Lecture # 07-a Consumer Choice (conclusion) Lecturer: Martin Paredes.
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Transcript of Lecture # 07-a Consumer Choice (conclusion) Lecturer: Martin Paredes.
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
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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*
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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?
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We have to solve a system of two equations for two unknowns:
1. MRSX,Y = PX PY
2. XY = 50
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MRSX,Y = MUX = Y MUY X
PX = 50 = 1PY 100 2
So X = 2Y
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Utility level: XY = U0 = 50
Then: (2Y) * Y = 50 Y2 = 25
=> Y* = 5=> X* = 10
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Y
X
• U = XY = 50
0
5
10
E* = 50X + 100Y = 1000
Example: Expenditure Minimization
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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
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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 €)
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1. Borrowing and Lending If the consumer cannot borrow or lend, he
will spend I1 in period 1, and I2 in period 2
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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
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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
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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.
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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.