Intro
Utility Function
Indifference Curves
Examples
Trinity
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Econ 4935 Urban Economics
Lecture CT2: Utility Function
Instructor: Hiroki Watanabe
Fall 2012
Watanabe
Intro
Econ 4935
Utility Function
CT2 Utility Function
Indifference Curves
1
Introduction
2
Utility Function
3
Indifference Curves
4
Examples
5
Trinity
6
Now We Know
Watanabe
Intro
Econ 4935
Utility Function
Indifference Curves
Introduction
Consumer Theory Overview
Task for Today
2
Utility Function
3
Indifference Curves
4
Examples
5
Trinity
6
Now We Know
Econ 4935
Trinity
CT2 Utility Function
1
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Examples
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Intro
Utility Function
Indifference Curves
Examples
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Consumer Theory Overview
1
What do consumers face?
2
What do consumers want?
3
How do consumers resolve conflict above?
4
How do consumers resolve conflict above in
conjunction with location choice?
CT1
CT2
CT3
from Lecture 1A onwards
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Utility Function
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Indifference Curves
Examples
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Task for Today
1
C = 1 vs yC = 10.
2
(C , T ) = (10, 3) vs (yC , yT ) = (3, 12).
Fact 1.1 (Comparing Bundles)
1
Numbers are easy to compare.
2
Bundles are hard to compare.
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Utility Function
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Indifference Curves
Examples
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Task for Today
Today’s focus: quantification of our preferences.
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CT2 Utility Function
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Intro
Utility Function
Indifference Curves
1
Introduction
2
Utility Function
Utility Function
Ordinal Property
3
Indifference Curves
4
Examples
5
Trinity
6
Now We Know
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Examples
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Examples
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Utility Function
Example 2.1 (Corona)
A bundle of six-packs and bottles of Corona:
 = (6 , 1 ).
One way to represent Liz’s preferences by a
number is to assign the total # of bottles
66 + 1 = T(6 , 1 )
to a bundle (6 , 1 ).
Is this assignment reasonable?
T(2, 0) =
T(2, 0) =
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Utility Function
T(1, 6) =
T(1, 3) =
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Utility Function
Definition 2.2 (Utility Function)
A utility function (C , T ) assigns a number (called
utility level) to a bundle (C , T ).
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Intro
Utility Function
Indifference Curves
Examples
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Ordinal Property
Utility is an ordinal concept.
In Corona example Example 2.1 , we could have
assigned 3 ☺’s instead of 1 ☺ for each bottle:
(6 , 1 ) = 186 + 31 .
Assignment is still reasonable:
(2, 0) =
(2, 0) =
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Intro
(1, 6) =
(1, 3) =
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Utility Function
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Indifference Curves
Examples
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Ordinal Property
Assigned values of utility level does not matter so
long as they represent what we prefer to consume.
Watanabe
Intro
Econ 4935
Utility Function
CT2 Utility Function
Indifference Curves
Examples
1
Introduction
2
Utility Function
3
Indifference Curves
Art of Drawing What We Cannot See
Example
4
Examples
5
Trinity
6
Now We Know
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Intro
Utility Function
Indifference Curves
Examples
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Art of Drawing What We Cannot See
While (·) does represent what Liz prefers, it is
hard to visualize.
We have a handy little device to show her liking on
a graph.
Definition 3.1 (Indifference Curves)
The indifference curve at a bundle  = (C , T ) is a
collection of bundles that is equally preferred to .
If a bundle y = (yC , yT ) and z = (zC , zT ) are on the
same indifference curve, then Liz is indifferent
between them.
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Indifference Curves
Examples
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Art of Drawing What We Cannot See
We can trace the indifference curve at  = (C , T )
by collecting the bundles that
yield the same utility level as (C , T ).
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Indifference Curves
Examples
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Example
Take
Example 2.1
.
Question 3.2 (Indifference Curve)
Liz’s utility function for (6 , 1 ) is
(6 , 1 ) = 66 + 1 .
Trace indifference curves at (6 , 1 ) = (3, 0) and
(3, 12).
