implications of income-based school assignment policies for racial

Educational Evaluation and Policy AnalysisSpring 2006, Vol. 28, No. 1, pp. 49-75

Implications of Income-Based School Assignment Policiesfor Racial School Segregation

Sean F. ReardonStanford University

John T. Yun

University of California, Santa Barbara

Michal KurlaenderUniversity of California, Davis

A number ofpublic school districts in the United States have adopted income-based integrationpolicies-

policies that use measures offamily income or socioeconomic status-in determining school assign-

ment. Some scholars and policymakers contend that such policies will also reduce racial segregation.

In this article this assumption is explored by computing upper and lower bounds on the possible and

probable levels of racial segregation that would result from race-neutral income-based school as-

signment policies. The article finds that, in general, income integration is no guarantee of even mod-

est racial desegregation. In particular, the extent of ancillary racial integration produced by an in-

come-integration policy will depend on the size of racial income disparities within a given district, the

specifics of an income-integration policy, and the patterns of racial and socioeconomic residential

segregation in a school district. Data on racial income inequality and income segregation in urban

districts throughout the United States indicate that very high levels of racial segregation are possible

under any practical income-integration policy. The authors conclude that, given the extent of resi-

dential racial segregation in the United States, it is unlikely that race-neutral income-integration poli-

cies will significantly reduce school racial segregation, although there is reason to believe that such

policies are likely to have other beneficial effects on schooling.

Keywords: income, race, school desegregation

1. Introduction

In the 50 years since the Brown v. Board of Ed-

ucation' decision, public school racial desegrega-tion efforts have been a prominent feature of manyurban school districts in the United States. How-ever, the use of race-conscious school assignmentpolicies has become more controversial over the

last decade. Both mandatory and voluntary school

desegregation plans have been increasingly-and, in some cases, successfully--challenged inthe courts and in public debate. As a result, manyschool districts have been seeking "race-neutral"methods to maintain some racial diversity in theirschools.

Given the uncertainty of the future legal sta-tus of race-conscious school assignment poli-cies, researchers and educational policymakers

This research was supported by a Spencer Foundation/National Academy of Education Postdoctoral Fellowship and a grant from

the NCES/AERA Research Program to the first author. We thank Claudia Galindo for research assistance and four anonymous

reviewers for their excellent suggestions. All errors remain our responsibility.

49

Reardon, Yun, and Kurlaender

are looking more closely at race-neutral schoolassignment mechanisms-such as socioeconomicintegration-for achieving the historical goals ofrace-based school desegregation: equalizing ed-ucational opportunity and avoiding racial isola-tion. Socioeconomic integration plans are at-tractive for several reasons. First, there is wideconsensus among researchers that concentratedpoverty influences future educational attainment,potential earnings, and societal liabilities such aswelfare dependency. In addition, because raceand class overlap so closely, some contend thatmixing students by economic status will alsoproduce acceptable levels of racial integration(Chaplin, 2002; Kahlenberg, 2001). However, itis not clear how much racial desegregation canbe attained using solely income-based school as-signment policies; nor is it clear how contingentracial integration may be on the specifics of suchincome-based policies. Previous work on thisquestion has been limited, offering some usefulbenchmarks for potential impacts of income-based integration on racial integration acrosslarger geographic units, such as counties andmetropolitan areas (Chaplin, 2002), and provid-ing infonnative case studies of specific districtsundergoing changes in student assignment poli-cies (Flinspach & Banks, 2005).

