Intro
Solving UMP
Extreme Cases
Σ
Econ 4601 Urban & Regional Economics
Lecture 5: Utility Maximization Problems
Instructor: Hiroki Watanabe
Summer 2010
1 / 39
Intro
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
Budget Line Meets Indifference Curves
Tangency
Find the Exact Solutions
3
Extreme Cases
Perfect Substitutes
Perfect Complements
4
Summary
Σ
2 / 39
Intro
Solving UMP
Extreme Cases
Σ
So far we have discussed:
$(Lecture 2)
(Lecture 3)
, (Lecture 4)
Now combine all the above to predict Greg’s
consumption behavior given his budget constraint
and preferences.
Situational background:
Greg wants to have as many cheesecakes and tea
as possible.
Greg’s budget constraint does not allow him to
choose (xC , xT ) =(infinite, infinite).
Greg has to make choices.
3 / 39
Intro
Solving UMP
Extreme Cases
Σ
We assume that Greg picks the most preferred
combination among what he can afford.
The framework to analyze Greg’s choice behavior is
called utility maximization problem (UMP for short).
4 / 39
Intro
Solving UMP
Extreme Cases
Σ
Greg’s decision making process is summarized as
follows:
Utility Maximization Problem
Greg chooses the bundle (xC , xT ) that gives him the
highest utility level , among the affordable bundles. In
other words, he
max u(xC , xT )
subject to pC xC + pT xT = m.
5 / 39
Intro
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
Budget Line Meets Indifference Curves
Tangency
Find the Exact Solutions
3
Extreme Cases
Perfect Substitutes
Perfect Complements
4
Summary
Σ
6 / 39
Intro
Solving UMP
Extreme Cases
Σ
Budget Line Meets Indifference Curves
How exactly do we find Greg’s optimal bundle
x∗ = (xC∗ , xT∗ )?
Consider the case when
u(xC , xT ) = xC xT
m = 60,
(pC , pT ) = (4, 3).
7 / 39
Intro
Solving UMP
Extreme Cases
Σ
Budget Line Meets Indifference Curves
20
Tea (xT)
15
10
5
0
0
5
10
Cheesecakes (xC)
15
20
8 / 39
1
Intro
Solving UMP
Extreme Cases
Σ
Budget Line Meets Indifference Curves
50
15
0
10
10
15
0
75
25
Tea (xT)
30
0
37
32 350 5
5
5
12
5
22 200
5
17
0
15
75
25
5
27 50
2
20
27
5
25
0
22
17
5
20
0
5
12
5
100
50
5
125
75
100
25
75
50
50
25
0
0
175
150
5
25
10
Cheesecakes (xC)
15
20
9 / 39
1
Intro
Solving UMP
Extreme Cases
Σ
Budget Line Meets Indifference Curves
5
17
0
15
125
75
25
5
27
0
25
5
22
20
50
30
0
0
75
25
50
Tea (xT)
5
12
5
15
0
10
200
5
25
125
100
50
75
50
25
5
175
150
0
75
0
0
5
27
25 5
0
22
5
17
10
32
20
0
10
15
35 37
5
0
10
Cheesecakes (xC)
25
15
20
10 / 39
1
Intro
Solving UMP
Extreme Cases
Σ
Budget Line Meets Indifference Curves
11 / 39
Intro
Solving UMP
Figure:
Extreme Cases
Σ
Tangency
Tangency Condition
At the optimal bundle, the indifference curve is tangent
to the budget constraint (for standard preferences), i.e.,
they both have the same slope at x∗ .
12 / 39
Intro
Solving UMP
Extreme Cases
Σ
Tangency
What does the tangency condition imply?
The slope of IC denotes MRS (MWTP): the cups of
tea Greg is willing to give up for one slice of
cheesecake.
The slope of budgte line denotes relative price (op.
cost): the cups of tea Greg has to give up for one
slice of cheesecake.
Greg’s idea of the tea’s worth coincides with
market’s idea of the tea’s worth when he chooses
the right amount.
13 / 39
Intro
Solving UMP
Extreme Cases
Σ
Tangency
What if the tangency condition is not met?
Suppose MRS is larger than the relative price (IC is
steeper than the budget line).
Greg is willing to give up 30 cups of tea for a slice of
cheesecake,
while he has to sell only 43 cups of tea to buy a slice
of cheesecake.
It’s wise to increase cheesecake consumption and
reduce cups of tea.
The current bundle is not optimal.
14 / 39
Intro
Solving UMP
Extreme Cases
Σ
Tangency
Suppose MRS is smaller than the relative price (IC is
steeper than the budget line).
Greg is willing to give up .1 cups of tea for a slice of
cheesecake,
while he has to sell 43 cups of tea to buy a slice of
cheesecake.
It’s wise to reduce cheesecake consumption and
increase cups of tea.
The current bundle is not optimal.
15 / 39
Intro
Solving UMP
Extreme Cases
Σ
Find the Exact Solutions
Where exactly is the bundget line tangent to the
indifferent curve?
We need a little bit of calculus to find Greg’s
optimal bundle.
UMP
max u(xC , xT ) = xC xT
subject to
MRS at (xC , xT ) is given by
The relative price is
4xC + 3xT = 60.
−xT
.
xC
−4
.
3
16 / 39
Intro
Solving UMP
Extreme Cases
Σ
Find the Exact Solutions
Tangency condition:
−xT
xC
=
−4
3
⇒ xT =
4
3
xC .
(1)
There are lots of bundles (xC , xT ) satisfying (1).
