Racism and urban structure Indonesia
JOURNAL
OF URBAN
ECONOMICS
2, 85-103 (1975)
Racism and Urban Structure
I
SUSAN ROSE-ACKERMAN
Institution for Social and Policy Studies and Department of Economics,
Yale University, New Haven, Connecticut 06520
Received October 12. 1973
This paper begins the task of integrating models of racist behavior into general
theories of urban land use. The paper derives equilibrium prices for a racist city
and demonstrates that the city is less dense at the core, more dense in the suburbs,
and covers a larger area than an unprejudiced city. A more complex theory of
housing supply is then developed, and it is shown that racism’s impact on the
ghetto depends upon the ease with which maintenance can be reduced, the cost
of replacing abandoned housing, and the nature of legal controls on housing
quality.
I. INTRODUCTION
With few exceptions, economic analysesof racial discrimination in housing
markets have not been systematically integrated into a theory of urban structure and land use.2Indeed, much of the literature assumeswithout sustained
analysis that blacks pay more for housing becauseprejudice restricts supply
and simply attempts to estimate the size of the price mark-up. Richard Muth
[ 12, pp. 106-l 121, following a suggestion of Martin Bailey [2], has made
one of the few attempts to develop a logically consistent model of the demand
side of a racist housing market.3 Using this model, Muth argues that racial
prejudice lowers the price of housing available to blacks. Muth, however,
fails to consider explicitly how suppliers respond to racial prejudice and hence
ignores racism’s broader consequencesupon the distribution and density of
the urban population. The first half of this paper (Sections II-IV) formalizes
Muth’s theory of racist behavior in the context of a general urban housing
1This paper was financed by NSF Grant No. GS-39,671. The author thanks Bruce
Ackerman, Edwin Mills, Richard Muth and Robert Pollak for helpful comments on an
earlier draft.
2The exceptions stress the importance of housing market discrimination in restricting
the employment choices of blacks. See [5] and [B].
3 Muth’s theory of racial discrimination is not the only one which is plausible. Other
theoretical attempts that take a different perspective include [3], [7], [15], [16], and [17].
For empirical work that is closely related to Muth’s model see [9] and [lo].
85
Copyright 0 1975 by Academic Press. Inc.
All rights of reproduction in any form reserved.
86
SUSAN ROSE ACKERMAN
model of the type commonly used in recent work. After showing how the
price-distance function for housing is affected by racial prejudice, the paper
demonstrates that a prejudiced city is less dense at the core, more dense in
the suburbs and covers a larger area than its unbigoted counterpart. In the
second half of the paper (Sections V and VI), a model is developed which
takes into account the fact that most of the housing stock consists of old
units depreciating over time. This model makes it possible to refine the analysis of supplier behavior and to discuss the impact of racism on the density,
quality and price levels of housing in the ghetto. Two variants of the model
are compared-one in which maintenance can be varied and one in which a
housing code determines the maintenance inputs required. It is shown that
when the law requires suppliers to maintain a minimum level of housing
services, blacks gain more from prejudice than when suppliers are free to
adjust maintenance levels or abandon properties.
II. THE MUTH
FRAMEWORK
Muth’s model (like all other theoretical discussions of racial prejudice)
assumes that bigotry per se has no impact upon the supply of housing. His
demand oriented model focuses upon two types of people: whites who have
an aversion to living near blacks and blacks who do not care where they live.4
Thus, whites bid more than blacks for housing in predominantly white areas,
leading to two racially distinct neighborhoods with a common border. Except
along the black-white border the black neighborhood is assumed to be
surrounded by land that cannot be used for housing. If quality adjusted
housing prices are initially equal in the interior of the white and black neighborhoods, blacks can outbid whites along the border. If landlords are profit
maximizers, and if there are no costs of converting housing from white to
black use and no collusion, housing changes hands and the border moves
away from the center of the black neighborhood until the prices that blacks
and whites are willing to pay are equal at the border. In equilibrium, when
prices are equal at the border, white interior prices must be higher than white
border prices since whites prefer to be surrounded by households of their
own race. It follows that rents in the black neighborhood are lower than in the
white interior, and lower also than equilibrium prices in the absence of
prejudice.
III. A MODEL
OF THE HOUSING MARKET
NO RACIAL PREJUDICE
WITH
In the present section, a simple model of the housing market in an unprejudiced city is developed to provide the groundwork for introducing
racism in Section IV.
4One could also assume, as Muth does [12, p. 1081, that blacks prefer to live next to
whites, but this assumption is not necessary if the black market area is restricted.
RACISM AND URBAN
STRUCTURE
87
A. Demand
The simple model explicated here uses some standard assumptions employed by many workers in the field and exemplified in [12]. Thus it is assumed that a household’s utility level depends upon its consumption of two
mutually exclusive groups of commodities-housing and all other goods
except location; that the prices of other goods do not vary by location; that
the variable cost of transport increasesat a nonincreasing rate with distance
from the central business district (CBD); that each household makes a fixed
number of trips to and from the CBD per unit of time; that transport costs
are the samein all directions, and that all jobs are located in the CBD within
a small circle of radius v miles.5 If housing services per unit of time can be
represented by a single variable, q, that is a weighted average of housing
attributesfi then the individual household chooses its residential location by
maximizing its utility function U = L/(x, q) subject to the constraint’
x + p(k)q + w, y) - y = 0.
where
= spending per unit of time on all commodities except housing and
transportation but including leisure.
= consumption of housing per unit of time.
4
k
= distance from V, the boundary of the CBD. Transportation is
assumedto be costlesswithin the CBD.
p(k) = price per unit of housing at k.
= cost of travel to and from the CBD per unit of time, Tk > 0,
T
(T includes both out-of-pocket costs and the opportunity cost
of time).
Y = income per unit of time.
The maximum occurs where
X
Uz = U,/p(k)
(1)
and
- pk = Tn or pk = (- l/q)T,
< 0.
(2)
In order for an equilibrium to exist, the price of housing must fall with distance, and stability requires that -qpk fall more rapidly then Tk. Otherwise,
households could benefit from moving either closer to or further from the
city [12, pp. 24-251.
6 Muth [12, pp. 17-211 drops some of these simplifying assumptions in later portions of
his book, but these modifications will not concern us here.
6This formulation follows Mills [ll, p. 2011 and Muth [12, pp. 18-191.
7 Muth [12, pp. 21-221. The following paragraph reproduces Muth’s results for readers
unfamiliar with his work. Muth’s formulation differs from the slightly more complex model
of Alonso [l] where distance, k, enters the utility function directly.
SUSAN
88
ROSE~ACKERMAN
The supply side is also derived from Muth [12, pp. 47-491 and is a simple
competitive market in which all housing inputs are variable. Suppliers at
each k combine three inputs: land, L, structural inputs, S, and maintenance
inputs, M, in order to produce housing. Housing is not divided into discrete
units, so that supplier i located at distance k can costlessly subdivide the
total @(k) he supplies to provide any q(k) desired by an individual household.
All land beyond v is zoned for residential or agricultural use. Since there are
no alternative usesfor land beyond v except farming, profits per unit of time
for producer i at k are
z-i(k) = p(k)Qi(k)
- r@&(k)
- s&(k) - mMi(k),
(3)
so long as r(k) > P
where
Qi(k) = Q&W), S(k), M(W),
r(k) = land rent per acre at k,
p
= maximum rent per acre obtainable from agriculture,
s
= price of structural inputs, and
m
= price of maintenance inputs.
The prices ?, s, and m are independent of k. Producers maximize profits by
equating the price of each factor to the value of its marginal product. Hence,
r(k) = Ak)QL,,
(da)
m = p(k)QM,,
(4b)
s = p(k)Qsi.
(4c)
Where QL~, QM~and Q,si are the marginal products to the ith producer of
L, M, and S, respectively. In a competitive market profits must be equal for
producers at all k, a fact which, as Muth has shown in a similar model
[12, p. 491, makes it possible to specify the relationship between land rent
and housing price as
rk/r = (dPL)(Pk/p),
C. Market Equilibrium:
where pi = land’s share.
