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Risk aversion for nonsmooth utility functions


  • Würth, Andreas
  • Schumacher, J.M.


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.

Suggested Citation

  • Würth, Andreas & Schumacher, J.M., 2011. "Risk aversion for nonsmooth utility functions," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 109-128, March.
  • Handle: RePEc:eee:mateco:v:47:y:2011:i:2:p:109-128

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    References listed on IDEAS

    1. Lars Nielsen, 2005. "Monotone risk aversion," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 25(1), pages 203-215, January.
    2. B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290,
    3. Ariel Rubinstein, 2006. "Lecture Notes in Microeconomic Theory," Online economics textbooks, SUNY-Oswego, Department of Economics, number gradmicro1, March.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680, June.
    5. repec:dau:papers:123456789/1531 is not listed on IDEAS
    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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    Risk aversion Nonsmooth utility;


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