STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 321
where V2 = V2(ct, c^ is a function of c2. Let then
* _ EctVt E«yi - ft)(l + r) + J^) t/2
C2 det EV2 EV2
Lemma C.2 then implies
= (y1-k)^ + r*) + y2* by (3.7).
concave <
/(c2) linear => J A(c2) df(c2) |/(c2*) J C/2 df(c2). (3.9)
convex "^
Now, \V2dV(cij > 0, and/(c,s) is a linear function of U-JU2, whose
concavity properties (in c^) are determined by the sign (assumed constant)
of S 2 (i/,/t/2)/0c2 2 ; consequently, (3.8) and (3.9) together imply
i^ujv^ < 0 ^ ^ > C2*)/Wi _ Cz*} _ (1 + r,} | 0
=> tfiCi. «.*) - (1 + i-*) ^ i . c2*) | 0. (3.10)
Assumption III and the definition of c* imply that
so that
tfnfe*. c2*) - (1 + r*) VtS.cS, c2*) < 0,
UJA. *•*) - (i + '*) J/iCi. ^*) 5 o o ex g Cl*.
The theorem then follows from (3.10). Q.E.D.
3.4. When there exist perfect markets for income insurance and for
assets, then (3.4)-(3.7) imply that y2* is equal to the insurance value of
future income (zj) and r* is equal to the market sure rate of return
(»•„).*> Suppose that d^VJV^/Bc^ = 0; Theorem 3.3 then implies that
current consumption tx is equal to the level (Cj*) that would be optimal
if all income were insured (y2 = y2*) and all savings were held in the safe
asset (r = /•*). This result holds independently of the actual insurance
policy and asset portfolio chosen by the consumer. Hence, endogenous
uncertainty has no impact on consumption. Furthermore, y2* (= z2) and
r* (= r0) being directly observable market values, ly (= cx*) may be
chosen first (as a function of z2 and r0), the optimal insurance policy and
asset portfolio being determined thereafter (jointly, for this given tj).
20 Some readers may find it more convenient to transpose this interpretation to the
situation where there is no uncertainty about future income, so that yt = y2*.
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472 PART V. DYNAMIC MODELS