The Envelope Theorem: Numerical Examples

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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.

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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.

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Contributed by: Jeff Hamrick (March 2011)
Suggested by: Fred Meinberg
Open content licensed under CC BY-NC-SA


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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.



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