Envelope Theorems for Arbitrary Choice Sets

2002/03/01 by Paul Milgrom, Ilya Segal · 61 citations
Decision Sciences · Business, Management and Accounting · #Auction Theory and Applications #Supply Chain and Inventory Management #Consumer Market Behavior and Pricing

paper · doi:10.1111/1468-0262.00296

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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