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Lesson 21: Partial Derivatives in Economics

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

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Lesson 21: Partial Derivatives in Economics

1. 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. 2. Part I Partial Derivatives in Economics
3. 3. Outline Marginal Quantities Marginal products in a Cobb-Douglas function Marginal Utilities Case Study
4. 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. 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. 6. Outline Marginal Quantities Marginal products in a Cobb-Douglas function Marginal Utilities Case Study
7. 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 .
8. 8. Outline Marginal Quantities Marginal products in a Cobb-Douglas function Marginal Utilities Case Study
9. 9. 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?
10. 10. Outline Marginal Quantities Marginal products in a Cobb-Douglas function Marginal Utilities Case Study
11. 11. 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.
12. 12. 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
13. 13. 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
14. 14. 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?
15. 15. Result Tim tried the function t(S + 1) σ(t, w , S) = w
16. 16. 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)
17. 17. 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
18. 18. 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
19. 19. Part II Linear Models with Quadratic Objectives
20. 20. Outline Algebra primer: Completing the square A discriminating monopolist Linear Regression
21. 21. Algebra primer: Completing the square
22. 22. Outline Algebra primer: Completing the square A discriminating monopolist Linear Regression
23. 23. 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?
24. 24. Outline Algebra primer: Completing the square A discriminating monopolist Linear Regression
25. 25. Example Suppose we’re given a data set (xt , yt ), where t = 1, 2, . . . , T are discrete observations. What line best ﬁts these data?