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 generation
 in proportion to its number, and otherwise seek neutrality
 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