«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 becoming
 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) assures
 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 eligibleattainable
 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 tentative,
 and generalize to an infinite-dimerisional space the idea
of a hyperplane separating attainable from better-than-optimally-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 consumption
 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