268 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA

The construction is illustrated in Figure 14. Since the las
inequality in (46) holds also for all subintervals of [T, T*],
we have from (52), (50), if x*=g(z*) and p<o,

T*
2Une(p) = — jeu (u(æ;) — u(æ*)) dt > — u'(x*) Y [te 2*) dt >

ve (z¥F— 1) .
> w'(x*) ve

since e>0 and #’(x*) y>o0. On the other hand, we have from
(52), (39) with x,>0, (36) and (38 a) that

dt< fg) dt < (1*—=) æ

whence 1* —- t™>z/%#>0 and

33)

— Uz (p)> o/(x*) y e2/2Za*>o.

Finally, we define the feasible path (x3, 2}) by (36) and

2, =max §2., z*} for 0=<t<Tkx, 2 =z, for t>Ty ,

subject to a later choice of Ty in such a way that 24. <z*,

Then

Ur (o)=— le <t (u(æ;) — u(æË)) dt > ny, ** , where 2*>o,

4) Koopmans - pag. 44