STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 327
A strong case may be made for regarding endogenous risk aversion as a
meaningful, operational concept. Arrow [2] argues as follows that absolute
risk aversion for total wealth may reasonably be expected to decrease
with wealth: "If absolute risk aversion increased with wealth, it would
follow that as an individual became wealthier, he would actually decrease
the amount of risky assets held" (p. 35). In that argument, wealth is used
as a primitive concept, and the increase in wealth is treated as exogenous.
The argument may, however, be reformulated for the case where assets
are acquired with savings and used to finance future consumption. One
would then say: "If absolute risk aversion for c2 increased with c2 along
a budget line, it would follow that as an individual accumulated more
wealth, he would actually decrease the amount of risky assets held." One
may thus consider that standard arguments invoked to discuss increasing
versus decreasing absolute risk aversion for "wealth" apply almost
verbatim to "risk aversion for c2 along a budget line," that is, to endogenous
risk aversion.
The arguments for decreasing absolute risk aversion are perhaps not
compelling (the argument quoted above lacks generality when there are
more than two assets), but it is a general conclusion that £t is less than,
equal to or greater than its "expected utility" certainty equivalent according
to whether absolute risk aversion for c2 decreases, remains constant or
increases with c2 along budget lines defined by r* — with some plausibility
arguments in favor of the "decreasing" case.
4.5. There remains now to relate the results of Sections 3 and 4. This
will be done in three steps.
(i) When there exist perfect insurance and asset markets, then y2*
(as defined in Theorem 3.3) < y2 f (as defined in Theorem 4.3). Indeed,
define y by
max Eu (c,, (yi - Cl)(l + r) + v°°) = U{ct\ (y, - q'Xl + r*) + y2*)
= r£ Wi , (yt - yl". Similarly, because the
chosen future income could have been exchanged against the certainty of
V. yf > y%*. It follows that cj* <