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Risk aversion for nonsmooth utility functions

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  • Würth, Andreas
  • Schumacher, J.M.

Abstract

Abstract This paper generalizes the notion of risk aversion for functions which are not necessarily differentiable nor strictly concave. Using an approach based on superdifferentials, we define the notion of a risk aversion measure, from which the classical absolute as well as relative risk aversion follows as a Radon-Nikodym derivative if it exists. Using this notion, we are able to compare risk aversions for nonsmooth utility functions, and to extend a classical result of Pratt to the case of nonsmooth utility functions. We prove how relative risk aversion is connected to a super-power property of the function. Furthermore, we show how boundedness of the relative risk aversion translates to the corresponding property of the conjugate function. We propose also a weaker ordering of the risk aversion, referred to as essential bounds for the risk aversion, which requires only that bounds of the (absolute or relative) risk aversion hold up to a certain tolerance.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Mathematical Economics.

Volume (Year): 47 (2011)
Issue (Month): 2 (March)
Pages: 109-128

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Handle: RePEc:eee:mateco:v:47:y:2011:i:2:p:109-128

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Web page: http://www.elsevier.com/locate/jmateco

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Keywords: Risk aversion Nonsmooth utility;

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  1. Nielsen, Lars Tyge, 1997. "Monotone Risk Aversion," CEPR Discussion Papers 1651, C.E.P.R. Discussion Papers.
  2. Zhegal, Amina & Touzi, Nizar & Bouchard, Bruno, 2004. "Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility," Economics Papers from University Paris Dauphine 123456789/1531, Paris Dauphine University.
  3. Mark J Machina, 1982. ""Expected Utility" Analysis without the Independence Axiom," Levine's Working Paper Archive 7650, David K. Levine.
  4. Ariel Rubinstein, 2006. "Lecture Notes in Microeconomic Theory," Online economics textbooks, SUNY-Oswego, Department of Economics, number gradmicro1, Spring.
  5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680, September.
  6. B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290, arXiv.org.
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