Fun with Rent Functions!

We derived a rent gradient Remember, slope was related to mgl transport cost. Let’s assume that we have an open city. What does that mean? A> People can migrate from elsewhere. Utility can’t increase. Rent Distance Ag.Rent

We derived a rent gradient Suppose, everywhere that transportation costs decrease. Open city! What happens at u = 0. A> Nothing What happens elsewhere? Rent Distance Ag.Rent Why?

We derived a rent gradient Suppose, everywhere that transportation costs decrease. Closed city! What happens at u = 0. A> Rent falls Why? What happens elsewhere? Rent Distance Ag.Rent Why?

Let’s get more analytical (Brueckner Handbook) Two Eq’m conditions (18) K = distance W = income t = mgl. Transp. Cost u = utility N = population (19)

Land price rises (18) k = distance W = income t = mgl. Transp. Cost u = utility N = population If R A , since W, t, and u are fixed, the only thing that can change is If land price rises, then people are worse off, so they move. City contracts until enough move out. Utility is constant.

Income rises (18) k = distance W = income t = mgl. Transp. Cost u = utility N = population What if W  ? u can’t increase, so price of land (housing) must rise, as people move in to take advantage of increased W. At there is excess demand so the city must expand. Higher rents  smaller housing  higher density.

Mgl. Transport cost rises (18) k = distance W = income t = mgl. Transp. Cost u = utility N = population What if t  ? u can’t decrease, so price of land (housing) must fall, as people move out. At there is insufficient demand so the city must contract. Lower rents  bigger housing  lower density.

Closed city (18) k = distance W = income t = mgl. Transp. Cost u = utility N = population Assume that N is constant. If R A , with N constant, If land price rises, less land for same number of people. Rents rise, housing prices rise.

Closed city (2) k = distance W = income t = mgl. Transp. Cost u = utility N = population What about a change in income W? Let p = price of housing -++ As W , people demand more housing, more land, and utility . Since you have the same number of people, if demand  further out, some of it must decrease further in. KEY: What happens at k = 0? Increase in t does the reverse.

Open v. closed? Costs of migrating may be high so utility differences may persist over time. BUT, migration flows ultimately must eliminate differentials. In real world, there is positive correlation between income and city population predicted by OC model. Within a city you almost always want to do “open” analysis. Suppose you build a small park. Who will benefit? Why?

Land andLabor Mkt. Eq’a

Revisiting Model We had business more centrally located. Then residential. At edge of the city, we get farmland. Distance Land Rent Business Residential Agric. City limits

What is Zoning? Zoning involves a set of restrictions on what people can do with their land. Generally imposes the restriction with some sorts of public good in mind. Discuss

Revisiting Model Suppose we forbid land development past a certain distance. What will the impacts be? Immediate impact? City ends at boundary! Distance Land Rent Business Residential Agric. City limits Service boundary

More general effects... Limiting size of city reduces labor supply –Wages rise, but, –This induces immigration from outside. Since we have a smaller city, rents and density MUST RISE Business sector bids less for land, because nonland costs have risen... Residential sector bids land away from the business sector. So we will see...

Revisiting Model Ultimate impact depends on whether change is “small” or “large.” If it is “small” residents can’t be better off, because others would migrate in. Distance Land Rent Business Residential Agric. City limits Service boundary

Ultimate winners and losers In an open city, residents neither win nor lose. Migration keeps their utility constant. Landlords outside the service boundary lose. Residential landlords win. Business landlords lose. Zoning is about land.

Labor Market We’ve talked about the land market. If people come in, what is likely to happen in the labor market? Wages will fall. Rents will rise and wages will fall. DLDL SLSL L0L0 w0w0 wage Labor S'LS'L L1L1 w1w1

Eq’m in Land and Labor Two sectors – Business, Consumers Business  =  (w, R)U = U (w, R) --+-

U > u*  < 0 U < u*  > 0 Equilibrium in the Land and Labor Markets Among urban areas, what must happen for business profits to be constant? Wage, w Rent, R Why?  = 0 Among urban areas, what must happen for consumer utility to be constant? Why? U = u* Eq’m where 2 curves cross! Why? ReRe wewe Increased R off- sets decreased w Increased R off- sets increased w U < u*  < 0 U > u*  > 0

U > u*  < 0 U < u*  > 0 Equilibrium in the Land and Labor Markets Suppose profits rise? From previous eq’m,  is now greater than 0. Wage, w Rent, R  = 0 In new eq’m, w' e > w e ; R' e > R e U = u* ReRe wewe Increased R off- sets decreased w Increased R off- sets increased w U < u*  < 0 U > u*  > 0 Other firms come in, demand land, bid up wages.  >0