This PDF is a selection from an out-of-print volume from the National
Bureau of Economic Research
Volume Title: Explorations in Economic Research, Volume 3,
number 1
Volume Author/Editor: NBER
Volume Publisher: NBER
Volume URL: http://www.nber.org/books/gort76-1
Publication Date: 1976
Chapter Title: Housing Demand in the Short Run: An Analysis
of Polytomous Choice
Chapter Author: John M. Quigley
Chapter URL: http://www.nber.org/chapters/c9079
Chapter pages in book: (p. 76 - 102)
I
JOHN M. QUIGLEY
National Bureau of Economic Research
and Yale University
Housing Demand in the Short
Run:
An Analysis of Polytomous
Choice
ABSTRACT:
In this paper the author presents a model of household
choice among types of residential housing that incorporates intramet-
ropolitan variations in housing prices arising from variations in work
site location. Under suitable assumptions, the prices that households
face in choosing among alternative types of residential housing are
deduced. ¶ The empirical analysis suggests that consumers are re-
sponsive to the systematic variation in these prices in their choices
among housing types in a metropolitan area. A model relating house-
hold choices among some 18 types of residential housing to intrarnet-
ropolitan price variation is estimated by maximum likelihood methods
using conditional logit analysis. The results of the analysis, which is
conducted separately for some
O stratifications of households by
income and family size, provide strong evidence of the importance of
these intrametropolitan variations in
relative
prices in motivating
choice among alternative types of residential housing.
NOTE: A previous version of this paper was presented at the winter meetings of the Econometric Societ
New York, December 1973. I am grateful to Bill Apgar. Jim Ohis, and William Weaton for helpful criticise
of an earlier draft, and to Wallace Campbell, Walter Fisher, and Philip Klutznick of the Board reading
committee for their comments on the final version of the paper.
76
rch
sity
old
et-
ork
olds
are
re-
ces
use-
met-
ocis
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by
e of
(I ng
ciey,
ticism
ading
77
F-
Housing Demand in the Short Run
Existing empirical studies of the demand for housing,
usually based on
aggregate cross-section data, ignore (Or
assume away) several crucial
features of the urban housing market. First, these studies
measure housing
consumption in a single dimension, rental
payments (Or housing values),
despite the obvious heterogeneity of the housing
stock. Secondly, these
studies either ignore housing prices completely
in focussing on the
income-expenditure relation, or they rely
upon crude measurements of
'average" housing prices in an entire metropolitan
area.'
The few analyses of the demand for housing based
upon micro units, i.e.
individual households and dwelling units, have established,
not surpris-
ingly, that specified types of housing
consumers demand particular com-
ponents of housing services. However, these recent studies have
only
analyzed the effect of housing prices upon household demand
under the
implicit assumption that components of housing services
may be pur-
chased quite independently of one another.2
Theoretical analyses of residential location and the demand
for housing
stress the importance of the work trip in determining the spatial location of
housing consumption and the quantity of "housing services" demanded,
Yet with very few exceptions, these theories ignore the existence of durable
and differentiated stocks of residential housing. These theoretical analyses
in effect assume that the urban area will be built de
novo during any
period of analysis.
Neglect of the heterogeneity of housing in both residential location and
housing demand studies is clearly justified in certain situations, notably in
the analysis of comparative statistics when the central focus of the investi-
gation is upon the long-run equilibrium of the entire market for "housing
services." Since in the long run housing can be converted or built anew at
any site, the convenient notion of undifferentiated "housing services,"
measured by total monthly expenditures, is appropriate in analyses of both
consumer demand and choice of location.
Yet it is equally clear that dwelling units emitting the same quantities of
"housing services," as measured by contract rent or monthly expendi-
tures, are often viewed as utterly distinct by both housing suppliers and
demanders. Indeed, both producers and consumers may view them as
much less similar than other units which differ substantially in price. The
substantial costs of transforming the characteristics of existing units implies
that housing units of various types may barn substantial locational quasi-
rents for long periods of time.
Indeed, the first attempts to incorporate distinct components of housing
services explicitly into consumer demand theory have already been under-
taken by Sweeney. In his insightful theoretical analysis, Sweeney defines a
"hierarchy" of housing commodities and derives the equilibrium condi-
tions for a market characterized by discrete housing types that can be
John M. Quigley
ranked identically by all consumers from the "most
preferred" to the "least
preferred" type. Sweeney also investigates changes in the
demand for all
housing types in response to a change in the price of any
single type. In
concentrating upon the "hierarchical" nature of the housing
commodity,
however, Sweeney ignores the spatial aspects of the housing market.
The polycentric nature of employment locations in real urban areas and
the importance of the work trip in determining both residential location
and the choice of housing type greatly complicate the problem. The
durability and fixity of residential housing suggests that households face
differing effective prices for the same types of housing depending upon
their work place locations, at least as long as transport is not costless.
This paper extends the theoretical analysis of the demand for housing to
incorporate the spatial dimension (and thus the residential location deci-
sion), as well as the choice of housing type. In particular, we address the
choice of housing type and residential location in a metropolitan area
which may have several work places. In this short-run analysis, the spatial
distributions of the stocks of various types of housing are given. Although
the monocentric assumption of traditional residential location models is
abandoned, the analysis relies upon the primary insight of residential
location theorythe willingness of consumers to substitute transport costs,
specifically work trip commuting costs, for housing prices in choosing
residential locations. The theoretical model indicates how choices among
housing are related to systematic variations in the relative prices faced by
households for the same types of residential housing. The model indicates
that these prices, in turn are heavily dependent on the interaction of work
place location, the spatial distribution of the stock of housing, and the
characteristics of the urban transport network.
The model is estimated empirically, by conditional logit analysis, based
upon the actual choices made by a sample of some 3,000 renter house-
holds in the Pittsburgh metropolitan area. The results provide rather
powerful predictors of the housing choices made by the sample of relocat-
ing households; yet the results are not necessarily consistent with the
notion of equilibrium in the housing market as a whole. In particular, the
results are generally consistent with the possibility, at given prices, of
excess demand or excess supply of particular types of housing at certain
locations.
In choosing a dwelling unit, households jointly purchase
a wide variety
of attributes at a particular location. Considerable effort
has already been
expended by researchers to isolate those attributes of the
housing "bundle"
that command prices in the market. Without
loss of generality, we can
classify units into housing "types"
or collections of attributes. Each housing
type is defined at specified values of the vector of attributes
that command
market prices. The set of mutually exclusive housing
types represents all
ast
all
In
ity,
t.
and
tion
The
face
pon
less.
ley
70
Housing Demand in the Short Run
possible choices that may be made by any housing
consumer. We assume
that each consumer will choose one (and only one) residence
from the set.
