ECN741: Urban Economics
The Basic Urban
Model: Solutions
The Basic Urban Model
Motivation for Urban Models
 Urban models are built on the following simple sentence:
 People care about where they live because they must
commute to work.
 This sentence contains elements of 6 markets:






Housing
Land
Capital
Transportation
Labor
Export good
The Basic Urban Model
Motivation for Urban Models, 2
 So now we are going to write down equations for these 6
markets.
 It is difficult to solve a general equilibrium model with 6
markets.
 That is why we rely on the strong assumptions discussed in
previous classes.
 Moreover, the best way to understand a complex system
is to write down a simple version and then try to make it
more general.
 That is what we will do later in this class.
The Basic Urban Model
Housing Demand
 A household maximizes
U  1    ln Z    ln  H  .
 Subject to
Y  Z  P u H  tu.
 where
t  t 0  t yY
The Basic Urban Model
Housing Demand, 2
 Recall from the last class that the Lagrangian for this
problem is:
 U Z , H    Y  Z  P u H  tu  ,
 And the first-order conditions for Z and H imply that
U / H
 MBH  P u .
U / Z
The Basic Urban Model
Housing Demand, 2
 With a Cobb-Douglas utility function,
U


H
H
and
so
U
1

Z
Z
 1 
Z 
 P{u}H .
  
The Basic Urban Model
Housing Demand, 2
 Now add the first-order condition with respect to λ:

 Y   Z  P u H  tu   0.

 Combining results:
  1 

Y  
 P{u}H  P u H  tu   0.
  

The Basic Urban Model
Housing Demand, 3
 These conditions imply that
H
 Y  tu 
P u
.
Z  1   Y  tu  .
The Basic Urban Model
Deriving a Bid Function
 A bid function, P{u}, can be derived in two different
ways:
 The indirect utility function approach, pioneered by
Robert Solow
 The differential equation approach, in Alonso, Muth,
Mills.
 The best approach depends on the context!
The Basic Urban Model
The Indirect Utility Function Approach
 Substitute the demands for H and Z into the
exponential form for the utility function:
U =
 where
k  Y - tu 

P u
k  1   
1


The Basic Urban Model
Indirect Utility Function Approach, 2
 All household receive the same utility level, U*, so
1
or
 k 
P u   * 
U 
Y  tu 
1
P u   Y  tu 
1
 The height of the bid function, γ, obviously depends on
the utility level, U*.
The Basic Urban Model
The Locational Equilibrium Condition
 Remember from last class: The price of housing
adjusts so that, no matter where someone lives, savings
in housing costs from moving one mile further out
exactly offsets the increased commuting costs.

The savings in housing costs is:
 P u H
 The increase in commuting costs is just t.
The Basic Urban Model
The Differential Equation Approach
 Thus, the locational equilibrium condition is:
t
P u 
H
 Now substitute in the demand for housing to obtain the
differential equation:
tP u
P u
t
P u 
or

.
 Y  tu 
P u  Y  tu 
The Basic Urban Model
Differential Equation Approach, 2
 This is an exact differential equation. It has the
function, P{u} on one side and the argument, u, on the
other.
 It can be solved simply by integrating both sides.
 The key integral is:

g  u
du  1n  g u  
g u
The Basic Urban Model
Differential Equation Approach, 3
 The result:
1
1n P  u    1n Y  t u   .
 
or
P u  e

Y  tu 
1
  Y  tu 
1
The Basic Urban Model
Housing Supply
 The housing production function is assumed to take the
Cobb-Douglas form:
H s u  AK u
1 a
L u
a
where the “S” subscript indicates aggregate supply at
location u, K is capital and L is land.
 Because this is a long-run model, the role of labor in
housing production is ignored.
The Basic Urban Model
Input Demand
 Profit-maximizing forms set the value of the marginal
product of each input equal to its price:
1  a  P u H s u  r  constant
K u
aP u H s u
L u
 R u
The Basic Urban Model
Note on Land Prices
 Note that the price of land is a derived land.
 In residential use, the price of land is determined by
the price of housing.
 Land at a given location has value because someone is
willing to pay for housing there.
 It is not correct to say that someone has to pay a lot for
housing because the price of land is high!
The Basic Urban Model
Solving for R{u}
 Now solve the input market conditions for K{u} and
L{u} and plug the results into the production function:
1 a
 1  a  P u H s u 
H s u  A 

r


1 a
 1 a 
 Aa 

r


a
 aP u H s u 


R u


 P u H s u 

.
a
 R u



a
The Basic Urban Model
Solving for R{u}, 2
 Now HS{u} obviously cancels and we can solve for:
R u  CP u
a
or
where
R u   CP u
1a
1 a
 1 a 
C  Aa 

 r 
a
The Basic Urban Model
Solving for R{u}, 3
 Combining this result with the earlier result for P{u}:
R u  C

