Spatial mismatch: An equilibrium analysis

"onal NcE

&urban 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
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
Ayb > t ( O N b - N W) dyb 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
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
(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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714 J.K. Brueckner, R.W. Martin I Regional Science and Urban Economics 27 (1997) 693-714 Ihlanfeldt, K.R., 1988, Intrametropolitan variation in earnings and labor market discrimination: An economic analysis of the Atlanta labor market, Southern Economic Journal 55, 123-140. lhlanfeldt, K.R., 1992, Housing segregation and the wages and commutes of urban blacks, The case of Atlanta fast-food restaurant workers, Policy Research Center Paper No. 30 (Georgia State University). Ihlanfeldt, K.R., 1993, Intra-urban job accessibility and hispanic youth employment rates, Journal of Urban Economics 33, 254-271. Ihlanfeldt, K.R. and D.L. Sjoquist, 1990, Job accessibility and racial differences in youth employment rates, American Economic Review 80, 267-276. lhlanfeldt, K.R. and D.L. Sjoquist, 1991, The effect of job access on black youth employment: A cross-sectional analysis, Urban Studies 28, 255-265. Kain, J.F., 1968, Housing segregation, negro unemployment, and metropolitan decentralization, Quarterly Journal of Economics 82, 175-197. Martin, R.W., 1996a, Job decentralization with suburban housing discrimination: An urban equilibrium model of spatial mismatch, unpublished paper (University of Illinois). Martin, R.W., 1996b, Job decentralization with costly outward commuting: An urban equilibrium analysis of suburban transportation improvement policies, unpublished paper (University of Illinois). McMillen, D.P., 1993, Could blacks earn more in the suburbs? Racial differences in intra-metropolitan earnings variation, Journal of Urban Economics 33, 135-150. Price, R. and E. Mills, 1985, Race and residence in earnings determination, Journal of Urban Economics 17, 1-18. Reid, C.E., 1985, The effect of residential location on the wages of black women and white women, Journal of Urban Economics 18, 350-363. Straszheim, M.R., 1980, Discrimination and the spatial characteristics of the urban labor market for black workers, Journal of Urban Economics 7, 119-140. Sullivan, A.M., 1986, A general equilibrium model with agglomeration economies and decentralized employment, Journal of Urban Economics 20, 55-74. Vrooman, J. and S. Greenfield, 1980, Are blacks making it in the suburbs? Some new evidence on intrametropolitan spatial segmentation, Journal of Urban Economics 7, 155-167. White, M.J., 1976, Firm suburbanization and urban subcenters, Journal of Urban Economics 3, 323-343. Zax, J.S., 1991, Compensation for commutes in labor and housing markets, Journal of Urban Economics 30, 192-207. Zax, J.S. and J.F. Kain, 1996, Moving to the suburbs: Do relocating companies leave their black employees behind? Journal of Labor Economics 14, 472-504.