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ELSEVIER
Regional Science and Urban Economics 27 (1997) 693-714
ECONOMICS
Spatial mismatch: An equilibrium analysis
Jan K. Brueckner
a:~
' , Richard
W.
Martin b
aDepartment of Economics and Institute of Government and Public Affairs, University of Illinois at
Urbana-Champaign, 1206 South Sixth Street, Champaign, IL 61820, USA
bDepartment of Economics, Agnes Scott College, Decatur, GA 30030, USA
Received 11 December 1996; accepted 8 January 1997
Abstract
The spatial mismatch hypothesis, first stated by Kain (1968) argues that job decentralization in US cities has contributed to low incomes and high unemployment rates for black
Americans. Decentralization relocates job sites to white suburban communities far from the
CBD, and housing segregation prevents blacks from relocating their residences near the new
workplaces. The purpose of the paper is to analyze an urban equilibrium with spatial
mismatch. Despite the existence of a suburban employment center, blacks in the model are
forced to live in the central zone they occupied in the original monocentric city, commuting
across the white residential area to access suburban jobs. This 'mismatch' equilibrium is
contrasted with an unrestricted equilibrium where blacks are free to locate wherever they
choose. ©1997 Elsevier Science B.V.
Keywords: Spatial mismatch; Commuting; Housing discrimination
JEL classification: R0; RI; J7
1. I n t r o d u c t i o n
The spatial m i s m a t c h hypothesis, first stated by Kain (1968), argues that j o b
decentralization in U S cities has contributed to low i n c o m e s and high u n e m p l o y m e n t rates for black A m e r i c a n s . T h e a r g u m e n t is that decentralization relocates j o b
sites to white suburban c o m m u n i t i e s far f r o m the C B D , and that blacks are unable,
*Corresponding author.
0166-0462/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved
PII S01 66-0462(97)00004-5
694 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
for a number of reasons, to move their residences near the new workplaces. The
principal barrier to relocation is thought to be racial discrimination in suburban
housing markets, but secondary reasons include the high financial cost of moving
and the psychic barrier to relocating into an unfamiliar suburban environment.
With their residences thus frozen in place in the central city, black workers must
endure a long and costly commute to hold suburban jobs. Inaccessibility also
raises the cost of job search, which lowers the chance that a black worker is even
able to find work in the suburbs. Both effects mean that job decentralization is
likely to reduce black employment. Furthermore, since higher-wage jobs may be
more likely to relocate to the suburbs, decentralization may also worsen the mix of
remaining jobs in the CBD, leading to a lower average wage for black workers
lucky enough to remain employed.
Worsening conditions in American central cities have led to a resurgence of
empirical research on the spatial mismatch hypothesis. Many studies have
demonstrated a positive relationship between employment prospects for blacks and
measures of job accessibility, providing a direct test of the hypothesis. These
studies include Ellwood (1986); Ihlanfeldt and Sjoquist (1990, 1991); Ihlanfeldt
(1992), (1993); Holzer et al. (1994); Zax and Kain (1996). Zax and Kain (1996)
present especially striking results by using the payroll files of a Detroit company to
show that black workers were more likely than whites to quit following the firm's
relocation to the suburbs.
Other studies ask whether job decentralization leaves low-wage jobs in the
CBD, usually reaching an affirmative answer. These papers include Straszheim
(1980); Vrooman and Greenfield (1980); Reid (1985); Price and Mills (1985);
Ihlanfeldt (1988); Ihlanfeldt and Sjoquist (1991); McMillen (1993). A third set of
studies asks whether the commuting costs of blacks are excessive, given the wages
they earn. Gabriel and Rosenthal (1996) show that, holding wages, house prices
and neighbourhood characteristics fixed, blacks endure significantly longer commute times than whites, confirming the existence of spatial mismatch. Zax (1991)
presents similar findings. ~
Although White (1976); Straszheim (1980); Sullivan (1986); Helsley and
Sullivan (1991) analyze the effects of employment subcenters, no theoretical
model has addressed the spatial mismatch hypothesis directly. The purpose of the
present paper is to develop such a model, with the goal of analysing the welfare
effects of mismatch. Since the hypothesis is inherently spatial, any model should
be constructed in a manner that takes space into account, following the long
tradition of urban models. The model must incorporate two spatial elements that
are central to the mismatch hypothesis: (i) job decentralization; (ii) a restriction
Similarly, Hughes and Madden (1991) find that changes in residential location would significantly
improve the welfare of black workers by reducing their commuting costs and housing prices.
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714 695
that prevents black households from relocating to the suburbs. This restriction
means that if blacks are to work at suburban jobs, they must undertake a long
reverse commute.
The mismatch hypothesis focuses on the impact of elements (i) and (ii) on
labor-market outcomes for blacks. Before this impact is analyzed, however, the
effect of spatial mismatch in the housing market must be understood. The reason is
that the restriction on suburban relocation of the black population generates a
severe housing-market distortion, which by itself can be expected to reduce black
welfare. Section 2 provides a partial analysis of the welfare impact of spatial
mismatch by focusing on this housing-market distortion. Labor-market effects,
which are introduced later, are temporarily suppressed through the assumption that
white and black wages are fixed.
In the model, a suburban employment center (SBD) comes into being at the
edge of a previously monocentric, linear city. Blacks and whites cluster around the
CBD and SBD in the absence of a locational restriction, but black residences are
frozen in place near the CBD under spatial mismatch. The analysis shows that,
under mismatch, blacks pay higher rent at a given commuting distance than in the
unrestricted case. This is a consequence of the distorted residential pattern, which
forces blacks to pay more in order to bid away enough land from whites to
accommodate their population. While higher rents translate into a black welfare
loss, the analysis shows that whites are unaffected by mismatch in the most
important case.
