«44 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
which differs from V(p) by a constant. As before, we shall 
write Ur(p) if the integral in (20) extends from o to T<{oo. 
The stipulation in (E) that keeps consumption from be- 
coming altogether too small is necessitated by (15), merely to 
prevent Uz(p) from diverging to - oo as T—oco. However, we 
shall for p>o define as the eligible-and-attainable set the set 
of all paths with the prescribed z, for which V(p) exists. (E) as- 
sures us that no paths worth consideration are excluded from 
the eligible set. If z, were to be very small, we could still 
allow for growth by taking x correspondingly smaller. 
In the following propositions (F) through (J) optimality 
is defined by maximization of (20) on the appropriate eligible- 
attainable set. It is assumed in propositions (F), (G), that an 
eligible-attainable path (£, Z,) is given, which is under scrutiny 
for its possible optimality. The propositions associate with 
such a path tentative implicit prices of the consumption good 
and of the use of the (identical) capital good. Once optimality 
of the path (£,, 2,) is confirmed, these prices are no longer ten- 
tative, and generalize to an infinite-dimerisional space the idea 
of a hyperplane separating attainable from better-than-optim- 
ally-attainable programs, illustrated in Figure 5. The (dated) 
price of the consumption good is defined from (20) by 
> 
fy 
{,=e-" X 
; u(x), 
the present value of the marginal instantaneous utility of con- 
sumption at time ¢ if the given path (%,, 2,) is followed. The 
price of the use of the capital good is similarly defined by 
2. 
i 
eo Ky 
4] Koopmans - pag. 20