SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 
~ 
ZI: 
(J) In the unique optimal path (X,, Z,) for any initial 
capital stock z, with 0<z,<z, z,#7(p), both =, an’ ~~ exhibit 
a monotonic and asymptotic approach to x» a - mt 
tively, from above if z_> 7z(p), from belo: 
For later discussion, we note from (27) that the asymptotic 
level £(p) of consumption per worker, while independent of 
the initial capital z,, is reduced as the discount rate is increased. 
The maximum of #(p) for p=o0 is attained at p=0 We shal 
not examine the cases where p=>f'(0) - A. 
Finally, a word about the case where one tries to apply 
a negative discount factor p<o. Writing - g=0, this means 
looking for a utility function extending the finite-horizon 
example 
et u(x,) dt 
to an infinite horizon. This is not as far-fetched as it may seem. 
After all, we have so far given no weight at all to mere numbers 
in comparing generations. If we were to weight each gene- 
ration in proportion to its number, and otherwise seek neutra- 
lity with regard to timing, the population growth parameter 
would take the place of ¢ above. 
In order to apply RaMSEY’s device in the present case, one 
would have to find a feasible path (x,, z,) such that 
(31) 
V" 
n 
, €” 
VU, 
— u(x) dt 
[4] Koopmans - pag. 27