(5)
Incomes Equal
Market equilibrium occurs when total demand for housing equals the
total supply. This equilibrium can be specified if strong assumptions are made
about consumers. If the number of workers is exogenous, if every household
contains a single worker, and if all households have equal incomes and identical tastes, then the equilibrium price at each k must be one such that households are indifferent between locations. If pi(k) is the pattern of prices which
RACISM
AND URBAN
STRUCTURE
89
would prevail if consumers attain utility level Uj, then the housing demanded
by each household at k, qj(k), is uniquely determined. Every household has
a family of bid-rent lines pi(k) with pk < 0, each corresponding to a different
level of utility. (The higher is pj(k) the lower is utility).8
The supply of housing at k when pj(k) representsthe market price is
where Qij(k) is the quantity supplied by the ith producer at distance kgiven
pi(k), and nk is the number of producers at k. The maximum number of
households that can be accommodatedat any k is then, Ni(k) = (Qj(k)/qj(k)).
Assuming that i& is the distance at which rj(L) = P when pi(k) holds, the
maximum number of households that can be accommodated for somepi(k) is
Lj
Nj =
s0
Nj(k)dk.
Equilibrium in the housing market occurs for pi(k) such that H = Nj where
H = number of households. If H > Nj, there is excess demand and pj(k)
must rise and conversely, for H < Nj.
IV. RACIAL PREJUDICE IN THE HOUSING MARKET
A. The Location of Blacks in a City Without Prejudice
In order to introduce racial prejudice into the model it is necessary to
specify the behavior of black households in a city with unprejudiced whites.
The notion of a “city without prejudice” is, however, ambiguous. On the one
hand, if no prejudice has ever existed in any area in life, and if blacks and
whites are equally productive, membersof both raceshave equal incomes, and
so long as they have equivalent tastes, blacks are distributed randomly
throughout the housing stock. On the other hand, if blacks are discriminated
against in education and in jobs but not in the housing market, they have
lower average incomes and hence trade off accessibility and housing differently from whites.
The impact of introducing prejudice in the housing market differs in each
case. In the first case since black income levels equal white levels, Muth’s
analysis implies that blacks occupy a self-contained section of the city which
could be either a pie shaped wedge or an annulus. Blacks, however, pay
lower rents at each distance than whites living at the samedistance but located
in the white interior. Although blacks are indifferent to the location of their
8The best discussion of bid-rent lines is found in Alonso [l].
90
SUSAN ROSGACKERMAN
self-contained neighborhood, whites prefer to have blacks living in a ring
around the CBD in order to minimize the length of the black-white border.
In the second case, even if the tastes of blacks and whites are identical,
blacks have different bid-rent functions from whites because of the blacks’
lower income. Empirical work indicates that the income elasticity of the
demand for housing is greater than one9 which, under plausible assumptions,
implies that the bid-rent functions of the wealthy are flatter (i.e., the absolute
value of the slope is lower) than those of the poor [12. p. 291. As a consequence, the blacks, because of their relative poverty, would live closer to the
center of the city than the whites. Hence blacks occupy a compact section of
the city even if no prejudice exists in the housing market. When white prejudice is introduced, the black neighborhood includes additional housing at
the border between the black and white neighborhoods. In this paper it is
assumed that blacks would live in the city center even if no prejudice existed
in the housing market. Such an assumption is reasonable since average black
income levels are lower than those of whites, with the difference plausibly
attributed to discrimination in education and jobs.
B. Market Equilibrium with Lower Class Blacks
(1) The unprejudiced city. In an unprejudiced city where low income blacks
occupy the city center, the equilibrium pattern of prices can be determined in
a simple manner. Given any bid-rent curve for blacks, pj”(k), with corresponding household demands, q,R(k), and housing supplies, Qj(k), we have:
Nj”(k) = Q,(k)/qJR(k), where N,R(k) is the number of black households
that can be accommodated at k. Given H”, the total number of black households, the edge of the black neighborhood is at bj (where bj is measured in
miles from the border of the CBD) such that
bi
H” =
NiB(k)dk.
J’0
Let pjW(k) be the white bid-rent function that passes through [bj, pjB(bj)].
This bid-rent function implies a particular qJU3(k),Qj(k) and Nj”(k) = Qj(k)/’
q”(k) for each k > bj. Because of higher white incomes, white demand is
greater at any price than black demand, and therefore density of settlement
falls discontinuously at the black-white border. Letting
ii
Njw =
.I’ bj
and P
Njlli(k)dk,
= number of white households, the bid-rent lines pjB(k), pj”(k)
are
9 Muth [ 131 and Reid [ 141 find income elasticities between one and two. Not all wealthy
households, however, have a high income elasticity of demand. See [4].
RACISM AND URBAN
STRUCTURE
91
the equilibrium price lines if NjzL’= Hw. If Njw > Hw, excesssupply exists
and p”(k) and pB(k) must fall (b must rise) to establish equilibrium. Conversely, if Njw < Hw excessdemand exists and prices must rise (b will fall).
(2) The racist city. If white households are prejudiced, they are both attracted to and repelled from the CBD. Although the utility function for blacks
has the same form as that assumed above, UB = .P(x, q), whites’ utility
functions include the distance to the black-white border as an argument:
Uw = Uw(x, q, k - b). The whites’ budget constraint is
x + p(klq + T(k, v”) - Y” = 0.
Maximizing Uw subject to the budget constraint yields
uz
=
uq/P
=
Uk-b/(pkq
+
(6)
Tk),
or
Tdq +
(7)
Since U&-b > 0 and U, > 0, prices in the white neighborhood fall more
slowly than in the unprejudiced city and may even rise with k over a certain
range beyond the black-white border. Ifpk > 0 at b, then white prices increase
with k until Tk = uk-b/uz. Beyond this point they fall with k, approaching
the rate of change of white housing prices in an unprejudiced city so long as
the marginal utility of moving an extra mile away from the black-white
border falls as k increases.If p?(k), pow(k), and b. in Fig. 1are the equilibrium
price lines and border with no prejudice and if poR(k) is the price schedule
for prejudiced whites such that at bo, poR(bo) = poB(bo) = pow(bo), then
NoR > Hw i.e., there is excesssupply ifpoR(k) holds, and equilibrium can only
be established through a fall in pB(k) and pR(k) and an increasein b to accommodate the increased black demand. The equilibrium white bid-rent function,
pk
I
=
-
[Uk-b/Uz]/q.
I
k
b.,
FIGIJRE
1.
SUSAN ROSE- ACKERMAN
92
p ik,
FIGURE 2.
plR(k) in Fig. 2, cannot, however, be below p,,“(k) for all k or elseNIR < Hw.
Therefore, plR(k) must cross pow(k) at some point, kr, and racist whites pay
lower prices in border regions than whites in a city without prejudice, but pay
higher prices in outlying areas. The price difference between a racist and a
nonracist city is larger the more intense is white prejudice and the more
slowly Uk-,, falls with distance from the border.
It follows that the area of a racist city is larger than that of a nonracist one
since the higher equilibrium level of housing prices beyond kl in a prejudiced
city implies that f is greater.‘OFurthermore density is lower in the ghetto of
a racist city and also in the white area between bl and kl, but is higher in
the white suburbs beyond kl, since prices are lower in the first two regions
and higher in the third. Density between b. and b1 is likely to be higher as
well because of the shift from high income white to lower income black
occupancy. However, the population could be lessconcentrated in this region
if blacks have a very high elasticity of demand for housing, or have incomes
close to white levels and if prices are much below those that would prevail
in an unprejudiced city.”
i” If the production function is Cobb-Douglas, the price of housing, p, where r(lr) = i is independent of the particular bid-rent line which prevails. Let Q,(k)=ao~,(k)Uls,(k)‘lM,(k)“3
where al + ap + a3 = I. Solving for the marginal products of L, S and M, and substituting
in 4a, b and c, gives expressions for p(k), S&r) and M,(k). Substituting for Si and Mi in
Qi and for Qi m p(k), we obtain
P(k) = (.,,~~:,::.~:~,,.,,,)r(k~l.
I* (5) implies that land rents follow the same pattern as housing prices. They rise if p”(k)
rises and fall if pH(k) falls. However, the land rent gradient is steeper than the housing price
gradient and is steeper the smaller is land’s share in producer’s revenues.
RACISM
AND URBAN
V. A MORE SOPHISTICATED
MODEL
93
STRUCTURE
OF THE HOUSING
MARKET
A. Introduction
The racist city was introduced in Section IV, using a simple model in which
all housing inputs are variable. At any point in time, however, most of the
structural characteristics of the housing stock are fixed by earlier construction
decisions.12 Thus owners of older housing are faced with the choice of varying
maintenance levels, demolishing their buildings, or abandoning their property.