During any given period only a small fraction of urban
households
become "movers" and actively search for
new residences in the urban
area. Typically these households include:
1.
additional workers induced to the urban
area;
2.
new households formed during the period;
3.
those whose preferences for housing attributes have
changed;
4.
those for whom the relative prices of housing
types have changed
appreciably.
g to
deci-
Since preferences for particular configurations of housing
are strongly
s the
related to family size, composition, and age as well
as family income, the
area
third category includes movers induced by life-cycle changes in house-
atial
holds. For reasons discussed below, the fourth category includes
house-
ough
holds whose work place has changed as well as those with unchanged
els is
work places who face changes :n relative prices. However, since moving
ential
within the urban area imposes economic and other costs
upon households,
costs,
we may suppose that for households with unchanged preferences and
sing
work places, appreciable changes in relative prices will be required
to
mong
induce intrametropolitan mobility.
ed by
In any period each household making a residential choice gathers
icates
information on the spatial locations of each type of housing and
on the
work
market prices of housing types at these locations. Since alternative spatial
d the
locations impose costs upon the household, each household similarly
gathers information on the accessibility costs associated with different sites.
based
These accessibility costs will reflect the out-of-pocket costs and the oppor-
tunity costs of the time expended in commuting and in travelling to other
points.
ouse-
rather
locat-
For an individual household, the choice of the best, or "optimal"
location, for any particular type of housing is straightforward, at least in
th the
principle. For each possible location the household adds the accessibility
ar, the
costs to the housing price schedule and calculates the total cost of
es, of
consuming that type of housing at that location. The site at which this total
ertain
cost is a minimum is the optimal location for consuming the partcular type
of residential housing.
variety
The household's ultimate choice among housing types is systematically
y been
related to this cost minimizing calculus. After calculating the optimal (i.e.
undle"
the minimum cost) location for each type of residential housing, the
e can
household chooses among locationally subscripted housing types on the
ousing
basis of its preferences for the underlying housing characteristics and the
rnand
relative costs (or effective prices) of the alternatives. Note that the total cost
ents all
I
S
80
John M. Quigley
of each housing type at its minimum priced location is
the relevant price in
considering the choice among housing types.
If, as the assumption of
residential location theory suggests, work trips are the most iniportant
component of accessibility costs, the effective price
facing different house-
holds for consuming a particular type of housing varies with the place-
ment of their work sites relative to concentrations of the available stock. If
travel time is related to alternative wages, the price will also vary for
households with different wages
In a city where work places and incomes are not identical and where
durable and heterogeneous residential structures exist, our theory suggests
that consumers' choices among housing types will be dependent upon
these relative prices.
For simplicity assume that each household entering the housing market
possesses perfect information about housing prices and the spatial distribu-
tion of housing units; that is, assume that each moving household knows
the surface of prices and housing stock densities in the urban area for every
housing type.
For most households, the single most important component of the
accessibility costs of any site is commuting expenditures. For example,
studies of household trip-making behavior indicate that work trips alone
account for 40-45 percent of total trips and account for more than twice as
many trips as any other class. In addition work trips are, on average, longer
than other types of trips, so their share of accessibility costs is much larger
than their share of total trips. Finally, work trips are typically made
on a
regular basis to particular sites and most other trips are made to diverse
destinations. It has been found, for example, that "the [accessibility] costs
to any single point [other than work place] are almost always trivial."6 In
contrast, journey-to-work costs are typically incurred to reach a particular
destination and their magnitude is substantial. These factors
suggest that
work trip costs are a good approximation to total accessibility
costs. In
particular, we will assume that households have
an inelastic demand for
trips to the work site and that all other trips
are made to ubiquitous and
substitutable destinations. This assumption is fairly
common in models of
residential location.
In contrast, however, to traditional residential location theory,
we do not
assume that all households have the same work place. We recognize the
polycentric nature of urban areas by assuming instead
that locating house-
holds have known and fixed work places.
Under these assumptions the household
can calculate the total cost of
consuming each type of housing at each location. By
searching for the
minimum, the household can discover the optimal
site and its associated
cost for each type of housing. As noted previously the
optimal site and the
cost associated with it will vary with work place
and wages or incomes.
e-
If
the
le,
ne
as
ger
ger
na
rse
osts
In
ular
that
In
for
and
s of
y
n
nt
or
re
sts
et
U-
ws
ery
not
the
use-
:t of
the
ated
the
es.
81
Housing Demand in the Short Run
(1)
= mm
= mm [R, -s-
Definitions for the variables appear in the following
list:
R1,
is the contract price (monthly rent) of housing
type i at residential site
m.
Tjm
is the (monthly) cost of work trips between work
place I and residence site
m for workers with income y.
P,,1
is the total (monthly) cost of housing type i
at location m for workers of
income y with work site j.
is the effective or minimum (monthly)
price of consuming housing type
I
for workers with income y and work site
j.
131ii,
I = 1, 2,
. . .
, I
identifies housing types;
m = 1, 2, . .
, M identifies residence sites
j = 1, 2.....I identifies work sites;
y =1, 2.....Y identifies incomes.
Households with given work places,
I. and income, y, face
a budget
constraint of the form
y = P2z + P
where z is the amount of other (nonhousing, nontransport) goods
con-
sumed at price P, and P41 is defined in equation 1 (with the work place
and income subscripts suppressed) as the cost of consuming housing
type i
at its minimum priced location.
For each of the I discrete types of residential housing
we define X. as the
vector of their underlying characteristics (x11, x21,
. . .
, x,), i = 1, 2, . .
, I.
Households are assumed to value the underlying
characteristics of the
housing types as well as other goods
z, i.e., they have utility functions of
the form,
U(X,z)
Since each locating household occupies but
a single housing unit, each
household makes one choice out of the
range of discrete housing bundles,
in addition to its choices of other (nonhousing,
nontransport) goods. For a
household of given income, knowledge of the housing
type consumed and
its effective price determines the
amount of other goods that may be
purchased. Thus for given incomes, each housing bundle and
its price
represent a complete choice over all goods, i.e., the mixed direct-indirect
utility function
(4) V(X1,P)
L
82
John M. Quigley
represents the budget-constrained level of utility derived by a household
with income y living in housing type i. The consumer's problem is to select
the housing type i which yields the highest level of utility.
Preferences for particular underlying characteristics defining housing
types depends upon certain attributes of the households, notably family
size and composition, or "life cycle" attributes. If we consider households
with common incomes, y, and life cycle attributes a, utility maximization
implies that housing type i
will be chosen if
U
(X,, P) > Up,, (Xi, P) for all j i
Since some of the influences upon consumer tastes are unobserved even
if households are stratified by income and household attributes,
the
deviations of individual preferences from the average of the socioeconomic
group (y, a) may be summarized in a stochastic component.8
U,,,, (Xi, P) = W,,a (X1, P) + Eva
where
represents the preferences of the "representative" consumer,
and EVa summarizes the influences
upon preferences of all factors which
are unobserved.
