STOCHASTIC

322 DREZE AND MODIGLIANI
Consumption and portfolio decisions may be taken sequentially and are
"separable", in that sense. 21
When 3 2 (f/1/C/2)/0c 2 2 ^ 0, then the sign of that quantity is also the sign
of the impact of endogenous uncertainty on current consumption. It is
noteworthy that U-JUt, hence its second derivative, is invariant under
monotonic transformations of the utility function, and thus independent
of risk aversion. There thus exist ordinal preferences, consistent with our
assumptions, such that endogenous uncertainty results in increased consumption,
and alternative preferences such that the opposite result holds.
In the latter case, the consumer chooses an uncertain prospect which
yeilds a higher expected utility than the sure prospect of identical market
value, but he simultaneously chooses to consume less in the first periodpostponing
the (uncertain) benefit to the second period. Such behavior
is consistent with risk aversion, in spite of the saying that "a bird in hand
is worth two in the bush."
3.5. It is appropriate at this point to inquire about the meaning of the
rather unfamiliar quantity d^UJU.J/Bc., 2 which controls the response of
consumption to endogenous risk, and to inquire whether there is ground
for supposing that some sign is more plausible than another.
First we recall that UJU2 is a familiar quantity, the slope of the indifference
curve; hence o(t/j/'(/2)/3r2 is the rate of change of the slope of the
indifference curves as we increase c2 for fixed ct. That derivative must
have a positive sign by Assumption III fo is not an inferior good). The
function (.S 2 (U1/U2)/8c2 i )i measures the curvature of UJU, as a function
of c2.
An intuitive explanation of the relevance of B 2 (U1/U2)/cc2 2 for consumption
decisions under uncertainty is provided in Appendix D, by means of
a simple graphical illustration.
More generally, the following can be said:
(i) d^Ux/U^/dc^ = 0 identically in rx and c, if and only if
U(ci» c2) = F(g(ci) + Kci) • fa). F' >0,h>0 (see Appendix B). That is,
d^UJU^/dc^ = 0 is the ordinal property of Uic^, c2) that is necessary
for risk neutrality in terms of c2, and sufficient for such neutrality to
obtain under a monotonic transformation of U.
(ii) given any r > — 1, y > 0 and u, there exist y1 and
j>2 = (y — y\)(\ + r) such that dcjdr = 0; when dcjdr = 0, then
d 2 cjdr 2 has the sign of a 2 (C1/(/2)/ac2 2 (see formulas A.7-A.8). That is,
d^UJU^/dcz 1 = 0 is the ordinal property of f/fe, c2) that is necessary for
al An extension of these propositions to an n-period model, n > 2, has been provided
by Pestieau [11], under the additional assumption of homothetic indifference surfaces.
1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 473