Lesson 21: Partial Derivatives in Economics

Lesson 21 (Sections 15.6–7)Partial Derivatives in Economics

Linear Models with Quadratic Objectives

Math 20

November 7, 2007

Announcements

I Problem Set 8 assigned today. Due November 14.

I No class November 12. Yes class November 21.

I OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)

I Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

Part I

Partial Derivatives in Economics

Outline

Marginal Quantities

Marginal products in a Cobb-Douglas function

Marginal Utilities

Case Study

Marginal Quantities

If a variable u depends on some quantity x , the amount that uchanges by a unit increment in x is called the marginal u of x .For instance, the demand q for a quantity is usually assumed todepend on several things, including price p, and also perhapsincome I . If we use a nonlinear function such as

q(p, I ) = p−2 + I

to model demand, then the marginal demand of price is

∂q

∂p= −2p−3

Similarly, the marginal demand of income is

∂q

∂I= 1

A point to ponder

The act of fixing all variables and varying only one is themathematical formulation of the ceteris paribus (“all other thingsbeing equal”) motto.

Outline

Marginal Quantities


Marginal Utilities

Case Study


Example (15.20)

Consider an agricultural production function

Y = F (K , L,T ) = AK aLbT c

where

I Y is the number of units produced

I K is capital investment

I L is labor input

I T is the area of agricultural land produced

I A, a, b, and c are positive constants

Find and interpret the first and second partial derivatives of F .

Math 20 - November 07, 2007.GWBWednesday, Nov 7, 2007

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Outline

Marginal Quantities


Marginal Utilities

Case Study

Let u(x , z) be a measure of the total well-being of a society, where

I x is the total amount of goods produced and consumed

I z is a measure of the level of pollution

What can you estimate about the signs of u′x? u′z? u′′xz? Whatformula might the function have? What might the shape of thegraph of u be?


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Outline

Marginal Quantities


Marginal Utilities

Case Study

Anti-utility

Found on The McIntyre Conspiracy:

I had a suck show last night. Many comics have suckshows sometimes. But “suck” is such a vague term. Ithink we need to develop a statistic to help us quantifyjust how much gigs suck relative to each other. This way,when comparing bag gigs, I can say,“My show had a suckfactor of 7.8” and you’ll know just how [bad] it was.

This is a opposite to utility, but the same analysis can be appliedmutatis mutandis

http://www.themcintireconspiracy.com/forum/viewtopic.php?p=12073

Inputs

These are the things which make a comic unhappy about his set:

I low pay

I gig far away from home

I Bad Lights

I Bad Sound

I Bad Stage

I Bad Chair Arrangement/Audience Seating

I Bad Environment (TVs on, loud waitstaff, etc.)

I No Heckler Control

I Restrictive Limits on Material

I Bachelorette Party In Room

I No Cover Charge

I Random Bizarreness

Variables

Tim settled on the following variables:

I t: drive time to the venue

I w : amount paid for the show

I S : venue quality (count of bad qualities) from above

Let σ(t,w ,S) be the suckiness function. What can you estimateabout the partial derivatives of σ? Can you devise a formula for S?

Result

Tim tried the function

σ(t,w ,S) =t(S + 1)

w

Example (Good Gig)

500 dollars in a town 50 miles from your house. When you getthere, the place is packed, there’s a 10 dollar cover, and the lightsand sound are good. However, they leave the Red Sox game on,and they tell you you have to follow a speech about the clubfounder, who just died of cancer. Your Steen Coefficient istherefore 2 (TVs on, random bizarreness for speech)

σ =100

500(1 + 2) = 3/5 = 0.6


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Result

Tim tried the function

σ(t,w ,S) =t(S + 1)

w

Example (Good Gig)

500 dollars in a town 50 miles from your house. When you getthere, the place is packed, there’s a 10 dollar cover, and the lightsand sound are good. However, they leave the Red Sox game on,and they tell you you have to follow a speech about the clubfounder, who just died of cancer. Your Steen Coefficient istherefore 2 (TVs on, random bizarreness for speech)

σ =100

500(1 + 2) = 3/5 = 0.6

Example (Bad Gig)

300 dollars in a town 200 miles from your house. Bad lights, badsound, drunken hecklers, and no cover charge. That’s a SteenCoefficient of 4.

σ =400

300(1 + 4) = 6.666

Part II

Linear Models with Quadratic Objectives

Outline

Algebra primer: Completing the square

A discriminating monopolist

Linear Regression


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Outline



Linear Regression

Example

A firm sells a product in two separate areas with distinct lineardemand curves, and has monopoly power to decide how much tosell in each area. How does its maximal profit depend on thedemand in each area?

Outline



Linear Regression

Example

Suppose we’re given a data set (xt , yt), where t = 1, 2, . . . ,T arediscrete observations. What line best fits these data?

Lesson 21: Partial Derivatives in Economics

Technology

Transcript of Lesson 21: Partial Derivatives in Economics