lesson 21 partial derivatives in economics 1194885782977551 4

7/29/2019 Lesson 21 Partial Derivatives in Economics 1194885782977551 4

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Lesson 21 (Sections 15.67)Partial Derivatives in Economics

Linear Models with Quadratic Objectives

Math 20

November 7, 2007

Announcements

Problem Set 8 assigned today. Due November 14.

No class November 12. Yes class November 21.

OH: Mondays 12, Tuesdays 34, Wednesdays 13 (SC 323)

Prob. Sess.: Sundays 67 (SC B-10), Tuesdays 12 (SC 116)
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Part I

Partial Derivatives in Economics
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Outline

Marginal Quantities

Marginal products in a Cobb-Douglas function

Marginal Utilities

Case Study
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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) = p2 + I

to model demand, then the marginal demand of price is

q

p= 2p3

Similarly, the marginal demand of income is

q

I= 1
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A point to ponder

The act of fixing all variables and varying only one is the

mathematical formulation of the ceteris paribus (all other thingsbeing equal) motto.
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Outline

Marginal Quantities


Marginal Utilities

Case Study
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Example (15.20)Consider an agricultural production function

Y = F(K, L, T) = AKaLbTc

where Y is the number of units produced

K is capital investment

L is labor input

T is the area of agricultural land produced A, a, b, and c are positive constants

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

Marginal Quantities


Marginal Utilities

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

x is the total amount of goods produced and consumed

z is a measure of the level of pollution

What can you estimate about the signs of ux? uz? uxz? 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
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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 youll know just how [bad] it was.
http://www.themcintireconspiracy.com/forum/viewtopic.php?p=12073http://www.themcintireconspiracy.com/forum/viewtopic.php?p=12073http://find/


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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 youll know just how [bad] it was.

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

mutatis mutandis
http://www.themcintireconspiracy.com/forum/viewtopic.php?p=12073http://www.themcintireconspiracy.com/forum/viewtopic.php?p=12073http://find/http://goback/


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Inputs

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

low pay gig far away from home

Bad Lights

Bad Sound

Bad Stage Bad Chair Arrangement/Audience Seating

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

No Heckler Control

Restrictive Limits on Material Bachelorette Party In Room

No Cover Charge

Random Bizarreness

V i bl


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Variables

Tim settled on the following variables:

t: drive time to the venue

w: amount paid for the show S: venue quality (count of bad qualities) from above

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

R l
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Result

Tim tried the function

(t, w, S) = t(S + 1)w
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R lt


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Result


(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, theres 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 is

therefore 2 (TVs on, random bizarreness for speech)

Res lt
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Result


(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, theres 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 is

therefore 2 (TVs on, random bizarreness for speech)

=100

500(1 + 2) = 3/5 = 0.6
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Example (Bad Gig)

300 dollars in a town 200 miles from your house. Bad lights, badsound, drunken hecklers, and no cover charge. Thats a Steen

Coefficient of 4. =

400

300(1 + 4) = 6.666
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Part II

Linear Models with Quadratic Objectives

Outline
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Outline

Algebra primer: Completing the square

A discriminating monopolist

Linear Regression


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Outline
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Linear Regression


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Example

A firm sells a product in two separate areas with distinct lineardemand curves, and has monopoly power to decide how much to

sell in each area. How does its maximal profit depend on thedemand in each area?

Outline
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Linear Regression


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Example

Suppose were given a data set (xt, yt), where t = 1, 2, . . . , T are

discrete observations. What line best fits these data?
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lesson 21 partial derivatives in economics 1194885782977551 4

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