«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