SEMAINE D'ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC.
mathematical situation, and the existence of an optimal program is
found to depend on a stronger restriction on the utility function used.
To facilitate analysis, I have studied this restriction only within the
class of stationary and additive per capita utility functions, expres-
sible as a sum of future per capita utilities derived from a constant
one-period utility function #(x) and discounted at a constant rate ç.
Within that arbitrarily chosen class, an optimal path is found to
exist if and only if g>0. The question of existence of an optimal
path is so far unresolved within the wider class of continuous but
not necessarily stationary or additive utility functions. It is plausible
to assume that within that class there will again be a subclass for
which, in a given technology, no optimal path exists
MORISHIMA
Your argument is based on the assumption that the rate ot
growth of population is constant. This, together with others, implies
the uniqueness of the Golden Rule path. Suppose, instead, that the
rate of growth of population is an increasing function of the con-
sumption per capita until it reaches a certain level, after which the
population growth-rate will decrease. Then there are possibilities
of multi-Golden Rule paths. You shall have to be concerned with
the comparison between Golden Rule paths (the best Golden Rule
path, the second best Golden Rule path and so on) and also with the
locality of the stability of the best Golden Rule path.
You treat capital and labour in an asymmetric way: capital may
be unused if too much capital is available, while labour is fully
employed throughout the whole process. May I say that a certain
degree of optimality has already been presupposed in your as-
sumption of automatic maintenance of the full employment of la-
bour? Is the full employment of labour maintained even if labour
is treated in the same manner as capital, i.e. if there is a possibility
of unemplovment of labour?
Koopmans - pag. 71