With the housing-market impact of spatial mismatch established, Section 3 adds
labor-market effects to the model. This is done by making SBD and CBD wages
endogenous for both groups and thus sensitive to labor supplies. All the other
elements of the fixed-wage model are retained, so that the analysis can build on
previous results. While the increase in the model's complexity means that
simulation methods must be used, the results show a striking qualitative similarity
to those of the fixed-wage analysis. Blacks are once again hurt by spatial
mismatch, while whites are largely unaffected. The black welfare loss is partly due
to a decline in the black CBD wage, which the literature identifies as an important
consequence of mismatch. It is important to note that since the qualitative welfare
conclusions are unaltered by the presence of labor-market effects, housing-market
analysis by itself provides a good prediction of the overall welfare impact of
spatial mismatch.
By capturing both housing-market and labor-market effects, the model's
treatment of the impact of spatial mismatch is nearly complete. However, since the
market-clearing paradigm rules out black unemployment as an effect, a major
focus of the empirical literature is absent from the model. Thus, an important goal
for future work would be to develop a model with equilibrium unemployment
where the effects of spatial mismatch could be explored. Such a model might also
include suburban job search by workers, which would be adversely affected by
mismatch.
696 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
2. Fixed-wage model
2.1. The s e t u p
To generate transparent results, the analysis is carried out using the simplest
possible model. The city is linear and has unit width, with the CBD located at one
end. Distance to the CBD is denoted by x. Since urban residents consume land
directly, the housing sector is suppressed from the model. Land consumption is
fixed for both the white and black groups, with white households each consuming
one unit of land and black households consuming 0 units, where 0 < 1 (reflecting
lower black incomes). The population sizes for the white and black groups are Nw
and Nb, respectively.
Initially, all employment is located at the CBD, where the whites and blacks
earn fixed incomes of yw
c and ybc respectively (yw
c >ybC). Because lower land
consumption for blacks means that their bid-rent curve for land is steeper than that
of whites (see below), blacks occupy centrally located land and whites live farther
out. In particular, recalling that land consumption levels are 0 and unity for the
two groups, the blacks live between x = 0 and x = ONb, and the whites live
between ONb and ONb + N w. The boundary of the city corresponds to the outer
edge of the white area, and the distance to the boundary is denoted f ~ ONb + N w.
The initial equilibrium is disrupted by job decentralization, which is incorporated into the model in a stylized fashion. In particular, it is assumed that a
suburban business district (SBD) comes into being at the edge of the city (at
x = f). If the city is viewed as a rectangular island with the CBD and SBD at
opposite ends, location beyond the SBD (at x values above f ) is infeasible. As a
result, the city's residential area corresponds to the interval (0, f ) both before and
after job decentralization. SBD employers pay fixed incomes of ySw and yS < yS to
the white and black groups, respectively. Although higher SBD incomes for both
groups might be a natural assumption, no such restriction is imposed.
If the residential location of the black population is unrestricted, then as jobs
decentralize, some of the city's black workers will move adjacent to the SBD, to
which they will commute. White SBD commuters will locate outside the blacks,
and this pattern will be repeated around the CBD. In contrast to this 'unrestricted'
case, spatial mismatch is generated by a restriction o n the r e s i d e n t i a l l o c a t i o n o f
b l a c k h o u s e h o l d s . In particular, it is assumed that black households c a n n o t m o v e
into the s u b u r b a n a r e a o f the o r i g i n a l city, which was occupied by whites [this is
the interval (ONb, f)]. If they wish to work at suburban jobs, blacks must commute
from locations within the original black area of the city [the interval (0, 0Nb)].
This locational restriction, which in effect freezes the residential areas of the black
and white groups in the original configuration, reflects the existence of racial
prejudice on the part of suburban landlords. Given that suburban job growth has in
fact led to some decentralization of black residences, landlord prejudice in the
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (I997) 693-714 697
suburbs is obviously not as strong as assumed. However, the resulting locational
restriction serves as a useful device for framing the analysis of spatial mismatch.
2.2. The unrestricted equilibrium
The main goal of the analysis is to evaluate the welfare effects of spatial
mismatch by comparing the unrestricted and restricted (or 'mismatch') equilibria.
To simplify the comparison, it is assumed that parameter values are such that the
unrestricted equilibrium has both groups commuting to both employment centers,
as shown in Fig. 1. In this case, black CBD workers live in the x-interval (0, £ c ) ,
white CBD workers live in the interval (i c, x*), white SBD workers live in the
interval (x*, i s ) , and black SBD workers live in the interval (i s, 2), where
0 < i c < x * < i s <2.
\
\
\
\
\\\\
0
~c
_
ONb
x*
~s
SBD
CBD
Fig. 1. Unrestricted equilibrium.
698
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 2 7 (1997) 6 9 3 - 7 1 4
The solid intersecting lines in the figure are bid-rent curves for the various
groups of workers (the dotted line is explained below). The bid-rent curves of
CBD workers slope up toward the CBD, while the curves of SBD workers slope
up toward the SBD. In equilibrium, the curves must intersect at the boundaries
between different residential areas, as shown. 2 In addition, rent must be high
enough to bid land away from nonurban use. Assuming that the opportunity cost of
land is zero, the white bid-rent curves must then fall to zero at x*, which is the
boundary that separates white CBD and SBD workers.
To compute the unrestricted equilibrium, the first step is to derive the bid-rent
curves. These come from the budget constraints of the two types of workers,
which are written e w + r w = Y w - tk and e b + Or b = Y b - tk. In these constraints,
e w and e b denote consumption of a numeraire nonhousing commodity, r w and r b
denote rent per unit of land, k denotes commuting distance, and t is commuting
cost per mile. Rearranging the budget constraints, the bid-rent curves of white and
black CBD workers are
c
rw = Y w - t X - e w
(1)
c
Y b -- t x -- e b
rb -
0
(2)
The bid-rent curves of white and black SBD workers are given by Eqs. (1,2) with
f - x in place of x and income superscripts equal to S. As usual, the bid-rent curves
indicate the rent payment at a given location are consistent with a particular level
of nonhousing consumption.
