Intro
Solving UMP
Extreme Cases

Econ 3023 Microeconomic Analysis
Chapter 5:
Utility Maximization Problem
Instructor: Hiroki Watanabe
Fall 2012
Watanabe
Econ 3023
Intro
5 UMP
Solving UMP
1
Introduction
2
Solving UMP
3
Extreme Cases
4
Now We Know
Watanabe
Econ 3023
Intro
Extreme Cases
5 UMP
Solving UMP
Extreme Cases
1
Introduction
Consumer Theory Overview
2
Solving UMP
3
Extreme Cases
4
Now We Know
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Intro
Solving UMP
Extreme Cases
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Consumer Theory Overview
1
What do consumers face?
2
What do consumers want?
3
How do consumers resolve conflict above?
Chapter 2 & 10
Chapter 3 & 4
Chapter 5
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Consumer Theory Overview
Consumers choose the best combination of
goods they can afford.
The framework to analyze their choice behavior is
called utility maximization problem (UMP).
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Consumer Theory Overview
Definition 1.1 (Utility Maximization Problem)
Liz chooses the bundle (C , T ) that gives her the
highest utility level among the affordable bundles.
In other words, she
max(C , T )
Watanabe
Econ 3023
subject to
pC C + pT T = m.
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Intro
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
Budget Line Meets Indifference Curves
Tangency
To Find the Exact Solution
3
Extreme Cases
4
Now We Know
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Extreme Cases
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Budget Line Meets Indifference Curves
How exactly do we find Liz’s optimal bundle
∗ , ∗ )?
∗ = (C
T
Consider a case when
Utility function (C , T ) = C T
Income m = 60,
Price (pC , pT ) = (3, 2).
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Extreme Cases
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Budget Line Meets Indifference Curves
30
Budget Line
Tea xT (cups)
25
20
15
10
5
0
0
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5
10
Cheese xC (slices)
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20
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Intro
Solving UMP
Extreme Cases
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Budget Line Meets Indifference Curves
30
5
Indifference Curves50
50
0
25
45
0
Tea xT (cups)
40
0
35
0
20
30
0
250
15
200
10
150
100
5
50
0
0
Watanabe
5
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Intro
10
Cheese xC (slices)
15
20
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Extreme Cases
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Budget Line Meets Indifference Curves
30
5
Indifference Curves50
50
Budget Line
0
25
45
0
Tea xT (cups)
40
0
35
0
20
30
0
250
15
200
10
150
100
5
50
0
0
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5
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Intro
10
Cheese xC (slices)
15
20
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Tangency
Fact 2.1 (Tangency Condition)
For standard preferencesa , the indifference curve is
tangent to the budget line at the optimal bundle, i.e.,
they share the same slope at ∗ .
a This condition does not apply to certain types of preferences.
See Exercise 3.2 for example.
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Solving UMP
Extreme Cases
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Tangency
What does tangency condition imply?
1
2
Trinity on the budget side
Trinity on the preference side
Liz’s idea of the tea’s worth coincides with
market’s idea of the tea’s worth when she
chooses the right amount.
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Extreme Cases
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Tangency
What if tangency condition is not met?
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Extreme Cases
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Tangency
30
5
Indifference Curves50
50
Budget Line
0
Relative Price 450
MRS
40
25
Tea xT (cups)
0
35
0
20
30
0
250
15
200
10
150
100
5
50
0
0
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5
10
Cheese xC (slices)
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20
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Intro
Solving UMP
Extreme Cases
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To Find the Exact Solution
Where exactly is the budget line tangent to the
indifferent curve?
We need two conditions to find the point of
tangency:
Fact 2.2 (Finding the Solution)
For standard utility functions, the solution can be found
using
1
tangency condition and
2
budget constraint
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Extreme Cases
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To Find the Exact Solution
Problem 2.3 (UMP)
max(C , T ) = C T subject to
MRS at (C , T ) is given by
The relative price is
1 In
Watanabe
3C + 2T = 60.
−T 1
.
C
−3
.
2
what follows, I’ll get you MRS except for perfect substitutes.
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Extreme Cases
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To Find the Exact Solution
Tangency condition:
−T
C
=
−3
2
⇒ T =
3
2
C .
(1)
There are lots of bundles (C , T ) satisfying (1).
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Intro
Solving UMP
Extreme Cases
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To Find the Exact Solution
30
5
Indifference Curves50
50
Budget Line
0
25
45
0
Tea xT (cups)
40
0
35
0
20
30
0
250
15
200
10
150
100
5
50
0
0
Watanabe
5
10
Cheese xC (slices)
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20
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To Find the Exact Solution
To pin down the solution, use budget constraint:
3
60 = 3C + 2 C .
2
Conclude ∗ = (10, 15).
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To Find the Exact Solution
Exercise 2.4 (UMP)
Liz spends her income ($5) on coins (C ) and tea (T ).
Price is given by (pC , pT ) = (2, 1). His preferences are
p
represented by (C , T ) = C + T .
1
Write Liz’s utility maximization problem.
2
What is the tangency condition in this example?
p
(MRS at (C , T ) is −2 T ).
3
Which bundle will Liz choose?
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Intro
Solving UMP
Extreme Cases
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To Find the Exact Solution
5
2.5
Indifference Curves
Budget Line
4.5
2.0
3
T
0
4.
Tea x (cups)
4
2
3.
5
5
