Envelope Theorems for Arbitrary Choice Sets

Posted: 17 Aug 2002

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.

Suggested Citation

Milgrom, Paul R. and Segal, Ilya, Envelope Theorems for Arbitrary Choice Sets. Available at SSRN: https://ssrn.com/abstract=312251

Paul R. Milgrom (Contact Author)

Stanford University ( email )

Landau Economics Building
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Stanford, CA 94305-6072
United States
+1-650-723-3397 (Phone)
+1-419-791-8545 (Fax)

HOME PAGE: www.milgrom.net

Ilya Segal

Stanford University ( email )

Stanford, CA 94305
United States
650-724-4905 (Phone)
650-725-5702 (Fax)

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