1a
*
 Y  tu  
1 1 a
Y  tu 
1 a
 This function obviously has the same shape as P{u},
but with more curvature.
The Basic Urban Model
Anchoring R{u}
 Recall that we have derived families of bid functions,
P{u} and R{u}.
 The easiest way to “anchor” them, that is, to pick a member
of the family, is by introducing the agricultural rental rate, R ,
and the outer edge of the urban area, u :

R u  R.
The Basic Urban Model
Determining the Outer Edge of the Urban Area
R(u)
_
R
CBD
u*
u
The Basic Urban Model
Anchoring R{u}, 2
 This “outer-edge” condition can be substituted into the
above expression for R{u} to obtain:
 
R
*
C
1a
Y  tu 
1 a
 With this constant, we find that
1 a
 Y  tu 
R u  R 

 Y  tu 
The Basic Urban Model
Anchoring P{u}
 Now using the relationship between R{u} and P{u},
1
 Y  tu 
P u  P 

 Y  tu 
where the “opportunity cost of housing” is
a
R
P
C
The Basic Urban Model
A Complete Urban Model
 So now we can pull equations together for the 6
markets






Housing
Land
Capital
Transportation
Labor
Export Good
The Basic Urban Model
Housing
 Demand
H
 Supply
 D=S
 Y  tu 
P u
H s u  AK u
1 a
L u
a
N{u}H {u}  H S {u}
where N{u} is the number of households living at location u.
The Basic Urban Model
Land
 Demand
aP u H s u
L u
 R u
 Supply
L u  u .
 [Ownership: Rents go to absentee landlords.]
The Basic Urban Model
The Capital Market
 Demand
1  a  P u H s u  r
K u
 Supply:
r is constant
The Basic Urban Model
The Transportation Market
 T{u} = tu
 Commuting cost per mile, t, does not depend on
▫
▫
▫
▫
Direction
Mode
Road Capacity
Number of Commuters
 Results in circular iso-cost lines—and a circular city.
The Basic Urban Model
Labor and Goods Markets
 All jobs are in the CBD (with no unemployment)
 Wage and hours worked are constant, producing
income Y.
 This is consistent with perfectly elastic demand for
workers—derived from export-good production.
 Each household has one worker.
The Basic Urban Model
Labor and Goods Markets, 2
 N{u} is the number of households living a location u.
 The total number of jobs is N.
 So
_
u
 N u du  N .
0
The Basic Urban Model
Locational Equilibrium
 The bid function
1
 k 
P u   * 
U 
Y  tu 
 The anchoring condition

R u  R.
1
The Basic Urban Model
The Complete Model
 The complete model contains 10 unknowns:
 H{u}, HS{u}, L{u}, K{u}, N{u}, P{u}, R{u}, N, u, and U*
 It also contains 9 equations:
 (1) Housing demand, (2) housing supply, (3) housing S=D,
(4) capital demand, (5) land demand, (6) land supply, (7)
labor adding-up condition, (8) bid function, (9) anchoring
condition.
The Basic Urban Model
The Complete Model, 2
 Note that 7 of the 10 variables in the model are actually
functions of u.
 An urban model is designed to determine the
residential spatial structure of an urban area, so the
solutions vary over space.
 In the basic model there is, of course, only one spatial
dimension, u, but we will later consider more complex
models.
The Basic Urban Model
Open and Closed Models
 It is not generally possible to solve a model with 9
equations and 10 unknowns.
 So urban economists have two choices:
 Open Models:
▫ Assume U* is fixed and solve for N.
 Closed Models:
▫ Assume N is fixed and solve for U*.
The Basic Urban Model
Open and Closed Models, 2
 Open models implicitly assume that an urban area is in a
system of area and that people are mobile across areas.
 Household mobility ensures that U* is constant in the system of
areas (just as within-area mobility holds U* fixed within an
area).
 Closed models implicitly assume either
 (1) that population is fixed and across-area mobility is
impossible,
 or (2) that any changes being analyzed affect all urban areas
equally, so that nobody is given an incentive to change areas.
The Basic Urban Model
Solving a Closed Model
 The trick to solving the model is to go through N{u}.
 Start with the housing S=D and plug in expressions for
H{u} and HS{u}.
 For H{u}, use the demand function, but put in P{u}=R{u}a/C.
 For HS{u}, plug K{u} (from its demand function) and the above
expression for P{u} into the housing production function.
The Basic Urban Model
Solving a Closed Model, 2
 These steps lead to:
H s u  DR u
1 a
 where
1 a
 1 a 
D  A

