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 there- 
fore 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 propo- 
sitions (H), (1). 
4] Koopmans - pag. 26