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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
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Thus, if utility is measured in conformity with (14), no path 
is « infinitely better » than the golden rule path. In particular, 
no feasible path x, can indefinitely maintain or exceed a level « 
of utility flow that exceeds #. Thus the golden rule path con- 
tinually attains the highest indefinitely maintainable utility 
flow. 
(B) For every feasible path, either lim U, exists (is a 
T+00 
finite number), or Ur diverges to — oo as T tends to oo. 
In the first case, we call the path eligible, in the second 
ineligible. Then (B) establishes a clear superiority of each 
eligible path over each ineligible one. On the eligible set we 
choose as the utility function 
ry 
U= Je {x;) — @) dt 
In propositions (C), (D), an optimal path is defined as a 
path maximizing U on the set of eligible and attainable paths. 
It is not hard to find eligible and attainable paths for 
every admissible initial capital stock z,. If z,>2, one only 
needs to refrain from net investment until the capital stock 
Z,=3L eM of the golden rule path has caught up with the 
given initial stock Z,=z,L,» and to continue along the golden 
rule path thereafter. If o<z,<É, one can through a finite 
period of tightening the belt arrive on the same path. 
(C) For any initial capital stock z, with o<z, SZ there 
exists a unique optimal path (X,, Z,) in the set of eligible and 
attainable paths. For z,#2, both &, and Z, exhibit a strictly 
monotonic approach to & and Zz, respectively, from below if 
0<2,<2, from above if 2<z,ZZ. For z,=2, the optimal path 
is X,=X, Z,=Z for all t, the golden rule path. 
Fal Koopmans - pag. 18