ECN741: Urban Economics
More General
Treatment of Housing
Demand
Housing Demand Theory
Class Outline
 Alternative Utility Functions
 An Exponential Density Function?
 The Envelope Theorem
 Maximizing a Bid Function
 Introduction to Comparative Statics with General
Functional Forms
Housing Demand Theory
Stone-Geary Utility Function
 One simple extension to the Cobb-Douglas utility
function is to a Stone-Geary utility function:
U   Z  SZ

1 
H
 SH


where SZ and SH are “survival quantities.”
 In this case, the demand functions reflect the
“required” spending on Z and H:
H  SH
 Y  tu  S z  P {u } S H 
 

P {u }


Z  S Z  (1   )  Y  tu  S z  P {u } S H

Housing Demand Theory
Stone-Geary Utility Function, 2
 Thus, the indirect utility function is:
 Y  tu  S z  P {u } S H 
U* k


P
{
u
}


 What a disappointment! This expression cannot be
solved for P{u}!
 So this utility function does not help.
Housing Demand Theory
CES Utility Function
 Another well-known utility function is the “constant
elasticity of substitution” or CES function, which is
U   (1   ) Z   H



1/ 
 Because demand functions are unaffected by a
monotonic transformation of utility, one can save a lot
of algebraic mess by simplifying this to



U  (1   ) Z   H
Housing Demand Theory
CES Utility Function, 2
 The demand functions for this utility function can be
derived and so can the indirect utility function.
 This indirect utility function can be solved for P{u}.
 I leave this derivation as an exercise.
 A CES utility function would be a good choice for a
simulation model or urban spatial structure, since it
includes the Cobb-Douglas utility function as a special
case.
Housing Demand Theory
An Exponential Density Function?
 Some scholars (Mills, Muth) have argued that an
exponential density function can be derived from an
urban model.
 This was a pretty amazing claim because the
exponential density function had been used in many
empirical studies—and it worked very well!
 Let’s see what assumptions are needed for this to be
true.
Housing Demand Theory
An Exponential Density Function, 2?
 Start with the standard locational equilibrium
condition, and assume that the demand for housing
takes a constant elasticity form.
 A constant elasticity form, by the way, can be derived
from an “incomplete demand system,” which has a
composite commodity essentially as a residual.
 See LaFrance, American Journal of Agricultural
Economics, August 1986.
Housing Demand Theory
An Exponential Density Function, 3?
 So we have
P {u }  
t
H

H  B (Y  tu ) P {u }

 Putting these together leads to the differential equation

P {u } P {u }  
t
B (Y  tu )

Housing Demand Theory
An Exponential Density Function, 4?
 The general solution to this differential equation is:
1 
P {u }
1 
Q
(Y  tu )
1
B (1   )
where Q is a constant of integration.
 This obviously yields a pretty messy nonlinear form for
P{u}, and hence for R{u} and D{u}.
 It does not lead to an exponential form!
Housing Demand Theory
An Exponential Density Function, 5?
 We can simplify this result by assuming that the price
elasticity of demand for H, μ, is -1. This leads to
P {u }
P {u }

t
B (Y  tu )

 The solution to this differential equation is
ln{ P {u }}  Q 
(Y  tu )
1
A (1   )
 This won’t lead to an exponential density, either!
Housing Demand Theory
An Exponential Density Function, 6?
 Now let’s change the demand function to:

H  BY P {u }

 This changes the differential equation to
P {u }

P {u }
t
BY

 and the solution to
 t 
ln{ P {u }}  Q   
u
 
 BY 
or
Q
P {u }  e e

 (t / BY )u
Housing Demand Theory
An Exponential Density Function, 6?
 Now remember from a basic urban model that
R {u }
1/ a
R {u }   C P {u } 
and D {u } 
a (Y  tu )
 In addition, the net income expression in the
denominator of D{u} comes from the housing demand
equation. With the same simplification (=no tu),
D {u } 
 C P {u } 
1/ a

e
Q / a
a Y
 An exponential form at last!

