:=C/Rc
CR
+
—~
x
_
334-4)
whence
sw
('} Relation (2°
?) Relation 1-irom
(251-6,
0... E
wi
(}) Relation (2s7-4)
Relation 251-;
Allais - pag. 143
340
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
It can thus be seen that if the adjustment items (i - P)Ce
‘upwards) and Jr (downwards) on Rg are taken into account,
the values of ©, 6 and ©, are some 159%, higher than when
the calculation is based on uncorrected figures of national insome
(1) (2).
2) In the case of the exponential model, it is of interest
to distinguish the contribution of the different primary inputs
to primary income, consumed income, capital and primary
capital, according to the period in which they appear in consumed
income. For illustration purpose, three intervals of
time (0, 0,), (©,, 6,) and (0,, ) will be considered.
The shares of primary income which correspond to the
intervals (o, 6,), (0,, 8,) and (0,, ©) are respectively (3).
(334-5)
(334-6)
354 /.
R° 9, 5
wo [ e(0)d0-1-e ©
9 e, 9, 0.
Ry, o=/ ©(0)d6=¢c ©, ©
Ry 0 7
82
2
¢ e
Jde= ©
(6
©
() § 320 to 326.
3 If military assets ($84.3 billion) are included in the evaluation of
reproducible capital (seg note (2) D. 130 above). higher values still are
“eached
AN
© = 5.14 © = 4.72 0. = 5:37
but the inclusion of these assets when valuing capital is open to auestion.
(3) Relation (251-3). It will be recalled that
Qo
[ c (6) da
11] Allais - pag. 144
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. Oa
Clearly, consumed national income can be divided up into
‘he corresponding shares, ‘or which the €: nressions ar- from
(334-8)
BE.
v
(334-0)
0, CU
Rae T°
334-10)
vith (1)
(334-11)
and
334-12)
1} Relation (2, ,
Allais - pag. 145
342 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
From relation (252-1), the corresponding shares for capital
are
1
(334-13) C!… Ro rele 74) ( 2
# \T rue 77° J)
(LX _uye (1 _y _ 21 _&
rR [lect [3-3]
9, = —{ — — — : ]
1 u 1-Ou
uo _ 22
(334-15) C° -R, lo] e 2
2 1-0 24 uw
nw is given by (334-16), whence
81 02 œo _. I Ro
(334-16) C=CH+CE+CR = He
45 Ro=ORo
=)
{
Clearly, from the discussion in § 252, the corresponding
shares of primary capital
&,
Cou
Ci io
m--Lo,
à
() Relations (251-9) and (25I-5).
11] Allais - pag. 146
SEMAINE D ETUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
can be obtained by putting i=o0 in the expressions for
breakdown of capital i.e. by putting
(334-17)
4
Bac
-
_L1€-in
the three relations (334-13) to (334-15).
Finally, the average amortisation periods for the three
sroups considered are
334-18)
1
:(e)de ‘
=
334-19)
«AV
334-20)
The last o: tiese c:xpressions is particularly simple.
aally
334-21)
‘7.1 Allais - pag. 1
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
3) As a further illustration, take
A, I vear
0,=75 years
From the estimates presented above
© =4.56 u=i-p=0.85%
61.7%
and the following table may be constructed from relations
(334-5) to (334-20).
TABLE 8
Characteristics for @=4.56
u=0.85% p=1.7%
6,=1 6,=5
Exponential hvpothesis
Share of Consumed Income
Breakdown of Re, Ry, C and C
œ
5 0< 00
D<BcT |
NN TOO
n.461 |
0.651
0.340 |
R
Share of Prime-+ Income
nN. 1Q7 |
0.460
nN RAG
0.334
Share of C- v+
Share c
Capita
Averag
+ +,
*imar
——
Capital
ation
Period
N.020
nN 0272
nN A180 |
3.48
0.272
0.203 |
2.601
2.71
0.202
O0.2I5
2.045
2.08
0.708
0.685
N.257 |
nN. 56
111 Allais - pag. 148
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. = 4!
[t can be seen that primary inputs emerging in consumed
income before a year has elapsed represent 209%, of primary
income, but only 29, of capital. On the other hand, inputs
which appear in consumed income after 5 years represent 719%,
of capital, but 339 only of primary income.
It will also be observed that by reason of the low values
of p and (: - ç) the difference between the shares corresponding
to consumed income and primary income remains relatively
small; this is also true of capital and primary capital.
For the same reason, the amortisation periods ©: differ
relatively little from the ratios C#/RE.
Finally, it can be seen that for small values of 8,, the contribution
of primary income to capital is practically negligible.
[f (334-13) is developed as a Taylor series, the main comnonent
of C% /C is seen to be 6; /2 ©2,
4) These theoretical results help to explain the empirical
figures obtained earlier (!) showing that the primary income
incorporated in investment at any given moment represents
32% of the total primary income of that moment, whereas
the contribution of the different inputs of primary income over
ime to reproducible capital is of the order of 90%, (?).
5) To make an approximate assessment of the relationship
between the empirical results and those in Table 8, we
adopt the following hypothesis (H), bearing in mind that it is
not rigorously verified, and indeed, can at best be only a
rough assumption. The hypothesis is that primary inputs
incorporated in investment can only emerge in consumed income
after the expiry of a period 0°, whereas the primary
‘nputs which enter stocks and goods in course of production
1 3336.
4) Each contribution representing 32°, of the primarv income of tha
noment
1 Allais - pag. 146
346 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
all appear in consumed income before the end of 6°. The
approximate nature of this hypothesis is evident from the fact
‘hat certain primary inputs which are incorporated in reproducible
capital in the form of maintenance outlays or repairs
appear in consumed income immediately, while at the same
lime the primary inputs which correspond to stock formation
may only reach consumed income long after the inputs corresponding
to certain types of investment.
If hypothesis (H) is taken as correct (which, rigorously, it
is not), the contribution of primary income to investment,
from the figures in table 8 and relation (334-15) should be
6’
Rw =¢ © =) =— 0.316
i.e., for ® =4.56
8’ =5.25
From the preceding discussion, this certainly is an overestimate.
Similarly, if hypothesis (H) were true, the contribution of
primary income to fixed investment (i.e. capital immobilised
in the form of equipment and structures) should be
co”
$_|,
C
5 ev’
u |
€
+
I
x
Or
\
=O0I
© = 4.56
u=0.0085
(!) See Chart II below.
11] Allais - pag. 150
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 47
the value of 60’ is determined as
—2.4C
The earlier discussion suggests that this estimate is an understatement.
Thus, if hypothesis (H) were verified, we should
have
2.46 < 6° < 5.25.
Since the exponential model is assumed to correspond tc
reality, the distance between the two values for 8” results from
‘he incorrectness of hypothesis (H); but the fact that these
wo values are of the same order of magnitude does suggest
‘hat hypothesis (H) does not depart too far from realitv
Taking the geometric mean of the two estimates,
nit
v-25=3-59 -
For this value of 6’, the following theoretical values are
found :
© See Charts III A and
“TL 7? below
i Allais - pag. 15.
348 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
instead of the observed values u= 0.32 and 1 - A=0.go0 respec-Hvely.
Here again, if the exponential model is assumed to be
correct, the divergences are the result of the incorrectness of
hypothesis (H), but the concordance of orders of magnitude
shows that the interpretation of reality corresponding to hypo-‘hesis
(H) is acceptable as a first approximation.
As might have been expected, the values of 0°=3.59 which
nas been derived is of the same order of magnitude as the
average production period, 0 =4.23. In point of fact, if the
supplementary hypothesis (H’) that all investments involve
a production period equal in length to the average production
period is made in addition to hypothesis (H), it follows necessarily
that
y =6=4.23
It may be concluded from this that if the formulation of
the exponential model can be considered as conforming with
reality, approximate assessment of the real situation can be
made by considering the hypotheses (H) and (H’) as valid.
Taking 0,=3.59, and using for illustrative purposes the
value of 8; =0.25 (3 months) the following table can be derived.
111 Allais - pag. 152
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 8490
TABLE Q
Characteristics for
D=4.56 i—p=0.85%
0=1.7% 6,=0.25
0.=3.59
Exponential Hvpothesis
Share of consumed income
Breakdown of Rg,
J<8<0.25
0.25<0<3.59
0.051
0.479
Km. C, ana
0< 83.30
0.531
4% -
3.50<08<00
0.470
R_.
Share cf ‘*im-' ' income
Share
Share
Car:
Average
riod
mr
0.053
0.0014
D.O0IA
0.125
0.124
0.492
0.170
0.106
[ 704
[.718
0.545
0.455
0.180 | 0.820
0.197
0.8073
1.554
7.056
1.562
8.15
6) Whatever the amortisation schedule (0), for low va.
lues of : -
334-22)
a
-C
It follows that if any two of the three quantities CZ , R
and ®* are known, the third can be estimated approximately.
1} Relations (112-1) and (123-8)
153
lais - pag.
1] Alla
1] A
350 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
If it is now assumed that the amortisation periods for equipment
and structures are respectively 5 and 20 years (1), and if
it is further assumed that the average production period for
equipment goods is (Hypothesis H”)
® = 4.23
‘his leads to assessing at 10 and 25 years respectively the approximate
length of the amortisation period of the primary
inputs incorporated in equipment and structural assets.
If it is assumed that all investment consists of durable
goods, and maintenance and repair outlays are abstracted,
the position can be summarised as in Table 10 below (in $ billion)
on the basis of the estimates in § 333-6 (2).
TABLE 10
(nventories
Zquipment
Structures
- +
. .
Total
Equipment and
Structures .
[22.3
340.8
726.6
[190.7
1077.4 |
RA
199.5
34.1
20.5
262.1
63.6 !
C/Ra
0.61
Io
26
4.56
16.04 |
C | Re:
n % in %
10.2
28.4
61.4
75-9
13.0
II.1
TON
ra
80.8 |
24.1
() These are the figures used by various fiscal administrative agencies.
(?) The values of R, for equipment and structural assets are obtained
y dividing the figures in the first column into those in the third. The
?, figures for inventories are then obtained bv difference
263. T — 34.1 — 20.5 = 100.4
11] Allais - pag. 154
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 851
It can thus be seen that the figure of 10% of capital corresponds
to 76% of consumed income for inventories, while
for structural assets the corresponding figures are 61% for
capital and 119%, for consumed income. For equipment and
structures as a whole 249%, of consumed income corresponds to
00% of capital.
This table has an evident relationship with tables 7, 8 and o.
The divergence between the amortisation period of some
17 years, found for equipment and structures considered as a
whole, compared with values of some 8 and 10 years in tables 8
and 9 is due to the fact that the assumption has been made in
the calculations in Table 10 that investment is constituted solely
of durable goods, and that (given the non-availabilty of valuable
data) repair and maintenance outlays included in the
value of equipment and structures have not been taken into
account. But the average amortisation period of repair and
maintenance outlays is probably of the order of two to three
years, and since their value is relatively high by comparison
with total investment in new goods, it is probable that in
a full calculation the g2-~ would be considerably reduced, or
might even disappear
) Let C and R be the capital and consumed income corresponding
to investment, C, and R, the share of that investment represented by new
goods, and C, and R, the shares represented by repair and maintenance
autlavs (these svmbols are valid only for the present note). Then
[f the assumption --Tv
R.C
R
‘hen
alan
Thus, if annual repair and maintenance outlays account for 1/4 of the
total annual resources devoted to investment, the average amortisation
period falls from 17 to about 13 vears. If it were assumed that repair and
Âllais - pag. 155
352 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Assuming that the expenditures required to maintain existing
reproducible capital in a usable state amont to one-third
of total annuel investment, and that the average amortisation
period for these outlays is 214 years, it is easy to arrive at
the following table (1).
TABLE II
+
[Inventories . . .
Repair and maintenance
. . - .
Equipment (1) . . |
structures () . .
‘22.3
73.0
317.6 |
486.8
Total . | 1T1Q0.7
Equipment and!
Structures . . .]
[077.4
Equipment and
Structures (Y) . .|1004.4
R
174.0
2G.2
31.8
27.5
262.1
R8.4
oz |
C/Re
0.70
2.5
Io
25
1.56 |
12.18
6 gb
C | Re
a % mn %
10.2
66.4
6.1
26.5
R7.2
11.1
12.1
10.4
roo
|
TOO
Ra 8
22.6
83.7 |
22.5
(!) Non-amortised value of initial installations, excluding the value of
anv repair and maintenancv charges incorporated.
maintenance outlays represent 1/3 of total annual investment, the amortisation
period would decline from 17 to 12 years approximately.
It can be seen from this calculation how much interest would attach to
availability of estimates of the sums required to maintain reproduction
capital in functioning order.
() In allowing for repair and maintenance outlays, the elements of Re
which correspond to equipment and structural assets respectively in Table 10
pecome 34.1 (I—#) and 29.5 (1—#), so that the share of the value of fixed
capital accounted for by repair and maintenance outlavs becomes
[T0 x 34.1+25 x 29.5] # =1077.4 x.
The corresponding share of Rg is 1077.4 x/®’, where ©’ is average amorrisation
period for repair and maintenance outlavs. If K is the percentage
11] Allais - pag. 156
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
855
Obviously, given the non availability of valuable data,
‘his can be no more than an indicative calculation. Nevertheless
its results are quite suggestive if they are compared with
‘he different items in Tables 7, 8 and 9 which relate to the
percentage contribution of primary income to capital and consumed
income.
Thus, the percentage 66.4 appearing in Table Ir in the
breakdown of Rc differs relatively little from the 68.4 in
Table 7, the 66.6 in Table 8 or the 54.5 in Table g. Similarly,
the overall percentage of 83.7 corresponding in the breakdown
of capital to equipment and structures after exclusion of repair
and maintenance expenditures is directly comparable to the
figures of 70.8 and 82.0 in Table 8 and g. Finally, the amorlsation
periods of 12.18 years for equipment and structural
assets in Table 11 is directly comparable to the corresponding
periods shown in Tables 8 and 9 as 9.56 and 8.15 respectively
(1).
7) In usual practise, only those outlays whose amortisaion
period is one year or longer are in principle considered as
nvestment.
However, in the first place, this limit is no more than a
which repair and maintenance outlays represent in annual investment, then
we should have
vhence
= 0.060
from which the various figures in Table II follow.
() If it is assumed that the cost of maintaining reproduction capital in
operating condition represents one half of the sums devoted to investment,
the breakdown percentages for C and R are found respectively as 10.2
11.5 - 24.8 and 53.6 for C and 59.0 - 20.0 - 11.3 and 9.7 for Rec.
11] Allais - pag. 15;
354
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
principle. Often enough, an outlay which will to amortised
over 18 months will still be charged against operating expenses.
Let 1; denote the practical limit.
Secondly, investments whose amortisation period is below
t; and which are accounted as operating expenses contain
both primary inputs and earlier inputs incorporated in the
existing supplies and equipment. Thus, minor maintenance
charges include at the same time wages, supplies produced
some time earlier and the reward for the use of equipment.
Thus the maintenance outlays which emerge into consumed
income before 1; include not only the primary inputs supplied
at moment ¢ but also primary inputs corresponding to goods
produced earlier. The primary inputs of moment £ enter consumed
income before 1, and at least as a first approximation,
the second group of inputs can be considered as entering consumed
income before (1, +6) where © is the average length
nf the production period with the year as time unit (%).
It may be assumed, at least as a first approximation that
the shares of the two categories of input are respectively proportional
to
(334-23) A -
Ro
N=[f
we consider the weighted average
(334-24) O'=AT,+ (1-1) (1, +0)=1, + (1-1)
1) 0 is defined by relation (112-2).
2) Relation (333-5).
11] Allais - pag. 158
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Etc. 855
it appears possible to consider as a first approximation that
investment goods, understood in the usual meaning of the
term, include all primary inputs which emerge in consumed
income after 6’
Since
9’ =3.59 © = 4.23
_ =0.É54
we find
-
3 59 - (1 - 0.657) x 4.23 — 2.74
a value which appears to be quite reasonable in the light ol
the discussion above.
8) It is also possible to derive an approximate representation
of reality by substituting the following hypothesis (H”,
for the two hypotheses (H) and (H’): the amortisation schedule
for the primary income embodied in stocks and goods in course
of production follows an exponentia’ law:
334-25,
ab!
r1] Allais - pag. 159
356
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2°
Thus, the share of primary inputs which have been incorporated
in stocks and goods in course of production and emerge
in consumed income between 0 and 0 +d0 is
334-26)
(I —_ À) Re
©. df
while the share of primary inputs which have been incorporated
in equipment and structures and emerge in consumed income
between 6 and 0 + 40 is
0 b
(aay ee (9 204
(334-27) Ro | g ——Ng |as
For the first of these two shares (stocks and goods in course
of production), the general formulae of the exponential model
may be applied by substituting ® by ©,, so that the capital
corresponding to expression (334-25) is (1)
(334-28)
1=and,
in the notation of § 333
(334-29)
C a — A 0,
X= CT I-G-00, 6
() Relations (251-5) and (251-9).
11] Allais - pag. 160
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 857
so that
(334-30)
The following values have been found earlier:
\ =0.316 % =0.10 i - p=0.85% O,=4.74
whence
© =0.69
Thus the value of ®, found here is fully comparable with
the values of 0.61 and 0.71 in tables 10 and IT.
For the value of 6°=3.59 found earlier
x =.0055
indicating that for 8=6’, the primary income corresponding
‘o stocks and goods in course of production is almost completely
amortised .
This calculation shows that the hypothesis (H”) does not
differ greatly from the hypothesis (H).
.1 Allais - pag. 161
358 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
If ©, is put equal to 0.69, the average amortisation period
for investment, ©,, is
(334-31) © Je | #
, = © De EE
0. df
© - (1-21),
4-56 - 0.684 x 0.69
0.316
whence
O,=12.0
T'his value must be compared with the values shown in tables 8,
9 and I1 of 0.56, 8.15, and 12.18.
9) Taking hypothesis (H) in conjunction with table o,
the average amortisation period of the primary inputs corresponding
to investment I is 8.15, whereas it is 12.9 in the
calculation immediately above. At first sight, these figures
appear to be rather low for the entire range of equipment and
structure (1).
() At all events, it should be borne in mind that, independently of the
basic assumbtions of the exbonential model, we should have (condition
(23-12)
Yc
ec
= I—vyc (i—p)
2 relation which holds for primary inputs taken as a whole whatever the
function ¢(6).
This condition depends onlv on the assumption of invariance over time
11] Allais - pag. 162
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 859
[t is quite easy to make a direct estimate of a reasonable
order of magnitude. From $ 333, the value of equipment and
structures (in $ billion) is
C,=(1-%) C=1100.7 - 122.3=1077.4
The annual amortisation of this capital is
a, =1-(1- dC
nes
I-(x1-%) pC
.e. from the estimate derived earlier
A, =121.8 - 0.9 x 0.017 x 1077.4
=121.3 - 16.5 =105.3
Thus, a first approximation to the length of the amortisation
yeriod for investments is derived as
®,=1077.4/105.3=10.23
of the function (60) i.e. the invariance over time of the composition of
the labor force. Thus, for the U.S.A. in 1956, we should have
4.56
T—0.0085 x 4.56 = 4.74 -
Thus there are structural reasons for the average amortisation period for the
whole range of primary inputs to remain relatively low.
() It will be recalled that for fairly low values of p and i—p, the
1verage amortisation period of an element of capital is approximately equal
to the ratio of its value to the annual amount of amortisation, whatever
the amortisation schedule
‘11] Allais - pag. 163
360 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - “
However, to obtain the average amortisation period for
the corresponding primary inputs, this figure must be increased
by the average length of the production period for investment
goods, whose order of magnitude is 8'=3.59 (!). The average
duration of the amortisation period for the primary inputs corresponding
to investment is then
®,=10.2+3.59=13.79
which may be compared with the estimates of 8.1 5 in Table g
and 12.9 in § 334-8. From this point of view, the approximative
hypothesis (H”), which leads to a value of 12.0 (?) is
to be preferred to the interpretation underlying hypothesis (H),
which leads to a value of 8.15.
Clearly, if it is assumed that the general formulation of
che exponential model is valid, the difference between the
theoretical value of 12.9 calculated under hypothesis (H”) and
the observed value of 13.8 may be explained by the inaccuracy
of the assumption in (H”). But in the light of the appealing
nature of this assumption, at least as a first approximation,
the concordance of the orders of magnitude of the two estimates
can be considered as underscoring the validity of the
exponential model. At the same time, the results derived using
interpretation (H) show that this hypothesis offers only a
roughest resemblance to realitv.
10) Since the average production period for consumed
income is ©, it may be assumed as a first approximation that
‘he average production period for investment goods is also 6.
It is then also possible to represent as a first approximation
the reality by the following model (Hypothesis H”). In each
(") § 334-5.
(3) § 334-8
11] Allais - pag. 164
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 861
period of time equal to Ô, total production is equal to consumed
income plus invested income, and the corresponding
investment is completely used up during the following period.
In the first period, the share of R,(¢#) which appears in consumed
income is (1 - A), while A is invested. In the second
period, the share of primary income R,(f) which appears in
consumed income is A(x - A) while 22 is invested. In the n°”
period, the fraction of primary income R,(f) which appears
in consumed income R” of that period is given by (!)
an-1 (1 - 2) -
Thus, as a result of this process, the amortisation schedule
of primary income is of exponential form, and that over a
period of length ®, the amount of non-amortised primary income
is reduced in the ratio of 1 to À. It follows that if the
formulation of the exponential model is taken as exact, the
share À of primary income incorporated in investment is
since © is the lengh of the amortisation per:oc.
i
|
=4..
Te
vi
50
Naturally we have
=),
A) G+FA+
4.
[11] Allais - pag. 165
362 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the value of A is found as
A =0.40
This figure is of the same order of magnitude as the 0.316
found in § 333-6. Here again, the divergence can be attributed
‘0 the approximative nature of the hypothesis (H”), on the
assumption that the exponential model is valid. But if this
approximation is admitted, the concordance of orders of magnitude
indicates that, if hypothesis (H"”) is taken to be approximately
true, the exponential model can be considered as not
departing very far from reality.
This reasoning shows, at least intuitively, how the assumption
of exponential amortization of primary income is justified.
If the general formulation of the exponential model is taken
as exact, it also supplies an estimate of the order of magnitude
of the proportion A of primary income incorporated in investment,
which agrees with observed data (1).
(!) Of course it would be of the greatest possible interest to generalise
‘he above reasoning to the case in which the lengths of the production
ind amortisation periods of different types of investment are no longer
considered as uniform, but the frequency of a production period extending
irom § to 6+ df would be (6) de, and the frequency of an amortisation
period for primary inputs extending from 6 to 0+d6 would be ¢(6)d6,
with the discrete model becoming continuous (See § 333-2 above). Up to
now, I have been unsuccessful in making a calculation of this kind. If it
could be done, it would doubtless enable the validity of the exponential
model to be verified efficiently, by determining the relations between the
quantities A, x and @ and the data Rc, C et p, using this model.
In present circumstances, and in the absence of a calculation of this
kind progress can onlv be made on the basis of rough estimates
11] Allais - pag. 166
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 863
Turning now to the overall amortization period for primary
income, here this period, given the assumptions made, is
equal to
334-32)
& -(; ; 8 axon’ A
LYR-1(s
3 ÎT | -
¥
whence
(334-33)
For
O=4.23,
A=0.316, (!) we find
®=4.05.
[his value is of the same order of magnitude as the values ©
® and 6, 4.56 and 4.23 respectively, found earlier (-,
11) Of course, the concordance of the orders of magnitude
of the different values of the quantities studied in the
preceding calculations cannot be considered as proof of the
validit~ "te exponential model. The successive hypotheses
(1
5 and
374-7
11] Allais - pag. 167
364 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(H), (H'), (H”) and (H””) cannot be considered as rigourously
valid, and the computations have involved a certain number
of further approximations. At the same time, the non rigorous
validity of hypotheses (H) to (H'”) may account for the
divergences noted between the various estimates that have been
derived.
Be this as it may, the rather striking coherence in the whole
set of results derived using these different assumptions and
applying quite rough calculating techniques, does seem to
show, given the apparently reasonable nature of the interpre-‘ahons
(H), (H”), (H”) and (H””) which have been succesively
considered, that the exponential hypothesis can be considered
as a fairly close reflection of reality.
12) Finally, the results of the analysis described above
are not out of line with data based on the composition of the
labor force.
a) Given the composition of the labor force, the estimate
derived from the above analysis that the share of primary
‘ncome devoted to investment is of the order of 30% to 35%
seems quite reasonable. Thus, in 1955, the composition by
sector of the working population in the U.S.A. was (1):
Agriculture
[ndustry .
Mining .
Construction .
Transport .
Trade . .
Services . .
(Government
Total
(*) Statistical Abstract, 1959, pp. 206 and 2ro.
of these figures is beyond the scope of this paper
11.8%
29.2%
1.4%
4-9%
7.2%
23.0%
10.4%
12.19%,
[OO
"J.
/
A detailed discussion
11] Allais - pag. 168
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 865
About one-half of industrial production consists of durable
goods (1), and it can be taken that about 4 of the output of
aon-durables is reinvested. Thus, it can be considered that
18.59% of the labor produces primary inputs for durable goods.
[t may also be considered that the construction sector’s activity
results for the greatest part in the production of durable goods,
while the corresponding figure for mining may be of the order
of 50%, so that the corresponding percentage of the labor force
for the two sectors is about 5%. The contribution of inputs
from the trade sector to the production of durable goods can
be estimated at some 209%, of its output, accounting for a
further 4 4 % of the labor force. In total, the share of primary
inputs in durables production arising in the industry, mines,
construction and trade sectors adds up to a total of some 28%
of primary income.
It may not be unreasonable to assume that activity in the
transport and governmental sectors (19.39) is distributed between
durables and non-durables in line with the average for
other sectors, so that the total contribution to production of
non-durables of sectors other than transport and government is
“30 —
[t follows from this set of estimates taken as a whole that
the share of primary income going to non durables and investment
may be respectively of the order of
. oO
65%
1
ER AA
ALLAIS.
rokvo Memorandum (1960 A)
1] Allais - pag. 169
366 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus, we find for the percentage of the primary income
which is embodied in investment a figure which is of the same
order of magnitude as the figure which is given by the preceding
analysis.
b) From Tables 8 and 9, some 5% and 20% of primary
income emerge in consumed income during the following quarter
and year. These estimates seem also perfectly compatible
with the figures for the composition of the labor force given
above.
But it should be stressed that analyses based on considera-Hon
of the breakdown of the labour force, given the present
state of the analysis must be considered as very rough, and
of indicative value only.
Determination of the Function «(0)
335. It does not appear to be absolutely impossible to determine
the function (6) from an analysis of the composition of
‘he working population and available data on the structure
of production, at least on an approximative basis.
Unfortunately, this calculation involves a great deal of
work, and up to now I have not been able to find the time to
do it, even though the results of such an analvsis would clearly
oe of the greatest interest.
It may well be, and in the author’s view it is quite likely,
that over the useful range of variation of 8 i.e. from o to 100
years, the exponential form
(335-1)
p(0)- Le ©
A
11] Allais - pag. 170
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Etc. 867
provides a good representation of the function which would be
erived empirically, at least as a first approximation.
Were it to appear that an exponential expression cannot
represent the function adequately, the assumption could be
made that the function e"*°¢(6), where p is an appropriate
constant, could be developed as a Taylor series and quite well
represented by its first terms. This is a relatively weak hypo--hesis,
and one which does not appear to be inacceptable. The
general properties arising when this hypothesis is postulated
are treated in the appendix ("
As an illustration, values ot
335-2)
335-3)
31
Ê :
(335-4)
335-5)
335-6)
X
ta
pa
_
«£
-t
Gu
(1) The study contained in the appendix considers the function [‘@
‘rom zero to infinity; but the characteristics at infinity of the functions ‘ ‘6)
and ¢(0) need not be taken into account in numerical applications. is
therefore possible to consider only a finite range of variation of 0. ‘or
example the interval (o, 100).
Allais - pag. 171
E.
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
‘OT
(335-7)
©,=4 years ® = 3.5 years u=4%
are given in the following table (!) (3).
The last three expressions listed represent respectively the
amortized and non-amortized components of primary income
and the share of capital corresponding to primary inputs which
are amortized after an interval of at least 6 (3).
TABLE 12
0
7ears
o
:/12
1/4
il?
LO
“5
20
25
50 |
"O00
B(6)
0.250
0.244
0.234
0.220
0.194
n.151
3.118
0.002
0.071
D.020
0.005
0.001
0.2 x 1073
O.I x 107%
0.2? x 10711
7(0)
0.285
0.279
0.266
0.247
0.214
0.161
0.121
0.091
2.068
0.016
0.003
0.0009
0.2 x 1073
0.2 x 1076
D.I x IO-!2
5
1a
0.976
0.931
0.866
0.751
0.564
0.424
0.318
0.239
2.057
0.013
0.003
3.8 x 1073
0.6 x 1076
O.4 x 1071?
J
0.023
0.068
0.133
0.248
0.425
0-575
0.681
p.760
2.942
2.986
2.996
2.990
2
©
TP/C
0.999
0.997
0.991
0.970
0.900
0.810
0.714
0.618
0.259
0.094
0.032
0.010
0.3 x 107%
AT x TN79
() The relations (335-2) to (335-6) are deduced from (250-5) (k=1),
(334-1), (334-5), (334-3) and (334-8). The relation (335-6) is deduced from
(334-8), (251-5) and (251-9).
(3) From (251-8) we have
[ I
0 0,
®,=4, ®=3.5 we derive u= 3.55% .
The values of ®, and © considered here correspond to the average
estimates there of ($ 321 and 326), and differ from those considered in
3% Three significant places have been kept
for
IT] Allais - pag. 172
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
LAC
The various results corresponding to different values of
®, ® and #=p are given in charts I, II, TIT A and III B.
The Exponential Hypothesis and Reality
336. The exponential hypothesis appears to be a very rest.
‘ive one, at least at first sight.
However, it is possible to verify that it may in fact provide
a quite fair approximation to real conditions.
From this point of view, there are three particularly important
coefficients to be considered. These are k, ©, and À
defined by the relations (!)
336-1)
336-2)
'336-3)
I" B(0)do
©,
a eB(e)de
A
—=— [ Ÿslo)ae
2k@ J
As it has already been noted, k should be taken as not
oreatly differing from unity (3).
M) Relations (211-3), (220-5) and (220-6)
2) § 311.
| Allais - pag. 173
A)
330}
4/6)
CT Cape 2
j sn id ‘ : £335
EXPONENTIAL MOOEL ~~ | d
REPRESENTATION OF FUNCTIONS ae) ano we)
Eo. FOR ®,z4 ®:36 YEARS … 3
|
; Rr
AB) ZL 2 2 Pa
oo iG
#l9-L. e 6 Spiess
Formulae. .{335,2) and (335,3}
k
=
=
‘ \
4
220 LA \_
B{8) À
340
>
4g)
N
x
18)
vears . 7% A
Lode.
orsame
years,
yo:
r
Chart H
[ | 5 335
IQ ENTIAL MODEL
AND NON AMORTISEO SHARE
PRIMARY INCOME
9
AMORTISED SHARE, 1e © 2
NON ANORTISED SHARE e ©
! !
(Relation (335,5) $ 335)
ri
—~
ul
;
+
or
-
T
cised shore
any income
.rging In consumed
“come after Oyeors)
T
106-anf
Ne
Rt \ de
Le ON
a
%
F Le] BEE M slaying ine. TE Eras;
4. TIAL MODES AT pass
EXPONENTIAL MODEL — 5
SHARE OF: CAPITAL oc ba REE
y CORRESPONDINS FO PRIMARY FMPUTS = ue
EMTERGING IN CONSUMED NATIONAL INCOME.
AFTER: A DELAY GREATER. THAN @ (= 1 "ES
FOR: BD» 8.5 TEARS i pa =
og hn. oily rom HE Ec
ef né ‘ ’
Ce, O4. cf ag 8 i For wed
Æ
—
N
0.25}
»
lai Ly Lig ELE
sented Fr SISTER EE Eg
sé CE ILES
a
«“
N
ry
15
4 a TE EE
EXPONENTIAL MODEL
; SHARE OF CAPITAL. 1
CORRESPONDING TO PRIMARY INPUTS |
RENÉ IN CONSUMED NATIONAL INCOME
NA DELAY GREATER THAN 6
FOR wu Ÿ
Chart ILE
§ 335
4
4
rf
; Relation {. 356) § 335
+
1
4
150)
J
=
nd
-
+
“0
9
AJ
874
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Since we have
(336-4) Ye=0,_ for 7-9:
4%.
(1
and ?- p is in general small, it follows that when an estimate
of 7, 1s available, it is possible to derive an order of magnitude
of ©.
Then if we consider all those models for which
(336-5)
E=1
®, vw.
they differ from each other by the ovder of magnitude of the
parameter A, which is equal to unity in the exponential model.
Since we have
(336-6)
0) B(o)e 77°
?(0)- k¢(i-p)
4
and
b(o)=T
f?
it follows that as a first approximation, ¢(0) differs relatively
ttle from $3(0) for k=1.
It then follows that, at least as a first approximation, A is
greater than or smaller than unity according to whether the
amortization is more or less vapid than that corresponding to
" Relation (240-10)
Relation (223-7).
Relation (220-2)
Allais - pag. 178
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
the exponential amortization which is a characteristic oi th
exponential model (1).
It is shown in the Appendix to the present study that under
the very general hypothesis that the function [3(8)e*’ can be
developed as a Taylor series, at least for a certain value of p,
the coefficient A (which is equal to 1 in the exponential model)
will not differ generally from unity by more than +o.5, for
hypotheses which can reasonably be accepted.
It follows from this that while the exponential model clearly
may not offer a perfect portrayal of real conditions, it doubtless
does not differ therefrom all that much, so that, at least in
terms of a first approximation, it is capable of depicting the
essential features of reality quite correctly.
If this approximation is deemed insufficient, the develop
ment of the function 3(6)e*’ as a Tavlor series of which the very
first terms only are retained, provides as far as can be judged
a reasonable representation of actual conditions (#
The Value of the Model
337. It follows from the preceding dimission that.
a) the model is consistent with the information which ca..
obtained from available statistical data:
b) it cannot be proved that the assumptions made are the only
ones which would give results consistent with available data
y From relation (z:zc
x
Thus if (0) declines more rapidly than
© © we have
since
for (60)=e ©/v.
(3) See Appendix
.| Allais - pag. 179
376 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Clearly, there exists an infinity of different mathematical
functions which in a restricted field give analogous results,
and this characteristic simply derives from the nature of things.
The point is that everything takes place as if the theory
presented and the model which illustrates that theory were correct.
As far as I know no other theory, no other model, have
aver been put forward in order to explain the same facts.
It may therefore reasonably be suggested that the theory
and the model presented here can be accepted and used, at least
as working tools, so long as no alternative theory or model is
advanced which leads to results which concord still better with
the real situation.
In any event, the theory and the model which have been
discussed here have the advantages that they represent analytical
tools and that they oblige the economist to reflect on a
large number of issues which hitherto have been insufficiently
studied, not to say completely neglected (1).
Finally, it may be observed that every theory has a twin
aim; on the one hand to describe and explain reality (2), but
at the same time to constitute a guide to efficient action. For
a given degree of approximation, the best theory at any given
moment is the one which fulfils the condition of being the most
convenient, or in other words, the simplest of all those which
represent reality with that degree of approximation. If this
criterion be admitted, and personally I know of no other (3,
the theory given here has the double advantage of being on
the one hand very simple, but also of describing and explaining
reality, so far as it can be comprehended with the information
presently available to us, i.e. of being compatible with
observed facts and simultaneously establishing coherent and
simple links between these facts.
(') Doubtless precisely because they were too difficult.
(3) In other words, to find relationships between the different aspects
of reality, expressing the most complex in terms of the most simple.
(3) See for instance HENRI POINCARE. The Value of Science (Flammarion)
:1] Allais - pag. 1R0
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
Ori
PART IV
PLICATIONS
400. The preceding theory admits of several particular
suggestive applications.
For suimplicity’s sake, the following exposition will be developed
in terms of the exponential model with first order homo
geneity.
A. — THE POSSIBILITY OF INCREASING REAL PER
CAPITA NATIONAL INCOME BY INCREASING CAPITAL
INTENSI
410.
From relation (251-15) we have for k=1
(410-1)
== 1_ ©
FE - © e © =[1+0, (2 ¢
1 ©
CM
el #
and from (z-,. .
, and (210-10)
1410-2}
® =Yc=C/Re
0,=Yc,
Allais - pag. 151
378
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
All the statistical data analysed lead to the conclusion that
in present conditions ®_ is of the order of marnitude
@ =
the time unit used being the year, and we have probably (4
38< O< 1.6.
The increase in real consumable income Rg which can be
had by moving from a value © to the value ®, corresponding
to the maximum value of Re is equal to
(410-3)
KR
= wo ee 8
g R°
or, from relation (410-1)
(410-4)
2
0
© eo
-—e
®
In each situation, the value of the difference 7 - ¢ is given
by the relation (251-5)
410-3,
1-6 rz 5
€
(1) This question cannot be taken up -here.
11 Allais - pag. 187
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
87
From the foregoing we derive the following table of values
of g and : (in °’)
TABLE
values
+
3= Y
4-4
4.2
+
3.8
3.6
3-4
3.2
3.0
2.5
2.0
1.5
1.0
0.5
1.0
0.5
0.I
0
0.2
0.6
1.6
3.1
10.7
29.5
82.9
332.8
957.4
—3.0
2.5
—I.3
0
I.5
3.1
4.9
7.0
13.7
23.7
40.4
73-7
173.7
0.4
D.T
2
0.1
0.6
1.4
2.7
4.7
13.9
35.9
98.6
402.1
1360.8
1.2
O
1.3
2.8
4-4
6.3
8.3
15.0
25.0
41.7
75.0
[75.0
2.1
0
0.1
0.5
1.3
2.4
4.2
6.6
17.5
43.1
116.0
484.1
[938.5
D
1.2
2.5
4.0
5.6
7-4
89.5
16.2
26.2
47.9
76.2
176.2
o.I
0.5
1.1
2.2
3.7
5.8
8.7
21.5
50.9
135.8
581.1
2762.5
2.3
3.6
5.1
6.7
8.5
10.6
17.3
27.3
43-9
77-3
177.3
The two charts IV and V which follow give the values of
Re/ Rom as functions of 7-¢ and 7c.
Real consumable national income is a maximum for ® =0,,.
We see that in an initial situation such as that of the United
States in 1956, in which the capital-output ratio y, is of the
order of 3.5, the maximum is very nearly attained, and that
for a corresponding value of ®, of 4, the possible increase is
only of the order of 1%.
The same is true for most western economies. The levels
of the capital-output ratio are such that they can be considered
as corresponding to a situation which is very near to the capitalistic
optimum.
Even in the case of economies in which the capital-output
ratio is only of the order of 2 (if there are any such), the
increase in real income to be derived from a lengtheni:is of
11
Allais - pag. 183
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
CHART IV
a, ++ | CONSUMED REAL INCONE i :
As A FUNCTION OF THE DIFFERENCE ip
IN THE CASE OF A HOMOGENEOUS PRODUCTION FUNCTION
"EXPONENTIAL MODEL |
Ê 4
hy
i -
7
Lb
10
pa rite of growth of om non
Ep vole of be capital - output ratio M2 C/R, For ii
Real fo A À
: co € id
= -[14üsi_qrle
H
3e
Fetations (4101) and (4192)
ovale
ue HY
3
‘111 Allais - pag. 184
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
CHART
88.
‘
Re CONSUMED peal INCOME | hl a
AS A FUNCTION OF THE CAPITAL-OUTPUT RATIO
IN THE CASE OF A HOMOGENEOUS PRODUCTION FUNCTION
{ EXPONENTIAL MODEL
100
LE
Lopitol - output rotio
-C/R
_ 1%
dae *
Rem 3
Repl income consumed
Relations (610 1) and ($70.2)
+
Allais - pag. 185
382 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the period of production, in the case where ®,6=4, is only of
the order of 36%, the difference i - p being some 25%.
It is only for capital-output ratios below 1.5, i.e. very low
ones indeed, that gains of real income could be very appreciable:
but in any case for which estimates has been available
no such low values of yc have been met.
Clearlv it follows that:
r) developed economies have relatively little to gain from an
increase in Savings,
2) that less developed countries, on the contrary, can realise
substantial gains from greater savings efforts when the marginal
productivity of capital is very high, but that these
gains are however not as large as is generally believed.
Significance of the Results Obtained
411. It can thus be seen that the real income gains which
can be had by lengthening the average duration of the capitalistic
process are in general overestimated, not to sav considerably
overestimated.
But a basic assumption of the model should be borne in
mind when making this interpretation. This is that there is
optimum management in all the situations considered, implying
that existing savings are utilised in the best possible way.
‘Paretian optimum hypothesis) (1).
This being so, although the results derived above show
that greater savings would not be very advantageous in an
economy in which there is optimum management, they in
no way imply that increased savings would not be very productive
if savings already accumulated had in the past been
invested in mistaken directions.
(1!)
§
II7
"111 Allais - pag. 186
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
883
In this latter case, a growth in new savings could have
major beneficial effects. They would however be due not to
the inadequate volume of savings which had been accumulated
in the past, but to their defective utilisation in the form of
economically erroneous investments.
In a situation of this kind, it is not the global volume of
accumulated savings which is at issue, but its use in the form
of non-productive investments. Thus inefficient management
vather than insufficient capital intensity must be considered
as providing the explanation of unduly low average productivity
(1).
[t is thus clear that the important factor is not so much
the volume of accumulated savings evaluated in terms of the
number of years national income it represents — which does
not vary greatly from one country to another, even including
underdeveloped countries - but the use that these savings
are put fo
(!) See my book: A United Europe, the Road to Prosperity (CALMANN-LÉvy,
1960), Chap. II, pp. 28 to 38 and 42 to 51 and Appendix € ‘ac:
to 207)
Allais - pag. 187
1
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
B. — PROCESS OF MAXIMUM GROWTH OF PRODUC-TION
PER UNIT OF PRIMARY INCOME FOR GIVEN
TECHNICAL KNOWLEDGE
Decrease in Real Consumed Income due to Growth in Primary
Income
420. We have seen [relation (233-2)] that in the case of the
general model we have
(420-1)
_ -kO,p
R_=K(t) e
where Rom is the maximum value of the real national consumed
income Re (1), K(#) is a certaih function of # independent of p,
is the coefficient of homogeneity, and © represents the average
(!) § 228 and 233 above. For a constant value of p we have
Roll
I
where Ra represents the national primary income ($ 110), œ(t) the effect
of technical progress (§ 211), k the homogeneity coefficient ($ 211) and k’
has the value indicated in § 228. For the exponential model we have [relation
(251-17)7
K(/) =o
and for E=1
K{() z
ec ,P
de
‘
7
111 Allais - pag. 188
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
DDL
amortisation period for
In other words, if we take the
condition
(420-2)
Thus Rey is maximised for 0: €
This means that of all growth processes, the most advan
tageous 1s that for which the rate of increase of primary incomes
IS zero.
This apparently paradoxical result is easily explained when
it is remembered that if population is increasing (and even supposing
that natural resources increase at the same rate, which
they don’t), this growth implies that every year some proportion
of gross national product must be allocated to maintain the
volume per capita of equipment and durable consumption goods
at the same level.
From the preceding relation, this diminution of consumable
production is equal in relative terms to
(420-3
ct
and we have for small values ot y
420-4,
r'6,
It further follows from relation (420-1) that a negative rate
of growth (population decline) would be advantageous. This
finding is valid so long as the assumptions, the consequence of
which is the constancy of (i ¢), remain valid. But these ass-{1:]
Allais - pag. 189
886 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2K
umptions would no longer be acceptable if the population
declined to a point where sectors experiencing increasing returns
became dominant (1).
The General Case
421. These results are completely independent of the hypotheses
concerning the function 3(8) providing that the assumption
be made that y is constant, at least as a first approximation.
In fact, it has been shown that for any quasi-stationary
process with steady growth. we have
(421-1) I dR 1 dc.
4 RU C7
&
so that from relation (110-1) We have
(421-2)
Ra=(1-0oy)R
whence for k=1 and at least as a first approximation (*)
(421-3)
R =(1 —0 Y)R
!) ALLAIS (1963): Some Analytic and Applied Aspects of the Theory
capital.
(*) Relations (123-2) and (132-2)
(3) Relation (229-2).
:1] Allais - pag. 190
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
887
and since from relations (251-12) and (251-6) we have
(421-4)
y » 6,
the preceding result relation (420-4) is found again.
Estimate of p for the United States
122. For the United States p and ©, may be taken ac.
0.017
oO,
and according to (4zu-4
The diminution ot real consumed income 1s thus «
order of 69%.
and 326.
i Allais - pag. 191
388
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
C. — COMPARISON OF PRODUCTIVITY FOR TWO
COUNTRIES
The Role of the Capitalistic Structure in the Relative Productivitv
of Two Countries
430. The theory presented enables the influence of the capitalistic
structure of two countries on their relative productivity
to be evaluated. For, from (410-1)
(430-1)
—_ 1 1
1 =1 ® + -_-)
RE
R- Ru 0,
where R! and R? are the real national consumed incomes of
the two countries considered.
According to (251-17) and for k=1 (8 211) we have
(430-2)
Bl 1 -0, (re -p,)
Rom _ R, (1) . 17%2
R® R(t)
where o! and ¢? represent the rates of growth of primary income
R. for the two countries
11] Allais - pag. 102
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 88
Finally we havc
(430-3,
of
The first factor represents the relative influence of the grow.
primary national income in the two countries 1).
The second factor
(430-4)
1
©
represents the relative influence of the capitalistic structures o
the two countries on the real consumed income per un:
primary income.
If ©, and ©, are of about the same values, it can vc con
cluded that this influence is negligible.
This will be so if y, and y, have similar values, since
AV
(430-5)
and since ç.
ana p, are generally small.
("Y See § 420 to 422 above
2) Relation (251-11)
ss, Allais - pag. 193
890 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
Comparative Productivity of the United States and French
Economies
431. The comparison between the productivity of the French
and American economies in 1957 can be summarized in the
following figures:
U.S.A. / FRANCE
Productivity: ratio
Equipment per worker: ratio .
23
2.4
Output per unit of equipment: ratio
Capital-Output Ratio
v=C/R
U.S.A.
3.3
{1056)
France
>3.3 (1
(1054)
From these figures it appears that output per working hour
in the United States is approximately twice as high as in
France, whereas the productivity of a unit of equipment is
about the same (2).
Certain analyst have sought to attribute greater American
productivity per man-hour to the larger volume of equipment
used there. However. it has been seen that
( N
431-1,
y £_8
R
fe
(1) As far as can be judged, y is of the same order of magnitude in
France as in the U.S.A. [ArraIs (1960 B), p. 29].
(?) ALLAIS (1960 B), pp. 28-32 and 105-207: ALLAIS (1062 A). p. 721.
(3) Relation (251-12)
rr] Allais - pag. 104
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
891
pd Trading GS TETE Tap WATT ORL Rn Tate ATT
= Since the French and American economies are characterized
by values of y of the same order of magnitude, it may be
concluded that the capitalistic structures of the two countries
are similar (!); they can therefore not play any role in accounting
for the average difference in productivity observed.
This average difference of productivity, of the order of 2.3
to I, should be attributed to other causes (3). It follows then
that with the same capital-output ratio y=C/R, the amount
of equipment per worker is approximately double.
In conclusion, the fact that the ratio between the physical
volume of industrial equipment per worker is 2.4 to I whereas
the value of capital relative to the national income is little different,
should not be considered as a cause in itself, but as an
effect of greater American efficiency. The fact that in physical
quantity American equipment per head is at present 2.4 times
as high as in France corresponds simply to the fact that, for
reasons other than the capitalistic structure, U.S.A. productivity
is about 2.3 times as high. It follows from this that for
the same value of capital per head measured in terms of hours
of work, the amount of equipment is about 2.3 times as high.
From this point of view, the concordance of the order of magnitude
of the two values 2.3 and 2.4, which were found as the
ratios of productivity and of the volume of output respectively,
is particularly striking.
Thus, if an explanation of greater American efficiency is
sought using the greater value of equipment per worker measared
in physical units rather then by: considering the capitaloutput
ratio as a means of estimating the influence of accumulated
capital, this is to treat as a cause a phenomenon which
in realit ‘s only an effect =~
at
(') Since they are characterised by the same function ¢() and © (0)
least as a first approximation (§ 111).
(?) ALLAIS (1960 B), Part I.
(3) ALLAIS (1048) and (1960 B), pp. 28-32
-1 | Allais - pag. 195
392
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Analysis of a Simble Model
432. These considerations can be illustrated by a simple
model which shows that the conventional explanation of greater
U.S. productivity as being due to the larger volume of equipment
per worker may be completely erroneous.
Consider an economy in stationary equilibrium characterised
by the three following relationships:
452-1)
"A=f [9X, E]
E=g [¢X,]
2
A is the quantity of consumption good (A) produced during
each period, E the output of an equipment good (E), X total
labour input of which a part X, is absorbed by industry (A)
while Xz represents the labour input of industry E.
The quantity E of equipment goods produced during a preceding
period is assumed to be used completely during the next
period.
Finally, gq is a parameter indicating quality.
The functions f and g are increasing functions of each of
the variables and are assumed to be of first order homogeneity.
We now consider two situations I and II characterised respectively
by two values ¢; and ¢, of q with
(43% 2.
the values of X, and Xi remaining unchanged.
“r1] Allais - pag. 106
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
843
Since the functions are assumed to be of first-order homogeneity,
we have
432-3)
[n situation Il
432-4)
because of two effects, one direct, since
432-5)
q, A.
the other indirect, due to
432-6)
The following
two situations
E-Sh
——
table summarises the comparison between
Situation
Situation
us
Productivity: ratio
Equipment per worker: ratio
Output per unit of equipment:
ratic
Capital-Output Rati
«a | Allais - pag. 197
304
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The case described here is completely analogous to the
American | French case examined above, and the fact that the
similarity is formalistic does not render it any less striking.
If he had no other facts than these, an economist from another
planet with no knowledge of the internal structure of the
model (!) would be tempted to aver that the higher productivity
in situation II resulted from the larger volume of equipment
in use.
But, by hypothesis, we know that such an explanation
woud be fofally misleading, since the higher productivity in
situation II is entirely due to the greater value g,=2q, of
the parameter q.
If, in this model, productivity is twice as high in situation
II, this is not because there is twice as much equipment
available, but because of a factor other than the capitalistic
structure. The fact that the volume of equipment per worker
1s twice as high follows from the fact that productivity is twice
as high while the capital-output ratio is the same.
In reality, it can be verified from the model that the volume
of equipment plays an intermediate role, and to clarify the
issue it should be left out of account. From relations (432-1)
we then get
(432-7)
A=flgXa, g(gXg)]
whence, because of the first-order homogeneity
132-8)
A gF| a , 2.
x X x
(") Characterised in particular by equation (432-2).
11] Allais - pag. 198
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8095
Thus we see that in the model considered, productivity depends
only on ¢ and on the coefficients X,,/X and Xg/E which
are moreover related to each other bv
(432-0)
As far as capital is concerned, productivity in the mod.
is proportional to 7 and from the capitalistic point 51 view
depends only on the allocation of labour between direc 7 |
and indirect (Xg) activity. This allocation is represente* hv
the « characteristic diagram
1
This diagram, and not th volume ot equipment, characterises
the capitalistic structure |
As between the two situations I and II above, the capitalistic
structure is the same.
It would, of course, be inaccurate to claim that the volume
of equipment is irrelevant to the higher productivity of situation
II, since it enters as an essential intermediary; but it would
be no less inaccurate to claim that the twofold productivity was
due to the use of a doubled volume of equipment.
--, Allais - pag. 199
396 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Naturally, this model proves nothing so far as the analysis
of the comparative productivity of the American and French
economies is concerned (!). It shows merely that a type of
reasoning which is generally considered as very attractive may
be completely false.
Analysis of the Difference in Productivity between the French
and American Economies
133. Equipment, as we have just seen, plays only an intermediate
role in the analysis of productivity differences between
the American and French economies. If is the capitalistic
structure which it is important to study.
The capitalistic structure is represented by its characteristic
function
|$(z,06)
D't 6)dh
which describes the time distribution of the inputs of primary
income used in the past emerging in the consumed national
income of instant /. There is everv reason to believe that this
() I should particularly stress that in my opinion greater American
productivity cannot be explained by a higher aualitv of the labour force
‘herve. (See AILAIS. 1060 B. p. 33).
‘111 Allais - pag. 200
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
curve is of exponential form (!). If so, it can be characterised
by a single coefficient, the average distance in time of the
primary inputs
(433-1)
~
J
¢\0)dd
whence it follows that if the values of O=v (*) are approximately
the same for the two countries, their capitalistic structures
are roughly similar and therefore cannot represent an evpla
natory factor of greater American productivitv
Relation
Relation
Relation
I-01 %
‘….
Gauss
“|
Allais - pag. 201
398 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
D. — POLICIES FOR CAPITALISTIC DEVELOPMENT
440. Productivity growth is an essential component of any
development programme. This growth may result from various
factors, such as an improvement of technical knowledge, better
training of executives and workers, etc. The purpose of the
present analysis is to estimate the influence on productivity of
the more or less capitalistic economic structure, in other words,
variations in the extent to which more or less use is made of
indirect production processes.
Evaluated in terms of years of national income, the capitaloutput
ratio in Africa or Asia is probably not of a very different
order of magnitude fro mits value in the United States.
Even assuming a value of only 2 (which seems very unlikely)
(!) against 3.5 for the U.S.A., this could not explain
more than a productivity difference of the order of 1 to 2,
whereas the difference is at least of 1 to 20. Thus, if American
production techniques were abruptly transplanted in Africa
(which would require enormous investment) without at the
same time remedying the existing causes of low productivity,
the African economy would not be able to replace completely
the investments which had been made in parallel with their
depreciation and obsolescence.
In fact, it has already been noted that the model implies
the apparently paradoxical relation
C=® R,
[4
M ALLAIS, 1960 A*, s 2!
2) Relation (251-0).
-1] Allais - pag. 202
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC
&_3
which nevertheless on the basis of the information available at
the present time appears to be verified by the available statistical
data (}).
As we have seen the economic significance of this relation
is that the value in wage units of reproducible capital does not
depend on the more or less capitalistic (i.e. more or less roundabout)
structure of the production process, as described by the
coefficient ®. The value in wage units of available capital is
independent of the rate of interest ¢ and of the rate of growth p.
This means that, evaluated in hours of work, the figures
for capital available per worker in so-called under-developed
economies should be comparable to the figures of the advanced
economies (1).
The implication of this is that the accumulated capital of
the advanced economies does not have any greater weight for
them than that represented for under-developed economies by
the capital they have available.
This finding shows that in formulating development policy,
it may be advantageous to adopt the most efficient production
techniques as quickly as possible. In this case, an attempt
should be made to keep interest rates so low that more round
about techniques can be applied.
However, it follows clearly from the preceding that the
more or less capitalistic structure of the production process
(O smaller or greater) does not suffice to explain recorded
differences in productivity levels.
In reality, the explanation of the enormous differences in
productivity observed as between the west and underdeveloped
countries ,half of the world’s population has an income less
than 1/20th that the average American (?) has much less to
do with below standard values of the capital-output ratio
than with:
a) differences in per capita availability of natural resources.
(M See ArLLAIS (1960 A), $ 25.
Arrais (1961 B) vol. T (n. 8), pp. 22-24 and vol. II (n. 7), pp. 39-80.
rr}
1
.. 1 Allais - pag. 203
900 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
b) differences in the level of technical education,
c) differences in economic management in general.
If follows immediately that the installation in Africa and
Asia of equipment analogous to that of the U.S.A. would not
be sufficient unless ways were found to deal with the other
factors which make for these countries’ lower productivity (1).
And this leads in turn to the conclusion that the general
emphasis placed in recent years on savings and investment as
the key to speedy development, a view held in respect of
Europe as much as for third countries, is based on an erroneous
position. The factors essential to development ave completely
different.
Fortunately, it appears that this point of view, which the
author has propounded continuously since 194%, is now shoved
more or less explicity, by an ever growing number of economists.
The present study basically constitutes its theoretical
justification.
(Y) Arrars (1061 B) and (1062 A)
‘11] Allais - pag. 204
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SJPENDIX
THE INFUENCE ON THE RESULTS OF A VARIATION
OF THE FUNCTION Ç(6)
It is very interesting to study what happens to the various
results when the function (8) is no longer of exponential form,
the aim being to see to what extent they are modified when
that function changes.
I will study first the general case in which, j being constant,
3(0)e*’ can be expanded as a Taylor series and second two
particular cases for which the calculations can be carried all
the way through.
It may perhaps be thought that the calculations of this
appendix are no more than an intellectual game. Yet they
are very important in the sense that they allow the demonstration
of the fact that under very general hypotheses, th- +arameter
À (!) does not differ very greatly from unity whence
it follows that the properties of the general model remain vzrv
similar to those of the exponential model
Relations (220-6), (220-14), (25i-,)
allu
anc
Allais - pag. 205
202
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
[. — THE GENERAL CASE
8(8)e** CAN BE DEVELOPED AS A TAYLOR SERIES
Hypothesis: The Development of B(8)e*® as a Tavlor Series
510. Assume that the function 3(8)e*’, in which y is a certain
constant, can be developed as a Taylor series (1). This is
a very general hypothesis. Then
n-1
(510-1) OBERT RL SL Pa
n-1!
Since
(510-2)
ref" u)_= [° e 2
-/ 8le)do of == / Bele do
we have
(510-3)
rab BL ms,
u pe pe
b 6 b
(510- b(u J [= 00 een =
510-4) ( ) R [nou + ms) +. aul +
(*) It goes without saying that for practical purposes it is only of use
to consider an interval for § of o to 25 vears. or at most of o to 100 vears.
(3 Relations (211-3) and (220-1)
[rr] Allais - pag. 206
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
This expression for ¢ enables all the
the case of the general model to be applied.
We have in particular
relations found in
(510-5) ¢(8,_ 043%4..
7
3
The Meaning of the
I.
903
The hypothesis made is equivalent to assuming that
(510-6)
B(6)= uw,
0 fo (8)+ w, B,(8)
3 0
{ ree 4 Uni 8a! 1+
with
(510-7)
Ono
‘> T
r
I
(510-8)
(510-9)
Dy,
-
Relation (223-7;
Lay Allais - pag. 207
2304
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(510-10)
wW + WU 40000 + ww _,+...
(510-11)
J
"7p, (0)d0=k
The function 3(0) then appears as the weighted average of
the functions B,_, (0), each satisfying the condition that its
integral over the range zero to infinity is equal to k, and that
the sum of the weights is unity.
Naturally we have
(510-12)
B =k
Thus, the larger w,, the more closely B(8) approximates to
an exponential form.
Convergence of the Series Exbansions
It can be seen directly from the expression for the functions
B,_1(6) that the convergence of their series expansions
depends essentially on the order of magnitude of T. The smaller
T, the more rapidly the exponentials converge.
From this point of view, the order of magnitude of the
w, , is of secondarv importance.
11] Allais - pag. 208
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
oc.
The Sign of the w_
The w,_, may be positive or negative. However, to simplify
the exposition, the following discussion will be limited tc
the case in which they are positive. It will be seen, however,
that the majority of the results which will be derived, in particular
those concerning the order of magnitude of the parameters
T, ©, and 4, hold in the general case where some
coefficient have a negative value.
Expression for :.
SU 4
511. In the same w--(SII-T)
vy
=
we have from _ _.
T1]
vo
1] — 1
whoa
-
WT
AY
with
511-2)
Ge
Finally, from the general formula
\
+
we have
(511-3,
Allais - pag. 200
006
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus if we put
(511-4)
— (i-o)e
8) _ B,.(6)e
Pri ( ) TRO (1a) 6, (i-¢)
then
(511-5) p(0)_ bo (i=P)20(0)4 +s 0g Dy (1-0) 9s (O)s
b (2-0)
LC
(511-6) (0) 9e Poli-P) Bo (0)s ov wy Oy ($20) By (0).
w, b, (3-P)+. + Ww, _, ¢, (1-0) + ese
The function (8) thus appears as a weighted average of the
function ¢,_,(0) with weights v,_, ¢,_,
We have
(511-7)
: I
Mall ea
iL
Thus providing (¢-p) is small and # not too large, the
weighting of the functions ¢, , (8) will differ little from the
weighting of the functions ¢, , (0).
"111 Allais - pag. 210
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 007
Representation of an Amortization Law ¢(0) by a Taylor Series
Expansion
It is clear that for a given interval of variation of 0, for
example
.00 years
it 1s always possible to represent any function (0) given b)
the statistical analysis with a sufficient degree * - “roxim-"‘on
by taking a limited number p of the terms y. T5" ex
pansion of 3(8)e*’. Thus it is essential to study * “<-+>ties
of this expansion.
Expressions for ©. and
512. Developing the different terms oi ,
have
ul Series
, WE
(512-1)
(') This is legitimate if U/1<1, a condition which is usually met by
observed values of i—p. According to (510-8) we have u=1/T and 7
is given below by (512-584 We will see that in any case we have T<E
‘relation (513-2)1
1
Allais - pag. 211
008 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
whence
LL oN
Daly
E(u
b
1
+
Ds
L° A
4
’
4.
Since
(512-3)
O(u)=1-0 u+rABO uw’...
we have
(512-4)
(5145)
1 I[b, 2b nb,_,
Or T[ Za 2a Bt
rp b n(n-1) b :
bo 301, 1 ) Busy,
4 2 ar
h 2b "
— — +- 0
f
(1Y Main text, relation (220-2)
11] Allais - pag. 212
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
50 that
512-0)
(512-7)
=
x
= W,+2 WwW,
),+3 W 4 --wh
n="
— LU,
w - 2 n
; W,+ere:+ 4
w
1°
OI again
512-8)
t/
nri
512-0)
subject, as I have shown, to the condition
(512-10)
LE
Ww, 4
d
Further, taking the values of the global quantities K, I...
