STOCHASTIC

324 DREZE AND MODIGLIANI
in (cj, c2)-space, one may wonder whether —(£/22/t/2) increases, decreases
or remains constant when the starting point is displaced in some particular
direction. Leland [8] considers a move along the (tangent to the) indifference
curve through (cj , c2): c2 increases and c^ is simultaneously decreased
to keep utility constant. Leland assumes that such a move decreases
absolute risk aversion, and derives as an implication that current consumption
diminishes if the variance of y2 increases, the expectation of y2 being
kept constant. In other words, such an "increase in risk" reduces current
consumption.
Sandmo [14] assumes that — U22/U2 decreases with c2 and increases
with c1 ; he then defines an "increase in risk" as a multiplicative shift in
the distribution of y2 combined with an additive shift that keeps the mean
constant. His assumptions imply that such an increase in risk reduces
current consumption. 22
We will now state and prove (Section 4.3) a theorem and a corollary
that generalize the analysis of Leland and Sandmo. An interpretation of
our results is given in 4.4, where it is also explained how Theorem 4.3
generalizes these related results. Finally, we come back in Section 4.5 to
the relevance of market opportunities for consumption decisions under
uncertainty.
4.3. The condition appearing in Theorem 4.3 refers to the behavior of
the absolute risk aversion function along budget lines with slope dc^dc^ =
—(1 + r*). Define indeed
The sign of R determines whether absolute risk aversion increases (> 0),
decreases (< 0) or remains constant (= 0) when cx increases and c2
decreases along the budget line c2 = (yt — Ci)(l + *"*) + y2 • In this
definition r* is still given by (3.7) and satisfies EU-ljEU2 = 1 + r*.
THEOREM 4.3. Let y^ be such that
max U(Cl, (y, - Cl)(l + r*) + >'2 + ) = EU(tx, (yx - c{)(l + r) + y2)
(4.2)
and let cx + be the value of cx maximizing the left hand side of (4.2). Then
R = 0 (identically in c2 given ^) implies £x ~ c/.
22 Related results have been established under the additional assumption of additive
(cardinal) utility, e.g., by Mirman [10] or by Rothschild and Stiglitz [13]. The latter
paper clarifies in a basic way the concept of "increase in risk."
1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 475