STOCHASTIC

310 DRfeZE AND MODIGLIANI
second order. The distribution function corresponding to (y2, r) will
be denoted by dc^jdyi > 0.
In the second place, we assume that the consumer's preferences among
consumption vectors are convex, and that his choices among uncertain
prospects reflect risk aversion, or possibly risk neutrality; that is, 5
ASSUMPTION IV. U is concave.
Various properties of V, derived from assumptions II, III and quasiconcavity
of U, are collected in Appendix A.
1.3. The (cardinal) indirect utility function corresponding to U(C\ , c2)
may be written V(y, r), where
V{y, r) =aet max Ufa , (y - Cl)(l + r)).
Let r = r° be the sure and only rate of interest at which a consumer may
lend and borrow; then V(y, r°), a function of y alone, is the cardinal
utility function for wealth relevant to the analysis of choices among timeless
uncertain prospects. In other words, if a cardinal utility function for wealth
were derived from observations about choices between timeless uncertain
3 For notational convenience, we use the integral symbol without introducing parallel
statements in the notation of discrete random variables; the standard symbol E is used
for the expectation operator when there is no ambiguity about the underlying mass or
density function.
* Thus, V is defined up to a linear increasing transformation; if the consumer were
to choose between the certainty of consuming 0^ , ca) and the prospect of consuming
either (c/, c2') or (cj, c'j) with respective probabilities -n and 1 — 17, he would never
prefer the former alternative if U(ct, fj) < ir£/(ci', c„') + (1 — ir)t/(cj, cj).
6 As argued elsewhere by one of us [4], risk preference may be excluded without loss
of generality, if one assumes the availability on the market of fair gambling opportunities.
TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION