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