A theorem determining the effect of a differential change in a parameter on the outcome of a maximization problem. Formally, consider the problem of choosing x to maximize a function f(x; a), where a is a parameter. Denote the solution to this maximization by x(a) and define the maximum value function V(a) by

The envelope theorem states that

so that the derivative of the value function is the partial derivative of the objective evaluated at the solution to the optimization.