Chapter 6 Demand Key Concept the demand function

• 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) • gives the optimal amounts of each of the goods as a function of • the prices and income faced by the consumer.

• x 1 (p 1, p 2, m) • We now change the arguments in the demand function one by one.

• ∆x 1/∆m > 0: a normal good ∆x 1/∆m < 0: an inferior good • It depends on the income level we are talking about. • bus, MRT, taxi

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

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

m 2 m 1 m 2

• Look at specific preferences.

• Perfect substitutes • p 1 < p 2 • IOC (x axis) • Engel (sloped p 1)

• Perfect complements • IOC (at the corner) • Engel (sloped p 1+ p 2)

• Cobb-Douglas • • x 1 = am/ p 1 and x 2 = (1 -a)m/ p 2 so x 1/x 2 is constant at ap 2/ (1 -a)p 1 IOC (line from origin) Engel (sloped p 1/a)

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

• They all are homothetic preferences. • If (x 1, x 2) w (y 1, y 2), then for all t >0 • (tx 1, tx 2) w (ty 1, ty 2) • (無異曲線等比例放大縮小)

• We want to show if (x 1, x 2) is optimal at m, then (tx 1, tx 2) is optimal at tm. • If we double the income, we just double everything.

• If (x 1, x 2) is optimal at m, then (tx 1, tx 2) is optimal at tm. • We care about the ratio of good 1 to good 2. (x 1, x 2) w (y 1, y 2) ↔ (tx 1, tx 2) w (ty 1, ty 2) • If we have found an optimal ratio, we just keep it when income is changed.

• If (x 1, x 2) at m, then (tx 1, tx 2) at tm. Suppose not, then (y 1, y 2) is feasible at tm and (y 1, y 2) s (tx 1, tx 2). Then (y 1, y 2) w (tx 1, tx 2) and it is not the case that (tx 1, tx 2) w (y 1, y 2). However, (y 1/t, y 2/t) is feasible at m, so (x 1, x 2) w (y 1/t, y 2/t). By homothetic preferences, (tx 1, tx 2) w (y 1, y 2), a contradiction.

• Reasonable? (toothpaste)

• • • Quasilinear preferences p 1 = p 2 =1 u(x 1, x 2) = √x 1 + x 2 MRS 1, 2 = -MU 1 / MU 2 = -p 1/ p 2 MU 1 = 1/(2 √x 1), MU 2 = 1 • MU 1/p 1 = MU 2/p 2 implies x 1 = ¼ is a cutting point

• 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

• We now change own price in x 1 (p 1, p 2, m) • ∆x 1/∆p 1 > 0: good 1 is a Giffen good • ∆x 1/∆p 1 < 0: good 1 is an ordinary good

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

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

p 1 p’ 1

• Look at specific preferences.

• Perfect substitutes • • POC p 1 > p 2: x 1 = 0 p 1 = p 2: all budget line p 1 < p 2: x 1 = m/ p 1 • draw demand curve

• Perfect complements • POC (at the corner) • demand (m/(p 1+p 2))

• Quasilinear • u(x 1, x 2) = v(x 1) + x 2 • good 1 is in discrete amounts

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

• Illustrate the demand curve for the quasilinear case

v(1)-v(0) v(2)-v(1)

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

• the inverse demand function • x 1 = x 1 (p 1), given p 1, how many x 1 that a consumer wants to buy • p 1 = p 1 (x 1), given x 1, what price of p 1 would have to be in order for the consumer to choose that level of consumption

• Cobb Douglas • x 1 = am/ p 1 • p 1 = am/ x 1

• Inverse demand has a useful interpretation • • |MRS 1, 2| = p 1/ p 2 p 1 = |MRS 1, 2| = ∆$/∆ x 1 How many dollars consumer is willing to give up to have a little more of 1 • marginal willingness to pay

• 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