### + + U

```ECN741: Urban Economics
The Basic Urban
Model: Comparative
Statics
The Basic Urban Model
Class Outline
 The point of comparative statics
 Open model comparative statics results
 Closed model comparative statics results
 Comparative statics tables
 Comparative statics graphs
The Basic Urban Model
Why Comparative Statics?
 Urban models describe urban residential structure
when (simplified!) models of 6 markets are combined.
 A key set of questions involves the way urban
residential structure changes when one of the
parameters of the model changes.
 CS involves finding the derivative of the variables in the
model with respect to key parameters—accounting for
interactions across markets.
The Basic Urban Model
Why Comparative Statics, 2?
 Key parameters include Y, t, R , U* (open), and N
(closed).
 With the basic model we can ask what happens to
urban residential structure when
incomes rise over time;
transportation innovation or investment takes place;
the cost of non-urban activity (e.g. agriculture) changes;
the situation in one city changes (or the opportunities in
other cities change); or
 population increases everywhere (or in one city where out
migration does not occur).




The Basic Urban Model
How to Do Comparative Statics
 Comparative statics are based on total derivatives, not
partial derivatives.
 CS results must account for all the variables in one of the
equations we have derived.
 For example, many of the equations in the model,
including equations for R{u}, P{u}, and D{u}, can be
expressed as a function of the parameters and only one
variable, namely, u .
The Basic Urban Model
How to Do Comparative Statics, 2
 In these cases, CS derivations are based on equation
like this one for R{u}:
dR u R u  R u   du 




d

 u   d  
 In this equation, δ can be any of the model’s
parameters (at least any of the ones that appear in the
R{u} equation!).
 A key to finding many CS derivatives, therefore, is to
find the impact of the relevant parameter on u .
The Basic Urban Model
Open Model Comparative Statics
 Most CS results are relatively easy to obtain with an
open model because of the form taken by the indirect
utility function:
k Y  tu 
k Y  tu 
*
U 


a 
R 
P
 
C 
 
or

P U
Y
k
u 
t
*
a
R U
Y 
C k

t
*
The Basic Urban Model
Open Model Comparative Statics, 2
 In this formulation, u is on the left side of the equation
and all the key parameters are on the right.
 So it is straightforward to derive CS results for the
impact of all these parameters on u .
 These results can then be inserted into the formula
given earlier to find CS results for R{u} and other
variables.
The Basic Urban Model
Open Model Comparative Statics, 3
 From

P U
Y
k
u 
t
*
a
R U
Y 
C k

t
*
 We can see that
du
1
 0;
dY
t
a
du
u
 0
dt
t
a 1
du
P
du
aU R

0;

*
dU
kt
k t C
dR
*
0
The Basic Urban Model
Open Model Comparative Statics, 4
 So the physical size of an urban area:
 Increases when income rises.
 Decreases when commuting costs rise.
 Decreases when opportunities improve elsewhere.
 Decreases when agriculture becomes more profitable.
The Basic Urban Model
Open Model Comparative Statics, 5
 Now take the expression for R{u}
1 a
 Y  tu 
R u  R 

 Y  tu 
 To find the CS derivative for Y, differentiate with
respect to Y, recognizing that Y affects u , and plug in
the above result for d u/dY. The result:
d R u
R u

0
dY
a Y  tu 
The Basic Urban Model
Open Model Comparative Statics, 5

 R  
Y b 1
Y  tu
N 

 u
 
b

t  b  1
 t   t  b  1Y  tu 

 Then we can differentiate with respect to Y, recognizing
that Y affects u , and plug in the above result for d u/dY.
This yields:

d N R   Yb
 2 
 1  0
b

dY
t  Y  tu 

The Basic Urban Model
Open Model Comparative Statics Table
Parameter
Variable
u
R{u}
or P{u} or D{u}
N
Y
t
R
+
-
-
-
+
-
0
-
+
-
-
-
U*
The Basic Urban Model
Closed Model Comparative Statics
 In a closed model, the derivatives of u with respect to
the parameters come from the population equation,
now with a bar over the N:

 R  
Y b 1
Y  tu
N 

 u
 
b

t  b  1
 t   t  b  1Y  tu 

 This equation is messier than the one for an open
model, but its nonlinearity does not get in the way of
CS as it does for solving the model.
The Basic Urban Model
Closed Model Comparative Statics, 2
 For example, after a little algebra one can show that:
t  b  1 N
du
1

0
b 1
2
dR
R b  Y 

 1
 Y  tu 
 Not surprisingly, increasing agricultural rents shrinks
the urban area.
 Recall: b = 1/aα
The Basic Urban Model
Closed Model Comparative Statics, 3
 To find the CS derivative dR{u} / dR , we must substitute
this result into:
d R u R u   tbR   du  

1  


d R
R   Y  tu   dR  
 The result (derive as an exercise) is a bit surprising
because R{u} decreases at small values of u and
increases at large values of u.
The Basic Urban Model
Closed Model Comparative Statics Table
Parameter
Variable
Y
t
R
N
u
+
-
-
+
R{u}
small u:- small u:+
or P{u} or D{u} large u:+ large u: -
+
+
U*
-
-
+
-
The Basic Urban Model
Comparative Statics Intuition
 We can develop an intuition for these results with some
simple graphs for R{u} (or P{u} or D{u}).
 To interpret these graphs note that
 Population depends on density and urban size (= u )
 A change in Y flattens P{u}and R{u} (remember, Pʹ{u}=
-t/H!)
 A change in t steepens R{u}.
The Basic Urban Model
CS Result for Y
Open Model
R{u}
Closed Model
R can’t drop at
u=0 because
utility is fixed
R{u}
With city size increase,
density cannot increase at
all locations
Density declines near
center
Density increases in suburbs,
which grow
R
R
u1
u2
u
u1
u2
u
The Basic Urban Model
CS Result for t
Open Model
R{u}
Closed Model
R does not change at
u=0 because tu=0.
R{u}
With city size decrease,
density cannot decrease at
all locations
Density goes up in the
center and down in the
suburbs, which shrink.
R
R
u2
u1
u
u2 u1
u
The Basic Urban Model
CS Result for R
Open Model
R{u}
Closed Model
Utility level
(indexed by
height of R{u})
cannot change
R{u}
R2
R2
R1
R1
u2
u1
u
City size and
density cannot
both move in the
same direction
Implies
higher
density
everywhere
u2 u1
u
The Basic Urban Model
Other CS Results
Open Model (U*)
R{u}
City shrinks and
becomes less dense
R
Closed Model (N)
R{u}
City grows and
becomes more dense
R
u2
u1
u
u1
u2
u
The Basic Urban Model
Informal Tests of CS Results
 These results predict that cities will get less dense in
the center, more dense in the suburbs, and larger as
incomes rise and transportation costs fall.
 Many estimates of population density functions for
cities around the world find this to be true.
 But the models have only one worksite and many other
simplifications.
 Is this just a lucky coincidence or do the models
capture something fundamental?
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