Envelope theorem

From Wikipedia, the free encyclopedia

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 ( $a$ ):

$M(a) = \max_{x} f(x, a)\,$

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 (arg max) value of $x$ that solves the maximization problem in terms of the parameter ( $a$ ), i.e. so that $M (a) = f (x (a), a)$ . The envelope theorem tells us how $M (a)$ changes as the parameter ( $a$ ) changes, namely:

$\frac{dM(a)}{da} = \frac{\partial f(x^*, a)}{ \partial a} \Bigg|_{x^* = x(a)}.$

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 $x * = x (a)$ .

[edit] Envelope theorem in generalized calculus

In the calculus of variations, the envelope theorem relates evolutes to single paths. This was first proved by Jean Gaston Darboux and Ernst Zermelo (1894) and Adolf Kneser (1898). The theorem can be stated as follows:

"When a single-parameter family of external paths from a fixed point O has an envelope, the integral from the fixed point to any point A on the envelope equals the integral from the fixed point to any second point B on the envelope plus the integral along the envelope to the first point on the envelope, J_OA = J_OB + J_BA." ^[1]