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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 products in a Cobb-Douglas function
Marginal Utilities
Case Study
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Marginal products in a Cobb-Douglas function
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 products in a Cobb-Douglas function
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 products in a Cobb-Douglas function
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/7/29/2019 Lesson 21 Partial Derivatives in Economics 1194885782977551 4
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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
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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
Tim tried the function
(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
Tim tried the function
(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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Algebra primer: Completing the square
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Algebra primer: Completing the square
Outline
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Algebra primer: Completing the square
A discriminating monopolist
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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Algebra primer: Completing the square
A discriminating monopolist
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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