In this article, we build on this work by inves-tigating the extent to which race-neutral income-based integration plans would produce ancillaryracial integration, particularly in large urbanschool districts, where racial segregation tends tobe high. We compute upper and lower bounds onthe possible and probable levels of racial segre-gation that might result from several types ofincome-integration policies. We find that therange of possible racial segregation consistentwith income integration is quite wide, and de-pends largely on the size of racial income dis-parities within a given district and on the opera-tional definition of income integration embodiedin a given plan. Specifically, income-integrationplans that use continuous or quasi-continuousmeasures of socioeconomic status (such as actualincome) and require strict income balance acrossschools ensure higher levels of racial integrationthan do plans that rely on dichotomous incomemeasures (such as poverty status or free andreduced-price lunch eligibility) or that requireonly approximate income balance among schools.In general, given current levels of racial income

50

disparities in the United States and practical con-straints on income integration plans, we find thatincome integration does not guarantee any re-duction in racial school segregation-very highlevels of racial segregation are possible underany practical income-integration policy.

In addition, we examine patterns of racialand economic segregation (both residential andschooling) in urban areas and conduct simula-tions to estimate the impact on racial segregationof an income-integration plan. These simulationsyield lower bounds on the probable levels of re-sulting racial segregation which, in conjunctionwith data on residential racial segregation pat-terns, suggest little ancillary racial integrationwould result from income-integration policies,except in districts with already low levels of racialsegregation. Taken together, these results indicatethat, although income-based integration may be aworthwhile goal in itself, it cannot be assumedthat income-based integration plans will neces-sarily lead to racially desegregated schools.

In Section I of this article we discuss the back-ground and rationale for the current interest ineconomic integration policies, and review priorevidence on their effects on racial segregation. InSection 2, we describe analyses to compute upperand lower bounds on the possible and probablelevels of racial segregation under several vari-ants of income integration policies. Section 3concludes.

1.1. Background: Why Income-Based SchoolAssignment Policies?

The move to socioeconomic integration planshas been motivated by both the legal and policycontext surrounding school desegregation, aswell as by an extensive body of literature onthe effects of racial and socioeconomic schoolcomposition on educational outcomes. High-profile legal challenges to school assignmentpolicies that include race-and an increasingreluctance among school boards and other pol-icymakers to use race in school assignmentpolicies-have led many school districts toconsider more race-neutral measures in design-ing student assignment plans. The factor mostcommonly used as a proxy for race is socio-economic status, which many argue may haveadvantages over race-based school desegregationin both the courts and in public opinion (Kahlen-berg, 2001).

In recent years, mandatory (court-ordered) ef-forts to maintain racial integration at the K-12level have met increasing resistance from schooldistrict officials and from judges who wish to endthe courts' longtime federal oversight of schooldesegregation plans. In addition, voluntary effortsto promote racial balance across schools throughrace-based school assignment plans have facedopposition from school leaders, policymakers,and the public. Despite recent federal court de-cisions in Lynn, Massachusetts, and in Seattle,Washington, 2 that have supported the right of dis-tricts to implement voluntary desegregation plansfor the purposes of integration and for promotingdiversity, the legal future of race-conscious stu-dent assignment plans is far from certain.3 TheK-12 implications of the Grutter v. Bollinger4

Supreme Court decision on the use of race in col-lege admissions are equally uncertain. AlthoughGrutter clearly suggests the state has a com-pelling interest in promoting racial diversity ineducational settings, and that race-conscious poli-cies can be used to do this-so long as they are"narrowly tailored" 5 -Grutter's relationship toschool desegregation policies has not yet beentested by the Supreme Court.

While the legal questions surrounding the useof race in K-12 voluntary school assignment plansremain unresolved, many school districts are al-ready turning toward socioeconomic school de-segregation plans as a viable race-neutral methodof school assignment. These include school dis-tricts in La Crosse, Wisconsin; Wake County,North Carolina; San Francisco, California; andCambridge, Massachusetts. Many of these dis-tricts are also gaining considerable attention overtheir unique assignment plans. Recent media re-ports attribute test score gains in Wake Countyto socioeconomic integration (Finder, 2005),although it is not clear that there is sufficient ev-idence of a causal warrant for this claim.

