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

u

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

u 'u

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

u 'u

Why?

Let’s get more analytical (Brueckner Handbook)

Two Eq’m conditions

ARutWkR ),,,( (18)

k

NuWtkkD0

),,,(

K = distanceW = incomet = mgl. Transp. Costu = utilityN = population

(19)

Land price rises

ARutWkR ),,,( (18)

k = distanceW = incomet = mgl. Transp. Costu = utilityN = population

If RA , since W, t, and u are fixed,the only thing that can change is

k

If land price rises, then people are worseoff, so they move. City contracts until enough move out. Utility is constant.

Income rises

ARutWkR ),,,( (18)

k = distanceW = incomet = mgl. Transp. Costu = utilityN = population

What if W ?

k

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 thecity must expand.Higher rents smaller housinghigher density.

Mgl. Transport cost rises

ARutWkR ),,,( (18)

k = distanceW = incomet = mgl. Transp. Costu = utilityN = population

What if t ?

k

u can’t decrease, so price of land(housing) must fall, as people move out.

At there is insufficient demand so thecity must contract.Lower rents bigger housinglower density.

Closed city

ARutWkR ),,,( (18)

k = distanceW = incomet = mgl. Transp. Costu = utilityN = population

Assume that N is constant.

If RA , with N constant,

k

If land price rises, less land for samenumber of people. Rents rise,housing prices rise.

Closed city (2)k = distanceW = incomet = mgl. Transp. Costu = utilityN = population

What about a change in income W?

Let p = price of housing

)())((W

p

W

u

u

p

dW

dp

- + +As W , people demand more housing, more land, and utility .Since you have the same number of people, if demand furtherout, 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

Lan

d R

ent

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

Lan

d R

ent

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

Lan

d R

ent

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.

DL SL

L0

w0

wage

Labor

S'L

L1

w1

Eq’m in Land and Labor

Two sectors – Business, Consumers

Business = (w, R) U = U (w, R)

0d dw dRw R

- -

0dR wdw

R

0U U

dU dw dRw R

+ -

0

UdR w

UdwR

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

Ren

t, 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?

Re

we

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

Ren

t, R

= 0

In new eq’m, w'e > we ;

R'e > Re

U = u*

Re

we

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