ppa786: urban policy

PPA786: Urban PolicyClass 3:Housing Concepts, Household Bids

Urban Policy: Housing Concepts Household Bids

•Outline of Class

▫Land concepts

▫Housing concepts

▫Housing bids and locational equilibrium


•Land Concepts

▫Land rent is the price for using one unit of land, say an acre, for one unit of time, say a year.

▫Land value is the price of buying one unit of land, again say an acre.


•Land Concepts, 2

31 2 42 3 5

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LL L LL

Lt Lt

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RR R RVi i i i

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•The Determination of Land Rent

▫Land is an input; the price of land (= annual rent) is a derived demand—derived from its role in producing an output, say Q.

▫In equilibrium, the price of an input equals the value of its marginal product:

( )( )L q L LVMP P MP R


•Land Rent, Continued

▫Now suppose that Q must be shipped to a market The distance to the market, designated u,

varies across firms. It costs $s to ship a unit of Q one mile. The marginal product of land equals a.

▫Then land rent is determined by:

{ }q LP su a R u


Land Rent and Distance from the Market

without with substitution substitution

< Figure 1 >

R(u) A R(u) B

Market u Market u

Urban Policy: Housing Concepts Household Bids• Housing Concepts

• Housing is measured in units of housing services = H

▫H= quality-adjusted square feet.▫Depends on housing characteristics (X1, X2, …)

•P = the price per unit of H per year.

•R = rent for a housing unit = PH.

▫If the unit is an apartment, R = contract rent.▫If the unit is owner-occupied, R is not observed.


•Housing Concepts, Continued

•V = the value of a housing unit = the present value of the rental flow (not observed for renters).

•So:1 2

1 1

{ } { } { , ,..., } { } { }

{ } { } { } { } { }{ }(1 ) (1 )

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t tt t

R u P u H X X X P u H X

R u P u H X P u H XV ui i i


•How Does a CBD Worker Decide Where To Live?

▫She compares the marginal benefit (MB) and the marginal cost (MC) of moving one mile farther from the CBD.

( { }) lower housing cost

( ) increased commuting cost

MB P u H

MC t u


•How Does a CBD Worker Decide Where To Live? (Continued)

▫She then keeps moving out until she comes to the location (u*) at which MB equals MC:

( { }) ( )P u H t u


Tradeoff Between Housing and Commuting Costs

< Figure 2 > $ MC = tu MB = -P(u)H CBD u* u


•The Twist: How Housing Prices Are Determined

▫Now suppose that all households are alike (an assumption to be relaxed!). Then they all pick the same u*!

▫This is impossible, so P{u} adjusts until people are equally satisfied no matter where they live.

This is called locational equilibrium.


•The Twist: How Housing Prices Are Determined (Continued)

▫Thus, P{u} adjusts until, at all locations,

▫that is, until the slope of the P{u} function equals –t/H.

{ }P u tu H

Urban Policy: Housing Concepts Household Bids• The Twist: How Housing Prices Are Determined

(Continued)

▫Because the slope is negative, P{u} is higher closer to the CBD than it is in the suburbs.

▫When P{u} is high, people substitute away from housing so that H is low.

▫When H is low, the slope of P{u}, namely, -t/H, is high in absolute value.

▫It follows that P{u} is steep near the city center but flattens as one moves out toward the suburbs.

Urban Policy: Housing Concepts Household BidsThe Bid Function for Housing

(Price per Unit of Housing Services)

Slope = ΔP/Δu

= -t/H

< Figure 3 >

P(u)

CBD u

ΔPΔu


Finding the Edge of the City

▫Urban activities must compete with rural activities for access to land.

▫Suppose P* is the opportunity cost of pulling land out of agriculture and into housing.

▫Then urban activities will take place out to the point, say, u*, at which the price of housing exceeds P*.


Determining the Outer Edge of the Urban Area< Figure 3A >

P(u)

P*

CBD u* u


Policy Questions and Bid Functions

▫Some policies affect a single urban area. If they make the area more attractive, people

move in; otherwise, people move out to other areas.

These policies are analyzed with an “open” model.

▫Other policies affect all urban areas. These policies do not give anyone an incentive to

move out of an area. These policies are analyzed with a “closed”

model.


The Height of the Bid Functionand the Size of the Area

▫To understand the distinction between open and closed models, recall that we derived a formula for the slope of P{u}, not for its height.

▫As the height of P{u}, goes up,

The level of satisfaction in an urban area goes down,

And the population goes up.

Urban Policy: Housing Concepts Household BidsThe Height of the Bid Functionand the Size of the Urban Area

< Figure 3B >

P(u)

CBD u


Open versus Closed Models

▫In an open model, one selects the height of P{u} that yields the same level of satisfaction as a household can obtain in another urban area.

At any other height, people would move in or out.

▫In a closed model, one selects the height of P{u} that makes the area large enough to fit all its population.


Open versus Closed Examples

▫Suppose one city in a regions cleans up its air and no other city does.

The impacts are given by an open model. People move in and housing prices go up until the higher

cost of living offsets the utility gain from cleaner air!

▫Suppose all cities in the region clean their air.

The impacts are given by a closed model. Nobody has an incentive to move out and utility goes up

due to cleaner air.

ppa786: urban policy

Documents

housing unit

housing prices

household bids urban

urban policy ppa786

unit of land

units of housing services

housing characteristics

price of land