If these facts are taken into account, the magnitude of the price benefits obtained by blacks depends upon the ease with which suppliers can react to
lower prices by spending less on maintenance or by abandoning unprofitable
units. To make this point precise, this section presents a model of the behavior
of owners of old housing in a city without prejudice. Prejudice is introduced
into the model in Section VI.
To develop the problem further, the operation of the housing market will
be considered under two hypotheses about supplier behavior. First, owners
are assumed to be legally permitted to reduce maintenance levels in any way
that maximizes profits. Second, the existence of a housing code is posited,
requiring landlords to maintain “quality” despite the aging of their housing
units. It is then possible to define the conditions under which a legal effort
to maintain housing quality redounds to the advantage of blacks. Throughout
the discussion an effort is made to show how consideration of the existing
housing stock modifies the earlier results using a more conventional supply
model.
B. Basic Assumptions
In order to contrast a housing code model with one in which maintenance
can be varied, a general framework is required for dealing with a housing
stock composed principally of old structures. In the model developed below
the age of dwelling units falls with distance from the CBD. The city is assumed
to consist of a central circle containing all jobs surrounded by a series of
rings, each ring containing housing built in a particular year. No vacant
land exists except in the urban fringe beyond some distance k*. There are
two groups of suppliers: landlords owning existing housing who (unless
prevented by law) choose the level of maintenance that maximizes profits, and
builders of new housing who operate beyond k*. All producers have CobbDouglas production functions for housing services in the three inputs:
I2 Long run equilibrium models such as those developed by Mills [I I] and Muth [12] do
not deal explicitly with the existing housing stock. The fullest current consideration of
existing supply is found in the urban simulation model in [6].
94
SUSAN
ROSE-ACKERMAN
L, S, and M. Output at time t for producer i at k is
Q,(k, t) = aoLi(k)ulS,(k)az(k,“Mi(k, t)as.
(8)
Producers of new housing at k > k* have identical constant returns to scale
(CRTS) production functions where & = 1 - aI - a:+ Owners of old housing have decreasing returns to scale production functions where al + az(k)
+ u3 < 1. The exponent on S(k), an(k), depends upon the age of the structure and falls as age increases.Since age and distance are inversely correlated,
this implies duZ/dk > 0. Producers of new housing maximize the discounted
present value of profits. Since L and S must be fixed when the housing is
built, if we assume that payments to L and S continue over time and that
builders expect all prices to remain constant, the present value of profits to
producer i at k is;13
cc
ri(k) =
p(k)Qi(k,t)eewldt - (rLi(k) + sSi(k)),!w
s0
m
Mi(k,t)e-wtdt, (9)
-m
i’ 0
where w = the interest rate.
C. The “Variable Maintenance” Model
1. Profit maximizing conditions. Equilibrium conditions can now be
specified for builders of new homes and for owners of older structures when
landlords are permitted to vary maintenance to maximize profits.
First, consider new construction. Builders beyond k* maximize (9) with
respect to (8). This maximum occurs where
Qi(k,t)eewtdt - *
= 0,
Wa)
W
u2(t)Q,(k,t)e-wtdt - ” = 0,
W
~&)Qi(k,t>
~i(k,t)
-
m = 0 for all t.
(lk)
Owners of old housing, however, need only consider returns in the current
time period if demolition and new construction are assumedto be impossible.
Since S and L are fixed, landlords maximize profits where (IOc) holds for
I3We ignoreherethequestionof the optimallife of a housingunit.
RACISM
AND
URBAN
STRUCTURE
95
t = 0. Equation (1Oc) implies that Mi(k) is an increasing function of p(k).
This can be proved by substituting for Qi(k) from (8) and solving for Mi(k):
l/a3
M,(k)
=-p(k)Li(k)"hsi(kp(")
1.
a3a1
[
(11)
m
The partial derivative of Mi(k) with respect to p(k) is obviously positive.
Having specified the landlord’s profit maximizing expenditure on maintenance, the returns to L can be determined given that the returns to S are
fixed at s&(k). If vigorous competition prevails throughout the housing
market so that profits are forced to zero on all housing, the returns to land are
r(k)L;(k)
= (1 - a&(k)Q@)
- LX(~).
(12)
2. The rate of change of land rent. Since land is not necessarily paid the
value of its marginal product, the relationship betweenland rents and housing
prices is more complicated than in the conventional model presented in
Section II.
From (12)
r(k) = (1 - a3)p(k‘e”o
- “sick>
L(k)
L(k) ’
Hence,
rk = (1 _ a3) pkG
[
I
+ p(k$“L’]
- $?!?.
dk
dk
And
rk
Pk
P dQiL
- = (1 - a: Qi(k), it is only necessary to consider the case in which the housing code
is binding. When no racial prejudice exists but blacks have lower incomes than
whites, the equilibrium bid-rent lines for the housing code model must be
below the equilibrium bid-rent lines when M is variable because &(k) > Q(k)
for all k < k* and hence, if prices were not lower under the housing code
model, supply would outstrip demand. l4
If, however, code enforcement generates abandonment, prices may be
higher under a housing code model, especially when the negative externalities
generated by vacant dwellings are taken into account. If housing adjoining
abandoned housing is judged by households to provide a fraction of the
housing services provided by other housing, then if housing at k - e is
abandoned, e a small number, the quantity of housing at k becomes A&(k),
0 < X < 1. If X is low enough i.e., if the externalities generated by vacant
units are high, then, in equilibrium, housing at k may be abandoned also
because the M(k) that would provide zero profits is less the A4 required to
produce &(k) units of housing.
Of course, if the cost of demolition is low enough, the externalities of
abandonment may be eliminated by new construction. This occurs if the r(k)
that satisfies
r(k)
wag(k)
= ~L,(k)
.=
Qt”(k,t)e-Wt
-
Dw
J o
is greater than or equal to zero.
I4 While black and white tenants will pay less for each unit of housing service, they may
pay more or less for each unit of srructural input provided (measured, for example, as rent
per square foot).
SUSAN
98
ROSE ACKERMAN
Regardless of whether housing codes generate higher or lower prices, they
induce a pattern of settlement that is different from that prevailing under the
variable supply model. When M is variable, blacks live in housing with higher
ratios of L and S to M relative to the ratios than would obtain with fixed Q.
Moreover, when code enforcement generates abandonment, regions close to
the CBD are vacant while blacks are densely settled in the adjoining sector in
old housing with relatively high levels of maintenance.
VI. THE IMPACT OF RACISM IN A CITY WITH AN
AGING HOUSING STOCK
A. Variable Supply
When landlords can vary M, the magnitude of the price benefits obtained
by blacks from racial prejudice depends upon the price elasticity of maintenance spending for older housing in ghetto and border regions. The more
landlords reduce maintenance in response to lower prices the less difference
there is between the equilibrium bid-rent lines for blacks with and without
racism.
The shape of the equilibrium bid-rent lines is the same as those under the
simple supply model (e.g., pP(k), plR(k) in Fig. 2). In the present case,
however, new construction only occurs between k* and L1. The ratios of L
and S to M are higher than those prevailing in the simple supply model, but
the basic result is analogous: less supply in the central section and more in
the suburbs. Speaking more loosely, under the “variable supply” model
racism leads to lower “quality” housing in the ghetto as well as lower prices
per unit of both Q and space.
B. Housing Codes
When housing codes require landlords either to maintain Q(k) at some fixed
level &(k) or to abandon their property, the price benefits of racism are smaller
the higher the cost of maintaining old housing to code levels, the greater the
expense of demolishing old units and replacing them with new construction,
and the larger the externalities from abandoned units. The shape of the
equilibrium bid-rent lines will be the same as pP(k), plR(k), but there may
be a zone of empty houses close to the CBD.
C. Variable Supply and Housing Codes Compared
While it is relatively easy to demonstrate that blacks pay less for housing
when racism exists in either the variable supply or housing code model, a
second issue requires a more sustained argument. This section specifies the
conditions under which housing code enforcement generates a larger welfare
RACISM AND URBAN
STRUCTURE
99
gain for black tenants than a free market regime under which landlords are
free to vary maintenance. The argument proceeds in three stages.First, it is
demonstrated that, when abandonment does not occur, equilibrium utility
levels under a housing code cannot be equivalent to those that prevail under
the “variable supply” model. Second, it is shown that, in the absence of
abandonment, blacks in a racist city are almost certain to be better off
under a regime of housing code enforcement while whites could be worse
off. Third, if abandonment with its accompanying externalities occurs, then
no definitive welfare statements are possible.