Thus if the preference functions
are interpreted as having a stochastic
component, the probability (pva,) that a particular household
of class (y.. a)
will choose housing type i
over all other types depends on the probability
that the utility of housing typei exceeds the
utility of each other type
j, i.e.
P,,aa = prob [Uya (Xi, P') > U,,a (K,, P)] for all I
I.
and
prob [Eyaj
Eval < W,,,, (K1, P)
t'',,,, (K,, P)] for all j
i
Equation 8 indicates that the
probability of choosing
any particular
housing type depends
on the vector of housing characteristics of a/I
housing types and their total
costs and on a vector of stochastic elements.
If
the vector of stochastic
terms follows some known distribution,
it
is
possible to derive an explicit formula
for p.
In particular, as McFadden has
dernonstrated,
if e, and
are statistically
independent with the reciprocal
exponential distribution
prob(
Zj)eZ.
then
1
prob(1 -
Z1) =
=
+ e'
eZi
1
uigley
ehold
select
Using
holds
zation
s,
j,
i.e.
n, it
83
Housing Demand in the Short Run
and
e'
v,
(11)
=
amily
In equation lithe probability of choosing any particular housing
type i
depends on the attributes and prices of each of the available
types. The
sum of the probabilities over the I housing types is
1 and the probability of
choosing any single type will lie between 0 and 1. In short,
equation 11
even
represents a well-behaved probability function. From equation ii, the
the
odds of choosing i over alternative j may be expressed
nomic
as
-
P
w
(
P)
Pya
-
(li,
,. P)
or
su mer,
which
log
= Wi,,, (X,P) W.,, (X,P)
chastic
Equation 13 implies that the choice between any two housing types is
s (y, a)
independent of the characteristics of the other housing types. Since, by
ability
definition, the set of housing types represents the entire range of choice,
an
individual's ranking of all possible housing types is completely determined
by a series of paired comparisons. This property, the so-called "indepen-
dence of irrelevant alternatives," implies that if those characteristics which
define housing types are chosen correctly, the analysis can be generalized
to address the probability of choosing "new" types of housing (i.e.,
combination, of housing characteristics which may not be observed in a
rticular
given sample).
of all
The logic of equation 11
also implies a separability property in the
erits. If
choice of housing characteristics. Even if housing characteristics are only
is
available in discrete bundles or types, for any given price vector a
household's probability of choosing specified levels of two characteristics
istically
can be decomposed into an independent marginal and a conditional
probability.
In the empirical analysis that follows, it will be assumed that W is
linear in
its parameters.'0 In this case,
W,,,, (X1, P) = b,,,, , +
P
the statistical model is a multinomial generalization of the logit model
often applied to situations involving binary choice, and the parameters can
similarly be estimated by maximum likelihood methods. In addition,
if
84
John M. Quiglev
preferences can be approximated by any
function linear in its parameters
McFadden has shown that the likelihood
function is concave, implying that
iterative estimation procedures converge upon
the unique maximum likeli-
hood estimator of the b parameters.'1
Equations 13 and 14 imply the
multinomial
logistic model to be
estimated separately for each stratification of income
and socioeconomic
characteristics (y, a).
(15)
log (p1/p,) = b, (x,, - x,) +
. .
.
+ b,,
(x,,, - x,,) + b,1 (P'
P)
Empirical estimates of the demand for housing types and individual
housing characteristics are obtained by using information from a large-
scale home interview survey conducted in 1967 in the Pittsburgh Met-
ropolitan Area.'2 The empirical analysis uses price and housing stock
information gathered on some 25,000 dwelling units to analyze the
housing choices made by approximately 3,000 rental households who
made location decisions within the seven year period 1960-1967.
The central hypothesis is that the multiplicity of work places interacts
with the location of durable stocks of differentiated housing types to create
systematic variation in the relative prices of housing types that confront
households in the urban area. These systematic variations in relative prices
are derived from variations in journey-to-work costs, and by hypothesis
they affect households' choices among housing types or housing configura-
tions.
Besides testing this hypothesis in some detail, the analysis allows empiri.
cal testing of several other hypotheses concerning housing market be-
havior. These hypotheses are developed following the definitions of the
particular variables used in the analysis. The operational definitions of the
types of residential housing, their component characteristics, and the
calculation of the effective prices facing each household are first discussed
in turn.
THE TYPES OF RESIDENTIAL HOUSING
As previous analyses have stressed, payment for housing
services includes
payments for a wide variety of qualitative and quantitative attributes of
residential structures, In defining discrete housing
types, or combinations
of these underlying attributes, theoretical
considerations suggest two rough
guidelines. On the supply side, the existence
of discrete housing types or
submarkets implies that it must be costly
to transform housing units among
submarkets. On the demand side, housing
units within any submarket must
be viewed as (virtually) identical, but
housing units in different submarkets
must be viewed as separate and distinct
entities.
y
SI
at
e
ic
al
et-
ck
he
ho
cts
ate
nt
es
S's
ra-
In-
be-
the
the
the
sed
des
of
ons
gh
s or
ong
ust
kets
85
Housing Demand in the Short Run
Both the empirical and theoretical literature
suggest that households
of
differing income and family size will choose
units of varying
residential
density (or lot size) and varying interior
size. In addition,
the qualitative
characteristics of residential Structures
are valued by households.
Based upon these considerations and
available sample
information 18
types or submarkets of rental housing
are defined by proxies for
residential
density, quality, and interior size. Residential
density (or effective
lot size) is
proxied by structure type, which is
reported in three
categories; single
detached units, common-wall units
(including row and
duplex houses),
and multifamily (apartment) units.
The age of the dwelling unit is used
as a proxy for housing
quality and
obsolescence.
Units are classified into
two categories: those
built before
1930 and those built after 1930. The cutoff
year for defining age
categories
was chosen from considerations of sample size with
respect to the data
source. It should also be noted that there
was relatively little
new residen-
tial construction in the Pittsburgh
metropolitan area during the
period
1930-1 945.
Although it would have been preferable
to use floor space in describing
interior size, the only available information
in the sample is the number
of
bedrooms in each dwelling unit. Interior
size is thus proxied by the
number
of bedrooms in the unit, reported in
three categories:
less than two
bedrooms, two bedrooms, and three
or more bedrooms.
The types of rental housing
are thus described by 1 8 combinations:
three
structure types by two quality levels by three
interior size
measures.
