Voting for Public Goods (Fisher, Ch. 3) © Allen C. Goodman 2015.

Voting for Public Goods

(Fisher, Ch. 3)© Allen C. Goodman 2015

Bread and Schools

• Suppose that we live in a suburb.

• Suppose there are 10 residents. Each one earns $60,000.

• They can spend it on bread, or schools.

60

60

Bread

Sch

ools

PrefersBread

PrefersSchools

Bread and Schools

• They have to pick a tax level that each one of them will pay.

• If they decide on $4,000, each will pay $4,000.

60

60

Bread

Sch

ools

PrefersBread

PrefersSchools

Bread and Schools

• Let’s add a few more “identical” people.

60

60

Bread

Sch

ools

• We have five possible levels of “schools”

s1

s2

s3

s4

s5

s1 > s2 > s3 > s4 > s5

Bread and Schools

• Alternatively, individuals 1-5 are willing to give up different amounts of bread to get school resources.

60

60

Bread

Sch

ools

• We have five different levels of taxes.

s1

s2

s3

s4

s5

t1 > t2 > t3 > t4 > t5

How do we decide?

• Consider a politician. She has to win an election, and she has to get enough votes by promising the “right” amount of school resources

60

60

Bread

Sch

ools

• Suppose she promises s5. Person 5 is happy (he didn’t want much). But everyone else wanted more. So politician loses election 4-1 to someone who promises more.

s1

s2

s3

s4

s5

1

2

3

4

5

RolePlaying

2 Candidates

How do we decide?

60

60

Bread

Sch

ools

• Suppose she promises s4. Persons 1, 2, and 3 are happier because they’re getting closer to what they want. But she’ll still lose 3-2. s1

s2

s3

s4

s5

1

2

3

4

5• Suppose she now promises s3.

She’ll win the election because Persons 1 and 2 are happier yet, and Person 3 is happiest, he’s getting exactly what he wants.

If you don’t believe me ...

60

60

Bread

Sch

ools

• Suppose another politician promises s2. Person 3 won’t be happy anymore because you’re providing MORE school resources than he wants … so he’ll vote against it.

• KEY POINT !!! The median voter is decisive. Eq’m school will be at s3. Each voter will pay (60 - b3) in taxes and get s3.

s1

s2

s3

s4

s5

1

2

3

4

5

b3

Median Voter and Optimum

• We’ve seen that we will agree on some particular amount, by majority vote.

• Is this amount economically optimal, based on the rule that we developed in the previous lecture?

• Why or why not!

What makes this work?

Single Peaked Preferences

• Suppose you had 3 people with indicated preferences.

• Suppose they’re not single-peaked, i.e.

• If they are single peaked, there is a clearly defined “winner” and the individual gets less satisfaction if he/she moves away from it.

First Second ThirdPerson Choice Choice Choice

A 4000 6000 8000

B 6000 8000 4000

C 8000 4000 6000

Level 1 4000Level 2 6000Level 3 8000

Less U Even less U

lowerlower

lower

Closer levelto 1st choice,but less preferredthan 2nd choice

$ spent

Another Way to Look at ItUA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

4,000 v. 6,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!Preferred!NOT

We prefer 4,000 to 6,000!

6,000 v. 8,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!NOT!Preferred

We prefer 6,000 to 8,000!

4,000 v. 8,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!NOT!NOT

BUT!!! We prefer 8,000 to 4,000!

So?• $4,000 beats $6,000• $6,000 beats $8,000, BUT• $8,000 beats $4,000!

Choices aren’t transitive

Choices aren’t transitive

Why does this happen?

• Person C does not have single-peaked preferences, but is rather an extremist! … alternatively he/she HATES mediocrity.

• C becomes less and less happy until spending gets very low.

• C is unhappy with moderate positions.• You can get fancier, but “single-peaked

preferences” are desirable if we want to reach voting equilibria.

How much should we worry?• If we believe in

standard demand theory, we shouldn’t worry too much about the single-peaked problem

• Why?

Price in $

Quantity

Q*

• At Q*, we have optimal amount of Q.

• As we go further away in either direction, happiness ↓.

Demand theory is broadly consistent with Single-Peaked preferences

Demand theory is broadly consistent with Single-Peaked preferences

Median Voter Theorem

• If voters’ preferences are single-peaked, • if the choice to be made by voting is

represented along a single continuum, • if all alternatives are voted on, and • if voters act on their true preferences, • THEN the choice selected by majority

vote is the median of the desired outcomes.

What does median voter model say?

• If you have some number of jurisdictions, one can argue that the levels of schools, fire protection, police protection are broadly consistent with consumer preferences.

• Is it perfect?– No, not all citizens vote.

– If there are a lot of issues, the same citizen is not likely to be the median voter on every issue.

Median and Optimum

• Median may be close to optimum as determined by theoretical criteria.

• BUT, it may be either greater than or less than the optimum.

• By the very nature of the equilibrium, some substantial number might want more … and some other substantial number might want less.

C1

Other Models

• Some (generally conservative) economists propose a model in which government officials wish to:– Maximize the amount of

government spending.– Remain in office.

• How does this work?

Expenditure

Ev

Price ($)

Ev is preferred bymedian voter

If referendum is rejected, Expenditures revert to Er.

Er

Loss of ConsumerSurplus of C1.

C1

C1

Other Models

• Instead, according to these models, government officials, in trying to maximize expenditures, pick as much as EM. Why?

• How does this work?

Expenditure

Ev

Price ($)

Ev is preferred bymedian voter

If referendum is rejected, Expenditures revert to Er.

Er

Loss of ConsumerSurplus of C1.

C1 C1

EM

Comments

• Median voter model seems to address consumer preferences, BUT a large group of people are dissatisfied.

• Monopoly models may appeal to some voters, BUT they don’t explain how a government that continually does not satisfy median voter stays in power.

• Michigan’s Headlee Amendment makes it very difficult to raise taxes. Suggests that at least a lot of Michigan voters are concerned about governmental powers.

NEXT!Chapter 4

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