Envelope theorem

The envelope theorem (also Envelope theorem or envelope theorem ) is a fundamental set of the calculus of variations, often used in micro- economics. He describes how the optimum value of the objective function of a parameterized optimization problem when changing the parameter behaves.

A distinction is usually between two versions of the envelope theorem: one for optimization problems and for those with secondary conditions, the first version is a special case of the second.

Representation

Optimization problem without constraints

(Envelope theorem for optimization problems without constraints :) Be a continuously differentiable function with and a scalar - in short. Where is the problem

Which has a solution, which is continuously differentiable. Then the so-called feed-forward function of added (i.e., the original function evaluated at the - in this case only a Dependent - point where it assumes its maximum ). The envelope theorem then states:

It is found that has the first order in the calculation of a variation of the change does not affect the effect.

Extension: The sentence also applies to multiple parameters. It is then valid for the maximization problem with (,) and for any ():

Optimization problem with constraints

( Generalized Envelope theorem for optimization problems with constraints :) Be a continuously differentiable function with and a scalar - in short. Where is the problem

Which has a solution, which is continuously differentiable. It is the corresponding Lagrangian. The Lagrange multipliers are continuously differentiable. Moreover, the Jacobian matrix possess the rank.

Then, an optimal value function of states and the envelope theorem:

Expansion: The proposition is also applicable in cases with multiple parameters. With analogous definitions then holds for arbitrary ():

Comments

Is the envelope of the family of curves, hence the name of the set.

Example without constraints

Be an example given the following problem:

First-order condition of the maximization problem is

Placing them to condition, it follows for the "optimal". Substituting this back into the original function, which returns the optimal value function. It is interested now how this value function changes when changing. This will then be shown "direct" first with the envelope theorem and illustration. With the envelope theorem immediately follows:

And

The same result could have been also calculate " directly ". For this one must the value function, however, be calculated explicitly:

And thus also

Application

An application can be found in microeconomics. There you can use the envelope theorem both in the theory of enterprises as well as in the theory of households.

In the field of the theory of the production companies amount designated in dependence on the input, it follows, by setting as the price vector for output and Inputgut, and as producers gain, Hotelling's lemma. However, it is also possible to use the envelope theorem in the cost reduction. This works similar to Shephard's lemma.

In theory, the households have the envelope theorem is used in conjunction with indirect utility functions. It can be easily analyzed using Roy's identity, what happens during an income or a change in price. For the indirect utility function is partially derived by income and price.