276 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

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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