Lecture # 06Lecture # 06

Consumer ChoiceConsumer Choice

Lecturer: Martin ParedesLecturer: Martin Paredes

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1. Motivation2. The Budget Constraint3. Consumer Choice4. Duality5. Some Applications

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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

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Example: Consumer Expenditures, US, 2001

Allocation of Spending

Category Income $20K-$29K Income over $70KFood $4,499 $9,066Housing $9,525 $23,622Clothing $1,063 $3,479Transportation $5,644 $13,982Health Care $2,089 $2,908Entertainment $1,187 $3,986

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• 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: PX . X + PY . Y

• The Basket is affordable if total expenditure does not exceed total income:

PX . X + PY . Y I

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Definition: The Budget Constraint defines the set of baskets that the consumer may purchase given the income available.

PX . X + PY . Y I

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Other Definitions:1. The Budget Set is the set of baskets that

are affordable to the consumer2. The Budget Line is the set of baskets that

are just affordable:

PX . X + PY . Y = I

=> Y = I — PX . X PY PY

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Example: Suppose I = € 10 PX = € 1 PY = €

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Budget line: 1. X + 2 . Y = 10

or: Y = 10 — 1 . X 2 2

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I/PX = 10

Y

X

•A

B

I/PY= 5

•

10

I/PX = 10

Y

X

•A

B

I/PY= 5

•

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I/PX = 10

Y

X

•A

B

I/PY= 5Budget line = BL1

•

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I/PX = 10

Y

X

•A

B

I/PY= 5Budget line = BL1

-PX/PY = -1/2

•

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I/PX = 10

Y

X

•

•

A

CB

I/PY= 5Budget line = BL1

-PX/PY = -1/2

•

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Change in Income: Shift of the Budget Line Suppose I = € 12 PX = € 1 PY = €

2=> Budget line: X + 2Y = 12

If the income rises, the budget set expands, and both intercepts shift out

Since prices have not changed, the slope of the budget line does not change

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Y

X

10

5

BL1

I = € 12PX = € 1PY = € 2

Y = 6 - X/2 …. BL2

Example: Shift of a budget line

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Y

X

10

5

6

12

BL2

I = € 12PX = € 1PY = € 2

Y = 6 - X/2 …. BL2

Example: Shift of a budget line

BL1

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Change in Price: Rotation of the Budget Line Suppose I = € 10 PX = € 1 PY = €

3=> Budget line: X + 3Y = 10

If the price of Y rises, the budget line gets flatter, and the vertical intercept shifts in

Since neither income nor the price of X have changed, the horizontal intercept does not change

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Y

X

5

I = € 10PX = € 1PY = € 3

Y = 3.33 - X/3 …. BL2

BL1

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Example: Rotation of a budget line

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Y

X

I = € 10PX = € 1PY = € 3

Y = 3.33 - X/3 …. BL2

BL1

BL2

3.33

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Example: Rotation of a budget line

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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) subject to: PX . X + PY . Y I X,Y

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There are two types of equilibrium:1. Interior Solution:

Consumer chooses a positive quantity of both goods

2. Corner Solution: Consumer chooses not to consume one of the goods.

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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

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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

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Y

XBL

0

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Y

X

IC1

BL0

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Y

X

IC3

BL0

IC1

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Y

X

•

Optimal choice (interior solution) at point A

IC2

BL0

IC1

IC3

A

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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 + PY . Y = I

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Example: Suppose U(X,Y) = XY

I = € 1000PX = € 50

PY = € 100

Which is the optimal choice for the consumer?

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MRSX,Y = MUX = Y MUY X

PX = 50 = 1PY 100 2

So X = 2Y

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Budget line: PX . X + PY . Y = I

=> 50 X + 100 Y = 1000

Then: 50 (2Y) + 100 Y = 1000 200 Y = 1000

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

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Y

X

• U* = XY = 50

0

5

10

50X + 100Y = 1000

Example: Interior Consumer Optimum

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The tangency condition can also be written as:

MUX = MUY

PX PY

Interpretation: At the optimal basket, the marginal utility per euro spent on each commodity is the same. “Each good gives equal bang for the

buck” Marginal reasoning to maximize

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Definition: A corner solution occurs when the optimal bundle contains none of one of the goods.

The tangency condition may not hold at a corner solution.

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How do you know whether the optimal bundle is interior or at a corner?

Graph the indifference curves Check to see whether tangency

condition ever holds at positive quantities of X and Y

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Example: Perfect Substitutes Suppose U(X,Y) = X + Y

I = € 1000PX = € 50

PY = € 100

Which is the optimal choice for the consumer?

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MRSX,Y = MUX = 1 MUY

PX = 50 = 1PY 100 2

So the tangency condition is not satisfied

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Y

X0

Example: Corner Solution – Perfect Substitutes

10

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BL: 50X + 100Y = 1000

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Y

X0

10

20

BL

U = X+Y

Example: Corner Solution – Perfect Substitutes

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Y

X0

10

20

Example: Corner Solution – Perfect Substitutes

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Y

X0

10

20•A

Example: Corner Solution – Perfect Substitutes

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Suppose now: U(X,Y) = X + Y I = € 1000

PX = € 100

PY = € 50

Which is the optimal choice for the consumer?

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Y

X0 10

20

BL: 100X + 50Y = 1000

Example: Corner Solution – Perfect Substitutes

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Y

X0 10

20

BL

•B

Example: Corner Solution – Perfect Substitutes