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Envelope Theorem without Differentiability

Author

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  • Yuntong Wang

    (Department of Economics, University of Windsor)

Abstract

We establish a new envelope theorem in which the choice variables are discrete and the objective function and the constraints are Lipschitz continuous with respect to the parameters. The parameters can be ?nite or in?nite dimensional vectors in a Banach space. In an application, we revisit the principal-agent problem and derive a weaker ?rst-order condition than the traditional one in the literature. In an insurance example, we use the condition to show an insurance contract that is discontinuous at some level of the loss.

Suggested Citation

  • Yuntong Wang, 2014. "Envelope Theorem without Differentiability," Working Papers 1404, University of Windsor, Department of Economics.
    • Handle: RePEc:wis:wpaper:1404
    as

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    File URL: http://web2.uwindsor.ca/economics/RePEc/wis/pdf/1404.pdf
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    References listed on IDEAS

    as
    1. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    2. Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
    3. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    4. repec:bla:scandj:v:90:y:1988:i:3:p:275-89 is not listed on IDEAS
    5. Caputo, Michael R., 1990. "How to do comparative dynamics on the back of an envelope in optimal control theory," Journal of Economic Dynamics and Control, Elsevier, vol. 14(3-4), pages 655-683, October.
    6. Grossman, Sanford J & Hart, Oliver D, 1983. "An Analysis of the Principal-Agent Problem," Econometrica, Econometric Society, vol. 51(1), pages 7-45, January.
    7. Sah, Raaj & Zhao, Jingang, 1998. "Some Envelope Theorems for Integer and Discrete Choice Variables," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(3), pages 623-634, August.
    8. Frank H. Clarke, 1976. "A New Approach to Lagrange Multipliers," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 165-174, May.
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    12. Silberberg, Eugene, 1974. "A revision of comparative statics methodology in economics, or, how to do comparative statics on the back of an envelope," Journal of Economic Theory, Elsevier, vol. 7(2), pages 159-172, February.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Envelope theorem; Discrete choice set; Lipschitz continuity; Generalized gradients; Principal-agent problem.;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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    This paper has been announced in the following NEP Reports:

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