Observe that the slopes of the white and black bid-rent curves are respectively
- t and - t/O for CBD workers and t and t/O for SBD workers. Recalling 0 < 1, it
follows that the black curves are steeper, justifying the close-in locations of the
black residential areas in Fig. 1. Observe also that for white or black workers to be
indifferent between CBD and SBD employment, nonhousing consumption must be
the same regardless of where they work. Hence, the consumption variables e w and
e b do not have workplace superscripts.
The conditions for the unrestricted equilibrium describe the bid-rent intersections as well as requiring that the groups fit in their respective residential areas.
Using Eqs. (1,2), the conditions are
c
Yb -- t~c -- eb
C
-Yw -txC-ew
(3)
2 Note that in contrast to previous models with racial discrimination, such as Courant and Yinger
(1977), white renters are not prejudiced in the present model. As a result, they do not require a rent
discount in order to live next to blacks. Instead, spatial mismatch is due to racial prejudice on the part
of suburban landlords, which prevents blacks from living near the SBD.
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
C
S
Yw - tx* -- e w = yw - t(Y - x * ) - e w
s
Yw - t ( £ -
s
Yb
A s)-
ew -
--
t(2 - A s) - e b
0
C
Yw - t x * - e w = 0
^S
x -x
^C
699
(4)
(5)
(6)
= N w.
(7)
Eq. (3) says that the bid-rents of black and white CBD commuters intersect at £ c ,
and Eq. (5) says that the bid-rents of black and white SBD commuters intersect at
£ s . Eqs. (4,6) indicate that the bid-rents of white CBD and SBD commuters
intersect at x* and that rent at the intersection equals zero. Eq. (7) says that the
white population fits between £ c a n d A s. Note that, along with £ = O N b + N w, Eq.
(7) yields i c + ( £ _ £ S ) = O N b ,
which ensures that the black population fits in the
black residential areas.
The conditions in Eqs. ( 3 - 7 ) can be solved for the unknowns ~c, x*, £ s, e w and
e b. Let
A y w =--yC - ySw,
Ayb ~ yCb -- y s
(8)
denote the C B D - S B D income differentials for the two groups, which are
unrestricted in sign. Then the solutions are
^C
x
= [ d y b - OAy w + tO(1 - O ) N b ] / 2 t ( 1 -- O)
x* = [Ay w + t Y ] / 2 t
^S
x =x
^C
+N w
e w = [yCw + y Sw
(9)
(10)
(11)
-tY]/2
eb = [Ybc + YbS - tO(Nw + N b ) ] / 2 "
(12)
(13)
The requirement O < . f C < x * < £ s < £ ,
which is reflected in Fig. 1, reduces to
four conditions involving A y w and A y b, as can be seen by manipulating Eqs.
( 9 - 1 1 ) , recalling that £ = ONb + N w. The conditions are:
( A y b / O ) -- t(1 -- O ) N b < Ay w < ( A y b / O ) + t(1 -- O ) N b
(14)
Ay b - t(1 - O ) N w < Ay w < Ay b + t(1 - O ) N w.
(15)
To illustrate these conditions graphically, suppose the second inequality in Eq.
(15) were to hold as an equality, with A y w - - A y b + t ( 1 - O ) N
w. In (Ayb, Ayw)
space, the graph of this equation is a line with unitary slope. Together, the
700 J.K. Brueckner, R.V~ Martin / Regional Science and Urban Economics 27 (1997) 693-714
inequalities in Eq. (15) then say that for a (Ay b, Zlyw) pair to be admissible, it must
lie between two parallel lines with slope one. Similarly, the inequalities in Eq. (14)
require the (Ay b, Ayw) pair to lie between two steeper parallel lines with slope
1/0 > 1. To satisfy both requirements, the (Ay b, Ay w) pair must therefore lie in the
diamond-shaped area defined by the two sets of parallel lines, as shown in Fig. 2.
A necessary (but not sufficient) condition for this outcome is that the pair lies
inside the box with its comers at the vertices Q and S of the diamond. This reduces
to the necessary conditions - t(ONb + N w) < Ay w < t(ONb + N w) and - tO(Nb + N w) <
AYb<tO(Nb+Nw), where the Q and S coordinates are taken from Fig. 2.
Intuitively, the C B D - S B D income differentials must not diverge too much from
Q
-to(~b + N~)
tO(Nb + N~)
~ --t(ONb + Nw)
1=MIXED
2=SBDWH
3=CBDBL
Ayw
Fig. 2. Admissible region.
4=SPLIT
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714 701
zero in order for each group to commute to both the CBD and SBD in the
unrestricted equilibrium. Other features of Fig. 2 are discussed below.3
2.3. Mismatch cases
In the mismatch equilibrium, as explained above, racial discrimination by
suburban landlords prevents blacks from living near the SBD. In this case, the
residential area for blacks is the same as the interval (0, ONb) they occupied in the
monocentric city, with whites again living in the interval (ONb, £). Since the
residential areas are thus predetermined, the endogenous elements in the mismatch
case are commute flows and the consumption levels of the groups.
There are three conceivable commuting patterns for each group: CBD commuting only, SBD commuting only, and commuting to both centers. For each group,
however, one of these patterns is inadmissible. The CBD-only case is inadmissible
for whites, and the SBD-only case is inadmissible for blacks. Thus, the mismatch
equilibrium cannot have each group commuting exclusively to the employment
center most distant from its fixed residential area. This is inconsistent with the
assumption that both groups commute to both centers in the unrestricted
equilibrium.
