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Duality mappings for the theory of risk aversion with vector outcomes

  • Sudhir A. Shah

    (Department of Economics, Delhi School of Economics, Delhi, India)

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    We consider a decision-making environment with an outcome space that is a convex and compact subset of a vector space belonging to a general class of such spaces. Given this outcome space, we de¯ne gen- eral classes of (a) risk averse von Neumann-Morgenstern utility func- tions de¯ned over the outcome space, (b) multi-valued mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) multi-valued mappings that yield the risk premia corresponding to a lottery, and (d) multi-valued mappings that yield the acceptance set of lotteries corresponding to an outcome. Our duality results establish that the usual mappings that generate (b), (c) and (d) from (a) are bi- jective. We apply these results to the problem of computing the value of ¯nancial assets to a risk averse decision-maker and show that this value will always be less than the arbitrage-free valuation.

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    Paper provided by Centre for Development Economics, Delhi School of Economics in its series Working papers with number 160.

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    Length: 26 pages
    Date of creation: Aug 2007
    Date of revision:
    Handle: RePEc:cde:cdewps:160
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    1. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good choice Axioms: When can many-good lotteries be treated as money lotteries?," Journal of Economic Theory, Elsevier, vol. 56(2), pages 313-337, April.
    2. Kihlstrom, Richard E. & Mirman, Leonard J., 1974. "Risk aversion with many commodities," Journal of Economic Theory, Elsevier, vol. 8(3), pages 361-388, July.
    3. Juan Martínez-Legaz & John Quah, 2007. "A contribution to duality theory, applied to the measurement of risk aversion," Economic Theory, Springer, vol. 30(2), pages 337-362, February.
    4. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good risks: An interpretation of multivariate risk and risk aversion without the Independence axiom," Journal of Economic Theory, Elsevier, vol. 56(2), pages 338-351, April.
    5. Levy, Haim & Levy, Azriel, 1991. "Arrow-Pratt Measures of Risk Aversion: The Multivariate Case," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(4), pages 891-98, November.
    6. Duncan, George T, 1977. "A Matrix Measure of Multivariate Local Risk Aversion," Econometrica, Econometric Society, vol. 45(4), pages 895-903, May.
    7. Richard E. Kihlstrom & Leonard J. Mirman, 1981. "Constant, Increasing and Decreasing Risk Aversion with Many Commodities," Review of Economic Studies, Oxford University Press, vol. 48(2), pages 271-280.
    8. Yaari, Menahem E., 1969. "Some remarks on measures of risk aversion and on their uses," Journal of Economic Theory, Elsevier, vol. 1(3), pages 315-329, October.
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