Housing Demand Theory and Utility Functions in Urban Economics
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ECN741: Urban Economics
More General
Treatment of Housing
Demand
Professor John Yinger, The Maxwell School, Syracuse University, 2020
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
5. Introduction to Comparative Statics with General
Functional Forms
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
Housing Demand Theory
Stone-Geary Utility Function
One simple extension to the Cobb-Douglas utility
function is to a Stone-Geary utility function:
where
S
Z
and
S
H
are “survival quantities.”
In this case, the demand functions reflect the
“required” spending on
Z
and
H
:
Housing Demand Theory
Stone-Geary Utility Function, 2
Thus, the indirect utility function is:
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
Demand functions are unaffected by a positive
monotonic transformation of utility, so if
δ
is positive
(which it may not be),
one can save a lot of algebraic
mess by simplifying this to
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
} with
the added assumption that
β
= 0.5
.
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, but the
β
= 0.5
assumption is pretty strong.
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
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 striking claim because the exponential
density function had been used in many empirical
studies—and it worked very well!
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
Putting these together leads to the differential equation
Housing Demand Theory
An Exponential Density Function?, 4
The general solution to this differential equation is:
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
The solution to this differential equation is
This won’t lead to an exponential density, either!
Housing Demand Theory
An Exponential Density Function?, 6
Now let’s use gross income in the demand function:
This formulation, which is used by Mills and Muth,
assumes that households do not consider commuting
costs when they decide how much housing to buy.
One interpretation is that households have separate
mental accounts for housing purchases and
transportation.
Mental accounts are discussed in Thaler (1999).
Housing Demand Theory
An Exponential Density Function?, 6
This step changes the differential equation to
and the solution to
Housing Demand Theory
An Exponential Density Function?, 7
Now combining the housing production function, the capital
market equilibrium condition, the housing
S = D
condition,
and the definition of density from a basic urban model yields
where
Φ
is a constant.
With the above expression for
P
{u
}
and
no commuting costs in
the denominator, this equation simplifies to
where
D
0
and
δ
are constants.
An exponential form at last!
Housing Demand Theory
An Exponential Density Function?, 8
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,
μ
,
is -1.
In other words, households consider
tu
when they
decide where to live, but the impact of
tu
on their net
income only affects their consumption of
Z
, not of
H
.
An alternative approach that leads to an exponential
density is to assume that
θ
= 0 and
μ
= -1
(Kim and
McDonald,
Journal of Regional Science
, May 1987).
The empirical evidence (covered in the next class) does
not support the first part of this assumption.
Housing Demand Theory
An Exponential Density Function?, 9
As it turns out, this analysis leaves out a step, because it focuses
on deriving a bid function, not a market price function with
heterogeneous households, which is what we see in an actual
urban area.
The only clean way to rescue the algebra is to assume that all
households are alike, which implies that the bid function equals
the market price function.
If households are not all alike, the above assumptions are not
sufficient to generate an exponential bid-function envelope.
In general, the form of the market price function (to be derived in
a later class) is more complex than the form of a bid function—and
hence is unlikely to take an exponential form.
Housing Demand Theory
Questions
•
Why are Cobb-Douglas utility functions so
common in urban models despite their strong
assumptions?
•
Are there tractable alternatives to a Cobb-
Douglas utility function?
•
Under what assumptions does the density
function in an urban model take an exponential
form?
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
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.
The history of this theorem is recounted by Silberberg
(
Journal of Economic Education
, Winter 1999)
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
The Lagrangian expression for this problem is
Housing Demand Theory
The Envelope Theorem, 4
Differentiating this Lagrangian leads to the following
first-order conditions
which lead, in turn, to the following solutions:
Housing Demand Theory
The Envelope Theorem, 5
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
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, 6
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
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
x
i
*
s and
λ
*
and then substitute
those values back into
y
.
The Envelope Theorem gives us a shortcut!
Housing Demand Theory
The Envelope Theorem, 7
To get to this shortcut, first write the constraint at the
optimal values of the
x
i
s.
