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Risk aversion for nonsmooth utility functions

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  • Wuerth, A.M.
  • Schumacher, J.M.

    (Tilburg University, School of Economics and Management)

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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Wuerth, A.M. & Schumacher, J.M., 2011. "Risk aversion for nonsmooth utility functions," Other publications TiSEM d948cfad-5e83-46ce-ae72-6, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:d948cfad-5e83-46ce-ae72-6239a2975baa
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    References listed on IDEAS

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    1. B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290, arXiv.org.
    2. Lars Tyge Nielsen, 2006. "Monotone Risk Aversion," Studies in Economic Theory, in: Christian Schultz & Karl Vind (ed.), Institutions, Equilibria and Efficiency, chapter 17, pages 317-329, Springer.
    3. repec:dau:papers:123456789/1531 is not listed on IDEAS
    4. Ariel Rubinstein, 2006. "Lecture Notes in Microeconomic Theory," Online economics textbooks, SUNY-Oswego, Department of Economics, number gradmicro1.
    5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    6. Machina, Mark J, 1982. ""Expected Utility" Analysis without the Independence Axiom," Econometrica, Econometric Society, vol. 50(2), pages 277-323, March.
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    Cited by:

    1. Schumacher, Johannes M., 2021. "Ex-ante estate division under strong Pareto efficiency," Mathematical Social Sciences, Elsevier, vol. 113(C), pages 10-24.

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