To establish this claim, suppose whites commute exclusively to the CBD. This
C
->
S
would require Yw-tx--Yw, indicating that the disposable income of a white SBD
resident is higher commuting to the CBD than working at the SBD. This inequality
reduces to Ayw>--t(ONb +Nw), using £ = ONb + N w, which in turn means that (Ay b,
Ayw) lies outside the diamond in Fig. 2. A similar argument establishes that blacks
cannot commute exclusively to the SBD.4
The elimination of these possibilities leaves four combinations of commute
patterns for the groups:
• MIXED: both groups go to both centers (each center is mixed).
• SBDWH: whites go to both centers and blacks go to the CBD (SBD has only
whites).
• CBDBL: whites go to the SBD and blacks go to both centers (CBD has only
blacks).
• SPLIT: whites go to the SBD and blacks go to the CBD.
3In contrast to the situation shown in Fig. 2, the diamond's other vertices could lie in the first and
third quadrants [this depends on the signs of tO(Nb-Nw) and t(ONb-N~), which determine their
coordinates].
4 For all blacks commute to the SBD, the inequality yS
b--t£>--yCb must hold, or Aye<_-t(ONb+Nw).
Noting that t(ONb+ N~) < - tO(Nb+ N~), this implies JYb <- tO(Nb+ Nw), which says that (Ayb, Ayw)
lies outside the diamond in Fig. 2.
-
-
702 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
To derive the conditions that characterize these cases, consider the M I X E D case.
For this case to obtain, a white w o r k e r residing at x = ONb, the boundary b e t w e e n
the black and white areas, must prefer to w o r k at the C B D , and a black worker at
c
this location must prefer to w o r k at the SBD. T h e s e conditions imply Y w - tONb
s
s
> c
Yw--t(x--ONb) and Yb--t(x--ONb) Yb--tONb, respectively. Rearranging these
inequalities, the conditions for the M I X E D case are
(MIXED)
d Y w > t ( O N b - Nw),
dyb < t ( O N b - Nw).
(16)
Since the black w o r k e r at ONb prefers C B D e m p l o y m e n t under the S B D W H case,
the second of these inequalities is reversed. Since the first inequality is reversed
under C B D B L , while both are reversed under SPLIT, the conditions for the
r e m a i n i n g cases are
(SBDWH)
(CBDBL)
(SPLIT)
Ayw>t(ONb-Nw),
Ayw<t(ONb-Nw),
Ay w < t(ONb - Nw),
Ayb > t ( O N b - N W)
dyb<t(ONb-Nw)
Ay b > t(ONb - Nw)
(17)
(18)
(19)
It follows f r o m Eqs. ( 1 6 - 1 9 ) that the point (Ayb,Ayw)=(t[ONb-Nw], t[ONb Nw])=--W in Fig. 2 divides the d i a m o n d into four quadrants in w h i c h the various
equilibria obtain. This d e c o m p o s i t i o n m a k e s sense intuitively. For e x a m p l e , a high
value of Ay w leads to the M I X E D and S B D W H cases, w h e r e the C B D has white
c o m m u t e r s , while a low value leads to the C B D B L and S P L I T cases, where no
whites c o m m u t e to the CBD. A l t h o u g h Fig. 2 shows all four m i s m a t c h cases as
possible outcomes, s o m e m a y be infeasible for particular parameter values. 5
2.4. Mismatch equilibria
The bid-rent curves in the M I X E D e q u i l i b r i u m are s h o w n in the upper left panel
o f Fig. 3. The black and white residential areas contain both upward and
d o w n w a r d - s l o p i n g curves, w h i c h indicate the areas o c c u p i e d by S B D and C B D
5 To see this, observe that while point W is always to the left of vertex Q, W may lie to the left of
vertex S. This occurs when t(0Nb-N~)<-tO(Nb+Nw), or when 20Nb--(1-8)Nw <0. In this case, the
CBDBL and MIXED quadrants lie outside the diamond, and these mismatch equilibria are infeasible.
Even when point W is to the fight of S, W may lie above the diamond, in which case the MIXED
equilibrium is ruled out. This occurs when 20Nb-Nw<O. Note that if the first inequality holds, the
second one must hold as well. The second inequality, however, does not imply the first. (The second
inequality is a restatement of the condition that point W lies above the uppermost of the steep lines in
Fig. 2. W can be shown to always lie between the flat lines and above the lower steep line, so that this
condition determines whether it lies outside the diamond).
J.K. Brueckner, R . ~ Martin / Regional Science and Urban Economics 27 (1997) 693-714
ONb
ON~
z*
MIXED
703
z"
SBDWH
I
0N~
CBDBL
SPLIT
Fig. 3. Mismatch cases.
commuters, respectively. A key feature of the equilibrium is that the black bid-rent
must exceed or equal the bid-rent of white CBD commuters at all points in the
interval (0, ONb). If this were not true, whites could outbid blacks for some of the
land in the fixed black residential area. To achieve this outcome, the low point of
the black bid-rent curves, which occurs at the black C B D - S B D commute
boundary (x=Y), lies on the extension of the white curve, shown as the dotted
line.
As can be seen, the equilibrium exhibits a dramatic discontinuity in land rent at
x = O N b, the b l a c k - w h i t e border. Such a discontinuity could not exist in an
unrestricted market because blacks would outbid whites for the land they occupy.
Spatial mismatch, however, is based on a fundamental asymmetry that makes this
704
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
an equilibrium outcome. In particular, while blacks must compete with whites for
the land inside x ON b, racial discrimination by landlords means that whites face
no competition from blacks for the land beyond O N b. As a result, rents in the black
area cannot fall below the floor provided by the white bid-rent curve, while no
such floor exists in the white area. Given the land-rent discontinuity that results,
landlords in the white area have a strong incentive to switch to black occupancy.