C, Y. etc., as they are given by the statistical analysis is
equivalent to fixing the value of ¢(i -p) for a given value of
11] Allais - pag. 213
910 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
i~p (!). Thus, from (511-1) and (511-2) the values of T and
of the w must satisfy the condition
aT) — 4 La 01 aie
(51211) To] "Ee ET * b=)
where the value of ¢(i - p) is given.
Thus, the weights w,_, are subject to the two conditions
(510-10) and (512-11). For given weights w,_, satisfying the
condition (512-10), the relation (512-11) determines T and the
relation (512-9) determines A. Relation (512-8) determines ©,
once T is known.
As the expression for A as a function of the w,_, is independent
of T, discussion of the value of A can be carried on
Independently of the conditions (512-8) and (512-11), taking
account only of the condition (512-9).
Equation (512-9) can be rewritten as follows (for simplicity
the subscripts of the © have been omitted)
512-12)
Mn? Ww, A,
eel
à]
Un Wy]
for nl
with
(512-13)
I
LS b Sl
2 n
where A, | denotes the value of A for the function 8, , (2.
(}) $ 312.
For example. we have
relation 226-2).
(3) Relations (220-6) and (=10-7)
1] Allais - pag. 214
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
Again, wr
hav
(512-14)
As the first term 1s positive and the second is necessa
greater than - w~ again find the general condition
(312-15)
The Order o} mMagnituc
513. It may be remarked that in general, and :* ti. €
ference (i -g¢) is sufficiently small, if vc is given (this
equivalent, as a first approximation, to the ©, being given (,.
It follows that the order of magnitude of T is determined, as a
first approximation ™ "he relation
f -
515 _
Consequently the less ranid the decline of the coefficients w,_;,
the lower the value of T, and the more rapid the convergence
of the series expansions (5I. _
() From the BUNJAKOWSKY-SCHWARZ Inequality (BRONsS-DJAJEW,
Taschenbuch der Mathematik, p. 135).
This inequality, moreover, follows irnmediately from
given in $ 517 below (relation 517-2).
(?) Main text, condition (220-13).
(3) Which, it mav again be recalled, is equivalent
(relation 226-2).
( We have according to relation (za
2 vy s@ -
1c
5)
Relation
51
2-SEMEN
-nternretation
veing given
11} Allais - pag. 21
312 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
In any case, we have according to (512-8) and (512-10)
(513-2)
r<o
Order of Magnitude of A
1) Case wm which the w,_, decline sufficiently quickly
514. In the first place, it can easily be seen that if the w,_,
decline sufficiently rapidly, A will have a value not verv different
from unity.
a) Decrease of the w,_; as 1/n9 (q=>4)
Assume first that the w, , decline as with 1/#9 so that for
example
‘514-I)
u A
n-1=-—7
n 4
From (512-14) we will have
(514-2)
pe Tat?
2a | DI. 1
TL TEL
with
(514-3)
A2!
nd =1
11] Allais - pag. 216
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
O..
We will then have
(514-4)
À
na
where {(p) is the Riemann function
(514-5) S(p)==As
g(p) is finite for p=2, 4
For g=4, we have
Aq is a decreasing function
“1, finite for €-a
we have
so that for the particular hypothesis envisaged,
3
(514-6)
These results are of course valid when the À, — ##e,.
corresponding to equation (514-1), although unequal, are nev
ertheless of the same order of magnitude.
b) Exponential
r, .
wror€dsc
the
to
515. Now assume instead that w,
so that for exa,
(515-1,
J),
“ik
. diminishes a.
Te
Allais - pag. 21;
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Then we will have
I=2w n1=\fI+e + e
A -
)
A
1-e¥
-K —2K -
Xn w,_,=A[1+2e +3e +
À -K -2K —3K
= —|e +2€ +3e Fo
»—K
A d I
eË dK 1.7K
(515-2)
c (r-e”%)
2 -K —2K “
Zn w,_,= Alr+4e + Qe or
A —K -2K —3K
=__[e +de +0e , .
ok.
7
“111 Allais - pag. 218
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
g.
so that accordins ‘” ‘51: 1°
(515-3)
le
(515-4,
whatever the value of IX. This result is a ve. -markable
one; it is due to. the fact that in this case the mod ‘ reduccs
to the exponential model
For, from (511-1) we have
+
Gu,
s(n
515-5)
which gives finally
(515-6)
-
11: Allais - pag. 219
916 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
which is effectively the function ¢ of the exponential model with
315-7)
®,
LR
r
n
Furthermore, from (510-6) we have
6
+ _ —K({n—1) 0. \n— CT
15-8 B(0)=k(1-¢ F)[1+e 3 es = (7) 5
(515-8) (=e) a 7
Pb
(515-0) 8(0)=klr-c)
or finally, taking (515-4) into account
(515-10)
6
B(o)=k © >
a.
which is effectively the function 8(8) of the exponential
model (3).
Naturally, these results will hold approximately if the A, _,
of the equation (515-1), although not equal, are nevertheless
of the same order of magnitude.
("
(2)
Relation (25:
Since
0
1,
4
r
(®) Relation (351-2)
rv Allais - pag. 220
SEMAINE D ETUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 9]7
¢) Decline of the w,_, as nd KX®-L
Again, it could be assumed that
(515-11) 0 ,=XAnte
For g =1 and using the same methods as above, we find for
exambnle
(515-12,
515-13)
(515-14,
>.
uv
z
—
so that, accordine
(515-15)
Allais - pag. 22.
018 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
When K grows from zero to infinity, A grows from 3/4 to 1,
so that
(515-16)
3/4
In all cases it can be seen that a sufficiently rapid decline
in the w,_; will result in a value of A relatively near to 1.
From this, it may be deduced that in this case the properties
of the model will not differ greatly from those of the exponential
model.
Case in which the w__, vanish bevond a Certain Rank
516. Now examine the case in which the »,_, vanish beyond
a certain rank and assume that
516-1)
w, 4 =0
for
n>b+l
a) Case in which all w__, are of the same order of magnitude
Assume firstly that all w,_, are of the same order of magnitude,
for example, that they are all equal.
Then from relation (512-14)
(516-2)
—
pi ple]
ES
2 vad J
(22)
'11] Allais - pag. 222
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
C
Le.
(516-3)
we have thus
(516-4)
A being equal to 1 for p=1 (exponential model).
An analogous result holds of course if the w, ,, althoug
not equal, are nevertheless of the same order of magnitude.
b) Case in which the vw, _, do not decline rapidly
517. It is again easy to see that if the quantities w_ _, dc
decrease too rapidly, A is below unity or very near to i
For, in this hypothesis and from (512-14), (1/[2Znu
which is in any case below 1/2, will generally be small.
As to the second term of (512-14), if the w,_,, are cons.
idered as masses and the » as distances, it can be written
(SI7-I,
T
2
2m, OM}
(2m, OM,”
so that
(517-2)
NG Im. + Im. Gh,
OG cs WW
r
> am,
11] Allais - pag. 223
320 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
in which G is the centre of gravity of the points M,,, assumed
to be of mass m,.
But the points M, can be classified in two categories M,
and M; according to whether they lie between O and G or
between G and P of the abcissa p
(,
vi
©
For the M; points
(517-3)
GM, < OG .
Again, if the 0, do not decline too rapidly, G will be to
the right of the centre of the segment OP, and the image O’
of O with respect to G will be to the right of P. Then we will
have
(517-4)
GM. < 0G
so that
(517-5)
1 Em, GM°
2 OG'Em.
Thus under the assumptions made. that is to sav if
(517-6)
[S58
"111 Allais - pag. 224
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 92]
we have certainly from “
(517-7)
and since à = A
‘517-8)
In an apparently paradoxical fashion we see that for this
condition to hold, the coefficients of high powers of 9 "a the
expansion of 3((8) and B(0)e"* must be sufficiently large
Naturally, it follows from the foregoing that in the general
case, in which the w,_, vanish only at infinity, the conditions
(517-8) will again be verified if
517-0)
which is a relatively weak condition.
Finally, we see that in the case of hypothesis (517-6), as
n the cac> ~f the weaker hypothesis (317-9), the inequalitics
Relation (220-14).
\*) In the light of the indications given above § will probably be neg
ative
111 Allais - pag. 225
922 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
(517-8) are verified, whence it follows that the limited expansions
derived in the case of the general model for the different
quantities, and given in § 240, may be used.
518. Secondly, since n >, it follows from (512-12),
(512-13) and (512-14) that
1 Ew'w,, Ac Into,
5181) 2 Eros 0 ro]
Assume that
(518-2) w, =0 for n-1>= 9p
so that the sums only cover p terms. Assuming further that the
nw, _, are never-increasing, then applying the Tchebicheff inequality
(1)
(518-3)
2 15 JF
n° w 2X n 2 nw.
nS sel n=1 n-1
whence
(518-4)
A
I En
p En a,
i.e.
(518-5,
N
1
13
nm
() BRONSTEIN-SEMENDJAJEW, 0d. cit., D. I35.
T11] Allais - pag. 226
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
IL
This condition shows that if the w,_, do not decline toc
rapidly, the upper limit of A will remain relatively low.
If, for example, we assume tk~*
(518-6)
no _«
then we will have from
(518-4)
x
£L
-K!
VI
wt
whence
(518-8)
and
(518-9)
nw, _ =u,
We will have therefore for anv values of w, and ;
(518-10)
so that ror instance
v |
»othesis (510-2
(58-11
21 | Allais - pag. 227
924 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
Thus we see that if w, is not too large, A will not differ
greatly from I.
On the contrary, if w, is near unity the superior limit of
A given by (518-10) becomes very large. For the values of w,
near unity the inequality (518-5) becomes
(518-12)
D+I
20.
It may at first sight appear to be a rather surprising
result that if w, is near unity, that is, if the relative weight
given to the term B,(8) corresponding to the exponential
model (!) is relatively large, A can reach very high values
whereas in the case of the exponential model, À = 1 (2).
(*) Relation (510-13).
(2) From (518-6) and (518-2)
EN O,-1=00+R (p—1) (1—1w,)
I +2
D RO,-1= 0. +R P ) ®+2) (1 —tn,)
Thus from (512-14)
Tn w, +2",
L an WŸ
2 2, )
(2)
3)
and
A)
Ww, +k
".-)
(+4) (1—wo)
I (1—y.)]2
For small values of «
5)
For small valves of
12
A vo.
6)
p=If
D
~~
T11] Allais - pag. 228
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
This finding will be verified later for a particular case
These results can be summarised in the following *-Lic.
On the whole, we see that in the majority of cases which can
reasonably be envisaged, A does not depart greatly from unity.
Since in all cases the integrals
LA
sq
"19 di
have values which are near r and vc respectively (?), and since
2E 07.
“0° 3(6)d0
it follows that in the majority of cases which can reasonal:,
be considered, the first three moments of the function J(
are near to the corresponding moments of the exponential
model. It then follows that the related properties differ little
from tho<e cf the exponential mod-'
VE
Thus if p increases indefinitely and if 1—w, tends to unity, A increases
indefinitely. However. for w,=1 the model reduces to the exponential
model and A=1 (see below Chart IV which corresponds to a particular case)
() $ 528 to 532 and $ 540.
(7) § 311 and 313.
3) Relation (220-6)
Allais - pag. 22,
326 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE 14
w,_,
Infinite
Ww ?
n-1=—4
a
_K{(n-t)
Ww, ,— Ae
1-1. 06
A
series
—K{(n—1)
w, =Ane
3
1
~~
A
1] —
Finite
series
of p
terms
¥
tw, —_
&
nw, p
Jw, | = 2
RA A _1
1 I
9 h
W,, OW, ....
so ey Wy,
nw
nn 1
never
increasing
nw, ,=K(r-w,)
Mn
2
à
A]
D+T
2 Env ,
{4.44
p+1 2 p
2 b-1 1-0
‘111 Allais - pag. 230
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 02"
It may be recalled that according as A is below or above
unity, amortisation is more or less rapid than in the exponential
model (1).
Value of €
SIQ.
(519-1)
From the limited expansion (314-6) we have
A -
fo> 872i -p,
[f it be assumed that
(510-2)
‘y
a condition which is met in the majority of cases, then
(510-3) ter Ze (imp <@ <r in
To see this more specifically, consider the case of the United
States for which
(510-4)
(>
336.
x1 | Allais - pag. 231
28 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Now
(519-5)
4)
50 that
519-0)
Ye=23.61
Xe (i ov —
> (?— p) = 0.04
Thus we see that under the assumptions made, ® differs
little from ve.
In any case we have from (123-12)
- Tc
OX Tho a
whatever the function (6).
The above results are illustrated in the following study of
two particular cases.
(1) Main text, relation (125-4).
11] Allais - pag. 222
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 929
1
PARTICULAR CASES
4. THE DEVELOPMENT OF {3(6)e*’ REDUCES TO THE n'® TERM
(n'>2) (PARA-EXPONENTIAL MODEL)
Hypotheses
520. With the exception of the exponential model (n=1,
the simplest variant of the general case which has just been
studied corresponds to the equalities
(520-1)
Rd
;
520-2)
For want of a better term, this will be denoted as the para
exponential model (1).
Expressions for the main quantities
521. We have
(521-1
)
u”
(') For n=1 these formulae correspond, of course, to the exponenttal
model ($ 250).
(11] Allais - pag. 233
330 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
whence according to (220-5) and (220-6)
(521-2)
521-3)
Ge _
A
a
n
L
oF
2 n
Consequently, applying the general formulae (!), we find
n-2
(521-4) B(6)= (2) 20
n-1! \@, 0,
521-5)
(521-6)
(521-7)
521-8)
(521-9)
(521-10)
Qu) T -
[rs Sul"
ol0)- 20 (2) a
“x! \®0/ ©
eo 5
I. (i-p)
R, Q n
EC. — [I4, —© [2-Be
fr. & i)
“mire _
(=p) | [ry © (i-p)]”
R n -9, (i-e)
Re. -[r, Q (i-p)]"e
Rom n
() Main text, relations (220-1), (220-5), (223-7), (227-3), (224-2), (226-2)
and (228-12).
‘111 Allais - pag. 234
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
93
All these expressions remain unchanged if, with n being
any positive number whatever (not necessarily an integer),
we have
(521-11)
B(9)=b
T(n
where I'(n) is the function Gamma (
The Value of the Functions 3(0) and +.
522. For given ® and small ¢-¢, © differs little from &,
and from relation (521-2) we see that | is greater * - 'ar~.
values of w. It follows that at infinity, for given 6, “ ern-7(6)
take smaller values as » becomes larger. The .. ‘3!
increasing # is therefore to concentrate the masses >
around the average values ©.
D
‘
a WS
bm
Ab,
Naturalis
. DEeCcoIr
1.7
Allais - pag. 235
332 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The function ¢(8) is zero for 8=0 and 0=0o0, and its
maximum 1S reached for
(522-1) 0=9,, H-T
q-T
0+
__. 1 @
+-p) n
‘n-I)T
_—Tu
4.
For infinite » we have
(522-2)
B,=0,
and
'522-3)
O=0
A
i
The whole mass of the 3(6)d6 is concentrated on the point
of the abscissa ©_ (1).
The influence of n
323. Assuming that, in conformity with the statistical data,
523-1)
Yc=13.5
1-0=49%,
(!) Since A=1/2 (see main text, § 220).
11] Allais - pag. 226
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 95.
the comparative values of the most interesting quantities are
summarised in the following table for n- 1. z, 3 and œ (") (à
, #0) a
(>
; 2(6) d6
[2 o(6) dé
À
JU
3.92
0.75
1.80
3.63
0.04
2.25
O.II
0.51
0.65
n.?z2
D.24
16
[.163
9.697,
co (0) we have
0.67
3.68
0.058
0.05
0.72
0.2
.I63
0.4%,
~
tor
2
L(Y)
/ b
“ ¥
(0)
d
U
su
zu
St 0
(®) For n=o00 the method of obtaining the limits is given iate.
523-2 to 523-9). oo
(°) Variations in this quantity corresponding to variations ir
completely negligible
©
(relations
ai
al.
Allais - pag. 237
334 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
This table shows clearly the influence of # on the results.
It can be seen that the average amortization period ® is in
all cases very near to the value of yc. The value of ®, remains
practically unchanged as is also the case for Rg/R,. The loss
of real income (Rom - Re)/Rou is very low in all cases, and
furthermore the value of this expression declines as # increases.
The only quantities to vary appreciably correspond to the
amortisation schedule, as can be seen from the values of the
integral [,, [7 and [3 of oe).
Charts I, IT and III represent the variations of the functions
30). 0), [* 90) db
for n equal to I, 2, 3 and 5. It will be seen that the elasticity
3(0) declines more rapidly for higher values of #. Similarly,
amortisation is the more rapid for high n, but while the differences
are real ones, thev are not substantial.
Very High Value of n
We can have a good idea of what happens when # takes on
high values by considering what the formulae already given
become when # is infinitely large. We have then (1)
(523-2)
523-3
olu)=e ©"
9 = @,
(1)
From
$
521
‘11] Allais - pag. 238
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
(523-4)
€
G-(523-5)
(523-6)
523-7"
I<
AN
Imm ~
‘CM
-
Further, from (521-4), (521-6) and Stirling’s formula
(523-8)
523-9)
IS VE 5)
vo
co) ee
Var le) e %»
0)
1
d,-8)(t
It can be verified without difficulty that for » infinite, (06)
and ¢(0) are infinite for 8 =0, and zero for any other value
of 6. The whole mass of the 3(6)d6 is concentrated on the
point 6=0_, When yc and + ) are given by observation,
©, is determined from equation (523-3).
‘11 Allais - pag. 239
“| PARA-EXPONENTIAL MODEL
iE | REPRESENTATION OF THE FUNCTION ff)
lil FOR. SELECTED. VALUES. OF mi”
AND FOR. Ka35 Ped tie
ag PRE toga ri
Chart !
fat
=
>
>
TV
O
=
3
=
T
=
3
q
ri
Ie .
i= =
0.20:
2k 1
. we A PS wn JE jf ci wi or
Exponential Model
Mare - exponential Model
I
Ti
A
“
-
=,
pe
BAC
RIDE
j
VE ÂGE ES A
=
>
T
4
>
J
=
>
1
N
OL
on FMW SO
years
;
PARA ~EXPONENTIAL MODEL
REPRESENTATION OF THE FUNCTION @/6)
FOR SELECTED VALUES OF m
AND FOR L=3.5 <p42
gp fb. 22
PO 5 (& “5
{ Pelation (521,6)
Chars Jd
6 521
J,
4
;
4
ve)
) 20
3
LEGEND.
Exponential (lade!
Para- exponential Mode
#
-’
7}
_
-
4
3
D
“40°4
2
»
3
01:
; Co ; dog 3 Chine Dei pas sie TRE Chore IT
3, Los a ar Jk} ; ay LE ei § 828
PARA- EXPONENTIA MoDer }—*
REPRESENTATION OF THE FUNCTION J, eféjdé UE ;
(FOR SELECTED VALU wy Jy pl) 6 EL ebues
AND FOR 4235 =: €-PusX pitt deep
by pa ey a a GE
re Yor ob lI im _- rie Lit
ng pouf ES [ES | ES
oN de ms 1 @Z 1° ® 00m .
- «Won amortised |share of primary income
>
A
20
’
358 :—
dagen pL
net | Exponentiai Model
mp2 Paro-exponeakial fiode)
0-3 1-4
0.25}
er
(ir fered
poss pr pI a 01 4 120
‘nk @f ae rfoe 30 se ©
Ve AE ee
a Jo are
a,
>
-
mn
x
>
1
VJ
x
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 934
Determination of ©. and n
524. In the present case, to determine the constants ©, and
n. we have the two relations
(524
(524-2)
whence
(524-3)
524-4)
and therefore
vs 24
| ey v © ,
524-5) [0 = 7
leu ‘
an implicit equation which determines »# as a function
and 6.
It can easily be verified that even in the very simple case
where 3(0) is assumed to be of the form (520-1), the determination
of ©, and n is subject to a very large margin of error if the
value of Yc is near that of ® and © itself is not very accurately
bnown
, Allais - pag. 243
340 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
The following table gives the values of ® as a function of #
for vc=3.5 and 7-p=4% (1).
The relation between n and © for Ye=3.5 and i—p=4%
»)
Jt
y.
3.70
2.77
This table shows that the slightest variation in © involves
a very large change in n.
The fact that © be given is thus an insufficient basis for determining
the constants of the model. The function ¢(8) must
be considered as a whole.
i For n infinite, relation (523-5) gives
-@f{i-0o)
‘a E—C,
11] Allais - pag. 244
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
C.
B. THE EXPANSIO. 27 [/3je'’ REDUCES TO THE FIRST AND n'
TERMS (MIXED MODET
Hypotheses
525. Assume now that
(525-1)
(525-2)
Since B(0) must be positive whatever 6, and is zero at infi
nity, we can write
(525-3)
Expressions for the Main Quantities
526.
Pur
(526-1)
11, Allais - pag. 245
14)
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
3
Then, from (510-3) and (525-2)
(526-2)
w+ =r1
and from (510-6)
(526-3) B(6)=(1-w)B,(0)+w8,_,(6)
(*)
According to (512-6), (512-7), (510-6), (510-7), (511-1),
(511-3), (227-3), (224-2), (226-2) and (228-12), we have then
(526-4)
(526-5)
526-0)
(526-7)
(526-8)
© =[1-w0 + on] T
. Eo
1—W+ Ww RAK T
b
I-0v+-wr
flo) =lewew 1 (0) EE
Gu) 9
©
Tu [1+Tu]"
[Tow 2 | np
H-I! i.
PO) bg Te -
T(i-p) [r+T(i-p|"
_[1+7(i-0)] &
(!) The present notation thus corresponds to that used in the previous
section [relation (&10-0)1 as follows
Ww = Oui
nm = I—m = t).
TT] Allais - pag. 246
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMEIRIOUE ETC. 943
+
526-0)
(526-10)
(526-11)
526-12)
k
[t is of course also possible to write
526-13)
*
526-14)
526-15,
J!) Naturally we find the general relation (226-9
Allais - pag. 24
526-16)
44
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
R, e% (2-p)
Rou 1-7, (i-p)
Fa
These relations are more convenient for use where yc is
given.
When w=o0, this model reduces to the exponential model
of the main text and when w =1, it reduces to the para-exponential
model examined in the preceding section (with n>2).
The case n=2 is particularly important. In this case, the
Taylor expansion of B(0)e"" can be limited to the first two
terms.
The Form of (0)
327.
©(0) is maximum for 6 =0,, such that
0
(527-1) =r [ral
T [1+T (i-p)] [2 22 [Tip n-2
#3) a) |
[In all cases we have
Te 7
2
0m
T
: FAT]
() Again we find the general relation (228-14) for E=1.
IT] Allais - pag. 248
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 04-and
there is no longer a maximum value for
(527-3) (ETS mz! | T( im
T w
6, decreases when w decreases, i.e. as the exponential modo]
is approached
Values ©
528. Although the model reduces to the exponential model
[b=c, A=1] for w=0 (!), and to the para-exponential model
la::01] with
| = (m2)
2 n
for w=1, it would be incorrect to deduce that in all cases the
properties of this model always lie somewhere between those
of the exponential and para-exponential models, and in particular
that we would alwave have
(528-1)
For, from (526-5), however small vw, there exist values of »
which are sufficiently large for nw greatly to exceed 1 - w.
Jr 10r
9 with nr
(1a) Allais - pag. 249
046 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
For these values. we have
528-2)
i
À > —
2t)
and I/2w can of course be very large when w is very small (1).
First Limitation of the Value of A
520. Providing w is not too small, À will remain at a relatively
low level. Indeed, it mav be noted in the first place that
() We have
=
on
an
>t
Hu
Thus A will differ from its limit value bv less than e, i.e. we will have
2 (1)
,
nt
æe if
17 >
rE
Thus, if for example w=1/100, A will differ from its limit value 50 by
less than 1/10 for n > 3000. It can thus be seen that A can only have
very high values if n is very large, and such values are incompatible with
observed reality, at least as far as can be judged
11] Allais - pag. 250
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 947
from condition (517-7) we have (
(529-1)
for
Second Limitation on the value o,
530. Moreover, since we have according . y..-,
(530-1)
L
we see that there are then three cases.
|
C
here we have
(530-2)
+
For here condition
\
I
cau b
Le
t) + N°
. 251
11] Allais - pag
J48 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
I: 1/3 <wW< 2/:
Tru
So that
1
(530-3) >
=x
2
LIT
I 2
— We
F5 3
[11: 2/3 < w ~
n
A
I+2 0
(T+w)?
20) !
Thus we have
+ <A cH
(530-4) 25
for
2
— << Ww
3
1
It follows from this that whatever the value of n we have
(530-5)
=
10Tr
I
~— (0
1
À
or
11] Allais - pag. 252
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
>
whence
530-6)
Third Limitation of the Value oj
531. From (526-4) we have
531-1)
and we can write
(531-2)
D+
so that we have
331-3)
() These results show that w has to be large enough for A to remain
sufficiently small. This is the general result given in § 518 of the Appendix.
It may be recalled th~t in all cas~
conditions (220-13) and
(520
Allais - pag. 253
350 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ie.
(531-4)
mn. n
Zu 2?
2
putting
(531-5)
po
uO,
The discussion can be summarised in the following table
1)
9
U,
ON 2 n
©
À.
*7
+
0
where v, and v, are the values of v for which À = 1/2 and where
(531-6)
A _ (n+2)?
mn
We verify A,,>1/2 whatever the value of #n (}).
Thus in every case we have
(531-7)
A
-.
(n+2)
Qn
(1) Condition (220-13).
11] Allais - pag. 254
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
(
For selected values of #, the values of A are as follows
Thus we see that when the value of # is not too large,
fairly near 1.
1
Mains
In any case
(531-8)
so that
(531-9)
whatever the value +
Ww
Recapitulatios
532. The foregoing resus are su
table
nariscu
1
ilo
svllowing
dilais - pag. 255
252 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Finally, it should be remarked that even for w<1/3 À differs
significantly from unity only for very small values of ©
and very large values of #. In fact, we have the following
fable (1) (3):
Values ni
A
0.05 | 0.10
0.25
-.00 |
“.03
.18
1.76
L I
0.99 0.96
1.04 1.00
1.22 | I.12
L.77 1.37
D.50
I
0.89
0.87
0.89
0.03
ND.78/
I
0.82
0.76
0.72
0.69
0.67
È
0.75
0.67
0.60
0-55
o.«
For the particularly important case, n=2, we have
A I+20
(I+w)?
whence
3/4 <A
It will be recalled that
A —
1
2
for
mn = OO
(3)
) Relation (526-5). .
2) The values of A for n=c are given only for indicative purposes.
si goes without saying that verv large values of n are completely unrealistic.
(®) Relation (528-2).
11] Allais - pag. 256
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 953
CHART IV
Chore [¥
4526
MIXED TIODEL
VALVES OF A
t+ zn
= +
J? a nw/?
[Re lotion (526,5)]
LEGEND...
w= or wn. 1 £xpomentral Mode,
wal | Myi Mixed Model
Para -exponentiol Mode:
Ne.
Allais - pag. 25,
354 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Chart IV represents the variation of À with n for different
values of w. The reader is reminded at this point that it follows
from the discussion in § 336 that amortization will be more
or less rapid than in the exponential model according as A is
less than or greater than unity.
Thus we see that over a wide range of variation of w, the
properties of the model considered are near those of the exprnenhal
model.
Analogous calculations could of course be undertaken for
the general case discussed in § 510. They would show that in
the majority of cases which can reasonably be envisaged, the
value of A remains fairly near unity.
Determination of w
533. In any case, it should be clearly understood that when
the value of Yc is given. the range of variation of u=T1/T is
limited.
For we have (!)
I I—U> w
_ Tos ms ts --- ss ue
(533-1) Yo (i—p) I+Ti7 0) 1+T(i-p)]
To simplify the discussion, take
533-2)
1 -0=0
which can be considered as a valid first approximation since
1 —p ts small. Then
555-3)
Yolt -0)= ©,
(1) Relation (526-11).
ir] Allais - pag. 258
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
955
and since necessar
(533-4)
it follows from the disc iiss:
gi
AY
_i0Uu.a hav
(533-5)
ie.
(533-6)
With
(533-7) 1
When # is large enoug.., 1 is possible vo write as a first ap
proximation:
(533-8)
nN
It can be seen that for sufficiently small values of 7-p,
the coefficient T can vary only within determined limits once
the value of Yc is given.
(") Condition (220-13).
(3) From (533-3), (533-6) and (555 7;
1:1 Allais - pag. 259
356 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Values of the Various Magnitudes for Given Values of Yo and
7 —
”
534. It is interesting to examine how the various parameters
vary with changes in w and # when yc and ¢ - 0 are given by
observation.
For example, suppose that we have
Yc=3.5
1-0=4%, .
lo calculate T and ©, we have the two relations (!)
(534-1)
(534-2)
1m + wv 0 ~1-7_(i-p)
1+T(i-p) [1+T (i-p)["
©,=[1-0w+wn]T
Values of T and ©,
535. The two following tables give the values of T and ©,
for selected values of w and n (?).
(") Relations (526-4) and (526-11).
() When # increases indefinitely (a situation which is clearly of theoretical
interest only), three cases are nossible
a) T does not tend to zero
Then we have
1+ T (i —p)
- 68
Chis relation defines the limit of T and .t implies
o<w< Yo 32
In this case wo is fairlv small.
for n=œ
11] Allais - pag. 260
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
GC
Values of T for y,=3.5 and i—p=4%
b) T tends .o
D.C~
9.1C
0.
0
0
alues of ©, for Ye=35 and
ow .
a.
Tr
e
- -
Lv
fr
I
ro with w-0.7
1
-3
n
n
ay
Then we have
5)
++ T
which implies
‘4)
c) w =v, -p.
In this case, we find
5)
T1+T
{2
_ or
Thus T tends to zero, so that relation 1,
Yoi—p) is thus a critical value. and for
J (i—0)=o0.14.
‘ For infinite
#)
« verified asymptoticalls
4%, we
nav
.rà from the preceding note, there are three cases:
111 AÂllais - pag. 261
958 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
These results are plotted in charts V and VI.
We see that if we consider only the values of # below 10,
‘he only values which it is realistic to consider, then, although
the auxiliary parameter T has a wide range of variation when
w and » change, the values of the parameter ©, (which are of
economic significance since ®, is the length of the amortisation
perlod for à -p=0) varv but little.
Here, T tends to a limit. Consequently ©, increases indefinitely and
ve have
I)
Ye (i—p)— ow] on
8 i — Yo en) (7—p)
b) yoli—p)<w <1.
Here, T tends to zero, and
2)
Now in this case.
3)
A wvnwnT.
nt 2
0 — Ye (2 -
LC
:0 that
LN
4j
0
Opp Tet
tn)
In this case it is clear that according to the value taken by w, the corr--sponding
value of ©, can take any valne greater than ve
For ev=1 we have
5)
0 —
PL = ye (i — g)]
For small (i—¢) this limit differs little from yc.
c) w = yc (i—p).
In this case we have
— C—C
6)
p hT(i-e)
e TG"
Ye (g— 2)
LIT
and since T tends to zero
~
4;
=)
UT
so that ©. increases indefinitely.
11] Allais - pag. 262
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 050
CHART v
+
-
RL
“METERT
PR 4
-e)
Chore ¥
£534
{Re lation (534,1)]
74
+
LEGEND
wks 3
v or nf Expenentiol Model
nye flxed Mod. :
Drre-» xpoeomendral! Model
lp
, Allais - pag. 263
J60 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
CHarT VI
Gh Than
“rp
sol
+251
+
DR EE SIE SY EERO.
MIXED MODEL | Ti
VALUES OF THE FARANÈTER @, °
Ce FOR Kad SE: AP=4%
Fe la-franan]T
Ctl weal
[Rkistions (534,1) ond (524,2)]
|
{
a
ni}
hd
375 |
LEGEND. Fe
DE po nl be neath I 4 CT SIA LE
=O. or m1. Eo enentisl Locks } < ”
wat | md Para-bxponential Mode:
— Pa
NIL
and
~
3, mm
3
.
111 Allais - pag. 264
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC
961
Furthermore, we find that T(: - ¢) remains effectively inferior
to unity, a necessary condition for the validity of the
expansion as a Taylor series of d(i-p) (").
Once the values of w and # become fairly large, T(i - p) is
relatively small and the expanded series converge rapidly.
Values of ©
536. From (526-9) and
(536-1)
4
+
‘
y
-
& 5
4’
or, again
536-2) @®=w
Whence the following table for ® can be constructed (
(') § 512, we have pu =1/T [relation (510-8)].
() The values of T are derived from (534-1) and (534-2).
{?) When = increases indefinitely, three cases are possible in line with
the preceding notes
a) o < w < Ycli—p).
In this case T tends to a limit define”
~£
‘ation
+;
and from this ana
1
de
. Allais - pag. 265
362
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
Values of @ for y,=3.5 and i—p=4%
“a
D
3.5
3.5
3.5
5
5
3.5
0.05 | 0.10 | 0.14 | 0.25
3-5
3-50
*.48
?,42
:-22
2,37
3-5
3-50
2.48
2.39
13
III
3-5
3-51
2.48
- 30
oO
3-5
3.52
2.50
= 42
Bi
0.5 | 0.75 ,
3.5
1.56
56
221
51
3.44
3.5
3.59
2.62
2.64
2.66
3.66
3-5
3.63
3.68
3.72
3.74
1.77
Le
3)
6
Yc (f—p)—ow
(1—w)(—p)
le __ _ ©
—w (1—w)(i—g)
whence
4)
0 < 0 .
I —
Yo (à — Ç)
b) yeli—p) < vw < 1.
T tends to zero and from (536-1), we have
0 — 7 —
9 PTIT ,T
I — (I —p)
5)
so that
I I—w i — p)l
((—p)L I—Yl—P) ©
Here, © can tend to any value according to the value of w.
For en=1 we have
(i-This
is the same limit as for ©.
7)
=
T
GLI 1e =o
c) w= Yc(i—p).
Here T again tends to zero and relation (536-2) gives
8) AewT
Thus ®, tends to zero.
i1] Allais - pag. 266
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 063
CHART Vil
| MIXED MODEL 1
VALUES OF THE AMORTISATION PERIOD ©
FOR X:3.5 <-p:4%
Chort YZ
$ 536
© afr (2-11-00) 7
fo [170-170 0) + 71-8)
[Relation (536,2) / (1)
Vi
~— LEGEND
w=0 or 7:1 Exponentio/ Model
132 fixed Model
ub
or
~~ Para-exponential SToders
Tin, w] are given by (534 1) ond
Allais - pag. 267
264 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
These results are plotted in Chart VII. It will be noted
that for n<10, the value of © is between 3.10 and 3.80 whatever
the value of w. Thus in all cases which are of practical
interest, ® does not vary greatly from its value in the exponential
model, i.e. 3.5.
Values of (Rem - Re)/Reom
537. From relation (410-3) and (526-16) we have (})
(537-1) &= 1 Re —I_ +f -6, (i-e)
R, 1-7 (2 ¢
TM (i-p)
Then, for given values of yc and (¢-p) and the values of ©,
calculated in $ 535, we can derive the following table.
Values of (Roy Ro) Roy pi percent for y,=3.5, i—n=4%
0.0c!
2
2
3
7
33
TOO
0.10
T.2
2
[.3
"7
3-5
IOO
0.14! 0.25
1.2
“2
£
OO
I.2
I.I
I.2
ve”
0.5 | 0.75
1.2
0.9
0.9
.0
TI
I.2
0.7
0.6
0.5
0.5
0.4
| 1.2
0.6
0.4
0.2
0.1
D
These results are plotted in Chart VIII. It will be noted
that in all cases which are of practical interest. the loss is less
) See , 410 above.
11! Allais - pag. 268
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 965
CHART V™ 7
0.04
Ren Re
Rem
MIXED MOLL
VALUES OF q= F' Res
>
)
1-8)
2371) #
yy
Xela.
2
Chore YT
é 538
46
0.03 |
un EGEND
wl or x.
+
Exponentnl Modes
Mixes Mode!
voonentof Model
002
4
Y
fa) €
Allais - vag. 260
266 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
than that given by the combination #=1, w=0 which corresponds
to the exponential model.
Use of the Tavlor Series Exbansions
538. It is interesting to confront the values which have just
been calculated, and those which would be derived using the
imited expansions obtained in the main body of the text (!)
538-1)
538-2)
% « © [TA @, (i-p)]
R 2 IV, 2
R =) P)
taking the exact value
538-3)
1-w+w (+1)
A_ 2
MT-m+nw)?
from (538-1) we deduce
(538-4)
© (ip) , TVI-4AT (8-6)
>
(!) Relation (240-10) and (240-13).
11] Allais - pag. 270
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETc. 067
whence
538-5)
|
[t is interesting to note that for
Zz,
y
ye,
3
the value obtained from this approximative formulation does
not differ from the exact value by more than 18%.
Study of a Particular Case
539. It is interesting to examine the variations of the essential
functions in the particular case in which
(530-1)
w
~
with
(530-2)
,.5 for u-|
Allais - pag. 27
268 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Then, from the results of § 526, we have
20
539-3) f= +
(539-4) AL
16
(539-5) Blo) = (14 2 9
9 3 @)
Nr
0
e
B(e) 7°
‘539-6) Ho) Ee
I, 3 O,u
5397) 6. 8 ~~ g
(Es Ie, (c+ 30 u)
2 4
(530-8) Re 1
R, I-uY,
(539-9) CRE) _ ham F
4
5 -0 u
R e °
( -IO) I__C _1_
(539 R_ Tu
I is determined by (534-1).
11] Allais - pag. 272
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETc.
36
The following table summarises the values of the principal
quantities for the exponential. mixed and para-exponential
models 7 £--Ye
=3-5
Yeo=3-5
u=- a9,
Exponential
Modes
1
Mixed
Para-exnonentia
Parameter
ve
n)= O
n-1)
=
fat
117
J.QF
2.25
2. 3d
7.000
0.23
,2
I)
rn
Ja
2.6.
NY DA
À
0
Fe
(14 (
11] Allais - pag. 273
970 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
as 0
In addition, three charts IX, X and XI are presented showing
the functions § (8), ¢ (68) and 1e ® (8) dO in the three cases.
The expressions for these functions are the following (1).
Mode!
Functions
>(0)
(0)
[52 o(6) db
1
Exponential
MWy=0
1 _ 8
0° ©
I .8
ne ©
9
a
il
Mixed
III
Para-exponential
m=1/2
W=1
46
8 2 eu
(+25 a
6
4 —
Se
(A + B6) +6
Sk
mê
A+ 2 pol” 14,20].
with
a
A ss
90, (T-%7T,)
(539-11)
B___ ©
27 0 (1-4 Y)
C= L[r 30,4
ra
YY Relations (251-2) for k=1, (251-3), (539-5), (539-6), (521-4) and
521-6).
1] Allais - pag. 274
MIXED MODEL
REPRESENTATION OF THE FUNCTION f3(8) |
FOR n-2 J 7 = 3.5 , 1e P=4A
| J w 6 "et
Chore IX
6 5.39
JRe lations (526.6) ond (533.5)
I
C
t
a
der
LEGEND.
W=0 T-407 Exponentio! oder
W-#3 T:302 Mixed Model
wt T=196 Para-exponentsol. ode
valu. T are deduced from (5341) § 534
539 Tables I ond IX]
t
E
x
+
î
Fol Ca
+
~
pre
VIIXED MODEL
REPRESENTATION THE FUNCTION (8)
FOR m2 5, £-pséX
fan, Balt
2(8)./- = ; of =
Ic as /
Share ¥
& 535
rT
[Relations (526,8) ond (506,11)]
=
~J
ND
n
ort
324
240
LEGENDE
Curve I w=0 T:=4.07 Exponential Modes
vie T3022 Mixed Model
T= 7.96 Parg-exponrerntrof Mode
ne values of . are deduced From, (534,1) §534
[5539 Tables Iond EX]
T
>
1
03
2
———
A
tT
JO 32 ~-.
>
T
8
— -
TD 700£L
NON ASTORTISED
ue £ plode
«17
Chore XI
6539
TRetations (526,8) and (526.11):
Ç
tr
=
oonentro/ fooe/s
+ Model!
-exponentiel Mode
n (5341) § 534
x
=
=
%
TET
[WY
374 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
It is really remarkable that the II curves, which relate to
he mixed model, stay very near to the I curves of the exponential
model, even though the weight attributed to the expoaential
element is but 2/3. This is due to the fact that the
coefficient A in the mixed model has a value very near that
which it takes in the exponential model, so that the 3(8) functions
of model I and II have approximately the same three first
moments (1).
General Comments
540. It is certainly very surprising at first sight to find that
for small values of w and for large values of n, the model
considered has very different properties from those of the
exponential and para-exponential models, though for w=o it
is identical to the exponential model and for w=1 identical to
the para-exponential model.
For the first and second, for example,
4
B ~ i.
for
1-0=4%
whereas for w=o0.10 and # sufficiently great, À has a value
very near 5, and ®_ can even become greater than any value ().
It may first be noted that no illusions should be had about
small values of w, since large values of n can have a greater
influence, with correspondingly greater deviation of the model
irom the exponential.
A 4 336.
(?) $ 532 and 535.
‘'11] Allais - pag. 278
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
975
Further, while 3(6) and ¢(x) appear as weighted averages
of the corresponding expressions for the exponential and paraexponential
models, this is not the case for ¢(8) and the other
parameters (1).
Finally, and most important, even if (8) and ¢(«), both
formally and effectively appear as weighted averages of their
expressions
for the exponential and para-exponential models (3), the values
of the parameter T, and therefore of the corresponding
functions 8 and ¢ are certainly not the same, since in each of
these cases, T is determined hs the equation
(540-1)
Relations (510-6), (50: .) and
Relations (526-6) and (526-7)
Relation es- +)
(Lx
1 Allais - pag.
270
076 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
which expresses the fact that theoretical value of Yc is equal
to its observed value, which is the same in all three cases (1),
and it is this which generates the large differences noted.
Nevertheless, very large values of n are of no more than
speculative interest, and from the foregoing discussion, it may
be considered that for n<7, the properties of the model studied
are extremely close to those of the exponential model, whatever
the value of w.
It follows from this that if we can consider only the first
few terms of the expansion of 3(0)e*® as a Taylor series, the
general properties of the model will be very close to those of
the exponential model.
“ For the exponential model, we must put w=o in relation (540-1);
or the para-exponential model, \=T. and for the mixed model &=u.
11] Allais - pag. 280
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
077
B1B3LIOGRAPHY
JEvons S. (1871), The Theory of Political Economy.
BoHM-BAWERK E. (Von) (1888), Positive Theorie des Kapitals
Bousquet G. (1936), Institutes de Science Economique: T. II
Paris, pp. 157-169.
ALLAIS M. (1943), Traité d'Economie Pure (Treatise on Pure Economics)
Imprimerie Nationale, Paris 1952 (2nd edition of Part. I, L’Economie
Pure (Pure Economics) of 4 la recherche d’une discipline économique
(In search of an economic discipline), Paris 1943}, 5 vol. in-4°
tooo pages (1).
Reviers
ArLars M. (1947 A), Economie et Intérêt (Economy and Interest). Imprimerie
Nationale, Paris 1947, 800 pages in 2 volumes in-8° raisin (on
deposit with the Librairie de Médicis).
AIrLaIS M. (1948), Pouvons-nous atteindre les hauts niveaux de vie Américains?
(Can we match high American living standards?). L: Monde
oth, 16th, and 30th October 1948 and 6th November 1948.
ROSTAS (1948), Comparative Productivity in British and American Industrv
Cambridge University Press.
Arrais M. (1954), Les fondements comptables de la macroéconomique - Les
équations comptables entre quantitiés globales et leurs applications (The
accounting basis of macroeconomics: accounting equations relating global
quantities and their applications). Presses Universitaires de France
Paris. or pp. in-1°
Arrais M. (1955), Observations sur analyse des relations entre le capital
et la production (Remarks on the analysis of the relations between capital
and output) in: « Travaux des Economistes de Langue Francaise »
1955. Ed. Domat-Montchrestien, 1956, pp. 188 to 223.
ALLaIS M. (1960 A), Influence du coefficient capitalistique sur le revenu
réel par tête (The influence of the capital-output ratio on real income
per capita).
(1060 *A). Memorandum presented to the Tokvo Conoress of the IST.
(') The second edition is identical to the first, differing only by the presence
of an introduction noting the new contributions which were made
in the first edition
Allais - pag. 281
)78 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Document 61, 70 pages. Because of copyright, only the theoretical part
has been published in the Bulletin of the I.S.I., Vol. XXXVIII, 2,
op. 3-27.
A integral version of 1960 A* will be published spanish in « Revista
de Economia y Estadistica » (January-March 1965). Faculty of Economic
Science, Cordoba University, Argentina.
ÂrLAIS M. (1960 B), L'Europe Unie, Route de la Prospérité (United Europe,
The Road to Prosperity). Calmann-Lévy, Paris, 1960, 369 p.
DESROUSSEAUX JACQUES (1961 A), Expansion stable et taux d'intérêt optimal
(Stable Expansion and the Optimal Interest Rate) in: « Annales des
Mines », Nov. 1961, pp. 31 to 46.
ALLAIS M. (19061 B), Le Tiers-Monde au Carrefour: Centralisation autoritaire
ou planification concurrentielle (The Cross-roads for Third Countries
- Authoritarian Centralisation or Competitive Planning) in: « Les
Cahiers Africains » Creation de Presse, Paris, 1961, n. 7 and 8.
ArLaIs M. (1961 C), La définition des fonctions caractéristiques et le problème
de l’imputation (The Definition of Characteristic Functions and
the Problem of Imputation) (to be. published).
ArraIrs M. (1962 A), The Influence of the Capital Output Ratio on Real
National Income. « Econometrica », vol. 30, n. 4, Oct. 1962, pp. 700-728
() (3). (Walras-Bowley Lecture, American Meetings of the Econonetric
Society, New York, 28th Dec. 1961).
BOITEUX M. (1962), Taux d'intérêt et optimum capitalistique d’une économie
en évolution (The Interest rate and the Capitalistic Optimum in an
Evolving Economy). Mimeographed paper. 16 p.
ALLAIS M. (1963), Quelques aspects analytiques et appliqués de la théorie
du capital (Some Analytical and Practical Aspects of the Theory of
Capital). Paper to the Congress of the International Economic Association,
Cambridge, July 1963, Mimeographed document, 71 pages.
ALLAIS M. (1964), A Theorem about the Optimum Accumulation of Capital
(To be published).
ArLrats M., The influence of the Volume of Capital on the Real National
Income. North Holland Publishing Companv (in preparation).
(') This study contains a large bibliography, presented in a systematic
fashion, which the interested reader may usefully consult.
(?) It is necessary to mention an unfortunate printers error which got
into the text of my Econometrica paper after correction of proofs.
The following correction was printed in a subsequent number of « Econometrica
» (vol. 31, n. 4, Oct. 1963, p. 784).
Instead of « ... the capitalistic process P (shown on the left on the next
page) tends to the asymptotic process Pa (shown on the right) with... »
read: « ... the capitalistic process P tends to the asvmptotic process Pa
with 5
‘I1] Allais - pag. 282
© LU 5SSION
FTISHFR
Professor ALLAIS’ interesting paper makes essentially one empirical
test. The model predicts two items to be fairly constant. One
of these is y, the capital output ratio, which the theory predicts
should vary rather weakly, and the other is 6, which the theory
predicts to be a constant. Indeed y is supposed to vary weakly
because it is essentially a function of 6, and some things that don’t
move very much. Professor ALLAIS secures numbers on these two
magnitudes.
Now, when a theory predicts something to be constant it is hard
to know how to evaluate the test that is used. One has to be able
to say how constant is constant and there is no standard for this.
The only possible comparison that can be made, I should think,
is a relative one. Professor ALLAIS’ theory predicts that 6 should
be constant and y should vary slightly; in fact, §_ varies substantially
more than does y. 6 moves by something of the order of
17.5 per cent as compared with 5.8 per cent for the figures for y
tor the United States. This takes the ratio of the range to the mid
point of the range as the basis for the calculations which are imprecise.
If the theory were right, it should be the other way around.
Now this is rather serious since it is almost the only empirical
1
Allais - pag. 283
080 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
test performed by Arrais. I can well understand that one cannot
rest much on an argument such as this one, since these numbers
are very rough and are subject to substantial errors of measurement.
Indeed, as ALLAIS has shown, even plausible errors of
measurement can lead to this result. On the other hand, I am
anable to understand why these numbers are appropriate for ALLAIS
to use and not appropriate for me to criticize. Professor ALLAIS
.nsists upon saying that everything goes according to his model.
In fact, this does not go according to his model, and while that
may be the result of measurement error, it may equally not be. It
s, of course, all too common for an investigator to claim that numders
which apparently point opposite to his theory are subject to
measurement error. Professor ArLals, however, is managing to do
more than that. First he puts forth these numbers as supporting his
theory; then, when I point out that in fact they do not support it,
he claims that is due to measurement error; finally, he claims that
n fact they do support it. Professor ALLAIS is working both sides
of the street here and I am afraid I cannot understand his somewhat
engthy argument on this point.
ALLAIS
First, in my opinion it is impossible to attach any importance
to these slight differences because the precision of the different
estimates is quite small and it is difficult to derive any conclusion
from small deviations which are probably of a random character.
For instance, if you consider the 58 Corin CLARK figures for
21 countries cited by me in Table V, p. 54 of my Tokyo paper
(ALLAIS, 1960 A), the range for y is quite large, from 0.83 to 8.48.
The median is 3.54 and the coefficient of variation about it is 22%.
If you consider each of the 58 figures, this is certainly wrong
and only the median of the lognormal distribution has any real
meaning.
11] Allais - pag. 284
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2. My second point is that the difference you underlined
as great as you say.
I gave the detailed figures in my « Econometrica »
October 7mA2, pages 714 and 77s They ar> «- follow:
pg”
knot
7
"ARI
Years
1880
18g0
1900
1906
1910
1913
1923
1929
1950
Ie =
"S
0
[=C0)
33
1.70
55
29
Tg
(1) For a quantity © he average relative deviation is v.
‘he average of the -
whet ~
TARIF
‘Tnited States Gireat Britain. France - 10
United State
France .
(;reat+ Brit»
Average relative devia
CNT
dllais -
pag.
NS
182 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus, the average relative deviation is respectively 5% and 89%,
for y and ©, and for the United States only and 4% and 3% for
the United States, Great Britain and France in 1913. In fact, in
‘he first case, the range of @, is slightly greater than the range of Y
for the first table. This order is reversed for the figures of the
second table.
I repeat that in my opinion we cannot attach any importance to
the decimals. Perhaps some years from now it will be possible to
have exact figures for the decimals but my conviction is that this is
impossible at the moment. And so far as the figures for the United
States are concerned, I must stress that there is some bias associated
with the estimate of the rate of interest. It is very difficult indeed
to estimate the pure rate of interest i. For example, after the war,
American monetary policy was such that the rate of interest was
maintained at artificially low levels and in this way the rate of
interest did not reach the value which it would have had it there
had been free play of the market mechanism. This explains the
slight tendency mentioned bv FISHER.
3. But that is not the real point. The real point is that the
results which are found show a striking agreement as far as orders
of magnitude are concerned; and at all events these results must
be explained. The proposed theory can predict small variations of
®, and y and in fact we verify that the estimates do not vary very
much.
4. My fourth point is the following. I am completely aware
that it is impossible for me to say that this model is the only one
which can explain the facts, because the empirical data we have are
insufficient for any final and definite conclusion to be derived.
I say only that the course of events is what it would be if this theory
were correct, and no more than this. This theory and the model
which illustrates it are compatible with all the known facts.
I don’t say that this theory will continue to hold. Nobody
knows this. I simply suggest that this theory can explain the fea-11]
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083
tures that we find in the empirical data in a simple way. This statement
is valid for any theory whatever. It is impossible to declare
for any theory « that is the right and definitive theory ».
We do not know if the Newton Theory of gravitation or the
Einstein Theory of gravitation is correct or not — we don’t know
that — what we can say is that the course of events is consistent
with their correctness, at least as first approximations, and I don’:
say anything more.
I would add that this theory and its associated may not be
right, but at least, as far as I have been able to judge, they have
the merit of having been able to force people to think about many
interesting features of reality (see my paper § 337).
Worp
[ am extremely impressed by the whole of Professor ArLLars
presentation. I have received many books and papers by ALLAIS,
but I have never had the opportunity to listen to an exposition like
this. Maybe this is the first time that Professor ALLAIS gives such
an integrated exposition, and if so I am very happy to be one of the
first listeners. It adds to my admiration that his material, from
the criteria of the Study Week, is very important. It is a question
of course to what degree his views will be accepted, but there is no
doubt that this is the type of material which we are supposed to
present and discuss.
My own comments and remarks are very small. My first question
mark in the margin refers to your notations for the capital.
If the total is divided in parts, then the usual notation is that the
parts have indexes, and the total has no index. In your system 1
find that the total has one index, and one of the parts has no index,
and this is a little confusing
Allais - pag. 287
284
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
ALLAIS
It was, I think, JEvoNs who used this characteristic curve for
the first time, but for the case of a stationary equilibrium and in
triangular form. At the end of the last century, BoERM-BAWERK
again used such a curve but in rectangular form. Subsequently a
German, STACKELBERG, and a Frenchman, BousQUET, again used
‘he concept of the characteristic curve for stationary process with
rectangular and triangular form. If they did not go farther, the
explanation is, I think, that they were unable to treat this quite
difficult question mathematically. I have given the references to
their works in the bibliography of my « Econometrica » paper with
some comment.
The concept of the characteristic curve is thus quite classic:
What I have done is first to calculate the different macroeconomic
quantities R, C, etc. and secondlv to generalize this concept for
“he dynamic case.
What is completely new is the calculation of the real consumed
national income, the principles of which I gave in my book « Economie
et Intérêt » in 1047. I stress that the hypotheses underlying
‘his calculation are quite natural and not very strong (see § 119 for
‘he general theory and § 221 for the general model).
I can underline one point more. In the present paper, because
‘he value of reproducible capital appeared so often, it seemed
better to me to use the letter C without a subscript to represent reproducible
capital and to use C, to represent the total value ot
reproducible capital and land. (On the contrary, in my book
:« Economie et Intérêt ». I have followed the ordinarv wav).
WoLD
Thank you for that explanation. Then 1 wonder whether it is
correct to understand from your analysis that thanks to the free
working of the forces in the capitalistic system, we are rather near
‘he optimum of maximal possibilities?
111 Allais - pag. 288
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AL LAIS
A complete explanation would be too long. My point is only
that if we assume that the total real value of capital is at a certain
level so the surface of maximal possibilities has a certain position.
as indicated on the following >
aptalistic optimum
TPIT
"ssrbliti.
Q, and Q, are the productions of two consumption goods, for
instance, butter, shoes, anything. If capital is varying, this frontier
is varying also and my point is that a maximum frontier does exist,
and that it characterizes an optimum capitalistic structure. This
question was studied quite extensively in my 1963 Cambridge paper.
In fact, the conclusion can effectively be that we are very near
the optimal situation as far as capital is concerned; that is, so far
as the capital distribution over time is concerned (see my paper
§ 410-411). But so far as other factors are concerned, this is not
so and specifically, the difference between French and American
productivity can be explained only if the other factors are taken
‘nto account (see my paper 8 --r 4-2"
[ FONTIEFR
In applying his analytical approach to interpretation of differences
and similarities in the relative magnitude of the labor ad capital
‘nputs in several countries. Professor ALLAIS apparent'v assumes
, AHais - pag. 280
286 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
25
direct comparability of the units in terms of which the labor inputs
are measured in different areas, Does this imply that a man year of
labor in the United States translated into relevant efficiency units
is equivalent to a man year of labor in Canada, in France or in
India? If it does not, how would a correction for differences in
abor efficiency affect his comparisons?
ALLAIS
I don’t suppose anything. The only relevant point is the fact
that there exist great productivity differencies. To explain them,
different factors must be examined: labor, natural resources and
capital. The present paper is limited to the study of the possible
nfluence of capital. In my book « L’Europe Unie », published by
Calmann Levy in 1960, I have presented for the United States and
France a general analysis taking into account the different factors,
the discussion of which is not possible in the limited framework of
this discussion.
Nevertheless as far as I can judge, my personal position differs
slightly from yours. I think you agree that capital is not playing
as great a role as many people believe and I am happy that you
have arrived at the same conclusion as I. But so far as labour is
concerned, it seems to me that may be we should be quite cautious
oecause there are two things to consider. The first one is the level
of the productivity of persons and the second is the efficiency of
the economic system. My own view on this questions, so far as
the United States and France are concerned, is that it is impossible
to conclude that American workers are superior to French,
Perhaps the best example I can give is that when I deliver a lecture
in France, people say « Oh, French workers are superior to the
American workers without any doubt » and in the United States
hey say exactly the contrary. That American workers are better.
in my opinion there is no difference at all. What is different is
the economic system, the pressure of competition. the price mechaitl
Allais - pag. 200
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< /
nism and so on. And my conclusion is that the main reason for the
difference in productivity between the United States and France, =c
far as I can judge, is the play of the economic system.
In any case I want also to answer some questions you raised
and you did not mention just now.
First I cannot affirm that there is only one explanation why
the capital output ratio is practically constant. In my book « Economie
et Intérét » (Chapter IX), I presented some other arguments
relating to the CASSEL theory and taking into account the length of
life, and I think these arguments are quite appealing. I would not
say that there is only one explanation. I would suggest only that
the stability, for practical purposes, of the capital output ratio can
be explained by the constancy of @, which represents the intel
lectual difficulty of conceiving roundabout production processes
My present theory is thus at least compatible with the facts.
Second, the coefficient ®, is, as I have already said, an index
of the intellectual difficulty of conceiving roundabout production
processes and psychologically, the thought that the coefficient ©
is constant over time because over time people have the same intellectual
capacity to conceive roundabout processes, is appealing
to me. Of course, I am suggesting an explanation, not offering a
proof.
Thirdly, LEONTIEF spoke about a possible cumulative effect.
[ would say that I have done some quite interesting statistical
research on the question of the intellectual capacity of people over
time. More specifically, I have tried to assess the rate of growth or
scientific and technological progress over time since the 12th century.
[ have considered the number of major scientific and technological
discoveries per century and the result of this statistical analysis,
which I cannot develop here, is that we can conclude that people’s
intellectual capacity has remained exactly the same since the 12th
century. But, if one consider real wages, there have been tremendous
fluctuations. Thus the commonly held view: that the extraordinary
development of the West in the last century is due to new inventions
is very questionable indeed. My opinion is that in fact the economic
svstem has played a very great role
Allais - pag. 201
388 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
~ R
Thus, if I could briefly answer to the LEONTIEF point, I would
say that there is no reason to believe that people are more intelligent
now than they have been yesterday.
In addition, as far as a cumulative effect is concerned I have
shown in my 1963 Cambridge paper that, subject to very general
and appealing conditions, it is impossible to expect an indefinite
increase of real national income to result from an indefinite increase
of real capital.
THEIL
Regarding the problem of the accuracy of statistical data, there
's a difficulty which econometricians have to face and which is due
to the considerable margin of uncertainty that economic statisticians
aave to face. For example, it may happen that the « best » estimate,
given the statistical evidence, which an economic statistician can produce
implies a capital-output ratio which is excessively large compared
with most published capital-output ratios. We should not
oe surprised when in such cases the statistician decides to replace
ais « best » capital by a lower figure and thus contributes to a too
nomogeneous picture of the stock of published capital-output ratios.
In the same way, if the economic statistician knows or feels that
he knows that the marginal propensity to consume is about 0.8, and
if he finds for a particular year that the consumption change is very
far from 80 per cent of the income change, he may decide to adjust
his figures. This is in my view one of the reasons why our picture
of published correlation coefficients tends to be too optimistic.
ALLAIS
These different observations are very interesting and I think
they should be discussed in the general meeting.
First, as far as the estimates of capital are concerned, it is cer-11]
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08G
tain that the estimates we have now are very questionable. But 20
or 30 years ago the same was true for national income figures and
many people were saying at that time that this sort of calculation
was useless and had no meaning at all, and so on.
My second point. What Professor THEIL has said is very interesting
indeed, as far as the proposal for some general statement concluding
this colloquium is concerned. I have read in the introduction
to the Study Week that econometrics has made much progress and
so on, and to-day we hear from a very competent colleague that
the work of every statistician in the world may after all be very
questionable, not only from the scientific point of view but from
the point of view of honesty. What you have said is terrible indeed.
Terrible. For if you were right, this would mean that statisticians
are dishonest. Personally I do not accept this point. I believe that
statisticians do not revise their estimates when they differ from
other estimates, and I can supply very good proof of it. For example,
there are differences between the various estimates CoLIN CLARK
has given for the capital output ratios of different countries at
different times. These estimates are indeed very different from one
country to another and from one time to another. As you know,
CoLIN CLARK has not made any personal calculations on this. The
figures published by him are estimates made by different statisticians
around the world, using very different methods and taking into
account statistical materials which are absolutely not comparable.
Not only are these figures not the same, but their order of magnitude
is absolutely different. We therefore cannot suspect the statisticians
of having modified their estimates in order to be in agreement witl.
the other evaluations. The contrary is the case.
But if these figures are absolutely different, they are distributed
lognormally and their median is practically the same as median for
the United States from 1880 to 1956. Thus we can conclude that
there is an unquestionable reality behind these very different estimates
and we can conclude that the differences are due to some
random influences
1 Allais - pag. 293
290 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
If I can answer to the FISHER point, I would say that the
agreement between the averages for the United States, for the world
and for 1913 (§ 321 and 323 of my paper) is in any case really
striking. Of course there are some differences but many other
factors are operating and what is astonishing is not that some small
differences exists, but that these differences are in relative terms
so small.
And finally I would add that some of the arguments I have
heard are very strange indeed. If the statistical data and my theory
were in total disagreement, nobody would have said, « Oh, it is
clear: if there is disagreement, the data are wrong ». They would
have said, and in my opinion rightly, « The theory is wrong because
it does not agree at all with the facts ». So, if there is concordance
here between the theory and the facts, we must recognise that there
is an agreement, and this is certainly a point in favour of any theory
at all times. Perhaps it could be argued that this agreement is due
co chance. I am very doubtful about such a proposition. It is
juite unbelievable that such coherence between figures for the United
States between 1880 and 1956 and for Great Britain or France in
1913 and for 58 values for 21 countries could arise bv chance.
