Public Goods Allen C Goodman 2002 Services in

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 person G = amount of public good Constraint is: G = F(X), where X = xi L = wi Ui(xi, G) + [G – F ( xi)]

Samuelson on Public Goods X G = F(X) U 1 X 1*+X 2* L = wi Ui(xi, G) + [G – F ( xi)] L/ xi = wi. Uix - F´ = 0. wi = F´/ Uix X 1 * L/ G = wi Ui. G + = 0. = [ F´/ Uix] Ui. G + = 0. = [F´/ Uix] Ui. G + 1 = 0. [Ui. G / Uix] = -1 /F´. MRS = -1 /F´ = MRT U 2 MRT – MRS 1 X 2 * G G*

X G = F(X) MRT MRSi U 1 PG= X/G X 1*+X 2* MRS 2 MRS 1 P*G= X/G U 2 MRT – MRS 1 X 2 * G G* G* 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 30 Prefers 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. Prefers Bread 30

Bread and Schools 30 Prefers 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. Prefers Bread 30

Bread and Schools • Let’s add a few more “identical” people. 30 s 1 • We have five possible levels of “schools” Schools s 2 s 3 s 4 s 5 Bread 30

Bread and Schools • We have five different levels of taxes. Schools • Alternatively, individuals 1 -5 are willing to give up 30 different amounts of s 1 bread to get school s 2 resources. s 3 s 4 s 5 Bread 30

How do we decide? 2 3 4 Schools • Consider a politician. He has to win an election, 30 and he has to get enough s 1 votes by promising the s 2 right amount of school resources s 3 • Suppose he promises s 5. s 4 Person 5 is happy (he didn’t want much). But s 5 everyone else wanted more. So politician loses election 4 -1. 1 5 Bread 30

How do we decide? 1 2 3 4 Schools • Suppose he promises s 4. Persons 1, 2, and 3 are 30 happier because they’re getting closer to what they s 1 want. But he’ll still lose 3 s 2 2. s 3 • Suppose he now promises s 3. He’ll win the election s 4 because Persons 1 and 2 s 5 are happier yet, and Person 3 is happiest, he’s getting exactly what he wants. 5 Bread 30

If you don’t believe me. . . 1 30 s 1 2 3 s 2 Schools • Suppose another politician promises s 2. 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 s 3. Each voter will pay b 3 in taxes and get s 3. 4 s 3 5 s 4 s 5 b 3 Bread 30

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

Is it optimal? MRSi 1 30 s 1 2 MRS, MRT 3 Schools s 2 4 MRT s 3 5 s 4 Possible Median MRS s 5 Mean MRT b 3 Bread 30 G* Public Good G

It may NOT be MRSi 1 30 s 1 2 MRS, MRT 3 Schools s 2 4 MRT s 3 5 s 4 Possible Median MRS s 5 Mean MRT b 3 Bread 30 G* Public Good G

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 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.