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On the Economic Meaning of Machina's Fréchet Differentiability Assumption

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Listed:
  • Zvi Safra

    (Tel Aviv University)

  • Uzi Segal

    (Boston College)

Abstract

This note shows that Machina's (1982) assumption that preferences over lotteries are smooth has some economic implications. We show that Fréchet differentiability implies that preferences represent second order risk aversion (as well as conditional second order risk aversion). This implies, among other things, that decision makers buy full insurance only at the absence of marginal loading. We also show that with constant absolute and relative risk aversion, expected value maximization, second order risk aversion, and Fréchet differentiability are equivalent.

Suggested Citation

  • Zvi Safra & Uzi Segal, 2001. "On the Economic Meaning of Machina's Fréchet Differentiability Assumption," Boston College Working Papers in Economics 511, Boston College Department of Economics.
    • Handle: RePEc:boc:bocoec:511
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    References listed on IDEAS

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    1. Machina, Mark J, 2001. "Payoff Kinks in Preferences over Lotteries," Journal of Risk and Uncertainty, Springer, vol. 23(3), pages 207-260, November.
    2. Machina, Mark J, 1982. ""Expected Utility" Analysis without the Independence Axiom," Econometrica, Econometric Society, vol. 50(2), pages 277-323, March.
    3. Uzi Segal & Avia Spivak, 1996. "First-order risk aversion and non-differentiability (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(1), pages 179-183.
    4. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    5. Gul, Faruk, 1991. "A Theory of Disappointment Aversion," Econometrica, Econometric Society, vol. 59(3), pages 667-686, May.
    6. Loomes, Graham & Segal, Uzi, 1994. "Observing Different Orders of Risk Aversion," Journal of Risk and Uncertainty, Springer, vol. 9(3), pages 239-256, December.
    7. Epstein, Larry G. & Zin, Stanley E., 1990. "'First-order' risk aversion and the equity premium puzzle," Journal of Monetary Economics, Elsevier, vol. 26(3), pages 387-407, December.
    8. Wang Tan, 1993. "Lp-Frechet Differentiable Preference and Local Utility Analysis," Journal of Economic Theory, Elsevier, vol. 61(1), pages 139-159, October.
    9. Hong, Chew Soo & Karni, Edi & Safra, Zvi, 1987. "Risk aversion in the theory of expected utility with rank dependent probabilities," Journal of Economic Theory, Elsevier, vol. 42(2), pages 370-381, August.
    10. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    11. Edi Karni, 1995. "Non-Expected Utility and The Robustness of the Classical Insurance Paradigm: Discussion," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 20(1), pages 51-56, June.
    12. Segal, Uzi & Spivak, Avia, 1990. "First order versus second order risk aversion," Journal of Economic Theory, Elsevier, vol. 51(1), pages 111-125, June.
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    Cited by:

    1. Freeman, David, 2015. "Calibration without reduction for non-expected utility," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 21-32.
    2. Alon Harel & Zvi Safra & Uzi Segal, 2003. "Ex-Post Egalitarianism," Boston College Working Papers in Economics 563, Boston College Department of Economics.

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    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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