Can someone explain to me the "Envelope Theorem"?

Question

In the language engish?

Answer 1

the simplified process of measuring the change in some parameter with respect to the change of the whole function.

english enough?

Answer 2

The envelope theorem is a basic theorem used to solve maximization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity. The statement of the theorem is:

Consider an arbitrary maximization problem where the objective function (f) depends on some parameter (a):


where the function M(a) gives the maximized value of the objective function (f) as a function of the parameter (a). Now let x(a) be the value of x that solves the maximization problem in terms of the parameter (a). Then M(a) = f(x(a),a). The envelope theorem tells us how M(a) changes as the parameter (a) changes, namely:


That is, the derivative of M with respect to a is given by the partial derivative of f(x,a) with respect to a, holding x fixed, and then evaluating at the optimal choice (x * ). The vertical bar to the right of the partial derivative denotes that we are to make this evaluation at x * = x(a).