262 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

28

We note that, by Assumption (c), both the feasible set and
the attainable sets are convex, and that the function g(z) defined
in (36) 1s strictly concave. Since g(z) vanishes for z=o0 and for
z=2, it reaches its maximum # in a unique point #, so that

(38) (38a) æ = g(2) > 9(2) for all 22, where o<z<2,
3 A A
(38b) 9(2)>9(2)= 0 > g'(z*) whenever 0 <Cz7<2<2*<Z.

From (35 a), (36), we have

‘
a

3:

2<x,+2 = g(z,

and hence for all feasible paths, using (35 a), (37), and the
fact that g(2)>0 only for 0<z<Z ,

40)

0<2,<z for all t=o0

Here o0=2z, has been ruled out because it would not allow the
positive consumption x, for #=¢ required by (35 a).

À 4.

A BASIC INEQUALITY AND ONE APPLICATION

The concavity Assumption (e) of u(x) implies that

41) u(x)- u(x*)<u (x*)e(x-x*) for all x, x*

4]

Koopmans - pag. 38