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.

We verify that the envelope theorem holds for the log-likelihood function when the underlying data are generated from a normal distribution. The evidence that the theorem is true is that the top and bottom pictures in the Demonstration are identical.

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.