This constraint must still hold when
α
changes,
which implies that
Housing Demand Theory
The Envelope Theorem, 8
Now if we multiply this result by
λ
and add it to the
messy comparative statics result derived earlier, we
find that
But, by the first-order conditions, the terms in parentheses in
the second line have to equal zero, so
Housing Demand Theory
The Envelope Theorem, 9
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
x
i
*
s!!
Of course this only applies to small changes, but, as we
will see, it is a very helpful result.
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
Housing Demand Theory
Maximizing Bids
In an urban model, the household budget constraint is
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
Housing Demand Theory
Maximizing Bids, 2
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, 3
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, 4
The Lagrangian is
The relevant first-order condition is:
and the result we need is:
Housing Demand Theory
Maximizing Bids, 5
Now we can use the Envelope Theorem to find the
impact of
Y, t, U*,
and
u
on
P
{u
}
using
The CS results are:
Housing Demand Theory
Maximizing Bids, 6
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:
which is, of course, the result we 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 1987
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.
Housing Demand Theory
Questions
•
How can the envelope theorem be used to derive
a household bid function?
•
What is the difference between bid functions
derived with the envelope theorem and other bid
functions?
•
Why is the envelope-theorem approach bid
functions used in analyses with general
functional forms?
Explore the concepts of housing demand theory, alternative utility functions, Stone-Geary utility function, CES utility function, and comparative statics in urban economics. Discover how these theories impact housing demand and pricing strategies.
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Presentation Transcript
ECN741: Urban Economics More General Treatment of Housing Demand Professor John Yinger, The Maxwell School, Syracuse University, 2020
Housing Demand Theory Class Outline 1. Alternative Utility Functions 2. An Exponential Density Function? 3. The Envelope Theorem 4. Maximizing a Bid Function 5. Introduction to Comparative Statics with General Functional Forms
Housing Demand Theory Class Outline 1. Alternative Utility Functions 2. An Exponential Density Function? 3. The Envelope Theorem 4. Maximizing a Bid Function
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 S = ) ( ) 1 H S where SZ and SHare survival quantities. Z H In this case, the demand functions reflect the required spending on Z and H: = + ) { } P u S Y tu S H S z H H { } P u ( = + tu S (1 ) { } P u S Z S Y Z z H
Housing Demand Theory Stone-Geary Utility Function, 2 Thus, the indirect utility function is: { } P u S Y tu S = * U k z H { } 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 ( (1 U = ) 1/ + ) Z H Demand functions are unaffected by a positive monotonic transformation of utility, so if is positive (which it may not be), one can save a lot of algebraic mess by simplifying this to = + (1 ) U 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} with the added assumption that = 0.5. 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, but the = 0.5 assumption is pretty strong.
Housing Demand Theory Class Outline 1. Alternative Utility Functions 2. An Exponential Density Function? 3. The Envelope Theorem 4. Maximizing a Bid Function
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: { } D u e = u , where , 0 This was a striking claim because the exponential density function had been used in many empirical studies and it worked very well! 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 t { } = P u H ) = ( { } H B Y tu P u Putting these together leads to the differential equation t { } { } P u P u = ( ) B Y tu
Housing Demand Theory An Exponential Density Function?, 4 The general solution to this differential equation is: 1 { } 1 + + 1 ( ) P u Y B tu = + Q (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, , is -1. This leads to { } { } P u B Y P u t = ( ) tu The solution to this differential equation is 1 ( ) Y A tu = + ln{ { }} P u Q (1 ) This won t lead to an exponential density, either!
Housing Demand Theory An Exponential Density Function?, 6 Now let s use gross income in the demand function: H BY P u = 1 { } This formulation, which is used by Mills and Muth, assumes that households do not consider commuting costs when they decide how much housing to buy. One interpretation is that households have separate mental accounts for housing purchases and transportation. Mental accounts are discussed in Thaler (1999).
Housing Demand Theory An Exponential Density Function?, 6 This step changes the differential equation to H = 1 { } BY P u and the solution to P u { } { } P u t = BY t = ln{ { }} P u Q u BY or = ( / t BY ) Q u { } P u e e
Housing Demand Theory An Exponential Density Function?, 7 Now combining the housing production function, the capital market equilibrium condition, the housing S = D condition, and the definition of density from a basic urban model yields + (1 )/ (1 ) a a { } ( P u = { } D u ) Y tu where is a constant. With the above expression for P{u} and no commuting costs in the denominator, this equation simplifies to = u { } D u D e 0 where D0 and are constants. An exponential form at last!
