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Chapter 6 Demand Key Concept: the demand function x 1 (p 1 , p 2 , m)

Chapter 6 Demand Key Concept: the demand function x 1 (p 1 , p 2 , m) Income m: normal good, inferior good Own price p 1 : Giffen good, ordinary good Other price p 2 : substitute, complement. Chapter 6 Demand The demand function x 1 (p 1 , p 2 , m)

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Chapter 6 Demand Key Concept: the demand function x 1 (p 1 , p 2 , m)

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  1. Chapter 6 Demand • Key Concept: the demand function x1 (p1, p2, m) • Income m: normal good, inferior good • Own price p1: Giffen good, ordinary good • Other price p2: substitute, complement

  2. Chapter 6 Demand • The demand function x1 (p1, p2, m) • gives the optimal amounts of each of the goods as a function of • the prices and income faced by the consumer.

  3. x1 (p1, p2, m) • We now change the arguments in the demand function one by one.

  4. ∆x1/∆m > 0: a normal good ∆x1/∆m < 0: an inferior good • It depends on the income level we are talking about. • bus, MRT, taxi

  5. Fig. 6.1

  6. Fig. 6.2

  7. Two ways to look at the same thing. • (1) At x1 – x2 space, connect the optimal bundles as the budget line shifts. This is called the income offer curve (IOC). • (2) At x1 – m space, connect the optimal x1 as the income changes. This is called the Engel curve.

  8. Draw a general preference to illustrate the income offer curve and the Engel curve.

  9. Fig. 6.3 m2 m1 m1 m2

  10. Look at specific preferences.

  11. Perfect substitutes • p1 < p2 • IOC (x axis) • Engel (sloped p1)

  12. Fig. 6.4

  13. Perfect complements • IOC (at the corner) • Engel (sloped p1+ p2)

  14. Fig. 6.5

  15. Cobb-Douglas • x1 = am/ p1 and x2 = (1-a)m/ p2 • so x1/x2 is constant at ap2/ (1-a)p1 • IOC (line from origin) • Engel (sloped p1/a)

  16. Fig. 6.6

  17. Notice any similarity among the three cases? • In the above three cases, m=c x1. • (∆x1/ x1) / (∆m/m) • = (1/c) / (1/c) • =1

  18. They all are homothetic preferences. • If (x1, x2) w (y1, y2), then for all t >0 • (tx1, tx2) w (ty1, ty2) • (無異曲線等比例放大縮小)

  19. We want to show if (x1, x2) is optimal at m, then (tx1, tx2) is optimal at tm. • If we double the income, we just double everything.

  20. If (x1, x2) is optimal at m, then (tx1, tx2) is optimal at tm. • We care about the ratio of good 1 to good 2. (x1, x2) w (y1, y2) ↔ (tx1, tx2) w (ty1, ty2) • If we have found an optimal ratio, we just keep itwhen income is changed.

  21. If (x1, x2) at m, then (tx1, tx2) at tm. Suppose not, then (y1, y2) is feasible at tm and (y1, y2) s (tx1, tx2). Then (y1, y2) w (tx1, tx2) and it is not the case that (tx1, tx2) w (y1, y2). However, (y1/t, y2/t) is feasible at m, so (x1, x2) w (y1/t, y2/t). By homothetic preferences, (tx1, tx2) w (y1, y2), a contradiction.

  22. Reasonable? (toothpaste)

  23. Quasilinear preferences • p1 = p2=1 • u(x1, x2) = √x1 + x2 • MRS1, 2 = -MU1 / MU2 =-p1/ p2 • MU1 = 1/(2 √x1), MU2 = 1 • MU1/p1 = MU2/p2 implies x1 = ¼ is a cutting point

  24. IOC: on the x-axis up to (1/4,0), then becomes vertical • Engel: sloped 1 up to (1/4, 1/4), then becomes vertical • “zero income effect” only after some point

  25. Fig. 6.8

  26. We now change own price in x1 (p1, p2, m) • ∆x1/∆p1 > 0: good 1 is a Giffen good • ∆x1/∆p1 < 0: good 1 is an ordinary good

  27. Two ways to look at the same thing. • (1) At x1 – x2 space, connect the optimal bundles as the budget line pivots. • This is called the price offer curve (POC). • (2) At x1 – p1 space, connect the optimal x1 as own price changes. • This is called the demand curve.

  28. Draw a general preference to illustrate the price offer curve and the demand curve.

  29. Fig. 6.9

  30. Fig. 6.11 p1 p’1 p1 p’1

  31. Look at specific preferences.

  32. Perfect substitutes • POC • p1 > p2: x1 = 0 • p1 = p2: all budget line • p1 < p2: x1 = m/ p1 • draw demand curve

  33. Fig. 6.12

  34. Perfect complements • POC (at the corner) • demand (m/(p1+p2))

  35. Quasilinear • u(x1, x2) = v(x1) + x2 • good 1 is in discrete amounts

  36. Start to buy the first unit of good 1 when p1 has decreased to • v(0)+m = v(1)+m-p1 • p1 has decreased to v(1) – v(0). • Startto buy the second unit of good 1 when p1 has further decreased to • v(1)+m-p1= v(2)+m-2p1 • p1 has decreased to v(2) – v(1).

  37. Illustrate the demand curve for the quasilinear case

  38. Fig. 6.14 v(1)-v(0) v(2)-v(1)

  39. We now change other price in x1 (p1, p2, m). • ∆x1/∆p2 > 0 • good 1 is a substitute for good 2 • ∆x1/∆ p2 < 0 • good 1 is a complement for good 2 • 像自己價格的改變

  40. the inverse demand function • x1= x1 (p1), given p1, how many x1 that a consumer wants to buy • p1= p1 (x1), given x1, what price of p1 would have to be in order for the consumer to choose that level of consumption

  41. Fig. 6.15

  42. Cobb Douglas • x1 = am/ p1 • p1 = am/ x1

  43. Inverse demand has a useful interpretation • |MRS1, 2| = p1/ p2 • p1 = |MRS1, 2| p2 • p1 = |MRS1,2| = ∆$/∆ x1 • How many dollars consumer is willing to give up to have a little more of 1 • marginal willingness to pay

  44. Fig. 6.15

  45. Chapter 6 Demand • Key Concept: the demand function x1 (p1, p2, m) • Income m: normal good, inferior good • Own price p1: Giffen good, ordinary good • Other price p2: substitute, complement

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