Lecture 06 Consumer Choice Lecturer Martin Paredes 1

Lecture # 06 Consumer Choice Lecturer: Martin Paredes

1. 2. 3. 4. 5. Motivation The Budget Constraint Consumer Choice Duality Some Applications 2

Example: Consumer Expenditures, US, 2001 • • Households with income $20, 000 -$29, 999 • Income (after tax): $ 23, 924 • Total expenditures: $ 28, 623 Households with income over $70, 000 • Income (after tax): $ 104, 685 • Total expenditures: $ 76, 124 3

Example: Consumer Expenditures, US, 2001 Allocation of Spending Category Income $20 K-$29 K Food $4, 499 Housing $9, 525 Clothing $1, 063 Transportation $5, 644 Health Care $2, 089 Entertainment $1, 187 Income over $70 K $9, 066 $23, 622 $3, 479 $13, 982 $2, 908 $3, 986 4

• Assume only two goods available: X and Y • Consumers take as given: • Price of X: PX • Price of Y: PY • Income: I • Total expenditure on basket: • The Basket is affordable if total expenditure does not exceed total income: PX. X + P Y. Y I 5

Definition: The Budget Constraint defines the set of baskets that the consumer may purchase given the income available. PX. X + P Y. Y I 6

Other Definitions: 1. The Budget Set is the set of baskets that are affordable to the consumer 2. The Budget Line is the set of baskets that are just affordable: PX. X + P Y. Y = I => Y = I — PX. X PY PY 7

Example: l Suppose l I = € 10 Budget line: PX = € 1 PY = € 2 1. X + 2. Y = 10 or: Y = 10 — 1. X 2 2 8

Y I/PY= 5 A • B I/PX = 10 • X 9

Y I/PY= 5 A • B I/PX = 10 • X 10

Y I/PY= 5 A • Budget line = BL 1 B I/PX = 10 • X 11

Y I/PY= 5 A • Budget line = BL 1 -PX/PY = -1/2 B I/PX = 10 • X 12

Y I/PY= 5 A • Budget line = BL 1 • C -PX/PY = -1/2 B I/PX = 10 • X 13

Change in Income: Shift of the Budget Line l Suppose I = € 12 PX = € 1 PY = € 2 => Budget line: X + 2 Y = 12 If the income rises, the budget set expands, and both intercepts shift out l Since prices have not changed, the slope of the budget line does not change l 14

Example: Shift of a budget line Y I = € 12 PX = € 1 PY = € 2 Y = 6 - X/2 …. BL 2 5 BL 1 10 X 15

Example: Shift of a budget line Y I = € 12 PX = € 1 PY = € 2 Y = 6 - X/2 …. BL 2 6 5 BL 2 BL 1 10 12 X 16

Change in Price: Rotation of the Budget Line l Suppose I = € 10 PX = € 1 PY = € 3 => Budget line: X + 3 Y = 10 If the price of Y rises, the budget line gets flatter, and the vertical intercept shifts in l Since neither income nor the price of X have changed, the horizontal intercept does not change l 17

Example: Rotation of a budget line Y I = € 10 PX = € 1 PY = € 3 Y = 3. 33 - X/3 …. BL 2 5 BL 1 10 18 X

Example: Rotation of a budget line Y I = € 10 PX = € 1 PY = € 3 Y = 3. 33 - X/3 …. BL 2 BL 1 5 3. 33 BL 2 10 19 X

Assumptions: 1. Consumers only choose non-negative quantities 2. "Rational” choice: The consumer chooses the basket that maximizes his satisfaction given the constraint that his budget imposes. Consumer’s Problem: Max U(X, Y) X, Y subject to: PX. X + P Y. Y I 20

There are two types of equilibrium: 1. Interior Solution: 1. Consumer chooses a positive quantity of both goods 2. Corner Solution: Consumer chooses not to consume one of the goods. l 21

l Graphical interpretation: The optimal consumption basket is at a point where the indifference curve is just tangent to the budget line. => MRSX, Y = PX PY 22

l Economic interpretation: The rate at which the consumer would be willing to exchange X for Y has to be the same as the rate at which they are exchanged in the marketplace => MRSX, Y = PX PY 23

Y BL 0 X 24

Y IC 1 0 BL X 25

Y IC 3 IC 1 0 BL X 26

Y Optimal choice (interior solution) at point A • A IC 3 IC 1 0 BL IC 2 X 27

l To find algebraically the quantities of X and Y in the optimal basket, we have to solve a system of two equations for two unknowns: 1. MRSX, Y = PX PY 2. PX. X + P Y. Y = I 28

Example: l Suppose l U(X, Y) = XY I = € 1000 PX = € 50 PY = € 100 Which is the optimal choice for the consumer? 29

l MRSX, Y = MUX = Y MUY X l PX = 50 = 1 PY 100 2 l So X = 2 Y 30

l Budget line: => l Then: PX. X + P Y. Y = I 50 X + 100 Y = 1000 50 (2 Y) + 100 Y = 1000 200 Y = 1000 => Y* = 5 => X* = 10 31

Example: Interior Consumer Optimum Y 50 X + 100 Y = 1000 5 0 • 10 U* = XY = 50 X 32

l The tangency condition can also be written as: MUX = MUY PX PY l Interpretation: At the optimal basket, the marginal utility per euro spent on each commodity is the same. l “Each good gives equal bang for the buck” l Marginal reasoning to maximize 33

l Definition: A corner solution occurs when the optimal bundle contains none of the goods. l The tangency condition may not hold at a corner solution. 34

l How do you know whether the optimal bundle is interior or at a corner? ð Graph the indifference curves ð Check to see whether tangency condition ever holds at positive quantities of X and Y 35

Example: Perfect Substitutes l Suppose U(X, Y) = X + Y I = € 1000 PX = € 50 PY = € 100 l Which is the optimal choice for the consumer? 36

l MRSX, Y = MUX = 1 MUY l PX = 50 = 1 PY 100 2 l So the tangency condition is not satisfied 37

Example: Corner Solution – Perfect Substitutes Y BL: 50 X + 100 Y = 1000 10 0 20 X 38

Example: Corner Solution – Perfect Substitutes Y BL U = X+Y 10 0 20 X 39

Example: Corner Solution – Perfect Substitutes Y 10 0 20 X 40

Example: Corner Solution – Perfect Substitutes Y 10 0 • 20 A X 41

l Suppose now: U(X, Y) = X + Y I = € 1000 PX = € 100 PY = € 50 l Which is the optimal choice for the consumer? 42

Example: Corner Solution – Perfect Substitutes Y BL: 100 X + 50 Y = 1000 20 0 10 X 43

Example: Corner Solution – Perfect Substitutes Y BL 20 B • 0 10 X 44