Racial prejudice is assumed to be powerful enough to restrain this impulse.
The equilibrium conditions for the MIXED case are as follows:
=
C
Yb --t2--eb
IS
)b -- t ( 2 - - 2 ) - - eb
0
(20)
0
C
Yb - t2 - e b
C
Yw S
Yw -
tx* -
C
- Y w - t 2 - ew
S
ew = Yw -
t(2 - x*)
t(2
(21)
- x*) - e w
(22)
(23)
- e w = O.
Eqs. (22,23) indicate that the bid-rent curves of white CBD and SBD commuters
intersect at x* and that rent at the intersection equals zero. Eq. (20) says that the
bid-rent curves of black CBD and SBD commuters intersect at 2, and Eq. (21)
indicates that the rent at this point is the same as on the extension of the white
CBD commuter's bid-rent curve, as shown in Fig. 3.
Solving Eqs. ( 2 0 - 2 3 ) yields the same x* and e w solutions as in the unrestricted
equilibrium (given by Eqs. (10,12)). The complete set of solutions for the M I X E D
case is
2 = [Ay b + tf]/2t
(24)
x* = [Ay w + tf l/2t
(25)
ew = [yC + y S _ t21/2
(26)
e b = [(1 +
O)y c +
(1 -
O)y s -
OBy w -
t21/2.
(27)
The bid-rent curves for the SBDWH, CBDBL and SPLIT cases are shown in the
remaining panels of Fig. 3, with the logic being the same as in the M I X E D case.
Note that unlike the M I X E D and C B D B L cases, the S B D W H and SPLIT
equilibria do not involve land-rent discontinuities. The equilibrium conditions for
all these cases come from applying various restrictions to the M I X E D conditions.
The conditions for the S B D W H case are derived by setting f = O N h in Eq. (21) and
deleting Eq. (20). The conditions for the C B D B L case are derived by setting
x * = O N b in Eq. (23) and deleting Eq. (22). The conditions for the SPLIT case are
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
705
derived by setting x* = f = ONb and deleting Eqs. (20,22). The solutions for these
cases are presented in Appendix A.
2.5. Welfare comparisons
The main task of the analysis, derivation of the welfare impacts of spatial
mismatch, can now be carried out. To begin, observe that the unrestricted
equilibrium is efficient: it maximizes total urban land rent plus the total
consumption o f urban residents. While this shows that the mismatch equilibrium
has lower aggregate welfare, our interest instead lies in the separate impacts of
spatial mismatch on white and black consumption. Evaluating these impacts
requires explicit comparison of the equilibrium consumption levels between the
unrestricted and mismatch cases.
Let /2iw denote the white consumption gain associated with mismatch equilibrium i, and let ~2ib denote the black gain (a negative value indicates a loss). In
particular, ~2iw=eiw-e~, where i denotes one of the mismatch equilibria and u
denotes the unrestricted equilibrium, and similarly for Oib" Consider first the
MIXED case, Since the e w solution in Eq. (26) is the same as in the unrestricted
equilibrium, it follows that whites are unaffected by mismatch when the relevant
equilibrium is MIXED. Thus,
oMIXED = O.
(28)
The change in black consumption in the MIXED case is found by subtracting Eq.
(13) from Eq. (27). This yields
~,~M1XED
h
=[O(Ayb--Ayw)--t(l-O)Nw]/2<O,
(29)
indicating a black loss from spatial mismatch. The inequality in Eq. (29) follows
from the first half of Eq. (15), which implies O(dy b - Ayw) - tO( 1 - 0)N w< 0. Since
0 < 1, satisfaction of this inequality means that Eq. (29) is negative, indicating a
black consumption loss.
Parallel analysis for the other cases is presented in Appendix A. It is shown that,
as in the MIXED case, whites are unaffected by mismatch when the relevant
equilibrium is SBDWH. However, whites benefit from mismatch when the
equilibrium is CBDBL or SPLIT. Thus,
f~SwBDWH= 0;
'o~CBDBL
SPLIT> O.
--,~
, ~Qw
(30)
Appendix A also shows that, in these other cases, the impact of spatial mismatch
on blacks is the same as in the MIXED case. Thus, when the relevant equilibrium
is SBDWH, CBDBL or SPLIT, blacks again suffer a consumption loss, so that
oSBDWH oCBDBL DSPLIT<0"
b
' ~b
' "~b
(31)
706 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
The results of all the preceding analysis are summarized in the following
proposition.
Proposition. If both groups commute to both employment centers in the unrestricted equilibrium, then the mismatch equilibrium is MIXED, CBDBL, S B D W H
or SPLIT. Regardless o f which mismatch equilibrium applies, black consumption
e b is lower than in the unrestricted equilibrium. If whites commute to both centers
(if the mismatch equilibrium is MIXED or SBDWH), their consumption e w is the
same as in the unrestricted equilibrium. If whites commute to the SBD only (if the
mismatch equilibrium is CBDBL or SPLIT), e w is higher than in the unrestricted
equilibrium.
To understand the black welfare loss, it is helpful to focus on the CBD resident,
who is black under both the restricted and mismatch equilibria. Since the CBD
resident incurs no commuting cost, the decline in his consumption (which is felt by
other blacks) means that CBD rent is higher in each of the mismatch equilibria
than in the unrestricted equilibrium. But since CBD rent determines the level of
rents paid throughout the black area, the welfare loss can be understood as a
consequence of a general rent escalation. This rent escalation occurs because racial
discrimination blocks the normal functioning of the land market, which causes
blacks to pay extra in order to bid away enough land to live on.
