STOCHASTIC

320 DREZE AND MODIGLIANI
(3.5) The expected marginal utility of a unit investment in every asset
is the same; the expected value of the rate of return on every asset, weighted
by the marginal utility of future consumption, is equal to the rate of return
on the safe asset;
(3.6) The expected marginal utility of a unit worth of insurance on
every source of earnings is the same; the expected value of the earnings from
any source, weighted by the marginal utility of future consumption, is
equal to the insurance value of these earnings.
Clearly, if there exist perfect asset markets, but no insurance markets,
the solution to (3.3) with all c^'s equated to one's is still given by (3.4)-
(3.5), and if there exist neither asset nor insurance markets, the solution
is given by (3.4).
Furthermore, if the rate of return on the entire portfolio, namely,
r« + ( Z (n - r o) x,j\x0 + £ x,\\,
is still denoted by r, and since y2 =def S«i Ja , (3.5)-(3.6) imply
m
EV2rjEU2 = r0 , EU2y2jEU2 = z2 = EU2 £ (a(^2 + (1 - a,) zf2) EU2 .
(3.7)
3.3 We now state and prove a theorem that has an immediate bearing
on consumption and portfolio decisions with perfect markets. It does,
however, admit of a somewhat broader interpretation, which justifies the
notation "y2*, r*, Cj*" introduced in the statement of the theorem.
THEOREM 3.3. Let y2* =det Ey2U2{£x, c^/EU^ , c2), r* =„„,
ErU2(^ , c2)/£I/a(4 , c2) and define cx* = ct*(r*, y2*) by
Then
%*, (* - Cx*Xl + r*) + J2*)
- (1 + /•*) t/2(Cl*, (yt - Cl*)(l + r*) + y2) = 0. 19
d 2 -zj~ 8c2 2 = 0 (identically in c2, given ^) implies (x = Cj*.
Proof The proof is based upon appendix Lemma C.2. Let
Wi. c2)/t/2(4 , c2) — (1 + r*) =ietf(c2); we may rewrite (3.4) as
0 = J UJ(c2) dV(c2) =