# 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:

http://emlab.berkeley.edu/users/webfac/card/e101a_s05/consumerenvelope.pdf

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:

http://blog.thisyoungeconomist.com/p/learn-microeconomics.html

Hernando Harker

April 27, 2010@ 11:51 amThanks for uploading. It´s hard to find this kind of study material in an

understandable way.

Eleni Kyrko

September 26, 2010@ 5:51 pmhelps a lot thx

Elisita

October 5, 2010@ 9:01 amThank you. You’ve made it a lot easier for me to understand.

Prektesh1

October 21, 2010@ 11:29 pmthat was great. You are a great instructor

toulouselucas

November 13, 2010@ 7:09 pmhelpful!!!!!!!!!

Oscar Galindo

November 19, 2010@ 12:31 amwould it kill you to slow down!

intromediateecon

November 19, 2010@ 10:16 am@Oscargs7 I agree. This video goes through the material quickly, but that’s

intentional. I do it to prevent boredom… watching someone write on the

board is boring. It is the explanation that is interesting. If I go through

something too quickly (i.e., jump frame to put some math on the board), you

are welcome to pause the video to spend some time absorbing the math.

jellyace1992

April 10, 2011@ 12:21 pm@Oscargs7 is it the emlab.berkeley thing that discusses consumer and

envelope theorem?

cbx3460

October 12, 2011@ 8:33 pmvery helpful. Thanks a lot. I understand that you wanna make your lectures

less boring by editing the steps. If you have time, why don’t you post two

versions: both the entire lecture and the edited lecture. Smart ones follow

the edited ones, slower ones like me follow the complete one.

intromediateecon

October 26, 2011@ 8:31 am@cbx3460 That’s a good suggestion. Unfortunately, for the old videos, I

don’t have the full footage anymore (just the edited files). I’ll keep this

in mind for future videos in this format.

martiong

December 17, 2011@ 12:40 pmdude, you’re good…!

Rullan Rinaldi

March 2, 2012@ 1:13 pmCool…. now i know where the Roy’s Identity came from keep on the great

works

FredRBP

June 10, 2012@ 6:16 pmyou know you can pause it…

Ben Tegethoff

September 2, 2012@ 1:54 amHessian matrices are just a 2nd-derivative test for constrained

optimization functions to see if you have a max, min, or saddle, though

since you posted this 9 months ago I’m sure you know that by now. Just

putting it out there for all the other mathletes.

amberzeliamelody

September 16, 2012@ 3:49 amFantastic, thank you, i appreciate the pace and the application. You’re an

excellent teacher.

Rubystache

September 24, 2012@ 1:17 amThank you so much for making/posting this video! Feeling much smarter/less

stupid now

Yuting Weng

December 19, 2012@ 2:47 amThanks that really helps a lot! After I watched the video I subscribe your

programme immediately:D Too bad that I study abroad in China where YOUTUBE

is not available if I don’t use vpn.

miloinindo

May 25, 2013@ 12:04 amFirst time I’ve had to pause a lecture to follow. You move through the math

a bit fast. Good explanation but it really would have been better if it was

a bit longer (in my opinion at least).

jinheff

October 5, 2013@ 2:06 pmThank you so much for making this video!

Akapulk01

October 29, 2013@ 6:45 pmThank you very much for this video also the part b is good!

Shuo Liu

December 10, 2013@ 4:07 pmso helpful ,thank you so much

shyamumich

March 27, 2014@ 2:58 pmVery useful video. One suggestion would be to have the statement of

envelope theorem up in the front.

Roshan Baskaran

May 3, 2014@ 7:10 amthank you sir

Julian Russ

August 19, 2014@ 2:16 pmAwesome stuff, thank you!

AngelusMortis1000

November 7, 2014@ 6:20 pmI forgot lamda :)