There are several arguments that underlie sup-port for socioeconomically based desegregationplans over race-based plans. Some policymakersand researchers argue that socioeconomic factorsare more reliable indicators of student and schoolperformance than race. This implies that achiev-ing socioeconomic integration will benefit stu-dents perhaps more than achieving racial inte-gration (Gottlieb, 2002; Rumberger & Palardy,2002). In addition, because race and social classare strongly correlated in U.S. society, some con-

Implications of Income-Based School Assignment Policies

tend that mixing students by economic status willalso ensure acceptable levels of racial integra-tion, thus providing students with the benefits ofboth racial and socioeconomic diversity (Chaplin,2002; Kahlenberg, 2001).

In the most comprehensive examination ofsocioeconomic desegregation, Kahlenberg (2001)outlines three main limitations of race-based de-segregation efforts. First, he argues, race is be-coming more of a legal liability than an advan-tage, as courts have become more receptive toFourteenth Amendment challenges to race-basedschool desegregation policies. Second, he positsthat it is social class composition-not racialcomposition-that determines school quality.And third, Kahlenberg contends that organizingefforts around improving educational opportuni-ties for all is more effective when inclusive ofWhites and middle-class parents, specifically,that "middle-class parents will use their politicalweight to bring greater equality of resources"(p. 83). Kahlenberg writes, "IT]he primary goalof socioeconomic integration is improved educa-tional achievement; economic balance is merelythe means to that end" (p. 111). He does not,however, argue that racial isolation is irrelevant;in fact, he contends that one of the benefits of so-cioeconomic integration is its capacity to reduceracial isolation.

The idea of socioeconomic integration is notnew. Since the Coleman report in 1966, consid-erable research has examined the relationshipbetween socioeconomic status and educationalachievement (Coleman et al., 1966). Such studiesconsistently find a sizable relationship betweenschool composition and educational achievementand attainment-in other words, in addition to in-dividual family background, a school's socioeco-nomic context is strongly related to students'educational outcomes (Anyon, 1997; Colemanet al., 1966; Entwisle, Alexander, & Olson, 1997;Mayer. 2002; Natriello, McDill, & Pallas, 1990).As a result, socioeconomic segregation betweenschools can contribute to lifelong educational(and economic) inequality.

Although research on the role of race andracial integration has been far less definite in itsconclusions, it has nevertheless pointed to someimportant benefits that can accrue in the deseg-regated or diverse schooling environment. Someof these benefits are directly connected to the so-cioeconomic integration rationale. For example,

51


previous studies suggest that part of what hasmade racial integration policies effective in in-creasing educational opportunities and in im-proving student outcomes is the access to the dif-ferent social networks present in the integratedschool or classroom versus segregated minorityschools (Wells & Crain, 1994). In addition, thereis evidence from the school desegregation litera-ture that desegregated schooling environmentsenhance African American students' educationalattainment (Boozer, Krueger, Wolkon, Halti-wanger, & Loury, 1992), and occupational aspi-rations (Dawkins & Braddock, 1994).

Yet, important racial and ethnic diversity ben-efits exist that are independent of income and thatmay be lost without some attention to racial com-position. In a recent study that disentangles racefrom socioeconomic effects, Hanushek, Kain,and Rivkin (2002) find that, net of individual andpeer socioeconomic background effects, higherBlack enrollment has a negative effect on mathe-matics achievement, and that these effects arestronger for higher-ability Blacks than for lower-ability Blacks or for White students. Moreover,these findings persist when controlling for differ-ences in school quality and the achievement ofpeers in the school and classroom. Rivkin (2000),however, does not find persistent effects of schoolracial composition on Blacks' educational attain-ment or earnings, when controlling for other in-dividual and contextual effects. Racial and ethnicdiversity also appears to produce social benefitsin domains other than achievement scores andeducational attainment. These benefits includethe ability to interact with members from racialand ethnic groups different than one's own(Kurlaender & Yun, 2003; Schofield, 1995), thepromotion of cross-racial friendships and peergroups (Hallinan, 1998; Hallinan & Williams,1989), and the ability to think more critically aboutsocietal and democratic issues (Duncan, Boisjoly,Levy, Kremer, & Eccles, 2003; Gurin, Dey,Hurtado, & Gurin, 2002). Moreover, there is evi-dence that racial segregation is self-perpetuating,and that school desegregation can break the cycleof societal racial isolation (Wells & Crain, 1994).Thus, the social benefits of integrated or diverseschooling environments can accrue not only to mi-nority students, but also their White peers.