e
 ( t / aAY ) u
a Y
 e 0e
c
c1u
Housing Demand Theory
An Exponential Density Function, 6?
 In short, to obtain an exponential density function, one
has to assume that the demand for housing depends on
Y, not on (Y - tu), and that the demand elasticity is -1.
 But if tu is not in the budget constraint, bid functions
do not arise in the first place!
 Consistency with bid functions requires the assumption
that θ = 0 (Kim and McDonald, Journal of Regional
Science, 1987). The empirical evidence does not
support this assumption.
Housing Demand Theory
The Envelope Theorem
 The Envelope Theorem is an important tool in
economics and several scholars have employed it to
good effect in the case of urban models.
 We will go over what the Envelope Theorem is,
including a proof, and then show how it can be used to
simplify urban model comparative statics.
 This is not the last class in which the Envelope
Theorem will appear!
Housing Demand Theory
The Envelope Theorem, 2
 Many of you may be familiar with the Envelope
Theorem, but in case you are not, I’m going to prove it
for you.
 The proof also serves as an explanation.
 Then I am going to restate the bidding problem in an
urban model and show you how you can do some
comparative statics derivations by applying the
Envelope Theorem to this re-formulation.
Housing Demand Theory
The Envelope Theorem, 3
 Let’s start with a standard maximization problem,
which can be written as
M axim ize
y  f { x1 , ..., x n ,  }
Subject to
g { x1 , ..., x n ,  }  0
 The Lagrangian expression for this problem is
£  f  g
Housing Demand Theory
The Envelope Theorem, 4
 Differentiating this Lagrangian leads to the following
first-order conditions
£ i  fi   g i  0
£  g  0
which lead, in turn, to the following solutions:
x  x { }
*
i
*
i
   { }
*
*
Housing Demand Theory
The Envelope Theorem
 By substituting the solutions back into the expression
for y, which is what we are trying to maximize, we find
can write the solution to the problem as
y  f { x , ..., x ,  }   { }
*
*
1
*
n
 Note that ϕ{α} is sometimes called the indirect
objective function; it is the maximum value of y for
given α using x’s that meet the constraint.
Housing Demand Theory
The Envelope Theorem
 Now suppose we want to know how this maximum
value changes when we change α, which is just an
exercise in comparative statics.
 Differentiating the above expression for y* yields
y
*






i
 xi
*
fi

 f
 This looks pretty complicated. It looks like we have to
re-solve the problem with the new values of α to find
the new values of the xi*s and λ* and then substitute
those values back into y.
 The Envelope Theorem gives us a shortcut!
Housing Demand Theory
The Envelope Theorem
 To get to this shortcut, first write the constraint at the
optimal values of the xis.
g { x , ..., x ,  }  0
*
i
*
n
 This constraint must still hold when α changes,
which implies that
g


g
i
x
i
*
i

 g  0
Housing Demand Theory
The Envelope Theorem
 Now if we multiply this result by λ and add it to the
messy comparative statics result derived earlier, we
find that *
*
*
y



i



 xi
fi
 f     g i
 g 


 i

 xi
f
i
i
  gi 
 xi
*

 f  g 
 But, by the first-order conditions, the terms in parentheses in
the second line have to equal zero, so
y
*

 f   g   £ 
Housing Demand Theory
The Envelope Theorem
 This is an amazing shortcut.
 To find the comparative static derivative of y* with
respect to α, all we have to do is differentiate y and g
with respect to α.
 We do not have to worry about changes in the xi*s!!
 Of course this only applies to small changes, but, as we
will see, it is a very helpful result.
Housing Demand Theory
Maximizing Bids
 In an urban model, the household budget constraint is
Y  Z  P {u } H  tu
 One way to think about the household problem is to say
that they have to figure out the most they are willing to
bid for housing at each location, given their budget
constraint and a fixed utility level. In other words, they
M axim ize
P {u } 
Y  Z  tu
H
S ubject to
U {Z , H }  U
*
Housing Demand Theory
Maximizing Bids
 The decision variables in this problem are Z and H, and
the parameters are Y, t, U*, and u.
 We are treating this as an open model.
 So we can use the envelope theorem to find the
derivative of P{u} with respect to each of these
parameters.
Housing Demand Theory
Maximizing Bids
 Although Z and H are formally choice variables, this
formulation cannot be used to find their demand
functions; this problem is to determine P{u}, not to
determine responses to P{u}.
 Nevertheless, one first-order condition is needed to pin
down the CS derivatives.
Housing Demand Theory
Maximizing Bids
 The Lagrangian is
£
Y  Z  tu
H
  U { Z , H }  U
 The relevant first-order condition is:
£
1

 U Z  0
Z
H
 and the result we need is:
 
1
UZH
*

Housing Demand Theory
Maximizing Bids
 Now we can use the Envelope Theorem to find the
impact of Y, t, U*, and u on P{u} using
y
*

 f   g   £ 
 The CS results are:
 P {u }
Y

1
H
0
 P {u }
t

u
H
0
 P {u }
U
*
   
1
UZH
0
Housing Demand Theory
Maximizing Bids
 In this setting, it is appropriate to treat u as a
parameter, and we can ask how bids changes when a
household moves to a more distant location:
 The envelope theorem reveals that:
 P {u }
u
 P {u }  
t
0
H
which is, of course, the result we have derived before.
Housing Demand Theory
General Comparative Statics
 One final point to make about this exercise is that these
CS results apply to any utility function.
 Hence they are part of a general CS analysis, such as
the one in Brueckner’s Handbook chapter.
 In fact, the CS results with general functional forms are
the same (i.e. have the same signs) as the results in the
table presented in the last class.