To understand the absence of a welfare impact on whites in the MIXED and
SBDWH equilibria, observe that the location x* at which the CBD and SBD are
equally attractive lies inside the white area in these cases. Therefore, whites
continue to work at both centers, and rent at x* remains zero. In cases SPLIT and
CBDBL, by contrast, x* lies inside the black area, so that at all locations in the
white area, SBD employment is preferred. The bid-rent curve of SBD workers can
then shift down so that it intersects the axis at x = ONb, and this shift raises white
welfare. Intuitively, the white gain emerges because mismatch eliminates black
competition for land around the center where suburban whites prefer to work.
The welfare effects can be seen clearly in Fig. 1, which allows a comparison of
bid-rent curves in the unrestricted and S B D W H equilibria. With all blacks working
in the CBD under SBDWH, the black CBD bid-rent curve must shift so that these
workers can outbid whites for all the land in the interval (0, ONb). Since the white
bid-rent curves are unaffected (this follows because x* is inside the white area),
the black CBD curve must shift up to the position shown by the dotted line. This
rent escalation, and the resulting black welfare loss, is a consequence of the
distorted residential pattern.
As a final point, observe that if SBD incomes are high relative to CBD incomes,
then both Ayw and Ay b are small, and case CBDBL is likely to emerge (see Eq.
(18)). Thus, high SBD incomes are associated with a white welfare gain from
spatial mismatch.
It should be noted that, because of different assumptions, the black welfare
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714 707
effect from mismatch is the reverse of that found in standard spatial models of
racial segregation (see, for example, Courant and Yinger, 1977). In contrast to the
present case, the size of the black area is unrestricted in these models, and white
renters (instead of landlords) are racially prejudiced, requiring a rent discount to
live adjacent to the black area. Depressed rent on the white side of the boundary
allows the black area to expand, which lowers the level of black rent and raises
welfare.
3. A model with flexible wages
Because CBD labor demand is perfectly elastic in the preceding analysis, job
decentralization has no effect on the availability of CBD jobs or on the wages they
pay. This scenario, however, overlooks an important element of the mismatch
story, which argues that spatial mismatch lowers black welfare partly through a
reduction in the wages of black CBD workers. To capture such an effect, the
previous model can be modified to include explicit CBD and SBD labor markets.
Although market-clearing assumptions rule out unemployment, the modified
model offers a more-complete picture of the impact of spatial mismatch by
capturing both housing and labor-market effects.
Suppose that output in the original monocentric city is produced using a
constant-returns Cobb-Douglas function with white and black labor as arguments.
Implicit in this specification is the assumption that white and black labor are
distinctly different inputs, which could reflect different levels of education in the
~1
B S
two populations. Output is given by Z = aL w L b, where Lw and Lb are white and
~TI 6 ~ z ~
black labor inputs. CBD output in the monocentric city is then equal to a~v w ~vb,
and white and black incomes are y w = a ( 1 - )(Nb/Nw) and y b = a 6 ( N b / N w ) a 1.
As before, production shifts to the SBD as jobs decentralize. Since wages are
flexible, both groups of workers divide equally between the CBD and SBD in the
unrestricted equilibrium, and commuting patterns adjust accordingly. The ratio of
white to black workers at each center is then the same as in the original CBD, so
that incomes for both groups equal those in the monocentric city.
The spatial-mismatch case again arises if black workers are unable to relocate to
the suburbs. In one respect, the analysis of this case is simpler than in Section 2
because all but one type of mismatch equilibrium can be ruled out. To see this,
observe that both types of labor are essential inputs given the Cobb-Douglas
technology. As a result, both types of workers must be employed in both centers if
any output is to be produced. This in turn implies that the mismatch equilibrium
must be MIXED.
Although isolation of the relevant case is immediate, characterization of the
equilibrium is more complex than before because of the endogeneity of incomes.
The following additional equilibrium conditions must be added to the previous
conditions Eqs. (20-23):
708 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
I ~ = x* - ONb
(32)
L Sw = £ -
(33)
X~
L~=2/O
(34)
L~ -- Nb - 2 / 0
(35)
c
c c 6
Yw = oL(1 -- 6)(L b/Lw)
(36)
s
S LS
Yw = a(1 - 6 ) ( L b / w)
(37)
C
C
S
S
C B-I
Yb = O~t~(Lb / L w )
S
Yb = °~6(Lb/Lw)
eS-1
(38)
(39)
Eqs. ( 3 2 - 3 5 ) express the white and black labor supplies at the two centers in
terms of 2 and x*, which in turn depend on incomes at the centers via Eqs.
(20-23). Eqs. ( 3 6 - 3 9 ) express these incomes as functions of the labor supplies.
Labor supplies thus depend on incomes and vice versa, and Eqs. ( 3 2 - 3 9 ) ensure
that these quantities are mutually consistent.
The equilibrium conditions are too complex to analyze in general, but numerical
examples can indicate the effect of spatial mismatch with flexible wages. The
baseline case has the following parameter values: a = 2 0 , 0 0 0 , 6 = 0 . 5 , 0 = 0 . 9 ,
t = 1.0, N W= 1000, N b = 1250. Although blacks and whites are equally productive 6
given 6 = 0 . 5 , the fact that N b > N w leads to lower black incomes in the original
monocentric city and in the unrestricted equilibrium.
Table 1 shows labor supplies and incomes at both centers, as well as
consumption levels, in both the mismatch and unrestricted equilibria. As can be
seen, white and black workers are equally divided between the CBD and SBD in
the unrestricted equilibrium, and incomes are the same across centers for each
group. In the mismatch equilibrium, black employment is slightly concentrated in
the CBD, which employs 54.5% of black workers, while white employment is
again almost evenly split, with 49.5% of whites employed in the CI3D. Since fewer
blacks work in the SBD in the mismatch equilibrium, yS is higher and yS is lower
than in the unrestricted equilibrium. Conversely, since more blacks work in the
CBD in the mismatch equilibrium, ybc is lower and yw
c is higher than in the
unrestricted case. Thus, black income is high at the SBD and low at the CBD, as
Note that equal productivity in this case does not mean that white and black labor are
indistinguishableas inputs. In addition, it does not mean that marginal products are equal. Rather, the
assumption 6=0.5 means that reversing the numbers of white and black workers has no effect on
output.
