Envelope Theorems for Parametrized Choice Sets

Existing envelope theorems apply to fixed choice sets or to convex maximization programs (Milgrom and Segal 2002). We derive envelope theorems for parametrized choice sets without imposing any convex or topological structure on the choice sets. We show that the traditional envelope theorem formula hold at any point where the generalized Lagrange multipliers and the derivative of the constraint are continuous. We provide conditions under which the value function is differentiable or absolutely continuous. We apply these theorems to topological spaces and mechanism design problems.