# Full text: Study week on the econometric approach to development planning

```276 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -
>
Proofs of (F), (G). These propositions express, and pro-
vide 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 con-
sumption 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 inequal-
ity 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 condi-
tions («), (8), as necessary and sufficient for the optimality of
a path (fe, &). We shall first look at the implications of con-
dition (B) in isolation. Called the Euler condition in the « cal-
culus 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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