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.
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.