The Effective Prices of Housing
Types
For each of the 18 types of residential
housing, the surface of
contract
prices (monthly rents) is estimated by
the average price in each of
50
locations (zones) in the metropolitan
area. The available stock of each type
01 housing is similarly described
by the number of units in each
zone.
Calculations made by households of the
costs of commuting to work are
facilitated by reference to
a set of 330 work sites (zones) and 130 residence
sites (zones).
Thus from equation 1 the
surface representing the total cost of
consum-
ing housing of type i
is
(16)
P,,, = R1,. +
where
= 1, 18
housing types
= 1, 330 work places
m = 1, 130 residence places
m' = 1, 50
residence places
y = 1, Y
incomes
To estimate the monthly cost of work trips we make two strong assump.
tions. First, we assume that households are free to choose the number of
hours they work; secondly, we assume that workers neither value the act of
traveling nor the intrinsic characteristics of travel modes. These assump.
tions imply that the time spent traveling is valued at the (marginal) wage
rate and that the choice of mode is made solely on the basis of time and
money costs.
Thus for an individual with (marginal) wage w, the monthly transpon
costs (TC) from fixed work place Ito residence place m will be equal to
the minimum of the cost of a single trip on public transit (TP,) or the
cost
of a trip by private auto (T%,) multiplied by the number of work trips
per
month (N);
i.e.,
T,
= N mm (TP,, TA)
The cost of trips by public transit is composed of out-of-pocket fares
(F)
and time costs. Let T'm be the elapsed time by public transit between
work
place j and residence site in.
TPjmw
'jm + Tm Wp
Similarly the cost of trips by private auto includes the out-of-pocket
cost of
fuel and maintenance'4 (expressed as E dollars
per minute), the cost of
parking at the destination (expressed
as half the costs of all day parking at
the work site, C) and the costs of time (where
Tm is the interzonal travel
time for an auto trip):
T,
L + T
(E + w)
The total expenditure required to
consume housing type i at any residential
location m may be computed
as
P
= R.,
mm
t(F,
+ Tm wy),
+ T,, [E + wJ)}
Figures
1
and 2 illustrate schematically
the spatial
distribution of
monthly contract prices and the
cumulative distribution of total housing
costs for a particular housing
type facing a particular worker. Figure 1
maps the surface of contract
prices R._, for a particular housing type in the
analysis area. As the schematic
is drawn, darker shades correspond to
higher monthly
rents for this type of housing
at different spatial locations.
Figure 1 in effect
presents the average monthly
rents of a particular housing
type in 50 zones in the
metropolitan area. Although the price
pattern
reveals some tendency for
prices to decline with distance from the Central
Business District (CBD), the
surface is characterized by
irregular peaks and
valleys and by
conspicuous "holes" where the
type of housing is simply
unavailable. Figure 2 plots
the ordered distribution
of the total costs of
p.
of
of
p-
)rt
to
st
er
k
of
of
at
el
of
Is.
al
id
ly
Jf
FIGURE 1
Schematic of the
Surface of
Monthly Rents for
New Two
Bedroom CommonWaIl
Units in the
tan Study
Pittsburgh Metropoli..
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II))III)), 11)1)211)112)21)1)51)1)1
2227)
77'7)) 1)111111)1111)271111121
''22:1'.SIl77(f.,-,
ff771) IIl)lI))IIlI III 1)) 222))
lIIllll)i)iij,,,,,,j,,,
l))))));))
II)))
111)2)2)1)1)
7'1175
llIl)flI,7, 111177777)
I II))J)))j),p) I) 1I)1)))ll)li,
II
II III) I) I))
II Ill II
consuming this type of housing
faced by an individual
with wage rate of
$7,000 employed in the CBD.
The figure was plotted
by applying equation
20, using the three travel
matrices, Firn, T,22, and T,,2
(130 x 330), and the
vector of parking costs C3 (1
x 330) aggregated from the
1 967 Pittsburgh
survey.
If households possessed
perfect information, the
optimal residential
location for this type of housing
for the individual
represented in Figure 2
and its effective price
to him would be the actual
minimum of the
cumulative price distribution,
$122 on the diagram.
Because housing market
information is costly and because
the indi-
vidual estimates of
total housing costs
are subject to measurement error,
the empirical analysis
does not rely
upon the single minimum total price
as
an estimate of the effective housing
price facing an individual. Instead the
average total price of the lowest
five percent of the stock of each housing
type is used to estimate
the effective price minimum. Figure
2 illustrates
this computation
and shows an estimated minimum
price of $128.
In addition
to these price estimates,
a variable measuring the total
number of units of each
housing type available in the metropolitan
area is
$
a
FIGURE 2
Ordered Distribution of Total Cost of Consuming New
Tø
Bedroom Common-Wall Units Facing a Household
in the CBD With an Annual Income of $7,000
P,ykm
250
225
200
175
*
plik
125
100-
0
0
10
20
30
40
50
60
70
80
Per cent of the housing Stock
included. This additional
measure is used to proxy for the information
available to consumers about the location
and prices of alternative housing
types.
The Complete Model and
Some Additional Hypotheses
As stated and developed
in previous sections, the model
to be estimated in
this section
is the multinornial logistic.
For each cross-classification of
income and family size, the
logarithmic odds of the choice between any
two types of residential housing
is a linear function of the attributes of each
housing type (in this
case proxies for residential density, interior size,
quality, and availdbility
in the metropolitan area and the eftective price Ot
each housing type
(which may
vary for particular households). From
equation 15, the specific
model is:
I
Housing Demand in the Short Run
89
log (pr/p,) = b1 (CW1
- CW3) + b2 (APT1 - APT) + b3 (BR
- BR)
+ b.1(ACF
GEJ) ± b5 (P
- P) -f b (ST1 -
i)
where
CW is a dummy variable with
a value of I
if j
is a common-wall unit
APT1 is a dummy variable with
a value of
I
if
i
is an apartment unit
BR1
is the number of bedrooms in
type I
AGEI is a dummy variable with
a value of 1
if I was built before
1930
P
is the effective monthly cost of
consuming housing type i
and
ST,
is the number of units of housing of
type i
in the sample
The parameters of equation 21
are estimated separately for each
of 30
combinations of income and family size. Equation
21, together with the
error term assumption in equation 9, define the likelihood
function (L)
whose logarithm is:
log L = -
jD1r log{
(CWkT - CW1r)
r=1 t=I
k=i
+ .
+ b6 (STkr
ST1r)jJ
where
R
is the sample size for each
stratification of income and family
size
and r
1, 2.....R is the index of
observations, and
Dir is a dummy variable with
a value of 1
if the rth household chooses
housing type I.
Maximum likelihood estimates of the
parameters of equation 22 are
obtained by an iterative
process.