Despite these positive findings, the researchon the benefits of racial integration remains con-tested and empirically unclear, in large part be-

52

cause it is difficult to disentangle racial composi-tion effects from other socioeconomic and schoolquality factors that impact educational processes.That said, enough evidence suggests the positivebenefits of racial integration that it is worth ex-amining whether socioeconomic integration nec-essarily produces racial and ethnic integration.

1.2. Current Income-Based SchoolAssignment Plans

The first school district to implement a so-cioeconomic desegregation plan was La Crosse,Wisconsin, which adopted its plan-based exclu-sively on students' free lunch eligibility-in 1992.La Crosse is a small, predominately White dis-trict. Although the 1990s brought an influx ofsoutheast Asian immigrants, today the schooldistrict is about 15% minority (predominantlyHmong and Laotian refugees). Only about 3%of students are Black, Latino, or Native American(Kahlenberg, 2001). The relative homogeneity ofLa Crosse's student enrollment makes it difficultto assess whether the socioeconomic-based planhas produced racial desegregation across schools,though a simple survey of the racial compositionacross schools of similar average socioeconomicstatus suggests that it has not. For example, in2002-2003, Asian (the largest minority group)enrollment shares across the district's 11 ele-mentary schools ranged from a low of 5% insome schools to a high of 40% in others.6

Other school districts are developing morecomplicated methods for determining students'socioeconomic status and for assigning studentsto schools. In 1999, Wake County, North Car-olina, removed the consideration of race from thedistrict's school assignment plan in favor of asocioeconomic and test-score-based assignmentplan. Wake County determines socioeconomicstatus based on neighborhood characteristics,as opposed to individual characteristics, and hasmapped out more than 700 "neighborhood zones"to classify students for assignment purposes(Silberman, 2002). Under this plan, no schoolmay have more than 40% of students from poorneighborhoods (as defined by aggregate neigh-borhood free and reduced-price lunch eligibility)and no more than 25% of students scoring belowgrade level on standardized tests (Jones, 2002;Silberman, 2002). A preliminary evaluation ofthe policy suggests that adoption of the class-based plan may have initially led to an increase in

I

racially identifiable schools (those with greaterthan 45% or less than 15% minority students-the guideline used under the previous race-baseddesegregation plan), but that this increase has sta-bilized in subsequent years (Flinspach, Banks, &Khanna, 2003).

In San Francisco, which has been under aschool desegregation consent decree7 for 2 de-cades, the school board several years ago adopteda socioeconomic integration plan based on a"diversity index." The move followed a con-tentious lawsuit in which a group of parentssued the district for restricting admission ofChinese students into the competitive magnetschools despite higher exam scores.8 The diver-sity index used for student assignment includesinformation on the following: free and reduced-price lunch eligibility; public housing assistance;low achievement as measured by scoring belowthe 30th percentile on the Stanford 9; limited ornon-English proficient; prior school's CaliforniaAcademic Performance Index; home language ifother than English; and residence (Flinspach,Banks, & Khanna, 2003; Kahlenberg, 2001).However, since the diversity index functions aspart of a choice system, it only helps to assignstudents to schools that have more students want-ing to attend than the school has seats (Commu-nity Advisory Committee on Student Assign-ment, 2005). Thus, the index does not addresssegregation in any schools with vacancies, whichinclude many of the city's most segregatedschools. Because the assignment process in ef-fect is relatively new, it is too soon to tell whetherit will sustain the racial and ethnic desegregationlevels achieved under the former race-based plan,but preliminary descriptive information suggeststhat it has not. In 1998-1999, the last year of theconsent decree, 64% of San Francisco schoolswere in compliance with the racial desegregationstandard of the decree; by 2002-2003, 2 yearsafter the start of the socioeconomic integrationplan, only 52% of San Francisco schools wereracially integrated, as measured by the samestandard (Flinspach, Banks, & Khanna, 2003;O'Keefe, 2005).