1. The nonequivalenceof equilibrium in the two models. Assume that plB(k)
and plR(k) in Fig. 3 are the equilibrium bid-rent lines under the “variable
supply” model. To prove that the utility levels representedby these price lines
cannot be the onesprevailing in equilibrium under a housing code model when
no abandonment occurs, this section assumesthat the contrary holds and
demonstrates that a contradiction exists.
Imagine, then, thatplB(k) is also the equilibrium bid-rent line in the housing
code model. Since abandonment does not occur, Ql(k) < &(k) for k < bl.
Therefore, with code enforcement, total black demand can be accommodated
between the border of the CBD and some b2 where bz < bl.
Consider now the white bid-rent line that passesthrough b2 which gives
whites the same level of utility as plR(k). This bid-rent line, pzR(k) in Fig. 3,
is above plR(k) for all k since at every k blacks are further away.
Given these bid-rent lines it is possible to establish the contradiction involved in assuming that plB(k) and p&k) establish equilibrium prices in
the housing code model. Beyond bl, supply to whites under the housing code
model is greater than under the variable supply model since the existence of
the housing code implies that &(k) 2 Ql(k) for k < k* and since the higher
prices beyond k* induce more new construction. White demand, however,
P k)
I
I
b,
b,
k’
FIGURE 3.
k,
i,
loo
SUSAN ROSE--ACKERMAN
is lower at every k > 61.“: Hence the total white population, H”‘, could be
accommodated between hl and some k < k,. Therefore, the contradiction
is established: either pi”(k) or p2R(k) or both must shift to establish equilibrium at some p,‘:(k), p,“(k) where am”
= plK(b,) and supply equals
demand.
2. Relative prices under the two models. It is now possible to specify conditions sufficient to guarantee that black tenants are better off under a housing
code than under a variable supply model provided that no abandonment
occurs. To do this it is only necessary to determine conditions sufficient for
pzR(bz) < plB(b2) (as shown in Fig. 3), since when this inequality holds,
equilibrium can only be established with a housing code at piB(k) < pP(k).
If only two assumptions are made, the desired outcome can be established.
First, assume that pBu‘ < pkB in a city without prejudice. Second, assume that,
when prejudice is introduced, the bid-rent functions of whites are “separable”
in the sense that they can be divided into a portion that depends on k - b and
a portion that depends on k, p”‘(k). The sufficiency of these conditions can be
shown by writing
pzR(bz) : plB(bz),
as
pgR(bz) - plR(bl) $ pP(bz) - ,oP(bl).
(16)
Since k - b is unchanged between bl and bS, we can write (16) as
pl”(bz) - pl”(bl)
: pP(bz) - pP(bl).
(17)
Since by hypothesis whites have higher incomes than blacks and have flatter
bid-rent functions in the absence of prejudice, this implies,
pl”‘(bz) - pl”‘(bl) < pP(bz) - plB(bl),
and thus,
pzR(bz) < plB@d.
(18)
The importance of this proof depends upon the plausibility of our two
conditions, especially the “separability” assumption. In fact pure “separability” does not obtain, although conditions can be defined in which it is
approached. From (7), the rate of change of white prices depends on a term
-Tkw/q(k), that equals pkW in the absence of prejudice plus a second term
(l/q(k))(U&U,)
that depends on k - b but is not independent of k.
The impact of changes in k on this second term, however, is smaller the lower
I5 We assumethroughout that given k and p(k), white households’ demand for q is
independent of the location of the black-white border although their overall satisfaction
level is lower the smaller is k-b. Furthermore, we assume that the impact of a housing code
on new construction is not substantial enough to revise the result in the text.
RACISM AND URBAN
101
STRUCTURE
k
b, h b.
FIGURE 4.
the elasticity of white housing demand and the smaller white households’
income elasticity of demand for x. Of course, even when separability is not
approximated, it is possible for (18) to hold. Ceterisparibus, thus occurs when
the difference in the slopes of the black and white bid-rent functions in the
absence of prejudice is relatively large.
The condition of the white tenancy must now also be considered. While in
a housing market without prejudice, both races gain from code enforcement,
once prejudice is introduced this result does not necessarily follow. Two
equilibrium cases are possible and are illustrated in Fig. 4 as 1 and II.16 While
whites benefit relative to a variable supply model in Case I, since prices are
lower and ba < bl, they do not necessarily gain in Case II for, although here
too whites gain from reduced prices, this may be cancelled by the fact that
the black ghetto is closer. (If whites are, in fact, worse off under Case II
than with a variable supply model, ~~fi(k) must intersect plR(k) at some
distant k). I7
3. Abandonment. If owners of old housing can choose either to provide &
or to abandon their property, then, as has been shown for the nonracist city,
the equilibrium price lines under a fixed supply model could be above or
below pP(k), plR(k). Blacks are therefore less likely to find a housing code
superior to a free market in which landlords can vary M.
I6 If ~@(6~)>~i~(b~), then a third case is possible in which equilibrium is established for
a pB(k) greater than plB(/r). In this situation blacks would be worse off and whites better
off under a fixed supply model than under a variable supply formulation.
I7 Under Case I density in the ghetto will be higher with a “housing code” model while
under Case II it will be lower. In both cases maintenance levels are necessarily higher at
every k although in Case I black families who would have lived in relatively new housing
between ba and bi are now accommodated in older housing at k _< bl.
102
SUSAN ROSE-ACKERMAN
VII. CONCLUSIONS
The limitations of the model developed here should be emphasized. The
paper deals only with equilibrium conditions and therefore does not contradict the empirical evidence indicating higher housing prices for blacks in
cities where disequilibrium conditions obtain as a result of a rapid black inmigration. Moreover, the model requires several assumptions which do not
necessarily hold in fact. First, the ghetto is assumedto be capable of expansion. It is not surrounded by nonresidential properties or by suburbs practicing exclusionary zoning. Second, suppliers are always willing to rent to the
highest bidder and are incapable of collusive activity. Finally, the fact that
many jobs are located in the suburbs far from concentrations of old housing
has been ignored.
Proceeding from this simplified model, the paper has developed Muth’s
theory of racist behavior by showing that the impact of prejudice on price,
quality and density dependsupon the nature of supplier response.The magnitude of black price gains from racism are smaller the higher the price elasticity
of maintenance spending for older housing in ghetto and border regions, the
higher the costs of demolition and new construction and the larger the externalities from abandoned or poorly maintained housing units. Nevertheless,
there does remain a basic asymmetry between the impact of prejudice in our
hypothetical urban housing market and discrimination in other areas, notably
the job market. While blacks benefit somewhat from white racism in housing,
they most certainly lose in the job market. Thus, the net impact of prejudice
upon blacks in the city hypothesized by Richard Muth may be either positive
or negative.
REFERENCES
1. W. Alonso, “Location and Land Use,” Harvard University Press, Cambridge, MA
(1964).
2. M. Bailey, Note on the economics of residential zoning and urban renewal, Larzd Econ.
35, 288-W (1959).
3. B. Cohen, Another theory of residential segregation, Land Econ. 47, 314-315 (1971).
4. M. Edel, Planning, market or welfare?-Recent land use conflict in American cities,
in “Readings in Urban Economics” (Edel and Rothenberg, Eds.), Macmillan,
New York (1972).
5. R. A. Haugen and A. J. Heins, A market separation theory of rent differentials in
metropolitan areas, Quart. J. Econ. 83, 660-673 (1969).
6. G. F. Ingram, J. F. Kain, and J. R. Ginn, with contributions by H. J. Brown, and
S. P. Dresch, “The Detroit Prototype of the N.B.E.R. Urban Simulation Model,”
National Bureau of Economic Research, New York (1972).
7. J. F. Kain, Effect of housing market segregation on urban development, Savings and
residential&awing:
1969 confhence proceedings,
U. S. Savings and Loan League.
8. J. F. Kain, The journey-to-work as a determinant of residential location, Papers of the
RegionaI Science Association 9, 137-160 (1972), Reprinted in “Urban Analysis,”
(Page and Seyfried, Eds.) Scott, Foresman. New York (1970).
