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Stephen ChiuUniversity of Hong Kong

Utility Theory

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Utility Theory

The cardinal approach The ordinal approach Consumer choice problem Intertemporal choice problem

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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

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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

U2=600

U3=610

Food(units per week)

Clothing(units

per week)

U1=500

W1=1000

W2=1M W3=1T Both John and Smith have the same indifference curve map!!!

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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

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Consumer Choice problem

Ordinal utility function indifference curve mapNumbering 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

Orange Juice(glasses)

Apple Juice

(glasses)

2 3 41

1

2

3

4

0

PerfectSubstitutes

PerfectSubstitutes

Two goods are perfect substitutes when the marginal rate of substitution of one good for the other is constant.

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Perfect Complements

Two goods are perfect complements when the indifference curves for the goods are shaped as right angles.

Right Shoes

LeftShoes

2 3 41

1

2

3

4

0

PerfectComplements

PerfectComplements

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Properties of ICs Map

More is betterTwo ICs do not

crossBending toward

origin

Y

X

A

C

U1U0

This is ruled out!

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Budget Constraints

Budget Line F + 2C = $80(I/PC) = 40

Food(units per week)40 60 80 = (I/PF)20

10

20

30

0

A

B

D

E

G

Clothing(units

per week)

Pc = $2 Pf = $1 I = $80

As consumption moves along a budget line from the intercept, the consumer spends less on one item and more on the other.

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Consumer Choice

Budget Line

U3

D Market basket D cannot be attainedgiven the currentbudget constraint.

Pc = $2 Pf = $1 I = $80

Food (units per week)

Clothing(units per

week)

40 8020

20

30

40

0

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Consumer Choice

Food (units per week)

Clothing(units per

week)

40 8020

20

30

40

0

U1

B

Budget Line

Pc = $2 Pf = $1 I = $80

Point B does not maximize satisfaction because there exist some point A which is attainable and yields a higher satisfaction.

-10C

+10F

A

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Consumer Choice

V

T

U3

U1

BU

Z

R

P

O S Q

A

Optimal consumption budget is found where budget line and an IC are tangential to each other

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coffee

teaU0

U1

U2

tea

coffee

Corner solutions are still possible

Tangency condition need not hold

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The cardinal approach

U2=600

U3=610

Food(units per week)

Clothing(units

per week)

U1=500

W1=1000

W2=1M W3=1T Despite different numbering of ICs, John and Smith both choose the same bundle

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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

C1

C2

1600

1000

500

Slope = -1.1

u(c1,c2)=const

Income in period 1

Intertemporal budget line

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1000-C1=S (1)

500+S(1+r)=C2 (2)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 2000 1000 5001500 1000

2 2500

10002

rrC

r r

r

Intertemporal choice problem

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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