# Public Goods © Allen C. Goodman, 2002 Services in an Urban Setting Lots of services are provided through public funds Schools, police, fire protection,

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Public Goods © Allen C. Goodman, 2002

Services in an Urban Setting Lots of services are provided through public funds Schools, police, fire protection, other govt services. Generally big tax users. Genlly in an urban setting? What do we want to explain?

Public Goods How much is provided? How is it paid for? Who gets it? Well use the model of a public good. Whats a public good?

Samuelson on Public Goods Look at a genl societal welfare function: W = w i U i (x i,G) W = welfare w i = individual weights x i = amount of good x per person G = amount of public good Constraint is: G = F(X), where X = x i L = w i U i (x i,G) + [G – F ( x i )]

Samuelson on Public Goods L = w i U i (x i,G) + [G – F ( x i )] L/ x i = w i U i x - F´ = 0. w i = F´/ U i x L/ G = w i U i G + = 0. = [ F´/ U i x ] U i G + = 0. = [F´/ U i x ] U i G + 1 = 0. [U i G / U i x ] = -1 /F´. MRS = -1 /F´ = MRT G X G = F(X) U1U1 MRT – MRS 1 U2U2 G* X 1 * +X 2 * X2*X2* X1*X1*

G X G = F(X) U1U1 MRT – MRS 1 U2U2 G* X 1 *+X 2 * X2*X2* G P G = X/G MRT MRS 1 MRS 2 MRS i 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 Bread Schools Prefers Bread Prefers Schools

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 Bread Schools Prefers Bread Prefers Schools

Bread and Schools Lets add a few more identical people. 30 Bread Schools We have five possible levels of schools s1s1 s2s2 s3s3 s4s4 s5s5

Bread and Schools Alternatively, individuals 1-5 are willing to give up different amounts of bread to get school resources. 30 Bread Schools We have five different levels of taxes. s1s1 s2s2 s3s3 s4s4 s5s5

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 Bread Schools Suppose he promises s 5. Person 5 is happy (he didnt want much). But everyone else wanted more. So politician loses election 4-1. s1s1 s2s2 s3s3 s4s4 s5s5 1 2 3 4 5

How do we decide? 30 Bread Schools Suppose he promises s 4. Persons 1, 2, and 3 are happier because theyre getting closer to what they want. But hell still lose 3- 2. s1s1 s2s2 s3s3 s4s4 s5s5 1 2 3 4 5 Suppose he now promises s 3. Hell win the election because Persons 1 and 2 are happier yet, and Person 3 is happiest, hes getting exactly what he wants.

If you dont believe me... 30 Bread Schools Suppose another politician promises s 2. Person 3 wont be happy anymore because youre providing MORE school resources than he wants … so hell vote against it. KEY POINT !!! The median voter is decisive. Eqm school will be at s 3. Each voter will pay b 3 in taxes and get s 3. s1s1 s2s2 s3s3 s4s4 s5s5 1 2 3 4 5 b3b3

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 Bread Schools s1s1 s2s2 s3s3 s4s4 s5s5 1 2 3 4 5 b3b3 Public Good G MRS, MRT MRSi MRT G* Mean MRT Possible Median MRS

It may NOT be 30 Bread Schools s1s1 s2s2 s3s3 s4s4 s5s5 1 2 3 4 5 b3b3 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 cant adjust at the margin like with private goods, but...

Tiebout Model They vote with their feet. If they dont like whats 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 doesnt 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. Peoples preferences are satisfied.

Tiebout Model Critique –People arent perfectly informed. –There may not be enough jurisdictions to meet everyones preferences. –Income matters. Someone from Detroit cannot move to Bloomfield Hills to take advantage of public goods in Bloomfield Hills. –Where you work matters. –Its probably a better model for suburbs than for central cities.

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