Econometrica

Journal Of The Econometric Society

An International Society for the Advancement of Economic
Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: Mar, 2002, Volume 70, Issue 2

Envelope Theorems for Arbitrary Choice Sets

https://doi.org/10.1111/1468-0262.00296
p. 583-601

Paul Milgrom, Ilya Segal

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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