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Differentiable von Neumann-Morgenstern utility

Author

Listed:
  • Lars Tyge Nielsen

    (INSEAD, Boulevard de Constance, F-77305 Fontainebleau Cedex, FRANCE)

Abstract

Differentiability is a convenient property of von Neumann-Morgenstern utility functions which is almost always imposed but has not been translated into behavioral terms. In applications, expected utility is usually maximized subject to a constraint, and the maximization is carried out by differentiating the utility function. This paper presents two sets of necessary and sufficient conditions for a risk averse von Neumann-Morgenstern utility function to be differentiable. The first of them is formulated in terms of the equivalent risk premia of small gambles. It says, in brief, that the equivalent risk premium is of a smaller order of magnitude than the risk itself, as measured by the expectation of the absolute value of the risk. The second set of necessary and sufficient conditions is formulated in terms of the probability premium of small lotteries. It says, essentially, that the probability premium for small binary lotteries goes to zero as the size of the lottery goes to zero.

Suggested Citation

  • Lars Tyge Nielsen, 1999. "Differentiable von Neumann-Morgenstern utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(2), pages 285-296.
    • Handle: RePEc:spr:joecth:v:14:y:1999:i:2:p:285-296
    Note: Received: May 11, 1997; revised version: May 14, 1998
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    Citations

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    Cited by:

    1. Matthias Lang, 2017. "First-Order and Second-Order Ambiguity Aversion," Management Science, INFORMS, vol. 63(4), pages 1254-1269, April.
    2. Wakker, Peter P. & Yang, Jingni, 2019. "A powerful tool for analyzing concave/convex utility and weighting functions," Journal of Economic Theory, Elsevier, vol. 181(C), pages 143-159.
    3. Nakamura, Yutaka, 2015. "Differentiability of von Neumann–Morgenstern utility functions," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 74-80.

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