THEIL
I am afraid that I did not succeed in making my point sufficiently
clear. What I was trying to say is that in a great many cases
economic statisticians have to face the problem that the data which
‘hey collect imply sizeable uncertainties. The uncertainty may either
be due to the inferior quality of the data themselves, or to their
imperfect relevance with respect to the variables for which they are
used, or to their sample character. When there is a large range of
uncertainty it is evidently rather arbitrary which particular point
estimate will be published. Under such circumstances it is not at all
unreasonable that other considerations are taken into account (besides
the uncertain numerical data) when a decision has to be made
irl Allais - pag. 204
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as to which figure will be published. If these considerations are
based on economic analysis we obtain the picture which I tried to
sketch
ALLAIS
[ agree with what Prof. THEIL has said. That is my vwr
position. But I am doubtful that there is any real tendenc for
statistician to adjust their estimates by taking other people’s results
into account.
3
ATT AT
In presenting my paper as briefly as possible I wish to stress
only some very important points on which remarks were formu-‘ated
in the LEONTIEF Group's discussion (1).
r) The main definitions are given on Table I.
2) The theory I present uses the concept of characteristic
curves (Table II) which was first developed by Jevons. There are
two curves, the first for the production process and the second for
the amortization process. The first represents the inputs 7, d6 of
primary income (the services of Labour and land) supplied at time
1-0 and emerging in consumed national income R at time #. The
second represents the inputs r,d8 supplied at.time # and emerging
n consumed income at time t+@.
In a stationarv process the two curves are simmetrical but in
(*) Discussion in plenary session.
(') During this discussion, large sheets were put on the blackboarc
main definition and equation were presented on these sheets.
The contents of these sheets is indicated in the tables in the abstract o
Professor ALLAIS’s paner, and the following discussion refers to these +*ables
Allais - pag. 205
392 PONTIFICIAË ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 23
a dynamic process they are different and the analysis is much more
complicated,
In the first part of the paper I calculate different macroeconomic
quantities, R national income, C reproducible capital, y and Ye
capital output ratios, as functionals of the two functions
(6) =
Ye
200)
The relations obtained (Table III) are simply accounting
identities.
It is assumed that there exists a valid index R,, of real consumed
national income such that
èR _, >yoY
R SyY
where the Y represent the primary inputs, the y their prices and
k the homogeneity coefficient of the production function.
From this hypothesis it is possible to derive the general formula
‘I17-18) of the paper expressing SR as a function of the Be
117-18)
1 SRY)
% RA)
roe
vis, 0) €
(udu 40
© (6,0) € Jt CO ZA
11] Allais - pag. 296
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003
From this relation it is possible to demonstrate that real consumad
national income is maximised when the rate of interest : is
equal to the rate of growth ¢ of primary income (Table III).
4) It is possible to illustrate the theory by a very general model
(Table IV). I assume only that the production function is logarithmicallv
linear and that the elasticities
3
can be considered over very large rang
independent of # but dependent on 6).
Thus we derive the expression for real consumed national
come as
in
LR (f) = La?) +
. A
a
la #
and we can calculate the different macroeconomic quantities
function of the Laplace transform d(u) of the elasticity B(A)
particular we finally reach the expressior
11.
n
where y is the capital output ratio C/R, : the rate of interest, p the
rate of growth of primary income R, (the Services cf Labor and
national resources), k the coefficient of homogeneity the production
function and ®, a constant equal to the vil. “ for i—o
Roy is the maximum value of R, attained °
xy
1 |
Allais - pag. 297
394 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ”R
For the maximum value Rg, of the real national income consumed
we find
(251-17) Rom = «(/) Safe —kBy0
where e is NAPIER’s constant, and o(#) and R,(f) are given. This
formula enables the influence of p on the maximum value Rom to
de studied (§ 420-421). |
The expression obtained for R,/Rg,, is very interesting since it
depends only on two unknown quantities k and ®, But it is easy
to show that the order of magnitude of k and ©, are respectively 1
and y. which is a statistical datum.
The expression for Rg, shows that of all the processes of
growth (p>o0) the most advantageous one is that for which the rate
of growth of primary income is zero.
For small values of the rates i and p it is possible to expand the
different macroeconomic’ quantities as TAYLOR series as functions of
hree constants k, 0, and A if the first terms of the TAYLOR expansions
only are considered (§ 220, 240 and 241).
In particular
Re k 2 (7 2
251-18) RIT 78 (7— p)
5) If we assume, as is quite natural, that the elasticity 8(0) is
decreasing exponentially i.e
(250-5)
8.
ke
11] Allais - pag. 298
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Q0QF
(exponential model, Table V) we find that every macroeconomic
quantity can be expressed as a function of R,(f), ¢(?), i(¢) and of
two constants only: 2 and ® In particular we have
251-6) and (251-12) y= =
—6-251-15)
ry 0,
— =
x
In this case we ha.
and in the appendix I have shown that under very general hypo
theses A remains in the neighbourhood of 1. Thus there is reason
to believe that, at least as a first approximation, reality can =~ =
presented by the exponential model.
If we consider the meaning of relation (250-5), ®_ appears as an
index of the intellectual difficulty of conceiving roundabout production
processes. Thus we can consider it as being practically constant,
and this hypothesis. seems to be confirmed by the statistical
data (§ 321).
6) With the formulae obtained it is possible to estimate
‘Table VI and VII) the parameter ® since
221-1)
We find that the order of marnitude o
{~
I]
Allais - pag. 200
296 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
For k there are many reasons for believing that its value does
not differ very much from unity.
À precise estimate of À appears to be particularly difficult to
make, but as I have already said, the analysis in the appendix
shows that under very general conditions, we can consider that
A col
7) The general model and its exponential variant appear to be
justified by the facts both as to their hypotheses and their consejuences.
:
The third part of my paper shows that the course of event to
what one would expect if the general model were be correct, and as
far as can be judged, reality does not differ very much from the
exponential variant of the general model.
I must particularly stress the striking agreement between the
conclusions of the theory and the practical constancy of the estimates
of the capital output ratios y and of the estimates of the coefficient
© ($ 321, 323 and 324) (Table VII).
8) Finally four different applications are given in the fourth
part of my paper (Table VIII). These are the following:
First, it is possible to estimate what could be obtained from an
ncrease in the capital output ratio y,=C/R,. It is shown that
developed countries are in the neighbourhood of an optimum capi-‘alistic
structure ($ 410-411).
Secondly, the diminution of the real national income as a consequence
of population increase is estimated to be of the order of
magnitude of kO p. For the United States, this figure is of the order
of magnitude of 79% (§ 420-421).
Thirdly, the model is used to study the problem of estimating
the influence of the capitalistic structure on differences in produclvity
between France and the United States. As far as can be
judged, existing differences must be explained bv other factors
(§ 430-433).
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Fourthly, it is possible to show the orientation which is required
in the development policy of underdeveloped countries.
9) I have finished my own exposition, but there is just one
more point, I would like to discuss. This point is related to some
of the other papers presented here, namely HAAVELMO’s paper,
LEONTIEF’s and THEIL’s. These three papers accepting as a valid
hypothesis that if real capital is increased there is a proportionate
increase in real national income.
This means that they assume the relation
pe
as a production function, where R, C and y are respect
national income, real capital and the capital output ict.
relation is apparently derived from the practical const:=-capital
output ratic
aval
Mis
he
but the deduction is in fact a very questionable one, empiricu!”
and theoretically.
First, as I said in the discussion on LEONTIEF’s and HAAVELMO’s
papers, from an empirical point of view this hypothesis is completely
unacceptable.
The only data we have, at least to my knowledge, are the results
obtained by DouGLas and his followers. According to them, the
elasticity of real income with respect to real capital is of the order
of magnitude of 0.2 to 0.25. If we use these values in the
Haavermo, LeoNTIEF and THEIL models we will find very very
different results. In particular LEONTIEF’s results are not as optimistic
as they appear in his paper
21 Allais - pag. 301
998 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
From a theoretical point of view it is possible to show that the
proposition: « real income is proportional to real capital » cannot
de derived from the proposition: « the capital output ratio is praccally
constant ».
[ can illustrate this proposition in using my own model which in
any event is mathematically consistent. In the exponential case of
this model we obtain two results simultaneously
a) the capital output ratio
323-1)
_C 6,
Y RT +67
changes little if we consider the usual range of variation of i:
5) real national income
(251-16)
6
De
y
has a maximum value R,, whatever the value of real capital C.
Thus, it is impossible to admit the derivation of relation (a) from
the observed constancy of the capital output ratio as a valid and
seneral proposition.
10) Finally, two main objections were raised in the LEONTIEF
group’s discussion.
The first was that one cannot have great confidence in the statisical
data so that the agreement between my theory and the facts
should be questioned. This argument is quite singular. In fact,
there is a striking agreement between three sorts of data.
‘I1] Allais - pag. 302
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For France, Great Britain and the United States in 1913 we
find an average value of y of 3.64. For the U.S.A. from 1880 to
1956, we find 3.46 and the median of 58 values given for 21 countries
by CoLIN CLARK is 3.54 (§ 323-1). Such agreement cannot
be due to chance.
Ordinarily if we have a theory and the facts are in disagreement
with this theory, the theory is rejected. But here the objection is
quite different. It is that if there is agreement we cannot infer
anything because the data are questionable. Perhaps we ought to
reject the facts because they are in striking agreement with the
model?
The second objection was rather more justified. Professor FISHER
noticed that the range of variation of ©, was a little greater than
the rate of variation of y. And, in the light of this theory. we
should have smaller variations for @®. because we have
325-1,
©, being a constant and y a function of :.
In fact, I have given the relative deviation of y and ©, from
their averages for the United States in my Econometrica “er
(1962). I found 5% and 8%; the difference is not so grc=* Jor
United States, Great Britain and France in 1913 we hav, and
3%. Here it is true that the first figure is greater than the second.
But in any case the extent of the range of ®_ can be ec" plained
by the difficulty of estimating the rate of interest correctl Tn fac
we have
| Allais - pag. 303
1000 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus, an error Ai has the same effect as Ayly® that is we must
compare y2A: with Ay. Thus the error in y can be 0.1 and the
error in ¢ can be 0.015 but
v2A; =O.I5 .
It can thus be seen that the influence of an error in ; can be great
and as you know it is very difficult ot estimate the pure rate of
interest.
We con derive the same conclusion in another way. For the
United States. we can calculate the theoretical value of :
where ©, is the average value 4.12 found for ®_,. Thus we find
the following table
ITnited State
t88o-190C
rf rr
107°
FOSO-TOGEH
Periods
Estim-*-d value |theoretical value
ty
difference
Ai = i —iT
[.2I
- 08
5
2.232
Thus, for the period 1880-1956, the differences A: are quite
small and do not exceed 1.5%. But for 1950-1956, the difference
is very great. The reason is, in my opinion, the fact that American
monetary policy has maintained the rate of interest on bonds at an
ir] Allais - pag. 304
SEMAINE D ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
100]
artificially low level which did not correspond to the general economic
situation. Thus, the rate of interest on investment was certainly
much higher than the rate of interest on bonds which I have
considered (Econometrica’s paper, p. 714). The conclusion is
that the bond rate is not a good indicator of the pure rate of
interest.
Thus, FISHER's objection might be valid if, and only il, we
know the right value of i. But we don’t know the right valu. of
and the errors on i can explain the greater range of variation of ©
In response to Professor FISHER, I must again stress that I did
not say that the facts prove the validity of my model, but only
that my model is in agreement with every known fact (§ 337 of my
paper), and nothing else.
HAAVELMO
I think we ought to distinguish clearly between what you say
about the shape of the production function and what you say ab:ut
the capital-output ratio, because the latter does not depend un ‘ie
production function alone. It depends on the whole pattern of
economic behaviour, that is, on manv other things besides the production
function
MALINVAUD
In his very stimulating contribution, Professor ALLAIS uses .
model of capital theory in which primary inputs are introduced
continuously and « mature » in final outputs after some time, the
delay, between input and output being subject to a fixed or varying
distribution. Such a model, that may be traced back to JEvons,
is rather neglected today. I am glad to take this opportunity to
express the view that we should probably use that model more
than we in fact do. It has rendered great services to capital theory
11] Allais - pag. 305
002 PONTIFICIAE ACADEMIAL SCIENTIARVM SCRIPTA VARIA - 28
in the past, and I am sure it can still render great services in the
ature. This is well illustrated by Professor ALIAIS’ results.
ALLAIS
As far as the HAAVELMO objection is concerned, I would say
that naturally every model involves certain simplifications. I do
not say that this model can represent absolutely the whole of reality.
But the hypotheses of the model seem to be quite appealing and
they are justified by the empirical research undertaken. Now the
conclusions of the model are in complete agreement with facts
Thus, perhaps we can accept this theory at least as a provisional
conclusion.
111 Allais - pag. 306
SPATIAL ORGANIZATION AND REGIONAL
PLANNING: SOME HYPOTHESES FOR
ECONOMETRIC ANALYSIS
WALTER (ISARD
Department of Regional Science - University of Pennsvlvania
Philadelphia. Penn. - U.S.A
INTRODUCTION
The objective of this paper is to raise certain fundamental
questions regarding the application of existing econometric and
regional science techniques to the problems of regional planning.
It is my hope that by raising these questions, J will be
able to stimulate the formulation of hypotheses which can be
tested with new kinds of data. Perhaps these hypotheses may
then lead to more effective theories and techniques with reference
to comprehensive planning for economic development.
The basic issue which I wish to confront is one which has
been avoided by econometricians and regional scientists. It is:
what are the properties of an optimal spatial allocation of decision-making
authority, planning functions, and other selected
governmental functions (inclusive of certain aspects of administration)?
Alternatively, for an hierarchical system of regions,
what should be the locational distribution of such authority
and functions? Furthermore, to what extent will the above
spatial allocation affect and be affected by the spatial distri
“121 Isard - pag.
1004 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
bution of industrial production, population, investment, consumption
and other economic magnitudes? The latter question
suggests the need to develop a more adequate general interregional
theory which encompasses the spatial distributions of
both economic activities and political-administrative-planning
functions and decision-making authority; and that these spatial
distributions be simultaneously determined (1).
In what follows, I can only make a small beginning at the
formidable task which confronts us. I shall suggest hypotheses
which relate to:
I) the advantage (positive or negative) of increased participation
potential with increase in the degree of spatial decentralization
of decision-making authority (utilizing concepts
based upon the gravity model as developed in the field of
regional science):
2) the advantage (positive or negative) of increase in the degree
of spatial decentralization with respect to:
a) cost of collecting information
b) cost of processing information
c) cost of transmitting information
d) time-cost of executives, representatives and officials spent
in reaching a decision based upon the processed information
made available;
3) the overview advantage (positive or negative) of increase in
the degree of spatial decentralization (where overview
advantage is defined in terms of the wisdom-full informationcoordination
factor in decision making as reflected in the
writings of Professor J. MARSCHAK and his associates).
(') For further elaboration of these question and for detailed exposition
of certain concepts utilized in this paper, the reader is referred to W. IsARD
and T. TuNc: Some Concepts for the Analysis of Spatial Organization,
« Papers and Proceedings of the Regional Science Association », Part. I,
Vol. 11, and Part. II, Vol. 12. Permission to reproduce without change
manv of the statements of these two mannscrpts is herehv acknowledged
21 Isard - pag. 2
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 1005
2. PARTICIPATION POTENTIAL AND DEGREE OF SPATIAL DECEN-TRALIZATION
Elsewhere I have proposed to measure participation potential
for a system of #» nodes (points. central places, regions) as
follows :
where d
Boo
Er
ow oP
3Gy 5
ab
d
JV; =
= distance between nodes à and 7, however defined,
on
M; = mass at j, however defined;
b and 3 = adjustment factors, whether constant or vari
able, applying to d;; and M,, respectively;
w. = a weight, however constructed, to be applied to M;
G;; = an appropriate constant applicable to potential inter
action between nodes 7 and 7;
;V; = participation potential of individuals (mass) at ÿ per
unit decision to be reached at 7: and
+
total participation potential of all nodes in the system
per unit decision to be reached at 1.
Now consider a hierarchy which may be defined by several
levels or orders of nodes. (Assume 4 orders of nodes, an order
being indicated by f or g; f, g=1, ..., A). In a simple hierarchy,
the single first-order node may be represented by the peak
of the pyramid or the top of the tree. An h'” order node is one
of the many at the base of the pyramid or tree. Intermediate
nodes may exist at one or more intermediate levels or orders.
The hierarchy may be regularly or irregularly structured, and
‘127 Isard - pag. 3
1006 PONTIFTCIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
may be symmetrical or asymmetrical. The flow of participation
or the exercise of influence may be upward throughout, downward
throughout, or both ways throughout; or it may be characterized
by any one of the many possible combinations of
npward and downward movements.
Take the symmetrical tree-like hierarchy of Figure 1. Each
node, starting from the peak, leads to four subordinate nodes.
The tree consists of four orders. the number of nodes of the
Erc
A regular Hievarchy of Nodes
CE à
20
£
2!
\ AN 85
first, second, third and fourth-orders being 1, 4, 16 and 64,
respectively. Their numbers run from 2 to 5 in the secondorder,
6 to 21 in the third, and 22 to 85 in the fourth. At the
start, let there be one person at each node, each person having
the same weight of unity. We also set d;=¢= = and let
G;, 8, b=1. When dz, represents the distance between two
nodes of orders f and g along any given branch from the peak
(the bar being placed over the subscript to indicate that we are
speaking about the order of node), d7=1 when f>g; and
d;; 0 when f<g. The sense of the last assumption is that
only upward flow of participation is permitted. A person can-‘12]
Isard - pag.
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. [007
not participate in decisions at any node of higher order than
the one at which he is located.
Participation Potential may be computed for various spatial
patterns of decision-making authority (with reference to a defined
set of decisions). If we normalize these patterns, so that
in each case the total authority is represented by unity, we are
in a better position to carry through the calculations (?). In
table 1 different degrees of spatial decentralization are indicated
by the several rows, row A representing 1009, centralization
and row Z representing a very high degree of, if not 100%,
decentralization. Here, we do not attempt to measure in
absolute terms the degree of spatial decentralization corresponding
to any allocation among nodes of decision-making authority.
Rather, we can state that according to certain readily accepted
criteria, some patterns are more spatially decentralized than
others. For example, A, B, C, D, E, F, P, Q, S, X and Z
is an ordering of patterns corresponding to increasingly higher
degrees of spatial decentralization (*). On the other hand, there
are no readily accepted criteria by which one can state that of
patterns H or G (or patterns S or T), one or the other involves
greater decentralization. Row A represents the highest degree
of centralization of decision-making authority, all decisions
being made at the single first-order node. Zero amount of decisions
are made at each 2nd-order, 3rd-order and 4th-order
node. Row B represents a somewhat less centralized <itua-()
This step precludes any effect of the variable, the spatial pattern of
decision-making authority, upon the total amount of authority that may
exist. (We may implicitly assume that the optimal total amount for any
given organization has already been determined, or that the total amount
is prescribed beforehand). In a more general statement this effect should
be encompassed.
(’) The readily accepted criteria are: 1) if there are two patterns different
with respect to the amount of decision-making authority at two orders
of nodes only, then the one having the larger amount of decision-making
authority at the higher order node is the more decentralized of the two:
2) if there are two patterns different with respect to three or more orders
of nodes, and if there is still another pattern which according to criterion (1)
is more decentralized than one of the two patterns but less than the other
then the latter is the more decentralized of the two: and so forth
el
Isard - pag.
1008 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE 1
Participation Potential for Patterns of Spatial Decentralization
Pattern of
Spatial
Decentralization
8
C
D
E
4!
G
€
I
x
L
N |
N
O
Q
>
S
T™
J
Vv
Ww
xX
Node of ! Nodes of | Nodes of | Nodes of
First Order | Second Order | Third Order Fourth Order |!
+
NONF NUMBER
(splees = C91)((99) …. (85
‘91
J
… 0
a. À
2 .…. 2
25 … .25
20 ... .20
J
0 …oU
9 .. 0
0 … 0
0 .. 0
05 ... .05
3
3
0
0
… 0
… 0
… 0
… 0
__ 0
75
75
.6
.6
5
5
5
2 … 2
2 … 2
5 .… 5
25 … 25
95 ..…. .95
2 … 2 |0 … 0
1 … 1 a … 1
0 …o |o … 0
| 25 … .25 (0 … 0 |
195 … .125| .125 … .125
2
Re
4
0
125 … .125| .25 … 25 | .125 … .125
125 ... .125( .125 ... .125| .25 … 25
6 … 6 jo …0 0 ..0
4 … 412 … 290 …O
20 3195 ala a
>
2...
2 … 2
1 … 1
Jd 1
4 ... 4
2 … 2
1 … À
2 … 2
1 … À
3 ... 3
2... 2
3 0.3
| 3 3
4 À
0 _- 0
i
4
2
J :
3 …
3 3 + J
2 2
1
TT ae @
2 … À
2 … 2
1 1
1. À
2 ... 2
3 ... 8
5 …_ _5
0
0
0
A
10
10
Total
Participation
Potential
Pp
43.33
47.83
52.27
54.50
61.30
88.40
130.00
65.67
99.67
151.67
168.67
220.67
70.13
97.33
152.53
207.73
249.33
262.93
304.53
115.40
170.60
212.20
267.40
264 20
640.00
12] Isard - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETc. 1009
tion. For every branch, the fraction of all decisions made at
the single first-order node is 0.9. Each 2nd-order node makes
0.1 of all decisions on any branch on which it is located. Each
3rd-order and 4th-order node makes zero fraction of all decisions.
In the situation represented by Q, 0.4 of all decisionmaking
authority resides at the top; 0.2 of all decision-making
authority along any branch resides in the 2nd order node of
that branch; o.1 of all decision-making authority along any
branch resides in the 3rd-order node of that branch; and 0.3
of all decision-making authority along any branch resides in
the 4th-order node of that branch. At the bottom of table 1,
row Z represents the case where all decisions for any branch
are made at the 4th-order node of that branch, none being
made at any node of any other order. Note that, because the
flow of participation and exercise of influence can only proceed
upward, no node can influence decisions at any other node of
the same or higher order.
Given our assumptions and this framework we compute
total participation potential, P, of all individuals over all decisions
for each of the situations of table 1. By definition, in
n
our normalized cases. P=27;-.V, where 7, is the fraction of all
i=1
decisions to be made at node ¢, this fraction being the same for
all nodes of the same order as 7 (9). The values for P are re-(*)
If there are s spatial patterns of decision-making authority to be
considered, each corresponding to a degree of spatial decentralization, then
for our set of assumptions the participation potential corresponding to each
degree is given by the single row vector [P.]
I 1 —I s
[P,] = 6-41 | EL fs ] 7=1)
! re, [Fou fe=1,.. À
where: Æ is the constant defining the number of nodes in the next higher
order directly linked to any given node; [k:-"), g=1, …, h is a single row
vector Ixh, h being the number of orders in the hierarchy; da is the
distance between a node nf order f and the node of order g to which it is
most directly connected; iz | is an A xh matrix (f, & , h); and
L fg
h< < matrix. each column of which describes a spatial pattern
av”
F121 Isard - pag.
7
1010 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
corded in the last column of table 1. As can be expected to
some extent at least, participation potential increases with increase
in degree of spatial decentralization. This is the case
for rows A to G where each row involves greater spatial decentralization
than the preceding row. When we come to row H
it is no longer clear, as indicated, whether G or H involves
a higher degree of spatial decentralization. Note that P falls
from 130.00 for G to 65.67 for H. From row H to and including
row L, there is once again a succession of patterns, each
one involving a higher degree of spatial decentralization than
the preceding one. The value of P also rises, without exception,
from one pattern to the next. At row M, once again it is not
possible to state that the pattern has a higher degree of spatial
decentralization than the preceding one (row L). We also note
that P at M is lower than P at L.
(To illustrate the dependence of P on the choice of values
for the basic parameters, we carry through several more computations.
The results are given in table 2. In column 1 of
or allocation of decision-making authority among the modes in the » order
hierarchy. In the calculation of the participation potentials of the last
column of table I, dzy = oo, whenever f < g since a downward flow of participation
or exertion of influence has been precluded by assumption, and
diu=e which we set equal to 1/10. The value of 2 is 4 so that the [ke]
vector is [1, 4, 16, 64]. The =| matrix becomes
d=Je
To 1 ï
to 7?
The product of [R#!] and z= is the row vector [43 1/3, 88, 224, 640]
zl
This row vector, when multiplied by the matrix [1,,] vields a row vector
which is the transpose of the last column of table 1.
[12] Isard - pag. 8
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETc. 1011
table 2 are reproduced the figures of the last column of table 1.
In column 2 are recorded the figures for the case as characterized
at the top of column 2 (which case is the same as that of
column 1 except that dy; is taken to be equal to 2 instead ot
unity). In columns 3 to 7 are recorded the figures for several
other cases, each case being characterized at the top of the
respective column.
The computations for many other cases might be made, for
example cases involving different values for ¢, 3, b, G;;, and
different assumptions on the number of individuals at each
node (all the cases of table 2 assume only one individual at
each node). These the reader may do for himself (°).
We now wish to proceed to the next step, namely, the transformation
of values for participation potential into values for
average productivity, i.e. the mapping of a participation potential
space onto an average productivity space.
Since we are unaware of any comprehensive empirical work
on this problem, we can only pose the question of the validity
of certain hypotheses which seem reasonable in the light of
social science knowledge. Can we state a certain functional
relationship between average productivity (dependent variable)
and P (independent variable) which can be represented by a
curve such as one of the curves on Figure 2 (6)? Starting with
the extreme at which P is at a minimum (which tends to correspond
to the pattern of 100%, spatial centralization), can
(®) He may also perform calculations for irregular tree structures as well
as for regular tree structures where the number of nodes at any given order
(except the first) is other than a k multiple of the number of nodes at the
next lower order.
() As previously noted, we do not treat explicitly the effect of P upon
the total amount of decision-making authority (which may be viewed as
control), and vice-versa. In a more general framework, this effect would
be recognized; and the influence of the variable, total amount of decision:
making authority, upon productivity would be recognized. In this connection,
see ARNOLD S. TANNENBAUM, Control and Effectiveness in a Voluntary
Organization, « American Journal of Sociology », Vol. 67, July
1961, pp. 33-46, and Control in Organizations: Individual Adjustment and
Organizational Performance, « Administrative Science Quarterly ». Vol. 7
September 19062. pp, 236-257
.1 Isard - pag. a
1012 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
TABLE 2
Total Participation Potential for Different Values of Basic Parameters
G, a, B, b. =
Pat tern of
Spatial
Decentrali- !
zation
BR
C
D
Rr
|
A
H
T
=
M
N
D
>
:
R
S
y
7
Ww
(1)
da], 439, da3=1 |
173, 433, dag =o!
d;; = 1/10
43.33
47.83
52.27 |
54.50
81.30
88.40
130.00
65.67
99.67
151.67
168.67
220.67
70.13
97.33
159.52
207.73
249.33
262.93
304.53
115.40
170.60
212.20
267.40
264 20
640 00
WE = We = Wa = Wa -
2)
(3)
same as
col. (1)
except
dy =2
same as col.|
(1) except | same as col.
dy] =2 (1) except
diz = 1/2 2423 dual
33.33 35.61
38.80 41.92
44.27 48.22
47.00 | 51.39
53.80 60.83
43.33
47.90
52.46
54.75
61.73
|
82.40
24.00
60.67
94.67
46 67
98.70
133.90
67.15
114.48
158 48
89.50
132.48
66.16
101.04
154.77
163.67
215.67
66.13
93.33
148 53
182.14
226.14
73.46
111.30
165 44
172.22
225.93
70.73
98.63
155 56
203.73
245.33
258.93
300.53
1192 40
219.57
254.71
273.71
308.91
1°26 54
212.50
255.48
269.43
312.42
117 15
167.60
209.20
264.40
361 20.
190.68
225.88
280.01 °°
269.35
174.08
217.07
274.00
273.99
R40 N00
640.00
658.32
f])
w = 3
otherwise
same as
col. (4)
63.33
66.10
68.86
70.25
77 18
102.10
145.05
77.17
111.80
165.49
182.81
236.49
79.93
107.63
184 4%
221.23
264.18
278.03
320.98
124.24
181.05
224.00
280.80
80 55
GEO O00 |
(6)
w = 3 |
We = 2 same as
wy = 1 col. (6)
wy = 1/2 | except
otherwise a3, dag,
same as | _
teol. (4) dig =
(7)
56.67
62.50
68.33
71.25
75.58
56.67
62.50
68.33
71.24
75.49
97.30
111.25
85.83
107.46
124 89
97.00
110.60
85.83
107,08
124.08
135.72
153.16
91.66
108.96
1231.56
134.71
151.71
91.66
108.66
130.76
154.16
168.11
176.76
190.71
192 45
152.86
166.46
174.96
188.56
123 00
146.05
160.00
182.60
210 15
145.10
158.70
180.80
216.50
241 00 |
226 00
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 1012
Average
"Huctity
"1 Yon Potentialwe
hypothesize that average productivity increases with in
creases in P (and presumably increase in spatial decentraliza
tion)? Would this increase of average productivity in the early
stage be at an increasing rate (curve BB) or decreasing rate
(curve AA)? If it is at an increasing rate, can it be presumed
to reach a point where it is at a constant rate (point X,
curve BB) and then a decreasing rate with still greater values
for P (and presumably still higher degrees of spatial decentralization)?
Further, at some later point, at a relatively high
value for P (and presumably a relatively high degree of spatial
decentralization), can average productivity be posited to fall off
fas in curve * °° ™
3
() For some relevant empirical findings, See T. Tomexovic,* Level of
Knowledge of Requirements as a Motivational Factor in the Work Situation,
« Human Relations », Vol. 15, No. 3, 1962, pp. 197-216; J.R.P. FRENCH,
Jr, J. ISRAEL and DAGFINN Âs, An Experiment on Participation in a Norwegian
Factory, « Human Relations », Vol. 13, No. 1, February 1960,
pp. 3-19. It is recognized of course that other variables are relevant. For
example, on the effect of personality upon the relation between participation
in decision making and productivity, see Victor H. Vroom, Some Personality
Determinants of the Effects of Participation, « The Journal of Abnormal
and Social Psychology, Vol. 59, November, 1959, pp. 322-327.
Also see RaLpH E. DAKIN, Variations in Power Structures and Organizing
Efficiency: A Comparative Study of Four Areas, « The Sociological Quarterly
». Vol. 3, July 1062. pp. 228-250: and Rensis LikKErT. 4 Motivation
Isard - pag. 11
1014 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
Just as in demand curve estimation, there are many complex
problems in the estimation of an average productivityparticipation
potential curve. We cannot discuss these problems
here. Once an average productivity-participation potential
curve is estimated, for certain economic-type organizations, can
such a curve be converted into an « extra-returns » curve? Can
a total output curve be constructed where total output of the
organization is in part a function of P? When price of the
output is unaffected by changes in total output associated with
variation in P, can such changes be multiplied by the constant
price to determine the extra-returns (positive or negative) corresponding
to each value of P (relative to some base value
of P)? When price of the output falls with increase in total
output ,then, given a price-output function, can change in total
revenue and thus extra-returns for any increase in P be determined?
With reference to a defined set of decisions or to
decisions on a concrete issue or set of concrete issues, can the
extra-returns curve be represented, for example, by a curve
such as curve CC in Figure 3? (8).
Approach to a Modified Theory of Organization and Management, and
CHRIS ARGYRIS, Understanding Human Behavior in Organisations: One
Viewpoint, both in Mason Haire (ed), « Modern Organization Theory »,
John Wiley, New York, 1959.
As VROOM states, after reviewing the literature:
« When the entire pattern of results is considered, we find substantial
basis for the belief that participation in decision-making increases productivity.
There is both experimental and correlational evidence indicating that
high levels of influence by workers in making decisions that they are to
carry out result in higher productivity than lower levels of influence. It
should be noted, however, that not all the findings are consistent with this
generalization. The results of both Lewin, LIPPITT and WHITE (1939) and
Morse and REIMER (1956) suggest that, under some conditions higher productivity
may be achieved with use of more autocratic methods ». .
VrooM urges careful investigation of the relationships between changes
in average participation in decision making and resulting changes in the
quality of decisions made, the strength of group standards regarding execution
of the decisions and the worker’s « ego involvement » in the decisions.
‘See Victor R. VwrRooM. Work and Motivation, Tohn Wilev. New York
forthcoming).
(8) Note that in the hierarchical tree-structure which has been assumed,
lateral communication is precluded. However, it may be hypothesized that
at least un to a point lateral communication is highly desirable and has a
‘121 Isard - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 1015
For non-economic or hybrid organizations, the question of
the conversion of an average productivity-participation potential
curve into an extra-returns curve is still more difficult. For
a system planning bureau, having jurisdiction over all local,
regional and national planning, change in average productivity
may be associated with change in the quality of the service
it provides (as measured by the effectiveness of its planning for
Dollars
. x{ra
Pelurns
social welfare). But, as we al! know, and as is the case for
many economic, non-economic and hybrid organizations, it is
exceedinglv difficult to measure changes in quality, to price
positive effect upon organization maintenance. (See James E. McNulty,
Some Economic Aspects of Business Bureaucracy, Wharton School, Univer
sity of Pennsylvania, 1962, mimeographed, pp. 43-53.) In this connection
McNULTy and others have suggested the following relationships:
1) when there is a highly centralized decision-making structure and all flows
are vertical, lateral communication (and thus morale and average pro
ductivity) is relatively low;
2) with decreases in centralization there tends to be increase in lateral com.
munication (and thus morale and average productivity);
however, when the decision-making structure tends to be highly decentralized
(approaches local autonomy) lateral communication (and hence
morale and average productivity) may tend to decline.
Consequently, the « extra-returns » function associated with participation
potential is to be adjusted for the lateral communications factor when this
factor is a variable for a set of possible decision-making structures
3)
21 Isard - pag. 12
1016 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
different qualities of a service, and thus to estimate an extrareturns
curve (°).
Where an organization is conceived to be society itself, can
change in average productivity of its labor force be translated
into changes in Gross System Product (inclusive of non-economic
type commodities appropriately priced)? Can an extrareturns
curve be constructed to relate changes in Gross System
Product to changes in participation potential?
3.
INFORMATION, COMMUNICATION AND OTHER DECISION-MAKING
COSTS
In the consideration of the spatial decentralization of decision-making
authority for an organization, it is important to
identify the differences among patterns in cost of information
collection, processing, and transmission, and of executive time
and other items. Unfortunately, this is another area in which
the accumulation of empirical materials and empirically-based
hypotheses is inadequate (1°). Nonetheless, it is essential to
treat explicity these costs. In suggesting hypotheses, we shall
draw heavily upon some of the pioneering thinking of
J. MarscHAK and T. MarscHak (1).
(°) Alternatively, extra-returns may be conceived as psychic income, or
good will, or political income based upon power, prestige. respect. affection
and other Lasswell-type commodities.
(1%) There are, however, interesting and somewhat, related materials on
administrative costs, such as in James McNuULty, Administrative Costs and
Scale of Operations in the U.S. Electric Power Industry - A Statistical
Study, « The Journal of Industrial Economics, Vol. 5, November 1956,
PP. 30-43; P.G. HERBsT, Measurement of Behavior Structures by Means
of Input-Output Data, « Human Relations », Vol. 10, No. 4, 1957, pp. 335-346;
and THEODORE R. ANDERSON and SEYymMoUrR WARKoOw, Organizational Size
and Functional Complexity: A Study of Administration in Hospitals,
« American Sociological Review », Vol. 26, February 1961, pp. 23-28. Also
see WiLLiam R. HucHEs, Short-Run Efficiency and the Organization of
the Electric Power Industry, « Quarterly Journal of Economics », Vol. 76.
November 1962, pp. 592-612.
(') See the works of Jaco MarscHAK already cited and THoMAs MAr-SCHAK,
Centralization and Decentralization in Economic Oreanizations.
« Econometrica ». Vol. 27. Tulv 1050, DD. 200-430.
121 Isard - pag. 14
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1017
For the purposes of this paper we shall consider four basic
cost components. The reader, of course, may wish to hypothesize
a smaller or greater number of cost components, and employ
a different set of definitions. Our four basic cost components
are:
a) cost of collecting information, where each node collects information
on conditions in its own tributary (jurisdictional)
area (12);
b) cost of processing and beneficiating information (inclusive
of computation costs and all research costs not elsewhere
covered):
c¢) cost of transmitting (transporting) information from one node
to another:
d) cost of the time of the executive, local representative and
other officials in reaching decisions individuallv or in conference
(13).
We now set forth, in the form of questions, some reason
able hypotheses concerning each of these components (1)
Cost component a): Can it be presumed that: 1) most of
the information needed for decision-making is on local condi-(1?)
Thus an h™ order node collects information on its local area, whereas
the single 1st-order node collects information that pertains to the system
of nodes (regions) as a system.
(**) At times, several or all of these cost items may be considered as
administrative cost, and less frequently as cost of « routine » administration.
We prefer to consider any administrative cost not covered by our classification
as belonging to the broad category of production costs.
Differences in costs which may arise from different degrees of articulation
and coordination of activities associated with different patterns of spatial
decentralization will be treated later as differences in « overview advantage. »
('*) It should be kept in mind that each of these components may be
directly related to various explicit rules that may be used in determining
the volume and nature of information to be collected, the wavs in which
information is to be processed, the type and volume of information to be
communicated, the media by which communications are effected, the kind
of information to be considered by decision makers, the form of its presentation
and so forth. Clearly, these rules will be related to attitudes and
sther variables which are to he considered in a later section.
121] Isard - pag.
1018 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
tions and must be obtained at local nodes; 2) a large fraction
of such information is visible to local people; and 3) the visibility
to non-local people of such information on a local node
rapidly falls off with distance from that node? If so, can we
conclude that the greater the distance a decision-making point
s from a given local node, the greater the volume of information
on that node and its tributary area which must be formally
collected and put in explicit form, ceteris paribus? Can we
hypothesize that for decisions on a representative concrete issue
or set of concrete issues cost component a) is some monotonically
increasing function of the degree of spatial centralization
as suggested by the curve of Figure 4? (9).
Dollars
Cost
Component
yey
Jagree
————
. . . ! 0 0
vabal Centrabization
Frc
Cost component b): When no slack facilities, labor and
other resources exist, and when inputs are costed in full at
prevailing rents and wage rates, do major scale economies
obtain in processing and beneficiating information? For processing
a given volume of information associated with decisions
on a concrete issue or set of concrete issues, can we hypothesize
that cost component b) is some monotonically decreasing func-(5)
In this function, should allowance be made for the likelihood that
important information pertaining to any f‘-order node and its tributary
area may be more visible to a person at the f*-order node than to a person
at an (f+ 1)" or higher order node?
Note that the functions underlying the curve of Figure 4 and the curves
of subsequent figures are taken to be continuous. In practice, they are
likely to involve steps and other discontinuities. The argument. however
remains unaffected
12] Isard - pag. 16
SEMAINE D’ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETc. 1019
tion of the degree of spatial centralization-as suggested by
curve M in Figure 5? (*). However, if one postulates that at
each local and regional node there exists at least some supply
of zero-cost decision-making labor, as is frequently the case
in the literature on socialist economics and centralized planning,
might the relevant function be represented by curve
N? (17) Can the spread between curves M and N at any
point along the horizontal axis be taken to represent the cost
savings from zero-priced computing and beneficiating resources
for the corresponding degree of spatial centralization? Further,
if it is accepted that local decision makers have greater local
visibility, and therefore need to collect less information, can
we hypothesize that their need for computation and data processing
is accordingly smaller? Would the dotted curve Q be
relevant rather than curve N?
Cost component c): In analyzing the costs of transmitting
information, is it valid to consider only two elements: terminal
costs (inclusive of coding and decoding costs) and operating or
« line-haul » costs which are usually a direct function of intervening
distance? Of course, both these costs should be taken
to cover maintenance expenses and fixed charges on investment
in communication (transportation) channels (!¥). When
operating costs rise less than proportionately with distance,
can the function relating transmission cost per standard unit
('*) After a point, it may also be hypothesized that scale diseconomies
arise because of congestion, overload of facilities, etc.
(') An advantage typically claimed for decentralized planning of the
economy is a personnel saving because the managers at each local node are
« a ready-made computational staff and all of them are required in any
case to be employed for the tasks of managerial supervision. In their
remaining time they simultaneously perform all the tasks of computing
each revised set of production decisions except for the computation of
prices (which is performed by the central agency). In the centralized solution,
on the other hand, the entire burden of these computational tasks
falls on the central agency, and the agency must hire a staff especially
for this purpose... » (T. MARSCHAK, op. cit., p. 400).
(*) Parallels to these cost elements mav be taken to exist when inform
ation is transported bv individuals
12 | Isard - pag. 17
1020 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
Dollars
Cost
Component
(6)
—
, 100
Yal Centralization
Dollar
Cost
Component |
fe)
per
Standard
Unit of
Information
Fig. 6
—
. . . 700
Degree of Spatial Centralization
of information to degree of spatial centralization be depicted
by a curve such as that of Figure 6 (where the y intercept is
taken to be terminal costs per standard unit of information)?
(¥) Can the product of (1) the function underlying the
curve of Figure 6 and (2) the function relating the total volume
of information collected to degree of spatial decentralization
(*¥) Can it be posited that on the average the higher the degree of sparial
centralization, the longer the distance of transmission of a standard unit
of information? We should recognize of course that certain units of ‘information,
for example, that generated at the single, 1st-order node, need
to be transmitted less with increase in degree of spatial centralization.
12] Isard - pag. 18
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1021
(which is fundamental to the construction of the curve of Figure
4) yield the relevant (total) function for cost component c)
as suggested by the curve of Figure ~?
Cost component d): Although the cost of the time spent by
decision makers in reaching decisions might be aggregated with
cost component b), for analytical purposes we prefer to keep
these two cost items separate. As with cost component 3), if
such time is fully costed, may we presume that significant scale
economies exist? Would the number of times at which a decision
on a concrete issue (or set of concrete issues) is to be taken
Dollars
Total
Transmission
Cost
Contre
a
-
Dollars
Total Cost
of Executive
Time Spent
- . at .
Decision-Making
on a
concrete
erie
+
fat
(Jia
Isard - pag.
TOG
1022 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
2
be the greater, the smaller the degree of spatial centralization?
Thus, would total time spent in reaching decisions on a concrete
issue be the greater, the smaller the degree of spatial
centralization? Could the relevant cost function of executive
time be depicted by a curve such as curve S of Figure 8 after
adjustment for different levels of efficiency and prices of executives
at different order nodes? On the other hand, if a supply
of slack executive (decision-making) labor is assumed to exist
at each node and if this labor is assigned a zero cost, then might
curve T of Figure 8 be considered relevant?
It is clear from the preceding discussion that a significant
amount of empirical research must be conducted to permit the
development of firm hypotheses on information and communication
costs within an organization. Pending such research
we may, for pedagogical purposes alone, set down a total cost
curve with reference to decisions on a representative concrete
issue (or set of concrete issues). If we postulate that no slack
resources exist and that all labor and facilities are properly
priced, we may let the curve in Figure g depict the relevant
total cost function. Other curves of different slope and also
of positive slope may of course be considered equally valid.
4. OVERVIEW ADVANTAGE (AND DISADVANTAGE)
We now turn to another set of significant factors which
vary with the degree of spatial decentralization. These factors
relate to the ability to make wise or good decisions. It has
generally been claimed that the decision-maker (the individual,
planning bureau, or government agency) at the Ist-order node
is able to make better decisions than others, ceteris paribus.
However, does the empirical evidence support this claim?
Within the tree-like organization of the preceding paragraphs,
does the decision maker at the 1st-order node have available
a much greater amount of relevant information than decision
12] Isard - pag. 20
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1023
makers at local nodes? (®) Is he in a better position to reach
a more consistent, or coordinated, or articulated set of decisions
pertaining to a representative concrete issue (or set of concrete
issues) than are the decision makers at higher-order nodes, who
at least to some extent act independently of each other? Can
he be said to have an overview advantage in decision making?
Thus, can we hypothesize that the more an organization’s decisions
are made by the decision maker at the first-order node
Dollars
Total
'nformation
and
Decision -
Making
Costs.
74) 5
"Oentralization
(or by decision makers at lower-order nodes), in general the
better the resulting set of decisions, ceteris paribus? In short,
can an overview advantage be said to exist, which varies di
rectly with the degree of spatial centralization in the decision
making structure of an organization? (*!)
There have been several hypothetical, highly simplified
examples presented to illustrate the nature of this overview
advantage. MARSCHAK has developed the case of a ship-build
{(**) In raising this question one must also keep in mind the extent to
which information may” flow from lower-order nodes to higher-order nodes,
and from one node along one branch to another node of the same order
along a different branch via a lower-order node.
(#) For some relevant discussion see J. MarscHAK in Haire, op. cit.;
Brau and Scott, op. cit, pp. 121-128; MARCH and SIMON, op. cit, Pp. 20I-210;
C. B. McGuire, Some Team Models of a Sales Organization, « Management
Science », Vol. 7, January 1961, pp. 101-130; ROY RADNOR, The
Application of Linear Programming to Team Decision Problems. « Management
Science ». Vol. s, Januarv 1959, pp. 143-150.
“121 Isard - pag. 27
1024 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
Pe
ing firm acting as a team and confronted with the possibilities
of either a centralized system of decision-making, or a decentralized
system. (?) His resulting figures would not be inconsistent
with the hypothesis which states that an overview
advantage does exist, and with the more concrete hypothesis
that states that this advantage can be roughly depicted by
curve CC of Figure 10. [In Figure 10, curve CC indicates a
disadvantage (a negative Figure) for any degree of spatial
centralization less than 100%]. The present author together
with Tung has developed another example involving the investment
by a large merchandising firm of $1 million at each of
several h*” order nodes (?). The resulting figures are also not
inconsistent with the hypotheses underlving curve CC of Figure
IO.
Clearly, neither of these two examples can be said to support
in any way whatsoever any hypothesis on overview
advantage. À considerable amount of spade work has yet to
be done in defining concents and terms before anv empirical
Dollars
+10,000
+ 6,666
SIG.
Al
DEGREE OF
20% SPATIAL
CENTRALIZATION
En
“TARSCHAK, Op. cit.
fsarp and TUNG. ob. cit., Part. II.
121 Isard - pag. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1025
test can be undertaken. Nonetheless, for pedagogical purposes
we shall utilize curve CC of Figure 10 to illustrate overview
advantages for ‘ifferent degrees of spatial decentralization.
5. THE NET EFFECT OF PARTICIPATION POTENTIAL, INFORMATION-COMMUNICATION-DECISION
COSTS AND OVERVIEW ADVANTAGE
We now examine the interplay of the several factors examined,
where only one state of the environment can occur. Such
examination immediately implies the hypothesis that the factors
examined are in effect the relevant factors. This hypothesis is
to be seriously questioned. However, if one does accept it, and
if one accepts the hypotheses implied by the extra-returns curve
of Figure 3, then in Figure 10 curve AA may be taken to
represent extra-returns with respect to decisions on a representative
concrete issue; however, the horizontal axis is taken to
measure degree of spatial centralization (rather than participation
potential). Also the pattern of 100 percent spatial central
ization is taken to be a point of reference for purposes of com
parison; its extra-returns is set at zero.
Further, if one were to accept the hypotheses underlying
the curve of Figure g, then in Figure 10 curve BB depicts
variation in total information, communication and decisionmaking
costs, again where this cost at 100 percent spatial centralization
is taken to be zero. Curve CC, as already indicated,
may be taken to portray overview advantage (disadvantage)
where such advantage 1s set at zero at 100 percent central
ization.
Since all advantages, disadvantages, and extra-returns for
the pattern of 100 percent spatial centralization are set at zero,
may it be hypothesized that the construction of a fourth curve
is meaningful? This curve, curve DD of Figure 10, at each
point of the horizontal axis indicates for the corresponding
pattern of spatial centralization the net advantage (disadvantf12]
Isard - pag. 23
1026 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
a
age) of this pattern relative to the pattern of 100 percent spatial
centralization. Where curve DD lies below the horizontal axis,
there is a net disadvantage for the corresponding pattern.
Where DD lies above the horizontal, there is a net advantage.
Obviously, where the entire set of hypotheses underlying he
curves of Figure 10 is accepted, the optimal degree of spatial
centralization would correspond to that point at which curve DD
reaches a maximum.
The curves of Figure 10, and the discussion thus far, relate
to decision making on a representative concrete issue (or set
of concrete issues) when it is hypothesized that only one, welldefined
state of the environment can occur. It is now appropriate
to relax this restriction and permit the occurrence of
several states of the environment.
Elsewhere, ISARD and Tune have extended the simple
example cited above so that the occurrence of each of three
states of the environment is possible (**). These states have
reference to the international situation and were designated:
I) good will, 2) usual, and 3) extremely tense, as indicated at
the head of the columns of table 3. In this example, only
overview advantage was permitted to vary with the state of
environment for the specific set of assumptions which was
adopted. The resulting figures for the case examined are recorded
in table 3. In net form, these figures constitute the
elements of the payoff matrix of table 4. It is immediately seen
that there is no one pattern of spatial decentralization which
can be designated optimal. In the light of modern decisionmaking
theory it becomes necessary to introduce still another
assumption which pertains to the attitude of the decisionmaking
individual or group (%).
(2) Isarp and Tung, op. cit., Part. IT.
(5) SEE W. IsSARD and M. Dacky, On the Projection of Individual Be
havior in Regional Analysis, Parts I and II, « Journal of Regional Science »
Vol. a. Nos. T and » respectively and articles cited therein
121 Isard - pag. 24
SEMAINE D'ÉTUDE SUR LE RUIE DE L’ANALYSE ECONOMETRIQUE ETC. 1027
CONCLUDING REMARKS
The purpose of this paper was simply to confront squarely
a basic problem in economic planning for regional development
— the problem of an appropriate spatial organization of planning
— and other governmental functions, and of the appropriate
spatial allocation of decision-making authority. Because
this fundamental question has been avoided by econometricians
and regional scientists in the past we have been able to present
only simple « reasonable » hypotheses. Admittedly, these hypothese
are weak, and their empirical testing will lead quickly
to new, superior hypotheses. Yet, these hypotheses represent
the best that can be culled from the existing voluminous, but
rigorless, literature. It is hoped that their presentation will
provoke major research effort and contributions to this area.
In closing, it may be mentioned that once empirical testing
has been conducted and a number of hypotheses established as
valid, it should be possible to estimate decision-making cost
differentials for each type of organization, firm, or industry.
With such cost differentials on hand, it would then be possible
to broaden location analysis in order to embrace the decisionmaking
function. Such broadening would involve essentially
the comparison of the decision-making cost differential with the
transport cost differential, labor cost differential, and other
cost differentials considered as relevant in a location investigation
of an organization. The necessary extension of location
theory based upon substitution points can easily be achieved.
Finally, with a superior classification of organizations, firms or
industries as national, regional and local, it would become possible
to introduce the decision-making function, at least in part,
into one or more of the channels of synthesis of regional techniques
which have been described elsewhere (?°),. For example
{(¥) SEE W. Isarp, et al., Methods of Regional Analysis, M.I.T. Press
Cambridge, Massachusetts. 1960. Chapter *2
«1 Isard - pag.
25
1028 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
it would be possible to develop a channel using comparative
cost, interregional linear programming, industrial complex
techniques centered around a balanced regional input-output
model for three or more orders of regions or nodes. Such a
channel could be utilized to project not only output, employment,
population, income, consumption and investment by
region, but also the spatial allocation of decision-making
authority and the spatial patterns of decisions, information
collection, processing and transmission. However, all this
development is for the future. The immediate task ahead is the
statement of relatively simple hypotheses in ways that can
be rigorously tested against empirical materials.
121 Isard - pag. 26
DISCUSSION
FISHER
[ have only a minor comment on Professor ISARD’s paper, it is
in connection with his use of weights, He uses an expression in
which W, and Ww, appear as weights; however, there is nothing about
W, which is specific to i or Ww. which is specific to 4 in that expression,
since the form of the expression in fact allows these to
be interchanged and the wrong weights applied to the two items.
The W, and Ww, therefore ought not to be referred to as weights in
the strict sense. On the other hand, IsARD uses these in obtaining
a weighted sum of items, the sum being taken over all pairs i
and 7. When that is done it is the case that W, appears in every
pair involving CD, and W, appears in every pair involving CD,
Thus each pair receives a weight equal to the product of W, and W,
and it is these products which should be considered as weights. It is
thus true that W. is specific to ‘ and W *o 4 although only the
pairs are use’
TSARD
Professor FISHER is correct in the context in which he speaks
However, in much of the literature using gravity models, 1! Wel us
W. and W, are used singly, and not ‘» produc form This + ‘he
reason I do so throuchout mv manuse=Isard
pag. 27
1030 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
HAAVELMO
I would first like to say that I am sure Prof. ISARD is right in
underlining the importance of the kind of problem here mentioned
and that also very little has been done about it. Now, the remark
I have refers to Prof. IsarD’s special model of interaction, and is
based on a recent experience I have had. One of my students came
to me because he was engaged in some study of retail trade in a
regional network of shopping centers. Some of these centers were
small and some larger, and the problem was how to locate stores,
parking space etc. We got into discussion about the following problem.
Consider the interaction formula that you have. In my case
it can be regarded as representing how many customers are drawn
from 7 to j and vice versa. We found that the symmetry of this
interaction formula was not adequate. We have a passive element,
so to speak, and have to distinguish between the active pull and
the response. We did not find the final answer, but I think it is
a general problem relevant also to your model.
[SARD
I agree that many, if not most situations in the real world involve
asymmetry. Many suggestions have been made about what the
measure could be in such cases. For any originating node (region)
the measure will differ according to type of receiving or terminating
point (node). Also, I have presented in my paper a regular hierarchy.
Actually, we know that nodes are differently distributed, and that
population is of different density in different regions and at different
nodes. But to introduce an irregular hierarchy causes all kinds of
problems. I do not have the time now to spell out some of the
interesting thinking on these problems.
‘127 Isard - pag. 28
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETc. 1031
WoLbp
When reading this paper in advance, I was quite fascinated by the
new vistas it opens up on the analysis of organization problems. At
the same time I was puzzled by some of the implications, and I am
not quite sure I have grasped the underlying basic hypotheses. This
is in particular so with your statement that neither zero nor
100%, spatial decentralization work at a disadvantage or at an advantage.
Am I right to understand that this result is based on an
assumption of complete insight into the future?
To emphasize this question I should like to tell what I once heard
from a friend of mine in the foreign service. We had talked about
the possibility of predicting developments about war and peace at the
upper decision levels of the great powers, and in particular I had
asked whether diplomats find it more difficult to predict such development
for the Soviet, Union than for the United States, My friend
said that the developments are quite unpredictable in both countries,
but for entirely different reasons. In the Soviet Union the decisions
of the inner circle are watertight secrets, and no outsider can get the
smallest clue to predicting what will happen. In the States, he
continued, every senator is an open book on his political views, and
his opinions can be read in any newspaper; different senators however
have different views, and the political decisions are made by committees;
what will happen in the future thus is a question of which
senators will be members of the committee at issue, and this depends
on so many intangible factors that no specialist in the world is able
to make valid predictions. Thus although the two systems differ very
much with regard to information about the inner circles, the situation
with regard to prediction is in reality very much the same, because
in the open system the complete insight is only apparent. What is
needed for genuine prediction is much more. The point I wish to
make is that the assumption of complete insight which is customary
in many scientific approaches is indeed very stringent.
:+ Isard - pag. 20
1032 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
FRISCH
1 have three suggestions to make. First, we must distinguish
between different things we may think of when we subdivide according
to regions or spatial distribution. For instance we may think
of regional flows of goods and services, from one region to another, or
from one centre to another, inc'uding in this flow of goods and services
also the flow of information leading to such concepts as noise
in information channels and the like. Second, we may think of the
pyramidation, or the regional distribution of decisional power.
Third, we must look a bit closer into the question of what we
really mean by a region or a center.
I think that when we speak of the first point of view, i. e. the
flow of goods and services between regions and centers even including
the flow of information, the viewpoint is not fundamentally new. In
a sense it does not introduce too much innovation in our way of
thinking. But, when it comes to distributing, decisional power, the
problem becomes extremely complex. We are facing the difficulty of
programming a decisional machinery at the top, and at the same time
elaborating rules and regulation for the working of the decisional machinery
in the individual regions and centres, which is such that the
ntentions of the centre are realized when certain decisions and a
certain amount of programming analysis is left to be done in the
regions and centres. Mathematically speaking, I have found it next
to impossible to attack this problem in a straightforward programming
way (1) and at the moment I can see no better way out than
to build up some kind of simulation or artificially constructed games
of decision at various centres. Then, third with regard to the definition
— what we mean by a region or a centre. I think it will be
a fallacy to concentrate too much on the traditional geographical
subdivisions, for instance administrative subdivisions in regions or
in states within a union or local! administrations within a state and
(') Note added July 1964: in the spring of this year I made an
attempt in a University of Pittsburgh memorandum, building on my non
complex method for enlving non linear programming problems
121 Isard - pag. 20
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 1033
so on. From the viewpoint of a complete programming problem
I think we must look first into what I have called the Problem of
Patterns of Centres, We may find a certain pattern of centres,
which is entirely different from the traditional administrative lines
of demarcation. It must be based on an entirely different complex
of relations that goes deeper down into the basic problems of economic
relations, transportations and so on. This problem of the
pattern of centres may be approached somewhat in the same way as
the construction of investment projects. In a complete development
and investment programming work, there are two stages. In the
first place, you must have a large list of investment projects to
choose from. The list must be as diversified as possible, each
project in the list being characterised by objective data of various
sorts: What the increase in capacity will be if and when we have
carried through the investments, what in put effects will be manifest
if and when you decide to go through with a specific group
of projects and so forth. I have said connection about this in my
paper. The essential point to retain in this comment is that when
you make this list of projects you must not think that the mere
entering of a project in the list is equivalent to saying « I am going
to carry this project out ». It is simply putting up an alternative
that may or may not be accepted and it is from this big list of
investments projects that we have to choose: the list is the basis of
the scientific programming of decisions on investments. Similarly
we should handle a list of Patterns of Centres
ATT AIS
I do not see at all what procedure professor FRISCH rer -r.mend._
If we have to choose investments or to decide what “> us
be followed for different regions, the problems to be solved are
absolutely different. A planning bureau can choose between different
investment projects according to different criteria - - that is
very simple and clear in principle, but if we have to decide what
policy to apply for different regions, the nature of the problem is
|
Isard - pag. 31
1034 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
completely different. In fact, who will decide? The Central Planning
Bureau? But if the people of the different regions are not
represented in this Central Planning Bureau, what will happen? I
will give only one example. In the different Western countries, we
have a tariff policy, and the effect of this tariff policy is that in
these countries the peripheral regions alone support the burden of
this policy. For instance in my own country, Brittany is bearing
the cost of protecting the french coal mines, and in Canada, it is in
the main the Province of Quebec which supports the Canadian
tariff policy of Canada and so on. So, in my opinion, it is impossible
to treat these two problems, investment on the one hand
and policy for each region on the other in the same wav.
FriscH
I am sorry that my explanation must have been too brief, because
Prof. ArLrars has completely misundertood what I meant to
say. When I spoke about analogy with investments selection in a
big list of projects, I was only referring to the formal construction
of what we are going to take as a pattern of centres. There may
be many alternative patterns of centres, or there may be many
investment projects in our list of projects,
I did not speak at all about how the decision of powers is to
be distributed. We have to use this big list of alternatives of centres,
and out of those make a choice. The choice is to be studied by
mathematical programming.
ALLAIS
Again there is a great difference but I do not think we can
discuss the auestion now.
21 Isard - pag. 32
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1035
[SARD
I think this discussion which has been going on is a part of the
general problem of deciding upon the criteria to be used in making
decisions. The criteria can be expected to, and should differ, from
one pattern of decision-making authority to another, from one regional
planning group to another, etc. Thus the question of how to
make decisions, how to conduct research, what techniques to use
must be related to the question of who is to make decisions. They
must be related to some system of regional planning authorities, to
some spatial pattern of decentralization in decision-making authority.
The question of the appropriate spatial pattern of decisionmaking
authority is the concern of my paper. It is a question which
must be answered if we are to resolve the issues raised bv FRISCH
and ALLAIS.
I agree with Professor WoLD that rigorously speaking, complete
information is implicitly assumed at certain points in my paper.
This is indeed a very stringent assumption. but is required if we
are to make anv headway on our basic nroblem.
DORFMAN
Prof. Isarp broaches some really frightening questions, and 1
am forced to drop from view all but one of them. He mentions that
it would be very economical to have a completely decentralized
system in which decisions could be made at each point in the light
of local conditions. But this is not necessarily so. It is not so if
the decision unit in each locality has to base its decision on imperfect
guesses about the decisions being made in other localities, which is
the normal case if the localities are bound together into a community.
An example is the difficulties faced by the independent highway
commissions of the United States in trying to establish an intellegible
system of highway codes that will not interfere with the flow
of traffic.
Isard - pag. 33
1036 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
Some of the experimental results of small group dynamics are
pertinent to this issue. If you decentralize the decisions in a group
too far, so that each member of a team is required to obtain a lot
of information from all the others in order to make his decisions,
the efficiency of the whole team is diminished. It is pretty clear that
the relationship between efficiency and centralization is not monotonic;
you run into serious inefficiencies at both extremes.
HAAVELMO
It seems to me that part of Prof. ISARD’s argument has to do
with the old question of whether errors committed by centralized
decisions will be bigger than the errors of decentralized decisions
because, allegedly, the errors in the latter case may tend to cancel
each other. But the answer is not so simple. It will depend essentially
on the kind of correlations that exist between the errors, and
this again depends on the network of relations in the economic
system.
