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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 98

The constant optimal path made possible by the initial capital
 stock z,=2(p) is found to be an asymptote for the optimal
paths associated with other values- of z,.

(I) For all z, with 0<z,=Z, the unique optimal path
(X,, Z,) is uniquely characterized by the two conditions

(2) lim 2:=2(p),
T=00

(B) the prices (21), (22) implicit in the path (X,, Z,)
satisfy the differential equation

a,
7
JJ
a

g,+p,=0 for all =o.

To interpret condition (B), let (x, z,) be a path which differs
 from the optimal path only slightly and only on a short
open interval IJ, on which 2,>2%, (see Figure 9 a). Then x,
will differ from #, first because the slightly higher capital stock
on J allows a slightly higher product, and secondly because
acceleration of investment during the first part of J and deceleration
 during the second part leads to some postponement
of consumption within J. In the light of (21), (22), the condition
 says that, for an arbitrarily small difference z,- 2, of
arbitrarily short duration, the utility effects of these two components
 of x,- x, must cancel if the path (£, 2,) is to be
optimal.
If condition (8) of Proposition (I) is satisfied, the inequality
between the first and third members of (25) becomes

— ~

x, — pl
Pe t ay) dt — PrlEr -- 21, ÉO .

r.] Koopmans - pag. 24