PPA786: Urban Policy
Class 3:
Housing Concepts,
Household Bids
Urban Policy: Housing Concepts Household Bids
• Outline of Class
▫ Land concepts
▫ Housing concepts
▫ Housing bids and locational equilibrium
Urban Policy: Housing Concepts Household Bids
• 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.
Urban Policy: Housing Concepts Household Bids
• Land Concepts, 2
RL 3
RL1
RL 2
RL 4
VL 



 ...
2
3
5
(1  i ) (1  i ) (1  i ) (1  i)

RLt
RL


t
i
t 1 (1  i )
Urban Policy: Housing Concepts Household Bids
• 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:
VMPL  ( Pq )( MPL )  RL
Urban Policy: Housing Concepts Household Bids
• 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:
 P  su  a  R {u}
q
L
Urban Policy: Housing Concepts Household Bids
Land Rent and Distance from the Market
< Figure 1 >
R(u)
A
Market
R(u)
u
without
substitution
B
Market
u
with
substitution
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.
Urban Policy: Housing Concepts Household Bids
• Housing Concepts, Continued
• V = the value of a housing unit = the present
value of the rental flow (not observed for renters).
• So:
R{u}  P{u}H { X 1 , X 2 ,..., X n }  P{u}H { X }

R{u}  P{u}H { X } P{u}H { X }
V {u}  


t
t
(1  i )
i
t 1 (1  i )
t 1
Urban Policy: Housing Concepts Household Bids
• 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.
MB  (P{u}) H  lower housing cost
MC  t (u )  increased commuting cost
Urban Policy: Housing Concepts Household Bids
• 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 )
Urban Policy: Housing Concepts Household Bids
Tradeoff Between Housing and Commuting Costs
< Figure 2 >
$
MC = tu
MB = -P(u)H
CBD
u*
u
Urban Policy: Housing Concepts Household Bids
• 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.
Urban Policy: Housing Concepts Household Bids
• The Twist: How Housing Prices Are
Determined (Continued)
▫ Thus, P{u} adjusts until, at all locations,
P{u} t

u
H
▫ that is, until the slope of the P{u} function
equals –t/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 Bids
The Bid Function for Housing
(Price per Unit of Housing Services)
< Figure 3 >
P(u)
Slope = ΔP/Δu
= -t/H
ΔP
Δu
CBD
u
Urban Policy: Housing Concepts Household Bids
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*.
Urban Policy: Housing Concepts Household Bids
Determining the Outer Edge of the Urban Area
< Figure 3A >
P(u)
P*
CBD
u*
u
Urban Policy: Housing Concepts Household Bids
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.
Urban Policy: Housing Concepts Household Bids
The Height of the Bid Function
and 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 Bids
The Height of the Bid Function
and the Size of the Urban Area
< Figure 3B >
P(u)
CBD
u
Urban Policy: Housing Concepts Household Bids
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.
Urban Policy: Housing Concepts Household Bids
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.