[SARD
I am glad to have Professor DorRFMAN’s comments. They indicate
to me that I have not been as clear as I should have been
on a number of points. First, I meant to suggest that only in some
situations and with respect to only some specific functions (e. g.
planning on local education) would a highly decentralized system
be desirable. In many other cases, such as in transportation planning,
a high degree of spatial decentralization of decision-making
authority is highly undesirable.
I also would subscribe to the view that in the operation of many
groups the relationship between efficiency and centralization is not
monotonic. Actually, I use a number of graphs to indicate the non-12]
Isard - pag. 34
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
1037
monoticity of this relationship; and most empirical investigation
would bear out this hypothesis. Again, I want to emphasize the
highly exploratory character of my paper, which represents only
an initial attempt to dig away at a problem which constantly plagues
every regional-national planner — and a problem which has been
by and large ignored by social scientists in their analytic frameworks
A &
Isard - pag. 35
THE RATES OF LONG-RUN ECONOMIC
GROWTH AND CAPITAL TRANSFER FROM
DEVELOPED TO UNDERDEVELOPED AREAS
WASSILY (LEONTIEF
Harvard University - Cambridge. Mass. - U.S.A.
1. The Underdeveloped Areas, which hold at least twothirds
of the entire population of the world, produce now only
about one-seventh of the world’s gross output of goods and
services; moreover, their rate of economic growth is at the
present time much lower — possibly only half as high — as
that of the advanced industrialized countries. That means that
the contrast between the richer and the poorer areas tends to
increase rather than diminish.
A rise in the rate of growth of the Underdeveloped Areas
would demand an increased volume of productive investment.
The additional capital could be created through stepped up
internal savings, or it might be obtained from abroad, that
is, transferred in the form of aid, foreign loans, or direct private
investment from the Developed countries.
How much additional investment would the Underdeveloped
parts of the world have to absorb if they were to raise their
average growth rate, over the next ten year period, up to the
average growth rate of the economically advanced industrial
countries? If capital transferred from Developed to Under
3
Leontief - pag.
1040 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
developed Areas were to constitute the principal source of such
additional investment, how large would this transfer have
to be?
The simple dynamic system presented below describes in
crude aggregative terms the relationships of the magnitude of
the capital transfer from Developed to Underdeveloped Areas,
and of the levels of saving and investment in both groups of
countries to their respective rates of growth. It was designed
so as to require not more factual statistical information than
is actually available. The over all capital-output and saving
ratios of the more and the less advanced countries as well as
the proportion (but not of course the absolute amount) of the
Gross National Product of the Developed Areas transferred to
the Underdeveloped countries are assumed to be constant over
the ten year period over which we project their future growth.
Since aggregative capital-output ratios (capital coefficients)
and saving ratios can be estimated — particularly for the
Underdeveloped Areas — only within a rather wide margin
of error and because our expressed purpose is to assess the
possible effect of changes in the amount of outside capital received
by Underdeveloped Areas on their rate of growth, not
one, but many alternative projections were made, all computed
from the same general formula, but each based on different
hypothetical combinations of the magnitudes of the
structural parameters enumerated above.
2. The following set of aggregative variables is used to
describe the state of the two groups of economics — Developed
and Underdeveloped — at any particular point of time, #:
12] Leontief - pag. 2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1041
VARIABLES
Gross National Product (domestically
produced)
Productive investment (total)
Capital transfer from Developed
to Underdeveloped
Areas . . .
Developed
Areas
+, 7
T 4
Eglv
Underdeveloped
Areas
\
N
vy
14
Growth rate of the domestically
produced Gross Na
tional Produc!
a)
“
J
The value of these seven variables given (that is, observed
or assigned) for the year 1959 constitutes the empirical basis of
a series of alternative projections of the economic growth of
both groups of countries over the ten year period ending at
taba
DEVELOPED AREAS
The following theoretical relationships are used to derive,
and to solve, the equations describing the growth of the Developed
Areas:
Savine Function:
I
Li) =4Y,,
, represents the fraction of the GNP allocated to invest:
ment
13) Leontief - pag. 3
1042 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Acceleration Relationships:
21
Y(t) = L®
b, is the capital coefficient (capital-output ratio) describing
the amount of capital required per additional unit of annual
GNP.
Growth Rate equation, obtained from (1) and (2):
St
——— oO
yo) =
A
V.(£) —
Ÿ,
Exponential Growth Function, obtained by solving (3):
4) Y,(®) = Y, (0) eMt , à, — 11
1x =
A.
where Y,(0) represents the level of the GNP in the base
year 0 and À, its growth rate, which remains constant as
long as :, and b, are fixed.
The amount transferred from the Developed to the Underdeveloped
Areas is assumed to constitute a fixed fraction, h,
of the GNP of the capital-exporting countries. Thus, the following
Transfer relationship, which is derived from equations
(4) above, implies that H(¢), the amount transferred, will
grow exponentially at the same rate as the Developed Areas’
GNP:
Transfer relationship:
1
H(#)=RY,({) = RY, (0)e
‘13] Leontief - pag. 4
SEMAINE D’ ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETc. 1043
[UNDERDEVELOPED AREAS
The productive investment in the Underdeveloped Areas is
being supported from two sources: The saved fraction, 1,, of
their own Gross National Product, ¥,(¢), and the capital-imports,
H(?):
Investment Function:
6)
L({)=i,Y,({) + H()=CAN
* A,
La
T
1 [we a
Acceleration Relationship:
=
Y, (HN
b, is the capital coefficient describing the amc... «. capiia
required per additional unit of annual GN.
Growth rate equation derived from ‘) and (7):
‘)
‘a)
Y.(5)
"+
Growth Function, obtained
tion (8):
r
Yh)=Y
ye
solving
Ir
‘he differential ecua
To verify the last equation one can substitute it and its
derivative in (0); the expression on the left hand side will
identically equal zero
-
1.3] Leontief - pag.
=
1044 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
~§
The growth of the Underdeveloped Areas turns out to be
described by a combination of two exponential terms. The
first reflects the effects of internal savings, the second the contribution
of investment financed through capital imports. Accordingly
the growth rate, A,, of the first component depends
on the magnitude of the domestic savings and capital-output
ratios while the second term grown at the same rate as the
GNP of the Developed Areas (1).
Equations (4) and (9) permit us to project forward the
growth of both groups of countries provided their base year
levels of their respective GNP’s, Savings, Growth Rates, as
well as the initial magnitude of the interregional capital transfer
is given. The corresponding values of the constants entering
into the two growth functions can be computed from the following
formulae:
pL Le) IL) Le
TY 0) Von PT ¥.0) Ya0)7.(0)
_0)
à, = À — Yalo) _ . __[L(0) — H(0)] Ya(o) _
BL Yo) MO) MST Lo) Lo)
H(o) | _
TO +o)
These relations are obtained by inserting the given base
year values of the variables in the appropriate Investment
Functions. Accelerations Relationships and Growth Functions.
(!) If the ratio of the saving to the capital coefficient in the Developed
and Underdeveloped regions happens to be equal. the solution of the
differential equation (8) is reduced to:
TIN
(0-2)
Y, (4) -
10)
nf
à -
SU
h,
2
b,
that is, the growth rates of bnth groups of countries would in this case
he reaual.
13]. Leontief - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
1045
3. The base year values of the variables used for projection,
presented in the following tables, are summarized below
The base year is 1950.
Gross National Product
(domestic) . . .
Productive Investment (total)
. . . .
Capital Transfer from Developed
to Underdevelobed
Areas . . .
Growth rate of the domestically
produced
Gross National Product
Developed Ateas
Underdeveloped Areas
$2,105 billion $195 billion
$228 billion $15 - $22 billion
p4 billion
+ 0-0 VA
A
Nv
The Developed Areas comprises Western Europe (exclud
ing Spain, Portugal, Greece and Turkey) United States, Canada,
Japan, Soviet Russia and other socialist countries; the
Underdeveloped Areas comprises all other countries. Since
these estimates, compiled from United Nations and other statistical
sources, are supposed to cover all countries, they obviously
are subject to a very substantial margin of error. The
estimates of the annual rate of gross productive investment in
the Underdeveloped Areas, which is particularly uncertain, is
presented not as a single figure but in terms of two figures —
a high and a low — which does not mean of course that the
true magnitude still might not lie outside of that range.
For similar reasons the base year estimate of the long-run
growth rates of the GNP’s is also presented for each area in
the form of two percentage figures — a higher and a lower one.
in
5. The calculations, the results of which are summarized
Tables T to VI, show how the future economic growth of
3] Leontief - pag. 7
1046 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
the Developed and Underdeveloped Areas might be affected
by changes in the fraction of the Gross National Product of the
first transferred for investment purposes into the second area
and also changes in the distribution of the GNP — in both
groups of countries — between current consumption and proluctive
investment.
The growth functions (4) and (9) are used to project the
GNP of the Developed and the Underdeveloped Areas over a
ten year period, 1959-1969.
The structural coefficients entering into these equations
depend — see (10) — on the base year magnitude of the GNP’s,
of the productive investments in both Areas and also on their
respective long-run growth rates in the base year.
Each table is based on a different combination of the estimated
1959 growth rates of both countries and of the estimated
amount of domestically financed investment absorbed by the
Underdeveloped Areas in that year. The estimates of the 1959
GNP’s of both Areas and of the amount of gross investment
absorbed in that year by the Developed countries remain the
same through all the computation.
The first column of figures in each table presents one particular
estimate of the base year state of both groups of countries
and also the levels of their respective GNP’s « ten years
later », projected from 1959 to 1969. This projection is made
on the assumption that the domestic saving ratios in both Areas
retain their base year magnitudes and that the economically
more advanced countries continue, throughout the ten year
period in question, to transfer to the less advanced the same
percentage (h) of their annual GNP as they did in 1950.
The three other columns show how hypothetically postulated
changes in original allocation of the GNP’s of both areas — if
introduced in the base year and then maintained over the ten
years covered by our projections — would have affected the
levels of their respective GNP’s « ten years later », i.e., in
1969. The corresponding average annual growth rates over
the period 1959-10609 are entered below
13] Leontief - pag. 8
SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 1047
The allocations of the GNP of the Developed Areas is described
by the constants ?, and h, i.e., the percentage of the
GNP invested domestically and the fraction of that Product
transferred to the Underdeveloped Areas and invested there.
In the Underdeveloped Areas it is described simply by ¢, — the
proportion of it saved and invested; the rest of the investment
being equal to the amount received from the Developed
rountries.
In the second column of Table I, the proportion of the
GNP of the Developed Areas transferred to the Underdeveloped
Areas is assumed to remain — over the ten year period covered
by the projection — the same (0.3%) as it was in 1959. However,
the domestic investment ratio is assumed to have been
raised in the Developed countries from 18.99, to 19.9% and
in the Underdeveloped countries from 9.2%, to 10.8%. As a
result of that the average growth rate of the two Areas — over
the ten year period, 1959-1969 — would go up, respectively
from 5% to 5.3% and from 3.29% to 3.5%.
In the third column the transfer coefficient, h, is stepped
up to 1.29%, and in the fourth column to 2.19%. That implies
a transfer of $15 billion and respectively- $25 billion in 1959,
and of correspondingly larger amounts with the growth of
the GNP of the Developed Areas in subsequent years.
Comparing the projected average annual growth rates shown
in columns 3 and 4 we find that even with the capital transfer
stepped up to $15 billion already in 1959, the Underdeveloped
countries would still continue to grow slower than the Developed;
with $25 billion they would begin to catch up. The
« break even point » at which the two growth rates become
equal would be reached with a base year capital transfer of
somewhere between $15 and $25 — probably around $20
billion.
The other five tables are organized on the same pattern as
the first. The difference between any two tables lies in the
assessment of the actual position of the two areas in the base
vear 1050
51 Leontief - pag. o
1048 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Thus, so far as the original 1959 growth rates are concerned
the projections shown on Table II are based on the same 5%
and respectively 3% figure as the projections presented on
Table I. The actual domestically financed 1959 investment
of the Underdeveloped countries is estimated in Table I at
$18 billion, while in Table II at only $11 billion. Accordingly,
the implicit estimate of the capital-output ratio, b,, falls from
3.76 on Table I to 2.56 on Table II.
With any given absolute amount of investment, the lower
is the capital-output ratio, the higher must be the rate of
growth. This explains why in the third projection on Table IT
with a base year transfer of $15 billion, the Underdeveloped
countries attained — over the ten year period, 1959-1969 —
an average annual growth rate of 5.4%, which is higher than
the corresponding growth rate of the Developed Areas, shown
to be 5.2%.
A comparison of the implicit capital-output ratios of the
two areas throws light on the plausibility of some of the base
year estimates from which the different sets of projections have
been derived. For example, in Table VI, b,=3.15 and
b, =5.64. It seems to be quite unlikely that the average capital
intensity of production would be so much higher in the less
advanced than in the more advanced areas. A combination
of a high growth rate for the Developed, with a low growth
rate and a high investment figure for the less developed Areas
must be rejected as implausible. This throws considerable
doubt also on the validity of all the four alternative projections
of future economic growth presented on Table VI.
For reasons similar to those described above, or for some
other reasons, a critical examination of the alternative factual
assessments of the state in which both groups of countries
actually found themselves in 1950 might lead some experts to
reject, out-of-hand, some other of the 24 different proiections
presented in these tables.
The examination of all the alternative projections of the
prospective growth of the two groups of countries over the
113] Leontief - pag. io
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETc. 1049
ten year period, 1959-1969, enables us in any case to answer
the two questions posed in the opening paragraph of this
paper. Whichever set of factual assumptions concerning the
base year situation one chooses to accept, one reaches the same
conclusion that the « break even point » between the rates
of the economic expansion of the two groups of countries could
not be reached before the Underdeveloped Areas would raise
their average annual growth rate to about five percent. To
accelerate their present much slower pace, they would have to
double the actual 1959 rate of investment in the very first
vear and then raise it progressively from year to year.
To make this possible the annual transfer of (productively
invested) capital from Developed to Underdeveloped Areas
would have to increase from $4 to around $15 or even $20
billion in the first year and then go up annually reaching the
level of between $28 and $35 billion in the tenth year.
An assessment of the feasibility of such an ambitious investment
program for the Underdeveloped countries supported by
a massive capital inflow from the Developed countries lies out
side the scope of this paper.
The factual conclusion drawn from the numerical results
of our computations has to be accepted, or rejected, in the light
of the plausibility of the analytical approach and the reliability
of the factual information on which they are based. Being
fully conscious of the uncertain and even controversial nature
of the statistical estimates, which have to be used in such
aggregative analysis, I purposefully presented not one or two
but a very large number of alternative projections reflecting a
wide range of possible initial conditions. The simple analytical
system developed for that purpose is constructed in such a
way that with a minimum of computational efforts the spectrum
of alternative projections can be expanded further through
insertion in the appropriate computational formula of still other,
different figures purporting to give a more correct assessment
of the base year situation. Thus, for example, the estimate
l13] Leontief - pag. I1
1050 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
of the amount of capital transferred in the year 1959 from the
Developed to Underdeveloped Areas might possibly be increased
from $4 to, say, $5 or even $6 billion. Projections
based on such revised descriptions of the base year situation
would incidentally show that a larger amount of additional
investment in Underdeveloped Areas and a correspondingly
greater increase in the level of capital imports from the Developed
Areas, than those that were mentioned above, would be
required to bring about an equilization in the growth rates of
these two groups of countries.
13] Leontief - pag. 12
-
4
A
ug
>
2
—-[ABLE
[
Economic growth and capital transfer
DEVELOPED COUNTRIES
(growth rate in base vear: 5%.
implicit b,=3.78)
Base Year (1950)
Gross National Product . . .
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (4)
Domestic investment . . . . . .
as per cent of national product (i,)
Total consumption . . . . . . .
as per cent of national product . .
len Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten vear period | .
Original
Allocation
%120%.0 bil.
B4.0 bil.
0.3%
$228.0 bil.
18.9%
B973.0 bil,
80.89,
$1086.77 bil.
2.0%,
UNDERDEVELOPED COUNTRIES
‘growth rate in base vear: 3%.
.mplicit b,= 3.76)
Base Year (1059)
Gross National Product . . . . .
Domestically financed investment . .
as per cent of national product (i,)
Total investment . . . . .
[otal consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten vear period
$195.0 bil.
$18.0 bil.
9.2%
$22.0 bil.
177.0 bil.
ao.8°/.
$264.7 bil.
2» 20/
from developed to underdeveloped areas
Alternative Allocations
1205.0 bil.
$1205.0 bil.
$1205.0 bil
$4.0 bil.
0.3%
$240.0 bil.
19.9%
Bg61.0 bil.
79.89%,
$15.0 bil.
1.2%
$237.6 bil.
19.89%,
$052.4 bil.
70.0%,
$25.0 bil.
2.1%
$235.6 bil.
19.5%
$044.4 bil
78.4%
$2039.0 bil.
$202q.2 bil.
$2019.9 bil.
5.3%
5.29
5.29,
$195.0 bil.
$21.0 bil,
10.89%,
$25.0 bil.
h174.0 bil
809.2°/.
$195.0 bil.
$21.0 bil.
10.89%,
$36.0 bil.
$174.0 bil.
8a.2°/,
$195.0 bil.
$21.0 bil.
10.8%,
$46.0 bil.
$174.0 bil.
80.29.
$275.7 bil
$319.6 bil.
$358.3 bil
3.5%
1.0%
6.1%,
x
—
-
-
_
TABLE li
~~ Economic growth and capital transfer from developed to underdeveloped areas
DEVELOPED COUNTRIES
(growth rate in base vear: 5%,
implicit b,=3.78)
Base Year (1950)
Gross National Product . .
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (4)
Domestic investment . . . . | .
as per cent of national product (i)
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten year period
Original
Allocation |
Alternative Allocations
$1205.0 bil.
$1205.0 bil.
$1205.0 bil.
$1205.0 bil.
$4.0 bil.
0.3%
$228.0 bil.
18.9%
$973.0 bil.
80.89,
$4.0 bil.
0.3%
$238.4 bil.
19.89%,
$962.6 bil.
79.9%
$15.0 bil.
1.2%
$236.1 bil.
19.6%,
$953.9 bil.
79.2%
$25.0 bil.
2.1%
$234.0 bil.
19.4%
$946.0 bil.
48.5%
$1088.72 bil.
$2036.4 bil.
$2026.8 bil.
$2018.4 bil.
5.09,
5.39%,
5.29
5.19
UNDERDEVELOPED COUNTRIES
(growth rate in base year: 3%,
implicit b,= 12.56)
Rase Year (1959)
Gross National Product . . .
Domestically financed investment . .
as per cent of national product (i,)
Total investment . . . . ,
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten vear neriod
$195.0 bil.
$11.0 bil.
5.6%
$15.0 bil.
$184.0 bil.
04.4%,
$195.0 bil.
$11.6 bil.
5.0%
$15.6 bil.
$183.4 bil.
04.1%
$195.0 bil.
$11.6 bil.
5.9%
$26.6 bil.
$183.4 bil.
04.1%,
$195.0 bil.
$171.6 bil.
5-9%
$36.6 bil.
$183.4 bil.
04.1%,
5266.6 bil.
5268.4 bil.
$331.3 bil.
$284.8 bil.
» 19/
, »0/
NLA
7.19,
=
a
NS
=
=
»
-
Y
H
A
ee
—
TABLE
III — Economic growth and capital transfer from developed to underdeveloped areas
DEVELOPED COUNTRIES
(growth rate in base vear: 4%,
implicit b,=4.73)
Base Year (1959)
Gross National Product
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (k)
Domestic investment . . . . . .
as per cent of national product (i,)
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . ©
Average annual growth rate over the
ten year period
Original
Allocation
Alternative Allocations
$1205.0 bil.
$1205.0 bil,
$1205.0 bil.
$1205.0 bil.
$4.0 bil.
15.0 bil.
0.3%
$228.0 bil.
18.9%
$973.0 bil.
80.89,
$4.0 bil.
0.3%
$239.0 bil.
19.8%
$962.0 bil.
20.9%
1.2%
$236.0 bil.
19.6%
$954.0 bil.
79.2%
$25.0 bil.
2.1%
$234.0 bil,
19.4%
$946.0 bil.
78.5%
$1784.0 bil.
$1818.0 bil.
$1818.0 bil.
$1801.0 bil
4.0%,
4.2%,
4.2%,
4.1%,
UNDERDEVELOPED COUNTRIES
‘growth rate in base year: 2%,
implicit b,= 3.85)
Base Year (1959)
Gross National Product . . .
Domestically financed investment . .
as per cent of national product (i)
Total investment . . . .
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten year period
$195.0 bil,
$11.0 bil.
5.6%
$15.0 bil.
$184.0 bil.
04.4%
$195.0 bil.
$11.6 bil.
5.9%
$15.6 bil
$183.4 bu
04.1."
$195.0 bil.
$11.6 bil.
5.9%
$26.6 bil.
$183.4 bil.
04.1%
$195.0 bil,
$11.6 bil.
5.9%
$36.6 bil
$183.4 hil
04.1%,
$240.0 Dil.
$240.9 D.
$275.0 bil.
$309.0 bil
rs
a
2.19
3.5%
4.7%
-
fi»
=
en
>
n
A
TABLE IV — Economic growth and capital transfer from developed to underdeveloped areas
x
J
DEVELOPED COUNTRIES
(growth rate in base year: 4%,
implicit b,=4.73)
Base Year (1959)
Gross National Product . . . .
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (4)
Domestic investment . . . . . .
as per cent of national product (i,)
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten year period . . .
UNDERDEVELOPED COUNTRIES
(growth rate in base year: 29,
implicit b,= 5.64)
Base Year (1950)
Gross National Product . . .
Domestically financed investment . .
as per cent of national product (i,)
Total investment . . . . . :
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate over the
ten vear period
Original
Allocation
$1205.0 bil.
$4.0 bil.
0.3%
$228.0 bil.
18.9%
$973.0 bil.
80.89,
$1784.0 bil.
4.0%,
$195.0 bil.
$18.0 bil.
9.2%
$22.0 bil.
$177.0 bil.
00.89%,
$228.0 bil.
2.0%,
$1205.0 bil.
$4.0 bil.
0.3%
$239.0 bil.
19.8%
$962.0 bil,
70.09%
$1818.0 bil.
4.29
$195.0 bil.
$19.0 bil.
9.7%
$23.0 bil.
$176.0 bil.
0.3%
242.0 bil
2 »0/
Alternative Allocations
$1205.0 bil.
$1205.0 bil.
$15.0 bil.
1.2%
$236.0 bil.
19.6%
$954.0 bil.
79.2%
$25.0 bil.
2.1%
$234.0 bil.
19.4%
$946.0 bil.
78.5%
$1818.0 bil.
$1801.0 bil,
4.2%,
4.1%,
$195.0 bil.
$19.0 bil.
9.7%
$34.0 bil.
$176.0 bil.
90.3%,
$195.0 bil.
$19.0 bil.
9.7%
$44.0 bil.
$176.0 bil.
0.3%,
$267.0 bil.
#z201.0 bil
2,20/
19/
—
©
Jt
Da
‘
>
J
Sa
N
De
TABLE V — Economic growth and capital transfer from developed to underdeveloped areas
+
DEVELOPED COUNTRIES
(growth rate in base vear: 6%,
implicit b,= 3.15)
Base Year (1959)
Gross National Product
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (4)
Domestic investment . . . . . .
as per cent of national product (1)
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1960)
Gross National Product . . . . .
Average annual growth rate over the
ten year period .
UNDERDEVELOPED COUNTRIES
(growth rate in base year: 29%,
implicit b,=3.85)
Base Year (1959)
Gross National Product . . .
Domestically financed investment . .
as per cent of national product (i,)
Total investment . . . . .
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1069)
Gross National Product . . . . .
Average annual growth rate over the
ten year period
Original
Allocation
€1205.0 bil.
$4.0 bil.
0.3%
$228.0 bil.
18.9%,
$973.0 bil.
80.8°/,
$2158.0 bil.
6.0%,
$195.0 bil,
$11.0 bil.
5.6%
$15.0 bil.
$184.0 bil.
04.4%,
$242.0 bil.
4 2
$1205.0 bil.
$4.0 bil.
0.3%
$239.0 bil,
19.8%
$962.0 bil,
29.6%,
$2220.0 bil
6.39%,
$195.0 bil.
$12.0 bil.
6.2%
$16.0 bil.
$183.0 bil.
93.89%,
$242.0 bil.
2.29
Alternative Allocations
$1205.0 bil.
$15.0 bil.
1.29;
$236.0 bil.
19.6%
$954.0 bil.
76.2%,
$2220.0 bil.
6.39%,
$195.0 bil.
$12.0 bil,
6.2%,
$27.0 bil.
$183.0 bil.
03.89
$286.0 bil.
3.9%
$1205.0 bil
$25.0 bil.
2.1%
$234.0 bil.
19.4%
$946.0 bil.
78.5%
$2199.0 bil.
6.2%,
$195.0 bil.
$12.0 bil.
6.2%
$37.0 bil.
$183.0 bil
03.8%,
$324.0 bil
5.29.
Tr
2
5
J
TABLE VI — Economic growth and capital transfer from developed to underdeveloped areas
DEVELOPED COUNTRIES
(growth rate in base year: 69%,
implicit b,=3.15)
Base Year (1959)
Gross National Product . . .
Capital transfers to
underdeveloped areas . . . . .
as per cent of national product (hk)
Domestic investment . . . . . .
as per cent of national product (i)
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1069)
Gross National Product . . . . .
Average annual growth rate over the
ten year period .
UNDERDEVELOPED COUNTRIES
growth rate in base year: 2%,
implicit b,= 5.64)
Base Year (1950)
Gross National Product . . .
Domestically financed investment . .
as per cent of national product (i,)
Total investment . . . . . © :
Total consumption . . . . . . .
as per cent of national product . .
Ten Years Later (1969)
Gross National Product . . . . .
Average annual growth rate aver the
ten vear period
Original
| Allocation
$1205.0 bil.
$4.0 bil.
0.3%
$228.0 bil
18.9%
$973.0 bil.
80.89
$2158.0 bil.
6.0%,
$195.0 bil.
$18.0 bil.
9.2%
$22.0 bil.
f177.0 bil,
q0.89/
8240.0 hil.
2.1%.
Alternative Allocations
$1205.0 bil.
$1205.0 bil.
$1205.0 bil.
$4.0 bil.
0.3%
$239.0 bil.
19.8%
$962.0 bil.
79.9%
$15.0 bil.
1.2%
$236.0 bil.
19.6%
$954.0 bil.
79.2%,
$25.0 bil.
2.1%
$234.0 bil.
19.4%
$946.0 bil.
78.5%
$2220.0 bil.
$2220.0 bil.
$2199.0 bil.
6.39,
6.29
6.2%,
$195.0 bil.
$19.0 bil.
9.7%
$23.0 bil.
$176.0 bil.
00.32%,
$195.0 bil.
$19.0 bil.
9.7%
$34.0 bil.
$176.0 bil.
90.3%,
$195.0 bil.
$19.0 bil.
9.7%
$44.0 bil.
$176.0 bil.
00.3%,
5242.0 hil
$272.0 bil.
$297.0 bil.
2.29/
2.4%,
1.39
>
SN
SN
=
=~
>
~~
Ssit”
FS
ALLAIS
Prof. LEONTIEF's paper is very interesting and the ideas he
expressed and his conclusions are very important and stimulating
from two points of view: first, theoretical and secondly political.
And in view of this importance I find myself obliged to express my
disagreement, in a very friendly, but very firm way. This disagreement
relates to equation 2, which is founded on the same hypothesis
as Prof. HaAvELMO’s paper yesterday. This ‘equation 2 has
practically underpinned Western policy over the last 15 years. In
every meeting of the United Nations, equation 2 has been implicitly
accepted and has thus had great importance. I think it is worth
while to discuss this equation 2 and the hypothesis on which it is
founded in detail. The hypothesis is that the real national income of
anv one country is proportional to its real capital.
I thin: this proposition can be accepted neither from a theoretical
nor an empirical point of view. We have to distinguish two different
points carrefully. The first is: we observe that the capital output
ratio is practically constant for a number of countries at different
times. That much is certain and I have personally given some data
to support this conclusion in my 1961 Econometrica paper. But it is
a completely different thing to accept that in every situation there
is proportionality between real income and real capital. Perhaps my
point of view is quite difficult to understand because it is not very
~Oomman
| Leontief - pag. 19
1058 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
I believe this point is absolutely independent of the model given
by my paper, but it is in the framework of this model that my view
can be put most clearly. In the model, the conclusion is reached that
the capital-output ratio must be practically constant. Nevertheless,
the conclusion is reached in the same model that there is a maximum
value for real national income, whatever the level of real capital.
Mav I recall that in my model. in the exponential case.
C9 (relation 251-6)
R I1+0 7
where y is the capital output ratio, C the nominal value of capital,
R the nominal value of the national income, à the rate of interest
and 0, a constant whose order of magnitude is 4. For small values
of 1, vy is practically a constant.
But we also have in the exponential case
K.
&
(relation 251-15)
where R is the real value of the national income, and k the coefficient
of homogeneity of the production function. Rg, is the maximum
value of R for T=0,
R has a maximum whatever the value of C. Nevertheless y
varies little over the usual range of variation of i.
Here, therefore, is at least one system of consistent hypotheses
and mathematical deductions in which there are at the same time
two valid propositions. The first is that when the rate of interest
is small, the capital output ratio is practically constant.
131 Leontief - pag. 20
SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
1054
At the same time there is a second proposition according to which
real national income has a maximum. whatever the level of real
capital.
Of course, this model in no way constitutes a proof, but at least it
supports the fact that the two propositions are absolutely different.
It also shows that it is impossible to derive a general property of the
production function, from the fact that the capital output ratio is
practically constant, according to which real national income would
be proportional to real capital. Thus from a theoretical point of view
it 1s impossible to deduce proportionality between real national income
and real capital from the fact that in certain circumstances we
can observe an apparent constancy of the capital output ratio.
From an empirical point of view the empirical researches of Douglas
and his followers have shown that we can write at least as a
first approximation
R = KLeCt
where R is real national income, L labor. C real capital, anc
x, {3 constants.
The Douglas results have been very much discussed but there
is one point which has never been contested This is that the order
of magnitude of the elasticits
41
Leontief - pag. 21
060 FONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
~
is 0.2 instead of 1 as'is implicitly assumed in equation 2 of LEon-IIEF’s
paper. In no case it is possible to conclude that there is
some proportionality between R and C which would mean B equal
to unity.
Now, in the Leontief calculations the hypothesis that real national
mcome tends to be proportional to real national capital plays a fundamental
role. If this hypothesis were abandoned, the results would
be completely different.
My argument is absolutely the same as yesterday. A given property
can appear to be reasonable without being right. I repeat
again. From an economic point of view, my model may be right
or wrong, this is as may be, but it is mathematically coherent. And
while it shows an approximate proportionality between income and
capital, nevertheless there is at the same time a maximum for real
national income whatever the value of real capital.
The political impact of the LEONTIEF paper, were its results
correct, would be of very great importance. The conclusion would be
that the West should very substantially increase its aid in capital.
My convinction is that this increase would have a much lesser effect
than that predicted by the LEONTIEF paper because the more important
factor is not capital but, in my opinion, technological
education.
Certainly, if we accepted a production function such as the Douglas
production function, which it seems very reasonable to accept,
at least as a first approximation, the result would be that the paper
would predict a much lesser effect. Thank vou.
LEONTIEF
I think I fully understand Prof. ALLAIS’ objection. To meet it
let me restate my position in respect to the point raised bv him in
the following way:
The stock of capital employed in productive process represents a
necessary but not a sufficient condition for attaining the level of
13] Leontief - pag.. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC
106
output corresponding to the fixed capital/output ratio entering in my
equation 2.
Accordingly, my computations show essentially the upper limits
of the future growth in national income that can be expected to be
brought about by various combinations of domestic savings and international
capital transfer, If the supply of other factor — such as
skilled labor, efficient management and natural resources — could
not expand proportionately the actual increase in income would in
each case fall short of that, in a certain sense, maximal rate.
I used the capital/output ratio essentially as a limiting factor.
Not saying that this amount of capital transfer is sufficient to produce
that output, I on the other hand imply that without the indicated
rate of capital transfer the corresponding rate of income growth —
computed on the basis of my formulae — could hardly be attained
Professor ALLAIS does not seem to disagree with this. So far as the
political conclusions — which he expects to be drawn from the results
of my analysis — are concerned, he might be right or he might be
wrong; it all depends on the additional — political — premises which
will have to be introduced before one would be justified to draw
them.
ALLAI.
I agree completely with what Professor LEONTIEF has said but :
wish to stress two points. First, precisely in this particular line, an
attempt is made to calculate some limits of the capital effect, and
this attempt is very interesting indeed. But if this is done, b should
be replaced in equation 2 by b/8 where 3 is equal to 0.2 and not 1,
if we accept the results of the Douglas researches and those of his
followers. My second point is that, so far as underdeveloped countries
are concerned, we must be very cautious in extrapolating data
which are valid for the West. This, I think, is the error that Douglas
himself has committed in assuming that his relation could represent
the production function over a verv large range. What he has shown
-31 Leontief - pag. 23
1062 PONTIFICIAE ACADEMIAE SCIENTIARVM: SCRIPTA VARIA - 2¥F
is that there is a correlation over time and in space between three
quantities: production, labor and capital. But it is impossible to
derive the conclusion that this function is valid over the whole range
of variation: it holds for the very restricted domain of data relating
to some Western developed countries. Thus, I could accept the
Leontief calculation, but only on two conditions. The first one would
be the replacement of b by b/B with BG equal to 0.2 in equation 2.
The second would be to say that this is a very tentative calculation
and that it is by no means certain that the results would remain as
valid as they may be for the West,
HAAVELMO
This exposé was very clear, there is little possibility of misunderstanding.
I am actually asking a question about an alternative hypothesis,
and whether the author has considered it. If you would
kindly turn to page 5, equation (6). I just wonder whether LEONTIEF
has considered the possibilities that equation (6) might have the alternative
form I,(#)=1,[ Y,(#) + H(#)]. That is to say, that investment
in the region is a fraction of regional income including what they get
from abroad. I think we see the possible implications. It means, for
example, that domestic savings could be negative if H is verv large.
LEONTIEF
I have not tried to perform an alternative set of computations
based on Professor HAAVELMO’s suggestion that the capital inflow into
a developing country be treated as a part of its total current income
of which a fixed fraction be allocated to productive investment. On
this assumption a still larger capital transfer from developed into the
less developed countries would be required to narrow down the gap
between the growth rates in their respective national incomes.
If T mav return with just one more remark to Prof. ALLAIS
13] Leontief - pag. 24
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIOUE ETC.
1063
comments: I did not make only one computation, I made very many
of them covering a wide range of different factual assumptions.
Examining the implicit magnitudes of capital coefficients reflecting
various estimates of the present rate of income growth and of productive
investment in underdeveloped countries, we find them to be in
some cases considerably higher, and in others much lower than could
be realistically expected. In most instances they appear to be quite
plausible. I am inclined to interpret such consistency as representing
indirect evidence in support of my conclusions.
FISHER
Professor LEONTIEF has presented a highly interesting paper.
find most attractive the idea of watching economic history with accompanying
illustrations. LEONTIEF has in fact produced interesting
pictures of what happens when the economy is open to foreign trade
and foreign trade either outgoing or ingoing is cut off. One might
also consider applying the same technique to the study of the dependence
of the economy on particular industries. For example, one
might open the system to the steel industry and see what would
happen if deliveries from or to that industry were cut off. The
pictures one would get from applying LEONTIEF’s technique to such
analyses would provide interesting insights into dependence of the
economy on particular industries or sectors.
ALLAIT
Prof. LEONTIEF has very rightly stressed that he has made calculations
with different values of the coefficients b, and b. But my
point is that it is not sufficient to consider values of the capital citput
ratios b, and b,. It is also necessary to take into account the coefficient
3 which is of the order of 0.2. Then the b would be replaced
| Leontief - pag. 25
1064 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
by values five times greater, and the range which has been considered
is not sufficient. With correct values of the b the results would
be much lower.
MAHALANOBIS
I think this is a very important question and I should like to
congratulate Professor LEONTIEF for the presentation of his paper.
But I find some difficulty because he has not stated explicitly the
population figures as my observations would depend possibly on the
factual basis of the partition into the developed and the underdeveloped.
Using a notional figure of three thousand million as the total
population of the world, I am asking whether China has been included
with six hundred million, again as a notional figure?
Secondly, assuming that some rates of growth of population have
been used, whether such growth was taken into account in estimating
the effect on income?
The conclusion is-that the rate will be such and such, and 20
or 25 billion (??) dollars of aid would be required per year. What
is the implication? Does the aid have to continue almost indefinitely?
What I am really trying to do is to bring in the well-known concept
of assisted take-off. Now supposing that there would be an assisted
take-off in 15 or 20 years, what would be the total effort required?
This seems to me to be a realistic and important question.
HEIL
I am much interested in Professor LEONTIEF’s paper, partly because
it is so close to an analysis which I carried out myself a number
of years ago (published in English in the International Economic
Review). ! was much‘impressed by the large differences in the per
capita incomes of the various countries. Accordingly, my criterion
was to reduce the variance of the logarithmic per capita income
13] Leontief - pag. 26
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1065
distribution by a certain amount, year after year. The criterion was
pursued by means of a system of aid from developed countries to
underdeveloped countries along the lines of a model which is close
to that of Professor LEONTIEF’s. The result was embarrassing. The
criterion implied that one country had to pay, viz., the United States,
and all others (including Canada, Switzerland, Sweden and Australia)
received aid. This unpractical result induced me to change the
criterion to that of aid to countries whose present per capita incomes
are below a certain level in such a way that they will reach a given
« marginal » level within a given period of time. This marginal
level was not considered to be constant over time. An exponentially
increasing function was preferred in view of the consideration that
our present standards for what is marginally acceptable will probably
he below the standards that will prevail after 2= vears.
AT LAIS
If we take as a provisional hypothesis that according to the Cobb
Douelas formulation
KLe(
where Ÿ is real national income. L is labor and C
real capital. then
(") Comments presented in plenary session.
were presented in the cenarate group
he preceeding comments
,1 Leontief - pag. 27
1066 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
4
where
C
{J
according to Professor LEONTIEF’s definition of page 3.
Then equation (2) must be written
12’)
dy, 8
i —_ 5 h (9)
instead of
2)
ayY- I
— = 5 100)
Now there is no doubt that the order of magnitude of § is 0.2 and
not I as in equation 2 of the paper. If 0.2 instead of 1 is taken for
8 the final results of the paper change very greatly.
In fact the range which Prof. LEONTIEF has ‘considered for b is
as follows: For b, 3.15 to 4.73 and b, 2.5 to 5.64, the range is about
I to 2. But if we take the coefficient § equal to 0.2 into account the
range which should be considered is five times greater since we must
consider b/f that is 5 b instead of b.
We must thus replace the coefficient b by the product b/ f=50b
in the calculations which follow. and this changes the final results
completely.
13] Leontief - pag. 28
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1067
[ think this is very important because, since, especially for India
in the last 25 years, western aid policy for underdeveloped countries
has been based on the hypothesis that real national income is proportional
to real capital. Nobody has questioned this and we have
spent many billions of dollars in a way which may have been less
efficient than for examples, educating the people. In my opinion, if
we consider all the variables which significantly influence develop:
ment, I would say that capital is less important than other factors,
and if this thesis were to be accepted, we should spend the greatest
part of our help in other directions.
[LEONTIEE
As author of a paper I would like to comment at length on the
interesting questions raised in the course of the subsequent discussion;
as Chairman, I have the responsibility for adjourning this meeting
within a few minutes. Thus. I will make these closing remarks very
brief.
In answering Professor MAHALANOBIS’ query, 1 must explain that
the United Nations figures which I used as a basis for my calculations
include Continental China as they do Soviet Russia among the
so-called centrally planned economies which I had to treat as belonging
to the group of industrialized countries.
All my computations were conducted on a total not-per capita
basis. Translated into per capita terms, the discrepancy between the
present growth rates of the underdeveloped and the industrialized
parts of the world would of course appear to be still greater and the
volume of the capital transfer required to eliminate, or at least to
reduce this difference within the next ten years — still larger.
So far as the concept of a take-off period is concerned, I suppose
it might be applied to the entire ten-year time span covered by my
hypothetical computation.
I fully agree with Professor ALLAIS’ emphasis on the importance
nf educational expenditure as a means of accelerating economic
“ y] Leontief - pag. 20
De
1068 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2X
growth. It has often been suggested that such expenditure should
be regarded as a part of productive investment. As a quantitative
measure of the total stock of such investment at any given point of
time one should possibly use the integral, i.e., the cumulated amount
of educational expenditure incurred annually over the preceeding say
thirty or fifty years.
+3] Leontief - pag. 30
THE SOCIAL TRANSFORMATION
FOR NATIONAL DEVELOPMENT
P. C. pMAHALANOBIS
Indian Statistical Institute Calcutta - India
. INTRODUCTIG
1.1. The problem of improving the material and cultural
conditions of the poorer countries of the world has been engaging
serious attention during the post-war period. The desire for
political independence is rapidly increasing and will continue
to grow in the countries still under colonial rule. Also, more
and more countries are becoming and will become politically
independent. With the gaining of independence, it is being increasingly
realized that political freedom is necessary, but is
not enough. In most of the underdeveloped areas, attention
is being given increasingly to economic development to improve
the level of living, by increasing the flow of goods and
services and by expanding facilities for gultural amenities. It is
also being increasingly appreciated that rapid economic growth
can be brought about only by an increasing accumulation of
capital to supply modern tools and machinery for new and
expanding productive activities which would, in time, solve
problems of unemployment or under-employment, and would
also continually improve the level of living. Such accumulation
of capital would call for increasing domestic savings, and the
utilization of such savings for productive purposes. The choice
of productive activities (that is, of investments) must also be
14} Mahalancbis I - pag.
1070 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
such as to secure the best possible rate of economic growth over
a time horizon of a generation or more.
1.2. How to bring this about? This is where the question
of social transformation becomes relevant. Some broad general
principles may perhaps be stated with confidence.
2. THE STRUCTURAL TRANSFORMATION
2.1. It is necessary to give opportunities for participation
in productive activities to the largest number of people, and as
soon as possible, to all such people as are capable of undertaking
such work, and also to utilize available resources in the
most effective way for the benefit of the nation as a whole.
To create fullest opportunities for rapid growth, it is necessary
to remove all barriers to the effective utilization of productive
forces, by the people, for the benefit of all the people of the
country.
2.2. There are many facets to the problem, some of which
are general and some peculiar to particular countries. It is not
possible to arrange them in any clear order of priority. In fact
the heart of the problem is to make changes in all necessary
directions at the same time, in a balanced way, so as to bring
about the structural transformation as quickly as possible.
2.3. The transformation of the social structure cannot be
an entirely internal process. Outside influences have been and
will continue to be at work. Colonial rule and economic exploitation
of the underdeveloped countries have themselves given
rise to reactions promoting the desire for political independence
and for improvement in the level of living in the underdeveloped
areas.
2.4. A new factor, of conscious international cooperation
in improving the social, political, and economic conditions of
the underdeveloped countries. has also emerced durine the last
14] Mahalanobis T - pag. 2
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ten or fifteen years through a quickening of the world conscience
on humanitarian grounds and also in the enlightened
self-interest of the more advanced countries. Isolation is no
longer possible, physically, psychologically or organizationally.
The influence of information, ideas, advice and aid from out
side would be an increasingly important factor.
2.5. A structural transformation of the whole society is,
nowever, indispensable to make conditions fit for rapid economic
growth. Without such transformation, any amount of
help from outside would be ineffective. The experience of many
countries during the post-war period would corroborate this.
3. THE ScienTIFIc REVOLUTIO
3.1. It is also necessary to develop the outlook of science
and the experimental attitude of mind in order to acquire
knowledge of natural and social forces and to invent new techniques
for initiating material and social changes. This is the
only way in which decisions can be made increasingly in a
rational manner, in accordance with principles of objective or
scientific validity based on relevant data and correct reasoning,
instead of on the sanction of authority based on status and
power or custom and conventional or revealed rule and laws
This may be called the scientific revolution.
3.2. The need of what I have called « the scientific revolution
» is recognized, but has not received sufficient attention.
[ have considered some aspects of this problem in an attached
note on « The Scientific Base of Economic Development ».
1. MODERNIZATION OF SoCII':
4.1. The social transformation and the scientific revolution
are both necessary. These are but two aspects of modernization
which can be distinguished but not separated. The social tran-141
Mahalanobis I - pag. 3
1072 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
sformation and the scientific revolution in combination leads
to modernization. The task of international cooperation is to
promote and help, in every possible way and in a peaceful manner,
the modernization of the underdeveloped countries.
4.2. Urgency of the task: The scientific and industrial revolution
took place in West Europe and North America roughly
over a period of three or four hundred years. It is not possible
to wait for such a long time for the underdeveloped countries
to attain a reasonable level of living. The historical process of
transformation must proceed five or ten times faster. Such
speeding up of the process of transformation has always been
a characteristic feature of biological evolution, and can be
achieved.
4.3. Different phases of the transformation: Some of the
newly independent countries are large, some are of medium
size, and some are extremely small in area, or in natural resources
or in population. They would have widely differing
needs. The particular form and contents and components of
each step of modernization would depend on the special conditions
of each country and the stage of development reached by
it, and would therefore, vary from one country to another or
from one region to another of the same country and also, over
a period of time, in the same country or in the same region.
4.4. International cooperation: The most significant fact
of the present age is the rapidly expanding contacts between
different countries of the world. This tendency is bound to
become stronger in future, increasing the scope of international
affairs in every direction. At the same time, what George
Washington had said about « no country being able to go
beyond its own self-interest in international affairs », would
continue to remain valid. The real need is, therefore, to discover
new areas of mutual self-interest, and to expand spheres
of common interest on both bi-lateral and multi-lateral basis
to the fullest extent.
(14 | Mahalanobis I - pag.
4
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4.5. It is also clear that even the most advanced countries
still have unlimited scope for both social and scientific procress.
For all countries, large or small, advanced or underdeveloped,
international cooperation is necessary and beneficial.
The smaller and the less-developed a country, the greater how
ever will be the need and importance of such cooperation.
5. PROBLEMS OF INTERNAL REFORMS
5.1. There is general agreement about some of the most
important contents or elements or aspects of the social transformation,
such as: — land reform; removal of social, economic
and political barriers; mass education and technical training;
increasing equality of opportunities; the possibility of a labourer
or an initiator securing the fruits of his labour; or the
need of medical and health services and cultural amenities etc.
There is much in common in respect of such components or
aspects of social transformation in the case of all under-developed
countries, with, however, the need of adaptations to suit
‘he special conditions of each individual country. Some of
these components or aspects are briefly considered below.
5.2. Land reform: Historically, land reform has been a
most important factor in the economic development of all
advanced or rapidly developing countries. Agriculture and
industry must advance at the same time. It is, however, generally
agreed that an agricultural surplus (or, the surplus
[rom extractives) is essential for industrial development. Changes
in land tenure and legislation would, therefore, be one of
the requirements of the highest priority in most, if not all,
1nderdeveloped regions.
5.3. The aim must be to secure the fruits of his labour to
the cultivator so that he has the incentive to improve the land
and to introduce more advanced technological methods. Tenancy
law should protect the tenant against eviction so long as
"14] Mahalanobis I - pag.
5
1074 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
he is using the land efficiently, and to secure to him the right
of fair compensation upon termination of the lease for all unexhausted
improvements made by him. It is also necessary to
eliminate the unproductive consumption of the surplus from
land by intermediaries and landlords, who have no productive
functions, by abolishing their rights.
5-4. The question of economy of scale of production may,
admittedly, introduce difficulties. The breaking up of large
farms may lead to a reduction of the surplus; however, the
beneficial effects of greater equality of income and wealth may
compensate for the other loss. Also, in countries where there
are too many cultivators, often with scattered plots, further
breaking up of the holdings may easily have adverse effects
on the efficiency of production. In such a situation it may be
necessary to promote consolidation of holdings either voluntarily
through cooperative, or by legislation, or both. Redistribution
of land has limits and is a complicated question. It
is wise to recognize that steps taken at one stage may have to
be reversed at a later stage. Appropriate measures must be
devised to suit the needs of each country at any particular stage
of development. The basic aim would always remain the same,
namely, to increase the agricultural surplus, and to use it for
productive purposes, as effectively as possible, in speeding up
the growth of the economy as a whole.
5.5 Removal of social, economic, legal and political barriers:
The underdeveloped countries have the very difficult
task of achieving a far faster rate of growth than had been
achieved by the most advanced countries during and after the
industrial revolution. It is indispensable that every one in the
working age-group should be fully utilized to increase the national
product. It is necessary, therefore, to remove all social,
economic, legal and political barriers which prevent individuals,
or groups and sections of individuals, to become fully productive.
Conditions are worst in a country stratified by caste,
colour, creed or language. and where whole sections of people
[14] Mahalanobis I - pag. 6
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107F
are sometimes deprived of opportunities by custom, law, or
social and political pressures by ruling groups. Removing all
such barriers is an essential condition for rapid growth.
5.6. Equality of opportunities and vertical mobility: Removal
of social and other barriers, in principle, is necessary
but not sufficient. It is essential to help every one to make
himself fit for the highest type of productive work of which
he is capable. Opportunities for education and training and
for productive work must be made as widely available as possible.
Great inequalities of wealth and income often lead to
denial of opportunities to the poorer people, and, unless removed,
give rise to a sense of frustration among the under-privileged
and hamper the growth of national solidarity. Sufficiently
rapid economic progress would be difficult or impossible in
societies in which there is lack of vertical mobility and where
small sections try to preserve their privileges based on heredity,
custom or law without any relation to their productive contributions.
5.7. Horizontal mobility: The social system may also hamper
the utilization of resources because customs or caste restrictions
prevent labour from moving into new occupations, or
labour is tied to the soil, or land may be concentrated in the
hands of small sections of the people who are unwilling to divert
it for more productive use for reasons of social or political prestige.
A small number of producers even in underdeveloped
countries may sometimes band together to prevent the free
entry of others or the introduction of new techniques. All such
restrictions must be removed to increase the horizontal mobility
of resources.
5.8. Possibility of securing fruits of labour and enterprise:
The elimination of concentration of social, economic or polilical
privileges in the hands of small sections of the people
would promote both vertical and horizontal mobility, and make
it possible for every one to secure a fair share of the fruits of
‘147 Mahalanobis I - pag. 7
1076 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VAR'A -
pK
his labour and enterprise. This is one of the most important
consequences of the social transformation and is particularly
helpful in promoting rapid economic growth. Appropriate legal
and institutional changes must be made to achieve this.
6. NATIONAL INTEGRATION
6.1. Sectional interests and barriers: A characteristic feature
of underdevelopment is the segmentation of the country
into innumerable regions, castes, tribes, languages, religious
communities, occupational and other groups which focus attention
on the welfare of small sections of the people without any
awareness of the best interests of the country as a whole. It
has to be recognized that rapid progress is impossible without
painful adjustments and damage to sections of the people whose
interests are based on special privileges or old techniques; and
that old beliefs, customs, and social institutions have to be
discarded, and all barriers of caste, customs, creed, colour,
language and sectional interests must be ruthlessly eliminated.
The greater the prevalence of such social barriers in the country,
the greater are the sectional rigidities within government
administration, and the fiercer are the inter-agency jealousies
and fights which continually delay decisions and hamper speedy
action.
6. . Integration: The removal of social and economic barriers
is an indispensable condition for the -emergence of the
sense of national solidarity without which national development
is impossible. It is not possible to isolate the scientific, or
the social, or the industrial aspects of the transformation from
one another. Advance must be made at the same time on all
fronts. This creates difficulties but also has its advantages.
Progress in one direction stimulates and promotes progress in
another direction. It is the task of leadership to maintain a
proper balance between the different aspects and phases of the
14] Mahalanobis I - pag. 8
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1077
process of modernization in its full sense. The aim continually
must be to create a society from which social, economic and
political privileges have been completely eliminated. To bring
about such a transformation would call for wise leadership
with a clear appreciation of aims and objective, a rational and
experimental attitude of mind with confidence in the outlook
of science, and willing to pay the price of much painful
adjustments.
7.
DANGER OF SUPERFICIAL IMITATION OF ADVANCED COUNTRIES
7.1. Because of the sense of urgency for economic growth
which is strengthening everywhere, there is a peculiar danger
of adopting, in a superficial way or at too early a stage,
methods, and forms and institutions, which are working successfully
in the advanced countries. It has to be kept in mind
that existing social and political institutions, or high levels and
standards of quality or performance, were established in the
advanced countries only with the gradual growth of the economy.
Such institutions may not be useful at an earlier stage
of development, and may even hamper progress. For example,
in underdeveloped countries, there is sometimes a tendency to
adopt too expensive or too sophisticated schemes of education,
care of health, public buildings and construction, wages or
salaries of government employees or labour legislation.
7-2. Education and training: Mass education to spread
literacy both among children and adults has special urgency;
here all possible help should be utilised, for example, by using
the services, for a small part of the day or the week, of those
who are already literate. Because the number involved are
very large, the adoption of too high a standard for teacher
qualifications, school buildings etc., at the primary level, would
make the cost prohibitive. At the secondary stage, more attenton
would have to be given to the qualification of teachers and
14] Mahalanobis I - pag. 9
1078 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
other educational aids; but scales of pay or cost of buildings
should still be kept in balance with the general level of living
of the students and the parents themselves. At the tertiary
level, still higher standards would have to be adopted for staff
qualifications and there would be need of more expensive
teaching aids; but the expenditure must be kept within the
limits of what the country can afford. It is at the stage of
advanced studies and research that standards should be really
high and comparable with the advanced countries; however, as
the number of advanced and research workers would be very
small in the beginning, this would not involve anv large total
expenditure.
7.3. The educational system should be viewed as a pyramid;
the lower the stage the wider should be the base (that is, the
number of persons under instruction) and the lower the scales
of expenditure compared to advanced countries, while at the
highest stage of advanced studies and research the number involved
would be extremely small but scales of expenditure may
approximate to those of advanced countries. Adoption, at too
early a stage, of standards and scales of expenditure of
advanced countries at lower levels would lead to severe restrictions
in numbers usually coupled with admission of students on
the basis of family income; this must have most undesirable
social and psychological consequences. When resources in men,
materials and money are inadequate, to increase the number
of students in accordance with the pressure on admissions, would
necessarily lead to window dressing and a dilution of standards
in practice. This can seriously hamper progress; the only remedy
is to adopt a system which would be in keeping with basic
aims and yet within the means of the country.
7-4. Medical care and technical services: A similar situatlon
can arise even more easily in the field of medical care.
Adoption of the high level of university education for physicians
in the advanced countries as the only standard at an earlv stage
1471 Mahalanobis I - pag. 10
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"079
would necessarily mean that most of the people will be deprived
of medical services in underdeveloped countries for a very long
time. A two-tier approach with a junior cadre of medical personnel
with, say, three or four years’ training, together with a
much smaller number of physicians with university training,
would make it possible to spread medical services much wider
and much faster. This would be equally true in many other
lines of technical work. A two or even a three-tier approach
with a higher, a medium, and even a third level of workers
who have had a very quick and specialized training, would be
not only within the means of the underdeveloped countries but
may be even more effective, because, in the still backward
conditions of the country, the lower level workers would be
much nearer to the general population and would be able to
work in closer touch with them. This would be particularly
true in agricultural extension and other services which would
bring the technical workers into contact with large sections of
‘he population.
7.5. Government expenditure: Government expenditure
often tends to become unduly large in underdeveloped countries
owing to the adoption of the much higher standards of advanced
countries. This leads to unnecessarily high scales of wages and
salaries for government employees or costly public buildings;
which, in its turn, would increase the feeling of separation
between government and the people, and hamper national integration.
7.6. Labour legislation: As production becomes modernized
and factories and enterprises grow in numbers and in size,
it would be necessary to develop labour legislation and regulatlons
to ensure that labour secures a fair share of the surplus,
and also to ensure working conditions being maintained reasonably
safe and healthy. Legislation in imitation of the more
advanced countries, at too early a stage, may, however, lead
to increasing inefficiency of performance, especially, in coun
vq) Mahalanobis I - pag. 11
1080 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA - 28
tries with surplus labour, and may increase costs of production
so much as to have serious adverse effects on exports. The
most important thing is to establish a definite link between
remuneration and output in the case of all types of work of
which the volume and quality can be estimated even roughly.
[t is necessary to recognise that trade union movements can
gain in real strength only on the basis of increasing productivity.
8. NATIONAL LEADERSHIP
8.1. The transfer of modern technology from the advanced
countries also calls for much adaptation to suit the needs and
local conditions of underdeveloped regions. To profit by the
experience of the advanced countries and yet to introduce
modern technology and modern social and political institutions
in a way suitable to the particular stage of development of the
country is a matter of crucial importance in the process of
modernisation. Ultimately, success would depend on the growth
of a rational outlook and the experimental attitude of mind,
first, among the leadership at all levels and then gradually
among the general mass of the people.
8.2. It is extremely important that advanced countries
should help and encourage in every way all progressive groups
within the country in promoting the process of modernization
and refrain from offering technical or economic aid in any way
which would hamper the social and scientific transformation.
(14) Mahalanobis I - rag. I2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 108:
THE SCIENTIFIC BASE
dr ECONOMIC DEVELOPMENT
PHASES OF ECONOMIC DEVELOPMENT
(.1. The essential characteristic of an underdeveloped
country is an extremely low level of living, that is, inadequate
supply of food, clothes, housing, drugs and other consumer
goods, and also lack of facilities for education, care of health,
social security, cultural amenities, etc., for the nation as a
whole. It is possible to make available small quantities of
consumer goods by direct imports or by domestic production,
on a small scale, with the help of imported machinery. In most
of the underdeveloped countries it is, however, not possible, for
lack of necessary foreign exchange, to import or to produce,
with imported machinery, enough consumer goods for the
people as a whole. In India, the first textile mill was established
in 1817; and India gradually became the second biggest
producer of textiles, next only to America. One hundred and
fifty years later, India would still remain underdeveloped. The
production of textiles or small quantities of other consumer
goods for a small part of the nation cannot, by itself, lead to
industrialisation and economic development.
i.2. Economic development can occur only by increasing
-he per capita production of the nation as a whole, through an
increasing use of machinery driven by steam or electricity as
a substitute for human and animal labour. In countries with
appreciable natural resources, it is necessary to establish the
basic engineering and power industries to enable the manufacture
of both consumer and capital goods within the country
"14] Mahalanobis I - pag. 13
1082 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
38
Establishing a minimum complex of such basic industries would
take at least ten o fifteen years, for which planning must start
ten or fifteen vears in advance.
1.3. To increase modern industrial production would call
for an increasing supply of engineers, technologists, and
technical personnel. The only way to ensure this would be to
establish and increase the number of schools, training colleges
and universities, and also to train teachers for such institutions.
This would take at least fifteen or twenty years; so that planning
for this purpose must start fifteen or twenty vears in
advance.
1.4. The best way of utilising the raw materials and natural
resources available within the country, for both domestic consumption
and for exports, can be found out only through
applied scientific research (*). Applied research, in its turn,
must be based on advances in fundamental research. Also, to
establish an adequate base for applied research it is necessary
to promote the spirit of pure research and supply the stimulus
of scientific criticism. This would be possible only when at
least a certain minimum number of scientists are engaged in
fundamental research, and opportunities for pure research are
becoming increasingly available. It is therefore necessary to
promote the advancement of both applied and fundamental
research. To establish a minimum base for scientific research
would take more than a generation of twenty-five or thirty
years; this, being the most slowly maturing sector, must be
given the highest priority.
(') Even the most advanced countries are obliged to devote large
resources to research for the improvement of products already being
manufactured and also to develop new products in order to hold their
position in the world export market. It is not possible for the underdeveloped
countries to start or expand the export of fully or partly
manufactured products by simply borrowing the. current technology from
advanced countries; it is essential also to develop applied research for a
continuing improvement of technolooical methods
‘14 Mahalanobis T - tag. 14
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2. THE SCIENTIFIC BASE OF THE ADVANCED COUNTRIES
2.1. The scientific base of the modern age can be appreciated
by even a brief review of the recent history of the
advanced countries. Four hundred years ago the generally
accepted view was that the earth was at the centre of the
universe; the position of human beings was unique and
supreme; and the highest sanction of truth was either divine
revelation or abstract logical reasoning in the mind of man. In
the sixteenth and the seventeenth centuries, there was a complete
revolution in the picture of the physical world; the earth
was seen as a small planet moving round the sun; and the
method of empirical observations and experimentation was
gradually established in both physical and life sciences.
2.2. Progress was at first slow in the sixteenth century. A
few selected names may be recalled to indicate the gradual
transformation of ideas. In astronomy, Nicholas Copernicus
(1473-1543) supported the view that the planets including the
earth itself were revolving in orbits round the sun; Tycho Brahe
1546-1601) supplied astronomical observations of unprecedanted
accuracy to make the next steps possible; JoHANN KEPLER
1571-1630) formulated the descriptive laws of planetary
motion; and GALILEO GALILEI (1564-1642) made conscious
propaganda in favour of the new philosophy of the universe.
[n anatomy, ANDREAS VESALIUS (1514-1564) published his
observations on the human body in 1543; in physics, WILLIAM
GILBERT (1544-1603) gave an account of magnetism based on
trustworthy experiments in 1600; in physiology, WILLIAM
HARVEY (1578-1657) described the circulation of the blood in
(628; JoHN NAPIER (1550-1617) supplied a convenient tool for
computation by the use of logarithms; and RENE DESCARTES
(1596-1650), a philosopher, contributed the powerful concepts
of coordinates for geometrical representation and of mathematical
functions. Francis BACON ( 1561-1626), firmly stated that
1 Mahalanobis I - pag. 15
1084 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
25
the only true method in science was to proceed from particular
sense observations to wider generalizations (Novum Organum,
Book I, xix), and clearly recognised that ’’the true and lawful
goal of the sciences is . . . that human life be endowed with
new discoveries and power.’’