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Intro
Utility Function
Indifference Curves
Examples
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Example
30
Indifference Curves
54
30
24
48
Bottles x1
24
18
42
18
12
36
12
6
6
0
0
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Intro
1
2
3
Six−Packs x6
Econ 4935
Utility Function
Indifference Curves
Introduction
2
Utility Function
3
Indifference Curves
4
Examples
Perfect Substitutes
Perfect Complements
Cobb-Douglas Utility
Quasilinear Utility
5
Trinity
6
Now We Know
Intro
Econ 4935
Utility Function
5
CT2 Utility Function
1
Watanabe
4
Examples
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Perfect Substitutes
Consider Liz’s preferences for Coke and Pepsi
 = (C , P ).
(C , P ) = C + P .
In this case, Coke and Pepsi are perfect
substitutes.
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Intro
Utility Function
Indifference Curves
Examples
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Perfect Substitutes
5
Indifference Curves
9
5
4
8
Pepsi x (oz)
4
P
3
7
3
2
6
2
1
1
0
0
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1
2
3
Coke xC (oz)
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5
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Examples
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Utility Level u(x)
Perfect Substitutes
20
18
16
14
12
10
8
6
4
2
0
10
9
8
7
6
5
4
Pepsi x (oz)
P
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Utility Function
3
2
1
0
0
1
2
3
4
5
6
7
8
9
Coke xC (oz)
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Indifference Curves
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Perfect Complements
Consider Liz’s preferences for cereal and milk
 = (C , M ).
C if C < M
(C , M ) = min {C , M } =
M if M ≤ C .
In this case, cereal and milk are perfect
complements.
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Intro
Utility Function
Indifference Curves
Examples
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Perfect Complements
5
Indifference Curves
Milk x (oz)
4
4
3
M
3
2
2
1
1
0
0
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Intro
1
2
3
Cereal xC (oz)
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5
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Indifference Curves
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Examples
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Utility Level u(x)
Perfect Complements
10
9
8
7
6
5
4
3
2
1
0
10
9
8
7
6
5
4
Milk x (oz)
M
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3
2
1
0
0
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Utility Function
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2
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7
8
9
Cereal xC (oz)
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Cobb-Douglas Utility
The most commonly used utility function is
Cobb-Douglas utility function:
(C , T ) = C bT ,
where  > 0 and b > 0.1
E.g. (C , T ) = C T .
1 Thanks to ordinality of utility function, you can use
() := log[()] =  log(C ) + b log(T ) to represent the same
preferences.
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Intro
Utility Function
Indifference Curves
Examples
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Cobb-Douglas Utility
10
90
Indifference Curves
9
80
8
70
Tea xT (cups)
7
60
6
50
5
40
4
30
3
20
2
10
1
0
0
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Intro
1
2
3
4
5
6
7
Cheese xC (slices)
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Examples
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Utility Level u(x)
Cobb-Douglas Utility
100
90
80
70
60
50
40
30
20
10
0
10
9
8
7
6
5
4
Tea x (cups)
3
2
1
T
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Utility Function
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0
1
2
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4
5
6
7
9
8
10
Cheese xC (slices)
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Indifference Curves
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Quasilinear Utility
Another commonly used utility function is a
quasilinear utility function of the form
(1 , 2 ) = 1 + (2 ).
E.g. (1 , 2 ) = 1 +
p
2 .
1 is the number of baskets for example.
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Intro
Utility Function
Indifference Curves
Examples
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Quasilinear Utility
10
Indifference Curves
Tea xT (oz)
8
6
12
4
2
2
Intro
10
Watanabe
2
8
6
4
0
0
4
6
8
Composite Goods xC (baskets)
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Examples
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Quasilinear Utility
Utility Level u(x)
12
8
4
0
10
9
8
7
6
5
4
3
2
Tea x (oz)
1
T
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Intro
0
0
Econ 4935
Utility Function
1
2
3
4
7
8
9
10
Composite Goods xC (baskets)
Indifference Curves
Introduction
2
Utility Function
3
Indifference Curves
4
Examples
5
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
6
Now We Know
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CT2 Utility Function
1
Watanabe
5
Examples
CT2 Utility Function
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Intro
Utility Function
Indifference Curves
Examples
Trinity
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Marginal Willingness to Pay
Just like the slope of the budget line has the
meaning (recall trinity),
the slope of the indifference curve carries two
important meanings.
