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