I had planned to something from Section 15.7 but this is mostly 15.6 plus completing the square

1. Lesson 21 (Sections 15.6–7)
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 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

2. Part I
Partial Derivatives in Economics

3. Outline
Marginal Quantities
Marginal products in a Cobb-Douglas function
Marginal Utilities
Case Study

4. Marginal Quantities
If a variable u depends on some quantity x, the amount that u
changes by a unit increment in x is called the marginal u of x.
For instance, the demand q for a quantity is usually assumed to
depend on several things, including price p, and also perhaps
income 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
= −2p −3
∂p
Similarly, the marginal demand of income is
∂q
=1
∂I

5. A point to ponder
The act of fixing all variables and varying only one is the
mathematical formulation of the ceteris paribus (“all other things
being equal”) motto.

6. Outline
Marginal Quantities
Marginal products in a Cobb-Douglas function
Marginal Utilities
Case Study

7. Marginal products in a Cobb-Douglas function
Example (15.20)
Consider an agricultural production function
Y = F (K , L, T ) = AK a Lb T c
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 .

11. Outline
Marginal Quantities
Marginal products in a Cobb-Douglas function
Marginal Utilities
Case Study

12. 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 ? What
formula might the function have? What might the shape of the
graph of u be?

15. Outline
Marginal Quantities
Marginal products in a Cobb-Douglas function
Marginal Utilities
Case Study

16. Anti-utility
Found on The McIntyre Conspiracy:
I had a suck show last night. Many comics have suck
shows sometimes. But “suck” is such a vague term. I
think we need to develop a statistic to help us quantify
just how much gigs suck relative to each other. This way,
when comparing bag gigs, I can say,“My show had a suck
factor of 7.8” and you’ll know just how [bad] it was.

17. Anti-utility
Found on The McIntyre Conspiracy:
I had a suck show last night. Many comics have suck
shows sometimes. But “suck” is such a vague term. I
think we need to develop a statistic to help us quantify
just how much gigs suck relative to each other. This way,
when comparing bag gigs, I can say,“My show had a suck
factor of 7.8” and you’ll know just how [bad] it was.
This is a opposite to utility, but the same analysis can be applied
mutatis mutandis

18. 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

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

20. Result
Tim tried the function
t(S + 1)
σ(t, w , S) =
w

22. Result
Tim tried the function
t(S + 1)
σ(t, w , S) =
w
Example (Good Gig)
500 dollars in a town 50 miles from your house. When you get
there, the place is packed, there’s a 10 dollar cover, and the lights
and sound are good. However, they leave the Red Sox game on,
and they tell you you have to follow a speech about the club
founder, who just died of cancer. Your Steen Coefficient is
therefore 2 (TVs on, random bizarreness for speech)

23. Result
Tim tried the function
t(S + 1)
σ(t, w , S) =
w
Example (Good Gig)
500 dollars in a town 50 miles from your house. When you get
there, the place is packed, there’s a 10 dollar cover, and the lights
and sound are good. However, they leave the Red Sox game on,
and they tell you you have to follow a speech about the club
founder, who just died of cancer. Your Steen Coefficient is
therefore 2 (TVs on, random bizarreness for speech)
100
σ= (1 + 2) = 3/5 = 0.6
500

24. Example (Bad Gig)
300 dollars in a town 200 miles from your house. Bad lights, bad
sound, drunken hecklers, and no cover charge. That’s a Steen
Coefficient of 4.
400
σ= (1 + 4) = 6.666
300

25. Part II
Linear Models with Quadratic Objectives

26. Outline
Algebra primer: Completing the square
A discriminating monopolist
Linear Regression

32. Algebra primer: Completing the square

33. Outline
Algebra primer: Completing the square
A discriminating monopolist
Linear Regression

34. Example
A firm sells a product in two separate areas with distinct linear
demand curves, and has monopoly power to decide how much to
sell in each area. How does its maximal profit depend on the
demand in each area?

35. Outline
Algebra primer: Completing the square
A discriminating monopolist
Linear Regression

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