If this model of housing choice is
appropriate, several hypotheses about the signs and
magnitudes of the
estimated parameters can be addressed.
First, from equation 11 the own-
price elasticity of choice
among housing types is
N1, = Pb5(1
- pi)
and the cross-price elasticity is
N. = Pb5p
To insure a negative own-price elasticity
and a positive cross-price elastic-
ity, the estimate of b, should be
negative for each stratification of income
and family size.
We should also
expect the parameter b4 to be negative since, ceteris
paribus, households prefer higher quality
dwelling units, that is, holding
structure type and size constant, housing
types indexed by quality form a
"commodity hierarchy." Similarly, holding
structure type and quality
e
90
John M. Quiley
constant, housing types indexed by size
form a "commodity hierarchy";
thus we expect the estimate of b
to he positive. The coefficient of the
housing stock term, b6, should be positive, since households can obtain
more information, at the same
search cost, for housing types in greater
supply.
Holding income constant, we should expect that larger families demand
larger units and more exterior space. Thus for larger families with the same
income we should expect that the estimate of b3 will be larger than for
small families. Similarly, the estimates for b, and b2 should be smaller in
magnitude (or more negative) for larger families than for smaller famjlips
Holding family size constant, we expect that higher incomes are as-
sociated with greater consumption of higher quality, larger units with more
exterior space. Thus for the same family size we expect that the estimate of
b3 will be larger for the higher income households than for lower income
households. Similarly, the estimates of b1, b2 and b4 should be more
negative for higher income households than for lower income households.
Table
presents the coefficients of the multinornial
logistic model,
1
estimated by the maximum likelihood method, for each of thirty combina-
tions of income and family size. The model is estimated separately for each
of five family sizes (corresponding to households of 1, 2, 3, 4, and 5
or
more members) for each of six income classes (corresponding to annual
incomes of less than $3,000$4,999, $5,000$6,999, $7,000$9,999,
$10,000$14,999, and $15,000 or more).
For each household in the sample, the total cost was calculated for each
of the 18 types of residential housing at each possible location by using
equation 20 and the mid-points of the income classes to derive hourly
wage estimate w;'5 the minimum total cost (P) including housing and
transport cost, was estimated for each housing type by calculating the
average price of the cheapest 5 percent of the stock for each household.
One type of residential housing was chosen as a numeraire; the prices
facing each household are relative to this numeraire.'6
For each of the 30 nonlinear regressions, the results reported in Table 1
were obtained by specifying a convergency criterion of .01. In most cases
five or six iterations were required. For each set of results the sample size is
noted and the asymptotic t ratios of the coefficients
appear in parentheses.
In 26 of the 30 equations the relative price coefficient has the antici-
pated sign; the estimated coefficient exceeds its standard
error
in 22
equations and it appears highly significant in 16 stratifications. The
ratios
of the relative price coefficients
are substantially lower for the two higher
income groups. For renter households earning between $10,000 and
$15,000 a year, three of the estimates
coefficients are significant at about
the .05 level and the other
two are insignificant. For renter households
earning more than $15,000 a year,
none of the price coefficients are
significant.
TABLE 1
Estimated Coefficients of the Multinomial
Logistic Model by Family Size and Income Class:
log (p,/p.) = b1 (CW1 - CVv) + h2 (APT1
- APT,) + b3 (BR - BR,) + b4 (ACE1 - ACE,) + b (P
- P) + b6 (STe - ST,)
Number
Number
Common
Apart-
of Bed-
Structure
Relative
Family
of Obser-
Wall
ment
rooms
Age
Price
Stock
Size
vations
(CW)
(APT)
(BR)
(ACE)
(P*)
(ST)
Income $3,000 and Below
48
2.560
1 .958
0.538
-1.743
1.392
0.0 15
(2.91)**
(3.69)**
(0.97) (2.20)*
(4,79)**
(0.51)
61
1 .045
2.379
-0.478
-1.428
-3.167
0.006
(2.34)**
(5.05)
(1 .69)*
(2.45)**
(1.62)
(6,34)**
3
41
-2.679
-0.902
3.309
-3.2 10
-4.400
0.136
(3.64)**
(1 .88)*
(5.25)**
(4.04)*
(2 .3 1) ** (55.52)**
4
15
-3.619
0.089
5.187
-0.201
-20.571
0.015
(1 .62)*
(0.09)
(2,38)**
(0.10)
(1.96)*
(2.1 1)**
5+
25
-22.89 1
2.280
50.0 13
-22.926
-86.251
0.162
(2.86)*
(0.59)
(3,55)**
(3 .2 7)**
(3,01)**
(3.50)*
Income $3,000-$4,999
1
1.023
104
2.438
-0.757
-0.650
-6.866
(3.02)**
(6.68)*
0.004
(3.54)
(1.64)
(5.50)**
(5.92)*
2
140
0.075
1.792
-0.261
-1.930
-2.170
0.007
(0.28)
(6.58)**
(1.27)
(4.82)
(2.08)**
(9.41)'
95
-1.523
-0.276
1,687
3
-2,548
-6.475
(459)*
0.010
(0.92)
(5.92)**
(5.70)**
(4.60)
(9.01)
0
'-fl. Saa*a
TABLE 1
(continued)
Number
Number
Common
Apart-
of Bed-
Structure
Relative
Family
of Obser-
Wall
ment
rooms
Age
Price
Stock
Size
vations
(CW)
(APT)
(BR)
(AGE)
(P1)
(ST)
4
88
-1,054
-0,810
1.770
0.455
-6.364
0.006
(3.00)1*
(2.40)1*
(5.20)"
(1 .20)
(3.87)"
(593)**
5+
87
-1.278
-1.930
3.282
-0.247
-3.998
0.010
(3.72)"
(4.34)1*
(8.91)1*
(0.67)
(2.90)"
(9.15)"
Income $5,000-$6,999
1
91
0.150
2.564
-1.222
-3.049
-1.888
0,009
(0.31)
(5.50)1*
(3.21)"
(4.26)"
(1.20)
(654)1*
2
223
0.291
0.874
-0.223
-0.223
-2.906
0.003
(1.65)*
(4.29)1*
(5.1 3)1*
(1.25)
(3.98)"
(7.79)"
3
224
-1.500
-0.849
2.020
-2.383
-4.465
0.010
(7.06)"
(445)1
(10.92)1*
(9.03)1*
(5.87)"
(14.89)"
4
194
-2.693
-0.903
3.823
-2.738
-4.140
0.014
(10.1 2)"
(4.23)1*
(13 .54)1 *
(8.86)"
(4.91)"
(14.16)"
5+
223
-1.591
-2.0 13
3.170
-0.893
-6.116
0.008
(753)1*
(7.32)1*
(13.46)1*
(3,89)**
(6.61)"
(12.16)**
Income $7,000.-$9,999
79
0.664
1
1.561
-0.586
-3.179
-0.491
0.006
(1.55)
(3.73)
(2.39)
(6.11)"
(0.44)
(5,54)1
C,.
