Communities and Goods © Allen C. Goodman, 2000

How do people sort themselves into communities? DW ( ) provide a simple model. Let’s assume that higher income people have higher willingness to pay for public services. Leads to figure on right. Service Level G House Price P Low value Medium value High value You get a mkt. valuation of services like Rosen’s hedonic functions.

How do people sort themselves into communities? Some math. Start with an asset offering y/yr of services. P = y/r – (tP/r) r = interest rate t = property tax  P[1 + (t/r)] = y/r P = y/(r + t). Discuss. Service Level G House Price P Low value Medium value High value

How do people sort themselves into communities? In the DW example: Community j: P j = y/r + R j G j /r – tP j /r R j = Value household places on services G j = Level of services Similarly for P k. P k = y/r + R k G k /r – tP k /r P j – P k = (R j G j - R k G k ) /(r +t) Service Level G House Price P Low value Medium value High value GjGj GkGk PjPj PkPk

Zoning and Communities In most places (in US) people must pay for their services directly out of property tax. Price of house type i in community j is: P ij = y i /r + (g j – t j P ij )/r Here, we’re defining g j = R j G j from before. Discuss. When we solve, we get: P ij = (y i + g j )/(r + t). Point here is that people are using the property tax as a user fee.

Zoning and Communities Point here is that people are using the property tax as a user fee. BUT … remember, everyone pays the same property tax rate whether they have a small house of a big house. P ij = y i /r + (g j – t j P ij )/r Suppose that y i = 10,000, g j = 3,000, and t i = 0.03, r = 0.1. If all of the houses in j are the same, then: P ij = 100,000. Important point here is that the property tax serves as a user fee for services. If you want more services, you pay more taxes.

Zoning and Communities Suppose, instead that half of the houses would cost 100,000 (if all were same in community) and half would cost 200,000. The small houses (100,000) are on 1 acre lots. They have 1 unit of capital costing 80,000. The big houses (200,000) are on 2 acre lots. They have 2 units of capital, costing 160,000. Let’s assume IN THIS COMMUNITY that r = 0.10, and t = This  property tax of 3000/house = 0.5*0.02*100, *0.02*200,000.

How much are they worth? When we solve, we get: P ij = (y i + g j )/(r + t). P small = (10, ,000)/( ) = 108,333. P big = (20, ,000)/( ) = 191,666. Value of land? Small  108,333 – 80,000 = 28,333/acre. Big  (191,666 – 160,000)/2 = 15,833. What will happen? ANSWER! – Would be owners of small houses will bid land away from owners of big houses to get fiscal surplus! Leads to unstable equilibrium.

How much are they worth? We’ve seen this in some of the presentations. If you build smaller units, owners of larger units may face capital losses. People in richer town may seek “minimum lot size” or maximum density. It implies that a development will use more land than it otherwise would, and it makes it more likely that the houses developed will be of higher value. This provides greater tax revenue (per new household) and less fiscal deficit for those who are there.

Costs of services What about public services? Baumol looked at this some time back. Group activities into 2 categories 1. “Progressive” ones in which innovations, capital accumulation and scale economies   output/hr. 2. “Traditional” ones. Hard to increase productivity. Three assumptions Ignore costs other than labor Assume wages in two sectors move together Money wages  as fast as output/hr in progressive sector.

Some analytics 1 = traditional; 2 = progressive Y 1t = aL 1t Y 2t = bL 2t e rt Wages: W t = We rt Proposition 1 – Relative costs C 1 will rise w/o limit. C 2 will be constant. C 1 = W t L 1t /Y 1t = We rt L 1t /aL 1t = (W/a)e rt. C 2 = W t L 2t /Y 2t = We rt L 2t /bL 2t e rt = (W/b) C 2 /C 1 = a/be rt

More analytics Proposition 2 – If we hold budget shares constant, good 1 might vanish. Suppose that prices are proportional to costs and price elasticities of demand = 1 (as close as he gets to demand). C 1 will rise w/o limit. C 2 will be constant. C 1 Y 1 = We rt L 1t C 2 Y 2 = We rt L 2t. Dividing these  C 1 Y 1 / C 2 Y 2 = L 1t /L 2t. Set this equal to A. Y 1 /Y 2 = A (C 2 /C 1 ) = Aa/be rt  0 as t .

Two more propositions Proposition 3 – If you freeze relative outputs just about all of the labor will end up in the traditional sector. Proposition 4 – If you freeze relative outputs, the growth rate of the societal output will fall. All of these follow from the difference in productivity. Explains situation in schools, arts, pretty well. Doesn’t do real well with medical care. Interesting to think about with the advent of the Internet.

Smith’s elaboration Smith noted that Baumol didn’t look at demand very closely. He translated Baumol’s model into a general eq’m framework with a transformation function and a community utility function. YAYA YBYB

YAYA YBYB Smith’s elaboration As economy grows PPF shifts out depending on growth rates r A, and r B. If A  traditional; B  prog.

YAYA YBYB Smith’s elaboration Where:  A,  B are income elasticities  r is substitution elasticity between Y A and Y B > 0. If  A =  B = 1, and  r = 1, we get:

YAYA YBYB More likely If  A >  B = 1, and  r < 1, we get: Setting income elasticities equal will not make outputs grow at equal rates unless  r =0.