Putting Risk in Its Proper Place

This paper examines preferences towards particular classes of lottery pairs. We show how concepts such as prudence and temperance can be fully characterized by a preference relation over these lotteries. If preferences are defined in an expected-utility framework with differentiable utility, the direction of preference for a particular class of lottery pairs is equivalent to signing the nth derivative of the utility function. What makes our characterization appealing is its simplicity, which seems particularly amenable to experimentation. JEL Code: D81 Keywords: properness, prudence, risk apportionment, risk aversion, stochastic dominance, temperance, utility premium. Louis Eeckhoudt Catholic Faculties of Mons University (FUCAM) Chaussée de Binche 151 Mons 7000 Belgium [email protected] Harris Schlesinger Department of Finance University of Alabama Tuscaloosa AL 35487-0224 USA [email protected] This paper was partly written while both authors were visiting at the IDEI, University of Toulouse. Financial support from the French Federation of Insurance Companies (FFSA) is gratefully acknowledged. We thank seminar participants at the EGRIE meeting in Zurich, CORE, DELTA, the University of Bologna, the University of Pennsylvania and Rice University for many useful comments on a draft version of this paper. Detailed comments from Georges Dionne, Neil Doherty, Christian Gollier, Miles Kimball, Achim Wambach, Claudio Zoli and three referees were especially helpful. Putting Risk in its Proper Place The concept of risk aversion has long been a cornerstone for modern research on the economics of risk. Ask several economists to de…ne what it means for an individual to be risk averse and you are likely to get several di¤erent answers. Some, assuming an expectedutility framework, will say that the second derivative of the von Neumann-Morgenstern utility function u is concave or, assuming di¤erentiability, that u00 < 0. Others might de…ne risk aversion in a more general setting, equating it to an aversion to mean-preserving spreads, as de…ned by Michael Rothschild and Joseph E. Stiglitz (1970). It is not likely that one would de…ne risk aversion via some behavioral consequence, such as the propensity to purchase full insurance at an actuarially-fair price. Although somewhat newer, the concept of "prudence" and its relationship to precautionary savings also has become a common and accepted assumption.1 Ask someone to de…ne what it means for the individual to be "prudent" and they might say that marginal utility is convex, u000 > 0, but they also might de…ne prudence via behavioral characteristics. For example, Christian Gollier (2001 p. 236), de…nes an agent as prudent "if adding an uninsurable zero-mean risk to his future wealth raises his optimal saving." In other words, unlike the case with risk aversion, prudence is often de…ned via an optimizing type of behavior, rather than some type of more primitive trait.2 More recently, some new concepts have entered the literature such as "temperance" (uiv < 0) and "edginess" (uv > 0), which arise as necessary and/or su¢ cient conditions for various behavioral results.3 But what exactly are these concepts and what do they imply