Fact 5.1 (Trinity on the Preference Side)
The following are the same:
1
The slope of indifference curve.
2
Marginal willingness to pay.
3
Marginal rate of substitution.
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Indifference Curves
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Marginal Willingness to Pay
Definition 5.2 (Marginal Willingness to Pay)
If Liz is just willing to give up  cups of tea for a slice of
cheesecake, then  is called her marginal willingness
to pay.
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Marginal Willingness to Pay
E.g., if (4, 10) = (5, 6), then Liz is just willing to
give up 4 cups of tea for an additional slice of
cheesecake.
She is indifferent between those two bundles.
Her MWTP at (4, 10) is 4.
If (8, 4) = (9, 3),
Her MWTP at (8, 4) is 1.
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Intro
Utility Function
Indifference Curves
Examples
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Marginal Willingness to Pay
MWTP is also referred to as marginal rate of
substitution (MRS).
The slope of indifference curves represents
the marginal willingness to pay.
Take
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Example 2.1
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Indifference Curves
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Examples
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Marginal Willingness to Pay
30
Indifference Curves
54
30
24
48
Bottles x1
24
18
42
18
12
36
12
6
6
0
0
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Intro
1
Econ 4935
2
3
Six−Packs x6
4
5
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Examples
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Marginal Willingness to Pay
Example 2.1
is a rather unusual case.
MRS usually varies depending on the bundle .
Consider two bundles:
 = (C , T ) = (1, 29384720396)
y = (yC , yT ) = (3240894603, 1).
MRS at  is pretty large (why?)
MRS at y is pretty small (why?)
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Intro
Utility Function
Indifference Curves
Examples
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Marginal Willingness to Pay
I.e., people prefer a balanced combination of two
goods rather than something extreme.
This tendency is called convex preferences.
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Indifference Curves
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Examples
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Marginal Willingness to Pay
10
90
Indifference Curves
9
80
8
70
Tea xT (cups)
7
60
6
50
5
40
4
30
3
20
2
10
1
0
0
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Intro
1
Econ 4935
Utility Function
2
3
4
5
6
7
Cheese xC (slices)
8
9
10
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Indifference Curves
Examples
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Comparison to Relative Prices
Notice the difference:
Relative price is the cups of tea Liz has to sell (give
up) to get one slice of cheesecakes to keep to her
budget.
Marginal willingness to pay is the cups of tea Liz
is willing to give up to get one slice of
cheesecakes to remain as happy as before.
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Intro
Utility Function
Indifference Curves
1
Introduction
2
Utility Function
3
Indifference Curves
4
Examples
5
Trinity
6
Now We Know
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Intro
Econ 4935
Utility Function
Examples
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CT2 Utility Function
Indifference Curves
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Quantification of preferences.
Utility functions for perfect substitutes, perfect
complements, Cobb-Douglass and quasilinear
preferences.
Trinity.
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Indifference Curves
Examples
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Map du Jour
Source http://bigthink.com/strange-maps/
312-the-population-of-chinas-provinces-compared
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Intro
Utility Function
Indifference Curves
Examples
Trinity
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Airline du Jour
Today’s color theme is provided by courtesy of
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Econ 4935
Utility Function
Northwest Airlines
CT2 Utility Function
Indifference Curves
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Index
Cobb-Douglas utility
function, 24
convex preferences, 37
indifference curve, 13
marginal rate of
substitution, 31, 34
marginal willingness to
pay, 31, 32
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Econ 4935
perfect complements, 21
perfect substitutes, 18
quasilinear utility function,
27
trinity on the preference
side, 31
utility function, 9
CT2 Utility Function
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