2.3. The concept of an objective world of physical reality
gradually took firm shape in the seventeenth century in the
hands of gifted astronomers, mathematicians and scientists. A
few names may be mentioned from among those who were born
in the first half of the century: PIERRE FERMAT (1601-1665),
CHRISTIAN HUYGENS (1629-1695), BLAISE PASCAL (1623-1662),
ROBERT BOYLE (1627-1601), JoHN RAY (1624-1705), ROBERT
FIOOKE (1635-1703), ISAAC NEWTON (1642-1727), and GoTT-FRIED
WILHELM LEIBNIZ (1646-1716). The rate of advancement
of science increased progressively in the eighteenth and the
nineteenth centuries, and during the last few decades has opened
new frontiers with almost unimaginable possibilities.
2.4. The advancement of science prepared the ground for
the industrial revolution in Europe in the eighteenth century,
first in spinning and weaving, next in the use of iron and steel,
and then of electricity in the nineteenth century, which stimulated
the growth of the capitalist economies in West Europe and
North America. The spread of the scientific outlook also prepared
the ground for the age of reason and the French revolution,
which occurred at the end of the eighteenth century, and
oromoted the growth of nationalism in Europe, in its modern
sense, in the nineteenth century.
2.5. The industrial revolution increasingly replaced human
and animal power by steam or electricity to drive machinery
for the increasing production of both consumer and capital
goods. The development of engineering techniques led to a
close linkage between science and technology; and during the
last hundred and fifty years. industrial development is being
14] Mahalanobis I - pag. 16
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stimulated by a scientific discovery or a scientific discovery is
being stimulated by industrial needs.
2.6. For the last five or six thousand years, or more, the
average per capita production remained more or less constant
or fluctuated within narrow limits. The industrial revolution
changed all this, and led to a spectacular increase in the variety
and volume of goods produced. As a consequence of such
increasing production, the standard of living of the advanced
countries of West Europe and North America reached a level
far higher than the rest of the world. Also, the advancement
of science, technology and industry, made it possible for the
western countries to become strong military powers; and,
because of such military supremacy, the west was able to bring
a large part of the world either into direct colonial rule or into
conditions of economic or political subjugation.
2.7. The last forty years have also seen the rise of U.S.S.R.,
as another world power, rapidly growing, through the promotlon
of science and technology, in economic, industrial and
military strength together with a continuing increase in the
level of living. The monopoly of scientific and technological
knowledge and the unchallengeable military supremacy of the
western countries have now gone. The increasing parity
detween the ’’western’’ and the ’’eastern’’ countries in science,
technology, industry, and military power is a most significant
fact of the present time. Because of the unprecedented
destructive power of atomic and nuclear weapons, it has become
absolutely necessary to avoid a nuclear war which would be
catastrophic for both sides and the whole world. Coexistence
of both the ’’western’’ and the ’eastern’’ powers has become
indispensable.
2.8. There is no intention on either side to make a direct
attack. The advanced countries pose no special problems
decause it is not possible to hold such countries indefinitely in
subjugation. However, so long as there are underdeveloped
*4+] Mahalanobis I - pag.
17
1086 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
areas, both power groups are likely to try to extend their
influence over the less advanced countries; and this would
remain a continuing source of potential conflicts. The very
existence of underdeveloped countries should, therefore, be seen
as a threat to peace. Rapid transformation of all the underdeveloped
countries into modern viable societies is an essential
condition for peaceful coexistence. Such a transformation would
promote the enlightened self-interest of both power groups, and
would also create conditions favourable for the advancement of
human and cultural values on a world-wide basis.
3. THE ROLE OF SCIENCE IN THE MODERNISATION OF THE LESS
ADVANCED COUNTRIES
3.I. Modernisation of the less advanced countries through
rapid industrialisation is thus an urgent need of the whole
world. Is such modernisation possible or can a modern society
with a viable economy, with expanding social and political
freedom, and cultural amenities, be sustained without
establishing a sound scientific base? This is a question of
crucial importance for the present age.
3.2. In order to answer this question, it is necessary to
appreciate the deeper changes in human thinking which were
brought about by the emergence of science. In every sphere
or organised activity in human society, authority has always
been associated, and must always be associated with a system
of hierarchical levels. This applies to primitive societies, matriarchal,
patriarchal, or tribal; successive levels of feudal
lords; organised churches and religions; military, police and
administrative systems; enterprises, business and commerce;
and law. A law court of appeal may reverse the decision of a
lower court; but the decision of the court of appeal is itself
subject to change by a still higher court. The decision of the
highest court, to which a case has been actually referred. has
14] Mahalanobis T - pag. 18
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC,
108;
to be accepted not because such a decision is necessarily right,
but because it is the decision of a superior authority (*). Society
must accept this authority principle for stability and orderly
progress, even in organised revolutionary activities.
3-3. This very authority principle must, however, be absolately
and completely rejected in the field of science. Modern
science is based on a patient accumulation of facts, on the study
of processes and their interrelations or interactions and a
stability or uniformity of nature (*) which can be discovered by
‘he human mind. The findings of the most eminent scientists
are subject to critical check by their professional colleagues and
by the youngest scientific workers, and must be rejected if there
is no satisfactory corroboration. Science can advance only
through free criticism on a completely democratic basis, with
every research worker of competence enjoying equal status.
The theoretical or conceptual framework of science must be
(’) It is possible, indeed, that this decision itself would have been
reversed if there had been a still higher court to which the case could be
referred. If a decision of a higher court of appeal is considered to be like
the turning up « heads » (in tossing an unbiased coin) when the decision
upholds the verdict of the lower court, and is considered to be like the
turning up of « tails » when the verdict of the lower court is reversed,
then the successive decision of the higher court would look like the
results of the tossing of a coin. This would be the real guarantee that the
system of law is functioning properly.
{} The phrase « uniformity of nature » must be, of course, interoreted
to include chance events and random processes. Although games of
chance were known and were widely prevalent in ancient times in Chira,
[ndia and other countries, it is important to note that the concept of
probability did not arise until the 16th and the 17th centuries, that is,
not until the emergence of modern science. This in easy to understand.
Before the emergence of the modern scientific view of an obiective world
of physical reality, all chance events would have to be necessarily ascribed
to the whims of gods, demons, or supernatural forces. After the emergence
of the scientific view of an objective world of physical reality, it became
necessary, both logically and psychologically, for the human mind to accommodate
the occurrence of chance event as an integral part of the uniformity
of nature. This could be accomplished only on the basis of the
theory of probability, or rather, as I should prefer to put it, only through
a statistical view of the world. It seems to me, therefore, that the con--ept
of probability, of the statistical view of the world did arise at the
same time as the emergence of modern science onlv because it could not
possibly have arisen earlier.
Mahalanobis I - pag.
x Ly
[088 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 76
continually revised to find a proper place for all known facts.
A single new observation may call for a more comprehensive
theory. The older accumulated knowledge continues to remain
valid; later discoveries must, however, be integrated with the
earlier knowledge. The accumulation of scientific knowledge is
inscreasing through the efforts of all the scientific workers of the
world. A new fact may be observed or a new theory formulated
by any worker, however young, and in any country where
research has been established. International collaboration is,
therefore, an indispensable condition for the progress of science.
3.4. Authority derived from status is irrelevant to science.
Science has introduced a new concept of ’’scientific’’, or
‘objective validity’’ which has its foundation in nature itself,
and which cannot be upset by any authority based on status
or by supernatural powers. The transformation of all the
advanced or rapidly advancing countries has been based on
accepting, in an increasing measure, a scientific or rational view
of life. This is the foundation of the modern age.
3.5. It is essential in every country to establish and
strengthen the outlook of science, a way of thinking which
becomes more and more powerful as it is more widely adopted,
and which replaces dogma, superstition, and outdated customs.
This scientific outlook cannot be established by force. It must
depend on acceptance through proper understanding. In practical
affairs, the important point is that a wise policy and
programme of action should be increasingly adopted on the
basis of rational argument, supported by relevant factual evidence,
and should not be rejected because of emotional bias or
formal dogmas or conventional rules of procedures. It is,
therefore, necessary continually to encourage and promote the
advancement of science in every country, large or small. Because
science is indivisible, and also because science must be
established in every country, it is also necessary, continually, to
promote scientific collaboration between all countries of the
world. large and small, and advanced or developing.
‘141 Mahalanobis T - pag. 20
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1080
3.6. It is scarcely necessary to point out that there is no
conflict between the scientific and rational view of life, on one
hand, and aims and objectives based on moral or cultural
values, on the other hand. On the contrary, moral and cultural
values which are truly universal, and are or narrowly sectarian
or nationalistic in a restricted sense, must have an objective and
rational basis.
3.7. The advancement of science and the growth of the
scientific outlook must be recognised as an essential condition
for the modernisation of the less advanced countries. It is necessary
for each country to have, as quickly as possible, a sufficient
number of men with a scientific outlook to influence the thinking
of the nation. How to attract and hold a sufficient number
of able persons to science is thus the crucial problem of national
and world development. This can be achieved only through a
proper and adequate social appreciation of science and
scientists. The actual transformation must be brought about
from within each country. Scientific aid from the advanced
countries can, however, be of great help in this process.
1. PRESENT PROGRAMMES OF TECHNICAL AID
4.1. The need of technical aid has been recognized for some
considerable time. Bi-lateral or multi-lateral and international
‘echnical aid has often taken the form of either offering educa-‘ional
and training facilities to young workers from the less
advanced countries or sending technical or scientific experts
to such countries. Considerable benefit has no doubt accrued
through such aid but it is necessary to recognise that much effort
has also been wasted.
4.2. Scholars from the less advanced countries are usually
selected on the basis of results of examinations; success in
examinations not being a necessarily reliable indicator of scien-14]
Mahalanobis I - pag. 21
1090 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
38
tific or technical ability, the very process of selection is inefficient.
Some of the young scholars have difficulty in adjusting
themselves to the pattern of living in the advanced countries.
Some of them do not do well in their studies. Some pass the
examinations successfully but have no aptitude for scientific
work. Some of the more able scholars prefer to live and settle
down in the advanced countries, especially in the U.S.A., because
of the higher level of living or greater opportunities for
scientific work. Some scholars of ability, when they return
to their own countries, are unable to find suitable openings for
a scientific career; and some of them go back to the country
where they were trained. In applied science and technology,
and especially in social sciences, many young scholars, who
had often studied problems or learnt methods which are appropriate
for advanced countries but totally irrelevant to their
own native countries, are unable to adapt or develop methods
to suit local conditions. Out of the large number of scholars
who go to advanced countries for training, only a very small
number of really able scientific workers ultimately become
available for fruitful work in their own country. The cost of
giving scientific or technological training in an advanced country
is also very high. Giving training to individual scholars
in advanced countries (whether the expenses are provided in
the form of foreign aid or met by the scholars themselves or by
the country of origin) have been, therefore, extremely wasteful
in terms of both men and money.
4.3. There has been also continuing difficulties in finding
suitable individual experts for the less advanced countries.
Competent scientific workers are reluctant to accept such assignments
partly because of the lack of facilities for their own work
in the less advanced countries and partly because their scientific
or academic career is likely to be adversely affected through
their absence abroad. In consequence, assignments sometimes
have to be given to persons who are not fully qualified for the
job, with unsatisfactorv results. To create suitable conditions
‘141 Mahalanobis I - pao. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC.
1091
for scientific work in the less advanced countries is an indispensable
condition for attracting competent scientists to go out to
such countries.
4.4. Programmed technical aid on a group basis has been
more effective. A team of young engineers from a less advanc-2d
country can receive most valuable training in an advanced
country when such training is oriented to specific technological
projects. Teams of experts from advanced countries have also
been of very great help in establishing factories or in starting
new projects in the less advanced countries. Such technical
aid, especially in engineering, technology and applied sciences,
should be continued and expanded. Special projects for establishing
technological and research centres in the less advanced
countries have also been taken up by some of the international
agencies. This type of aid can be of great value provided a
sufficient number of scientific workers in the less advanced
countries can be trained to work in such centres, and also
provided necessary conditions are established to enable them
‘o do their work properly.
5. SCIENTIFIC EDUCATION AND RESEARC,
5.1. It has been argued in the earlier sections that for modernisation
it is necessary to establish a foundation for scientific
research and the social appreciation of science in the developing
and less advanced countries. Every path-finder in a new
field of research must work in the first instance by himself; if
ne is successful, other persons gradually get interested in the
subject. Such path-finders always had, and will always have
to overcome much opposition, and even hostility, until the new
subject becomes a recognised part of the « established » field
of science. But it is only a few scientists of outstanding ability
who can work in isolation. Most research workers require the
14] Mahalanobis I - pag. 23
1092 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
stimulus of free interchange of views and ideas and of appreciation
among professional colleagues.
5.2. The community of scientists has a structure of a series
of widening circles similar to the structure of scientific subjects
or of science as a whole. When a top scientists speaks appreciately
of some work in his special field, other scientists or
lay men accept his evaluation and pass on the information to
others. The social appreciation of science gradually emerges
as a result of the diffusion, in widening circles, of the views
of scientists, who are experts in specialised fields of research,
to scientists in related and associated fields, then to scientific
workers generally, and finally, through persons of position and
standing who have contacts with scientists, to the general
public. The speed with which such appreciation can spread
increases rapidly with the increase in the number of scientific
workers and improvements in the channels of communication.
In the advanced countries, the awareness of the importance of
science is increasing rapidly which, in its turn, is raising the
social status of scientists and is promoting an increasing flow
nf resources for research.
5.3. The whole process is extremely slow in underdeveloped
countries. The number of research scientists is very small, and
channels of scientific communication are non-existent or
meagre. Scientific workers usually receive lower pay and have
a lower status than the administrative staff in government or
in business concerns; and have to work in a rigid system of
hierarchical authorities. Promotion may depend, not so much
on the high quality of the scientific work done, but on success
in pleasing those who are higher up in the official hierarchy.
Even permission to apply for posts elsewhere is subject to
the discretion of superior officers. There is a continuing tendency
to bring scientists and scientific work under stricter control
of the administrators, partly, perhaps. from an unconscious
fear of rivalry of power. Even if the right of criticism
‘14 Mahalanobis I - pag. 24
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
,0072
is accepted in principle, it is restricted in practice because scientific
workers are often afraid, rightly or wrongly, of giving
offence to persons holding higher posts. In consequence, many
scientists in underdeveloped countries suffer from a lack of
self-confidence, and are afraid to take up original lines of investigation.
There is little possibility of a proper evaluation
or appreciation of scientific work within the country. This
leads to an exaggerated dependence on the opinion of foreign
scientists and gives rise to much imitative work. Also, when
‘here is lack of appreciation or criticism from the advanced
countries, there is sometimes a tendency to ascribe the unfavourable
view to racial or national prejudices, and there is
resistance against collaboration with foreign scientists.
5.4. In underdeveloped countries there are very few, some-‘imes
only one or two, individuals of outstanding ability in
scientific research or in any other scientific field. As leadership
can be supplied only by individuals of high ability, and as
such persons are few in number, it is much more difficult in
underdeveloped countries to utilise the services of individuals
of average ability and qualification. The advanced and advancing
countries have a double advantage. They have a large
number of persons with qualities of leadership and can, therefore,
utilise in a fruitful way larger numbers of persons of
average ability. This is why many scientific workers from
anderdeveloped countries, who are unable to do much useful
work in their own native country, can often do very good
work in the environment of a higher state of organisation of
research in an advanced country.
5.5. The aim of scientific aid must be to create in every
anderdeveloped country, as quickly as possible, a sufficient
number of research scientists to form a community of professional
workers which would be sufficiently large to facilitate
an independent evaluation of scientific work through free criricism
and frank exchange of views. It is, therefore, necessary
Mahalanobis I - pag. 25
1094 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
to focus attention on identifying and giving support to persons
who have the ability to undertake research work of high quality,
and to try to increase their number as quickly as possible,
and at the same time to offer opportunities for training
to persons of average ability whose services would be equally
essential in supplying a wide base for the pyramid of scientific
work.
5.6. There is urgent need of fostering the spirit of objective
scientific criticism through free expression and exchange of
views and opinions. One effective way of promoting this would
be to make it easy for scientific workers to migrate from one
post to another and give an absolute guarantee of such freedom
to migrate. Any scientific worker who feels, rightly or
wrongly, that he has not enough opportunities for fruitful
work in one institution would be free to migrate to some other
institution. Such migrations or the possibility of such migra
tions would have an indirect but most important selective effect
on scientist at all levels.
5.7. It is necessary to recognise that the social value of an
individual scientist of high ability is far greater in a developing
country because of the leadership he may be able to supply.
[t is only scientists engaged in fundamental research who can
function as the eyes and ears of the nation in making the nation
appreciate and identify urgent needs of applied research. The
emergence of even one or two outstanding research scientists
can enhance the prestige of the nation in a most significant way
at the international level and promote the growth of self respect
and self confidence of the nation. This is why it is particularly
important in developing countries to identify such individuals,
at first very few in number, and give them all possible facilities
and encouragement to continue their work in their own country.
5.8. In the highly developed countries science advanced
both from progress at the highest levels of research, at the top,
and from the wide diffusion of education. at the bottom. The
‘147 Mahalanobis I - pag. 26
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1095
same strategy may be adopted with advantage in the less advanced
countries. What is urgently needed is to lay the foundations,
with as wide a base as possible, for a country-wide
system of school education oriented to science and, at the
same time, to develop advanced studies of science and technology
and research at the highest level. The school system
must fit into the economic life of the general mass of the people
and have its grass roots in the villages. It must offer facilities
for training technicians and technical personnel for science and
‘echnology and also supply candidates of outstanding merit for
admission to higher scientific and technological institutions.
5. NEED OF DIRECT AID FOR SCIENCE
6.1. I shall offer, briefly, a few suggestions for giving direct
aid for the development of science in the less advanced coun-‘ries.
I have stressed the need of building up a system of school
education with a definite orientation to science. It would be,
however, a fatal mistake to establish an expensive system of
sducation on the model of the advanced countries which would
have little relevance to local needs and would be beyond the
means of the national economy. It is necessary to evolve a
system, through experimentation and trial and success, which
would be within the means of the national economy. The approach
must be therefore to use teaching aids which are easily
available or can be made available on a large scale and at a
low cost. As most of the pupils will be living in villages, it
would be of great advantage if agriculture and some of the rural
industries can be adopted as a base for the teaching of science.
The programme may consist largely of nature studies, observations,
and experiments which can be done with the help of
simple articles, specimens, etc., likely to be locally available
or which can be constructed with local materials.
ig] Mahalanobis I - pag. 27
1096 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
2€
6.2. There would be still some need of supplying teaching
aids and materials from outside which would have to be specially
designed to reduce costs. It is essential also to prepare
books of instructions and text books to suit a fairly wide range
of needs. These are difficult tasks which would call for extended
study and research by scientists of high calibre with a serious
interest in problems of science education. As basic conditions
in underdeveloped countries are likely to be similar to a large
extent, it may be possible to evolve broad general methods for
science education which would be capable of being adapted
without much difficulty to suit differing local conditions.
6.3. A great deal of pioneering research would be necessary
for this purpose for which the help of advanced countries is
indispensable. A good deal of experimental studies will have
to be undertaken under conditions actually prevailing in underdeveloped
regions. In the beginning, the studies would have
to be organised on a small scale with the help and support
of the local authorities and of such teachers and scientists as
may be available to cooperate of the local authorities and of
such teachers and scientists as may be available to cooperate
in the venture in the underdeveloped country itself. The project
can be gradually extended, in the light of experience, to cover
different subject fields at different educational levels, and also
[rom one underdeveloped country to another. Fortunately,
even one or two scientists can start the work in one single
country. The important point is to make a beginning at the
earliest opportunity.
6.4. I may now mention a second type of programme. Certain
facilities for scientific research are already available in
India and other developing countries. In most of these countries,
scientific work is being hampered for lack of small replacement
parts, additional accessories and instruments, and supply
of essential consumable stores which have to be imported
from the advanced countries. It is often difficult to secure
147 Mahalanobis I - pao. »8
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 1097
import licences on account of shortage of foreign currency. This
difficulty can be overcome through a simple plan of gifts in
kind of replacement parts, instruments and equipment, stores,
books and journals and reprints or microfilms of scientific papers
etc., to be arranged through non-governmental committees
of scientists. Such committees, which can be set up in the
advanced countries through or in cooperation with appropriate
scientific organizations or societies, would try to secure suitable
grants from Government and other sources. In developing
countries where scientific research has already started, the
counterpart committees of scientists would also be set up, preferably,
at a non-governmental level and with a majority of
members from universities and non-governmental scientific
mstitutions. All arrangements would be made with the concurrence
of the government of the less advanced country concerned,
but decisions relating to gifts for scientific work must
be made by direct consultations between the scientific committees
themselves. A scheme of this type can be usefully
started, on an experimental basis, for a few selected countries,
at a low cost, with gifts to the total value of perhaps one or two
hundred thousand dollars per year. The amount can be increased
if the experiment proves successful.
6.5. Another important form of scientific aid would be to
arrange for competent research scientists from the advanced
countries to work for a year or two in existing research units
in the less advanced countries or to help in establishing high
level research units in such countries. The less advanced countries
can offer challenging problems and opportunities for research
in many fields of science, which cannot be duplicated
in the advanced countries, for example, in geology meteorology
and geography; biology, botany, and zoology; agriculture;
medical science and public health; economics of development;
linguistics, archaeology; and historical and cultural studies of
various kinds. In some of the developing countries there would
41 Mahalanobis I - pag. 29
1008 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
be also increasing opportunities for active participation in research
in mathematics and statistics, and physico-chemical and
technological sciences. In establishing research units in underdeveloped
countries it would be desirable to keep one broad
aim in view, namely, to encourage joint studies by active collaboration
between different research units. This would help
in developing a community of research cells or units which, in
its turn, would foster the growth of the spirit of scientific criticism
and appraisal among wider circles of scientific workers.
6.6. To attract competent visiting scientists it is necessary
to offer them facilities to pursue or start fruitful research in the
less advanced countries; sometimes special equipment may
have to be provided for this purpose. Secondly, the assignment
in a less advanced country, would have to be treated as deputation
in the same way as participation in scientific expeditions,
and which would be recognised as a part of normal
duties and also as a possible qualification for promotion. The
visiting scientist must receive sufficient compensation in his
aome currency to meet his continuing home commitments
during his absence abroad. Living and other local expenses
should be normally met by the institution or by the government
of the country in which he would work. Such sharing
of costs would promote effective cooperation in the less advanced
countrv, and would also reduce the total cost appreciably.
6.7. An important part of the responsibilities of a visiting
scientist would be to give training to the scientific workers of
the underdeveloped countries. When necessary, the visiting
scientists would be able to select, for further training in an
advanced country, the right type of persons who can be depended
upon to go back to their own country after the completion
of the training abroad. It would be also possible to give aid
in the form of equipment and instruments in an effective way
on the basis of objective appraisals of needs and possibilities
by the visiting scientists.
141 Mahalanobis I - pag. 20
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ ANALYSE ECONOMETRIOUE ETC. 109G
6.8. A tourth programme could be to send from advanced
countries young scholars, who have just finished their educadon
in universities or higher educational institutions or have
already done some research, to start or continue suitable lines
of research for about two years or so in existing institutions
or in research units to be established for this purpose in underdeveloped
countries. The common participation in research
projects of young scholars from the advanced and the underdeveloped
countries would be of great help in establishing
scientific traditions and an atmosphere of scientific criticism.
It would promote self-confidence among the scientific workers
of the underdeveloped country, especially, if the visiting scholars
from advanced countries take higher degrees from insti
tutions in the less advanced countries.
6.9. All the above forms of scientific aid can be started, on
a small scale and at low cost, and, if successful, can be expanded
in the light of experience. Also, these forms of scientific aid
would not in any way overlap or hamper bigger programmes
for gifts of expensive equipment or large projects for the setting
up of national or regional centres and institutes for scientific
research in the less advanced countries. On the contrary, the
modest programme described in this note would prepare the
ground for bigger projects.
7. CONCLUSION
7-I. In conclusion I may refer, very briefly, to some recent
developments. After the second world war the movement for
‘erminating colonial rule gained rapidly in strength, and one
country after another in Asia and Africa has won political independence.
It is being increasingly realised, however, that
independence is not enough for economic development. The
need of economic and technical aid is also being increasingly
14] Mahalanobis I - pag. 31
1100 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
28
appreciated. Both the « western » and the « eastern » powers
have started helping in the economic development of the less
advanced countries in Asia, Africa and Latin America, but
still without an adequate impact. The time has come to recognise
that economic aid is essential but is also not sufficient.
7.2. Revolutions to capture political power have been occurring
throughout human history and are even now occurring
in many of the politically independent countries in Latin America
or in most of the newly independent countries in Asia and
Africa. Such revolutions do not automatically promote rapid
economic development, because purely political revolutions do
aot lead to any fundamental transformation of the old society
based on the principle of authority associated with levels of
status. It is becoming increasingly clear that rapid economic
development cannot be achieved without developing a structure
of society in which decisions would tend to be made more and
more on grounds of reason, that is, in accordance with the
orinciple of objective validity instead of authority. It is relevant
to note that the French Revolution was preceded by the
age of reason; the American War of Independence had the
support of influential leaders inspired by the spirit of science;
and the socialist government, which was established after the
October Revolution in 1917 in Russia, made great efforts to
build up a countrywide system of science-oriented education
and to promove scientific research and, in this way, succeeded
in modernising the whole society leading to rapid economic
development.
7-3. One thing is clear. In the absence of rapid economic
development, political conditions in the less advanced countries
would remain unstable. In many or most countries there would
be one revolution after another tending to get the two power
groups involved “directly or indirectly in the struggle. The
world must get out of this vicious circle. - There are only two
possibilities. One is for a violent type of revolution to occur
141 Mahalanobis I - pag. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 110,
which would suddenly change the whole structure of society
to make it fit for rapid development of science and economic
progress. The other alternative is deliberately to build up the
foundation of science-oriented education and research to promote
the modernisation of society in a peaceful way, and make
conditions favourable for economic development.
7-4. Aid for scientific and economic development from either
the western or the eastern countries, even when given in a spirit
of competition, would be cooperative in effect. In any event,
competition in constructive tasks of building up scientific foundations
in developing countries is less dangerous and is likely
to be far more useful than competition in the methodologies
of warfare. Also, collaboration in promoting education and
research in pure science can be pursued without any threat
t0 national security or national interests, and would be of great
help in promoting a rapid advance of the underdeveloped countries
and in fostering better understanding among the nations
of the world. The advanced countries have a great opportunity
for peaceful cooperation in giving aid for science
This pamphlet has not yet been published in any journal and is being
printed for private circulation. Certain aspects of these problems were
discussed by me in articles and addresses between 1955 and 1959 which
were reprinted in Talks on Planning (1961), and in other articles such as
A Note on Problems of Scientific Personnel (1959), Recent Developments
in the Organisation of Science in India (1959), and a lecture at Sofia
University in December 1961. Some of the ideas given in the present
pamphlet were presented by me before a Conference on International
Cooperation in Salzburg-Vienna in July 1962.
Professor P. M. S. Blackett in his presidential address to the British
Association for the Advancement of Science in 1957 and in other articles
in Nature, (3 February, 1962; May 1962 etc.) has considered various problems
from the point of view of the advanced countries. Professor Stevan
Dedijer made a penetrating analysis in article in Nature (6 August, 1960)
and in another article published recentlv in Stockholm, TVF, 22, (1962)
14]
Mahalanobis I - pag. 33
STATISTICAL TOOLS AND TECHNIQUES
IN PERSPECTIVE PLANNING IN INDIA
P. C.yMAHALANOBIS
Indian Statistical Institute - Calcutta - India
INTRODUCTION
The phrase ‘perspective planning’ is being used in India
since about 1954 or 1955 in the field of National Planning in
which long range targets have to be set up 10 or 1 5 or 20 years
in advance. The object of the present paper is to explain why
perspective planning is essential in the case of under-developed
countries and give some illustrative examples of the statistical
information and methods which have been found useful for
this purpose in India. This is not the occasion to attempt a
comprehensive discussion of techniques of perspective planning.
It is useful to make a distinction between projections and
targets. The word ‘projection’ is used in the same way as
in advanced countries to refer to the value of production, or
of consumption or of other variates, at a specified date in
future, estimated on the basis of historical records. Projections
are essentially estimates obtained on the basis of analysis of
lime series or some kind of extrapolation in time. It is convenient
to use the word ‘target’ as the value of production, of
consumption, or of other variates of interest which is desired
to be attained on a specified date in future, through the pro-+51
Mahalanobis II - pag.
1104 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
cess of implementation of an economic plan. The word ‘target’
would be used consistently in this sense.
Objects of planning in India: The ultimate objects of
planning are to improve the level of living, and expand facilities
for education, care of health, cultural amenities gtc.
for all the people of the country. A spectacular improvement
in the level of living of the advanced countries has been possible
in the past, and a similar improvement would be possible
in the less advanced areas in future, only through a continuing
increase in the per capita production of all the people of the
country. Such increase in per capita production can be attained
only through a continuing substitution of human and animal
power by machines, driven by steam or by electricity, for
productive purpose of all kinds including industry, agriculture,
rransport and distribution.
Changes in the level of living: As our chief concern is
with the improvement of the level of living, a continuing National
Sample Survey was started in India in 1950 which is
collecting comprehensive information on various aspects of
the level of living in rural and urban areas with a view to
assessing the change over time. The total per capita expenditure
per month on all consumer goods and services of each
household has been used as a rough indicator of the level of
living of the household. The method of fractile graphical
analysis (!) has been used to study the distribution by size of
total per capita expenditure per month of households. Studies
are also being made of the relationship between the total per
capita consumer expenditure and the per capita consumption of
individual items in terms of money and also in physical quan-‘ities
where possible.
À study of the distribution of per capita total consumption
expenditure by decile groups of households (arranged in ascend-(')
See A method of fractile graphical analysis, « Econometrica », vol. 28,
PP. 325-351, 1960; also A preliminary note on the consumption of cereals in
India, « Bulletin of the International Statistical Institute ». vol. 38, pp. 53-76,
1062.
15] Mahalanobis II - pag. 2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1105
ing order of the total per capita consumer expenditure in each
household), shows that for the data collected in the National
Sample Survey Round 8 (covering the period July 1954-March
1955), the percentage share of the lowest decile group was 3.01
in rural areas and 2.65 in urban areas, and of the second
lowest decile group (between the tenth and the twentieth percentiles
of households ranked by per capita expenditure) was
4.09 and 3.90 for rural and urban areas respectively. For
purposes of perspective planning, four per cent may be used
as the share of the second lowest decile group of households (?)
Targets of planning: The average per capita expenditure
in the second lowest decile group was a little over Rs. 10 per
month (*) in 1960-61. For purposes of illustration, it is possible
to adopt a target of raising, over a period of 1 5 years,
the average per capita consumption expenditure in the second
lowest decile group of households from Rs. 10 to Rs. 20
per month (or fifty dollars per capita per year). This amount,
at 1960-61 prices, would provide only a very modest level of
living in terms of food, clothing and other essential goods or
services and amenities.
Doubling the per capita expenditure in fifteen years implies
a rate of growth of nearly five per cent per capita per year.
[t is of interest to note in the present connexion that the per
capita income in USA has increased sevenfold in the cours ot
[20 years or at a rate a little over 1.69, per capita per year.
A reasonable target of planning in India would thus call for
a rate of increase of income at a rate nearly three times greater
than the actual rate of increase attained in the USA during
the last 120 years. The above comparison would supply a
rough idea of the dimension of the efforts required for economic
development in India.
(*) The lowest decile group has not been used because it may be a sonewhat
heterogeneous category comprising vagrants, persons living in isolaton,
tribal people, households in a transient income group etc. many
vhom would require special ameliorative measures.
* One rupee = 1 shilling 6 pence — 0.21 U. S, cent approximately
3)
151
Mahalanobis II - pag.
[106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
For purposes of planning it is necessary to deal with actual
figures and not merely in percentages. The population of India
is expected to increase to about 650 million compared with
about 430 million in 1960. (This, of course, is the population
projection for 1975 on plausible assumptions, and not a target;
in fact, if it were possible to bring about a reduction in the
rate of growth of population, Indian planners would no doubt
adopt a much lower figure as a target). The number of second
lowest decile group of households in 1975 would be about
12.5 million on the basis of about five persons per household.
To attain a target of Rs. 1,200 (or $ 240) per year per household,
the aggregate income of the second lowest decile group
of households would have to be Rs. 15,000 million. If it is
assumed that this group would still continue to have a four
per cent share of the total expenditure of the households (*)
then the aggregate national consumption expenditure of households
would be 25 times greater or Rs. 375,000 million.. The
aggregate national income of India in 1975 would have to
be somewhat larger to allow for investments and certain other
items. The level of national income to be attained in 1975
would have to be somewhat more than double the target of
income at the end of Third Five Year Plan in 1966. The rate
of growth would have to be about seven per cent per year.
Need of rapid industrialization: Such a rapid change (at
a rate three times greater than that of USA) would call for
rapid industrial expansion over a period of 15 years.
The ultimate aim is expanding continually the production
of consumer goods and services. It is necessary to increase
the supply of machinery and energy for this purpose. In
(*) In India the distribution of consumption expenditure of households
by size of expenditure has been found to be steady (with some small fluctuations
probably due to the effect of changes in prices) over the last ten
years. The pattern of distribution of income of households by size of
income has also been found generally to change only very slowly over time
in most countries of the world. The assumption that the share of the
second decile group (or of other fractile groups) of households would remain
practicallv the same in India in 1075 is plausible
‘151 Mahalanobis IT - pag.
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 1107
India, and in most of the other underdeveloped countries, it
is not possible continually to import machinery for production
of goods or of fuel on account of shortage of foreign currency.
[t is essential to establish and expand industries to manufacture
machinery, electricals, transport and construction equipment.
To increase the capacity for the production of capital
goods and energy would be thus the only sound foundation for
the expansion of consumer goods and services in future.
At the same time, in all underdeveloped countries it is
possible to increase the production of consumer goods with
small tools by using traditional methods. This type of production
is labour intensive and would give gainful employment
to a large number of people who would otherwise remain
idle for a good part of their time.
In India a dual strategy was adopted from 1956 in the
Second Plan to expand, on one side, the strategic heavy industries
for steel, metals, machinery, electricals and chemicals,
etc. to build up the foundations of industrial progress,
and at the same time also to expand the traditional cottage
industries and small scale production.
Targets of capital goods: It is therefore necessary to expand
and set up not only targets of income or of consumer
goods but also of machinery, steel and other metals, electricity,
transport, etc. which would be used for the production of the
desired volume of consumer goods and services.
Targets of scientific and technical .personnel: - To achieve
the targets of production, it would be necessary rapidly to
Increase the technical staff to prepare and implement an increasing
number of projects. Training facilities must be
expanded sufficiently quickly to turn out technical and scientific
personnel in adequate numbers at all levels. Scientific
and technological research would have to be expanded and
oriented to serve the needs of national development in an effective
manner. Fundamental research as well as training in
+5, Mahalanobis II - pag.
3
1108 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
research must also be encouraged and developed at the same
time to foster the accumulation of basic knowledge and to
supply a sound foundation for national decisions being made
increasingly on rational grounds.
Balances at the stage of production and utilization: An
essential condition for successful planning is to estimate in real
terms the requirements of each project to ensure that right
quantities of materials, machinery and men are available at
the right time at every stage of the implementation of the
project. Also, products and services resulting from the completion
of each project must be promptly and effectively utilised
to promote the execution of other projects and for the progress
of the plan as a whole.
The physical targets of production must be balanced in
terms of physical quantities of raw materials, machinery,
energy, transport etc., and also in terms of man power and
of the flow of money. Incomes are generated in the very process
of production; and supplies are utilised through market
operations. Planning requires that aggregate incomes should
be balanced with expenditure, savings should match investments,
and the supply and demand of individual goods and
services should be balanced in real terms so as to avoid any
inflationary rise of prices or undesirable shifts in prices. Physical
and financial planning are different aspects of the same
reality.
In India a perspective view of development over a long
period of years began to be taken from the end of 1954. It was
recognised that the targets and the balances of materials and
of man power would be only approximate partly for lack of
information and partly for defects in organisation and implementation.
It was therefore recognised that planning would
have to remain flexible and to enable necessary adjustments
being made almost continuously. At the same time it was
essential to keep in view a wide time horizon of 15 or 20 vears
NT more.
‘151 Mahalanobis II - nag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1106
The use of simple models: In 1954-55, some simple models
were used to work out the basic strategy of the Second Five
Year Plan. The total investment was divided into two parts,
one À; as the fraction used for investments for the production
of capital goods, and the other À, as the fraction used for
investments for the production of consumer goods (A; +X, =1).
If the corresponding net output-investment ratios for the production
of investment goods and for the production of consumer
goods respectively are 3; and 3, then the total net outputinvestment
ratio is $=2; 3;+2, 8... By using the following
two sector model, and using numerical values for the total
investment, and estimated values of 3; and j3_, suitable values
of À; and À, were selected so as to enable the economy to grow
at the target rate of five per cent per year or so. In order
lo estimate the volume of employment, the capital investment
required per worker, say 0, was also used.
The growth of national income (Y) in the two sector model
s given by the following formula:
in which Y, is the national income in the base year, Y, the
national income in the #-th year, and «, the rate of investment
in the base year.
On this basis, a Draft Plan-frame for the Second Plan was
prepared in March 1955 (°). Values of the different parameters
as used in the Draft Plan-frame, the Second Plan (1956-61)
as actually realised, and the Third Plan (-061-66) as estimated,
are shown in the Table given below
(*) The methods used have been described in The approach of operational
research to planning in India. « Sankhva ». vol. 16, PP. 3-130, 1955.
Mahalanobis II
pag.
1110
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE I
Investment allocation, capital per worker and
net outbut-investment ratio
slan
ercentage allocation ; ; .
p of sdlage au for capital net output-investment ratio
— -- per
invest- consumer worker
ment goods (Rs.)
goods
invest- consumer total
ment goods (AiBi+AcPe)
goods
m
8
Second Plan:
Draf plan-frame
(1955) - . .
Second plan: actual
(1956-61) . . -
Third plan: estimate
(1061-66) . . 39 61 6.000
0.21
2C
0.67 0.51
0.53
0.38
0.63
0.47
Many changes were made in the targets and allocations
of the Draft Plan-frame at the stage of the preparation of the
Second Plan; the values of the parameters of the Second Plan
as actually realized and the values given in the Draft Planframe
are therefore not strictly comparable. The interesting
point to note is that the estimated parameters for the Third
Plan are fairly close to the parameters used in the Draft Planframe.
The rate of investment («,) in the first year of the Second
Five Year Plan was 9.8 per cent, the initial national income (Y,)
was Rs. 108.0 billion and the values of the other parameters
were A; =36%, A, =64%, B;=0.11, B.=0.53, as given in the
second row of the above table. Using these values in the above
expression, the estimated national income comes out as
Rs. 129.7 billion for 1960-61 against an actual figure of
Rs. 130.1 billion both expressed at 1952-53 prices. In the case
of the Third Five Year Plan, using the parameters given in
the third row of the table, an initial income of Rs. 145.0 billion
(at 1960-61 prices) and an initial rate of investment of II per
"15] Mahalanobis II - pag. 8
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1113
cent, the estimated income for 1965-66 on the basis of the twosector
model is Rs. 188.9 billion against an estimate of
Rs. 190.0 billion given in the Third Five Year Plan on the
basis of detailed sector-wise calculations.
It may be concluded, therefore, that the two-sector model
can supply a fairly reliable method for estimating future income.
Values of the parameters used for the base period are
no doubt subject to errors of estimation; but this would be
true in the case of other methods also. The two sector model
sives realistic estimates presumably because it has reasonably
correct structural relations between relevant variables.
Values of output-investment ratios: Output-investment ra
tios 3; and 3, determine, together with the chosen values of 2,
and À, and the total amount of investment, the rate of increase
of income and have an important role in planning. These
two coefficients of net output-investment ratios were calculated
from technological and statistical information in respect of
hundreds of enterprises combined with appropriate weights.
The calculated values for manufacturing industries are given
in Table A 1.
Need of perspective planning: Steel: The need of looking
2 long way ahead was learnt in India through experience.
[ shall give one example. In 1949 when preparatory work had
just started for the First Five Year Plan, a decision was pracically
reached to increase the capacity for the production of
steel from a little less than one million ton per year to two milion
tons per year in the course of five years. However, a careful
survey was made of the current demand as in 1949. It was
found that the maximum demand would be about 1.5 million
fon per year. With marginal expansion of existing steel plants,
t was possible to produce about a million ton per year within
the country. Owing to the wide prevalence of the views of
short-range economic theory, it was therefore decided that it
would be inadvisable to include a new million ton steel plant
in the First Five Year Plan of India.
15 Mahalanobis II - pag. y
112 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
In consequence, great difficulties began to be experienced
from the early years of the First Plan. Practically all the estimates
for investments had been made in purely financial terms
and a sizable increase in investments had been approved purely
on a financial basis. As soon as the investment projects began
to be implemented, there was a sharp and continuing increase
in the requirements of steel and other goods and services.
Very soon the demand for cement increased to nearly three
times the domestic supply. There was also a continuing and
large expenditure of foreign currency for the import of steel,
which added up to something like 2,000 million dollars in
the next ten years or so. In 1050 it would have been possible
to establish a new million ton steel plant with perhaps about
150 million dollars of imported machinery. Had this project
deen started at that time an additional supply of one million
ton of steel (worth more than one hundred million dollars per
year) would have been available from the early years of the
Second Fide Year Plan, and would have resulted in a very
large and continuing saving of foreign exchange. The decision
to drop the million ton steel project from the First Plan
was due to attention being focussed only on the current demand
in 1949, that is, due to a complete failure to appreciate the
need of looking ahead to get ready to meet the demand for
steel which was certain to increase rapidly in future.
Targets of steel in 1970: At heavy cost we had learnt the
lesson of not proceeding with the building up of capacity for
steel production 12 or 15 years ago. Much attention is now
oeing given to advance planning for steel. A detailed analysis
of the requirements of steel is made, where possible, by individual
items of production. With a given set of production
targets for, say, 1970, it is possible in this way to prepare
useful estimates of the requireménts of steel. Some illustrative
figures for the transport equipment industry is given in the
following table.
‘151 Mahalanobis I] - pag: 10
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1113
TABLE 2
Steel requirements for transport equipment industry in 1970
ndustries
[. Steam locomotives
2. Electric locomotive
3. Diesel locomotive-1.
Wagons . .
5. Passenger coache
h. Automobiles
7. Motor cycles.
8. Bicycles
o. Ships (GR”
tons of rolled steel require.
production steel required ment in 1970
target in 1970 per unit of (in thousand
output tons)
J
0
"5.0
22.0
Ny
RC. Naf
dv
nn, €
5.0
80.0
04.0
«
Source: Demand for steel, special steel and pig iron. India: 1060-1970
Perspective Planning Division. Plannine Commission
The transport equipment industry would thus require about
[.34 million ton of steel per year. Requirements of other industries
were estimated in the same way; the grand total for
'ndustries came to about 8 million tons of rolled metal.
In other cases a different approach is necessary. The steel
requirement per rupee of net investment has been estimated
for different types of activities. For example, the consumption
of steel is 40 tons per investment of Rs. 100,000 in railways;
the corresponding figure is so low as only 5 tons in large and
medium scale irrigation. The total steel requirement for a
target of investment in the Fourth Plan amounting to
Rs. 170,000 million can be estimated at 20 or 21 million tons.
Also, on the basis of the investment outlay for the last
year of the Fourth Plan, one can estimate the steel require:
ment at about 5 million tons at the end of the Fourth Plan
Mahalanobis Il - pag.
1114 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
Adding to this the current requirement of 8 million tons for
industries, the total demand would be about 13 million tons
of steel in 1970-71. In the same way it has been estimated
that the requirement of steel would reach 18 or 19 million
tons In 1975.
Balance of electricity: It is possible in the same way to
estimate the requirements of electricity from the physical targets
of production for any given year. For example, the production
of ferro-manganese in 1960-61 was 100,000 tons for
which the electricity consumed was 500 million kwh. For a
target production of 385,000 tons for ferro-manganese in 1970-1971,
the requirements of electricity would be 1,952 million
kwh. A similar method of calculation was used for different
types of industries. Table A-3 in the Appendix gives the
details. Steel and electricity are typical illustrations of the
material balances which have been prepared in India for important
commodities and energy for perspective planning of
the economy 15 or 20 years ahead.
Perspective planning of fertilisers: The population of India
ls growing roughly at the rate of perhaps g million per year.
The additional quantity of food grains required for these 9
million people would be about 1.5 million tons a year. This
would add up to 22.5 million tons in the first five year period
not to speak of 60 million tons in the second five year period).
At an average price of go dollars per ton, the cost of importing
22.5 million tons in a five year period would come to about
2.000 million dollars.
On the other hand, if imported ammonium sulphate is used,
each ton on an average should increase the yield of food grains
by about 2.2 tons. On this basis, roughly 10 million tons of
:mported ammonium sulphate would enable the domestic production
of food grains being increased by about 22 million tons
in a five year period. At an average price of 70 dollars per
ton of fertilisers, the cost in foreien currencv would be onlv
"151 Mahalanobis II - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
LHD
about 700 million dollars or a third of the cost of imported
food grains.
Imported foodgrains can be quickly distributed and it is
possible to make necessary arrangements for such imports at
short notice in the course of a year or so under normal conditions
of easy availability of foodgrains in the world market.
{The lack of foreign currency is the only limitation in a country
like India). The import of fertilisers, however, require placing
of orders a year or two or even more years in advance because
the supply position is not so easy as in the case of foodgrains.
Such a plan would, therefore, require taking a view of future
needs two or three years ahead.
A third possibility would be to set up a new factory every
year for the production of 750,000 tons of ammonium sulphate
per year. At the cost of about go million dollars for each
factory, the total expenditure would come to 450 million dollars
of which, however, only 250 million dollars would be the
foreign exchange requirement. The setting up of a new fertiliser
factory would require at least five or six years; the process
of planning must therefore start something like 10 vears in
advance.
Finally, it is also possible to manufacture in India machinery
for the installation every year of a new fertilizer factory
with capacity to produce %50,000 tons of ammonium sulphate
per year. The foreign exchange requirement for this purpose
would be less than 100 million dollars, to be spent once and for
all. However, the installation of a plant to manufacture machinery
for the production of fertilisers would take at least five or
six years. When the first batch of machinery is produced, it
would take another five years or so to complete the construction
of a fertiliser factory. Such a plan would require a view being
taken of future requirements at least 12 or 15 years in advance.
Consumer goods: In the case of consumer goods the increase
in demand is estimated on the basis of the increase of
ncome accepted as a target. Standard methods are used to
15] Mahalanobis II - pag. 13
1116 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
calculate the elasticty of demand from information regarding
expenditure (and consumption in physical terms, where possible)
of a large number of commodities and services which is
being collected every year by the National Sample Survey
(NSS) of India. In the NSS, the design of interpenetrating
net-work of sub-samples (IPNS) is always used providing at
least two independent estimates of each variate. It is, therefore,
possible to estimate the elasticity of demand on the basis
of each sub-sample and also on the basis of the combined
sample of the twe sub-samples pooled together. Table A-2 in
the Appendix gives estimates of percentage increases in demand
over the five-year period of the Third Plan. The two
independent sub-sample estimates supply useful information
nn the margin of uncertainty of the estimates.
In a planned economy it is not possible to allow the supply
to increase with the demand without any restriction. It is
necessary to increase domestic savings by restricting the consumption
of non-essential or luxury goods. It is therefore necessary
to impose excise and saies tax or controls on imports
or on production to bring about a balance between the planned
supply and the estimated demand.
Recently the method of fractile graphical analysis is being
used for estimating elasticities of demand for households having
different values of total per capita consumer expenditure (which
is a rough indicator of the level of living). This approach has
the great advantage of showing, in a very simple way, the
pattern of change of the elasticity of demand with a change
in the level of living. Analysis by fractile groups is particularly
useful in studying the effect of excise and sales tax in balancing
supplv and demand.
Perspective planning of man-power: It is only with the
help of skilled workers, technicians, technologists and engineers
that raw materials can be converted into machinery, electricity
and power which can then be used for the production of both
capital and consumer goods. A rapidly increasing supply of
"151 Mahalanobis IT - pag. 14
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
ill
engineers and technical personnel is essential for economic
development. It is necessary to establish and broaden the base
of primary and secondary education and to establish technical
and scientific institutions and increase their number rapidly.
The most serious difficulty is the lack of trained and ex
perienced teachers at all levels. To build up a sound founda-‘ion
for the outturn of technical personnel would take a great
deal of time; it is a much more slowly maturing process than
establishing heavy machine building, steel, heavy electrical or
heavy chemical industries. Perspective planning is indispensable,
and it is necessary to have targets twenty years or more
in advance.
Scientific and technical manpower: From about 1955 a
great deal of attention is being given in India to the question
of technical manpower. The method used for estimating the
requirements of technical personnel is simple and straightforward.
Information relating to manufacturing industries for the
reference period 1956 was collected as a part of the National
Sample Surveys and was analysed in detail to ascertain the
number of professional and technical workers (including engineers
and scientists) employed in manufacturing industries.
Estimates for a number of selected industries are given in
Table 3 in the form of percentages of total employment (that is,
number of engaged persons) in different industries. Separate
figures are given in col. (2) for the proportion of professional,
technical and associated workers taken together, in col. (3) for
the proportion of engineers, architects and surveyors, and in
col. (4) for the proportion of scientists including chemists, physicists,
geologists and other physical scientists.
There are wide variations in requirements of professional
and technical personnel or of engineers or scientists from one
industry to another. In chemicals, and aircraft assembling
and repair, the proportion of professional and technical staff
is about 10 per cent. The chemical industries, naturally, require
5 per cent of scientists (no doubt, mostly chemists) anc
v5] Mahalanobis II - pag. 15
1118 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
only 0.6 per cent of engineers. In contrast, aircraft assembling
and repair requires a high proportion of about 5.5 per cent of
engineers but practically no scientists.
TABLE 3
Technical personnel in selected industries: samble survey of manufacturing
industries, 1956
industries
percentage of total employment
professional engineers scientists
2)
(3)
(4)
1. Rice milling . .
2. Cotton textiles . . . . |
3. Glass and glassware . . .
4. Tea manufacturing . . . . .
5. Aluminium, copper, brass: secondary
products . . .
0.87
0.90
0.99
2.3G
0.08
0.12
O.IC
"1
0.00
0.5T
0.18
0.03
2.49
2.65
2.58
0.51
0.05
0.471
6. Sugar . . . . . . . . .
7. General engineering and electrical
engineering . . . . . . . .
Paints and varnishes . . . . .
Cement . . . . . ..-Petroleum
refining
rr. Electricity generation and transmission
. . . . . . . . . 6.50
12. Iron and steel: primary products 5.70
13. Railway wagon manufacturing . 8.46
I4. Aircraft assembling and repair . 9.03
15. Chemicals (including drugs) . 0.90
2.02
0.31
0.89
1.55
0.0I
3-47
I.12
2.40
4.79 0.04
2.86 0.58
3.02 0.21
5-47 0.00
0.62 2.06
Source: Occupational Pattern in Manufacturing Industries, India 1956 by
PITAMBAR PANT and M. VAsUDEVAN with a foreword by P. C. MAHALAvols.
Planning Commission, Government of India, 1959.
In col. (2) ‘professional’ stands for all professional, technical and
related workers. In col. (3) ‘engineers’ cover architects and surveyors.
[n col. (4) ‘scientists’ stand for chemists, phvsicists. geologists and
yther phvsical scienticts
4]
Mahalanobis II - pag 16
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1116
With any assumed target of production for any particular
industry in any given year, it is possible to estimate the total
number of engaged persons and hence the number of professional
staff, engineers and scientists. Requirements of engineers
and technical personnel were estimated in this way for
purposes of perspective planning.
Expansion of technical staff: Appropriate action was taken
to expand the capacity of existing scientific and technological
institutions and to establish new institutions all over the country
to ensure a sufficiently rapid expansion of scientific and technical
personnel. The following table shows the new admissions
into universities and higher educational institutions of the
university standard in science and technology.
TABLE
1dmissions into higher degree level institutions in science ana
technology
subject
[. Science.
2. Engineering
3. Medicine .
+. Agriculture
5. Total
1950-51 1960-61 1965-66 1975-76
n thousand:
J
1950-51 1960-61 1055-68
as percentage of 1950
700
00
530
227
356
24C
396
625
320
“ec
1975 76
figures
518
i,750
800
750
20
On the whole the planning for scientific and technical manpower,
particularly for engineering, has been quite satisfactory
in India. For example, the new admissions in engin:zring
‘ncreased from 4,000 a vear in 330-51 to 300 Tr in
_15] Mahalanobis II - pag. 17
1120 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
1960-61. Also, the target is about 25,000 new admissions in
1965-66, and 70,000 in 1975-76.
Outturn of engineers: The Appendix Table A-4 gives the
outturn of scientists and engineers in India from 1915 to 1960.
It would be seen from Table A-4, line 8 and col. (4), that
the number of degree level engineers turned out between
1915 and 1947 was 14,984 in 33 years before independence.
This was practically matched by a turnout of 14,385 in five
vears during the period of the First Plan (1951-56). The outturn
increased much further to 24,166 during the five-vear
period of the Second Plan.
The outturn for individual years between 1951 and 1960
also shows a very rapid increase. The outturn of degree level
engineers was 1,700 in 1951 which was nearly doubled in
three or four years. Perspective planning of technical personnel
was seriously started from 1955; the effect became visible
after four years in 1959 when the outturn rose to 6,770 against
3,689 in the previous year, that is, an increase of more than
three thousand in one vear.
Scientific Research: Although the intake and outturn of
scientists also has been increasing fairly rapidly, I am sorry
to say that perspective planning of scientific research has not
yet started seriously. The emergence into the modern age of
any underdeveloped country would be possible only with the
building of the base of science education and scientific research.
Certain compelling reasons can be appreciated very easily.
Natural resources are not identical everywhere; there are wide
variations from one country to another. Resources available
within any country can be used most effectively only through
continuing applied scientific and technological research in
which use is made of basic scientific knowledge to solve practical
problems. It is also necessary to provide facilities for
fundamental research not only for the accumulation of scientific
knowledge but also to supply scientists who would ‘be
able to diagnose problems properly and identify how such
15] Mahalanobis II - pag. 18
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. lizi
problems should be handled or what kind of help should be
obtained from abroad. There is also a deeper need of replacing
the traditional pattern of making decisions on the basis of
authority by decisions to be made increasingly on objective
grounds based on scientific and rational thinking.
Perspective planning is indispensable. The need of perspective
planning, especially in underdeveloped countries, may
be stated very briefly in conclusion. It is necessary to increase
the supply of consumer goods. To do this it is necessary to
expand continually the production of capitel goods. Both
would require an increasing supply of engineers and technicians.
Industrial and technological developments would call
for a rapid expansion of applied research which, in its turn,
would require a sound foundation of basic research.
The factor of time may be next considered. Factories for
the production of practically any kind of consumer goods
can be established in a year or two with the help of imported
machinery or fuel. To develop the production of capital goods
and energy would take it at least 10 or 15 years. To secure
an adequate supply of engineering and technical personnel
would require 20 or 25 years. To have enough scientists ot
ability for both applied and basic research would take at least
a generation or even more. It is clear that perspective planning,
looking 15 or 20 or 30 years ahead, is indispensable for
all underdeveloped countries
-y) Mahalanobis II - pag.
A
1122
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§&
APPENDIX
TABLE A-1
Estimates of 3 and © for major groups of manufacturing industries
with 1057 and 1060-61 weights
f (thousand Rs.)
1957 1960-61 1957 1960-61
weights weights weights weights
I. Metallurgical industries . .
2. » semi manf. .
3. Mechanical and general engineering
. . . . . . .
4. Transport equipment . .
5. Electrical equipment . . .
6. Industrial machinery (I) .
7. » » (II) .
8. Chemicals . . . .
9. Textiles . . . . . . .
ro. Rubber and leather products
11. Food industries . . .
12. Mining industries . . .
13. Timber and cellulose
dustries . . . . .
14. Mining and oil industry . .
15. All industries . .
(2)
1)
0.19
0.47
0 20
0.4%
0.66
0.45
0.50
0.6"
0.45
0.4C
0.62
0.47
0.35
0.28
0.62
0.7
0.2
0.2%
0 28
0.671
0.30
0.33
ù -0
0 -
0.33 0.31
0.43 0.39
0.26 0.25
: A
(5)
"78.9 172.3
75.5 10.0
A
10.9
15.3
18.5
22.7
20.1
30-3
10.5
14.8
12.9
20.5
10.C
el
20.
10.
IA.5
I°.0
Tr
21.2
g.5
12.7
12.3
II.I
15.3
“ote: The coefficients are obtained from detailed industry-wise information
compiled by the Perspective Planning Division of the Planning
Commission in collaboration with the Planning Unit of the Indian
Statistical Tnstitute
‘151 Mahalanobis IT - pag. 20
SEMAINE D'ETUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 112.
TABLE A-2
Estimates of percentage increase
vear dlan beriod ~~" 7
demand
DUEFr
che
inird
five
nercentage increase in der-1ame
of item
urban Tra
crea] Teed
“nef
{. Pulses . .
2. Vegetables
3. Spices .
1. Edible oil .
6
9
J
5. Sugar . . . . 38
6. Milk and milk
products . . . 46
7. Meat, fish, eggs . 38
R. Fruits and nuts . 46
43
4G
45
Beverage and refreshments
.
to. Tobacco
II. Kerosene .
12. Fuel and light
12
41
-
13. Cotton clothing
(mill-made) .
i4. Washing soap
(5. Toilets . .
40
78
45
43
39
AI
16. Railway .
17. Conveyance
(8. Cinema . .
to. Domestic utensils
53
49
39 138
20
44
22
41
oR
Nz
48
39
AI
36
30
22
53
53
40
46
-Q
’
28
40
33
27
29
3
20
36
cl
34
30
22
3I
37
eu
39
33
22
4G
ex
35
37
:
;-51
38
27
32
34
DA
2
57
33
30
>
<
Vote Based on elasticities calculated in the Indian Statistical Institute
Calcutta from 10th round NSS data relating to December 1955
Mav 1956
Mahalanobis II
pag. 2
~onsuming industry
Iron and steel
I finished steel
2 pig iron . .
3 steel re-rolling
2. Ferro-manganese
3. Ferro-silicon
4. Alloy steel .
5. Aluminium
6. Copper
7 Zinc
€ Coal
I hituminnnus
unit
m. tc:
077 tons
m. ton
TABLE A-3
Electricity balance
volume of production
electricity consumption
in m. kwh
1970-71 1960-61 1965-66 1970-71
capacitv
1960-61
production
1965-66
capacitv
8)
(6
(
7:
"208
T 100
3,750 6,750
30 60
550
I,IOO 1,025
320 480
250 625
2,500 5,000
I
2
T)
227
TC
oN
387
2N0
“00
—
<.
~
N30
THT A
“80 0
Tof
I,OQAC
5,004
electricity
consumption
per unit of
production
0)
500 kwh /ton
20
5,000
8,000
1,250
20.000
)
))
5,00C
4,20
—À
Ne
N
«
3
NoTEs: Source: Perspective Planning Division paper: Demand for Electricity, India 1960-1970. The table covers
all industrial uses of electricity. The norms of eletricity requirement in different industries, given in col. (9).
have been used to work out the consumption of electricity in 1960-61. 1965-66 and 1070-71 given respectivels
in cols. (6). (7) and (8)
[ABLE A-3 (continued
Electricity balance
onsuming industry
ini
1960-61
production
rolume of production
1965-66
eapacity
1970-71
capacity
1960-<
electricity consume
nm. kwh
a.
electricity
consumption
per unit of
production
3
J
2 brown . . . .
petroleum (refining) .
Fertilizers
1 nitrogeneous, elec
trolytic process .
2 nitrogeneous, res’
3 phosphatic
Heavy chemical
ulfuric acid
soda ash .
caustic soda, ci.
‘cal process
m. Or
CE
=n
“A
qr
a.
20 kwh /ton
J
))
\
320.4
300
220.0
WE
x“
t
190
se
J
Source: Perspective Planning Division paper: Demand for Electricity, India 1960-1970. The table covers
all industrial uses of electricity. The norms of electricity requirement in different industries, given in col. (9),
have been used to work out the consumption of electricity in 1060-61, 1065-66 and 1070-71 given respectively
in cols. (6). (7) and (8)
-4
-
‘
2
T]
3
4
3
ND
on
consuming industry
4 caustic soda, elec
trolytic process .
12. Plastics . . .
13. Soap . . . .
14. Synthetic rubber .
15. Paper and pape.
board . . .
16. Newsprint and securitv
paper
unit
9"
200 ton
TABLE A-3 (continuec
Electricity balance
volume of nroductinn
electricity consumption
in m. kwh
1970-71 1960-61 1965-66 1970 -
capacity
1960-61 1965-66
production capacity
(AN
f
(6)
(7)
42"
100.C
. rr
350.0
85.G
500.0
50
610.0
250.0
700.0
40
Ar
4
GO
L40
sd
,.C
820.c
- 3.0
-
J
sc
i -
Ay |
24.0 I5I.5
240.0
16
ra
- «6
electricity
consumption
per unit of
production
0)
4,200 kwh /ton
60 »
200 »
70C »
Ror
NN
~
Cry
—
NJ
~~
NoTEs: Source: Perspective Planning Division paper: Demand for Electricity, India 1960-1970. The table covers
all industrial uses of electricity. The norms of electricity requirement in different industries, given in col. (9),
have been used to work ont the consumption of electricitv in Tofo-fI, 1065-A6 and T070-71 given respectively
in cols. (6). (7) and (8)
TABLE A-3 (continue?
Electricity balanc
consuming industry
init
volume of production
1960-61
production
1965-7"
capacity
1970- 1
capacity
electricity consumptier
in m. kwh
1$-0-61
19¢
‘C7
electricity
consumption
per unit of
production
1
4
Ç
17. Cement . . .
(8. Cotton textiles
19. Jute . .
- Rayon and sta
ple fibre
I rayon fl:
ment
2 staple ©.
3 chemic "
A
in. tons
metre.
orn “ns
ade
LAF
oD
375
15.0
,,300.C
[,200.°
J
sf»
26.1
ce
Ow
LU.
Das
44.
yu
~0 kwh/ton
- kwh/oo0 metres
kwh/ton
+5 kv'h/oo0 lbs.
rc »
500 kwt/ton
,,000 kwh/000 Ibs.
-
ot
ool
x.
4
Source: Perspective Planning Division paper: Demand for Electricity, India rg6o-1970. The table covers al!
industrial uses of electricity. The norms of electricity requirement in different industries, given in col. (9),
have been used to work out the consumption of electricity in 1a60-61. 1065-66 and 1070-71 given respectively
in cols. (6), (7) and (8).
“
consuming industry
1)
22. Silkk...
23. Sugar | .
24. Vegetable oil .
25. Vanaspati ghee
26. Bicycles . .
27. Sewing machines
. . . .
28. Electric fans .
29. Electric lamps .
30. Matches . . . nm.
init
m. yds.
m tons
»
000 tons
m. nos
000 Nos
m nos
- boxe
TABLE A-3 (continued
Electricity balance
electricity consump iin .
volume of production in m. kwh electricity
consumption
per unit of
production
1960-61 1965-66
pruduction capaeity
1970-71
capacity
AN
(a
ne
0.0
Ran
Tog
765
740
300 kwh/ooo yards
60 kwh/ton
25 »
220 »
~~
“A.
30.0
, +
us - [
2.5
15 kwh /nos.
508
60 kwh/nos.
20 kwh /nos.
150 kwh/000 nos.
600 kwh/o00 gross
hoxes
57
p—
—
N
ne
nN
~
NOTES
Source: Perspective Planning Division paper: Demand for Electricity, India 1960-1970. The table covers all
industrial uses of electricity. The norms of electricity requirement in different industries, given in col. (9),
have been used to work out the consumption of electricity in 1960-61. 1965-66 and 1970-71 given respectivelv
in cols. (6). (7) and (8)
TABLE A-3 (continuec
Electricity halar-‘onsuming
indusiry
1960-productio
volume of nroductio-Ni
! 0-pacity
electricity consumption
in m. kwh
ch
electricity
consumption
per unit of
production
fn
4
4
y
4
J)
Plywood
ww
‘J
gd. me-32.
Calcium
33. Autome’ ©
res
*.temol
,5C
a.
nm
07 kwh/nos.
“20 kwh/nos.
xwh/ton
0
PR
3,770 27.672
1/.
Source: Perspective Planning Division paper: De nand for Electricity, India 1960-1970, The table covers all
industrial uses of electricity. The norms of electricity requirement in different industries, given in col. (9),
have been used to work out the consumption of electricity in 1060-61. 1965-66 and 1070-71 given respectively
in cols. (6), (7) and (8)
J]
J
4
4
n
-
ND
Te)
199 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
TABLE A-4
Outturn of scientists and engineers in India
number of persons graduatire
master’s degree
in natural science
total
average
per year
engineering
degree diploma total average
per year
(7)
A
+
1915-19
[020-24
[925-29
1030-34
1935-39
1040-44
1045-7
822
G17
r,072
2,782
2,938
3,378
2,511
Ho
182
=f
568
771
1,619
2,190
2,901
3,765
3,170
1,703
1,902
4,322
5.397
5,331
6,280
4,538
2,271
2,673
5,941
7,587
8,232
10,045
7,708
454
535
1,188
1,517
1,646
2,009
2,569
1,347
-
508
676
27
IQI5-47 — 15,283 463 14,084
rR
20, 473
44,457
)- 1048-50 2,047 — 0982 4691 4,623 9,314 3,105
1051-55 9,062 1,812 14,385 11,629 26,014 5,203
(1st Plan)
1956-61 15709 3,160 24,166 27,037 51,203 10,241
(2nd Plan)
0.
I.
12.
£3.
id.
[5.
rh.
1951
1052
1052
1,409
1,680
-,694
2,0 3€
2
1,409
1,980
-,694
2,068
2, TI
2,301
2,559
2,926
3,238
2,267
1,700
2,049
1,603
2,837
23,35,
4,001
4,608
4,619
6,071
6,715
4,001
4,608
4,619
6,071
6,715
7,587
7,920
9,633
12,961
12,102
7,587
7,920
9,633
12,961
r2,IN2
20
«iy
mm
50e
6,187
6,41 i
— 2
<, 571
3,071
3.735
'~* Figures are taken from Recent developments in the organization of
science in India by P. C. MAHALANOBIS, Engineers in India by Scientific
and Technical Manpower Division, Planning Commission; Education
in India by Ministry of Education, and also direct information
from the Resources and Scientific Research Division of the Planning
Commissinn
51 Mahalanobis TT - pag 28
TABLE A-5
Average per capita consumer expenditure in Rupees per month (30 days), percentage share of total consumer expenditure and
limiting values of consumer expenditure “fractile groups for ihe 8th round of the National Sample Survey, July 1954-March
1955, all-India: Rural and Urban
fractile
group
(percentage)
average per capita consurexpenditure
(Rs ‘
2oole
poolec
(|
FN
[ lowest
2. o— 10
3}. I0— 20
1. 20— 30
5. 30— 40
6. 40— 50
7. 50— 60
8. 6bo— 70
g. 70— 80
10. 80— 90
11. ÿG0—IOO
12. 0—100
13. Number
of villages
or
blocks .
Number
of households
4.21 4.64 4.48
6.25 6.56 6.42
7.72 8.24 7.99
9.26 9.52 9.37
10.91 10.89 10.90
12.01 12.05 12.63
4.82 15.12 14.94
7.72 18.54 18.17
12.42 23.80 23.04
16.44 39.00 42.16
14.03 14.08 14.06
6.20 6.68 6.54
9.30 9.96 9.59
[1.62 12.83 12.06
3.75 15.19 14.28
:6.00 18.56 16.94
:9.00 21.62 20.11
12.68 26.82 23.860
27-20 33.52 29.52
37-56 43.54 39.00
55.20 88.22 76.75
22.44 27.69 25.214
O3T
018
ISÉ,
OT.
Al
14
Source
'ndian Statistical
"netititte
Calcite
percentage share
urel
pooled
10)
ere
noo!c
9)
JE
1?)
1, 03 3.01 2.96 2.46 .65
.07 1.41 1.09 4.27 3.48 3.90
5.05 5.57 5.33 4.56 4.72 4.85
b.11 6.19 6.18 6.73 5.34 5.70
7.18 7-43 7.27 7.05 06.86 6.78
8.27 8.35 8.35 8.29 7.82 8.01
9.51 10.00 9.74 9.98 9.71 9.64
11.78 12.23 11.95 11.65 12.16 11.806
14.78 16.18 15.53 15.43 15.78 15.61
30.50 26.61 28.55 29.08 31.67 30.99
(OO0O.00 100.00 100.00 100.00 100.00 100.00
gi
0938
860
063
802
[855
limiting values Re
tri
pooled
16)
pooled
19
15)
2.19 2.20 .19
5-48 5.89 5-70
7.01 7.39 7.18
8.26 8.93 8.67
10.15 10.18 10.17
11.74 11.75 11.74
13.73 13.70 13.73
16.23 16.69 16.42
19.72 20.83 20.22
26.79 28.03 27.55
239.25 112.96 239.25
239.25 112.06 239.25
«76 2.43 2.43
8.14 8.52 8.30
10.22 11.24 10.82
12.68 14.02 13.29
14.65 16.91 15.30
17.42 20.17 18.50
20.91 23.79 21.73
24.45 29.65 26.08
30.92 37.20 33.48
46.71 53.61 46.65
525.07 333.92 525.07
525.07 333.02 525.07
©)
8
166
931
938
1860
963
892 1855
25>. JSSION
SCHNEIDER
I quite agree that, as you say, planning is necessary for under
developed countries. But: is it necessary because you want to have
a development in a very short time, or because you believe that a
market system would not function in India, or is it both?
MAHALANOBIS
[ think that the answer is that a market system has not been
tound sufficient even in the U.S.A. where the largest volume of
production is of free enterprise. The Anti-Trust Law was introduced
about 60 years ago. The Agricultural Adjustment Act came, I believe,
about 30 years ago. In the underdeveloped countries it is not
possible to rely on the market system alone for improving the conditions
of those who are now underfed. Also, disparities between
countries are increasing because of the faster progress of industrialisation
which has taken place and is continuing to take place in the
advanced countries. We can not wait for another hundred years
for economic development through the market system alone; we must
try to introduce other methods.
Also, there is increasing interlocking of international affairs with
domestic affairs. I was happy this time during my visit to ihe U.S.A.,
.,. Mahalanobis Il - pag. 3:
1134 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
to see that the desegregation movement is gathering strength. The
tragedy of the civil war had occurred hundred years ago; it has taken
hundred years for this to happen. I see one factor in the increasing
appreciation in America of the fact that the foreign policy of U.S.A.,
will collapse, so far as Africa or Asia is concerned, unless something
is done to remove the colour bar within the U.S.A. The coloured
people in the U.S.A., knowing what is happening in Africa, is also
no longer content to put up with segregation. I shall go a step
further; I am eager, intellectually and emotionally, that there should
be increasing relaxation of the tension between U.S.A. and U.S.S.R.
[ am happy about the Test Ban; I do hope that it will make further
progress. But I am aware that economic aid on both sides had
started in the spirit of cold war. In 1046 when the Atlantic Charter
was extended in the United Nations to cover the whole world, there
was hope of multi-lateral or, rather, international aid, for the underdeveloped
countries. This was turned into bi-lateral aid after the
Marshal Plan. Then the Cold War started; first the military phase,
and then economic aid much of which has been given as a kind of
political bribery, or in political rivalry. Some deeper thought has
to be given to this. I believe that world peace requires, and relaxation
of tension requires, rapid transformation of the underdeveloped
world. Economic cooperation would be of great help in bringing
this about. Such cooperation does not in any way jeopardise the miitary
security of the great powers. Also steel produced in India in
factories set up with the help of the West German, or the British or
‘he Russian Governments, find their way into the same piece of
machinery; all aid would be necessarily cooperative in physical fact.
I have spoken of the social and political aspects of the problem of
economic development because it is only through cooperative efforts
of all the advanced countries that a solution can be found. The use
of econometric methods must also suit the needs of the underdeveloped
countries.
‘151 Mahalanobis II - pag. 32
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONGOMETRIQUE ETC.
=
115.
KOOPMANS
We have covered a great deal of ground in the last ten minutes.
But my comments really go back to our earlier discussions on the use
of techniques and planning. I have the impression that Professor
MAHALANOBIS is using the term « sophisticated » with disapproval,
and « mathematical » with approval. I would like to express my
own belief that if you look at planning from the time at which it
starts off, the greatest return on economic analysis is at the early
« arithmetical » stage when small comparisons and computations of
the kind that have been illustrated are made; and that the time for
sophisticated analysis comes later whereas the payoff to it is probably
substantially smaller; I think I understand the sense in which the
terms « sophisticated » and « mathematical » are used in this way
and take that sense over for these comments. But I do not see this
as a reason to refrain from working on sophisticated methods — besides
having a certain direct payoff the sophisticated methods also
have the advantage of introducing rationality and objectivity in the
decision process, an advantage distinct from the direct payoff itself.
I also have a more detailed comment about the fertilizer calcula:
tion, which may illustrate this general comment. There is a comparison
here between different ways of supplying fertilizers. These
ways differ in three respects. One is timing of availability of the
fertilizer, another is rupee cost, and the third is dollar cost, or foreign
exchange cost, and the timing of these. Now I would submit that to
make the comparison one needs not only the dollar figures that are
stated here, one also needs a shadow price on foreign exchange, and
a (shadow) social rate of discount, in order to make the comparison
between these three alternatives. The shadow price of foreign
:xchange would itself have to be determined in a wider calculation,
which includes not only fertilizers but many other instances of decisions
where foreign sources and domestic sources are considered as
alternatives. In CHENERY’s work on Southern Italy (1) the shadow
nterest rate came out at something like 35°, per vear. This is to
CHENERY and KRETSCHMER, « Econometrica », 1950
05-99
-5 Mahalanobis 11 - pag. 3,
1136 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
be interpreted as the real rate of interest, on loans from outside
Southern Italy, that applied to the planning of the Southern Italian
economy as of that time, calculated on the assumption that the
production relations and planning objectives are adequately estimated
in the study.
MAHALANOBIS
To reply briefly, the fertilizer question has been studied by competent
foreign economists and it has been found that it would be a
good choice in India to build machinery for the manufacture of
fertilizers. My own position is simple, When a better solution is
available I shall adopt it. But I shall not wait to find the best solution.
In India, we can use 30 million tons of ammonium sulphate
every year, at a very low rate of dressing and without any fear of
decreasing return, We should start increasing the capacity to produce
fertilizers, and not wait until an optimum solution has been
astablished.
However, at a higher level, I completely share your views on the
value of sophisticated analysis. It is my firm belief that as the
structure of an economy becomes mature and stable, sophisticated
analysis would become more and more capable of being used effectively.
I can give one example; one of the dramatic successes of linear
programming was in oil refinery because organic chemistry has stable
structural relations. Sophisticated analysis would be increasingly
used in the advanced countries. Also, I should think that such
methods would be used more effectively in a planned economy like
U.S.S.R. because factors there are more under control. On the other
hand, sophisticated analysis, if it loses a sense of realism and is
primarily imitative, as is likely to happen in a country like India,
‘hen it is not only useless but a menace. .
I am not against sophisticated analysis in its proper setting. Even
at a low level of economic development, I am in favour of introducing
automatized production at the earliest opportunity and to the largest
"151 Mahalanobis II - pag. 24
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 113
extent possible in a country like India, and at the same time to gc
on with traditional methods of production. The danger lies in sophi
sticated analysis which are unsuited to the underdeveloped countries.
But we have to take the risk. In India we are going on with inpnt/
output analysis and all that kind of work in the hope that gradually
his approach will become useful. In applied science the important
point is to know what factors are relevant and what are their priocities.
Such understanding of relative priorities exists in the more
advanced countries but is difficult to acquire in an underdeveloped
environment. The methods have to be suited to the stage of development.
DORFMAN
The conclusion of Professor MAHALANOBIS’ fertilizer example was
‘hat the best solution for India would be to use the most indirect
method of production, that is, to import the machinery for setting
up a plant to manufacture machinery for fertilizer plants, This policy
would require deferring the output of the different fertilizers for
some ten or twelve years. If that be so, it would seem to me that
the cost of importing food during this long interval to meet the annual
population increase of some 5 million people should be counted as
part of the cost of this roundabout method of production, because
those food imports could be avoided by using some of the quicke:
methods, for example, by importing fertilizers or importing fertilizer
plants directly
\IAHALANOBIS
I apologize. I should have stated that for 15 years our people wil
10t be starving. In the meantime we are importing foodgrains fo:
current consumption as necessary; we are importing fertilizers; and
we are importing machinery to set up new fertilizer factories. Bui
15] Mahalanobis II - pag. 35
1138 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
we are also now establishing a factory to produce machinery which,
after four or five years, would enable us to set up a new ammonium
sulphate factory every year. I hope the position is clear. This policy
may not be suitable for a small country. I am not generalizing.
But in India as I have already mentioned we have 50 million hectares
of land which is sure of having enough water; we can use 25 or 30
million tons of ammonium sulphate per year at a very low rate of
dressing of 25 kilos per hectare; we may be able to use much bigger
quantities with advantage by increasing the dressing. To set up one
heavy machine building factory at a cost of foreign exchange of not
more than 100 million dollars once and for all is not taking a very
big risk.
[SARD
I just have a brief question. Do the materials that you have presented
have reference to a large population spread over a large area?
[ am sure you are aware of the thinking which emphasizes that industrialization
proceeds from relatively concentrated cores of activity.
To achieve major industrialization you must select a relatively small
number of points at which to start. I was wondering to what extent
this consideration enters into all the estimates and plans upon which
you report.
FISHER
In the discussion of Professor MAHALANOBIS’ paper Professor
DORFMAN raised a point which I would like to emphasize and to
which I do not think I reallv understand Professor MAHALANOBIS”
answer.
Professor DORFMAN’s point was that a proper analysis of the cost
of the program which imports machinery to make machinery to make
fertilizer must take into account the cost in foreign exchange of im-‘151
Mahalanobis II - pag. 26
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 1136
porting fertilizer and possibly of importing grain to feed the popu.
lation during the period before the constructed fertilizer plant comes
‘nto production. This is an appropriate cost of the program; it is
an opportunity cost and must be added to the cost of the program
n analyzing what constitutes an optimal policy. It is not clear that
Professor MAHALANOBIS has in fact done this.
MAHALANOBIS
I have only five minutes. I should again stress that it is not
logical to wait an indefinite time for a true optimum solution. As the
people of my country are hungry we have to import food stuff. We
have been doing this for some time. As a matter of fact we have
a long history of not producing a single kilo of fertilizer; we have
the experience of a million and a quarter of our countrymen dying
of famine in 1942-43. We are of course importing foodgrains and
fertilizers for current needs. But we have to take a long view and
we are also setting up fertilizer factories with imported machinery.
We have spent during the last fifteen years perhaps two thousand
million dollars to import foodgrains, fertilizers, and machinery to set
up fertilizer factories. Looking 15 years ahead, we think it is worth
spending one hundred million dollars to establish a factory to produce
machinery for new fertilizer factories. This I think is the real
argument.
ALLAIS
I have only two points. The first one has been already stressed
oy Prof. DorFMAN. 1 am not convinced by Prof. MAHALANOBIS’s
answer but I think we have not enough time to discuss this very
interesting problem. My second question relates to how the equation
of page 7 was derived? Did you assume that real national income is
proportional to real capital for this derivation or not?
+5] Mahalanobis II - pag. 37
1140 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2&
MAHALANOBIS
Please I do not quite understand. There is a footnote on page 7
which perhaps could give vou the right answer.
ALLAIS
Yes, but if you could give this explanation in a few words. This
point is quite important because for reasons I have already explained,
it is not possible to admit that real national income is proportional
to real capital.
MAHALANOBIS
The equation of the model? How we derive it? (1)
() The’ two-sector model is discussed in a paper entitled Some _observation
on the process of growth of national income in « Sankhva, the
Indian Tournal of ‘Statistics ». vol. 12. Parts 4. 1042
151 Mahalanobis IT - pag. 28
ECONOMETRIC ANALYSIS
AND
AGRICULTURAL AND DEVELOPMENT PLANS
D. GALE (JOHNSON
University of Chicago - Chicago - U.S.A.
In this paper I use the term agricultural development plan
as though it were synonymous with agricultural policy. I believe
that this is in keeping with the usage that prevails in the
world today. If one defines a plan as a statement of achievable
objectives, an indication of the means or resources that
will be available to achieve the objectives, and the institutional
arrangements that will be used to relate the means or resources
to the objectives, one can find few examples of agricultural
plans. Though the details have never been published, I am
reasonably confident that a plan approximately fulfilling the
conditions of the above definition existed for the virgin and
idle land program of the Soviet Union. In the same sense, it
can be said that plans have been evolved for the development
of new lands through irrigation or drainage in the United States
and in many other parts of the world.
But at national levels there have probably been no development
plans for agriculture that would satisfy the above deinition
* There is, of course, no denving that governments
() I do not believe that the agricultural components of the Soviet
Union's economic plans, either the five-year plans or the current seven
vear plan, meet the criteria of a plan specified in the text. In general. the
Johnson - pag.
1142 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
have attempted to intervene in a wide variety of ways to
affect agricultural development. In the United States there has
been systematic intervention in agricultural development
through policies for the settlement of the frontier, the establishment
of the land grant colleges and the associated experiment
stations and extension services. The British Corn Laws, which
have recently been reincarnated across the English Channel as
the Common Agricultural Policy, could be described as an
agricultural development plan in the loose sense in which the
term is often used today.
The past three decades has witnessed a significant change
in the scope and content of agricultural plans or policies. All
of the major industrial nations have evolved numerous measures
and programs in an effort to achieve a variety of objectives
in agriculture and for the farm population. Many of the less
developed countries have enunciated far reaching agricultural
development plans. One thing that is clear from the events of
the last three decades, at least to me, is that there is no magic
that follows from a development plan or policy. Order and
progress have not been created out of chaos. It cannot be said
that today the agricultural problems confronting the world or
any large part of it are any nearer solution than was true three
decades ago. In fact, I believe that in most of the industrial
countries the magnitude of the very difficult adjustment problems
now facing agriculture is to a considerable degree the
consequence of the agricultural plans and policies that have
been followed.
The preliminary outline of the Study Week included the
following sentence: « The development of economic theory and
recent experience of various economic svstems have bv now
objectives have not been achievable given the quantities of resources implied
in the plans. In addition, past performance has indicated that goals for
certain kev input aguantities such as fertilizers would not be achieved.
161 Johnson - pag. »
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. ]143
given sufficient proof that modern economies cannot be left
to the free play of market forces, but must be controlled and
directed both to anticipate undesirable economic situations and
repercussions and to achieve certain of the community’s objectives.
» While I hope that I will learn what developments in
aconomic theory and what recent experiences of various economic
systems result in so striking a conclusion, for the moment
I would only like to suggest the following paraphrase:
« The application of economic analysis and recent experiences
of various economic systems have by now given sufficient proof
that modern or underdeveloped agricultures can not be subjected
to agricultural development plans that impede necessary
adjustments, result in exploitation of the farm population, fail
to recognize the potential high returns to certain investments
in agriculture, or restrict the potential gains from international
specialization in production, but development plans must be
consistent, efficient in the use of scarce resources, and not
result in undesirable economic situations. »
In this paper I shall try to do four things. First, I shall
present a simplified skeleton of some of the interrelationships
‘hat exist between agriculture and the rest of the economy as
sconomic development occurs. Second, some of the major differences,
as well as the similarities, in the agricultural problems
of selected important areas of the world will be indicated
Third, a review of plans or projections that have been made
n two countries — the Soviet Union and the United States —
will be presented to indicate some of the major sources of errors
in projections and goals. Finally, I will emphasize some of
the major analytical and statistical problems that arise in ma-«ing
projections or creating development plans.
-, Johnson - pag. 3
144 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
[. AGRICULTURE AND ECONOMIC DEVELOPMENT (!)
A. General Framework
One of the important interrelationships between agriculture
and the rest of the economy during the process of economic
development has been called the law of declining importance
of agriculture. This « law » has both very strong empirical
support and a firm analytical base (?). The major industrial
countries of the world, with the exception of the United Kingdom,
had at the beginning of the 19th century approximately
the same percentage of the labor force engaged in agriculture
as is now true in the low income countries of the world. Except
for the Soviet Union, all of the major industrial nations now
have 20%or less of their labor force engaged in agriculture,
with the United Kingdom at about 5% and the United States
at 8%.
The theoretical analysis that explains the empirical decline
in the relative importance of agriculture as a source of employment
as economic growth occurs is relatively simple. The
analysis depends upon well. substantiated effects of per capita
income changes upon the allocation of consumer expenditures
(!) This section and certain other parts of my paper are based in part
upon my paper, The Role of Agriculture in Economic Development,
which was presented at the 1963 Resources for the Future Forum on the
Role of Natural Resources in Economic Development. in Washington. D.C.
on January 28 and 29, 1963.
(*) The empirical support can be found in Corin CLARK, The Conditions
of Economic Progress, 3rd ed., 1957, Ch. 10 and Simon KUzNETS, Quantitative
Aspects of the Economic Growth of Nations: II. Industrial Distribution
of National Product and Labor Force, « Economic Development and
Cultural Change », supplement to vol V. No. 4, July. 1057
16] Johnson - pag. 4
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1145
and upon changes in the real factor costs of agricultural products
that are required if economic growth is to occur (1).
When the level of per capita income exceeds some very low
level, perhaps $50, the income elasticity of demand for food
is less than unity. Since the income elasticity of demand for
all goods and services is unity, the income elasticity of demand
for nonfarm goods and services exceeds unity. An increase in
real per capita income, ignoring for the moment the source
of the increase, will result in a larger increase in the demand
for nonfarm than for farm products. Thus the relative output
of farm products will decline as a share of the total output.
However, the assumptions made are not sufficient to show
that the share of the labor force engaged in agriculture will
lecline. If the growth in the net marginal product of labor
engaged in nonagriculture is enough greater than the increase
in the net marginal product of labor engaged in agriculture,
the proportion of the labor force engaged in agriculture could
remain unchanged. As a rough approximation, if there is no
change in population, if the ratio of the change in the marginal
product of labor in agriculture to the change in marginal
product of nonfarm labor is equal to the ratio of the income
elasticies of demand for food and nonfood products, the sectorial
distribution of employment will remain unchanged. However,
this rough approximation will hold only if there are bar
ers to the movement of labor from agriculture into nonagricultural
pursuits. The larger increase in the marginal product
of labor in the nonagricultural sector than in the farm sector
means that the return to labor in agriculture will fall in a reative
sense. Thus labor will move from agriculture into the
nonagricultural sector in response to a change in the earnings
(") HERBERT SIMON, Effects of Increased Productivity upon the Rati
of Urban to Rural Population. « Econometrica » XV No ‘Januarv
1947), 31-42.
F.H. GRUEN, Agriculture and Technical
Économics », XLIII, No. + (November 1061)
io] Johnson - pag.
_ 4
1146 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA - 2&
differential and the percentage of the labor force engaged in
agriculture will decline. Obviously, if the relative increase
in the marginal product in thé nonfarm sector is large enough,
a rise in the relative price of farm products could offset the
decline in the earnings to farm labor and maintain the same
distribution of employment. However, this consequence will
almost certainly place a severe restraint upon the rate of growth
of per capita income. It means that virtually all of the increased
productivity must come in a sector of the economy that
employs a very small fraction of total labor.
If population increase is introduced into the picture, the
relative growth of demand will not be a direct function of
relative income elasticities. The relative growth will depend
upon the annual population growth plus the change in per
capita income weighted by the income elasticities. But the
growth "of population means that a significant part of the
increase in agricultural output is required to maintain per capita
consumption of food. Unless additional land of quality
equal to that already under cultivation is available, a change
in methods of production will bé required to offset the effect
of diminishing returns as the amount of labor applied to each
unit of land increases. Unless such a change in methods of
production occurs, the real cost of food will increase and resources
that could be used to increase nonfarm output will be
shifted to agriculture and the rate of growth of total output
will decline.
Significant rates of economic growth can occur only if there
are increases in productivity in a sector that employs the major
fraction of all of the labor of an economy. Fortunately, increases
in resource productivity in agriculture are possible and
the same set of forces that result in higher productivity in
nonfarm sectors appears to have been roughly as important in
agriculture as in the rest of the economy.
The factors that make it possible to increase agricultural
output per capita in the face of increased population. which
161 Johnson - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1147
will be discussed in some detail below, are the factors that contribute
to the decline in the relative importance of agriculture.
These factors resvlt in an increase in output per unit of input
in agriculture; combined with the lower income elasticity of
demand, subject to the conditions noted above, relative labor
employment in agriculture will decline.
As real per capita income increases, the rate of the transfer
of labour out of agriculture required to maintain an equilibrium
in returns to farm and nonfarm labor will almost certainly
increase. One reason is that the disparity between the income
slasticities of demand for farm and nonfarm goods increases.
À second reason is that the change in ratio of output per unit
of input in agriculture is more likely to approach the change
in the ratio in the nonfarm sector as agriculture becomes more
commercialized and the quantity and variety of producer goods
available to agriculture increase.
There is no reason to believe that there are any forces to
halt the decline in the share of labor employed in agriculture.
There is only one industrial country, the United Kingdom, for
which there is any evidence that the percentage of total employment
in agriculture has been approximately stabilized for any
period of time. Since the mid-thirties approximately 59% of
‘he total labor force has been engaged in agriculture. But
during that period, the government has followed a conscious
policy of inducing an increase in agricultural output. It is
almost certain that in the absence of that policy relative farm
smplovment would have declined.
B. Economic Development and the Demand for Agricultural
Products
It is unfortunate but apparently true that the growth of
demand for agricultural products is likely to be greater during
the early stages of economic development than during the later
5]
Johnson - pag.
1148 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
stages. Two factors that affect the demand for agricultural
products are responsible for this — population growth and the
income elasticity of demand. One of the well known effects
of the early stage of economic growth is a decline in mortality.
The improvement of sanitation, the availability of medicines
and medical facilities, the improvement of the quantity
and quality of the food supply, the reduction in famines due
to improved transportation, and the control of certain endemic
diseases have a significant and short run effect upon death rates,
especially of infants and children. There is almost nothing that
occurs during the early stages of economic development that
has an effect on fertility. Thus there is a very strong probability
that the rate of population growth will be higher during
the early stages than in the later stages of economic development
and that at a given time higher for low income countries
than for high income countries (1).
It has been noted above that the income elasticity of demand
for food is higher at low than at high income levels. Thus
if a high and low income country have the same growth rate
of per capita income, the income effect on demand will be substantially
greater for the latter than for the former. The two
effects are illustrated by Table 1, which presents certain projec-‘ions
made by FAO for a period of roughly a decade.
The estimates of population increase are relevant to the
comments made above. The three low income areas have
(!) This statement is based on the assumption that the growth in food
output is enough to sustain the growth in population. If the conditions
implied by a Malthusian model of economic growth apply, the growth of
population will depend upon the growth of food output.
The statements in the text about the relationship between population
growth and income levels are admittedly very crude and approximate generalizations.
Obviously there are many factors that influence population
growth. The difference in population growth rates in Western Europe
and the United States can not be explained by the difference in the level
of per capita income. An interesting article by IRMA ADELMAN (An Econometric
Analysis of Population Growth, « American Economic Review »,
vol. LIII, No. 3, June 1063, 314-30) provides some support for the view
expressed in the text
16] Johnson - pag. 8
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
: ]4C
projected annual rates of population growth ranging from 2.3
to 2.7%. The two high income areas have lower rates, namely
0.7 and 1.8%. The difference in estimated income elasticities
are of the general order that would be expected, except that
the elasticity for the EEC appears to be somewhat high, but is
apparently explained by the high income elasticity of demand
for meat.
TABLE I
Projected Annual Growth of the Potential Demand for Food Se
lected Areas, 1057-50 through 1069-71 ‘1)
Basic assumptions
Population (™
GNP/capita
Income elastici
Potential increase
Total demand .
Per capita demanc
GNP/capita (8) . . $165
(1957-59)
Asia and
Far Fast!
Near East
and
Africa(3
$260
Latin
America!
E
Nort
America(s
s
A
Food and Agriculture Organization, Agricultural Commodities
tons for 1970, p. A-2.
<xcludes Japan and Mainland China.
Excludes South Africa.
Excludes Argentina and Uruguay; includes Mexico and Central America
European Economic Community.
Canada and the United States.
Per cent increase per year; compound rate. Low income growth assump
cion used.
*) Converted into U.S. $ at 1955 prices
_ The increases in total demand for food are projected at
annual rates ranging from 3.4 to 3.79, for the low income
countries and 1.9 to 2.19, for the high income areas. Thus
the expected growth of demand for food in the low income
10] Johnson - pag. 9
1150 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
areas is approximately #59, greater than for the industrial
areas. In the low income areas, despite the high income elasticities
and modest increases in per capita income, most of the
increase in demand results from population growth. If Latin
America were to have a 39, annual increase in GNP per capita,
the demand growth for food would be in excess of 49,
annually. This would require a doubling of food output in
about 18 vears.
C. Economic Development and the Supply of Agricultural
Products
In the discussion of the general frame-work of the relationship
between agriculture and economic development the discussion
of changes in supply were largely in terms of output changes
that had to occur if economic development were to occur.
[t is time now to turn to a discussion of the major factors that
lead to output increases. The following classification includes
nothing that is new, but is useful for organizing our discussion.
Increased agricultural output occurs as a result of one
or more of the following: 1) increased use of inputs; 2) improving
the quality of inputs; 3) increased knowledge or a
change in the production function, and 4) a change in incentives
for farm operators and their families.
If all of the increased output must come as a result of
increased labor and land, it is unlikely that agriculture will
be able to make any significant contribution to economic development
(1). If this is the case. the growth of the agricul-(')
This observation may well have been correct for the United States
in the period before 1860. Given the large quantity of land that was available
for settlement, it was possible to expand output of farm products
without changing the relative combination of land to labor. Admittedly
scanty evidence for the period from 1820 through 1860 indicates the follow-161
Johnson - pag. 10
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 115]
tural population in a low income or underdeveloped economy
will have to occur at approximately the same rate as the growth
of total population unless per capita food consumption is to
decline. Labor cannot be released to other sectors of the economy
for their development and real factor costs of food would
not decrease. In fact, unless additional land can be brought
ander cultivation or land developed through irrigation or reclamation
at the same rate as the population grows, average output
per worker in agriculture may decline and real factor costs of
food may increase.
There are circumstances in which an unchanged production
function is consistent with increased output and lower marginal
costs of production. If the marginal product of labor is substantially
higher in certain nonfarm pursuits than on the farm,
the availability of certain manufactured inputs that may be
ased in agriculture may permit an increase in output per unit
of farm labor and a transfer of labor from agriculture to the
nonagricultural sector. This is a possible explanation of the
impact of a new and cheaper source of fertilizer (the partial
replacement of manure, fish, fish meal, and cottonseed meal
by chemical fertilizers in Japan), the replacement of feed by
petroleum products to produce power, or the transfer of certain
functions performed on farms to firms that can specialize to
a greater extent (churning butter, tanning hides. retting flax,
or butchering livestock).
There is evidence that for the half century from 1880 to
[930 in the United States the ratio of total output to input did
not change significantly, while there was an increase in the
output per unit of labor and per unit of land (real estate)
ng: Farm employment increased from 2.2 to 6.6 million; farm employment
1s a percentage of total population changed only from 21.5 to 19.79.
Estimated gross output per farm worker increased by only 11% in the
to-year period. There are some indications, admittedly crude, that national
der capita income increased onlv moderately during the four decades
>|
Johnson - pag.
I)
1152 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
The increases in the two average products were apparently
due to increased purchases of nonfarm products, which were
apparently cheaper than the farm inputs they replaced. While
farm labor increased absolutely over the period, farm labor
declined as a percentage of the total labor force from 49 to 21%.
The United States situation as of 1880 is one that is not
likely to be duplicated in any of the current low income areas
of the world. U.S. per capita income was already at a relatively
high level (about $400 in 1929 prices) and the income
elasticity of demand for farm products was probably 0.5 or
lower. About a fifth of farm output was exported, thus domestic
consumption could increase more than domestic output.
TABLE 2
Indexes of Output, Inputs, and Productivity, United States Agri
culture, 1880-1020 (!)
(1847-49 = 100)
Indexes
Farm Output . .
Production Inputs .
Farm Labor .
Farm real estate
All other .
Productivity
Output/labor .
Output/real estate
1880
7
.
+
1890
43
€
27
62
1900
56
72
Ge
1910
61
22
1920 1930
70 72
23 97
Ia% 137
v 96
53 62
75 74
45 48 52
6h 72, 75
M) Source: RarrH A. Loomis and GLEN T. BarToN, Productivity of Agri
culture, United States, 1870-1058. USDA. Tech. Bul. No. 1238 (April.
[961), pp. 57, 38. and 6o-6T
But when per capita incomes are low and the income elasticity
of demand for food is almost unity, agricultural employment
must increase absolutely and remain a relatively stable
percentage of total employment unless technological change
occurs.
16] Johnson - pag. 12
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 1153
It is quite likely that improved quality of inputs and a
change in the production function (technological change)
usually occur together. In an underdeveloped economy the
most important input — as measured by the share of the total
product paid to it — is labor. Fortunately, labor is an input
whose quality can be changed by well known procedures. Unfortunately,
these procedures involve the expenditure of significant
amounts of scarce resources. But the history of Japan
from 1870 through 1gr0 indicates that a poor country, with
very limited natural resources, can organize its resources to
provide for nearly universal education for the farm population,
for agricultural research, and for education of the adult farm
population in new and improved methods of production.
There is now rather general agreement among competent
scientists that the technological possibilities for increasing food
output in the low income areas are very substantial. Many
of the low income areas — Africa, Latin America, and Southeast
Asia — have room for substantial expansion of the cultivated
areas. But some of the most densely populated areas
— India and China, for example — are not in this fortunate
position. Again reference may be had to the Japanese experience.
Japan at the time substantial economic progress began
had little possibility of increasing the sown area. In the half
century from 1881-90 to 1931-40 the cultivated area of six
major crops in Japan increased by only 18% (}). However,
vields increased by 66% and production by 95%.
The excellent Report on India’s Food Crisis and Steps to
Meet It prepared by the Ford Foundation team clearly indicates
that it is technically possible to substantially increase food
production in India (3). Once of the major conclusions of the
(') Bruce F. Jounson, Agricultural Development and Economic Transformation:
Japan, Taiwan and Denmark ,Mimeo., 1960, p. 28.
(?) Published by the Government of India, Ministry of Food and Agriculture
and Ministry of Community Development and Cooperation, April
1959. The chairman of the team was Mr. Sherman Johnson of the United
States Department of Agriculture
116] Johnson - pag. 15%
1154 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 72¢
study should be noted, however. In the opinion of the team,
major increases in output can be achieved only as a result of
the simultaneous introduction of a variety of improvements (3).
For example, a substantially higher yield for a crop would
require new varieties, more fertilizer, heavier rates of seeding
or planting, measures for controlling insects and diseases, and
changed methods of cultivation. Any one of these changes
may be largely ineffective or may even have a negative in-Aucnce
in specific cases.
That it is technologically possible to increase food output is,
of course, only a necessary condition for a solution of the
agricultural problems of the low income areas. Much must be
done before the technological possibilities can be translated into
food available for an increasing population with a rising per
capita level of consumption. One of the major problems in
making the projections necessary for a development plan is
that of estimating when and to what degree the political and
economic circumstances will result in the realization of the
technological possibilities.
The last of the four factors that influence the level of agricultural
output is the nature of the incentives available to farm
families. The incentive structure has two major elements —
the terms of trade between agriculture and the rest of the economy
and the relationship between effort and reward as affected
by institutional factors as codified in the tenure system and the
structure of the farm organization, such as individual proprietorship.
collective or cooperative farms.
In a growing economy, starting from a low income level,
savings in or from agriculture are an important source of investment
funds for the entire economv. The extent, and the method
, @) Ibid. p. 18. « À few improved practices can be effective if adopted
singly, but the full benefit from most improvements can be obtained only
if they are adopted in combinations snitable for specific soil and climatic
conditions. »
“16] Johnson - pag. 14
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE Etc. 1155
of extracting such savings from agriculture, through taxes,
cents, low prices or forced deliveries at nominal prices, is
clearly a factor in the incentives available to farm workers. In
my opinion, the relatively poor output performance of agriculture
in the Soviet Union from 1928 through 1953 was due
lo ineffective and insufficient incentives imposed by a combination
of the price structure, forced deliveries and institutional
arrangements. On the other hand, during the late nineteenth
century Japan apparently transferred significant amounts of
savings from agriculture to the rest of the economy and still
achieved rapid modernization of agriculture and output growth.
Thus it is probable that not only the total amount of savings
extracted from agriculture, but the method of extraction is im
portant in determining output responses.
A part of the support for land reform in underdeveloped
areas is the hope that reform will lead to an improvement in
the incentive structure and thus to increased investment in
agriculture and greater interest by the farm operator in increasing
output. Some land reforms seem to have achieved
this result (the short lived reform in the Soviet Union after
World War I and the post World War II reforms in Japan and
Taiwan), but many others appear to have failed to have any
significant effect.
The increase in the ratio of output to input, which is a
measure of the effect of the change in quality of inputs as well
as of technological change, makes it possible to transfer labor
from the farm to the nonfarm sector. Rapid economic growth
requires such a transfer. However, rapid economic growth
does not require an absolute decline in farm population or farm
employment. If farm employment is to be stable in an absolute
sense, the annual change in the ratio of output to input in
agriculture in low income countries must be a very large
change if population and income growth per capita are of the
order of 2% per annum. In this situation the increase in
16] Johnson - pag. 15
156 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
demand for farm products (assuming little international trade)
will be about 3.5% per annum. Even if half of the increased
output is due to increased quantities of nonlabor inputs, the
ratio of output to input must increase by about 1.5%, each year.
This implies a doubling of the output-input ratio in about 45
years; in the last 50 years the output-input ratio has increased
by about 75% in the United States.
I should hasten to add that it is not essential for the absolute
level of farm employment to remain stable during the early
period of industrialization. The usual pattern has been for the
farm labor force to increase absolutely for several decades
after per capita incomes have started to increase. In fact, if
an economy starts from a position of, say, 80% of its labor
force engaged in agriculture and if total population is increasing
by 29%, per annum, it would be nearly impossible for nonfarm
employment to absorb all of the increase in labor force.
Under these assumptions, the nonfarm labor force would have
to grow at a 10% annual rate or double about every 7 years (!).
In this section I have tried to indicate briefly some of the
interrelationships between the supply of agricultural products
and the process of economic development. It has been indicated
that an improvement in the output-input ratio is required
before rising per capita incomes can be achieved in a low
income area. It has also been indicated that if population
growth is moderate or high and the income elasticity of demand
is high that the increase in the output-input ratio must
be a significant one if agriculture is not to act as a restraint
on economic growth. It also was indicated that while farm
employment must decline as a share of total employment, it
may increase absolutely during the early period of industrialization.
(') Such a growth cannot be said to be impossible; between 1926 and
1934 the annual increase in nonfarm emolovment was about 09% in the
Soviet Union
16] Johnson - pag. 16
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. ,
3 S51
I. MAJOR AGRICULTURAL PROBLEMS
Governments have various types of concern about the agriculture
of their respective nations. In many of the low income
countries there is a concern about the level of agricultural
output relative to what would be required to achieve satisfactory
nutritional levels for the present population and to meet the
demands that will come from a growing population. In most
of the high income countries this concern arises with respect
to the level of returns to resources engaged in agriculture. In
the Soviet Union and other bloc countries in Eastern Europe
the basic agricultural problem is to achieve a rapid increase
in agricultural output, especially marketed output, to provide
a growing urban population with a diet that compares more
favorably with that available in the Western countries and
to do this without requiring a significant diversion from the
ndustrial and military sectors.
Table 3, prepared by the U.S. Department of Agriculture,
summarizes the available data on changes in total and per capita
production of agricultural products for the past quarter
century. In general, the Southern Area includes most of the
underdeveloped areas of the world, while the Northern Area
includes the medium and high income countries. With respect
to total production, the increases are of the same general order
of magnitude. In fact, if adjustment were made for the very
unsatisfactory weather conditions in North America during
[935-39 and the very good weather conditions from 1958
through 1960 in the same area, total output probably increased
more in the underdeveloped areas. It is true, of course, that
within each of the two areas and within each of the regions
there were significant differences from country to country in
the growth of output. However, the point that I wish to make
here is that the types of agricultural problems faced bv the
10
Johnson - pag. 17
pt
TABLE 3
Indices of world agricultural production: Total and per capita, by region, average 1935-39 and annual
1958-59 fo 1060-61 (1) (Average 1952-53 to 1954-55 = 100)
Total production
Per capita production
Region
or
country
Average
Average annual Average annual
percent change percent change
PRAT vATRE A pee meer
1935-39 1952-54 Verage 1935-39 1952-54
to to to to
11960-61 1960-61 | 1935-39 1958-59 1959-60(2) 1960-61(* ‘1960-61 1960-61
1935-39 1958-59 1959-60(2) 1960 61(#)
Southern Area Percent Percent
Latin America . . . 72 121 123 124 3.I 3.4 103 107 106 104 0.04 0.6
Africa and West Asia 77 117 118 121 25 3.0 I00 I06 105 I05 0.2 0.7
Far East, less Japan
4 . . . . 8 117 119 IIQ IS 27 III I0o4 I06 105 —0.3 0.7
Communist Asia . . 096 120 IIS 1II7 IO 24 IIZ I0g I02 102 —0O.4 0.3
Total. . . . 8 118 118 120 17 2.0 106 106 105 104 —0.00 0.6
Northern Area
Western Europe . . 81 110 112 115 18 2.1 gz 106
Eastern Europe (°) . 108 132 130 I31 9 44 106 123
United States and Canada
. . . . .
Japan . . . . .
Australia and New
Zealand . . . . 74 120 ar
Total. . . . © 7118 In
World total . Be 7118 ro
34 I00 107 106 107 0.3 1.0
_- 1.a 3.I 93 III 110 iIr oO. 1.6
21 I.8 30 101 108 10ÿŸ 107 0.3 1.0
—
—
19)
ox:
(") Value of production at constant prices. Revised. Crops included in the index are harvested mainly between
July 1 of the first year shown and June of the following year. For a few crops and most livestook production,
estimates are for the calendar year of the year shown. - (3) Preliminary. - (3) Estimated. - (4) Includes Pacific Islands
(®) Includes Soviet Union.
Source: U. S. Department of Agriculture, The Wnrld Fand Rudoot Tnh> and tnhh TForaign Agrienltnral Tne
namic Renort Na a Revieed Tanitarv vols nN HE
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
1159
three major areas do not seem to be a function of differences
in output growth, but in the differences in demand changes
compared to changes in output. On a per capita basis, the
Southern Area has apparently had a much slower output growth
than the rest of the world.
The remarkable similarity in the growth of output in the
various areas of the world means that the underlying causes
of their quite different problems must be somewhere else. In
the case of the low income countries, the source of the problem
is not difficult to detect. If there is rapid economic growth,
the demand for farm products will increase at an annual rate
of 3.5 to 4%. This increase may be compared to an estimated
annual growth rate of output of about 1.5 to 3.0% for the
past quarter century and 2.7 to 3.49% for the past decade.
The nature of the problem in Eastern Europe is similar to
areas with rapidly growing demand for food, though there is
an important distinguishing element. In Eastern Europe the
growth in the demand for food will be substantially greater
‘han in Western Europe or the United States. While the growth
of population may be expected to be somewhat less than in
the United States, it will probably be much more rapid than in
Western Europe (!). The income elasticity of demand for
food in Eastern Europe — based on scanty information from
the Soviet Union — is relatively high; the income elasticity is
probably about 0.75 and almost certainly greater than o.s5.
Thus rising consumer incomes, perhaps at an annual per capita
rate of about 3%, combined with an annual population growth
rate of 1.4% would result in a growth of demand for food of
3 to 3.5% annually.
But the growth in demand is here estimated on the assump-‘on
that there now exists an equilibrium between the demand
(") U.S. Department of Agriculture, The World Food Budget, 1962 and
1966, Foreign Agricultural Economic Report No. 4, Revised, Tanuary 1062,
3. 7%
v| Johnson - pag. 19
1160 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and supply for food. In the Soviet Union, for most food products
there is excess demand at the fixed prices in the state
store. Generally the prices in the free market are substantially
higher than those in the state stores, for several important products
50 to 100% more. The increase in the retail price of
meat by 30% and of butter by 259% on June 1, 1962 without
eliminating all of the excess demand (reports of queues and bare
shelves persist) indicates that some of the food markets are
substantially out of equilibrium. While it is true that state food
store prices are generally substantially below prices in the free
market, the state food store prices are not low by comparison
with Western Europe and the United States nor are they low
compared to the income of the population. Approximately
50% of consumer income is expended for food, of which about
90% is purchased in state stores at controlled prices.
In Western Europe and the United States the « agricultural
problem », as viewed by most governments, is to maintain a
satisfactory level of farm incomes. Generally speaking, the
growth of demand is less than the potential increase in production.
Rapid adoption of new methods of production by
farmers means that the demand for farm labor at current returns
to labor is declining absolutely. When one adds to this
the fact that alternative earnings in the rest of the economy
are increasing, substantial reductions in labor inputs are required
in most of the countries. While some of the countries
have had programs designed to facilitate the transfer of labor
out of agriculture, most have tried to solve the problem of
declining demand for farm labor by various forms of subsidies
and price maintenance or increasing measures.
The agricultural programs of the higher income countries
have certain direct implications to the agricultural development
plans or policies of the low income countries. Most of the low
income countries depend upon a limited number of agricultural
products for the bulk of their export earnings. If the major
industrial countries follow policies that result in expanding
16] Johnson - pag. 20
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 116)
agricultural output, the export earnings of the low income countries
are adversely affected. It is sometimes argued that the
agricultural policies and the related import and export policies
of the industrial nations have little effect on tropical products
and, since most underdeveloped areas are tropical, very little
effect on the exports of underdeveloped areas. It is true that
most of the restrictive measures undertaken by industrial countries
affect cereals, which are grown mainly in temperate zones
and in developed nations. But sugar is heavily protected in
most industrial countries and cane sugar is a tropical product.
And the grain policies of industrial countries do have an impact
upon the market for rice, which is primarily a tropical
product. Underdeveloped areas are a major source of vegetable
fats and oils, which are subject to import duties in
some industrial countries. Sugar, rice, fats and oils are major
sources of foreign exchange for many underdeveloped areas;
the ability of such countries to import capital equipment and
other requirements for economic growth are affected directly
by the agricultural and trade policies of the industrial nations.
In summary and stated in broad terms, development plans
or policies for agriculture are primarily concerned with achieving
a rate of growth of output that is approximately the same
as the rate of growth of demand. In the underdeveloped areas
of the world and in Eastern Europe, the plans or policies must
concentrate upon measures that will increase the agricultural
output growth rate. Obviously this must be achieved with an
expenditure of resources that permits attaining other important
objectives. In the United States the policies should create a
situation in which output growth is no greater than the growth
of demand and at the same time achieve a level of return to
resources engaged in agriculture approximately equal to the
returns received by comparable resources engaged elsewhere.
In Western Europe the substantial imports of food give the
region the choice of expanding food output to suppl te re
‘16, Johnson - pag. 21
1162 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
gional demand or of continuing to import substantial quantities
of food. The first alternative appears to imply a substantial
increase in the cost of food, while the second alternative has
important implications to a number of exporters of farm products.
The first alternative implies a protectionist policy while
the second alternative would permit the region to realize significant
gains from international trade and specialization.
ITI. A REVIEW OF SOME AGRICULTURAL PROJECTIONS
A governmental plan or policy depends upon projections
of future events. Plans by private firms and individuals obviously
require projections of the same general kind, though
the degree of detail required may be substantially less. There
are, of course, significant differences in some of the effects of
errors in projections or forecasts when made by governments
and private individuals. For one thing, errors made by private
individuals may be offsetting. For another thing, errors made
by private individuals may bring into play forces to correct
the error, such as a decline or increase in market price, while
a government price policy, subject to rather more slowly functioning
political processes, may compound the consequences of
projection errors. This will happen (and has) if a farm product
price is established at a high level which results in attracting
additional resources into the production of the product and
the bidding up of the price of certain resources. In order to
avoid economic distress to resources engaged in the production
of the product, prices may not only be maintained at the previous
level but may be increased in order to provide a satisfactory
income to the overexpanded sector of the economy.
In this part of my paper I shall review agricultural projections
for the United States and agricultural goals for the Soviet
16] Johnson - pag. 22
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETc. 1163
Union (!). For the United States we have projections made
for both consumption and production. Only one aspect of the
production projections will be reviewed, namely crop yields.
It is assumed that the goals for the Soviet Union represent
projections of attainable output. For both the United States
and the Soviet Union the projections will be related to actual
experience.
A. Projections for the United State
I. Consumption of agricultural produc.
[ shall describe and comment upon only two of the numerous
projections of consumption of agricultural projects. One
projection was made in 1948 by the U.S. Department of Agriculture
and was for the average of the period 1955-65. I have
assumed that the data for the year 1960 represent the average
for that period. The other projection was made by REx F. DALY
of the U.S. Department of Agriculture and was published in
1956 and the projections were for 1960; this study also included
projections for 1975 but these are not considered here.
Basically the two studies projected the consumption of agricultural
products on the basis of a model in which population
and per capita income were projected for the specified period.
Then the consumption of agricultural commodities was estimated
from studies of income elasticities of demand and other
information concerning trends in relative demands. In order
to eliminate the effects of errors in population projections, all
data on consumption are here presented on a per capita basis.
In Table 4 I present certain of the underlying data on po-()
T wish to acknowledge my obligation to Professor JaMEs BONNEN
of Michigan State University who brought together much of the material
on the U.S. projections
3
Johnson - pag. 23
>
~
y
TABLE 4
Population, income, output, employment and per capita food consumption projects for 1960
Item _
Population . .
Labor Force . .
Employment
Unemployment
GNP in 1947 prices
GNP in 1953 prices
GNP in 1960 prices
GNP per capita, 1960 prices . . .
Personal Disposable Income (1960 prices) .
Per capita (1960 prices) . . . .
Per capita food consumption . 2.
Unit
or Base
Projections for 1960
1048 Studv(m 1956 Study(®)
Actual
1960
Million
Million
Million
Million
Billion §
Billion $
Billion §
$
Billion
178.6
72.0
68.5
3.5
179.9
72.8
68.9
3.0
-
430
SOI
- 805
1)
ye
Lo"
503
2,795
354
1,969
113.6
1035-30 = 100
T273 .
NY Long-Range Agricultural Policy: A Study of Selected Trends and Factors Relation to the Long-Range Prospects
for Agriculture, Committee on Agriculture, U. S. House of Representatives, 8oth Congress, 2nd Session, Washington,
D.C., March, 1948, pp. 28 and 34. Projections were for average of 1955-65 and for high employment conditions.
Rex F. Dary, The Long-Run Demand for Farm Products, « Agricultural Economics Research ». Vol. VIII
No. 3, July 1956, pp. 7.
——et
—
oO
A
N
x
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 1165
pulation, employment, GNP, and per capita income for the
projections with comparisons with actual data for 1960.
The 1948 study underestimated 1960 population by 22 million
or 12%. The relative error in the estimate of the labor
force was much smaller, though still very substantial since
all persons in the labor force in 1960 were alive in 1047 (the
last year of available data for the 1948 study). Gross national
product was underestimated by about 20%, though GNP per
capita was in error by only 89%.
The 1956 study projections for 1960 were remarkably accurate
for the underlying data on population, labor force, total
and per capita gross national product. This study covered a
period of approximately seven years since few, if any, data
were used for any year after 1953.
Projections of per capita consumption of a number of agricultural
commodities are compared with the actual consumption
in 1960 in Table 5. The 1948 study had quite accurate projections
for total meats, potatoes, and sweetpotatoes, and fats and
oils. The difference between projection and actual was substantial
for lard, poultry, eggs, total milk, wheat, and cotton. The
differences between the 1956 study projections and actuals were
not as great as in the former study, but were still substantial
for certain individual commodities — beef, veal, turkeys, eggs,
milk, and cotton.
Table 6 presents comparisons between projections and
actual yields. The 1948 study projects yields for 1965; the
1956 study was only for demand. In Table 6 I have included
projections from several other studies in addition to the 1948
study. These include projections for 1950 which were published
in 1945, a U.S. Department of Agriculture projection
for 1965 published in 1961, the Paley Commission projections
for 1975, and U.S. Department of Agriculture projections for
1975. Actual yields are given in the table for two three-year
periods - 1954-56 and 1960-62.
The ~~ vield projections from the 1948 study were
[15] Johnson - pag. 25
i166 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
TABLE 5
Projected and actual per capita food consumbtion, United States,
1060
Projections for 1960
1948 Study) 1956 Study(®) Actual
(pounds) 1960
Meats
Beef
Veal .
Lamb and mutton
Pork (excl. lard)
Total
155-160
TP
[Lard
Poultry
Chicken .
Turkey .
Total
22 (3
Eggs (number) . .
Total Milk Equivalent .
Fluid milk and cream . .
Condensed and evaporated
360
885
403
Cheese . . . .
Butter
Potatoes and Sweetpotatoes . . 105-10
Fats and Oils
74-0
9.5
4-5
68.0
156
85.0
6.2
4.8
65.2
161.4
7.9
24.0 28.2
4.5 6.2
28.5 34.4
380
698
205
324
653
351
7.5
8.4
7-5
108.2
_7
44.7
45.4
(') See footnote 1, Table 4, p. 32.
?) See fotnote 2, Table 4, pp. 10 and 12.
3) Adiusted from dressed weight to ready to cook
16] Johnson - pag. 26
TABLE 6
‘ojected and actual acre yields of grains, U.S.. 1950, 1954-56, 1960-62, 1965 and 1075
1950 1954-56 1960-62 1965 1965 77
Proj.(1) Actual Actual Actual Proj.(2) Proj.(3) Proj.(*
«<B
Wheat (bu.) . .
Corn (bu.) .
Oats (bu.) .
Barley (bu.) .
Gr. sorghum (bu
Soybeans (bu.}
Potatoes (cwt.)
Cotton (lbs.) .
Tobacco (lbs.)
NA
*
Le]
Lod
0
+04
J
1975
Proi.(® Proj.(6
Attain- Maxiable
mum
Jl
52
12
12
30
276
616
1,541
Gil
!) U. S. Dept. of Agriculture, What Peace Can Mean to American Farmers :
Misc. Pub. 562, May, 1945.
Long-Range Agricultural Policy: A Study of Selected Trends and Factors Relating to the Long-Range Prospects
for Agriculture, Committee on Agriculture, U. S. House of Representatives, 8oth Congress, 2nd Session, Washington,
D.C., March, 1948, p. 46.
U. S. Department of Agriculture, Farm Production Trends, Prospects, and Programs, Agri. Inf. Bul. No. 239,
May, 1961, p. 92.
Resources for Freedom: A Report to the President’s Materials Policy Commission (The Paley Commission), June,
1952, Vol. V, p. 66. The « A » yield estimate « based on the assumption that all commercial agriculture of the
U.S. is organized and managed to make full use of all available technology where such use would add more to
farm receipts than to expenses. » The « B » estimate was formulated on the basis of « a projection to 1975 of
the yield likely to come about from such application of available techniques as can reasonably be expected on
the basis of past experience. »
U.S. Department of Agriculture, Our Farm Production Potential, 1975, Agri. Inf. Bul. No. 233, Sept., 1960,
p. 6. « The economic maximum yield is based on full, efficient economic application of presently known technology
under assumed economic conditions. Economic attainable yields are yields that would be expected, by
1975, from actual application bv farmers of presently known technology. » (Ibid., p. 3)
1168 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 76
ceeded by 1960-62 yields in every case where comparison is
possible; the percentage excess ranged from 19 to 68 (!). It
may be noted that the 1965 projections from a study published
in 1961 were equalled or exceeded in six out of seven cases
by 1960-62 yields.
At the time they were made, the projections of the Paley
Commission were criticized for being unrealistically optimistic.
It may be noted that the lower of the two estimates (the « B »
projections), except for soybeans, have already been exceeded
by the 1960-62 yields. The last set of projections included in
the table were published in 1961. Most of the « attainable »
projections have been exceeded or equalled.
It should be noted that the yield projections have represented
what might be called judgment estimates. In some
instances past trends have been extrapolated, but often the
prospective yields have been estimated by agronomists and
other agricultural scientists who are well acquainted with a
specific crop and area. The projections have been the work
of competent scientists. It is clear that the yield projections
have missed or will miss the mark by a substantial margin.
It is also fairly clear that if development plans had been based
upon these projections, serious shortcomings would have
emerged in the prosecution of the plans or certain objectives
would not have been fulfilled. For example, the number of
farm families that could earn a given level of net income in
agriculture would have been much greater than the number
that can now earn that amount if the actual yields of crops
had been at levels consistent with either of the two sets of yield
projections for 1965. Prices received by farmers would have
been substantially higher than at present. Had farmers been
induced to make their plans in terms of anticipated prices
substantially above realized prices, a large economic loss would
have been realized
Q] The 1965 yield projections from the 1948 study were exceeded in
1954-56 in five out of seven cases and almost eaualled in another.
16] Johnson - pag. 28
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
B. Projections for the
Soviet Union
116%
In the Soviet Union the announced goals for agriculture
represent goals both for production and for consumption, after
adjustment for exports or imports and changes in stocks. Food
consumption will also differ from production because some
part of certain products is used for industrial purposes or part
of the output is used as an intermediate product (feed or seed).
For this brief discussion, the goals will be considered solely
as production projections. The goals for the Sixth Five Year
Plan (1956-60) and for the 1958-65 plan will be presented and
compared with actual performance for the 1956-60 period and
the first four years of the present plan.
Table 7 presents information for each of ten agricultural
commodities - output in 1955 and 1960, the goal for 1960,
the actual increase and the planned increase, and the actual
increase as a percentage of the planned increase. Of the ten
goals, only one was fulfilled. This was the goal for sugar beets
and it was overfulfilled by a substantial margin; in fact, by
such a wide margin relative to the fulfillment of the other goals
that it might be said that there was an error in the execution
of the plan ('). Of the other nine goals, the actual increase in
output as a percentage of the planned increase in output ranged
from 17% for cotton to 48% for wool. Instead of gross agricultural
output increasing by 70%, the increase was only 32%
or 46% of the planned goal.
The above comparison of goals and performance may be
considered by some to be irrelevant since the Sixth Five Year
Plan was abandoned in midstream. The Sixth Plan was replaced
by the Seven Year Plan for the period 1959-65. While the
Seven Year Plan is not yet completed — it is now in its fifth
(') It should be noted that the procurements in 1960 were 5.2 million
tons less than output. In 1955 the difference was only 0.4 million tons.
The difference probably consists of both waste and feed. In addition, 5.9
million tons of sugar beets were produced for feed in 1960.
5
Johnson - pag. 2c
170
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
TABLE 7
Output of agricultural products, 1955 and 1960, goals for 1960, and
increases in output and planned increase, Soviet Union
Product
1055
Output Goal
1960 1960
Actual incre
ase as per.
cent of
Increase planned
actual plan increase
(million metric tons)
Meat .
Milk .
Grain .
Cotton .
Potatoes .
Sugar beets .
Vegetables .
Wool . . .
Flax-fiber .
45-0
r0o7.0
8 +
61 34.2
134.4 20.0
3-6 4.5 0.2
71.8 84.4 126.5
31.0 57-7 47.2
14.1 16.6 28.3
0.26 0.36 0.47
0.28 0.42 0.51
(billion units)
46 =
fogs
28.
