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Envelope Theorems for Arbitrary Choice Sets

Paul Milgrom and Ilya Segal
Econometrica
Vol. 70, No. 2 (Mar., 2002), pp. 583-601
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/2692283
Page Count: 19
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Envelope Theorems for Arbitrary Choice Sets
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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.

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