An additional variant of income-based schoolassignment is used in the Cambridge, Massachu-setts, public school district, which began in 2001to use income in addition to race to determineschool assignments (Fiske, 2002). The new pol-icy does not remove race from consideration in


school assignments, but reduces its relative im-portance in assigning students to elementaryschools. As with the income-based plans in placein Wake County and San Francisco, it is too soonto tell whether this plan will sustain the racial in-tegration attained under the prior policy, whichrelied more heavily on race.

These plans represent a trend in student as-signment toward using socioeconomic factors inplace of race (or in place of using race alone) inschool assignment policies designed to establishdiverse schools. Although the approach and im-plementation of such plans vary across districts,they share a common emphasis on socioeconomicfactors (especially family income and factorsrelated to income, such as free and reduced-pricelunch eligibility, neighborhood poverty, and pub-lic housing assistance). To date, however, there islittle empirical evidence on the implications ofsuch school assignment policies for racial segre-gation. Our goal in this article is to provide someanalysis that will be helpful in determining theimplications of such policies, as well as to pro-vide some guidelines that may aid educationalpolicymakers in identifying optimal income-based policies.

1.3. Prior Research on the Impact of EconomicIntegration on Racial Segregation

Some researchers claim that decreases in so-cioeconomic segregation will necessarily lead todecreases in racial segregation due to the high cor-relation between income and race (Chaplin, 2002;Kahlenberg, 2001). Chaplin's (2002) study, how-ever, is the only systematic study to test this as-sertion empirically, and so is worth examining insome detail. Chaplin's study begins by computingthe overall racial segregation among all publicschools in the United States, and then asks howmuch this figure would be reduced through race-neutral income-integration policies. Chaplin' s op-erational definition of income integration requiresthat no school have more than 50% of studentseligible for free or reduced-price lunch. This is arelatively weak income integration requirement,because it would allow some schools to remainpoverty-free, while other schools would have asmany as 50% poor students.

Chaplin finds, first, that even if all racial segre-gation within every district were completely elim-inated, overall racial school segregation in theUnited States would be reduced by only 7-11%,

53


depending on which race groups are considered(Chaplin, 2002, Table 2). This is because mostU.S. school segregation results from between-district differences in the racial composition ofstudent populations rather than from uneven pat-terns of within-district assignment to schools(Reardon, Yun, & Eitle, 2000). Next, Chaplinfinds that if the minimum number of studentswere reassigned-without regard to race or trans-portation distances-within each district to cre-ate, where possible, schools with fewer than 50%poor students, then overall racial school segrega-tion would be reduced by only 1-4%, dependingon which race groups are considered (Chaplin,2002, Table 2). While this represents a verysmall fraction of all racial school segregation, itis a nontrivial proportion of the within-districtportion of school segregation. Finally, Chaplinasks what would happen to racial integration ifincome integration were implemented across met-ropolitan areas, rather than districts. This wouldinvolve the transfer of some poor students fromdistricts with poverty rates greater than 50%(where students could be reassigned in such away that all schools would have poverty ratesless than 50%), to schools in other districts withpoverty rates below 50% (and correspondingtransfers of nonpoor students in the opposite di-rection). Under this scenario, Chaplin finds thatoverall racial segregation would be reduced by5-15%, again depending on the racial group in-volved (Chaplin, 2002, Table 2).

Chaplin's study is informative, but also leavesa number of important questions unanswered.First, the study does not indicate what happens toracial segregation in individual districts. Sinceincome-integration policies are most likely to beenacted at a district level, we (and district offi-cials) would like to know the potential effects ofincome integration on racial integration withinindividual districts as well as on national pat-terns. Chaplin's approach averages over all typesof districts-large and small; urban, suburban,and rural; poor and nonpoor, racially diverse andhomogeneous-and so is uninformative regard-ing the local conditions under which income-integration policies might produce some ancil-lary racial segregation.