RACISM
AND URBAN
STRUCTURE
103
9. A. T. King and P. Mieszkowski, Racial discrimination, segregation, and the price of
housing, J. Polit. Econ. 81, 590-606 (1973).
10. V. Lapham, Do Blacks Pay More for Housing ?, J. Polit. Econ. 79, 1244-1257 (1971).
11. E. S. Mills, An aggregative model of resource allocation in a metropolitan area,
American Economic Review, Papers and Proceedings 57, 197-210 (1967).
12. R. Muth, “Cities and Housing,” University of Chicago Press, Chicago (1969).
13. R. Muth, The Demand for Non-Farm Housing, in “The Demand for Durable Goods”
(Harberger, Ed,), University of Chicago Press, Chicago (1960).
14. M. Reid, “Housing and Income,” University of Chicago Press, Chicago (1962).
15. T. Schelling, “Models of Segregation,” Memorandum RM-6014-RC, The RAND
Corporation (1969).
16. T. Schelling, Neighborhood Tipping, in “Racial Discrimination in Economic Life,”
(Pascal, Ed.) D. C. Heath, Lexington, MA (1972).
17. E. Smolensky, S. Becker, and H. Molotch, The prisoner’s dilemma and ghetto expansion, Lund Econ. 44, 420 (1968).
OF URBAN
ECONOMICS
2, 85-103 (1975)
Racism and Urban Structure
I
SUSAN ROSE-ACKERMAN
Institution for Social and Policy Studies and Department of Economics,
Yale University, New Haven, Connecticut 06520
Received October 12. 1973
This paper begins the task of integrating models of racist behavior into general
theories of urban land use. The paper derives equilibrium prices for a racist city
and demonstrates that the city is less dense at the core, more dense in the suburbs,
and covers a larger area than an unprejudiced city. A more complex theory of
housing supply is then developed, and it is shown that racism’s impact on the
ghetto depends upon the ease with which maintenance can be reduced, the cost
of replacing abandoned housing, and the nature of legal controls on housing
quality.
I. INTRODUCTION
With few exceptions, economic analysesof racial discrimination in housing
markets have not been systematically integrated into a theory of urban structure and land use.2Indeed, much of the literature assumeswithout sustained
analysis that blacks pay more for housing becauseprejudice restricts supply
and simply attempts to estimate the size of the price mark-up. Richard Muth
[ 12, pp. 106-l 121, following a suggestion of Martin Bailey [2], has made
one of the few attempts to develop a logically consistent model of the demand
side of a racist housing market.3 Using this model, Muth argues that racial
prejudice lowers the price of housing available to blacks. Muth, however,
fails to consider explicitly how suppliers respond to racial prejudice and hence
ignores racism’s broader consequencesupon the distribution and density of
the urban population. The first half of this paper (Sections II-IV) formalizes
Muth’s theory of racist behavior in the context of a general urban housing
1This paper was financed by NSF Grant No. GS-39,671. The author thanks Bruce
Ackerman, Edwin Mills, Richard Muth and Robert Pollak for helpful comments on an
earlier draft.
2The exceptions stress the importance of housing market discrimination in restricting
the employment choices of blacks. See [5] and [B].
3 Muth’s theory of racial discrimination is not the only one which is plausible. Other
theoretical attempts that take a different perspective include [3], [7], [15], [16], and [17].
For empirical work that is closely related to Muth’s model see [9] and [lo].
85
Copyright 0 1975 by Academic Press. Inc.
All rights of reproduction in any form reserved.
86
SUSAN ROSE ACKERMAN
model of the type commonly used in recent work. After showing how the
price-distance function for housing is affected by racial prejudice, the paper
demonstrates that a prejudiced city is less dense at the core, more dense in
the suburbs and covers a larger area than its unbigoted counterpart. In the
second half of the paper (Sections V and VI), a model is developed which
takes into account the fact that most of the housing stock consists of old
units depreciating over time. This model makes it possible to refine the analysis of supplier behavior and to discuss the impact of racism on the density,
quality and price levels of housing in the ghetto. Two variants of the model
are compared-one in which maintenance can be varied and one in which a
housing code determines the maintenance inputs required. It is shown that
when the law requires suppliers to maintain a minimum level of housing
services, blacks gain more from prejudice than when suppliers are free to
adjust maintenance levels or abandon properties.
II. THE MUTH
FRAMEWORK
Muth’s model (like all other theoretical discussions of racial prejudice)
assumes that bigotry per se has no impact upon the supply of housing. His
demand oriented model focuses upon two types of people: whites who have
an aversion to living near blacks and blacks who do not care where they live.4
Thus, whites bid more than blacks for housing in predominantly white areas,
leading to two racially distinct neighborhoods with a common border. Except
along the black-white border the black neighborhood is assumed to be
surrounded by land that cannot be used for housing. If quality adjusted
housing prices are initially equal in the interior of the white and black neighborhoods, blacks can outbid whites along the border. If landlords are profit
maximizers, and if there are no costs of converting housing from white to
black use and no collusion, housing changes hands and the border moves
away from the center of the black neighborhood until the prices that blacks
and whites are willing to pay are equal at the border. In equilibrium, when
prices are equal at the border, white interior prices must be higher than white
border prices since whites prefer to be surrounded by households of their
own race. It follows that rents in the black neighborhood are lower than in the
white interior, and lower also than equilibrium prices in the absence of
prejudice.
III. A MODEL
OF THE HOUSING MARKET
NO RACIAL PREJUDICE
WITH
In the present section, a simple model of the housing market in an unprejudiced city is developed to provide the groundwork for introducing
racism in Section IV.
4One could also assume, as Muth does [12, p. 1081, that blacks prefer to live next to
whites, but this assumption is not necessary if the black market area is restricted.
RACISM AND URBAN
STRUCTURE
87
A. Demand
The simple model explicated here uses some standard assumptions employed by many workers in the field and exemplified in [12]. Thus it is assumed that a household’s utility level depends upon its consumption of two
mutually exclusive groups of commodities-housing and all other goods
except location; that the prices of other goods do not vary by location; that
the variable cost of transport increasesat a nonincreasing rate with distance
from the central business district (CBD); that each household makes a fixed
number of trips to and from the CBD per unit of time; that transport costs
are the samein all directions, and that all jobs are located in the CBD within
a small circle of radius v miles.5 If housing services per unit of time can be
represented by a single variable, q, that is a weighted average of housing
attributesfi then the individual household chooses its residential location by
maximizing its utility function U = L/(x, q) subject to the constraint’
x + p(k)q + w, y) - y = 0.
where
= spending per unit of time on all commodities except housing and
transportation but including leisure.
= consumption of housing per unit of time.
4
k
= distance from V, the boundary of the CBD. Transportation is
assumedto be costlesswithin the CBD.
p(k) = price per unit of housing at k.
= cost of travel to and from the CBD per unit of time, Tk > 0,
T
(T includes both out-of-pocket costs and the opportunity cost
of time).
Y = income per unit of time.
The maximum occurs where
X
Uz = U,/p(k)
(1)
and
- pk = Tn or pk = (- l/q)T,
< 0.
(2)
In order for an equilibrium to exist, the price of housing must fall with distance, and stability requires that -qpk fall more rapidly then Tk. Otherwise,
households could benefit from moving either closer to or further from the
city [12, pp. 24-251.
6 Muth [12, pp. 17-211 drops some of these simplifying assumptions in later portions of
his book, but these modifications will not concern us here.
6This formulation follows Mills [ll, p. 2011 and Muth [12, pp. 18-191.
7 Muth [12, pp. 21-221. The following paragraph reproduces Muth’s results for readers
unfamiliar with his work. Muth’s formulation differs from the slightly more complex model
of Alonso [l] where distance, k, enters the utility function directly.
SUSAN
88
ROSE~ACKERMAN
The supply side is also derived from Muth [12, pp. 47-491 and is a simple
competitive market in which all housing inputs are variable. Suppliers at
each k combine three inputs: land, L, structural inputs, S, and maintenance
inputs, M, in order to produce housing. Housing is not divided into discrete
units, so that supplier i located at distance k can costlessly subdivide the
total @(k) he supplies to provide any q(k) desired by an individual household.
