48 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 98 
The constant optimal path made possible by the initial ca- 
pital 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 dif- 
fers 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 de- 
celeration during the second part leads to some postponement 
of consumption within J. In the light of (21), (22), the con- 
dition says that, for an arbitrarily small difference z,- 2, of 
arbitrarily short duration, the utility effects of these two com- 
ponents 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