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+\topmatter
+
+\title{Estimating Risk Aversion from \cr
+ Arrow-Debreu Portfolio Choice}
+
+\runningname{Hal R. Varian}
+\runningtitle{Estimating Risk Aversion}
+
+\thanks{This research was supported in part by the National Science
+Foundation. I would like to thank Richard Green for helpful remarks. I am
+especially grateful to an anonymous referee whose comments significantly
+improved the statements and proofs of the results.}
+
+\author{Hal R. Varian}
+
+\affil{University of Michigan}
+
+\date{October 27, 1984}
+
+\version{\today}
+
+\abstract{This paper derives necessary and sufficient conditions for
+Arrow-Debreu choices of contingent consumption to be compatible with the
+maximization of a state independent expected utility function that exhibits
+increasing or decreasing absolute risk aversion, or increasing or
+decreasing relative risk aversion. The conditions can be used to bound
+different measures of risk aversion based on a single observation of
+Arrow-Debreu portfolio choice.}
+
+\keywords{Revealed preference, expected utility, risk aversion, portfolio
+choice.}
+
+
+\address{Hal R. Varian, Department of Economics, University of Michigan,
+Ann Arbor, MI 48109}
+
+\endtopmatter
+
+\document
+
+\noindent {\scten The expected utility} hypothesis forms the basis for much
+of our understanding of investor behavior under uncertainty. It is
+commonly agreed that a well-behaved expected utility function should be an
+increasing and concave function of wealth, or, equivalently, that its first
+derivative should be positive and its second derivative should be negative.
+It is also widely accepted that the Arrow-Pratt measure of absolute risk
+aversion should be declining with wealth. There is much less agreement
+about the behavior of the Arrow-Pratt measure of {\it relative\/} risk
+aversion, although some investigators have argued that it should increase
+with wealth.
+
+In this note I derive necessary and sufficient conditions for choices of
+contingent consumption across states of nature to satisfy various
+hypotheses about the behavior of these measures of risk aversion. If the
+portfolio choice behavior of the consumer is consistent with the conditions
+I derive, then the conditions can be used to bound the Arrow--Pratt
+measures of absolute and relative risk aversion. The conditions are
+derived using methods of the ``nonparametric approach'' to optimizing
+behavior introduced by Afriat (1967) and extended by Diewert (1973),
+Diewert and Parkan (1978), and Varian (1982), (1983a). Applications of
+these methods to choice under uncertainty include Dybvig and Ross (1982),
+Green and Srivastava (1983), and Varian (1983b).
+
+\section The Maximization Problem
+
+Consider an investor who chooses a pattern of consumption across states of
+nature to solve the following problem:
+ $$\max \sum_{s=1}^S \pi_s u(c_s) $$
+
+\Refs
+
+\ref
+\by{Afriat, S.} \yr{1967a} \paper{The Construction of a Utility
+Function from Expenditure Data} \jour{International Economic Review}
+\vol{8} \pages{67--77}
+\endref
+
+\ref
+\by{Afriat, S.} \yr{1967b} \paper{The Construction of Separable
+Utility Functions from Expenditure Data} \paperinfo{mimeo, Purdue}
+\endref
+
+\ref \by{Breeden, D. and R. Litzenberger} \yr{1978} \paper{Prices of
+State-Contingent Claims Implicit in Option Prices} \jour{Journal of
+Business} \pages{621--651}
+\endref
+
+\ref
+\by{Diewert, E.} \yr{1973} \paper{Afriat and Revealed Preference Theory}
+\jour{Review of Economic Studies} \vol{40} \pages{419--426}
+\endref
+
+\ref \by{Diewert, E. and C. Parkan} \yr{1978} \paper{Tests for Consistency
+of Consumer Data and Nonparametric Index Numbers} \paperinfo{Working Paper
+78-27, University of British Columbia}
+\endref
+
+\ref \by{Dybvig, P. and S. Ross} \yr{1982} \paper{Portfolio Efficient Sets}
+\jour{Econometrica} \pages{1525--1546}
+\endref
+
+\ref \by{Green, R. and S. Srivastava} \yr{1983} \paper{Preference
+Restrictions, Asset Returns, and Consumption} \paperinfo{mimeo,
+Carnegie--Mellon University}
+\endref
+
+\ref \by{Varian, H.} \yr{1982} \paper{The Nonparametric Approach to Demand
+Analysis} \jour{Econometrica} \vol{50} \pages{945--973}
+\endref
+
+\ref
+\by{Varian, H.} \yr{1983a} \paper{Nonparametric Tests of Models of
+Consumer Behavior} \jour{Review of Economic Studies} \vol{50}
+\pages{99--110}
+\endref
+
+\ref
+\by{Varian, H.} \yr{1983b} \paper{Nonparametric Tests of Models of
+Investor Behavior} \jour{Journal of Financial and Quantitative Analysis}
+\vol{18} \pages{269--278}
+\endref