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.