# The Envelope Theorem: Numerical Examples

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The envelope theorem is used to solve maximization problems in the fields of microeconomics and finance. It is a fundamental result in the calculus of variations and is therefore often used in large deviations research.

[more]
Contributed by: Jeff Hamrick (March 2011)

Suggested by: Fred Meinberg

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

We explain the envelope theorem by way of a concrete example.

Define .

The log-likelihood maximization problem is to find parameters such that

.

Fix and define a new function such that

.

Now we view . Then the envelope theorem says that

.

The functions and are plotted in the top and bottom parts of the Demonstration output, respectively.

Roughly speaking, then, the envelope theorem says that fixing , maximizing over , and then taking the derivative with respect to is the same as taking the derivative with respect to , then fixing , and then substituting for the fixed the particular that maximizes.

## Permanent Citation