All land beyond v is zoned for residential or agricultural use. Since there are
no alternative usesfor land beyond v except farming, profits per unit of time
for producer i at k are
z-i(k) = p(k)Qi(k)
- r@&(k)
- s&(k) - mMi(k),
(3)
so long as r(k) > P
where
Qi(k) = Q&W), S(k), M(W),
r(k) = land rent per acre at k,
p
= maximum rent per acre obtainable from agriculture,
s
= price of structural inputs, and
m
= price of maintenance inputs.
The prices ?, s, and m are independent of k. Producers maximize profits by
equating the price of each factor to the value of its marginal product. Hence,
r(k) = Ak)QL,,
(da)
m = p(k)QM,,
(4b)
s = p(k)Qsi.
(4c)
Where QL~, QM~and Q,si are the marginal products to the ith producer of
L, M, and S, respectively. In a competitive market profits must be equal for
producers at all k, a fact which, as Muth has shown in a similar model
[12, p. 491, makes it possible to specify the relationship between land rent
and housing price as
rk/r = (dPL)(Pk/p),
C. Market Equilibrium:
where pi = land’s share.
(5)
Incomes Equal
Market equilibrium occurs when total demand for housing equals the
total supply. This equilibrium can be specified if strong assumptions are made
about consumers. If the number of workers is exogenous, if every household
contains a single worker, and if all households have equal incomes and identical tastes, then the equilibrium price at each k must be one such that households are indifferent between locations. If pi(k) is the pattern of prices which
RACISM
AND URBAN
STRUCTURE
89
would prevail if consumers attain utility level Uj, then the housing demanded
by each household at k, qj(k), is uniquely determined. Every household has
a family of bid-rent lines pi(k) with pk < 0, each corresponding to a different
level of utility. (The higher is pj(k) the lower is utility).8
The supply of housing at k when pj(k) representsthe market price is
where Qij(k) is the quantity supplied by the ith producer at distance kgiven
pi(k), and nk is the number of producers at k. The maximum number of
households that can be accommodatedat any k is then, Ni(k) = (Qj(k)/qj(k)).
Assuming that i& is the distance at which rj(L) = P when pi(k) holds, the
maximum number of households that can be accommodated for somepi(k) is
Lj
Nj =
s0
Nj(k)dk.
Equilibrium in the housing market occurs for pi(k) such that H = Nj where
H = number of households. If H > Nj, there is excess demand and pj(k)
must rise and conversely, for H < Nj.
IV. RACIAL PREJUDICE IN THE HOUSING MARKET
A. The Location of Blacks in a City Without Prejudice
In order to introduce racial prejudice into the model it is necessary to
specify the behavior of black households in a city with unprejudiced whites.
The notion of a “city without prejudice” is, however, ambiguous. On the one
hand, if no prejudice has ever existed in any area in life, and if blacks and
whites are equally productive, membersof both raceshave equal incomes, and
so long as they have equivalent tastes, blacks are distributed randomly
throughout the housing stock. On the other hand, if blacks are discriminated
against in education and in jobs but not in the housing market, they have
lower average incomes and hence trade off accessibility and housing differently from whites.
The impact of introducing prejudice in the housing market differs in each
case. In the first case since black income levels equal white levels, Muth’s
analysis implies that blacks occupy a self-contained section of the city which
could be either a pie shaped wedge or an annulus. Blacks, however, pay
lower rents at each distance than whites living at the samedistance but located
in the white interior. Although blacks are indifferent to the location of their
8The best discussion of bid-rent lines is found in Alonso [l].
90
SUSAN ROSGACKERMAN
self-contained neighborhood, whites prefer to have blacks living in a ring
around the CBD in order to minimize the length of the black-white border.
In the second case, even if the tastes of blacks and whites are identical,
blacks have different bid-rent functions from whites because of the blacks’
lower income. Empirical work indicates that the income elasticity of the
demand for housing is greater than one9 which, under plausible assumptions,
implies that the bid-rent functions of the wealthy are flatter (i.e., the absolute
value of the slope is lower) than those of the poor [12. p. 291. As a consequence, the blacks, because of their relative poverty, would live closer to the
center of the city than the whites. Hence blacks occupy a compact section of
the city even if no prejudice exists in the housing market. When white prejudice is introduced, the black neighborhood includes additional housing at
the border between the black and white neighborhoods. In this paper it is
assumed that blacks would live in the city center even if no prejudice existed
in the housing market. Such an assumption is reasonable since average black
income levels are lower than those of whites, with the difference plausibly
attributed to discrimination in education and jobs.
B. Market Equilibrium with Lower Class Blacks
(1) The unprejudiced city. In an unprejudiced city where low income blacks
occupy the city center, the equilibrium pattern of prices can be determined in
a simple manner. Given any bid-rent curve for blacks, pj”(k), with corresponding household demands, q,R(k), and housing supplies, Qj(k), we have:
Nj”(k) = Q,(k)/qJR(k), where N,R(k) is the number of black households
that can be accommodated at k. Given H”, the total number of black households, the edge of the black neighborhood is at bj (where bj is measured in
miles from the border of the CBD) such that
bi
H” =
NiB(k)dk.
J’0
Let pjW(k) be the white bid-rent function that passes through [bj, pjB(bj)].
This bid-rent function implies a particular qJU3(k),Qj(k) and Nj”(k) = Qj(k)/’
q”(k) for each k > bj. Because of higher white incomes, white demand is
greater at any price than black demand, and therefore density of settlement
falls discontinuously at the black-white border. Letting
ii
Njw =
.I’ bj
and P
Njlli(k)dk,
= number of white households, the bid-rent lines pjB(k), pj”(k)
are
9 Muth [ 131 and Reid [ 141 find income elasticities between one and two. Not all wealthy
households, however, have a high income elasticity of demand. See [4].
RACISM AND URBAN
STRUCTURE
91
the equilibrium price lines if NjzL’= Hw. If Njw > Hw, excesssupply exists
and p”(k) and pB(k) must fall (b must rise) to establish equilibrium. Conversely, if Njw < Hw excessdemand exists and prices must rise (b will fall).
(2) The racist city. If white households are prejudiced, they are both attracted to and repelled from the CBD. Although the utility function for blacks
has the same form as that assumed above, UB = .P(x, q), whites’ utility
functions include the distance to the black-white border as an argument:
Uw = Uw(x, q, k - b). The whites’ budget constraint is
x + p(klq + T(k, v”) - Y” = 0.
Maximizing Uw subject to the budget constraint yields
uz
=
uq/P
=
Uk-b/(pkq
+
(6)
Tk),
or
Tdq +
(7)
Since U&-b > 0 and U, > 0, prices in the white neighborhood fall more
slowly than in the unprejudiced city and may even rise with k over a certain
range beyond the black-white border. Ifpk > 0 at b, then white prices increase
with k until Tk = uk-b/uz. Beyond this point they fall with k, approaching
the rate of change of white housing prices in an unprejudiced city so long as
the marginal utility of moving an extra mile away from the black-white
border falls as k increases.If p?(k), pow(k), and b. in Fig. 1are the equilibrium
price lines and border with no prejudice and if poR(k) is the price schedule
for prejudiced whites such that at bo, poR(bo) = poB(bo) = pow(bo), then
NoR > Hw i.e., there is excesssupply ifpoR(k) holds, and equilibrium can only
be established through a fall in pB(k) and pR(k) and an increasein b to accommodate the increased black demand. The equilibrium white bid-rent function,
pk
I
=
-
[Uk-b/Uz]/q.
I
k
b.,
FIGIJRE
1.
SUSAN ROSE- ACKERMAN
92
p ik,
FIGURE 2.
plR(k) in Fig. 2, cannot, however, be below p,,“(k) for all k or elseNIR < Hw.
Therefore, plR(k) must cross pow(k) at some point, kr, and racist whites pay
lower prices in border regions than whites in a city without prejudice, but pay
higher prices in outlying areas. The price difference between a racist and a
nonracist city is larger the more intense is white prejudice and the more
slowly Uk-,, falls with distance from the border.