2
228
0.196
0.350
0.479
-1.881
-1.403
0.006
(0.87)
(1 .65)*
(339)**
(7.81 )
(2.26)"
(11.67)"
3
218
-1.242
-0.490
2.272
-2.955
-3.490
0.010
(543)**
(2.37)"
(11 .92)*
(10.95)**
(495)**
(14.24)"
4
166
-2.340
-0.616
4.193
-4.304
-3.565
0.015
(9.1 8)**
(1 .85)**
(11 .70)**
(354)**
(9.63)"
(12.50)"
5+
173
-2.096
-1.561
4.159
-0.750
-5.035
0.009
(8.46)**
(5,1 7)**
(1 2.12)**
(3.00)"
(4.90)**
(11.82)"
Income $1 0,000-s 14,999
24
-'1.267
-1.736
0.493
1
-4.651
1.105
0.020
(0.92)
(1.16)
(0.68)
(2.92)"
(0.22)
(3.44)"
2
153
-0.108
0.391
0.060
-1.503
-1.262
0.003
(0.40)
(1.51)
(0.44)
(6.46)"
(1 .83)*
(6.59)"
3
83
-0.640
-0.284
1.368
-1.051
-1.711
0.004
(1 99)*
(0.82)
(5.72)"
(3.25)"
(1 .81)
(5.52)"
4
56
-2.566
2.134
7.477
-6.366
-2.031
0.023
(4.63)"
(2.72)"
(6.89)"
(5.49)"
(1.10)
(6.94)"
5+
67
-2.511
-3.769
4.743
-2.674
-3.348
0.010
(5.78)"
(4.40)"
(7.39)"
(4,99)"
(1 .76)
(6.20)"
Income $15,000 and Above
17
-19.346
-10.694
1.256
-3.686
-3.321
0.040
(0.03)
(0.02)
(0.01)
(0.01)
(0.00)
(0.03)
TABLE 1
(concluded)
Number
Number
Common
Apart-
Family
of Obser-
of Bed-
Structure
Relative
Wall
ment
rooms
Age
Size
vations
(CW)
Price
Stock
(APT)
(BR)
(AGE)
(P*)
(ST)
2
50
-1.656
-0.230
1.926
-3.534
0.084
0.011
(2.31)**
(0.47)
(4.28)**
(5.48)"
(0.06)
(5,68)**
3
24
-3.605
1.531
3.466
-0.093
-1.572
(345)**
0.000
(1.15)
(4.21)"
(0.11)
(0.63)
(0.00)
4
14
-2.324
-1.390
3.347
-3.446
-2.458
0.012
(2.1 0)*t
(1.32)
(3.3Ø)*
(2.46)
(0.79)
(2.77)"
5+
10
-0.251
9.609
13 .027
-0.867
2.343
0.002
(0.13)
(0.09)
(0.12)
(0.50)
(0.21)
(0.31)
NOTE:
Asymptotic t ratios in parentheses.
"indcates coefficient different from
zero at .01
level,
indicates coefficient different from
zero at .05 level.
95
Housing Demand in the Short Run
The patterns of significance
suggest that the choices of
housing types for
the overwhelming proportion of rental
households (i.e. those
lower and
middle income rental households that,
for this sample,
comprise 85
percent of the rental market), are strongly
influenced by relative
prices. The
table also suggests a clear pattern in the
magnitude of the relative
price
coefficients for families of different sizes.
Within each income
class, the
magnitude of the price coefficient
increases with family
size. larger
families with greater demands for
necessities are more
responsive to
relative prices in their choices
among housing types.
The estimated coefficient of the
structure age variable has the
anticipated
sign in 29 of the 30 equations and is
highly Significant
in 22 of the
stratifications. Again, the t ratios
suggest that renters in the highest
income
class are least Sensitive to structure
age, but there does not
seem to be a
strong pattern in the magnitudes of the
estimated coefficients
across
income classes and family sizes.
The coefficients of the number of
bedrooffis indicate
a systematic pattern
across income classes and family sizes. The
coefficients are statistically
significant with the correct sign in 1 9 of
the stratifications. For
each of the
six income classes, the magnitude of
the coefficient
on the bedroom
variable increases with family size.
There is also a tendency
for the
coefficient to increase with income level
for a given family size.
The coefficient of the variable for
commonwall units is statistically
significant
in 20 of the 30 equations; the coefficient
of the variable
representing apartment units is significant in 15
stratifications. The pattern
of coefficients suggests, ceteris paribus, that
single detached units are
preferred to either of these types of families
with three or more members.
Holding family size constant, the coefficients
also indicate that single
detached rental units
are preferred by those of higher incomes.
In general, the model performs less well for
renters of the highest income
class, those earning
more than $15,000 a year. In part, this may be a
reflection of the smaller sample sizes for
households in this category.
However, the results may also
suggest that the definitions of the housing
types are inadequate to model the behavior
of the highest income group;
the aspects of housing which
motivate the residential location and housing
choices of the highest income households
are not well represented by only
18 types of residential housing.
Tables 2 and 3 illustrate the differences
in housing consumption attribut-
able, ceteris paribus,
to variations in the socioeconomic characteristics of
households. The tables indicate the predicted
probabilities of consuming
several housing characteristics
using the coefficients of Table 1 and assum-
ing each household
in the metropolitan area faced the same effective
housing prices (P). These
probability estimates may be interpreted as
those observed under
the monocentric or equilibrium assumptions of the
Predicted Probabilities of Housing Type Choice, p,1,,
TAULE 2
for Selected Incomes Across Family Sizes:
=
where
I'J =
h,,
!i
,
Family Size
1
Type of Dwelling
2
4
Income $3000 -$4,999
Common-'alI units
.11 .11
.39
.69
Apartments
.84 .82
.38
.27
.06
Single detached
.04 .07
.25
.34
.25
One bedroom
.78 .79
.35
.12
.o
Two bedrooms .20 .20
.53
.42
.48
Three bedrooms .02 .01
.12
.46
.49
Income $5,000-$6,999
Common-wall units
.03 .22
.54
.52
.52
Apartments
.94
.60
.23
.11
.oa
Single detached
.02
.17
.24
.37
.40
One bedroom
.95
.66
.23
.10
.03
Two bedrooms
.05
.28
.33
.61
.61
Three bedrooms
.00
.06
.16
.29
.64
classical theory. If all households were employed at
a single work site they
would, of course, face identical effective prices for the
same type of
residentia! housing.17 The probability estimates
were obtained by substitu-
tion into equations 11 and 14 and by then forming the marginal totals.
