Voting for Public Goods

(Fisher, Ch. 3)© Allen C. Goodman 2009

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.

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!

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

Even less ULess U

What about this?

$ spent

Another Way to Look at It

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Spending Spending Spending

4,000 v. 6,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!Preferred!NOT

6,000 v. 8,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!NOT!Preferred

4,000 v. 8,000

UA

4 6 8

UB

4 6 8

UC

4 6 8

Singlepeak

Singlepeak

? ?

Preferred!NOT!NOT

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.

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, 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 explain multiple suburbs.

• Does the monopoly model explain central cities?– How does one explain the Detroit Schools?

– How does one explain the Detroit City Council?

• These are NOT unique to Detroit.

• Many large cities have these problems.

• Are they unique to large cities?

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

NEXT!Chapter 4