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
709
Table 1
Comparison of mismatch and unrestricted equilibria in the baseline case
Blacks in CBD (L c)
Whites in CBD (L c)
Black CBD income (y~')
Black SBD income (ybs)
White CBD income (ywc)
White SBD income (ySw)
Black consumption (e~)
White consumption (ew)
Unrestricted
Mismatch
Change
625 (50%)
500 (50%)
8944
8944
11 180
11 180
7932
10 118
682 (54.5%)
496 (49.5%)
8526
9423
11 729
10 613
7005
10 108
+8.4%
-0.8%
-4.9%
+5.1%
+ 4.7%
-5.3%
- 13.2
-0.1%
e n v i s i o n e d in the u s u a l m i s m a t c h scenario. Finally, b l a c k c o n s u m p t i o n is 13%
l o w e r in the m i s m a t c h e q u i l i b r i u m , w h i l e w h i t e c o n s u m p t i o n is v i r t u a l l y the s a m e
as in the u n r e s t r i c t e d e q u i l i b r i u m ( l o w e r b y 0 . 1 % ) . T h u s , d e s p i t e the a d j u s t m e n t o f
w a g e s , the w e l f a r e i m p a c t s in the b a s e l i n e c a s e are q u a l i t a t i v e l y the s a m e as in the
M I X E D case u n d e r fixed wages: b l a c k s lose w h i l e w h i t e s are u n a f f e c t e d b y spatial
mismatch]
Table 2 provides sensitivity analysis by showing how these welfare impacts
v a r y in r e s p o n s e to c h a n g e s in the l a n d c o n s u m p t i o n a n d l a b o r p r o d u c t i v i t y o f
Table 2
Effect on consumption ratios of variation in 8 and 0
0.3
0.4
0.5
0.4
0.5
0.6
0.7
0.8
0.9
0.4
0.5
0.6
0.7
0.8
0.9
0.4
0.5
0.6
0.7
0.8
0.9
0.903
0.884
0.864
0.840
0.812
0.779
0.930
0.917
0.903
0.887
0.868
0.846
0.946
0.936
0.925
0.913
0.899
0.883
0.986
0.989
0.992
0.995
0.997
0.999
0.984
0.988
0.991
0.994
0.996
0.999
0.981
0.986
0.989
0.993
0.996
0.999
7 Because total output is highest when the groups are split evenly between the employment centers,
another cost of spatial mismatch is a reduction in the city's output.
710 J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
blacks. The Table depicts plausible variation in the parameters, with black land
consumption 0 ranging as low as 0.4, and the labor productivity parameter 6
ranging as low as 0.3 (reflecting lower black education levels). The welfare impact
is captured by tabulating the ratio of consumption between the mismatch and
unrestricted equilibria, which is computed for each group. The black consumption
ratio, denoted e b / e b, equals 0.883 in the baseline case, while the white ratio,
denoted ew~/e~, equals 0.999 (these values are shown in the last line of Table 2).
All the consumption ratios in Table 2 are uniformly less than one, indicating
losses from spatial mismatch. However, the white ratios are never smaller than
0.98, showing virtually no effect on whites. The black consumption ratio is smaller
than the white ratio in each case, and the ratios tend to be substantially below one,
indicating an appreciable welfare loss from spatial mismatch. Thus, despite the
presence of flexible wages, the results mimic those of the fixed-wage analysis of
the MIXED case. Whites are unaffected by spatial mismatch while blacks are hurt. s
The qualitative welfare results are therefore the same regardless of whether
labor-market effects are considered. This is an important conclusion because it
shows that housing-market effects by themselves provide a good prediction of the
overall welfare impact of spatial mismatch. For more discussion of the flexiblewage model, see Martin (1996a).
4. Conclusion
This paper presents the first formal analysis of the welfare effects of spatial
mismatch, providing a theoretical counterpart to the large empirical literature.
Although the paper's analytical results are unconnected to most existing empirical
studies, the results are in fact consistent with the findings of Gabriel and Rosenthal
(1996). Their study is based on the recognition that, among employed blacks, the
effects of spatial mismatch show up in disadvantageous commute times and
housing prices, as in the present analysis. Gabriel and Rosenthal's estimates
indicate that, holding housing prices fixed, commute times are longer for blacks
than for whites. Their estimating equation (which uses observations on individuals)
is T I M E = a o + a j * P R I C E + a 2 * B L A C K + m o r e
terms, with a2>0 (BLACK is a
race dummy). To see the connection to the present analysis, rewrite this equation
as P R I C E = - a o / a 1 + ( 1 / a j ) * T I M E - ( a 2 / a l ) * B L A C K + m o r e
terms. Although
the sign of a~ is not directly revealed by Gabriel and Rosenthal's analysis 9 it
should be negative, indicating an inverse relationship between housing prices and
8When 0 is smaller than shown in Table 2, the black and white consumptionratios become closer in
magnitude. In fact, when 0 = 0.1, both ratios are close to unity, indicating small welfare losses, and the
black ratio exceeds the white ratio for each of the three 6 values, indicating a larger white loss. This 0
value, however, seems empirically unreasonable.
9 Since house prices are represented by neighbourhood dummy variables, their effect is not directly
measured.
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
711
commute times. Given a z > 0 , the coefficient on BLACK is then positive. This
shows that, holding commute time fixed, blacks pay higher housing prices than
whites.