27.4
Index numbers (1955= 100)
Gross agr.
output . . 100 I22 I7O
“a
€
2.4 6.4
18; quo
“7-4 75.0
0.4 2.3
12.6 54.7
26.7 16.2
2.5 14.2
0.10 0.21
0.04 0.12
8.9 26.7
rn
38
46
38
17
23
165
18
48
3I
33
16
Sources: « Pravda », February 26, 1956; « TSU, Narodnoe khoziaistvo
SSSR v 1060 godu » (Moscow. 1061) pp. 262, 374-75 and 378
16] Johnson - pag. 20
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC,
117]
year — the performance for the first four years can be compared
with what would have been required if the goals were
to be achieved. This is done in Table 8 by assuming that the
output path between 1958 and 1965 was to have been linear Mh.
For the four years ending in 1962 the annual increase in
gross agricultural output was only 189%, as large as that required
to achieve the 1965 goal. Of the ten commodity groups,
there were two with output declines. For the other eight, the
ratio of the actual to the planned increase ranged from 21 to
93%, with only one of the ratios being in excess of 50%.
For the commodity with the highest ratio of actual to planned
increase — sunflowers — the planned increase for the period
was only seven per cent.
The record of output performance of Soviet agriculture is
somewhat better than that indicated by the estimated seven
per cent increase in output between 1958 and 1962. In 1958
climatic conditions were very favorable and agricultural output
was probably five per cent greater than it would have been
under average climatic conditions. However, in the published
discussions of the Seven Year Plan I have seen no evidence
that the effect of the favorable conditions in 1958 was taken
into account in establishing the 1965 goals. I suspect that the
modest increase in planned output for sunflowers and other
oil-bearing seeds reflected the high 1958 yields, but I have seen
no place where this was recognized.
A brief comment concerning the decline in sugar beet output
between 1958 and 1962 may be in order. There is little doubt
that the output of sugar beets for refining could have been
larger in 1962 than it was. The sugar beet goal for 1965 appears
to be an instance where the goal was set too high in terms
(') According to data presented in a speech by N.S. Krushchev
(« Pravda », March 5, 1962) it appears that the annual goals for agricultural
products for 1958-65 are approximately linear interpolations of the 1958
output and the 1965 goal. For two examples given — grain and milk —
almost exact linear interpolation was involved: for the third example —
meat — output was supposed to have increased at a greater absolute
amount in earlier than in later vears.
[16
Johnson - pag. 31
[172
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
TABLE 8
Output of agricultural products, 1958 and 1962, goals for 1065,
and average annual increase 1958-62 and planned increase,
1958-65
Output Goal
10958 1962 1965
Actual increase
Average as percent of
increase Annual planned
actual plan increase
(million metric tons)
Meat . .
Milk . . . 53.9
Grain . . 141.2
Cotton . . 4.34
Potatoes . 86.5
Sugar beets 54.4
Wool . . 0.3%
Flax fiber . 0.44
Sunflowers 4.6
Eggs
’
_
9.4 16.0
64.2 100.0 ("
147.5 164.0 (#
4.50 5.7 €
68.6 (4) 147
47.2 (5) 76 (°)
0.37 0.55
— 0.58
4.8 4.95 (9)
(billion units)
61
3
0.42
0.07
5 44
a
“+
9
- 00
5.03
LT
0.04
0.05
1.80 5.42
43
26
46
21
— (8)
- (8)
33
93
33
Index numbers (1958 = 100)
Gross agr.
output . 100 107 (IY 170
1.75 10.00
TR
Sources: « TSU, Sel’skoe, khoziaistvo SSSR » (Moscow, 1960), pp. 27.
iI, 202-03, 332-33 and « Pravda », January 26, 1963.
“ Minimum goal; maximum goal is 105 million tons.
Minimum goal; maximum goal is 180 million tons.
Minimum goal, mamixum goal is 6.1 million tons.
in terms of seed cotton.
(*) Output in both 1960 and 1961 was 84 million tons.
(*) 22.9 million tons of sugar beets were used for feed; these amounts probably
could have been processed for sugar.
() Minimum goal; maximum goal is 84 million tons.
(” Estimated from official data on output by commodity groups; weighted
by 1958 prices paid to collective farmers.
(?) Output declined between 1958 and 1962.
®) The plan specified a goal for oil-bearing seeds; I have assumed the goal
for sunflowers was 909% of the goal for all oil-bearing seeds on the
basis of 1958 output of sunflowers and other oil-bearing seeds.
16] [Johnson - pag. 32
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 1173
of consumption requirements or demand when account is taken
of the quantities of other foods available. This probably explains
why there was a substantial diversion of resources from
producing sugar beets for refining to sugar beets for feed. The
large imports of sugar from Cuba have probably had a significant
influence on the decision to reduce the refining of sugar
beets.
In 1961 the Soviet Union imported 3.34 million tons of
sugar (refined basis). Total granulated sugar production in
1962 was 7.8 million tons, of which 6.0 million tons was from
sugar beets. Total sugar refining in 1962 was 7% below 1961.
The decline in refined sugar from 1961 to 1962 is consistent
with the view that the 1965 sugar goal was set at too high a
level relative to other farm products.
One of the factors responsible for the modest increase in
Soviet agricultural output since 1958 has been the failure of
industry to deliver the inputs for agriculture that were implied
in the plan. In March, 1962 Krushchev remarked as follows
(1):
The seven-year plan called for increasing the output of
mineral fertilizers from 12,000,000 to 35,000,000 tons, an increase
of 23,000,000 tons. Three years have passed, and the
production of fertilizers has increased by only 2,900,000 tons.
In the first three years of the seven-year period the plan for
new capacity fulfilled only 44%. The same thing is happening
with the organization of herbicide production. Two years ago
the Central Committee and the government adopted a decision
on this question. Time is passing but there are no herbicides. »
The 1962 fertilizer output was 17.3 million tons or 13%
greater than in 1961. Fertilizer output in 1958 was 12.4 million
tons, instead of the 12 million tons implied by the above
quotation. Thus in four years the increase in output was only
4.9 million tons.
(
Pravda », March 6, 1062
Johnson - pag.
1174 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
In his March, 1962 speech KRUSHCHEV also noted that the
deliveries of certain farm machines declined substantially between
1957 and 1960. However, there appears to have been
a substantial increase in farm machinery production since
1960 - perhaps by 50%.
C. General Comment
It is fairly obvious from the above brief discussion of
projections, goals, and performance that much remains to be
achieved before specific projections of output and consumption
are to provide an adequate base for development plans. It can
be argued, of course, that either inadequate analytical and
statistical methods were used or that the goals were politically
motivated. Such an argument could be correct, but it implies
that more adequate methods are available and could be applied
in specific cases and that the political elements can be eliminated.
IV. SOME ANALYTICAL AND STATISTICAL PROBLEMS
For short periods of time, say three to five years, it appears
that fairly accurate projections of changes in the demand
for agricultural products are possible. However, even this
statement must be qualified to exclude such effects as the Korean
War or a significant change in the policies of other importers
or exporters where international trade is involved.
Projections of changes in output or of the effect of specific
policy measures upon output are subject to much greater uncertainty.
Even if we exclude the problems that may arise because
of climatic variations, other determinants of output do
not appear as yet to be subject to reasonably adequate projec-16]
Johnson - pag. 34
SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
117.
tion. This is true both of aggregate agricultural output and
of the output of individual farm products.
The factors that affect changes in output are often extremely
complex and not too well understood. In fact, some
are of a sort that are virtually unknowable. How do we know
when a new variety of seed is to be available? How do we
know how rapidly it will be adopted by farmers? How do
we know how quickly it will be improved by more adequate
adaptation to local conditions? In the United States we have
learned a substantial amount about the answers to the last
two questions from studies that have been made of the adoption
of hybrid corn ('). While we knew that a new hybrid for
another crop would probably be adopted much more quickly
than the 13 years it took before corn hybrids changed from
I to 809% of the seed used, we were not able to predict that
the same change would occur in grain sorghums in about 3
years. Nor did we predict that the yield advantage would be
as great as it was. Our original expectation was that the yield
differential between hybrid and ordinary grain sorghums
would be about 25% ; the actual yield differential appears to
exceed 50% and may be as much as 75%.
The use of trends to project output appears to be of little
value. Starting with the second decade of this century, the
decade increases in U.S. farm output have been as follows
(in percent):
IQIO-IG
IQIQ-2G
1920-3
G70 .
(") Zvi GriLicues, Hybrid Corn: An Exploration in the Economics
Technological Change, « Econometrica », 25 (a), October 1057
7
"161 Johnson - pag. 35
1176 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 7
The thirties was a period of adversity on agriculture —
drought, low prices, low incomes — yet output increased
more than during the 1910-19 period and almost as much as
during the twenties. Then during the forties output increased
almost as much as during the two previous decades. While
the growth of output was smaller during the fifties than during
the forties, the growth was greater than in any of the first
three decades included in the tabulation and despite certain
efforts to restrict production.
The same kind of seeming discontinuity in output growth
can be illustrated by changes in output in the Soviet Union.
Between 1953 and 1958 the official index of gross agricultural
output increased by 49% (!). This was an annual compound
rate of growth slightly larger than 8%. Between 1958 and
1962 gross output increased by approximately #9, or at a
rate of less than 29, annually. While most Western observers
did not expect the growth rate achieved between 1953 and
1958 to be maintained, I do not believe that there was any
one who predicted a growth rate as low as the actual one.
Mr. ARCADIUS KAHAN and I predicted that output might increase
by about 24% between 1058 and 1965 or at an annual
rate of about 3%. Our projection should be compared with
“he increase of 70% indicated by the Seven Year Plan.
Ex post we can say a great deal, both for the United States
and the Soviet Union, as to why there have been such changes
in the rate of growth. But in making projections we are still
not in a position to do very much in the way of predicting
changes in methods of production or the effects of changes in
incentives. For example, not all of the differences in views
concerning the effects of the output price level on production
in the Common Market or in the United States are due to self
interest or political views: a large part of the differences exist
(') I do not believe that the official output data for 1958 and 1953 are
strictly comparable, but the relative overestimation of output in 1958 is
not so great as to negate the point made in the text
‘161 Johnson - pag. 36
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
because we do not yet have an adequate analytical or statistical
framework for estimating aggregate supply response of agricul
tural products.
In many of the underdeveloped areas of the world, data are
either lacking or of unknown reliability. If it has so far proved
difficult to make reasonably accurate projections for the United
States or the Soviet Union, one can hardly be sanguine about
the possibilities for Africa, Asia, or South America.
[ do not want my remarks to be interpreted as implying
that all efforts at projections are futile nor that projections even
though subject to substantial error are without value. It is
certainly important for a nation that it be aware that if present
policies continue then it is quite likely that farm output
will grow no more rapidly than population, for example. Such
a projection should not be interpreted that food output will
not grow more rapidly than population, but that to assume
otherwise may well result in an undesirable consequence in
the future and that alternative policies should be considered
and evaluated.
But I do not believe that projections that can now be made
possess the necessary degree of accuracy to make it feasible
to rely upon detailed planning procedures which largely supplant
the operation of the market. There are many limitations
in the way the market functions, but there is abundant evidence
that there are also many limitations involved in the
execution of detailed and centralized plans.
There is a strong and convincing case for many types of
governmental action that will make economic growth more
rapid and less costly in terms of current consumption. Included
in such actions are certain obvious candidates — primary
and secondary education for the rural population, research
and adult education, improved market information,
sanitation and health measures. These are measures that improve
the quality of the human agent and provide the rural
population with the means for rational decisions. In particular
"16°
Johnson - pag. 3:
1178 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
situations, land reforms, governmental measures to improve
and expand credit facilities, subsidization of specific production
practices as a means of speeding their adoption (including the
creation of the capacity to produce such items as fertilizer),
special measures to bring new lands into cultivation through
irrigation, drainage, clearing or through provision of roads and
other facilities that would make it possible to settle new areas
or price supports at moderate levels as a means of encouraging
the expansion of commercial production merit appraisal and
consideration.
The underdeveloped areas are primarily concerned with
achieving an increase in the rate of growth of agricultural
output. This must be accomplished in a setting in which the
resources under the control of the government are relatively
limited. The funds that can be invested and the trained personnel
available for carrying out a plan or policy are clearly
very limited. In many countries there are not enough trained
economists and other specialists to develop the analyses and
estimates required for the formulation of a detailed development
plan or policy. In some cases, foreign specialists can
assist in such formulations, but the value of foreign experts can
de easily overestimated.
One of the most important contributions that econometric
analysis can make to the underdeveloped economies might well
be a series of studies that will aid in the decisions involved in
the allocation of trained personnel. Will such personnel have
a higher marginal product in establishing a research and extension
program or in organizing an irrigation project? Will the
marginal product be greater in developing a series of investment
priorities in the agricultural sector or in analysis of the
tenure system and other factors affecting incentives? These,
and similar questions are extremely difficult to answer and
are generally not the type of problems tackled by econometric
methods. But such questions may be more important than
some of the questions that we are ordinarily interested in.
16] Johnson - pag. 38
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC.
11
In the United States and Western Europe the major policy
objective concerns the level of farm incomes. The major analytical
questions are not the same as the ones discussed above.
The questions relate to the social costs of alternative policies
and the effects of alternative policies upon the level and distribution
of farm income and the size of the farm population.
There does not now exist, for example, a convincing analysis
of the effects of reducing the wheat price paid to German farmers
by 150 Dm per metric ton upon the average level of
return to labor and capital engaged in agriculture. Nor does
there exist an adequate analysis of the effect of the U.S. farm
programs upon the level and distribution of farm incomes or
of the effects of substantially reducing that level of expenditure.
Basically, our difficulty is that we have not as yet been
able to satisfactorily estimate the aggregate supply function for
agriculture. Given the many variables that affect the aggregate
supply function, when input prices must be allowed to change
as output prices change, when factor supply functions are shifting
over time and adjustments to changing factor and output
prices do not occur instantaneously, it is perhaps understand.
able that we have thus far made so little progress
johnson - pag.
MAHAL ANOBT
[ am practically in agreement with all the points made in thi
paper. I should just stress two or three points which seem to me {uv
de of particular importance. Firstly, industrialisation as an essential
condition for improvement of agricultural production. There has
been a lot of unnecessary controversy in a country like India by
raising the question which must you emphasise more, industry or
agriculture? Both of course; there is no conflict between industry
and agriculture. It is a simple but an extremely important point.
Secondly, I was very much interested to see, on page 19, the
sstimate of rate of growth of 3.5 per cent to 4 per cent in agriculture;
such a rate of growth is absolutely necessary for economic development.
In India it is necessary to have a rate of growth of the order
of 7 per cent per year for the economy as a-whole. With population
increasing more than two per cent per year, the per capita income
would increase at the rate of 5 per cent per year and would double
roughly in 15 years which would give some cause for hope. To
achieve this it is necessary to have a rate of growth of the order of
4 per cent per year for agriculture.
I should also entirely agree that demand estimates can be extremely
useful; we have found this in India. In calculating income
statistics, we have used a special method of arranging ‘he sample
Johnson - pag.
[182 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
households in ascending order of level of living, as measured, roughly,
by per capita consumption expenditure, and dividing the sample (or
the estimated population) into a suitable number (say, 100, 50, 25,
20, 10 etc.) of groups, called « fractile groups » each consisting of
the same number of percentiles (1%, 2%, 4%, 5%, 10% etc.) of the
sample, and then finding elasticities in terms of the fractile groups.
These elasticities (or their ratio) seem to be fairly steady; by putting
in current prices one can then immediately get income-elasticities in
money term at any point of the range.
Some interesting points have come out in India about the pattern
of change of consumption (in physical or money terms) of certain
commodities with increasing levels of living as measured by the per
capita consumption expenditure. For certain commodities this pattern
is quite different in urban and rural areas; for example, for
foodgrains. The interesting point is that, over a period of 10 years,
the pattern of consumption of foodgrains in urban areas and the
pattern in rural areas have been entirely different, but both have
remained practically steady. With the increase of income (in the
sense of national income or of households) in a country like India
which is very poor, or with an improvement in the supply position
of foodgrains, the per capita average consumption of foodgrains of
the whole population may go up or some times, with increasing prices,
may go down, but the pattern remains the same. Of course we are
interested in this, particularly from the point of view of inequalities.
The National Sample Survey of India, which covers the whole of
India, has at least two inter-penetrating sub-samples giving two independent
estimates; and also, of course, a combined estimate by
pooling the sub-samples. Using a graphical representation (in the
form of distribution of per capita consumption, or concentration curves
or in other ways) the difference between the curves based on the
two sub-samples gives a non-parametric and completely invariant
error area with which the analysis can be carried very far; I discussed
it in another paper. From our own experience I fully agree with the
points made by Prof. Jonson in his paper.
As regards the supply position, I think in an underdeveloped
16] Johnson - pag. 42
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 116.
country like India, the official figures are often unreliable for foodgrains.
There is no concept of checks and cross-checks. It is due to
treating statistics as legal evidence, that is, to consider that its sanction
depends on the level of authority which gives administrative
approval. The Food and Agriculture Ministry in India gives some
estimates of foodgrains; as this Ministry is in administrative charge of
matters relating to foodgrains, its estimate is the only one which must
be accepted (in the sense of a law court accepting legal evidence).
Even if another Ministry or Department of Government independently
make some other estimate (say, from consumption side) then
that estimate is not « official » in the legal sense and cannot be
used. In fact, in India, it has been urged that only one agency
should collect each type of statistics because multiplicity of agencies
might lead to differing estimates which would be confusing; in other
words, the very possibility of having checks and cross-checks must
be eliminated. It is a somewhat paradoxical situation. India has
some kind of reputation outside India for its statistics; but Indian
statistics remains weak because statistics is treated as a matter of
legal evidence and not of scientific validity. The official figures of
foodgrains in India (based on so-called complete enumeration of all
agricultural holdings), in my personal judgement, may be underestimates
by some thing of the order of 20 or 25 per cent. The
production of wheat and rice and other foodgrains can be estimated
through samples on the basis of the area sown together with the
yield per hectare ascertained by crop-cutting experiments. The consumption
of foodgrains can also be ascertained through a sample
of all households in the country. A direct ‘comparison is then possible
between such estimates of consumption with the independent
estimates of production after allowing for seeds, inventory, loss in
storage etc. As there would be at least two sub-samples and two
independent estimates of both consumption and production, such
comparisons can be carried out in an entirely scientific way. In
the case of a cash crop like jute, during the war, the production
estimates given in October of one year could be verified about 15
months later from the utilization estimates based on figures for
57 J
0
hnso
n
pag
. 42
1184 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA
shipping of both raw jute and manufacture and changes in stock
in the hands of jute mills; the production and utilization figures
were in agreement within a margin of about 2 per cent. Although
such an objective method can be used, the so-called « official »
estimates of foodgrains in India are based on the perfectionist theory
of figures being obtained for every field in the village; because this
is a so-called complete enumeration, and because such a system
has been in use for centuries, therefore it has high legal status like
custom immemorial; and also no independent check should be permitted,
because it would be confusing if the results of such a check
differed from the official estimate. So the point stated by Mr.
JOHNSON is well taken and extremely important. It also raises the
question whether a lot of sophisticated acrobatics on econometrics
on the basis of such unreliable statistics in the underdeveloped coun-‘ries
would not be a complete waste of scarce resources.
I entirely agree that without progress of agricultural output no
rapid economic development is possible. I do not clearly understand
the last paragraph at the end of page 37 «I do not believe that
projections that can now be made possess the necessary degree of
accuracy to make it feasible to rely upon detailed planning procedures
which largely supplant the operation of the market ». I understand
what this sentence means, but on the other hand, I take it that the
next paragraph is making out a case for State intervention. I find
a little gap here. Can you always leave it to market operation as
such in an underdeveloped country? I should strongly differ. In
my own country in 1943, during the war, because of that particular
doctrine (I may mention that the Economic Adviser of Government
at that time was Dr. GREGORY, the author of a standard book on the
Gold Standard) there was no attempt at any control or rationing of
foodgrains with the result that a famine broke out in Bengal. An
official commission appointed by the British Government found that
ome and a quarter million people had died directly from famine,
which is about twice the total casualties of the U.S.A. and U.K.
taken together during the Second World War. This happened.
Therefore, I do believe that government intervention is essential but
should be a minimum.
16] Johnson - pag. 44
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC, 1185
With the points made in pp. 37-38, I am in complete agreement
We are rapidly promoting the use of fertilizers in India; we are also
starting the manufacture of machinery to set up new fertilizer plants.
But there is one point which has not been brought out, namely,
the need of price support, or more precisely, having both floor and
ceiling prices because consumers must also be protected. There
should be buffer stocks and open market operations to the largest
extent possible. In India, Government could hold 5 or 6 million
tons of foodgrains in suitably located warehouses. There would be
pre-determined floor prices which may vary for different crops in
different areas; when the price touches the pre-determined low point
in any area, Government would start purchasing immediately with
a guarantee to continue without limit; this would give necessary
price support to growers. On the other hand, once the ceiling price
is touched, Government would start selling and this would protect
the consumers. Such a system, I think is necessary in countries
where there is a scarcity of food but not in countries like Thailand
or Burma where there is a surplus. I think it would be of great
help if the advanced countries would try to persuade an underdeveloped
country to adopt the above policy (using if necessary United
States PL 480 arrangements) to build up buffer stocks and have
open-market operations with not too large a gap between the floor
and ceiling prices so that speculators would not come in. This
should be adequate; other methods of physical control may have to
be used if the scarcity becomes very acute. I am amplifying the
remedies but not really differing from Mr. JoHNSON.
[ could not agree more with the statement on page 38, that there
are many questions which are not now within the range of econometricians,
or may be they can never be — I am leaving that
question open; I am simply agreeing that they are now beyond the
techniques which have been so far developed and which may be
entirely suited to the needs of the advanced countries. There is a
great danger of highly trained econometricians in my country going
on solving problems which may be entirely valid in respect of advanced
countries but which have no relevance to the problems
[16] Johnson - pag. 45
1186 PONTITICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - Z°
an underdeveloped country. I see a great danger here. On the
other hand, there are very important questions, some of which have
been raised by Mr. JoHNSON, about how the econometricians in
advanced countries can help the underdeveloped countries. This is
a question of vital importance, not only for the underdeveloped
countries, but also in the enlightened self-interest of the advanced
countries.
KOOPMANS
A brief question — in these elasticity estimates, does food include
the tin can and the tip to the waiter, or not?
JOHNSON
What I talked about was food at the farm-gate level
KOOPMANS
The other question is about the need for these projections — so
to say the marginal productivity of improvements in the accuracy
of the projection. What decisions do depend very much on having
accurate projections and what harm is done if projections are inaccurate?
If no great harm is done it is a nice purpose just the same
to improve accuracy, but it wouldn’t rank- with as high a priority
in econometric work
JOHNSON
I think this is a very pertinent and relevant question and I
would try to answer it in terms of the difference in plans in the
planned and the essentially free-market countries. I think that in-"167
Johnson - pag. 46
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1184
correct projections, if they affect policy decisions, can sometimes
have serious consequences. I would like to give two examples, one
drawn from the U.S. and the other from the U.S.S.R. In the early
part of the last decade, about 1950, a series of projections about
future demand/supply in agriculture were made for the U.S. The
projections implied that the growth of demand over the next decade
was going to be very rapid and that there seemed to be little possibility
of rapidly expanding production beyond what had been
achieved during World War II. It is impossible to say the exact
connection between that series of projections and what was actually
done in the policy field; it at least was used as an excuse in the
discussion for increasing the support prices in the early 1950s; this
action had very serious economic consequences, later, in inducing
too many resources to remain in agriculture. Obviously if nobody
had paid any attention to these projections, they would have done
no harm, but in this case attention was given to them and we arrived
at the wrong decision.
In the U.S.S.R. the Seven Year Plan for 1958-65 involved
projections for agriculture. These projections or plans were made
with great enthusiasm, but I assume were believed to be realistic.
A large increase in output was to be achieved by 1065, largely
through capturing what Soviet officials call « reserves » which didn’t
require additional resources; thus it wasn’t really necessary to push
the fertilizer plan or other plans to increase the resources used in
agriculture. The errors in the projections have had a serious impact.
Had the official projections for the. U.S.S.R. been similar to
those made by Kahan and me which indicated that output would
increase by less than 25 percent instead of 70 percent in 7 years, I
suspect that several very different policy decisions would have been
made during the past few years. Some of the difficulties that have
resulted from a reduction of the grain crop by perhaps a quarter
in 1963 compared to 1962 would not have occurred. The U.S.S.R.
would not now be using perhaps a billion dollars in foreign exchange
to purchase grain during the coming year (1963-64); they would
167
Johnson - pag. 47
1188 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
have taken policy measures that would have prevented these expenditures.
MALINVAUD
Professor JoHNSON has shown us that long-term forecasts in
agriculture were often very poor. I should submit that this is not
surprising. The future is difficult to predict anyway; and quantitative
economics is not an old science. Economist, who only recently
started to work with real data, have still much to learn in
order to predict better.
Even considering the present achievements, I would not accept
the conclusion that we should no longer try to make long-term
projections, Whoever has to decide for the future, must have some
idea of the future. Thus, the real alternative is between individual
long-term projections and what I would call centralized long-term
projections. Professor JoHNSON has not convinced me that we
should only rely on individual projections,
One of his claims is that individual errors are offsetting. But,
in the first place, I doubt whether they really are. Great mistakes
have been made during the past 15 years in various industries. My
‘feeling is that, in most cases, the centralized projections were less
wrong than the average individual projection. The centralized projection
had to fight against the common belief that was generally
much too extreme. Such was the case for fuel during the European
coal shortage, and again shortly after at the time of the Suez crisis.
According to me, econometric studies are usually bringing into the
discussion about the future some rational elements which are not
taken into account by individuals who have no time to go into a
serious analysis.
In the second place, even if the errors were offsetting, we should
not necessarily be satisfied with the situation. Offsetting errors do
not imply good decisions. Each individual decides on the basis of
his own mistakes. Decisions will then be inconsistent with one
16] Johnson - pag. 48
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118C
another to the extent that they are based on different assumptions
about the future. To convince me, one ought to prove that such
nconsistencies have no detrimental consequences.
In line with Professor MAHALANOBIS, I should add that I do not
like much the expression « to supplant the operation of the mar-Ket
» because it implies more than it really says. Most people would
agree that we have not to supplant the operation of the market if
the market operates well. The only question is to know whether it
does operate for the long-term allocation of resources. Are longterm
decisions in agriculture enlightened by the operation of the
market? So far as I know, they are not. Nowhere in the market
process will an individual banker who considers making a loan to
a farmer, find when the hybrid corn will reach the 80% level of
diffusion, or what the price of corn will be in five vears from now.
JOHNSON
There is no question that some of the remarks in the last three
or four pages are quite cryptic and perhaps warranted more detailed
exposition. My excuse is that I thought the paper was already too
long. I would like to explain and — in some degree — defend the
position that I took with respect to the usefulness of agricultural
projections and that such projections do not now « possess the necessary
degree of accuracy to make it feasible to rely upon detailed
planning procedures which largely supplant the operation of the
market ». Both Prof. MAHALANOBIS and Prof. MALINVAUD were
somewhat concerned about that statement.
What I meant by « detailed planning procedures » were quanity
allocations, of inputs or outputs, or price determinations by
sovernments that effectively determine the prices that affect producer
and consumer decisions. Obviously, there are a variety of
nterventions — planning — where I feel that we know enough. to
rely on governmental actions; these are listed in the paragraph
‘ollowing the sentence that has elicited these reactions. I there refer
"161 Johnson - pag. 49
1190 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
PRY
to education, research, land reform, building roads. These are
actions that supplant the market; when a government builds a road,
this supplants the private market for the road.
With respect to Prof. MALINVAUD’s other points, their basis may
oe in the brevity of my presentation but I am not sure. On page 37
[ state specifically, « I do not want my remarks to be interpreted
as implying that all efforts at projections are futile nor that projections
even though subject to substantial error are without value ».
This is rather different than what it has been implied that I said.
[n the paragraph from which the above sentence was taken, I noted
that if projections indicated that food output was likely to grow
less rapidly than population, governments should clearly act in
response to such a projection if it is believed to have a sound basis.
In the United States, all projections indicate that if present policies
are continued, output is going to grow more rapidly than consumption.
Such a forecast has significant policy implications.
I did not say that « individual errors are offsetting ». What I
did say (on page 22) was the following: « For one thing, errors
made by private individuals may be offsetting. For another thing,
errors made by private individuals may bring into play forces to
correct the error, such as a decline or increase in market price,
while a government price policy, subject to rather more slowly functioning
political processes, may compound the consequences of
projection errors. »
I am particularly concerned that Prof. MALINVAUD has interpreted
me as saying that simply because private forecast errors may
be offsetting, the resulting resource allocation is an efficient one.
[ did not say this; in fact, I once wrote a book (Forward Prices for
Agriculture) about the problem and concluded that a type of governmental
price forecasting for a production period could lead to an
improvement in resource efficiency.
But the point I have tried to make was that where errors are
made by private individuals, and this is particularly true in agriculture,
the market does bring into play forces that correct the
16] Johnson - pag. 50
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9
errors, whereas, in most cases where government policy is involved
‘he impact may well compound the effects of the previous error.
German agricultural policy today is an example of such a response,
Farm prices have been too high in the past and are too
high now. The high prices are impeding certain adjustments that
could well occur within the German economy in transferring labour
out of agriculture. However, since income growth in agriculture
was not quite as rapid as in the rest of the economy, the solution
adopted two or three years ago was to increase farm prices even
further. The basic problem is that there are too many people
engaged in German agriculture. So the error that was made by not
crying to make the adjustment through helping to transfer resources
out of agriculture has resulted in a compounding of the difficulties
VIALINVAUL
As I see it, that example does not show the government should
not interfere. It illustrates a rather different point, namely that it
is sometimes a difficult affair to determine first what a wise policy
should be, then to enforce it consistently, and also to know how to
revise it when necessar:-ALLAIS
I have only two very brief points. The first one is that I would
like to stress the interest of the work done by Professor JoHNsON
in recent years about agriculture in Soviet Russia. This work is
very useful and I had in the past the occasion to use it and I think
that is a type of study which should be multiplied. My second point
is, I think, quite important, in view of the discussion we must have
for the final statement. I must say that I agree completely with
what Prof. JOHNSON said on page 3 of his paper, namely, that it
appeared to him that it was necessary to paraphrase the intro-“16]
Johnson - pag. 5I
(192
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2§
duction fo this colloquy and I feel great sympathy with what he
says. To quote him: « The application of economic analysis and
recent experiences of various economic systems have by now given
sufficient proof that modern or underdeveloped agricultures cannot
be subjected to agricultural development plans that impede necessary
adjustments, result in exploitation of the farm population, fail
to recognize the potential high returns to certain investments in agriculture,
or restrict the potential gains from international specialization
in production, but development plans must be consistent
and efficient in the use of scarce resources and not result in undesirable
economic situations. » I would say this judgment is valid
not only for agriculture but also for industry and for the whole
economy, and I hope that we could stress this in our final statement.
HAAVELMO
I would like to ask Prof. Jomnson if the need for the kind of
research that he has suggested, might not be somewhat different in
areas with a low density of population and in areas with a high
density of population? It seems to me that the kind of production
policy needed may be different in the two cases.
WOLD
What I want to say has partly been said by Prof. Arras. First
of all I wish to compliment Prof. JounsoN for his superb paper,
and to emphasize that his general conclusions make important ma-“erial
for our final statement.
The eminent qualities of Prof. JoxNsoNn’s paper invite to general
comments on the position of present day econometrics. Economerics
is still a young science. Speaking broadly, there has been a
gradual evolution and expansion from micro to macro approaches.
Demand analysis and the ‘assessment of cost functions and pro-"16]
Johnson - pag. 52
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_-93
duction functions are key areas of theoretical and applied work
where econometric methods have been well established for a long
time and where they are in every day use both at micro and macro
level. Macro problems, again speaking broadly, are on the whole
more complex and difficult, and the econometric methods are here
far from having the firm grasp on the problems as at the micro
level. The theme of our Study Week is an inspiring challenge to
report on the present status of econometrics in the treatment of
economic growth and business cycles, the two major areas of macro
analysis. It is safe to say that as the avenues of quantitative ana-‘ysis
and econometric techniques have broadened in these areas, it
nas gradually become more and more clear that the two groups
of problems involve a highly complicated network of interrelated
phenomena, the conclusion being that the problems are not amenable
to simple and easy solutions. The spearheads of research and
applied work have moved forward in significant steps, marked by
successful attacks on a number of partial problems, but I think
everybody will agree that the successes have as yet only been
partial. More specifically, no integrated model of business cycles or
economic growth has as yet been reported which has been successful
in the qualified sense of passing the purgatory of a strict predictive
test (see the third section in my report « Toward a verdict on ma-“roeconomic
simultaneous equations » to the Study Week). Well to
note, the emphasis of this statement is not on the shortcomings of
existing models — personally I am convinced that the road is well
paved for continued progress, and the goal of reliable predictions
will be reached in due course — on the contrary, the emphasis is
on the partial results thus far established. It seems to me that econometrics
by now has reached an intermediate stage between micro
and macro. Micro analysis is well consolidated with regard to problems,
methods and results; Professor JoHNSON’s brilliant report
's ample evidence that econometrics is now mastering macro analysis
in the sense of economy-wide approaches, and it is appro:
priate to see this as an intermediate level of macro analysis since
ne is dealing with the agricultural sector, thus making a partial and
‘16] Johnson - pag. 53
1194 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - z2§
not a total analysis of the economic system. In the programme of
the Study Week there are several other important contributions at
an intermediate level between micro and macro analysis. Specific
reference is made to reports that focus on methods and techniques
of general scope; a typical case in point is the device of shadow
prices dealt with by Professor DORFMAN.
KoorMmANS
I have a question to Prof. JoHNSON in response to his comment
on investment in research. Research and development in industry
is often exportable to underdeveloped countries. My question to him
is whether this is also true in agriculture, or whether because of the
specificity of climate, soil and varieties, agricultural research has to
be essentially done over for each agricultural area.
LEONTIEF
Professor JOHNSON’s penetrating observations on the role which
econometric models might play in advancing the growth of underdeveloped
areas and possibly even of developed countries naturally
lead to the question of choice between special and general purpose
models. Throughout our present discussion much stress was put on
the necessity of building special models for special purposes on the
one hand, and on the other hand it was emphasized that the same
model can serve several different purposes.
Most of the difficult economic problems are those which involve
discovery and tracing through unsuspected secondary relationships
between the different parts of the economic system. A policy maker,
as a rule, is able to assess correctly — even without any help from the
econometrician — the obvious direct effect of measures which he is
about to recommend. Quite often he does neglect or disregard, however,
the indirect effects which might be unimportant from the point
‘161 Johnson - pag. 54
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1195
of view of his immediate objectives, but which might be in conflict
with different but possibly equally important objectives pursued si:
nultaneously by other policy makers.
The principal advantage of general planning and overall projecsions,
as compared with haphazard promulgation of special purpose
policy measures, is that it permits us to avoid such conflicts,
Special purpose models which permit us to estimate the principal.
or should I say, most desired effects of one particular policy measure
but slurs over its secondary repercussions is too blunt an instrument.
Consistent policy formulation requires general purpose, which is es
sentially general equilibrium models.
JOHNSON
I think there are only three of the statements that necessitate a
response from me. The first was Prof. HAAVELMO’s question, asking
whether the research needed would be somewhat different for areas
of low density than high density populations. I would argue the
difference would be fairly small and the reason is the following: in
the low density areas, which include large parts of South America
and Africa, an expansion of output which is based upon using addi-‘ional
land which can be brought into cultivation without any change
in technology will not lead to any significant increase in per capita
tevels of income. In my paper I described what happened in the
United States from 1820 to 1860, when the U.S. clearly had lowdensity
of population; agricultural output grew rapidly but as near as
we can tell output per worker changed very little in agriculture; and
also, although this does not necessarily follow, this was a period when
per capita income in the United States did not increase to any significant
degree. But to repeat, I would say that in the low population
density countries they also need substantial increases in productivity
per man and per acre and that many of the factors that are crucial to
prevent a decline per capita income in the high density countries also
apply there.
v, Johnson - pag. 55
1106 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 7
Mr. KooPMANS’ question is a very pertinent one and that is,
whether research in agriculture is exportable; can the research done
in the United States or Western Europe be exported to India? 1
would say the fundamental scientific research such as the theory
that has led to hybridization is clearly exportable, but I fear we
cannot go very far beyond that. It is necessary to adapt varieties,
different types of plants and even the methods of fertilization to
local conditions, There is also one other factor that is important
here and this is that in agriculture there is a continual fight between
man and nature. I would say that upwards of 3/4 of all the research
on the small grain in the United States done today is purely an
action required by man’s fight against nature. The resistance of
various plants to insects, disease, and viruses disappears over time
and there is a continual fight just to maintain yields; in wheat, for
example, in the United States, almost all research over the last thirty
years has been to fight against nature. A wheat variety that is very
popular in the United States and very high yielding might give a
zero yield after two or three years in India because it became subject
to an insect or to a disease.
I am not quite sure how I should respond to Mr. LEONTIEF; perhaps
no response is required other than that I agree with his view
that a general purpose or general equilibrium model that would do
all that special purpose models do — and much more besides —
would constitute a major contribution to both analysis and policy.
But after saying this, I must note that I do not believe that as a
profession we have progressed so far in the empirical application of
general purpose models to permit us to abandon the generation of
special purpose models,
Obviously special purpose models must be used with care if important
policy conflicts are to be avoided. However, two decades of
observation of agricultural policy in the industrial countries does not
convince me that the present unsatisfactory state of agricultural policy
has been due to a failure to understand the « indirect effects » of the
measures adopted.
‘16] Johnson - pag. 56
SELECTION AND IMPLEMENTATION
THE ECONOMETRICS OF THE FUTURE
RAGNAR qFRISCH
Tniversitetet à Oslo - Sosialakonomisk Institutt - Oslo - Norge
What I am going to present to you today is in all humility
a frontal attack on a ghost that has been haunting all of us for
the last generations, whether we want to be classified as belonging
to the West or to the East or to the uncommitted countries,
which, with a few exceptions, are the countries in Afro-Asia
and partly in Latin-America that are now striving towards
rapid economic and social development.
I can do nothing better than to begin by quoting the introductory
part of the program of this Study Week. And, inci
dentally, this introduction is a significant indication of the profound
understanding of the basic problem of our times which
the organizers of this Study Week, with the blessing of His
Holiness the Pope, have had. The introduction begins with
‘hese words, and I quote:
« Modern economies are extremely complex and both theory
and practice show that the free play of individual choice does
not guarantee, as used to be thought, favourable results for the
community.
Once this is admitted it is obviously necessary to provide
suitable informative and control instrument and fix the targets
which the economy is aiming at ».
| Frisch - pag.
1
{198
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
I think it is fair to say that the free market system has two
advantages: (1) its simplicity and (2) its effort-releasing effect.
But it has one fundamental shortcoming: it does not assure the
realization of specific preferences, such as a high rate of economic
growth, a distribution of income and wealth based on
social justice, aid to special social groups, economic development
of lagging regions within the country, development of
special agricultural and industrial sectors (for defense, health
or humanitarian reasons) etc. The purpose of wise planning is
to realize many such special goals, while retaining as many as
possible of the advantages of the competitive system.
We wish to search for some better economic system to
replace the time-honored system of the free market economy.
But in that search, we encounter a ghost that has been haunting
all of us for the last generations. It has been the same
ghost we have encountered regardless of the direction we have
chosen in our search for a better economic system.
This ghost is human nature itself. Some people are alert,
full of initiative and driving force, full of the active and unselfish
desire to apply all their abilities to the economic and social
betterment of their country and to that of mankind as a whole.
But, alas, the percentage of people possessing these virtues is
small, very small indeed. Many people are, more or less, dull
and selfish and can be induced to make a personal effort only
if thereby they can obtain some tangible advantage for themselves
or to the people close to them. In this connection, the
economic advantage will often stand in the foreground.
Therefore, the historical challenge, facing us as economists
and social engineers, is to help the politicians work out an
economic system built upon a set of incentives, under the impact
of which the economic activity will be satisfactory from
the viewpoint of the economy as a whole, even if the behaviour
of many individuals is essentially selfish. We must find a means
of circumventing the human obstacle to human progress.
17] Frisch - pag. 2
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
[t is this means about which I am going to speak today.
To size up the nature of the problem, let us review briefly
the directions that have been explored in the search for a better
economic system. Ronghiy speaking they may be classified
into three groups.
The first direction, one characterized by a very mild deviation
from the traditional market type of economy, consists of
admitting only monetary and fiscal instruments in the attempt
at steering the economy. The human behaviour patterns at the
various levels of society are such that, in this mildest form of
attempt to steer the economy in a desirable direction, one faces
a fundamental choice between inflation accompanied by fairly
full employment or a reasonably stable price level accompanied
by less than full employment of labour and other resources.
A precise description of the situation would, of course, necessitate
specifications of a number of details, but the choice 1
have mentioned indicates the essence of the matter. This
choice is strikingly illustrated in the famous Samuelson-Solow
menu. This menu consists, as you know, of a curve applicable
to the United States’ economy and showing how rapid an increase
in the price level we must be willing to accept in order
to reduce the unemployment percentage to a given level. An
even more important fact is that monetary and fiscal instruments
alone are not sufficient to assure the fulfillment of the
aighly specialized preferences we may have regarding the results
to be obtained from the community’s economic activity.
The mild form of steering about which I am now speaking
might perhaps be described by saying that it is a timid attempt
to introduce a small amount of enlightenment into that which
[ have called, on several previous occasions, the unenlightened
financialism.
The second direction in the search for an improved economic
system deviates a little more from the traditional market
conomy. It consists of admitting state intervention of various
sorts, aiming at influencing directly the quantities of goods
“171 Frisch - pag.
3
1200 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
and services produced or consumed. Among the attempts in
this direction fall numerous sorts of quantity regulations — in
particular, state control of investment in physical capital. In
some cases, this has been combined with nationalization of
existing big enterprises and/or with the establishment of new
state-owned or state-controlled enterprises which are to operate
alongside the still remaining private enterprises. A common
characteristic of all these arrangements has been their operation
under a monetary and financial machinery which, in all
essentials, was to remain of the traditional type, meaning that
we are still confronted with the Samuelson-Solow menu and
facing tremendous administrative problems, including problems
of loyalty and morale. These, I think, are in a nutshell the
characteristics of the mixed economies seen emerging in many
countries today.
The ghost has performed in his typical manner in all these
pursuits. Quantitative regulations of the prescriptive type have
a tendency to kill initiative and make the activity inflexible,
inefficient and stationary. And state administration, because
it takes away both the stick and the carrot which function
under a hard competitive system, has a tendency to eliminate
a large part of the driving force for personal effort.
Finally, the third direction, in search of a better economic
system, is represented by the more spectacular deviation from
‘he traditional market economy which is found in the centrally
steered economies of the East.
This more radical departure from the traditional market
economy has produced signal results in economic development
that cannot be explained away by any amount of ingenuity
and mental effort on the part of conservative economists and
statisticians. But the same ghost has acted in his typical manner
also in these more determined attempts to escape the shortcomings
of the free market system. There exists, indeed, an
overwhelming amount of evidence from centrally planned economies
showing that the active and positive participation of
171 Frisch - pag. 4
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120.
individual enterprises is not released through a system of quantity
targets (gross output measured in volume indices) established
from above. Nor, is it nossible to achieve the desired
results by such a simple system of incentives as that of paying
a premium to an enterprise according to its overfulfillment of
‘he centrally established quantity targets or according to its
ability to increase its volume index of actual gross output over
that of the preceding year.
There are several reasons for the failure of such incentive
systems. One is that quantity targets, established from above,
may induce the enterprise directors to conceal their true production
potential. Another is that they do not encourage directors
to use imagination and effort in economizing of input
elements. A third reason is that these systems do not induce
these directors to help rationalize production and realize desirable
investment within their specific fields. And fourth, they
lo not offer the inducement to improve the quality of the products,
because the establishing of quantitative targets of goods
and services can only, to a small degree, cover the infinite
variety of improvements in quality that constitutes a basic element
of economic progress.
A few examples will suffice to indicate the nature of the
experiences one has had. In the Soviet Union, in the period
before 1957, one worked according to what may be called the
ministerial system. There was one all-Union central ministry
for each group of goods. Because of frequent uncertainties of
supply from other ministries, each minister was tempted to set
ap his own ministerial factory for the component parts he
needed. This led, of course, to inefficiencies of various sorts.
There were also bureaucratic delays in settling questions due
to the scattered locations of enterprises over the whole country.
This motivated the abolition in 1957 of the central ministerial
system and the introduction of a territorial system. However,
his reform only replaced one type of difficulty by another. The
wishes and plans of the different regions were difficult to recon
“171 Frisch - pag.
5
1202 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - .
cile and make consistent from an all-Union viewpoint. These
difficulties encountered in the regional compartmentalization
have recently released a desire to revert, at least in part, to
principles of a more all-Union character.
As a final example, we may mention the Soviet attempt
at circumventing human shortcomings by separating the industrial
steering problem from the agricultural steering problem.
This attempt, as could have been safely predicted, has been a
failure. It constitutes a flagrant violation of the basic condition
that a steering system must be comprehensive, i.e. that it must
embrace simultaneously all facets of the economy. It is obvious,
for instance, that agricultural production depends essentially
on agricultural machinery and fertilizers, and both these means
of production are industrial products.
The suggestion I have to make regarding ways and means
of finally killing the ghost, or at least subduing him to some
extent, is not presented, of course, in a naive belief that here
is an « open sesame » that will, in one stroke, solve all difficulties.
Rather, it is a suggestion as to a way of thinking
which I believe is a conditio sine qua non for real progress in
our search for a solution.
We must begin by making a clear-cut and precise distinction
between two phases of the steering work: the selection and the
implementation.
The selection analysis is a study of what can be obtained
or ought to be obtained if only one considers the following:
first, such basic conditions for the economic activity as the
technological relations and the most deep-rooted relations governing
human behaviour, e.g. utility and its effects on demand;
and second, the preferences regarding the results to
be obtained in the nation as a whole, or in the world. In the
selection analysis we pay little or no attention to the system of
economic institutions under which the economic activity of
the nation or that of the world takes place or ought to take
place.
17] Frisch - pag. 6
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implementation analysis is a study of the kinds of
national or international institutions most helpful in bringing
about that particular constellation of the national or world
economy which has emerged as the optimal one in the selection
analysis, or at least to bring about a close approximation to
‘hat constellation.
The selection analysis must precede the implementation
analysis. If we go about it the other way, we would be putting
the cart before the horse. The selection analysis must be built
on a quantitative decision model, as distinct from an explanatory
model or a forecasting model. This will, I believe, be a
distinctive feature of the econometric planning work of the
future, since our main concern will be research work on how
‘he economy can best be steered.
In the technical part — which I shall not discuss in detail
here — a serious warning regarding a very popular « planning
procedure » is in order. It is a procedure that owes its popularity
more to its simplicity than to its real relevance for true
olannine.
I am referring to the popular procedure of initially guessing
at a « reasonable » national growth rate that « could probably
be obtained », and from this assumption drawing conclusions
regarding the production needed in special sectors of the economy,
the size of needed investments, etc. The reason why
this method has become so widely used is to be found, I think,
in its simplicity rather than in the fact that it is realistic and
rational.
The special aspects of the economy, such as production in
the various sectors, the size of investments, etc. are, in fact,
not determined even if the national rate of economic growth is
given. There may be many different development patterns that
all give the same rate of growth of GNP (The Gross National
Product) or of some other statistical measure of which one
mav think.
[17] Frisch - pag.
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1204 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Nor have we any assurance that the growth rate guessed
at is the optimal one, i.e. the best growth rate obtainable when
the structure of the economy, as well as the special preferences
which have been put up for the course of the economy, are
given. The optimal growth-rate will only emerge as a consequence
of a rational decision analysis which takes as its starting
point the preferences and the fundamental data describing
the structure of the economy. To start by target setting, whether
it be a specific figure for the growth rate of GNP or some
other specific target, is again to put the cart before the horse.
Certainly, we must end up by formulating targets; but
before this, there is a long way to go — namely through the
entire selection analysis.
Note. — The verbatim record of the technical parts of Professor FRISCH’s
presentation is not given here. Nor are the several mimeographed documents
distributed by Professor FriscH reproduced. This material will
subsequently be coordinated by Professor FriscH and published by him
separately.
"1771 Frisch - pag. 8
“TSCUSSION
ALLAIS
I said we must be very grateful to Prof. FriscH for his very clear,
and, I would say, provocative exposition of the future of econometrics.
I would also say that I have had much occasion to admire
Prof. FRISCH’s work. From a technical point of view, I completely
agree in general with his position, but his paper also expresses many
views which rest on value judgments and have evidently many political
implications. Professor FriscH has given his point of view in
a very excellent way as far as clarity is concerned, but I cannot follow
him insofar as fundamental questions of applied political economy
are at issue. His paper raises many questions which are connected
with the ordinary work of the econometrician, but which, it must be
recognised, have a high content of a political nature.
I am sorry that my knowledge of English is not such as to permit
me to express myself with all the nuance of meaning which is desirable
and this makes it difficult for me to specify my personal views clearly
‘0 you. But I think the purpose of this meeting is to bring out
divergences of opinion very clearly and it may perhaps not be without
utility to pur forward a different view from that of Prof FRISCH.
*) Comments on the FriscH's paper presented to the Study-Week.
Only a small part of this paper has been maintained and Professor ALLAIS
nas not had the possibility of reading the revised and reduced paper
printed on the preceding pages.
Frisch - pag. g
1206 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - ZC
1) First, as I already stressed on the first day of this meeting,
I think econometrics should remain neutral, i.e. we must avoid introducing
political views into our discussions. Personally, I would say
that I am a neoliberal, but I think political views should remain
outside the technical discussion of econometric problems. I do not
accept at all that Prof. FRISCH’s paper can be regarded in any way
as specifying the main lines of the future of econometrics. Econometrics
is a very powerful tool of analysis but nothing more. In
itself, it cannot determine what economic policy should be, but only
analyze observations and derive, in a rigorous way, the consequences
of specified hypotheses. Had Professor FriscH said in his paper:
« I admit as hypotheses, first, that a competitive system cannot realize
the « high goals of rapid expansion, growth, and social justice,
and second that these goals can be effectively realized in a central
planned economy », I would not have said anything because from a
scientific point of view, it is always possible and legitimate to make
hypotheses and to discuss the results. But instead Professor Friscu
has spoken of these two hypotheses as if they were well established
‘acts.
Econometrics must remain limited to the discussion of technical
questions. Certainly it is possible and admissible to discuss scientifically
the consequences of hypothesis of a political nature, but it is
necessary to avoid connecting them with political and ethical views
and with value judgements.
2) Professor FriscH spoke of the « simplicity » of the competitive
system but, I think, the same judgment can also be made on
FRISCH’s proposals for realizing justice and rapid growth, looking
for example at the first lines of § 5.1 (page 5).
Taking FriscH’s paper as a whole, I would say that things are
much more complex than his paper makes them out to be, and I
think we must be very cautious about all the statements made. As
an illustration of this complexity I will put forward for discussion
some very important questions on which it is evident that it is
impossible to follow the FrISCH conclusions,
I do not say that FriscH’s ideals should be criticised. On the
“171 Frisch - pag. 10
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contrary, I have the same social ideals as Professor Frisch, but 1
don’t think that his practical proposals would in any way forward
their realisation. My purpose is to comment on some postulates
which are generally accepted by those favour central planning, but
which are really very questionable.
3) First question: what exactly are the social preferences Protessor
FRISCH spoke about? Should there be a steering committee
‘0 decide what they must be? But we must be very cautious here.
[ think the proposition that some political steering committee should
1ave power to decide the aims to be or pursued by society is in fact
1 very questionable one. Prof. FriscH felt that it was necessary to
correct this position, and he says in his paper (page 2) that plebisci-‘es
could be used,
But, from a democratic point of view, the use of plebiscites is
itself very questionable. There are many difficulties which I cannot
discuss here in detail. But, for instance, who will have the right to
write the text of the referendum? This is very important because
‘he answers can evidently be biased. Many Frenchmen are very
sceptical about plebiscites to-day. Personally I consider them
a very dangerous procedure and in any case I think it is impossible
using plebiscites to decide what national preferences are in fact. The
definition of social preferences is a very difficult and complex problem
and I think the purposes of the society cannot be decided in a definite
way from a national point of view. There are millions of people who
have their personal and very legitimate preferences. Certainly, there
are many decisions which only a government can take, but these
decisions are only one element of what Professor FriscH calls « so-~ial
preferences ».
And we must consider not only the central government but public
agencies of any kind. It is imposible to reduce the problem of social
preferences to the problem of defining one preference function and
one preference function only. The problem is much more complex
ndeed and from this point of view the procedure which Prof. Frisch
has proposed is not at all satisfactarv
{/
Frisch - pag. 11
1208 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - Zu
4) The interest of the community is referred to in the introductory
text of this study week. But what is the interest of the community?
Who will decide what the real interest of the community is? Professor
FriscH would answer: the majority. But in a real democracy
there are many questions which cannot be decided by majority votes.
Minorities have rights which must be respected. So this question
s again very complex, and personally, I cannot see in any community
anything other than the superposition of individual interests.
And if this position is correct, it would appear to be impossible to
replace individual preferences by a single preference function for a
whole society.
5) Prof. FriscH has formulated excellently the stress he puts
on the relations which must obtain in all cases between the political
authorities and the people responsible for planning and he has said
that the political authorities can correct the plan or use it in some
way. But I would say that, from a practical point of view, this in
quite impossible, because politicians are incompetent in econometrics.
Who is to decide what the fundamental variables of the model are?
Who will decide if the calculations have been made in the right way
or not? All these questions are very difficult indeed, What the poli-‘ical
authorities can validly decide is the general rules governing the
decisions to be taken. But to the extent that millions of decisions
are in question, they cannot be taken by any central agency. And
the specific value of a market economy is that it provides a very
valuable tool for the organisation of decentralised decisions.
At any event, a political assembly can only discuss questions of
principle, and decide the general rules governing the decisions to
be taken. It can discuss technical plans and economic calculations
neither validly nor efficiently.
6) Professor FriscH’s starting point is that the overall purpose
of social policy should be the human personality. I agree completely
with this principle, but the question is: what are the different aspects
of respect for the person? Again, who will decide what exactly is
"17] Frisch - pag. 12
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20C
respect for the person, the rights of the majority, or the rights of the
minority? Whatever the solution to this fundamental problem may
se, 1 do not see it in FRISCH’s paper. In any case, majority rule
cannot give a valid and acceptable answer to every question.
What exactly is social justice? To speak about social justice is
quite appealing in public discussion. Everyone is for social justice.
But my own experience has convinced me that social justice is
accepted by different people in very different ways, and as far as
[ can judge, evoryone accepts what is in his interest as just, but
considers anything contrary to his interest as unjust. Reality, unfor--unately,
is such we must appreciate that there is no objective concept
of «social justice » at all, but only the conflicting interests of
millions of people.
My convinction is that FRISCH’s paper oversimplifies very complicated
and complex questions and that it is only in this way that
1e can justify the central planning procedure proposed by him. But
as he said himself at the beginning of his paper, simplicity can be
questioned, and to simplify problems is not to solve them.
7) In FriscH’s paper, everything is derived from the consider
ation of a single preference function only. But as I have already
said, we cannot consider only one preference function. And if we
agree that in parallel with the social preference function we must
‘ake individual preferences into account in some way, then there is
10 longer a single preference function, but ten million, a hundred
million, and, for the world, three billion preference functions, and
‘rom this point of view I cannot see at all how FRISCH’s paper could
work in reality. The question is much more complex, much more
difficult. We econometricians must recognize that it is impossible to
reduce the whole problem of social organization to a problem of
central planning.
In fact and in my opinion, only a decentralized organisation in
an appropriate framework taking parallel account of a market economy
and some central decision making by the government in its own
sphere can provide a correct solution to this very complex problem
"17] Frisch - pag. 13
1210 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
Only in totalitarian societies can the problem of social and economic
organisation be reduced to the formulation of a preference
function.
8) Professor FRISCH has said that there are two things which are
different, and which require to be treated separately. The first is
what he called the selection problem, and the second the implementation
problem. In my opinion, this is absolutely impossible. I am
an engineer, and I can give you a very good example of such an
‘mpossibility. In coal mines we don’t know at all what the production
functions are exactly. No engineer exists who can specify what
the production function is in a concrete situation. What engineers
do is to choose between different projects in comparing their
discounted net present value (in french: leur valeur nette actualisée).
And for this purpose they use a system of prices which only a market
economy functioning in an appropriate framework can provide. Thus
it is impossible to separate selection and implementation. From this
point of view, I cannot see at all how the FRISCH system could work.
We don’t know the production functions, and correct decisions cannot
only be taken in a decentralised system with the help of an appropriate
price system.
9) And again, what should be considered as desirable growth?
Growth of population, growth of efficiency — is growth really desirable?
Some people prefer stability to growth. Personally, I am for
efficiency, but that is a personal and subjective view. Other people
may prefer the stability of their jobs; they can definitely prefer stability
to efficiency. In fact, growth is not such an unquestionable
goal as Professor FRISCH was suggesting at the beginning of his
exposé.
For me what seems in fact desirable is not growth but simply
people’s happiness.
10) Professor FriscH has suggested that the centralized economies
have grown faster than the market economies. In fact, at the
least this statement is open to question and personally I think that
it does not conform to the real facts
‘17] Frisch - pag. 14
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t1) Professor FriscH has made some very strong criticisms of the
market economy, and I agree completely with him. At least in some
aspects, I recognize that the market economy has many, many drawbacks.
But such criticism should not remain limited only to some
points. It should be extensive.
Rightly, Professor FriscH has critised private monopolies, but
he did not say anything about public monopolies. He did not say
anything about trade union monopolies, the impact of which may
se much greater. Personally, I think that the unions play a very
useful role but it seems impossible to me to criticise only those
leviations which relate to the market economy, without simultane
ously analyzing the other deviations.
[f we want to be objective, we must compare the drawbacks of
‘he market economies with the drawbacks of the politically and ecoaomically
centralised systems. One needs only to look at the history
of the collectivist society in the U.S.S.R. to be convinced that central
planning has some drawbacks which can do more damage to social
justice than can those of the market economy. The millions of unemployed
in the United. States in the thirties can be compared with
the millions of dead in the U.S.S.R. during the same period.
Is it scientific to give an idealistic view of the planned economy
by comparing it with the reality of the market economy? If we want
‘0 compare, we must compare things which are comparable. In other
words, if we look at the reality of market economies, we must at the
same time look at the reality of the collective and centralised econonies.
And if we discuss what a collectivist and centralised economy
would be ideally, we should discuss what a market economy working
n an appropriate institutional framework could be, not what it is.
But it is not fair to compare the real aspects of a free economy with
the ideal aspects of a collective and centralised economy.
[t is absolutely scientific to stress the aspects of the market economy,
but if so one is faced with the necessity of stressing at the
same time what happened in U.S.S.R. in the thirties and in the
forties and what has happened in communist China in recent years.
71 Frisch - pag. 15
1212 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
12) If I can add one word about the market economy, I would
say one thing, namely that the people I know who are in favour of
the market economy are not at all in favour of the market economy
as such, as it is. They are in favour of a market economy operating
in an appropriate institutional framework, and if we want to do
justice to the market economy, I think it is necessary to study the
institutions of the framework of the market economy very carefully,
hefore coming to any conclusions.
As I have read in the FriscH paper, we should not rely on the
market economy at all, but my conviction is, on the contrary, that
it is impossible to solve the very difficult and complex problems we
have to face without some reliance on the market economy.
It is my convinction that with appropriate rules and in an appropriate
framework, a market economy can give reasonable participation
in the decision-making process to everybody, and to every
minority.
13) In conclusion, I greatly admire the scientific work of Professor
FRISCH, but I cannot follow him so far as the main themes
of his paper are concerned.
For me, FriscH’s paper appears as a long and convincing demonstration
of the practical impossibility of planning in Friscw’s
sense. Nevertheless, I recognize that this isa very personal and
subjective view.
But what appears to me as indisputable is the necessity for the
sconometricians to remain neutral.
From an objective point of view, it is absolutely impossible to
define the econometrics of the future bv reference to Professor
FRISCH’s paper.
In FRISCH’s sense there are in reality at least two, three or may
be ten econometrics of the future: the STONE future, the WozD future,
the ArrAIs future, and so on.
Thus, in my opinion, it is not desirable to connect econometrics
with a social philosophy of any kind whatever respectable it may
be. We Econometricians must, as such, remain neutral, we must
limit ourselves to the studv of econometrics in itself.
17] Frisch - pag. 16
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This does not mean that political science is without interest but
t does mean that we economists must remain very cautious about
what the interest of the community, social justice and so on really
are. Excuse me for having been so long and thank you.
DORFMAN
I do not wish to enter into the important and heated discussion
detween Professor FRISCH and Professor ALLAIS on the role and
propriety of social planning, but merely to point out the relationship
petween Professor FRISCH’s paper and my own in two respects.
However one feels about social planning, one must concede that
political decisions that influence the development of an economy have
io be made. For this purpose one needs some criterion for judging
whether social effects are good, or moderately good, or bad and this
s what Professor FRISCH’s capital F function does. One must have
something of this sort or no social decisions can be made on any
rational basis.
[ do not believe that this social preference function can be ascer
tained, however, by the method that Professor Frisch proposes,
namely by artfully constructed interviews designed to disclose how
people evaluate various possible states of their societies. In general,
people do not know how they will make an important decision until
‘hey are confronted with it, and they cannot tell you. Therefore ]
proposed that we attempt to determine social welfare functions by
inspecting how people have decided in the past rather than by asking
them how they would decide in the future. My purpose, however,
was the same as FRISCH’s: to determine a scale of social values.
