STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 325
Proof. The proof is based upon appendix Lemma C.l. For convenience,
it is broken into three easy steps.
with
(i) We first notice that
Vxfa, c2) - (1 + r*) U,fa, c2) =det hfa) = f(Ufa, c*)), (4.3)
f'(U) = Wu - (1 + r*) t/22]/C/2. (4.4)
Indeed, differentiating both sides of (4.3) with respect to c2, we verify:
dh/dc, = Ult - (1 + r*) £/22 = /'(#) • tf8 , which satisfies (4.4);
d*hldc2* = Ula - (1 + /•*) tf22! =/"(C/) • 0,R = 0 implies that /is a linear
^ convex
function of U. Let c2' be such that Ufa , c2') = J Ufa , c2) dY(cJ;
Lemma C. 1 then implies
R | 0 => J" Afe) «/y(Cl) I A(c,'). (4.6)
By (3.4) and the definition of h, j hfa) cPPfa) = 0; therefore,
R | 0 => hfa) = U,fa , c,') - (1 + r*) U,fa , c,') | 0. (4.7)
(iii) By definition,
£/(^ , c2') = Ufa\ (* - Cl0(l + /•*) + jV)
= max Ufa , (yx - c,)(l + r*) + ;-2 f )-
PART V. DYNAMIC MODELS