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Duality and equilibrium prices in economics of uncertainty

Author

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  • Adi Ben-Israel

  • Aharon Ben-Tal

Abstract

A random variable (RV) X is given aminimum selling price $$S_U \left( X \right):=\mathop {\sup }\limits_x \left\{ {x + EU\left( {X - x} \right)} \right\}$$ and amaximum buying price $$B_p \left( X \right):=\mathop {\inf }\limits_x \left\{ {x + EP\left( {X - x} \right)} \right\}$$ whereU(·) andP(·) are appropriate functions. These prices are derived from considerations ofstochastic optimization with recourse, and are calledrecourse certainty equivalents (RCE's) of X. Both RCE's compute the “value” of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of theCsiszár φ-divergence $$I_\phi \left( {p,q} \right):=\sum\limits_{i=1}^n {q_i \phi \left( {\frac{{p_i }}{{q_i }}} \right)} $$ a generalized entropy function, measuring the distance between RV's with probability vectors p and q. The RCES U was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCEB P , and apply it to problems ofinventory control (where the attitude towards risk determines the stock levels and order sizes) andoptimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage. Copyright Physica-Verlag 1997

Suggested Citation

  • Adi Ben-Israel & Aharon Ben-Tal, 1997. "Duality and equilibrium prices in economics of uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(1), pages 51-85, February.
    • Handle: RePEc:spr:mathme:v:46:y:1997:i:1:p:51-85
    DOI: 10.1007/BF01199463
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    References listed on IDEAS

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    1. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    2. Aharon Ben-Tal, 1985. "The Entropic Penalty Approach to Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 263-279, May.
    3. Aharon Ben-Tal & Marc Teboulle, 1986. "Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming," Management Science, INFORMS, vol. 32(11), pages 1445-1466, November.
    4. George B. Dantzig, 1955. "Linear Programming under Uncertainty," Management Science, INFORMS, vol. 1(3-4), pages 197-206, 04-07.
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