There is another significant divergence between us. Professor
FRISCH distinguishes sharply between the problems of selection and
implementation. But I do not feel that this distinction can be
maintained. The preference ordering of two economic policies depends
not only on their consequences in terms of rate of economic
growth, per capita income, level of employment, and so on, but also
on the implementation side of the policies themselves, for example
[17] Frisch - pag. 17
1214 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
on the extent to which they restrict economic freedom. Thus one can
compare only states of the economy that can be attained by instruments
that are similar in their social impacts and this implies that
constraints have to be brought into the picture from the outset for
they describe the states that can be attained by means of a limited set
of economic instruments.
KooPMANS
I am approaching my comment here in the same vein as Prof.
DORFMAN. I am not addressing myself to the question whether society
is or should be moving in the direction that Prof. FriscH’s
paper indicates. Rather, for the purposes of the discussion, I am
accepting his assumption that this is the direction, and speaking
more technically to the point whether the particular layout and
scheme of Prof. FriscH’s ideas is efficiently designed to achieve the
purpose that he has in mind. I have really only two points that I
would like to raise in this connection. One concerns the very strict
separation of the determination of structure from the determination of
preference. On this point I find my thinking to be somewhat related
to that of Prof. ArLAIS. It would seem to me that the policy maker
who is being interviewed in order to obtain a representation of his
preferences will be neither able nor willing to be too specific about
these preferences as long as he does not know what the implications
of his indications are. He is likely to be pragmatic — not only a
man who thinks abstractly about his own preferences. To stay in
his position he must respond to pressures and perhaps even threats
in order to be effective over a period of time. Even a man of great
wisdom would still have to be aware of what he is expected to do
by a number of groups who have ways of making their desires effective.
Therefore if he is presented by an econometrician with
questions « what are your preferences » or « what is the form in
which you would mould what you regard to be the preferences that
should guide this planning », he may feel that he is being tricked
even though without such intent. He is not an econometrician, and
17] Frisch - pag. 18
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1 4
in fact in order to feel comfortable in the situation in which he is
placed, he would have to be not only an econometrician but an
electronic computer as well. He would want to know what quantitative
consequences his indications of preferences would have in
order to feel comfortable about the answers he is giving, I wonder
whether a better and a more effective procedure, which would get
more effective cooperation, would not be an iterative one. While the
policy-maker is being asked to state his preferences, he would be
assured that this is for a trial run only and that the outcome of the
zomputation is to be presented only to him or to his associates before
‘hese preferences would be considered firm.
My second point has to do with the question of preferences being
1ecessarily temporary and having to evolve by experience. It is
stated that the broad purpose of planning is, among other things,
‘0 realise the high goals of rapid economic growth and social justice.
Of these if would seem to me that social justice is the more permanent
one whereas rapid economic growth is a more temporary one.
Finally I have a point which is put more in the nature of a
question to Prof. FriscH. This has to do with the way in which
‘he steering prices would be used in order to steer the economy. If
the production system has constant returns to scale it would seem
‘hat prices are a poor instrument by which to bring about moderate
“hanges in quantities. The production set is, in a two-dimensional
case, the set of all points that are on or below a way out of the
origin. Them for certain price ratios profit maximizing quantities
are found in the origin. For other slightly different price ratios,
profit is the higher the larger the output.” So in a strictly linear
‘echnology, the response to prices is likely to have a flip-flap character.
Now, one would ask, if the competitive market system that is
prevalent in many countries is, in a way, a model for the steering ot
the economy by prices, why does that flip-flap behaviour not manifest
itself so clearly in the competitive market as I am concerned
that it might manifest itself in the steering mechanism that we are
discussing. I believe that in the competitive market system prices
4) Frisch - pag. 19
216 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
are very effective in the long run as conveyors of information and of
incentives, but that in the short run a good deal of quantity information
goes back and forth between suppliers and demanders in all
markets. My question to Professor FriscH is whether in his bleuprint
for the operation of a planning system he does not need also to
provide for circulation of quantity information at all levels, or
whether he thinks prices alone will be sufficient.
VMAHALANOBIS
I should like to make a few observations at two levels from the
point of view of a country like India. First, I share the doubt
expressed by Dr. Koopmans whether a political leader or an administrator
or someone who is responsible for decision-making would be
able to understand the implications of econometric choices. Secondly,
even if he does understand some thing at a technical level, whether
he would be able to influence the political or social decisions in
accordance with econometric consideration. And thirdly, whether
such a leader would not make mistakes which would have their own
consequences.
I believe there is a good deal of validity in the doubt to which
Professor Koopmans has given expression. I have myself continually
faced the type of question asked not only by Professor FRISCH when
he was in India several years ago, but also continually since then
in connexion with planning. Even when somebody would like, would
have felt it advisable, to make a decision one way, he might have
to remain silent because of uncertainties of the political consequences.
This is a serious difficulty. I am speaking from experience. This is,
however, only one level.
At another level, I should very warmly welcome the outlook of
Professor FRISCH because I believe this would be of great educative
value. I welcome this imaginative approach, not because I think a
push-button type of dcision can be achieved immediately or even in
a few years but because I believe his outlook and his approach can
be very important factors in an educative process, at the decision-17]
Frisch - pag. 20
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making level and at the level of national planning. I myself have
seen during the last ten years in India the beneficial effects which
are coming out of such efforts, not only on the part of Professor
FRISCH and other specialists like him, but also on the part of those
who are working on such methods in India. I see a beneficial effect
— not directly, but in influencing the thinking of individuals and
groups and making them more sophisticated in their nutlock.
I also see a danger if such studies are taken superficially or are
imitated, as such things are apt to be very often in the underdeve-‘oped
countries, due to fashion or because of the high prestige and
authority of the advanced countries. A superficial imitation of
advanced countries at too early a stage may and have often become
‘he most serious obstacle *. progress in the under-developed
rountries.
ALLAIS
Can I stress some technical points? In a mixed economy there
are two sectors: the private sector and the public sector. So, che
first question I would raise is: could the state formulate a preference
function for the private sector? And if so what would this preference
‘unction be? So far as the public sector is concerned, we meet the
same difficulty. We must consider not only the state but also regions,
cities, public and semi public agencies and so on. Is it possible to
‘ake account of these different operators in one single preference
function?
[ have many doubts about this possibility.
[ do not see at all how it could be possible or desirable to represent
finally different and probably conflcting views by one and only
ne preference function.
Would it not be better to allow every operator some purchasing
power and to leave him free to use it as he sec.
‘17] Frisch - pag. 21
1218 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
FISHER
I have one or two points to make. Both of these are on issues
brought up during the discussion, the one by Professor Koopmans
and the other by Professor DORFMAN.
Professor KoopmaNs raised the point that small changes in prices
might induce large changes in action. This is, of course, a possibility,
but it is less serious the more different ways there are of doing things.
For example, even if the technology is a linear programming one, the
production possibility frontier will approach a continuous surface if
there are many different activities which cover the entire nonnegative
orthant.
Now the question of discontinuities of this type is also relevant to
Professor DORFMAN’s remarks and indeed my comment here should
pe taken as a comment on his paper rather than on FriscH’s. Professor
DORFMAN wants to present policy makers with shadow prices
when they are at a particular vertex and see what they will do. The
discontinuity problem arises in this connection because there will generally
be more than one set of shadow prices at a vertex. Each set
will be associated with movement from the vertex one is at to another
particular one. The alternatives must be presented to the policy
maker therefore in a form which insures that once he has said he
will move to another vertex he will not then also want to move back
at a different but still appropriate set of shadow prices. One must
‘herefore ask questions which bracket the range of admissible shadow
prices. Once again, if there are numerous activities, this is not a
serious problem because the sets of shadow prices associated with a
given vertex will not be very wide.
WOLD
It seems to me that Professor ALLAIS is dramatizing the argument
a little. It has not occurred to me that Professor FriscH nor anybody
else believes that it is possible to arrive at something like the actual
truth when setting up a utility function for a political decision at the
17] Frisch - pag. 22
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1216
macroeconomic level, The purpose of the device is much more pragmatic,
in the direction of clarifying the problem, articulating different
political views by specifying alternative targets and alternative instruments,
targets and instruments that can be modified from time to
:ime, the limited aim not being solutions of absolute optimality, but
rather solutions that make workable compromises. If I understand
correctly, Professors DorrFMAN and FriscH both adopt this prag
matic view. Thus it is seen that the pragmatic approach is far from
uniform, and that it leaves room for widely divergent views about
fundamental problems.
Coming to my second point, I am not altogether happy about
‘he points in Professor FRISCH’s papers and his presentation where
ne limits himself to deterministic approaches. It is always necessary
to simplify in model building, and it is also granted that deterministic
assumptions are sometimes adequate; however, I feel uncomfortable
when Professor FRISCH says that deterministic assumptions are a dash
forward — I think it is a dash backward. It is an illusion if you
selieve that you can get rid of difficulties in economy-wide models by
supposing that you are so precise that there are no unexplained
residuals in your approach. To reduce the residuals you must stratify
in great detail so as to obtain homogeneous cells in your statistical
tables; when the cells become small, however, the law of large numbers
ceases to work and instead of more deterministic regularity you
will run into more randomness and irregularity. The problem of the
model builder is to strike a sound balance between the gain in information
given by a finer stratification, and the loss in information
when the law of large numbers is weakened. In a sense the residuals
are the back side of the medal of large numbers; they will however
do no harm if they are treated as stochastic variables, and if the
model takes them into account in such manner that the operative use
of the model is in accordance with the mathematical rules for operating
with random variables. This last principle takes care of the
pitfalls when a deterministic model is stochasticized. Now as far as I
can see the deterministic models considered by Professor FRISCH are
not in the danger zone in this respect. At least for ordinary input
“17] Frisch - pag. 23
1220 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2%
output models it is known that they can be stochasticized in accordance
with this principle (see my report to the Tokyo session 1060
of the International Statistical Institute).
LEONTIEF
In order to understand somewhat better the implications of Professor
FrRiSCH’s practical proposals, I would like to ask whether there
would be an objection against allowing people to trade goods and
services among themselves at what one might call black market prices.
In principle at least, the difference between these and the
official shadow prices would reflect the discrepancy between the
actual shapes of preference function of consumers and their official
estimates prepared by the planning authority.
FRISCH
I have about ten pages of notes that I have taken during the
discussion. I will try my best to do justice to everybody, but you
will understand that this is a very difficult task in view of the long
discussion and the complicated points at issue.
One of the Prof. Arrais points was that we are discussing the
econometrics of the future and in this connection he said that econometrics
should be neutral. Everything, of course, hinges upon what
is meant by neutral. You know that for centuries there has been a
tendency to define neutrality in economics by saying that any analysis
which takes the free market system as an axiom, is « neutral »,
but any analysis that has the audacity of questioning the free market
system is not « neutral », but « political » and should therefore not be
allowed to enter into the ivory tower of the scientist. This has been
the situation in economics for a couple of hundred years but this is
not the situation any more. Today we have to recognise the fact that
there are also other economic system that are « in the air » and must
be discussed by us as social engineers. I must add that not only
17] Frisch - pag. 24
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are these systems “in the air” today, but I think they will continue
to be so, I even think they will constitute the main object of our
discussion in the future. If we are discussing the econometrics of the
future we have to recognise this, and I will state a personal belief
that 100 years from now our grand-children will devote practically
all their efforts to the study of those models that deviate from the
free market system. They will use only an infinitesimal amount of
their energy discussing such things as, say, the stability of the
2quilibrium in a free market system. This is my conception of the
>conometrics of the future. A second point that was raised by Prof
ALLAIS was regarding social preferences. He objected to these preferences
being decided by the responsable political authority. To this
{ can simply answer that so far there has not been invented any other
machinery than the political one, for steering an economy. We have
‘© accept this as a basic datum in our scientific researches.
The purpose of this conference is not to go into a complete
discussion of political theory and describe the whole list of political
system that are conceivable. But some political system there must
de and some authority has fo decide in the end. This I take as my
starting point. I simply accept the existence of a political authority
vhatever its nature may be.
A third point mentioned by Prof. Arrais: Who can decide on
vhat magnitudes ought to be attributed to the variables? The
politician cannot do it because he does not know econometrics. The
answer to this question hinges upon the distinction between the gross
ind the net form of the preference function. Of course the politician
does not know the depth of econometrics. This is precisely why the
scientific analyst in his interview with the policy maker has to concentrate
on the gross form of the preference function, on the « Santa
Claus » form of the preference function. From the purely
psychological point of view you may, of course, interpret the word
gross as anything under the sun. But this is not what I have done.
l have put up a well-defined model, I have defined my concept of
gross and net. So therefore I must say that I find Prof. ALLAIS
remark in this particular connection absolutely irrelevant to the di
scussion of mv paner.
17) Frisch - pag. 25
(222 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2¢
With regard to Prof. DorFMAN’s intervention I believe I got the
sense of what he said when he compared my approach with his own.
He would rather prefer to take as a starting point some sort of
optimal solution, and then subsequently proceed to a discussion with
the politicians perhaps using a sort of iterative procedure in this
discussion. On this score there is, I think, a very little difference
between Prof. DORFMAN’s point of view and mine. It is simply a
question of how best to shape the interviewing technique.
Prof. DorFMAN also mentioned the possibility of presenting different
described alternatives to the politician and letting him
choose between them. Possibly, that is not precisely Professor
DorFMAN’s point of view, but it is certainly the point of view of
somebody else around this table. So I think it merits being drawn
into our discussion. This is a very natural view point, a very simple
one. The idea is that the experts should work out different alternatives.
These alternatives should be listed on a big sheet of paper or
perhaps each alternative on its own sheet of paper. And then all
these sheets of paper should be put on the politicians table and the
scientific expert should say: « Now, please, out of these alternatives
choose the one you like. »
To me, this is an absolutely impossible procedure and I will
explain why. You can use this method if you have a very very
small model with two, three or four variables, because then the
aumber of possible alternatives is so small that the situation can be
grasped. But if you have a real programming problem with hundreds
of variables and thousands of possible alternatives, as you will have
for instance in an under-developed country that strives towards industrialization
with a long list of investment projects, you will find
that the method of listing alternatives can produce nothing but complete
confusion. You would simply be facing what an expert mathematical
programmer would call information death. You would be
killed by the amount of information.
You must proceed in another way, you must proceed in such
a way that the computing machine takes over the task of keeping
:race of all the alternatives. If you are going to do that, there is
17] Frisch - pag. 26
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absolutely no other way to proceed than by starting to define a gross
oreference function, a preference function in the « Santa Claus »
sense. That 1s the only way to proceed.
This is of course not saying that any solution that comes out of
the computing machine is going to be presented to the politician as
the sole and final solution. It will certainly be necessary to proceed
through a number of iterative steps, with contact between the polilician
and the scientific expert. I have frequently spoken and written
about the need for a continuous cooperation between the politician
and the scientific expert. The iterative steps must be in the following
sense. You start with a first and tentative formulation of the policy
naker’s gross preference function. You run the solution and you
come back to him and say « Now, this is what I found — is this
what you want? — Then the politician will scratch his head and say:
« Oh, no this is not really what I had intended. » Then you will
have to start a conversation with him trying to find out more precisely
what he actually desires. You proceed to a second approximation
to his preference function. And so on until you finally arrive
at a preference function and the corresponding solution both of which
‘he policy maker can accept.
Prof. KooPMANS had an important point when he said that the
policy maker is not willing to give up his own ideas regarding the
structure of the economy. This is — as I said very explicity in my
Arst presentation — precisely the point on which you must concenrate
most of your attention in the discussion with the politician. You
must make him give up his own ideas, about the structure. Why?
Because if he does not do that, you will not be able to help him. It is
ike a patient coming to a doctor. The patient has some preconceived
deas about how to solve the problem of his sickness. The first thing
‘he doctor must do is to try to get these ideas out of his head. If
somebody is suffering from a psychiatric disease and he wants to
jump out of the window and kill himself, what am I going to do?
{ drag him by the neck and give him an injection. That is the first
part of the treatment. I am not starting to argue with him at that
moment. But when he is quietened down, I start to talk kindly toc
yi Frisch - pag. 27
1224 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2
him and try to explain to him that his own way of solving his problems,
namely to jump out of the window, is not a good solution.
The solution that is really good for him is of an entirely different
nature. And I will try to help him find that solution.
KOOPMANS
There has been a misunderstanding, and I must express myself
sufficiently clearly. I was not pleading that a policy maker's own
idea of the structure be respected. I was pleading that when a policy
maker is asked for his preferences, that he be given a chance to see
the implications of confronting those preferences with the econometrician’s
model of the structure, so that he knows what is implied
in the first indication of his preferences — and I think that in your
answer to Prof. DorRFMAN you have already dealt with my point.
FRISCH
[ think this has cleared the question up perfectly. What Prof.
KooPMANS says right now is just what I said a little while ago about
the iterative process in discussion with the policy maker. But it is
essential that the process be iterative in terms of complete optimal
solutions. And if you are going to have any complete solution, you
must start by a preference function, trying to lead the politicians
mind completely away from his preconceived ideas of the structure.
Prof. KooPMANS had a second point which is really covered, 1
think, by what we have already discussed. « The whole thing must
evolve by experiments ». I think that those were the words used
by Prof. Koopmans, and of course my answer is absolutely « yes ».
A third point raised by Prof. KooPMANS was about prices, how
they can be used as means of implementations. He draws particular
attention to the fact that you may have an economy with flip-flap
>ffects of price changes. This is completely correct, And, as a matter
of fact, I think that I have myself in some econometric papers poin-17]
Frisch - pag. 28
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ced out this fact. In linear models the problems may look very dif-‘erent
from what they look in classical production theory. So there
is absolutely no difference of opinion on that. But in my system
the optimal prices have to be applied to specific enterprises and, ot
course, in any given enterprise you certainly have very definitely an
economy of scale, hence a non linear effect.
You have to a large extent a continuous and non linear process.
You do not have a constant return to scale ad infinitum.
There are quite a number of pages of the paper I circulated,
where I speak about non linearity, but unfortunately there was no
‘ime to go into it in my oral presentation
Let me now say a few words about an aspect of non linearity.
Suppose you consider a plant that uses either one of two factors of
production. You want to influence the director of the plant by
formulating incentives in such a way that: 1) The plant produces a
certain amount of product; 2) It uses one of the two alternative
‘actors of production rather than the other. Then you may consider
1sing two non-linear incentives. You may define a premium as a noninear
function of the amount produced, and with a sharply-defined
maximum at the point corresponding to the quantity which you want
‘o see produced. At the same time you may use an other premium
regarding the choice of factors of production. Now this is, of course,
a very simple example, but something similar can be used in other
and more complex cases. You will readily recognize, however, that
to work with a non-linear premium is a rather complex affair. You
might perhaps do it in certain cases of very great importance but if
is absolutely impossible to do it in all the micro-economic details
which you have to face when you want to steer an actual economy
[hat is why I put so much emphasis on this specific accounting medium
which is derived from the system of optimal prices. Such a
system of incentive is, of course, linear but it may to a large extent
be protected against flip-flap effects for the reasons I mentioned.
Prof. ISARD at this conference always reminded us that we
had to think in terms of regional problems, and I am glad that he
has always insisted on that. For my own sake I have not disregarded
F171 Frisch - pag. 29
1226 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -~ 28
this problem entirely in my previously writings. I have here for
distribution some big mimeographed sheets which show a work we
have been doing at the Oslo Institute of Economics, on the principles
of how to study the interrelations between three countries. This job
was done primarily by a Swedish economist, Mr. Tom Kronsjo, who
has been working with us for quite some time. He uses my system
of notation.
If any of you are particularly interested, I have a few more copies
for distribution.
Prof. MAHALANOBIS, I must excuse myself if I have not got your
point quite straight. There is quite a distance along this table and
nisunderstandings may arise. As I understood it your first point
was this: the politician will not be able to understand a scientific
question. My answer is: of course he will not. If you put up to him
a description of some 450 dependent variables and 31 degrees of
freedom he will be entirely lost. But I don’t suggest that you are
:o put up to him such a system. You should only put up to him
very simple questions, one at a time. That is the basis of the interviewing
technique. I am absolutely certain that if you can have a
quite and not too rushed conversation with the policy maker, being
careful that he understands your questions correctly then very
meaningful results will emerge. I am not stating a theoretical
hypothesis, but basing my opinion on actual conversations with
leading politicians including the Chief of Prof. MAHALANOBIS’
country.
Prof. Arirais spoke about the difficulties of constructing a preference
function.
His main point here was that we have many different preference
functions, the preference function of different groups, oi different
persons and so on. Of course we have, I have, for instance, found
in my interviewing of high-ranking officials that the Minister of Agriculture
will have different preferences from the Minister of Education
or the Minister of Industry. Similarly there will be differences between
regions of the country and, perhaps even more important, diferences
of opinion of what should be done in a group of countries
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‘hat are entering into some from of co-operation. Should some industry
be located in country No. 1 in this group; or in country No. 2:
Which one of the countries should concentrate on agriculture, etc.
There may certainly be differences of opinion in these questions and
[ think, I have fully realized this problem and I have made at least
one attempt to overcome it. This is presented in pages 28 to 37 in
‘he blue pamphlet. I have written nine pages about this particular
problem but you will forgive me for not having been able to squeeze
this into my oral presentation. I am referring to the section on the
construction of the coalition preference function. So my feeling is
that IT have not neglected the question of differences of desires. But
somehow such differences must be ironed out if the economy of the
country is to be steered and not be left to drift whither the wind
blows. It is the task of the political machinery to solve the problem
of differences of opinion. We as economists have to accept that
solution.
Prof. ALLAIS said he had the feeling that this decision problem
is solved in the free market system. My answer is that the optimal
solution corresponding to the desires of the political authority can
never be realized by the free market system, or more precisely there
is a probability of measure o that it will be. Why is it so? Because
the free market system does not open any possibility of expressing
political preferences or preferences of a very special sort. There is a
great number of special political preferences. I have mentioned
some, I could have mentioned many more. The problems are specific
and they are great in number. I would like to see somebody sit down
and list a number of these specific preferences and then set out to
prove that the free market system will realize all of them simul
‘aneously.
Prof. FisHER when he went to the blackboard referred to some
specific vertex of the admissible region, and he suggested that we
may take this vertex as the starting point for a marginal analysis
and discuss the matter of optimality with the policy-maker by startng
from this vertex. At least that is what I got out of Prof. FISHER’s
‘ntervention. This is very much against what I have tried to say
1 Frisch - pag. 31
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PONTIFICIAE ACADEMIAE SCIENWNTIARVM SCRIFTA VARIA - 28
The politician will not even understand what is an admissible region.
You must put up to him some questions that are much simpler.
[hese simpler things are precisely what is included in the interview
technique I have suggested.
Prof. Worp, in the first part of his intervention, was so much
in line with my own thinking that there is no need for me to go
into that question at all.
Subsequently Prof. WoLD mentioned the deterministic approach
as compared to the stochastic approach. As I have said, I am absolutely
in agreement with him on the ultimate need for introducing
‘he stochastic viewpoint. WoLD rather had the feeling that we had
to introduce the stochastic viewpoint already from the beginning.
Then he said that if we don’t do that our analysis won’t be a
dash forward but rather a dash backwards. My answer is: if we try
to introduce the stochastic viewpoint from the beginning in these
immense models we are facing an impossible task. There will be no
dash at all, whether forward or backward. The minimum factor at
this stage is a technique of handling a great number of variables. Regarding
the ultimate goal there is complete agreement between Prof.
WorLp and myself. Ultimately we will need multivariables techniques
combined with probability theory.
LEONTIEF mentioned a very important point. He spoke about
olack markets. There is always a danger of developing black-markets
or grey markets. Particularly so if you stick to the traditional
money economy with prices that move according to where the wind
blows. If there is an excess supply, the prices decline, and if there
is an excess demand the prices rise. And that is that. If you use
some sort of price regulations, you are bound to encounter a tendency
forward grey markets and black markets. My answer then
will be this: In the first place, you must remember that I am not
speaking about trading prices in the ordinary monetary sense. Certainly
you must have money to buy the small everyday things, but
this after all is a minor problem. I am speaking about the accounting
medium derived from the selectionally optimal prices to be applied
to the big questions of influencing the policy of individual plants, and
17] Frisch - pag. 32
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in this context, the questions of black markets and grey markets
emerges in a different setting. It is not saying that you will not have
some form of tensions. This is implicite in the concept of « steering »
as distinct from « letting it drift ». Certainly you must try to constitute
your system of incentives in such a way that there is as little
rension as possible But it is impossible to avoid tension all together
\LLAIS
[ apologise for taking the floor again, but it is only for a few
minutes, for clarity. In Prof. FriscH’s paper, there were in fact two
sort of things, firstly political views and value judgements relating
1 particular to the market economy and, secondly, technical analysis
of certain points. I would not have raised any questions about the
frst part if Prof. FriscH had said the view he expressed were
personal views. But they were presented in such a way that the
aninformed reader might think that these views are indispu:able and
correspond to well established facts.
I appreciate that Professor FRISCH can give a personal definition
of growth, but in fact he says that a free market system cannot
ensure the realization of rapid economic growth. In this context
‘he word growth is clearly used in the usual sense and not by
reference to another particular definition.
As far as the future of economic planning is concerned, personally
I am not against any study of the theory of planning. On the
contrary, I am very much interested. I think this type of work can
be very useful, but on one very definite condition, that is that this
‘heory be expressed in a very neutral form, without expressing definitive
and dogmatic value judgments about what a market economy
really is or what a central planned economy really is. These questions
are details and I don’t wish to make too much of them.
But y put four fundamental questions which are very technical
and I apologize if I was not able to express my thinking sufficiently
early. I think Professor FriscH has not answered these four
fundamental questions in a satisfactorv wav
Frisch - pag. 33
1230 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
First question: when Prof. FRISCH spoke a few minutes ago of
‘he question of preference functions, he said that he had dealt spoken
with it in his paper and mentioned especially pages 36 and 37. I have
ead them again. If I understand Prof. FriscH’s point of view
correctly, what he is proposing in fact is to construct a preference
function (page 37) but, again, he esteems it desirable to have only one
preference function and I personally consider that this reduction is
not possible and not even desirable. If we consider different preference
functions, for instance two or three only, although the number
has no importance, the nature of the optimum problem does change.
In any case, from an ethical point of view, this point is also quite
'mportant, because if there is but one preference function, this means
that some people will have power to define this preference function
according to their own preferences. Many criticisms can then be
raised. Thus my first question is: Does Professor FRISCH propose
consideration of only one preference function or is he ready to consider
a plurality of preference functions without attempting to reduce
this plurality to one only?
The fact is that he seems to imply that it is possible to reduce
a plurality of preference functions to one only which would be
maximised. If this possibility does not exist the paper’s entire reasoning
is deprived of its foundations.
My second point was that — and perhaps I was not sufficiently
clear this morning — in general we don’t know the production
functions at all and if we don’t know them, I cannot see how
FriscH’s system can work.
My third point is that it is not possible to separate selection and
implementation, because in general we don’t know the production
‘unctions.
And fourthly, it appears to me that it is impossible to reduce the
whole process of decision-taking to a dialogue between politicians
and planners. Thus, my question is: Does Professor FriscH intend
to plan the whole economy or only one part of it in the manner
he described? In the first case, decentralisation would be impossible
and the economic svstem would be verv inefficient.
17} Frisch - pag. 34
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FRISCE
First point raised by Prof. ALrLais: It is obvious that my paper
presents what I personally think is the most appropriate procedure
for constructing a national economic decision model. It represents
my personal creed at this stage. I thought that this was so obvious
‘hat there was no need for making it more explicit at this conference.
Second point, Prof. ALLAIS objected against my use of the
term « gross » in connection with the preference functions. I cannot
accept this criticism because the sense in which I used the world was
precisely defined in mathematical terms, and I always used the term
consistently in that sense,
Third Point. Prof. ALLAIS mentioned that different people may
have different preference functions (and he might have added that
one person may have different preference functions at different points
of time). All this is, of course, quite true. This problem is essentially
a political issue, and must be handled as such. We have no other
way of finally deciding about political issues than to use the political
machinery existing in the country in question.
It is not our task at this conference to discuss various political
theories and types of political machinery. We as economists simply
have to take for granted that somehow the nature of what we designate
as the national preference function, is arrived at. I have been
liscussing this before in several connections and this is the last time
[ shall repeat it at this conference.
THEIL
Several discussants have argued that it is not easy to construc
social preference functions and I can agree with this. Nevertheless,
[ would like to suggest that the exercise be carried out. The reason
s not that I believe that within a few years this kind of decisionmaking
will be put into practice. It is that the procedure provides
1s with a method which enables us to find out which parts of the
Frisch - pag. 3s
1232 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25
model have to be improved first and which are of less importance
This method proceeds in principle as follows. When we mis-specify
the model which underlies the contiaints and when the decision
maker maximizes the preference function subject to these constraints,
his decision will in general deviate from the optimal decision and
‘hus depress the utility level below the attainable maximum. This
atility reduction is an appropriate measure for the seriousness of
specification errors. For the special case of quadratic preference
functions and linear constraints this idea was applied on a fairly large
scale in my recent Optimal Decision Rules for Government and
T'ndustry.
FRISCH
Regarding the point raised by Prof. THEIL, I think there is no
serious or fundamental difference of opinion between us. Let me only
remind you that I have not described the desires of the politicians,
solely by means of the preference function. There are also political
bounds coming into the picture. I have explained this rather fully :n
‘he paper, but unfortunately I did not have time to insist very much
on this in my oral presentation.
171 Frisch - pag. 36
THE ECONOMIC FRAMEWORK OF REGIONAL
PLANNING
JAN} TINBERGEN (+)
Vederlandsche Economische Hoogeschool
Rotterdam - Nederland
1. Development planning has become an established activity
in many countries after World War II and both a practice
and a theory have emerged. It is only natural that practical
devices were in use before a somewhat integrated theory was
available and it also stands to reason that both practice and
theory are still changing. Most of the common body of theory
which is nowadays accepted by a majority of development
planners deals with an economy subdivided into sectors, that
is industries or activites — depending on the type of analysis
used. One of the main reasons for working with such models
is no doubt that the proper choice of sectors promising « comparative
advantages » in international competition is of para
mount importance for the success of any development policy
Even so, however, another « dimension » in economic development
is increasingly requiring attention, namely the spatial
or ve “onal aspect. Many governments. especially those
*) Professor Jax TINBERGEN was unable to attend although he had
noped to be one of the participants in the Study-Week up to the last
ninute. We are happy to include in the present volume his communication
which he sent in advance
Tinbergen - pag.
1234 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2K
of large countries, feel they must pay some attention to the
regional distribution of well-being and hence of new investments
and a large number of studies and activities are in operation
for the preparation of regional development. Since this
dimension brings into the realm of planning some other aspects
known as physical planning and more heavily depends on some
social issues, these studies and activities are carried through
by a variety of different experts among which the general economist
plays a minor role. Together with the need for very
detailed decisions involved all this has led to a situation in
which the main economic interdependencies are not as much
observed as in macro or sectorial planning. There is a clear
need for an economic framework for regional planning, that is
for models, methods and procedures of such planning. It is
the purpose of this paper to draw a few lines along which it is
hoped practical work can be done.
2. The addition of the dimension of space or distance to
economic analysis signifies a considerable increase in the number
of variables and of equations to be included in the models
of economic behaviour. Moreover their practical use requires
the knowledge of a large number of new « data », that is,
coefficients occurring in such models. Many of these data are
not readily available. In order nevertheless to arrive at workable
models, we must simplify as many less relevant elements
as possible: the old art of science building. It goes without
saying that various types of simplifications will have to be
tried out and compared before some standard approach can
be attained. It also stands to reason that one concrete situation
will not necessarily require the same type of simplification as
another concrete case. We propose to make a number of
suggestions meant as challenges for further discussion. These
suggestions may be seen as a further elaboration of previous
suggestions by the author [4].
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3. The first possibility for simplification consists, of course,
of a limitation of the number of sectors and of regions. This
means that the model we are going to discuss must be seen
as one indicating the main links only; one, therefore, of a
first orientation and a stage only in the complete process of
planning. From what follows it will be seen that we will
listinguish nevertheless between a few categories of sectors,
meaning that we have in mind a number of around ten sectors
still, maybe a few more even. As for the number of regions
anything between two and five may do. This implies that we
do not propose to deal with the micro-economics of regional
policy. It remains conceivable, however, that once models
will be manageable consisting of much larger numbers of sectors
and regions. For the time being the precise delimitation of the
regions will not constitute too much of a problem; it is selfevident
that as much homogeneity in well-being and sectoral
composition or demographic characteristics as possible must
be striven for; a further criterion will be that within a region
transportation costs must be clearly lower than between regions.
[mportant natural barriers to transportation therefore represent
natural frontiers between regions. Even with the small number
of regions proposed meaningful problems may be approached;
well-known examples of two-region problems are those of Italy
or of Belgium; examples of three-region countries are Peru
and Ecuador; problems of urbanization can be dealt with by
the assumption of « regions » consisting of groups of cities of
various size classes and a rural region and so on. We must,
however, reserve one region of our model to represent « foreign
countries » or the « rest of the world » in order to deal with
‘he phenomena of imports and exports. That then means that
in the case of Italy or Belgium, just quoted, we have three
« regions » and in those of Peru and Ecuador four. We need
not emphasize that for problems concerning individual giant
projects a subdivision « ad hoc » of the economy can be
chosen.
+, Tinbergen - pag. 3
1236 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 22
4. Important simplifications can be obtained by a type of
approximation well known from the theory of international
trade: the assumption that for some sectors transportation costs
can be neglected within a certain area and considered prohibitive
outside such an area. This is what the theory of international
trade does with regard to the factors of production. We
propose to do this for some products. In this train of thought
we will speak of a regional sector whenever its product cannot
be transported from one region to another. We think building,
‘he operation of buildings, retail trade and some more services
‘education, for instance) are good examples. We may occasionally
lump these sectors together or only distinguish two
of them, a labour-intensive and a capital-intensive one. The
consequence of our assumption is that any increase in the production
of other sectors in a given region will imply an increase
un production in the regional sectors in the same region.
5. A similar assumption will be made with regard to some
other sectors, to be called national sectors. Their products
cannot be moved between nations, that is, cannot be imported
or exported. Examples are government services, energy and
inland transportation, the latter by definition. As a first approximation
often building materials production may be added.
Energy will not be an example wherever interconnected electricity
networks exist, as in Europe.
For several of the regional and national sectors we may
neglect transportation costs if no transportation actually
takes place; for some others it will be very essential to take
into account transportation costs. This applies to building materials
and probably to some agricultural and mining products.
6. The remaining sectors will be called international sectors.
All commodities appearing in the balance of payments
are necessarily of this category; the inverse is not true, namely,
that they must appear in the balance of payments. It is
181 Tinbergen - pag. .
SEMAINE D'ÉTUDE SUK LE ROLE DE L ANALYSE ECONOMETKIOUE ETC. 1237
characteristic for these sectors that their production need not
be equal to their disappearance into national final demand and
Inter-sectoral deliveries. It is by manoeuvring with the productive
capacities of these sectors that the economy must try
to attain the highest comparative advantages, that is, to maximize
national product — maybe within limits set by regional
policy. As another simplification the transportation costs of the
oroducts of these sectors may be neglected.
7. It seems proper to use the method of input-output analysts
for the description of the necessary inter-sectoral deliveries.
Occasionally choices between alternative production processes
may have to precede and for some particularly important cases
even to be built in into the model making it a linear programming
model. It is characteristic for the subdivision of an economy:
into regions that there may be relevant differences in
production costs of the same commodity between regions. This
will translate itself into differences between one or more of the
input coefficients for the corresponding sector. This may simplify
itself up to the point where production in some sectors is
only possible in a limited number of regions, a good example
being mining or energy. We do not call such a sector a regional
sector, as the reader might suppose; its product will serve many
regions.
In some sectors indivisibilities may play a relevant role:
this is true for irrigation, energy and heavy industries. It
may be important for some services including university education.
We may follow one of two ways of representing this
feature: either we may assume a minimum value for investments
to be made in such a sector, or we may assume curvilinear
relations such as the well-known .6™ power of production
determining investment inputs. The latter approach cannot
be chosen too often if we do not want to make the system of
equations unmanageable.
ol Tinbergen - pag.
-
1238 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
8. Transportation problems may play a crucial role for
some national (and some international) products of a relatively
heavy nature (agricultural and mining products). Here the
interplay between location and costs at other locations may
have to be brought in. For these products the sub-division
of the economy into regions implies that we introduce prices
for the same product in different regions or even that we distinguish
between alternative regions of origin for one product.
The latter assumption may mean that we have, in each region,
a set of price notations for the same good with different origin
and at the same time implies the assumption of quality differences
and imperfect markets. The latter approach, though seemingly
more complicated, may actually be simpler since it permits
us to assume a finite elasticity of substitution between
products of different origin. It also permits the assumption
that the price of a good outside its region of production equals
its production cost in its region plus transportation costs to the
region of destination.
The alternative assumption of a uniform price in each region
for each product implies the use of inequalities among the
restrictions, usually mathematically more cumbersome to
handle.
9. For the type of commodities just discussed a key must
be used for the distribution, by the buyers, of total needs over
the conceivable regions of origin. These needs, to be indicated
as disappearance, equal the sum total of final demand in the
region (consumption plus investment), possible exports to foreign
countries and inputs used in the region by other sectors.
This disappearance itself originates from production within the
region and imports from other regions (including, possibly, foreign
countries). In loose terms, it will be bought wherever it
is cheapest, taking into account transportation. More accuracely
the key of distribution must be formalized and two alternative
keys have been discussed elsewhere [1]. The remarks
‘181 Tinbergen - pag. 6
SEMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIOUE ETC.
23C
made in the previous sector may be considered preparatory
remarks to the application of one or the other of these keys.
Clearly there are more possibilities [3]. Mathematically the
simpler key is the one where we assume an imperfect market
and finite elasticities of substitution between the imports from
alternative regions.
10. The preceding sections are a sketch only of a type of
model which may be used as a framework for regional development
planning. The vague way in which it has been described
illustrates the large number of possibilities of adapting any
concrete model to the particular structure of the country it has
to serve. Thus, the number of regions and their frontiers, the
number and nature of sectors and their distribution over regional,
national and international sectors may be so chosen as
to approach reality as much as possible. Even so hardly all
the coefficients it contains will be available from statistical
measurement and some of them will have to be chosen rather
arbitrarily. This applies especially to the substitution elastici-‘ies
for the products whose transportation costs cannot be neglected.
Probably it also applies to whatever cases of indivisibilities
will be introduced.
Even when the model has been established another choice
has to be made: the one of the type of development policy one
wants to analyse. It is necessary to follow the habits developed
in the theory of economic policy and to define the aims and
means of such a policy. The aims may either be chosen as a
set of numerically given targets or as the maximization of some
social welfare function. For regional policy the welfare function
will depend also on regional variables such as income per head
of the various regions and the most common policies usually
imply a reduction of the income differences. Many specificalions
are possible. To laymen such criteria as first raising the
income per head of the poorest region and after a certain proportion
to the next poorest region has been reached raising the
"181
Tinbergen - pag.
~
1240
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
income per head of the two poorest, and so on, has a certain
appeal. To the mathematician other formulations may be more
attractive, such as a welfare function depending on national
income and (negatively) on the standard deviation between
regional incomes per head; or a weighted average of the per
capita incomes of regions as a welfare criterion, where the
weights may be indicative of the importance attached to raising
the particular region’s income they refer to. Of course there
must also be a dynamic aspect to the welfare function, that is
that future incomes also influence it. We may take care of that
aspect in the macro phase of planning, however, and use regional
welfare distribution in the annual plans only.
As for the means of development policy we may think of
a wide variety again, from very few up to a large number of
them. We may also think of intermediary targets temporarily
considered as means. It is customary to think of investments
in a number of sectors as means in this sense; the main problem
being in which sectors and in which regions to take them so
as to serve the aims of the policv chosen in the best wav.
IT. For practical purposes development policy planning
will often have to be undertaken in successive steps (stages) [5].
We assume that in a macro stage the rate of development
and the corresponding volumes of total investment have
already been chosen for a series of years. We now concentrate,
in another step, on the distribution of this investment
volume over sectors and regions. We assume that the welfare
criterion is a weighted sum of regional incomes. For the time
being we disregard economies of scale. Each international sector
is eligible for an additional investment of standard size (say
one million currency units). Moreover, such investment can be
made in any of the R regions. After it has been chosen in region
7, it will require additional investments in regional and
national sectors. The former must take place in the same region
». The latter may again be undertaken in anv of the R
18] Tinbergen - pag. &
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC.
1254
regions. This means that each project can be presented in
R? alternative versions. With the aid of our model we can
determine the additional incomes it creates in the R regions
and hence also the weighted sum of regional incomes representing
the increase in welfare. We can also determine the total
capital to be invested in the bunch of projects consisting of the
investment of one million in the international sector plus the
investments in the regional and national sectors. The method
to be used is that we put equal to zero the investment in all
other international sectors. I have proposed to speak of the
semi-input-output method [1]. Thus the income-capital ratio
for all R? versions of the project can be found. The version
with the highest ratio is the « best version ». We may now
make a list of the ratios obtainable for the best version of investment
in each of the international sectors.
The use we can make of this list is that we select the international
sector with the highest ratio and invest all available
capital in this sector. With regard to international sectors this
will mean complete specialization. To be sure there will be
also additions to production in regional and national sectors
involved.
This complete specialization will be avoided whenever we
add restrictions to the additional quantities that can be exported
in any one international sectors. Whenever such a bound has
been reached, the second-best international sector gets its turn.
Complete specialization may also be avoided if we introduce
an element of decreasing returns, not yet discussed, but realistic
n agriculture and mining. It is not to be expected that it will
be avoided by the introduction of indivisibilities, or increasing
returns. But it may be avoided also if instead of restrictions
to exportable quantities we introduce a price level of export
goods negatively depending on quantities produced "z7
The introduction of non-linear inputs brings in the possibility
that the effects on regional income of two projects are not
31 Tinbergen - pag. g
1242 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARTA -
2.
additive. This calls for much more complicated methods of
selection of the best set of projects.
The problem of distributing a given volume of investment
over a number of sectors and of regions also takes a different
mathematical shape when we adhere to the policy criterion
called the layman’s (cf. section 10). In the beginning of our
selection process we are now only interested in the poorest
region’s income; that is, our weighted sum takes an extreme
form, where all other regions have zero weights. After the
poorest region’s income has reached a certain level we change
the weights. Meanwhile we have already added to other regions’
incomes, since as a rule each project bunch will add to
the incomes of several regions. With the changed weights our
preference for the projects not yet chosen may change. This
may make for diversification (that is, non-specialization in the
international sectors). We may have to stop our selection of
projects at the moment that all available capital has been used.
Up to that moment we have not only added, by our choice, to
the income of the preferred regions, the poorest, but also to
other regions’ incomes. It is not excluded that this « loss »
could have been less if we had chosen another order of selection.
We only want to mention this point, but we are not going
to treat it.
Finally there are problems created by the time structure
of the investment program. The selection rules we have discussed
will not, as a rule, simultaneously exhaust the capital available
in consecutive years. If this phenomenon assumes dangerous
dimensions we are confronted with the problem of a variety
of scarce factors — not so far discussed in this paper —
requiring the introduction of shadow prices for these factors
or equivalent methods. It is beyond the scope of this paper
to deal with this aspect.
12. A few concluding remarks will be made on the organization
and procedure of regional development policy. These
18] Tinbergen - pag. 10
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
1245
have to be more sketchy even than the preceding sections. We
will concentrate on two aspects, namely the possibilities of
decentralization of policy decisions and the question of the time
order in which decisions must be made. Both depend on the
mathematical structure of the model describing the relationships
between the unknowns, that is, the policy parameters or instrument
variables [6]. Decentralization, either with regard to
sectors or with regard to regions, will be possible whenever a
number of sectoral or regional parameters appear in a set of
equations not containing parameters of other sectors and regions
and sufficient in number to solve for the unknowns concerned.
If it so happens that after a decision on some instruments
further instruments occur in the same position, decisions
on the latter must be taken after decisions on the former. Since
the structure of the equations (or, if we like, the matrix of their
coefficients) does not only depend on the relationships between
the variables but also on the role given to some of them, it
depends on the target and instrument variables, that is, on the
type of policy, what possibilities for decentralization and what
necessities as for time order there are. A simple example may
illustrate our point. If targets are set for regional incomes —
a case different from the ones so far discussed — the production
volume of regional industries may be derived at once, provided
we can assume a direct relation between regional income and
production of regional industries.
Among the most fascinating and pressing problems oi
sectoral-cum-regional planning is the one whict forms of
decentralization — whether sectoral or regional and what
order of decisions is optimal [2]. Models of the class discussed
in this paper mav teach us about these questions
3 |
Tinbergen - pag. 11
244
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
REFERENCES
i. H.C. Bos and J. TINBERGEN, Mathematical Models of Economic Growth,
New York, 1962.
2] W. Isarp and E. SMOLENSKY, Application of Input-Output Techniques
to Regional Science, in: « Structural Interdependence and Economic
Development ». (Tibor Barna, ed.). London, 1963, p. 105.
3] W. LEONTIEF and A. STROUT, Multiregional Input-Output Analysis in:
« Structural Interdependence and Economic Development ». (Tibor
Barna, ed.). London, 1963, p. 119.
4] J. TINBERGEN, Regionaal-economische planning, Seminarie voor Toegepaste
Economie bij de Rijks-Universiteit te Gent, Gent 1961.
5] J. TINBERGEN, Planning in Stages, Statsgkonomisk Tidsskrift, 1962, p. 1.
6] J. TINBERGEN, De optimale organisatie der economische beslissingen.
Mededelingen der Kon. Ned. Akad. v. Wetenschappen. afd. Letterkunde,
Nieuwe reeks - Deel 24 - No. =~
1o| Tinbergen - pag. 12
a
5st OMS
INAL STATEMENT
At the conclusion of this Study Week, we should like tu
thank the President, Members and Chancellor of the Pontifical
Academy of Sciences for their initiative in arranging an international
discussion of the problems of Econometrics and for
their generosity and hospitality in putting at our disposal their
exquisite and unique Casina di Pio IV.
In the course of this Study Week, we have had the opportunity
to review and discuss many recent developments in
‘he following branches of our subject: macro-economic decision
models and development planning; optimal growth models;
the problem of uncertainty in development programming; the
influence of real capital on the growth of the real national
income; fiscal policy and economic growth; regional planning;
cost-benefit analysis; statistical tools useful in econometric
planning; the foundations of dynamic econometric models in
probability theory, and estimation procedures for econometric
models.
In addition to discussing the prepared papers, we also
considered the paths that future research might profitably
follow. Throughout our meetings, many different points of view
were expressed and opinions varied considerably on a number
of subjects; but general agreement was reached on the following
statement and proposals.
Although, as an organised science, Econometrics is barely
a generation old, it has already made substantial progress and
is attracting more and more attention and interest throughout
248
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 7
the world. In the last analysis, this achievement is probably
due to the effective learning process which results from combining
theories developed by means of mathematics, and thus
sharing in the clarity, rigour and power of that subject, with
observations of the real world, reduced as far as possible to
quantitative terms. Iteration between theory and observation
is leading in Econometrics, as in other branches of science,
to a systematic body of verifiable knowledge about the real
world.
Since the Study Week was concerned with applications of
econometric analysis, our thoughts on the desirable directions
of future research necessarily ranged over a wide area. They
can conveniently be summarised under three heads: the analysis
of the economy; economic objectives: and instruments
of control.
A number of important areas of analysis were mentioned
as particularly deserving of further study; the role of capital
accumulation in economic development; the relationships of
education and of scientific research to economic growth; the
desirability of introducing to a greater extent than hitherto a
regional dimension into econometric models so as to connect
the economic structure of a nation with that of its constituent
regions; and the urgency of developing techniques of quantitative
analysis suited to the less advanced areas of the world.
Emphasis was also placed on the need for the systematic testing
of theory against facts in the construction of explanatory,
forecasting and planning models, and on the importance of
publishing the results of such tests. (Only in this way can
the experience of one country serve as a guide to further studies
in other countries.)
Our discussion brought forcefully to our attention the need
for both empirical and theoretical analysis of the social objectives
of economic development, comparable in purpose and
quality with current research into technological conditions and
economic relationships. Social objectives cannot be deduced
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC,
124C
scientifically, but they are data that science must take into
account in fostering economic development. We still have
anduly primitive tools and methods for ascertaining and
describing even approximately the objectives of any country
or any group within it. For example, we have no effective
way of determining how a community acts when it tries to
reconcile two such competing goals as a high rate of future
economic growth and a high level of present consumption.
Progress in this direction would not only be conducive to
better planning for given goals, but would also contribute to
a clearer formulation of these goals and to an improved level
of political discussion of them. This in turn would lead to a
more intelligent and satisfactory selection of such goals.
Our discussions also made clear the need for a better under.
standing of the capabilities as well as the limitations of various
instruments of economic policies which governments can use
in the pursuit of their short and long-run goals. Research on
the nature of the instruments available has been neglected in
favour of research on more narrowly economic problems, such
as production functions and market behavior in the private
sector. This neglect has led to the adoption of goals that could
aot be attained by means of the available instruments and to
overestimating the effectiveness of some instruments. In short,
more research is needed into what governments can and cannot
do in trying to foster economic development and stability.
A side of economic development which we feel cannot be
overemphasised is the race between increasing productivity and
increasing population. Most of the research discussed at the
Study Week was concerned with one aspect or another of
productivity; yet measures for influencing the rate of population
growth may contribute at times even more to human
welfare than measures for influencing productivity. Much
work, theoretical and empirical, sociological, physiological and
economic, is needed on the population problem. Econometricians
can contribute especially through theoretical studies on
1250 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 26
the relationships between economic development and population
growth and empirical studies on the effectiveness of various
measures for influencing population growth. They can also
contribute by assisting in the improvement of vital statistics
and other demographic data under the difficult conditions prevalent
in under-developed countries.
Econometrics is a powerful tool of scientific analysis. It
cannot in itself determine what economic policy should be,
but it can work out in a rigorous way the consequences of
specific hypotheses and observations. Thus it can help considerably
in the successful functioning of an economic system
by bringing about a better understanding of the system and
an increase in the flow of information available to those who
operate it. This is true whatever the political environment in
which that system functions and whatever degree of development
it has reached. Naturally, continued improvement depends
on the general recognition of the new science by society
at large and on the additional resources, both moral and
material, that may be expected to accompany this recognition.
ALLAIS, BOLDRINI, DORFMAN, FISHER, FRISCH,
HAAVELMO, ISARD, JOHNSON, KoOPMANS, LEONTIEF,
MAHALANOBIS, MALINVAUD, MORISHIMA, PASINETTI,
SCHNEIDER, STONE, THEIL, WOLD.
NOTE COLLECTIVE FINALE
A la fin de cette Semaine d’Etudes, nous tenons à remercier
le Président, les Membres et le Chancelier de l’Académie Pon-‘ificale
des Sciences d’avoir pris l’initiative d'organiser une
discussion internationale sur les problèmes de l’économétrie et
d’avoir mis à notre disposition, avec une généreuse hospitalité,
leur exquise et unique Casina di Pio IV.
Au cours de cette Semaine d’Etudes, nous avons eu l’occasion
d'examiner et de discuter de nombreux développements
apportés récemment dans les branches suivantes de notre discipline:
modèles macroéconomiques de décision et planification
du développement; modèles pour une croissance optimale;
problème de l’incertitude dans la programmation du développement;
influence du volume du capital sur la croissance du
revenu national; politique fiscale et croissance économique;
planification régionale; calculs de rentabilité; instruments sta-‘istiques
utiles pour la planification ‘économétrique; fondements
probabilistes des modèles dynamiques et procédures
d'estimation dans les modèles économétriques.
Alors que nous discutions les mémoires présentés, nous
avons aussi considéré les voies que la recherche future pourrait
suivre avec profit. Tout au long de nos séances, des points
de vue différents furent exprimés et les opinions variaient
considérablement sur de nombreux sujets; mais la déclaration
et les propositions suivantes firent l’unanimité.
1252 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
=
Bien qu’elle soit tout juste âgée d’une génération en tant
que science organisée, l’économétrie a déjà fait des progrès
substantiels; elle suscite de plus en plus d’attention et d’intérêt
à travers le monde. En dernière analyse, ce succès est probablement
dû à l’efficacité du processus de découverte qui consiste
à combiner des théories, développées au moyen des mathématiques
et partageant la clarté, la rigueur et la puissance
de cette science, avec des observations sur le monde réel, traduites
autant que possible en termes quantitatifs. Le dialogue
entre la théorie et l’observation conduit en économétrie, comme
dans les autres disciplines scientifiques, à un ensemble
systématiquement organisé de connaissances vérifiables sur les
phénomènes réels.
Puisque la Semaine d'Etudes concernait les applications de
l’analyse économétrique, nos pensées sur les directions souhaitables
des recherches futures ont nécessairement couvert un
vaste domaine. Elles peuvent être commodément résumées sous
trois grands thèmes: l’analyse de l’économie; les objectifs économiques;
les instruments.
Nombre de sujets importants d’analyse ont été mentionnés
comme méritant particulièrement des études plus poussées: le
rôle de l’accumulation du capital dans le développement économique;
les relations de l’enseignement et de la recherche
scientifique avec la croissance économique; l’avantage que présenterait
l'introduction plus systématique dans les modèles économétriques
d’une dimension régionale grâce à laquelle la
structure économique d’une nation serait reliée à celle de ses
régions constituantes; l’urgence de la mise au point de techniques
d'analyse quantitative adaptées aux pays les moins développés
du monde. L’accent fut mis également sur l’importance
qu'ont des tests systématiques de la théorie par les faits
dans la construction de modèles explicatifs, prévisionnels et
décisionnels, comme aussi sur la nécessité d’une publication
des résultats de ces tests. (De cette manière seulement l’expé-
SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 1253
rience d’un pays peut servir de guide à des études ultérieures
menées dans d’autres pays).
La discussion a imposé énergiquement à notre attention le
besoin d’une analyse à la fois empirique et théorique des objecfifs
sociaux du développement économique, analyse comparable
dans ses intentions et dans sa qualité aux recherches actuelles
concernant les conditions technologiques et les relations
Sconomiques. Les objectifs sociaux ne peuvent pas être déduits
scientifiquement, mais ils constituent des données que la science
doit prendre en compte quand elle veut favoriser le dévelopnement
économique. Nous avons encore des outils et des méthodes
indûment primitifs pour constater et décrire, même de
manière approchée, les objectifs de n’importe quel pays ou de
n'importe quel groupe à l’intérieur d’un pays. Par exemple,
nous n'avons pas de moyen efficace de déterminer comment
une communauté agit quand elle essaie de réconcilier deux buts
aussi concurrents qu’un rythme élevé de croissance économique
future et un haut niveau de consommation présente.
Tout progrès dans cette direction non seulement permettrait
une meilleure planification à buts donnés, mais encore contribuerait
a une formulation plus claire de ces buts et à une amélioration
dans la qualité de la discussion politique les concernant.
Ceci conduirait alors à un choix plus intelligent et plus
satisfaisant des objectifs.
Nos débats ont aussi rendu clair le besoin d’une meilleure
compréhension des possibilités comme des limitations des divers
instruments de politique économique que les gouvernements
peuvent employer dans la poursuite de leurs objectifs
à court et à long termes. La recherche sur la nature des instruments
disponibles a été négligée au profit de la recherche
sur des problemes plus étroitement économiques tels que les
fonctions de production et les caractéristiques des marches
dans le secteur privé. Cette attitude explique à la fois l’adop-
1254 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2.
tion de buts qui ne pouvaient pas étre atteints au moyen des
instruments disponibles et le fait que I’efficacité de certains
instruments ait été surestimée. En bref, il convient de consacrer
davantage la recherche à l’étude de ce que les gouvernements
peuvent et ne peuvent pas faire pour favoriser le développement
et la stabilité économiques.
Un aspect du développement économique qu’à notre avis
on ne saurait surestimer est la course entre la croissance de la
productivité et la croissance de la population. La plupart des
recherches examinées à la Semaine d’Etudes traitaient de la
productivité sous un aspect ou sous un autre; cependant les
mesures destinées à influencer le rythme de croissance de la
population peuvent à certains moments contribuer au bien-être
de l'humanité plus même que les mesures visant à relever la
productivité. Il faut consacrer au problème de la population
de nombreux travaux, théoriques et empiriques, sociologiques,
physiologiques et économiques. Les économètres peuvent apporter
leur contribution en particulier par des études théoriques
sur les relations qui lient le développement économique à la
croissance de la population et par des études empiriques sur
l'efficacité de diverses mesures visant à influencer l’accroissement
de la population. Ils peuvent aussi aider à l’amélioration
des statistiques de l’état civil et des autres données démographiques
sous les conditions difficiles qui prédominent dans les
pays sous-développés.
L’économétrie est un outil puissant d’analyse scientifique.
Elle ne peut pas ÿl’elle-même déterminer ce que la politique
économique devrait être, mais elle peut établir d’une manière
rigoureuse les conséquences d’hypothèses et d’observations
spécifiques. Ainsi, elle peut aider considérablement à l’heureux
fonctionnement d’un système économique en apportant une
meilleure compréhension de ce système et en accroissant le flux
des informations parvenant à ceux qui le conduisent. Ceci est
Vrai quels que soient l’environnement politique dans lequel le
3EMAINE D'ÉTUDE SUR LE ROLE DE L'ANALYSE ECONOMETRIQUE ETC. 1255
système fonctionne et le degré de développement qu’il a at-‘eint.
Naturellement, des progrès continus supposent que la
nouvelle science soit reconnue du grand public; ils dépendent
des ressources additionnelles, à la fois morales et matérielles,
qui devraient normalement accompagner cette reconnaissance
ALLAIS, BOLDRINI, DorFMAN, FISHER, FRISCH
HAAVELMO, ISARD, JOHNSON, KOOPMANS, LEONTIEF,
MAHALANOBIS, MALINVAUD, MORISHIMA, PASINETTI,
SCHNEIDER, STONE, THE Won
~DEX
LE ROLE DE L’ANALYSE ECONOMETRIQUE DANS LA FORMULATION
DE PLANS DF DEVELOPPEMENT
LA SEMAINE D’ETUDE SUR LE ROLE DE L’ANALYSE ECONOME-TRIQUE
DANS LA FORMULATION DE PLANS DE DEVELOPPEMEN"
L’AUDIENCE ET LE DISCOURS DU SAINT-PERE
LES « SEMAINES D’ETUDE » ET LEUR REGIEMENT
t RAVAUX SCIENTIFIOUES
7] The analysis of economic systems (R. |STONE)
Discussion
Toward a verdict on macroeconomic simultaneous
equations (H.G A Worn)
Discussion
Econometric analysis for assessing the efficacy of puilic
nvestment 7 | DORFMAN)
Discussion
1258 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28
[4] On the concept of optimal economic growth
(T.C. Koopmans) . .
Discussion. .
225
280
5]
Croissances optimales dans un modèle macroeconomique
(E. 4 MALINVAUD) :
Discussion.
301
379
[6]
Dynamic structure and estimation in economy-wide
econometric models (F.M. FISHER) i
Discussion.
385
449
7
Decision rules and simulation techniques in development
programming (H.{THEIL) . .
Discussion . . .
465
495
+
[8]
Some observations on countercyclical fiscal policy and
its effects on economic growth (T.; HAAvELMo) . . 503
Discussion . . 517
9
Balanced growth and technical progress in a log-linear
multisectoral economy (M. MORISHIMA) .
Discussion. .
/
529
557
[101
A new theoretical approach to the problems of economic
growth (L.L. ) PASINETTI)
Discussion .
571
680
[11]
The role of capital in economic development (M.4AL-LAIS)
rx ® 8 1s =
Discussion —. ; . . .
697
979
27
Spatial organization and regional planning: some hypotheses
for econometric analysis (W. (IsARD) =
Discussion
1003
1020
w
, The rates of long-run economic growth and capital
transfer from developed to underdeveloped areas
(W. |LEONTIEF)
Discussion
1039
1057
SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC.
14] The social transformation for national development
(P.C. (MAHALANOBIS, n. M
15]
Statistical tools and techniques in perspective planning
in India (P.C. MAHALANOBIS, n. II). .
Discussion 1.
16]
Econometric analysis and agricultural and development
plans (D.G., JoHNsoN)
iscussionn
17]
Selection and implementation the econometrics of the
future (R. | FRISCH) ;
Discussion
1250
106Q
IIO3
1133
1141
1181
1197
1205
18] The economic framework of regional planning (J. {TIN-BERGEN)
. ---"-oP
CONCLUSIONS
Final Stateme..
“
Note collective finale
5
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year — the performance for the first four years can be compared
with what would have been required if the goals were
to be achieved. This is done in Table 8 by assuming that the
output path between 1958 and 1965 was to have been linear Mh.
For the four years ending in 1962 the annual increase in
gross agricultural output was only 18% as large as that required
to achieve the 1965 goal. Of the ten commodity groups,
there were two with output declines. For the other eight, the
ratio of the actual to the planned increase ranged from 21 to
93%, with only one of the ratios being in excess of 50%.
For the commodity with the highest ratio of actual to planned
increase — sunflowers — the planned increase for the period
was only seven per -cent.
The record of output performance of Soviet agriculture is
somewhat better than that indicated by the estimated seven
per cent increase in output between 1958 and 1962. In 1958
climatic conditions were very favorable and agricultural output
was probably five per cent greater than it would have been
under average climatic conditions. However, in the published
discussions of the Seven Year Plan I have seen no evidence
that the effect of the favorable conditions in 1958 was taken
into account in establishing the 1965 goals. I suspect that the
modest increase in planned output for sunflowers and other
oil-bearing seeds reflected the high 1958 yields, but I have seen
no place where this was recognized.
A brief comment concerning the decline in sugar beet output
between 1958 and 1962 may be in order. There is little doubt
that the output of sugar beets for refining could have been
larger in 1962 than it was. The sugar beet goal for 1965 appears
to be an instance where the goal was set too high in terms
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(') According to data presented in a speech by N.S. Krushchev
(« Pravda », March 5, 1962) it appears that the annual goals for agricultural
products for 1958-65 are approximately linear interpolations of the 1958
output and the 1965 goal. For two examples given — grain and milk —
almost exact linear interpolation was involved; for the third example —
meat — output was supposed to have increased at a greater absolute
amount in earlier than in later vears
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