 ar 
L u
The Basic Urban Model
Solving a Closed Model, 3
 Now plug in the supply function for L{u} and the
“anchored” form for R{u} into the above. Then the ratio of
HS{u} to H{u} is:
N u 
R  u Y  tu 

a Y  tu

1 a  1
1 a
The Basic Urban Model
Solving a Closed Model, 4
 Substituting this expression for N{u} into the “adding
up” condition gives us the integral:
u

0
R  u Y  tu 
a Y  tu 
1 a  1
1 a
du  N
 Note: I put a bar on the N to indicate that it is fixed.
The Basic Urban Model
The Integral
 Here’s the integral we need:
 u c  c u 
1
2
n
du 
1
 c2   n  1 n  2 
2
 c1  c2u 
u
n 1

 c1  c2u 
c2  n  1
where c1 = Y, c2 = -t, and n = [(1/aα)-1].
n2
The Basic Urban Model
The Integral, 2
 Thus the answer is
u
R b
 N u du  Y  tu 
1 a
0
  Y  tu 
u Y  tu 


2
tb
  t  b  b  1
b 1
b
u

 ,
 0
where b = 1/aα and the right side must be evaluated at 0
and u.
The Basic Urban Model
The Integral, 3
 Evaluating this expression and setting it equal to N
yields:

 R  
Y b 1
Y  tu
N 

 u
 
b

t  b  1
 t   t  b  1Y  tu 

 A key problem:
 This equation is so nonlinear that one cannot solve for u
(the variable) as a function of N(the parameter).
The Basic Urban Model
The Problem with Closed Models
 One feature of closed models is convenient:
 The utility level is not needed to find anything else.
 But another feature makes life quite difficult:
 As just noted, the population integral cannot be explicitly
solved for u .
 This fact (and even more complexity in fancier models)
leads many urban economists to use simulation methods.
The Basic Urban Model
Solving an Open Model
 The equations of open and closed models are all the
same.
 However, one equation plays a much bigger role in an
open model, namely, the key locational equilibrium
condition, because U* is now a parameter (hence the
“bar”), not a variable.
k Y  tu 
k Y  tu 
*
U 


a 
R 
P


C 


The Basic Urban Model
Solving an Open Model, 2
 This equation can be solved for u as a function of
parameters of the model.

P U
Y
k
u 
t
*
a
R U
Y 
C k

t
*
 This makes life a lot easier! This expression can be
plugged into the solution to the integral to get N, which
is now a variable.
The Basic Urban Model
The Problem with Open Models
 Open models are much easier to solve than are closed
models.
 The problem is that they address a much narrower
question, namely what happens when there is an event
in one urban area but not in any other.
 Be careful to pick the model that answers the question you
want to answer—not the model that is easier to solve!!
The Basic Urban Model
Density Functions
 A key urban variable is population density, which can be
written D{u} = N{u}/ L{u}.
 Our earlier results therefore imply that:
R u
D u 
a Y  tu 
 This function has almost the same shape as R{u} and, as we will
see, has been estimated by many studies.
The Basic Urban Model
Building Height
 The model also predicts a skyline, as measured by
building height—a prediction upheld by observation!
 One measure of building height is the capital/land
ratio, or K{u}/L{u}, which can be shown to be
K u
L u
where
 qR u
 A 1  a  
q

 Cr 
1a