This empirical finding is validated by the present analysis. For example,
consider the CBDBL equilibria in Fig. 3. At the black-white border ONb, black
and white commute distances (to the SBD) are equal, but blacks pay much higher
land rent than whites. A similar comparison can be made for each mismatch
equilibrium. By contrast, in the unrestricted equilibrium of Fig. 1, blacks and
whites pay the same land rents at locations where their commute distances are
equal (~fs and £c). Thus, by showing that blacks receive unfavourable commute
time/house price combinations, Gabriel and Rosenthal's findings match the
predictions of the present analysis.
Given the intensity of interest in the spatial mismatch hypothesis, further
theoretical work is warranted. Future research could relax several of the model's
assumptions. One purely technical modification would drop the 'island-city'
assumption, allowing residential location outside the SBD. Another technical
modification would allow land consumption to be endogenous and to vary with
location, as in the usual urban model. As noted above, a more substantive
modification would allow equilibrium unemployment in the labor market, perhaps
in combination with job search. Another substantive change would put more
structure on commuting costs, formalizing the common assertion that poor public
transit service from the central city to the suburbs deters blacks from seeking
suburban employment. To add this feature to the model, outward commuting for
blacks could be made more costly on a per mile basis than inward commuting.
White commuting costs would be equal in both directions and lower than those of
blacks, reflecting greater automobile usage. Simulation analysis of such a model is
presented in Martin (1996b).
•
•
•
10
Acknowledgments
We wish to thank Dennis Capozza, Donald Haurin, Robert Helsley, Stephen
Ross, Richard Stanton and several referees for helpful comments• However, any
errors or shortcomings in the paper are our responsibility.
Appendix A
This appendix presents the equilibrium solutions for the SBDWH, CBDBL and
SPLIT cases and derives welfare impacts.
~oFor example, in the MIXEDcase, the welfare of a black SBD commuterwould be the same if he
lived at x* and paid a rent given by the extension of the black SBD bid-rent curve. This rent is much
higher than the white rent paid at x*.
712
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714
The SBDWH
case
The solutions for x* and e w in the SBDWH case are again given by Eqs.
(25,26), and the solution for e b is
e b ----
[2ybc --
OAy w -
t0{(2 -
O)N b +
Nw}]/2.
(A1)
Since white consumption is the same as in the unrestricted equilibrium, ~2SwBDwH=
0. The black consumption impact is found by subtracting Eq. (13) from Eq. (A1).
This yields
~SBDWH
b
= [dy b -
O d y w - tO(1 - O ) N b ] / 2
< 0.
(A2)
The inequality in Eq. (A2) follows directly from rearranging the first half of Eq.
(14).
The CBDBL
case
The solutions for the CBDBL case are given by Eq. (24) and
S
(A3)
ew = Yw - tNw
e b = [(1 +
O)y c +
(1 -
O)y s - 20Ay w -
t{0(l -
O)Nb +
(1 +
O)Nw}]/2.
(A4)
These solutions pertain to a situation like that shown in Fig. 3, where the row point
on the black bid-rent curves lies on the white CBD bid-rent curve. Observe that
even though there are no white CBD workers in the CBDBL case, blacks must still
offer a rent at least as high as these workers would pay. Another possibility, not
shown in the figure, is that the bid-rent of white CBD workers is negative at Y, in
which case Eq. (21) does not apply. The black bid-rent curves then intersect the
axis at i, so that Eq. (21) is replaced by
c
Yb - - t.~ - - e b
0
- 0.
(A5)
When Eq. (A5) applies, the e b solution is replaced by
eb =
(yC + Ybs -- t Y ) [ 2
(A6)
When Eqs. (A4) is relevant, the black consumption impact is found by
subtracting Eq. (13) from Eq. (A4). This yields
f~CBDBL
b
= [O(AYb -- 2 A Y w ) + t ( O 2 N b --
Nw)]/2 < 0.
•
(A7)
•
c
The inequality in Eq. (A7) is established using the condmon Yb --tY--e b > 0 , which
indicates that rent is positive at Y. Substituting the Y and e b solutions from Eqs.
J.K. Brueckner, R.W. Martin / Regional Science and Urban Economics 27 (1997) 693-714 713
(24), (A4) and multiplying through by 0, this condition reduces to O(Ay b 2 A y w ) + t ( O 2 N b - O N w ) < O . Since 0 < 1 , this inequality implies that Eq. (A7) is
negative. When the e b solution is given instead by Eq. (A6), subtracting Eq. (13)
yields
~CBDBL
= -- t(1 -- O ) N w / 2 < 0.
b
(A8)
The white consumption impact in the CBDBL case is found by subtracting Eq.
(13) from Eq. (A3). This yields
,QCwBD~L= It(ONb -- Nw) - A y w ] / 2 > 0.
(A9)
where the inequality follows from the first part of Eq. (18).
The S P L I T case
The solutions for the SPLIT case are given by Eq. (A3) and
c _ tONb"
eb : Yb
(A10)
Since the e w solution is the same as in the CBDBL case, O~Lux is given by Eq.
(A9). The black consumption impact is found by subtracting Eq. (13) from Eq.
(A10). This yields
g2~PL'T = [Ayb - tO(Nb - Nw)]/2 < 0.
( a l 1)
The inequality in Eq. ( A l l ) is established by first noting that - A y w > - t ( O N b Nw) must hold in the SPLIT case, as can be seen by rearranging the first part of
Eq. (19). In contradiction to Eq. ( A l l ) , suppose A Y b - - t O ( N b - - N w ) ~ O . Adding
these inequalities yields A y b - t(1 - O)N w >--Ay w, which violates the first half of Eq.
(15). Thus, Eq. ( A l l ) must hold.
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