Lecture # 07-aLecture # 07-a

Consumer ChoiceConsumer Choice(conclusion)(conclusion)

Lecturer: Martin ParedesLecturer: Martin Paredes

2

1. Consumer Choice (conclusion)2. Duality3. Composite Goods4. Some Applications

3

Example: Perfect Complements Suppose U(X,Y) = min {X,Y}

I = € 1000PX = € 50

PY = € 100

Which is the optimal choice for the consumer?

4

Y

X0

10

20

BL: 50X + 100Y = 1000

Example: Corner Solution – Perfect Complements

5

Y

X0

10

20

Example: Corner Solution – Perfect Complements

BL

U = min{X,Y}

6

Y

X0

10

20

Example: Corner Solution – Perfect Complements

BL

U = min{X,Y}6.6

6.6

7

The mirror image of the original (primal) constrained optimisation problem is called the dual problem.

Min PX . X + PY . Y subject to: U(X,Y) = U0

X,Y

where U0 is a target level of utility.

8

Y

X0

U0

9

Y

X

E1 = PXX + PYY0

U0

10

Y

XE2

0E1

U0

11

Y

X

•

Optimal choice (interior solution) at point A

E*0

A

E2E1

U0

12

Suppose U0 = U*, which is the level of utility that solves the primal problem

Then an interior optimum, if it exists, of the dual problem also solves the primal problem.

13

Y

X

• Optimal choice (interior solution)

U = U*

PXX + PYY = E*

0

Y*

X*

14

Example: Suppose U(X,Y) = XY

PX = € 50

PY = € 100

Which is the basket that minimizes the expenditure necessary to attain a utility level of U0 = 50?

15

We have to solve a system of two equations for two unknowns:

1. MRSX,Y = PX PY

2. XY = 50

16

MRSX,Y = MUX = Y MUY X

PX = 50 = 1PY 100 2

So X = 2Y

17

Utility level: XY = U0 = 50

Then: (2Y) * Y = 50 Y2 = 25

=> Y* = 5=> X* = 10

18

Y

X

• U = XY = 50

0

5

10

E* = 50X + 100Y = 1000

Example: Expenditure Minimization

19

Consumers usually purchase more than two goods.

Economists often want to focus on the consumer’s selection of a particular good.

What to do? Use a composite good in the vertical

axis, that represents the amount spent on all other goods combined.

By convention, the price of a unit of the composite good equals 1. (Pm = 1)

20I/PX

m

X

•A

I/Pm= I

-PX/Pm = -PX•

Preference directions

IC

21

1. Borrowing and Lending Consider a consumer that lives for two

periods. Suppose the consumer has an income of

I1 in period 1, and I2 in period 2 To represent the consumption choice in

each period, define the composite goods C1: consumption in period 1 (in €)

C2: consumption in period 2 (in €)

22

1. Borrowing and Lending If the consumer cannot borrow or lend, he

will spend I1 in period 1, and I2 in period 2

23

Example: Borrowing and Lending

C1

C2

24C1* = I1

C2* = I2 • A

Example: Borrowing and Lending

C2

C1

25C1* = I1

C2* = I2 • A

Example: Borrowing and Lending

C2

C1

IC0

Preference direction

26

Suppose the consumer can put money in the bank and earn an interest rate r. If he decreases his consumption in

period 1 by € X (so C1 = -X), he will increase his consumption in period 2 by C2 = (1+r) * X

Slope of budget line: dC2 = - (1+r)

dC1

He will be able to spend up to I1 * (1+r) + I2 in period 2, while consuming nothing in period 1

27

Suppose the consumer can borrow money from the bank at the same interest rate r. If he increases his consumption in

period 1 by borrowing € X (so C1 = X), he will have to pay back € (1+r) * X. Hence his consumption in period 2 decreases by C2 = - (1+r) * X

Slope of budget line: dC2 = - (1+r)

dC1

He will be able to borrow up to I1 + I2 / (1+r) in period 1, while consuming nothing in period 2

28I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

29I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

30I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

31I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•Slope = -(1+r)

32I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

Case 1: Borrowing in period 1

IC0 Slope = -(1+r)

33I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

C2B •B

C1B

Case 1: Borrowing in period 1

IC0

IC1

Slope = -(1+r)

34I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

Case 2: Lending in period 1

Slope = -(1+r)

IC0

35I1

I2 • A

Example: Borrowing and Lending

C2

C1

I2+ I1(1+r)•

I1+I2/(1+r)•

Slope = -(1+r)

C2D • D

C1D

Case 2: Lending in period 1

IC2

IC0

36

2. Quantity Discounts Suppose a consumer spends his income of

€ 550 on electricity and other goods Suppose the power company sells

electricity at a price of € 11 per unit.

37

Composite Good

0

Example: Quantity Discounts

Electricity

38

Composite Good

0

Example: Quantity Discounts

550

40

Slope = -PE = -11

Electricity

39

Composite Good

0

Example: Quantity Discounts

550

40

Slope = -PE = -11

•

18 Electricity

A

IC0

40

Suppose the power company offers the following quantity discount: € 11 per unit for the first 10 units € 8 per unit for additional units.

41

Composite Good Example: Quantity Discounts

550

40

Slope = -PE = -11

0

•

1810 Electricity

A

42

Electricity

Composite Good Example: Quantity Discounts

550

40

Slope = -PE = -11

0

•

1810 75

Slope = -PE = -8

A

43

Electricity

Composite Good Example: Quantity Discounts

550

40

Slope = -PE = -11

0

•

1810 75

Slope = -PE = -8

A

IC0

44

Electricity

Composite Good Example: Quantity Discounts

550

40

Slope = -PE = -11

0

•

1810 75

Slope = -PE = -8

A B

30

IC1

45

 1. The budget line represents the set of all

baskets that the consumer can buy if he spends all of his income

2. The budget line rotates as prices change and shifts when income changes.

3. If the consumer chooses the consumption bundle by maximizing utility given his budget constraint, the optimal consumption basket will lie at a tangency between an indifference curve and the budget line or at a corner point.