24a. Proving The Envelope Theorem

In this video, I provide a loose proof of the Envelope Theorem (a very important result in mathematical economics). This theorem is foundational to microeconomic analysis. After deriving the envelope theorem, I also present two examples from consumer theory (1) Marginal Utility of Income, and (2) Roy’s Identity.

This is the first of two videos on The Envelope Theorem and applications. The next one is coming soon.

Note: This video assumes a mastery of calculus. Don’t attempt watching unless you have taken at least a year of calculus, and are familiar with the method of Lagrange.

For those of you who appreciate written explanations, there is a helpful pdf typeset tutorial here:


The typeset pdf (not written by me) starts from specific cases of the envelope theorem, building up to a general statement of it for consumer theory. It is nice if you still struggle with the material after studying this video.

For a list of videos and links to these videos (organized by topic), check out the Intromediate Microeconomics video web page:


25 thoughts on “24a. Proving The Envelope Theorem”

  1. Very useful video. One suggestion would be to have the statement of
    envelope theorem up in the front.

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