Envelope theorem

In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models. The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions that are optimized. Let and be real-valued continuously differentiable functions on , where are choice variables and are parameters, and consider the problem of choosing , for a given , so as to: subject to and . The Lagrangian expression of this problem is given by where are the Lagrange multipliers. Now let and together be the solution that maximizes the objective function f subject to the constraints (and hence are saddle points of the Lagrangian), and define the value function Then we have the following theorem. Theorem: Assume that and are continuously differentiable. Then where . Let denote the choice set and let the relevant parameter be . Letting denote the parameterized objective function, the value function and the optimal choice correspondence (set-valued function) are given by: "Envelope theorems" describe sufficient conditions for the value function to be differentiable in the parameter and describe its derivative as where denotes the partial derivative of with respect to . Namely, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to holding the maximizer fixed at its optimal level. Traditional envelope theorem derivations use the first-order condition for (), which requires that the choice set have the convex and topological structure, and the objective function be differentiable in the variable .