It follows that the area of a racist city is larger than that of a nonracist one
since the higher equilibrium level of housing prices beyond kl in a prejudiced
city implies that f is greater.‘OFurthermore density is lower in the ghetto of
a racist city and also in the white area between bl and kl, but is higher in
the white suburbs beyond kl, since prices are lower in the first two regions
and higher in the third. Density between b. and b1 is likely to be higher as
well because of the shift from high income white to lower income black
occupancy. However, the population could be lessconcentrated in this region
if blacks have a very high elasticity of demand for housing, or have incomes
close to white levels and if prices are much below those that would prevail
in an unprejudiced city.”
i” If the production function is Cobb-Douglas, the price of housing, p, where r(lr) = i is independent of the particular bid-rent line which prevails. Let Q,(k)=ao~,(k)Uls,(k)‘lM,(k)“3
where al + ap + a3 = I. Solving for the marginal products of L, S and M, and substituting
in 4a, b and c, gives expressions for p(k), S&r) and M,(k). Substituting for Si and Mi in
Qi and for Qi m p(k), we obtain
P(k) = (.,,~~:,::.~:~,,.,,,)r(k~l.
I* (5) implies that land rents follow the same pattern as housing prices. They rise if p”(k)
rises and fall if pH(k) falls. However, the land rent gradient is steeper than the housing price
gradient and is steeper the smaller is land’s share in producer’s revenues.
RACISM
AND URBAN
V. A MORE SOPHISTICATED
MODEL
93
STRUCTURE
OF THE HOUSING
MARKET
A. Introduction
The racist city was introduced in Section IV, using a simple model in which
all housing inputs are variable. At any point in time, however, most of the
structural characteristics of the housing stock are fixed by earlier construction
decisions.12 Thus owners of older housing are faced with the choice of varying
maintenance levels, demolishing their buildings, or abandoning their property.
If these facts are taken into account, the magnitude of the price benefits obtained by blacks depends upon the ease with which suppliers can react to
lower prices by spending less on maintenance or by abandoning unprofitable
units. To make this point precise, this section presents a model of the behavior
of owners of old housing in a city without prejudice. Prejudice is introduced
into the model in Section VI.
To develop the problem further, the operation of the housing market will
be considered under two hypotheses about supplier behavior. First, owners
are assumed to be legally permitted to reduce maintenance levels in any way
that maximizes profits. Second, the existence of a housing code is posited,
requiring landlords to maintain “quality” despite the aging of their housing
units. It is then possible to define the conditions under which a legal effort
to maintain housing quality redounds to the advantage of blacks. Throughout
the discussion an effort is made to show how consideration of the existing
housing stock modifies the earlier results using a more conventional supply
model.
B. Basic Assumptions
In order to contrast a housing code model with one in which maintenance
can be varied, a general framework is required for dealing with a housing
stock composed principally of old structures. In the model developed below
the age of dwelling units falls with distance from the CBD. The city is assumed
to consist of a central circle containing all jobs surrounded by a series of
rings, each ring containing housing built in a particular year. No vacant
land exists except in the urban fringe beyond some distance k*. There are
two groups of suppliers: landlords owning existing housing who (unless
prevented by law) choose the level of maintenance that maximizes profits, and
builders of new housing who operate beyond k*. All producers have CobbDouglas production functions for housing services in the three inputs:
I2 Long run equilibrium models such as those developed by Mills [I I] and Muth [12] do
not deal explicitly with the existing housing stock. The fullest current consideration of
existing supply is found in the urban simulation model in [6].
94
SUSAN
ROSE-ACKERMAN
L, S, and M. Output at time t for producer i at k is
Q,(k, t) = aoLi(k)ulS,(k)az(k,“Mi(k, t)as.
(8)
Producers of new housing at k > k* have identical constant returns to scale
(CRTS) production functions where & = 1 - aI - a:+ Owners of old housing have decreasing returns to scale production functions where al + az(k)
+ u3 < 1. The exponent on S(k), an(k), depends upon the age of the structure and falls as age increases.Since age and distance are inversely correlated,
this implies duZ/dk > 0. Producers of new housing maximize the discounted
present value of profits. Since L and S must be fixed when the housing is
built, if we assume that payments to L and S continue over time and that
builders expect all prices to remain constant, the present value of profits to
producer i at k is;13
cc
ri(k) =
p(k)Qi(k,t)eewldt - (rLi(k) + sSi(k)),!w
s0
m
Mi(k,t)e-wtdt, (9)
-m
i’ 0
where w = the interest rate.
C. The “Variable Maintenance” Model
1. Profit maximizing conditions. Equilibrium conditions can now be
specified for builders of new homes and for owners of older structures when
landlords are permitted to vary maintenance to maximize profits.
First, consider new construction. Builders beyond k* maximize (9) with
respect to (8). This maximum occurs where
Qi(k,t)eewtdt - *
= 0,
Wa)
W
u2(t)Q,(k,t)e-wtdt - ” = 0,
W
~&)Qi(k,t>
~i(k,t)
-
m = 0 for all t.
(lk)
Owners of old housing, however, need only consider returns in the current
time period if demolition and new construction are assumedto be impossible.
Since S and L are fixed, landlords maximize profits where (IOc) holds for
I3We ignoreherethequestionof the optimallife of a housingunit.
RACISM
AND
URBAN
STRUCTURE
95
t = 0. Equation (1Oc) implies that Mi(k) is an increasing function of p(k).
This can be proved by substituting for Qi(k) from (8) and solving for Mi(k):
l/a3
M,(k)
=-p(k)Li(k)"hsi(kp(")
1.
a3a1
[
(11)
m
The partial derivative of Mi(k) with respect to p(k) is obviously positive.
Having specified the landlord’s profit maximizing expenditure on maintenance, the returns to L can be determined given that the returns to S are
fixed at s&(k). If vigorous competition prevails throughout the housing
market so that profits are forced to zero on all housing, the returns to land are
r(k)L;(k)
= (1 - a&(k)Q@)
- LX(~).
(12)
2. The rate of change of land rent. Since land is not necessarily paid the
value of its marginal product, the relationship betweenland rents and housing
prices is more complicated than in the conventional model presented in
Section II.
From (12)
r(k) = (1 - a3)p(k‘e”o
- “sick>
L(k)
L(k) ’
Hence,
rk = (1 _ a3) pkG
[
I
+ p(k$“L’]
- $?!?.
dk
dk
And
rk
Pk
P dQiL
- = (1 - a: Qi(k), it is only necessary to consider the case in which the housing code
is binding. When no racial prejudice exists but blacks have lower incomes than
whites, the equilibrium bid-rent lines for the housing code model must be
below the equilibrium bid-rent lines when M is variable because &(k) > Q(k)
for all k < k* and hence, if prices were not lower under the housing code
model, supply would outstrip demand. l4
If, however, code enforcement generates abandonment, prices may be
higher under a housing code model, especially when the negative externalities
generated by vacant dwellings are taken into account. If housing adjoining
abandoned housing is judged by households to provide a fraction of the
housing services provided by other housing, then if housing at k - e is
abandoned, e a small number, the quantity of housing at k becomes A&(k),
0 < X < 1. If X is low enough i.e., if the externalities generated by vacant
units are high, then, in equilibrium, housing at k may be abandoned also
because the M(k) that would provide zero profits is less the A4 required to
produce &(k) units of housing.
Of course, if the cost of demolition is low enough, the externalities of
abandonment may be eliminated by new construction. This occurs if the r(k)
that satisfies
r(k)
wag(k)
= ~L,(k)
.=
Qt”(k,t)e-Wt
-
Dw
J o
is greater than or equal to zero.
I4 While black and white tenants will pay less for each unit of housing service, they may
pay more or less for each unit of srructural input provided (measured, for example, as rent
per square foot).
SUSAN
98
ROSE ACKERMAN
Regardless of whether housing codes generate higher or lower prices, they
induce a pattern of settlement that is different from that prevailing under the
variable supply model. When M is variable, blacks live in housing with higher
ratios of L and S to M relative to the ratios than would obtain with fixed Q.
Moreover, when code enforcement generates abandonment, regions close to
the CBD are vacant while blacks are densely settled in the adjoining sector in
old housing with relatively high levels of maintenance.
VI. THE IMPACT OF RACISM IN A CITY WITH AN
AGING HOUSING STOCK
A. Variable Supply
When landlords can vary M, the magnitude of the price benefits obtained
by blacks from racial prejudice depends upon the price elasticity of maintenance spending for older housing in ghetto and border regions. The more
landlords reduce maintenance in response to lower prices the less difference
there is between the equilibrium bid-rent lines for blacks with and without
racism.