1.
1
3.
0
1.0
0
0
Watanabe
0.5
2.5
0.5
Econ 3023
Intro
1
1.5
Coins xC
2
5 UMP
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
3
Extreme Cases
When Tangency Condition Does Not Hold
Perfect Substitutes
Perfect Complements
4
Now We Know
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When Tangency Condition Does Not Hold
There are some exceptions where tangency
condition does not apply.
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Intro
Solving UMP
Extreme Cases
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Perfect Substitutes
Consider six-packs and bottles of Corona (6 , 1 ).
Lizs MRS is 6 everywhere (why?)
Consider three cases:
1
2
3
Watanabe
Market rate of exchange is higher than 6.
Market rate of exchange is lower than 6.
Market rate of exchange is exactly 6.
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Extreme Cases
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Perfect Substitutes
Market rate of exchange is higher than 6.
1
Let’s say the market rate of exchange is larger than
6, say 12.
Liz will spend all her income on bottles.
This type of solution is called a corner solution
(e.g., (8, 0), (35, 0), (0, 68) etc.)
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Extreme Cases
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Perfect Substitutes
60
Indifference Curves
84
Budget Line 7
8
48
72
Bottles x1
66
60
36
54
30
24
48
24
42
18
12
36
12
6
0
0
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2
3
Six−Packs x6
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5
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Intro
Solving UMP
Extreme Cases
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Perfect Substitutes
If market rate of exchange is lower than 6...
2
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Extreme Cases
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Perfect Substitutes
60
Indifference Curves
84
Budget Line 7
8
48
72
Bottles x1
66
60
36
54
30
24
48
24
42
18
12
36
12
6
0
0
Watanabe
1
Econ 3023
Intro
2
3
Six−Packs x6
4
5
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Perfect Substitutes
This type of solution is called a corner solution
(e.g., (8, 0), (35, 0), (0, 68) etc.)
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Intro
Solving UMP
Extreme Cases
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Perfect Substitutes
Market rate of exchange is exactly 6...
3
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Intro
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Solving UMP
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Extreme Cases
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Perfect Substitutes
60
Indifference Curves
84
Budget Line 7
8
48
72
Bottles x1
66
60
36
54
30
24
48
24
42
18
12
36
12
6
0
0
Watanabe
1
Econ 3023
Intro
2
3
Six−Packs x6
4
5
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Perfect Substitutes
Exercise 3.1 (Perfect Substitutes)
Liz has $12 and spend his income on Coke & Pepsi
(C , P ). Suppose Coke is sold for $4 while Pepsi is sold
for $2.
1
Find Liz’s optimal bundle.
2
Confirm your answer using indifference curves and
the budget line.
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Intro
Solving UMP
Extreme Cases
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Perfect Substitutes
6
Indifference Curves
Budget Line
6
5
4
Bottles x
1
21
3
3
2
1
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Intro
1.5
Six−Packs x6
18
0.5
15
0
0
Watanabe
12
9
1
2
2.5
3
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Extreme Cases
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Perfect Complements
For perfect complements, MRS is not defined.
What to do with them then?
Consider left gloves (L ) and right gloves (R ).
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Extreme Cases
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Perfect Complements
12
4
Indifference Curves
Budget Line
10
8
8
2
Right Gloves xR
10
6
6
4
4
2
2
0
0
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6
8
Left Gloves xL
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12
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Intro
Solving UMP
Extreme Cases
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Perfect Complements
Optimal bundle ∗ for left gloves and right gloves is
1
2
Watanabe
on the budget line and
on the 45 degree line.
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Extreme Cases
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Perfect Complements
Exercise 3.2 (Perfect Complements)
Consider a bundle (cereal, milk)= (C , M ).
Liz says he can’t have cereals without milk and the
only time she drinks milk is when she eats his
cereals.
Liz’s preferred cereal-milk ratio is 2 to 3.
(pC , pM ) = (6, 4).
m = 24.
1
Draw her budget line.
2
Draw indifference curves at (2, 3) and (4, 6).
3
Mark the bundle Liz will choose on your graph.
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Extreme Cases
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Perfect Complements
6
Indifference Curves
Budget Line
Right Gloves x
R
5
10
4
8
3
6
2
4
1
2
0
0
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2
Left Gloves xL
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4
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Intro
Solving UMP
Extreme Cases
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Perfect Complements
Liz’s preferred bundles are lined up on the ray
M = 23 C .
The optimal bundle is
Watanabe
on the ray M =
2
on the budget line M =
Econ 3023
Intro
1
Introduction
2
Solving UMP
3
Extreme Cases
4
Now We Know
Econ 3023
Intro
and
−3

2 C
+ 6.
5 UMP
Solving UMP
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3

2 C
1
Extreme Cases
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Solving UMP
Extreme Cases
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How to set up the utility maximization problem.
Tangency condition for standard cases.
Optimal bundles for extreme preferences.
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Intro
Solving UMP
Extreme Cases
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Index
budget constraint, 16
corner solution, 26, 30
optimal bundle, 8
tangency condition, 12, 16
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utility maximization
problem, 5
∗ , see optimal bundle
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