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Envelope Theorems for Arbitrary Choice Sets

Author

Listed:
  • Paul Milgrom

    (Department of Economics, New York University, 269 Mercer Street, New York, NY 10003, U.S.A.)

  • Ilya Segal

    (Department of Economics, New York University, 269 Mercer Street, New York, NY 10003, U.S.A.)

Abstract

The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative. This paper studies optimization with arbitrary choice sets and shows that the traditional envelope formula holds at any differentiability point of the value function. We also provide conditions for the value function to be, variously, absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to mechanism design, convex programming, continuous optimization problems, saddle-point problems, problems with parameterized constraints, and optimal stopping problems. Copyright The Econometric Society 2002.

Suggested Citation

  • Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    • Handle: RePEc:ecm:emetrp:v:70:y:2002:i:2:p:583-601
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