Table 2 illustrates the probabilities for two income classes
over the five
family size categories. The table indicates that,
as family size increases,
households are less likely to choose multifamily
units and are more likely
to choose common-wall units and single detached
units, For income levels
of $3,000 to $5,000, the probability
of choosing apartment dwellings
declines from .84 for
one-person households to .06 for five-person house-
holds; the probability of choosing
common-wall units increases from .1 Ito
.69. Similarly the probability of choosing
single detached units increases
from .04 to .25
as family size increases from one to five members.
For a higher level of income, the
table indicates (hat larger family sizes
also systematically choose less
dense housing configurations. In contrast to
the lower income
group, households earning between $5,000 to $7,000
have higher probabilities
of consuming larger effective lot sizes at each
TABLE 3
Predicted Probabilities
of Housing
Type Choice,
p,
for Selected Family
Sizes Across
Income Classes.
where
W11,(X1, P) =
b,,1
+
I
Less than $3,000-
$5,000-
7 flflfl-
TypeolDwelling
$3,000
$4999
$6,g9
$9,999
$14,999 $15,000+
Three-person Families
Common-wall units
.54
37
54
.36
Apartments .01
.24
.38
.23
.23
.32
Singledetached
.21
.82
.25
.24
.18
.32
One bedroom
.18
.18
.35
.23
.18
iwo bedrooms
.00
.64
.53
.61
.62
.39
Threebedrooms
.18
.03
.12
.16
.20
.47
Five-or-more-person Families
Common-wall units
1.00
.69
.52
.46
.49
Apartments
.37
.00
.06
.08
.13
Singledetachd
.00
.25
.02
.13
.40
.41
.49
.50
Onebedroom
1.00
.03
.03
.01
.00
.00
Two bedrooms
.00
.48
.33
.24
.19
Three bedrooms
.00
.49
.64
.03
.75
.80
.97
family size. Holding
family size constant,
higher income households
systematically choose less dense
types of residential housing.
Within each income
class, Table 2 indicates that
increased family size is
associated with the choice of
housing types with larger
interior sizes (as
measured by numbers of bedrooms).
However, the comparison for the
two
income classes reveals that
at the same family size, higher
income house-
holds are generally
more likely to choose two and three bedroom
units
than lower income
households.
Table 3 illustrates the
differences in housing consumption
across the six
income classes for
two stratifications of family size. The
table indicates that
for three-person
families, the probabilities of
consuming units with larger
interiors are very similar
for households earning less than
$7,000 a year.
(The predicted
average numbers of bedrooms
are 2.0, 1.8, 1.9 and 2.0,
respectively for the lowest four
income classes.) Only for the two highest
income classes, where
the predicted average number of bedrooms
in-
creases to 2.3 and 3.0
respectively, does higher income increase the
likelihood of choosing
housing types with larger interior sizes.
FIGURE 3
Frequency Distribution of
Relative Prices for
Two lypes
of Housing Facing
Households Earning
Less Than $3,000
a Year
25
Effective price of old,
I bedroom common
wall units
20
Effective price of old,
r---i3 bedroom common wall units
-
II
I
r_-h
I_______
o
L___
0.6 07
.8
0.9
1.0
I
U
1.2
1.3
1.4
1.5
Relative price, P
NOTE:
Both prices are relative to the effective price of
new one bedroom, common-
wall units.
earning between $5,000 to $7,000
a year, the probability of choosing
apartments declines systematically for four-person households
whose work
place is more distant from the Central Business
District. Similarly, the
probability of choosing single detached housing
types increases systemati-
cally for tour-person households with less central
work places. For five-
person households. The same regular pattern of structure-type choices
is
revealed for the four work places. For larger
households at the same
income, however, those employed
at noncentral places are more likely to
choose larger effective lot sizes than
smaller households.
For five-person households of
a lower income class, the probability of
choosing single detached housing increases with less
central employment
locations, but the probabilities
are substantially lower than for households
with larger incomes. The probability of choosing
common-wall units
similarly declines at noncentral employment
sites, hut the probabilities are
uniformly higher than for households with larger incomes.
Even at the same family size, variations in the effective prices affect
households' choices of the interior size of units. For four-person families,
there is a small but
systematic increase in the probability of choosing
housing types with
more bedrooms at less central work sites. For five-
person households of both income classes this tendency is more pro-
nounced.
TABLE 4
Predicted Probabilities of Housing Type Choice,
for Otherwise Identical Households at Four Work Sites
Work Places
Inner
Central
Type of Dwelling
CBD
City
City
Suburbs
Four-person Families--Income $5000-$6
999
Common-wall units
.51
.54
.50
.42
Apartments
.40
.29
.19
.11
Single detached
.09
.17
.30
47
Onebedroom
.16
.13
.13
.14
Two bedrooms
.63
.63
.61
Three bedrooms
.21
.23
.26
.28
Five-person Families-Income
$5,000-$6 999
Common-wall units
.58
.51
.36
.19
Apartments
.28
.15
.07
.02
Single detached
.15
.33
.53
78
One bedroom
.05
.05
.06
.07
Two l)edrooms
.46
.43
.37
.33
Three bedrooms
.49
.53
.57
.60
Five-person Families-Income
$3,000-$4,999
Common-wall units
.74
.69
.60
Apartments
.14
.10
.06
.48
.04
Single detached
.i 2
.21
.33
.48
One bedroom
.57
.04
.05
.06
Two bedrooms
.39
.55
.52
.47
Three bedrooms
.09
.41
.43
.47
Table 4 clearly shows how
variations in the intrametropolitan
costs of
configurations of residential housing
affect households' choices
of consum-
ing several attributes of
the residential housing
"bundle".
The theory of the housing
market and the
computation of the effective
prices of housing units used
in the empirical analysis
suggest that these
price variations arise because:
existing housing units
are costly to transform
and the spatial distribution
of housing types changes
slowly in response to
market forces; households
employed at different sites face
different acces-
sibility costs to the
available supplies of durable
housing units. By neglect-
ing these considerations
many analyses of household location
and de-
mand for "housing" have
overlooked a crucial link
in understanding why
households choose
particular spatial locations
and why households choose
components of the bund!e of housing
services.
