Concavity, stochastic utility, and risk aversion

Abstract

This paper studies the relation between concavity, stochastic or state-dependent utility functions, and risk aversion. Using the common definition of risk aversion, but modified for state-dependent preferences, we show that concavity does not imply risk aversion. Instead, it implies a weaker version of risk aversion, defined herein, and called risk aversion for independent gambles. Furthermore, to characterise the economic meaning of concavity, we define two new risk aversion notions, called uniform risk aversion and uniform risk aversion for independent gambles, respectively. We show that concavity is equivalent to uniform risk aversion for independent gambles, and that concavity plus some additional conditions are equivalent to uniform risk aversion.

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Acknowledgements

We should like to thank Alexander Schied (the Co-Editor), an Associate Editor and two anonymous referees for comments and suggestions that greatly improved the article. We also thank Tommaso Denti, Qingmin Liu, Jaden Chen, Haokun Sun and Yizhou Kuang for insightful discussions. All errors are our own.

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Correspondence to Robert Jarrow.

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Jarrow, R., Li, S. Concavity, stochastic utility, and risk aversion. Finance Stoch (2021). https://doi.org/10.1007/s00780-021-00448-5

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Keywords

  • State-dependent utility
  • Stochastic utility
  • Risk aversion
  • Uniform risk aversion
  • Pointwise concavity
  • Uniform risk aversion for independent gambles

Mathematics Subject Classification (2010)

  • 91B08
  • 91B16

JEL Classification

  • D11
  • G40
  • D81