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