The bundle also has to be affordable:
4
4xC + 3xT = 60 ⇒ 4xC + 3 xC = 60.
3
xT = 7.5 and xC = 10.
Conclude x∗ = (10, 7.5).
17 / 39
Intro
Solving UMP
Extreme Cases
Σ
Find the Exact Solutions
Exercise
Greg spends his income ($5) on coins (xC ) and tea (xT ).
Price is given by (pC , pT ) = (2, 1). His preferences are
p
represented by u(xC , xT ) = xC + xT .
1
Write down the budget line.
2
Write Greg’s utility maximization problem.
3
4
What is the tangency condition in this example?
p
(MRS at (xC , xT ) is −2 xT ).
Which bundle will Greg buy?
18 / 39
Intro
Solving UMP
Extreme Cases
Σ
Find the Exact Solutions
6
5
7
4
4
2
5
T
6
Tea (x )
3
3
2
4
2
5
3
6
1
1
0
0
2
1
3
2
3
Coins (xC)
4
5
4
5
19 / 39
Intro
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
Budget Line Meets Indifference Curves
Tangency
Find the Exact Solutions
3
Extreme Cases
Perfect Substitutes
Perfect Complements
4
Summary
Σ
20 / 39
Intro
Solving UMP
Extreme Cases
Σ
There are some exceptions where tangency
condition does not apply.
21 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
Consider six-packs and bottles of Corona (x6 , x1 ).
Greg’s MRS is 6 everywhere.
Consider three cases:
1
2
3
Market rate of exchange is higher than 6.
Market rate of exchange is lower than 6.
Market rate of exchange is exactly 6.
22 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
1
Market rate of exchange is higher than 6.
Let’s say the market rate of exchange is 24 (i.e., 24
bottles of Corona is traded for 1 six-pack).
Greg is willing to give up 6 bottles for 1 six-pack.
Greg will spend all his income on six-packs.
This type of solution is called a corner solution (e.g.,
(8, 0), (35, 0), (0, 68) etc.)
23 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
24
Bottles (x1)
18
48
12
36
6
24
12
0
0
1
2
3
4
Six−Packs (x6)
5
6
24 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
2
Market rate of exchange is lower than 6.
Let’s say the market rate of exchange is 1 (i.e., 1
bottle of Corona is traded for 1 six-pack).
Greg is willing to give up 6 bottles for 1 six-pack.
Greg will spend all his income on bottles.
25 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
24
24
36
48
12
12
Bottles (x1)
18
24
0
0
36
6
1
2
3
4
Six−Packs (x6)
5
6
26 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
3
Market rate of exchange is exactly 6.
The market rate of exchange is 6 (i.e., 6 bottles of
Corona is traded for 1 six-pack).
Greg is willing to give up 6 bottles for 1 six-pack.
Greg can choose any bundle on his budget line.
27 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
24
48
24
36
Bottles (x1)
18
12
12
6
0
0
1
2
3
4
Six−Packs (x6)
5
6
28 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
Discussion
If a bottle of Corona is sold at $2, a six-pack is
usually sold at less than $12.
Does that mean nobody buys Corona by the bottle?
Dharma’s MRS might be less than 6.
She is willing to give up 1 six-pack for 4 bottles
(MRS= 4).
So, there are Corona sold by the bottle, and there
are people who buy Corona by the bottle.
29 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
Exercise
Greg has $12 and spend his income on Coke & Pepsi
(xC , xP ). Suppose Coke is sold at $4 while Pepsi is sold
at $2.
1
2
Find Greg’s optimal bundle.
Confirm your answer using indifference curves and
the budget line.
30 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Substitutes
6
11
5
10
7
P
Pepsi (x )
4
4
9
3
8
2
6
3
2
1
5
1
0
0
1
2
3
Coke (xC)
4
5
6
31 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
For perfect complements, MRS is not defined.
That does not mean the solution to UMP does not
exist.
Consider left gloves (x1 ) and right gloves (x2 ).
32 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
6
3
5
2
4
4
4
1
Right Gloves (x )
2
5
3
3
2
2
1
1
0
0
1
2
3
2
1
3
4
Left Gloves (x1)
1
5
6
33 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
Optimal bundle x∗ for left gloves and right gloves is
1
2
on the budget line and
on the 45 degree line.
34 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
Exercise
Consider a bundle (cereal, milk)= (xC , xM ).
Greg says he can’t have cereals without milk and
the only time he drink milk is when he eats his
cereals.
Greg’s preferred cereal-milk ratio is 2 to 3.
(pC , pM ) = (6, 4).
m = 24.
1
Draw his budget line.
2
Draw indifference curves at (2, 3) and (4, 6).
3
Mark the bundle Greg will choose on your graph.
35 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
16
4
6
2
14
6
12
M
Milk (x )
10
8
3
6
4
2
0
0
2
4
Cereals (xC)
6
8
36 / 39
Intro
Solving UMP
Extreme Cases
Σ
Perfect Complements
Greg’s preferred bundles are lined up on the ray
xM = 32 xC .
The optimal bundle is
1
2
on the ray xM = 32 xC and
on the budget line xM =
−3
x
2 C
+ 6.
37 / 39
Intro
Solving UMP
Extreme Cases
1
Introduction
2
Solving UMP
Budget Line Meets Indifference Curves
Tangency
Find the Exact Solutions
3
Extreme Cases
Perfect Substitutes
Perfect Complements
4
Summary
Σ
38 / 39
Intro
Solving UMP
Extreme Cases
Σ
How to set up the utility maximization problem.
Tangency condition for standard cases.
Optimal bundles for extreme preferences.
39 / 39