The shape of the equilibrium bid-rent lines is the same as those under the
simple supply model (e.g., pP(k), plR(k) in Fig. 2). In the present case,
however, new construction only occurs between k* and L1. The ratios of L
and S to M are higher than those prevailing in the simple supply model, but
the basic result is analogous: less supply in the central section and more in
the suburbs. Speaking more loosely, under the “variable supply” model
racism leads to lower “quality” housing in the ghetto as well as lower prices
per unit of both Q and space.
B. Housing Codes
When housing codes require landlords either to maintain Q(k) at some fixed
level &(k) or to abandon their property, the price benefits of racism are smaller
the higher the cost of maintaining old housing to code levels, the greater the
expense of demolishing old units and replacing them with new construction,
and the larger the externalities from abandoned units. The shape of the
equilibrium bid-rent lines will be the same as pP(k), plR(k), but there may
be a zone of empty houses close to the CBD.
C. Variable Supply and Housing Codes Compared
While it is relatively easy to demonstrate that blacks pay less for housing
when racism exists in either the variable supply or housing code model, a
second issue requires a more sustained argument. This section specifies the
conditions under which housing code enforcement generates a larger welfare
RACISM AND URBAN
STRUCTURE
99
gain for black tenants than a free market regime under which landlords are
free to vary maintenance. The argument proceeds in three stages.First, it is
demonstrated that, when abandonment does not occur, equilibrium utility
levels under a housing code cannot be equivalent to those that prevail under
the “variable supply” model. Second, it is shown that, in the absence of
abandonment, blacks in a racist city are almost certain to be better off
under a regime of housing code enforcement while whites could be worse
off. Third, if abandonment with its accompanying externalities occurs, then
no definitive welfare statements are possible.
1. The nonequivalenceof equilibrium in the two models. Assume that plB(k)
and plR(k) in Fig. 3 are the equilibrium bid-rent lines under the “variable
supply” model. To prove that the utility levels representedby these price lines
cannot be the onesprevailing in equilibrium under a housing code model when
no abandonment occurs, this section assumesthat the contrary holds and
demonstrates that a contradiction exists.
Imagine, then, thatplB(k) is also the equilibrium bid-rent line in the housing
code model. Since abandonment does not occur, Ql(k) < &(k) for k < bl.
Therefore, with code enforcement, total black demand can be accommodated
between the border of the CBD and some b2 where bz < bl.
Consider now the white bid-rent line that passesthrough b2 which gives
whites the same level of utility as plR(k). This bid-rent line, pzR(k) in Fig. 3,
is above plR(k) for all k since at every k blacks are further away.
Given these bid-rent lines it is possible to establish the contradiction involved in assuming that plB(k) and p&k) establish equilibrium prices in
the housing code model. Beyond bl, supply to whites under the housing code
model is greater than under the variable supply model since the existence of
the housing code implies that &(k) 2 Ql(k) for k < k* and since the higher
prices beyond k* induce more new construction. White demand, however,
P k)
I
I
b,
b,
k’
FIGURE 3.
k,
i,
loo
SUSAN ROSE--ACKERMAN
is lower at every k > 61.“: Hence the total white population, H”‘, could be
accommodated between hl and some k < k,. Therefore, the contradiction
is established: either pi”(k) or p2R(k) or both must shift to establish equilibrium at some p,‘:(k), p,“(k) where am”
= plK(b,) and supply equals
demand.
2. Relative prices under the two models. It is now possible to specify conditions sufficient to guarantee that black tenants are better off under a housing
code than under a variable supply model provided that no abandonment
occurs. To do this it is only necessary to determine conditions sufficient for
pzR(bz) < plB(b2) (as shown in Fig. 3), since when this inequality holds,
equilibrium can only be established with a housing code at piB(k) < pP(k).
If only two assumptions are made, the desired outcome can be established.
First, assume that pBu‘ < pkB in a city without prejudice. Second, assume that,
when prejudice is introduced, the bid-rent functions of whites are “separable”
in the sense that they can be divided into a portion that depends on k - b and
a portion that depends on k, p”‘(k). The sufficiency of these conditions can be
shown by writing
pzR(bz) : plB(bz),
as
pgR(bz) - plR(bl) $ pP(bz) - ,oP(bl).
(16)
Since k - b is unchanged between bl and bS, we can write (16) as
pl”(bz) - pl”(bl)
: pP(bz) - pP(bl).
(17)
Since by hypothesis whites have higher incomes than blacks and have flatter
bid-rent functions in the absence of prejudice, this implies,
pl”‘(bz) - pl”‘(bl) < pP(bz) - plB(bl),
and thus,
pzR(bz) < plB@d.
(18)
The importance of this proof depends upon the plausibility of our two
conditions, especially the “separability” assumption. In fact pure “separability” does not obtain, although conditions can be defined in which it is
approached. From (7), the rate of change of white prices depends on a term
-Tkw/q(k), that equals pkW in the absence of prejudice plus a second term
(l/q(k))(U&U,)
that depends on k - b but is not independent of k.
The impact of changes in k on this second term, however, is smaller the lower
I5 We assumethroughout that given k and p(k), white households’ demand for q is
independent of the location of the black-white border although their overall satisfaction
level is lower the smaller is k-b. Furthermore, we assume that the impact of a housing code
on new construction is not substantial enough to revise the result in the text.
RACISM AND URBAN
101
STRUCTURE
k
b, h b.
FIGURE 4.
the elasticity of white housing demand and the smaller white households’
income elasticity of demand for x. Of course, even when separability is not
approximated, it is possible for (18) to hold. Ceterisparibus, thus occurs when
the difference in the slopes of the black and white bid-rent functions in the
absence of prejudice is relatively large.
The condition of the white tenancy must now also be considered. While in
a housing market without prejudice, both races gain from code enforcement,
once prejudice is introduced this result does not necessarily follow. Two
equilibrium cases are possible and are illustrated in Fig. 4 as 1 and II.16 While
whites benefit relative to a variable supply model in Case I, since prices are
lower and ba < bl, they do not necessarily gain in Case II for, although here
too whites gain from reduced prices, this may be cancelled by the fact that
the black ghetto is closer. (If whites are, in fact, worse off under Case II
than with a variable supply model, ~~fi(k) must intersect plR(k) at some
distant k). I7
3. Abandonment. If owners of old housing can choose either to provide &
or to abandon their property, then, as has been shown for the nonracist city,
the equilibrium price lines under a fixed supply model could be above or
below pP(k), plR(k). Blacks are therefore less likely to find a housing code
superior to a free market in which landlords can vary M.
I6 If ~@(6~)>~i~(b~), then a third case is possible in which equilibrium is established for
a pB(k) greater than plB(/r). In this situation blacks would be worse off and whites better
off under a fixed supply model than under a variable supply formulation.
I7 Under Case I density in the ghetto will be higher with a “housing code” model while
under Case II it will be lower. In both cases maintenance levels are necessarily higher at
every k although in Case I black families who would have lived in relatively new housing
between ba and bi are now accommodated in older housing at k _< bl.
102
SUSAN ROSE-ACKERMAN
VII. CONCLUSIONS
The limitations of the model developed here should be emphasized. The
paper deals only with equilibrium conditions and therefore does not contradict the empirical evidence indicating higher housing prices for blacks in
cities where disequilibrium conditions obtain as a result of a rapid black inmigration. Moreover, the model requires several assumptions which do not
necessarily hold in fact. First, the ghetto is assumedto be capable of expansion. It is not surrounded by nonresidential properties or by suburbs practicing exclusionary zoning. Second, suppliers are always willing to rent to the
highest bidder and are incapable of collusive activity. Finally, the fact that
many jobs are located in the suburbs far from concentrations of old housing
has been ignored.
Proceeding from this simplified model, the paper has developed Muth’s
theory of racist behavior by showing that the impact of prejudice on price,
quality and density dependsupon the nature of supplier response.The magnitude of black price gains from racism are smaller the higher the price elasticity
of maintenance spending for older housing in ghetto and border regions, the
higher the costs of demolition and new construction and the larger the externalities from abandoned or poorly maintained housing units. Nevertheless,
there does remain a basic asymmetry between the impact of prejudice in our
hypothetical urban housing market and discrimination in other areas, notably
the job market. While blacks benefit somewhat from white racism in housing,
they most certainly lose in the job market. Thus, the net impact of prejudice
upon blacks in the city hypothesized by Richard Muth may be either positive
or negative.
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STRUCTURE
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