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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 = €
2
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
•
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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
10
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
10
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
20
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