Public Goods

© Allen C. Goodman, 2002

Services in an Urban Setting

• Lots of services are provided through public funds

• Schools, police, fire protection, other gov’t services.

• Generally big tax users.

• Gen’lly in an urban setting?

• What do we want to explain?

Public Goods

• How much is provided?

• How is it paid for?

• Who gets it?

• We’ll use the model of a public good.

• What’s a public good?

Samuelson on Public Goods

Look at a gen’l societal welfare function:

W = wi Ui(xi,G)W = welfare

wi = individual weights

xi = amount of good x per personG = amount of public good

Constraint is:

G = F(X), where X = xi

L = wi Ui(xi,G) + [G – F (xi)]

Samuelson on Public Goods

L = wi Ui(xi,G) + [G – F (xi)]

L/xi = wiUix - F´ = 0. wi = F´/ Ui

x

L/G = wi UiG + = 0.

= [F´/ Uix] Ui

G + = 0.

= [F´/ Uix] Ui

G + 1 = 0.

[UiG / Ui

x] = -1 /F´.

MRS = -1 /F´ = MRT

G

XG = F(X)

U1

MRT – MRS1

U2

G*

X1*+X2*

X2*

X1*

G

XG = F(X)

U1

MRT – MRS1

U2

G*

X1*+X2*

X2*

G

PG= X/G

MRT

MRS1

MRS2

MRSi

G*

P*G= X/G

What if you call out P*G ?Will you get the optimal amount of G*?

How do we do this in an urban area?

• Within an area, citizens are taxed, typically with a property tax.

• They pay the taxes, and then they have to decide how much they want.

• They all get the SAME amount

Bread and Schools

• Suppose that we live in a suburb.

• Suppose there are 10 residents. Each one earns $30,000.

• They can spend it on bread, or schools.

30

30

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 $2,000, each will pay $2,000.

30

30

Bread

Sch

ools

PrefersBread

PrefersSchools

Bread and Schools

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

30

30

Bread

Sch

ools

• We have five possible levels of “schools”

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.

30

30

Bread

Sch

ools

• We have five different levels of taxes.

s1

s2

s3

s4

s5

How do we decide?

• Consider a politician. He has to win an election, and he has to get enough votes by promising the right amount of school resources

30

30

Bread

Sch

ools

• Suppose he 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

How do we decide?

30

30

Bread

Sch

ools

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

s2

s3

s4

s5

1

2

3

4

5• Suppose he now promises s3.

He’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 ...

30

30

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 b3 in taxes and get s3.

s1

s2

s3

s4

s5

1

2

3

4

5

b3

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.

Is it optimal?

30

30

Bread

Sch

ools

s1

s2

s3

s4

s5

1

2

3

4

5

b3

Public Good G

MRS, MRT

MRSi

MRT

G*

Mean MRT

Possible Median MRS

It may NOT be

30

30

Bread

Sch

ools

s1

s2

s3

s4

s5

1

2

3

4

5

b3

Public Good G

MRS, MRT

MRSi

MRT

G*

Mean MRT

Possible Median MRS

Tiebout Model

• You have a bunch of municipalities.

• Each one offers different amounts of public goods.

• Consumers can’t adjust at the margin like with private goods, but ...

Tiebout Model

• They vote with their feet.

• If they don’t like what’s being provided in one community, they move to another.

Tiebout Model

• Assumptions

– Jurisdictional Choice -- Households shop for what local governments provide.

– Information and Mobility -- Households have perfect information, and are perfectly mobile.

– No Jurisdictional Spillovers -- What is produced in Southfield doesn’t affect people in Oak Park.

– No Scale Economies -- Average cost of production does not depend on community size.

– Head Taxes -- Pay for things with a tax per person.

• We get an equilibrium. People’s preferences are satisfied.

Tiebout Model

• Critique– People aren’t perfectly informed.

– There may not be enough jurisdictions to meet everyone’s preferences.

– Income matters. Someone from Detroit cannot move to Bloomfield Hills to take advantage of public goods in Bloomfield Hills.

– Where you work matters.

– It’s probably a better model for suburbs than for central cities.