Housing Demand Theory An Exponential Density Function?, 8 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. In other words, households consider tu when they decide where to live, but the impact of tu on their net income only affects their consumption of Z, not of H. An alternative approach that leads to an exponential density is to assume that = 0 and = -1 (Kim and McDonald, Journal of Regional Science, May 1987). The empirical evidence (covered in the next class) does not support the first part of this assumption.
Housing Demand Theory An Exponential Density Function?, 9 As it turns out, this analysis leaves out a step, because it focuses on deriving a bid function, not a market price function with heterogeneous households, which is what we see in an actual urban area. The only clean way to rescue the algebra is to assume that all households are alike, which implies that the bid function equals the market price function. If households are not all alike, the above assumptions are not sufficient to generate an exponential bid-function envelope. In general, the form of the market price function (to be derived in a later class) is more complex than the form of a bid function and hence is unlikely to take an exponential form.
Housing Demand Theory Questions Why are Cobb-Douglas utility functions so common in urban models despite their strong assumptions? Are there tractable alternatives to a Cobb- Douglas utility function? Under what assumptions does the density function in an urban model take an exponential form?
Housing Demand Theory Class Outline 1. Alternative Utility Functions 2. An Exponential Density Function? 3. The Envelope Theorem 4. Maximizing a Bid Function
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. The history of this theorem is recounted by Silberberg (Journal of Economic Education, Winter 1999) 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 Maximize Subject to = , } = { ,..., f x y x 1 n , } { ,..., g x 0 x 1 n 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 i f = + = 0 g i = = 0 g which lead, in turn, to the following solutions: x = *{ } i x * i = * *{ }
Housing Demand Theory The Envelope Theorem, 5 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 = x = { } * * 1 * n { ,..., f x , } y Note that { } is sometimes called the indirect objective function; it is the maximum value of y for given using x s that meet the constraint.
Housing Demand Theory The Envelope Theorem, 6 Now suppose we want to know how this maximum value changes when we change , which is just an exercise in comparative statics. Differentiating the above expression for y* yields * y * i x = = + f f i i This looks pretty complicated. It looks like we have to re-solve the problem with the new values of to find the new values of the xi*s and * and then substitute those values back into y. The Envelope Theorem gives us a shortcut!
Housing Demand Theory The Envelope Theorem, 7 To get to this shortcut, first write the constraint at the optimal values of the xis. * { ,..., i g x x = * n , } 0 This constraint must still hold when changes, which implies that * i x g = + = 0 g g i i
Housing Demand Theory The Envelope Theorem, 8 Now if we multiply this result by and add it to the messy comparative statics result derived earlier, we find that * * i i i * i x x y = + + + f f g g i i * i x ( ) = + + + f g f g i i i 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, 9 This is an amazing shortcut. To find the comparative static derivative of y* with respect to , all we have to do is differentiate y and g with respect to . 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 Class Outline 1. Alternative Utility Functions 2. An Exponential Density Function? 3. The Envelope Theorem 4. Maximizing a Bid Function
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 Y Z H U tu = Maximize { } P u = * Subject to { , } U Z H
Housing Demand Theory Maximizing Bids, 2 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, 3 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, 4 The Lagrangian is Y Z H tu ( ) = + * { , } U Z H U The relevant first-order condition is: Z 1 H = + = 0 U Z and the result we need is: 1 = U H Z
Housing Demand Theory Maximizing Bids, 5 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 U 1 { } P u Y 1 H { } P u t u H = = = = 0 0 0 * U H Z
Housing Demand Theory Maximizing Bids, 6 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 t { } = = 0 P u H which is, of course, the result we 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 1987 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.
Housing Demand Theory Questions How can the envelope theorem be used to derive a household bid function? What is the difference between bid functions derived with the envelope theorem and other bid functions? Why is the envelope-theorem approach bid functions used in analyses with general functional forms?
