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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 continually
 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