Utility Theory
Stephen Chiu
University of Hong Kong
1
Utility Theory




The cardinal approach
The ordinal approach
Consumer choice problem
Intertemporal choice problem
2
The cardinal approach
 In the 18th century, Bentham proposed that the
objective of public policy should be to maximize the
sum of happiness in society
 Economics became the study of utility or happiness,
assumed to be in principle measurable and comparable
across people
 Marginal utility of income was higher for poor people
than for rich people, so that income ought to be
redistributed unless the efficiency cost was too high
3
The ordinal approach
 Lionel Robbins (in 1932) argued that,
 Comparability of utility across people is not needed so
long we are concerned about predicting choices
 Economics is about “the relationship between given
ends and scarce means”, and how the “ends” or
preferences came to be formed was outside its scope
 Only stable preferences are needed
 Robbins didn’t think that public policy could be analyzed
within a formal economic framework
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The cardinal approach
 An agent’s utility level is like length or weight of an
object that is objective and measurable
 An agent with utility level 3,000 is happier than another
agent with utility level 200
 But … John always looks happy and enthusiastic, and
Smith unhappy and worrisome…
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The cardinal approach
 They both come to class...
 … given the same income and prices, John
always spends his income the same way as Smith
does
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The cardinal approach
Clothing
(units
per
week)
W2=1M
W3=1T
W1=1000
U3=610
Both John
and Smith
have the
same
indifference
curve map!!!
U2=600
U1=500
Food
(units per week)
7
Why diversity in consumption?
 Cardinal approach – diversity because of diminishing
marginal utility
 Ordinal approach – diversity despite no diminishing
marginal utility; what is needed is MU/$ being equalized
8
Consumer Choice problem
 Ordinal utility function
 indifference curve map
 Numbering of ICs unimportant, as long as they are
order preserving
 Some regularity conditions (a.k.a. axioms) on ICs
 Budget constraint
 The problem becomes to max utility subject to budget
constraint
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Perfect Substitutes
Apple
Juice
(glasses) 4
Perfect
Substitutes
3
Two goods are perfect
substitutes when the marginal
rate of substitution of one good
for the other is constant.
2
1
0
1
2
3
4
Orange Juice
(glasses)
10
Perfect Complements
Left
Shoes
4
Perfect
Complements
3
Two goods are perfect
complements when
the indifference
curves for the goods
are shaped as right
angles.
2
1
0
1
2
3
4
Right Shoes
11
Properties of ICs Map
Y
U0
 More is better
 Two ICs do not
cross
 Bending toward
origin
U1
A
C
X
This is ruled out!
12
Budget Constraints
Clothing
(units
per week)
(I/PC) = 40
Pc = $2
Pf = $1
I = $80
Budget Line F + 2C = $80
A
B
30
As consumption moves along a
budget line from the intercept, the
consumer spends less on one item
and more on the other.
D
20
E
10
G
0
20
40
60
80 = (I/PF)
Food
(units per week)
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Consumer Choice
Clothing
(units per
week)
Pc = $2
Pf = $1
I = $80
40
Market basket D
cannot be attained
given the current
budget constraint.
D
30
20
U3
Budget Line
0
20
40
80
Food (units per week)
14
Consumer Choice
Clothing
(units per
week)
Pc = $2
Pf = $1
Point B does not maximize satisfaction
because there exist some point A which is
attainable and yields a higher satisfaction.
40
30
I = $80
B
-10C
A
Budget Line
20
U1
+10F
0
20
40
80
Food (units per week)
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Consumer Choice
T
Optimal consumption
budget is found where
budget line and an IC
are tangential to each
other
P
B
U
A
R
U3
U1
O
S
Z
Q
V
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Corner solutions are still possible
coffee
coffee
U2
U1
U0
tea
tea
Tangency condition need not hold
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The cardinal approach
Clothing
(units
per
week)
W2=1M
W3=1T
W1=1000
U3=610
U2=600
U1=500
Despite
different
numbering
of ICs, John
and Smith
both choose
the same
bundle
Food
(units per week)
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An application: Intertemporal Choice
 Our framework is flexible enough to deal with questions
such as savings decisions and intertemporal choice.
 Suppose you live two periods: period 1 and period 2
 You earn an income of 1,000 in period 1 and a pension
of 500 in period 2
 Interest rate r. That is, by saving $1 in period 1, you get
back $(1+r) in period 2
 You consider period 1 consumption and period 2
consumption perfect complement
 Question: how much should you save now?
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Intertemporal choice problem
Income in period 2
C2
u(c1,c2)=const
1600
Slope = -1.1
Intertemporal
budget line
500
1000
C1
Income in period 1
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Intertemporal choice problem
1000-C1=S
500+S(1+r)=C2
Substituting (1) into (2), we have
500+(1000-C1)(1+r)=C2
Rearranging, we have
1500+1000r-(1+r) C1=C2 > C
Using C1=C2=C, we finally have
(1)
(2)
1500  1000r  2000  1000r   500
C

2r
2r
500
 1000 
2r
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Conclusions
 Ordinal utility theory is good enough so long as we
want to study choice
 Cardinal utility theory is needed if we want to study
public policy
 Happiness = subjective well being
 Happiness survey shows that average happiness in a
nation remains the same level once per capita income
reaches a certain level
 More on happiness if time permitted
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