Download PPA786: Urban Policy

PAI786: 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
• Some Perspective
▫ We are going to focus on concepts today (and in
the next few classes)—not on policy.
▫ But these concepts help us to think more clearly
about urban housing markets,
▫ And to think more clearly about urban policy
when we focus on it a few classes from now.
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.
▫ Land is an asset; like any asset its price
(=value) is the present value of the benefits
(=net rent) from owning it.
Urban Policy: Housing Concepts Household Bids
• Land Concepts, 2
Value of a
Land
Parcel
Net Rent of
Land in Year 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 )
Discount
Rate
Assumes RL is constant over time
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, which sells for PQ.
▫ In equilibrium, the price of an input equals the
value of its marginal product, so:
VMPL  ( PQ )( MPL )  RL
Urban Policy: Housing Concepts Household Bids
• Land Rent, 2
▫ One important lesson:
 The price of land is high in some locations because
people are willing to pay a lot for housing or
commercial activities at that location
 It is not correct to say that the price of
housing is high because land is expensive.
Urban Policy: Housing Concepts Household Bids
• Land Rent, 3
▫ Now consider this highly simplified example:
 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
Q
 su  a  RL {u}
Net Price of
Product
Marginal Product
of Land
Urban Policy: Housing Concepts Household Bids
• Land Rent, 4
▫ With a constant a, land rent is a linear function of
distance, u.
▫ But when the price of land goes down, firms are
likely to substitute land for capital.
 This increase in the amount of land at greater
distances from the market leads to a lower MPL at
greater distances
 And hence to a land-rent function with a slope that
gets flatter as distance from the market increases.
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, 2
• The price, P, of housing, H, depends on many
locational traits.
• Let’s start with distance to a worksite, u. More
later!
• People will pay more per unit of H if they have
good access to a worksite than if they do not.
• So P = P{u}.
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, with a long lifetime, T, for housing:
R{u}  P{u}H { X 1 , X 2 ,..., X n }  P{u}H { X }
T
R{u} T 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.
Drop in Price per QualityAdjusted Square Foot
MB  (P{u}) H  lower housing cost
MC  t (u )  increased commuting cost
Increase in
Commuting Distance
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:
Lower Housing Cost
Higher Commuting Cost
(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
Note: This MB curve is just hypothetical; we do not (yet) know its shape.
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, 2
▫ In other words, P{u} adjusts until MB=MC at all
locations, or
P{u} t

u
H
▫ that is, until the slope of the P{u} function,
ΔP{u}/Δu, equals –t/H, the transportation
cost/housing trade-off.
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
• The Twist: How Housing Prices Are
Determined, 3
▫ 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 Twist: How Housing Prices Are
Determined, 4
▫ This relationship between housing prices and access to
jobs is not just hypothetical.
▫ Many studies have found that, controlling for other
things, the price per quality-adjusted square foot is
lower at greater distances from a worksite.
▫ The following figure gives an example from a study of
mine on the Cleveland area; the price per unit of H is
35% higher next to the center than at the outer edge.
Urban Policy: Housing Concepts Household Bids
Housing Prices and Distance from Downtown, Cleveland, 2000
Log of Price
per QualityAdjusted
Square Foot
0
5
10
15
20
25
30
Distance from Downtown Worksite (miles)
35
40
45
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
resources 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)
Lower
Utility
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
▫ The height of the bid function is like the cost of
living.
▫ If the height is higher, housing costs more and
income does not go as far.
▫ If the heights of the bid functions are different in
two cities in an open model, incomes must be
higher (or amenities better) in the location with
the higher prices—or else nobody would live there!
Urban Policy: Housing Concepts Household Bids
Open versus Closed Examples, 2
▫ Suppose one city in a region 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—but of course everyone is better off.