101
Housing Demand in the Short Run
NOTES
Aggregate studies whi&h neglect housing prices
in focusing
on uic.ome expenditures
include: Margaret Reid, Housing and Income
(Chicago: Universits
of Chicago Press
1962); Alan R. \'\'inger, 'Housing and Income,"
pp. 226-232. Muth's study includes an index of
Western Economic
Journal June 1968,
construction costs (tli
across cities, and de Leeuw's intercity analysis
Boeckh index)
uses the Bureau of Labor
Stjjk-5
city-worker budget to provide an
average price for a
'standard" bundle of housing
services. See Richard F. Moth, "The Demand for
Non-farm Housing," in
The Demand
for Durable Goods, Arnold C. Harberger,
ed. (Chicago: University
of Chicago Press,
1962); Frank do Leeuw and Nkanta F. Ekanern,
"The Demand for
Housing: A Reviexx' of
the Cross-Section Evidence," Review of
Economics and Statistics,
February 1971,
pp.
1-10.
See Mah!on R. Siraszheim, "Estimation of the
Demand for Urban Housing
Services from
Household Interview Dat,i," Revieis of
Economics and Statistics
February i
pp.
1-8; Mahlon R. Straszheim, An Econometric
Analysis of the Urban Housing
Market
(New York: National Bureau of Economic
Research, 1975); John F,
Kain and John M.
Quiglev, Housing Markets ann Racial
Discrimination: A MicIOcConornic
Analysis (New
York: National Bureau of Economic Research,
1975); John M. Quigey,
"Racial Dis-
crinhination and the Housing Consumption
of Black Households," in
Patterns of Racial
Ljiscrirriin,stion
Vol.
1: Housing, George M. Von
Furstenberg ed. (Lexington,
Mass.:
D.C. Heath, 1974); A. Thomas King,
"Households in Housing Markets' The
Demand for
Housing Components" (College Park, Md.:
Bureau of Business and
Economic Research,
University of Maryland, 1973).
The classic references include: Richard F.
Muth, Cities and Horning (Chicago:
Univer-
sity
of Chicago
Press,
1969); Lowdon Wingo,
Transportation and Urban Land
(Washington, DC,: Resources for the Future,
1961); William Alonso, Location and
Land
Use (Cambridge: Harvard University Press,
1964).
James L. Ssveeney, "Quality, Commodity,
Hierarchies, and Housing Markets," Stanford
University, Department of Engineering.Economic
Systems, mimeographed, October
1972.
These estimates of the implicit prices of housing
attributes are derived from Lancaster's
analysis of hedonic goods. See Kelvin
J.
Lancaster, "A New Approach to Consumer
Theory," Journal of Political Economy,
April 1966, PP
132-156; Sherwin Rosen,
"Hedonic Prices and Implicit Markets: Product
Differentiation in Pure Competition,"
Journal of Political Economy, January/February
1974, pp. 34-55. For a recent survey of
this literatwe as related to housing markets
see Michael J. Ball, "Recent Empirical Work
on the Determinants of Relative House Prices," Urban
Studies, June 1973, PF) 213-233.
John F. Kain, "The Journey
to Woik as a Determinant of Residential Location," Papers
and Proceedings of the Regional Science
Association
1962, pp. 137-161.
7.
It may he that the! types of residential
housing form a "hierarchy" in the sense defined
by Sweeney, i.e. that
(NI)
UX11, z0) > U(X, z0)
tor all coilsurners. More generally,
since x is multidimensional, it is likely that only some
housing types are strictly hierarchical.
For example, it the components of x include
"housing quality" and "size," it
may be true that all consumers prefer higher quality to
lower quality units and larger dwelling
units to smaller units; consumers may have
mixed preferences, however, regarding the
tradeoff between larger, lower quality units
and smaller, higher quality
units.
a
I
'5
.5'
102
Iohn M. Quiglc'y
ft
H. Bluch arid 1. Marschak, ''Randwii Ordeiines and Stodistiu 11
Rvsponc"
(ontribut,n
to Probabtht', and Stati.stiis,
I. 01km. ed. (St,inlnrd: Stanford
University
Press, 1960).
Daniel McFadden. 'The Revealed Preferences of
,i Government Bureaucracy" 1'echq.
cal Report W. 17, Institute of International Studies, University of
California_Berkrlt.s
November 1968; Charles River Associates, "A Disaggregated
Behavioral Model of
Urban Travel Demand," Report CRA-156-2, March 1972; Daniel
McFadden "Conch.
tional LogO Analysis of Qualitative Choice Behavior,'' in Frontiers
in F(oflorne(rjCs
Zaremka, ed. (New York: Academic Press, 1974).
p.
Although this assumption (as well as (he assumed
error term distribution in equation 9)
is made solely in the interest of tractability, it is not quite
as restrictive as it may appear,
since a wide variety of functional forms may, in principle, be
accommodated by dummy
variabies ad piecewise linear approximations.
11
McFadden, 1968 (sec note 9).
Details concerning the survey instruments and the underlying
data may be found in John
M. Quigley, "Residential Location with Multiple Workplaces
and a IIeterugenc'ous
Housing Stock," Discussion Paper Numbe, 80, Program
on Regirniat ,mrt Urban
Economics, Harvard U niversity, September 1972.
For evidence on the relationship between housing
age and "objective nhc'asur'
of
housing quality," see John F. lOin and John M.
Quigley, "Evaluating the Quality of the
Residential Environment" Lnvironrrrent and P!annin,
Vol. 2, 1970,
p. 21-32
Cost estimates were obtained from John B.
Lansing aiid G. Hendricks, "How
People
Perceive the Cost of the Journey to Work," No.
197, Highway Research Board, 1967,
Pp. 44-55.
For the six income classes the (assumed) midpoints
and the associated hourly
wages
(based upon a 40 hour week for 50 weeks
per year) are:
income class
)
income mid-point
hourly wagss (Wy)
$
0-2,999
$ 2,500
$1.25
3,000-4,999
4,000
2.00
5,000-6999
6,000
3.00
7,000-9,999
4.24
10,000-14,999
t2,500
6.25
15,000-
17,500
8.75
Although the methodology
can be briefly stated, the calculation of the
effective prices
involved estimating the entire surface
of total housing costs lacing each
household for
each type of housing and
"scanning" each surface to find the
average price of the
cheapest fise percent of the stock of
each type. For each household,
its work place and
income class thence
an estimate of its wage rate)
are sufficient to calculate the
accessibility cost of each residential
location. Knowledge of this
cost plus the estimate of
contract prices at each residential location
for each housing type allowed
a surface of
total housing Costs to be defined
for each type of housing. For
each type of housing, the
prices and the number of
units at each residential location
were scanned to estimate the
average total cost of the cheapest five
percent of the stock when viewed from the work
place of each household
at its wage rate. The price of
one bedroom common.wall units
built after 1930
was used as the numerajre
Alternatively, if several work
places existed and the markets
for each type of residential
housing were in equilibrium,
differences in the effective prices
facing similar households
could arise only if
wages for identical labor inputs
varied by ss'ork place.