The Envelope Theorem: Numerical Examples

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+