### 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.

Original language | English |
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Pages (from-to) | 109-128 |

Journal | Journal of Mathematical Economics |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 |

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## Cite this

Wuerth, A. M., & Schumacher, J. M. (2011). Risk aversion for nonsmooth utility functions.

*Journal of Mathematical Economics*,*47*(2), 109-128. https://doi.org/10.1016/j.jmateco.2010.10.003