Second, Chaplin's operationalized definitionof income integration is both too weak and toostrong. It is too weak because, in practice, dis-tricts are likely to require somewhat more strin-

gent income integration than the 50% poverty ratemaximum (though perhaps not too much morestringent-Wake County, for example, has a 40%maximum enrollment from high-poverty neigh-borhoods). Thus, Chaplin's method may under-estimate the potential impact of income integra-tion on racial segregation. On the other hand, hisoperational definition is too strong, because inter-district income-integration policies are unlikelyto be widely adopted; thus his within-district re-sults are far more plausible than those that relyon inter-district student assignments. Finally,even Chaplin's within-district estimates certainlyoverstate the impact of income integration onracial segregation because they depend, as henotes, on the assumption that the reassignment ofpoor and nonpoor students within a district is ran-dom. In practice however, as we discuss below,in large urban and county-wide school districts,transportation distances make it impractical toassume a random reassignment of students amongschools; students would likely be reassigned toschools nearer their residence. In a district withsubstantial racial residential segregation, this willresult in students of the same race being dispro-portionately reassigned to the same schools, re-sulting in higher levels of racial segregation thanChaplin estimates.

Chaplin's paper is useful, however, in thatit sets lower bounds on the levels of overall(national) racial segregation that might resultunder district- and metropolitan-level income-integration plans. These lower bounds are quitehigh, suggesting little overall racial dividend fromincome-integration policies. Nonetheless, it maybe that income integration produces substantialracial integration in some types of districts orunder some types of operationalized policies.

In addition to Chaplin's study, a case study ofsocioeconomic integration in two districts sug-gests that the impacts of new socioeconomic in-tegration policies may in fact lead to increasedracial segregation or, at best, leave it unchanged(Flinspach, Banks, & Khanna, 2003). Specifi-cally, in San Francisco, preliminary evaluationsuggests that the plan has led to an increase inracial and ethnic segregation, while in WakeCounty the income-integration policy appearsto have initially produced a slight increase inracial segregation that has subsequently leveledoff. (Flinspach, Banks, & Khanna, 2003). Inseveral years, trend data from these and other

54

districts will shed more light on these outcomepatterns.

One additional source of potential evidence onthe likely effect of a race-neutral income policyon racial outcomes comes from the research onincome-based admissions policies in higher edu-cation. Kane (1998), for example, finds that class-based affirmative action does not provide a reli-able substitution for race-conscious admissionspolicies. His conclusion is based in large part on arelatively simple empirical observation that suc-cess in enrolling more Black students using class-based admissions policies depends on the ratio be-tween poor White and Black students in the uppertail of the achievement distribution. Althoughthere are some parallels from this work to theK-12 school integration context, the problem issubstantively different. The extension of thesefindings to socioeconomically based school de-segregation policies relies on the overlap betweenrace and class in the middle of the income distri-bution, not the tails. In addition, there are not a fi-nite number of spots in a school district, so thechallenge is how to allocate seats to all studentsmost efficiently. These differences suggest thatvery different outcomes may be possible in schoolassignment versus college admissions. A betterparallel for Kane's higher education work in K-12school assignment might be choice or magnet pro-grams that rely on socioeconomic factors to allo-cate a finite number of spots in desirable schoolsto students of different racial groups, a situationthat is beyond the scope of this article.

Given the relatively thin nature of the empiricaland analytic evidence regarding the likely effectsof income integration on racial segregation, thisarticle investigates the possible effects of incomeintegration in large, diverse school districts-thetype of district where racial segregation is gener-ally most pronounced-and investigates the con-ditions under which such policies might producesubstantial ancillary racial benefits. Moreover, wefocus solely on within-district income-integrationpolicies, as these are the most practical approachto income integration, Kahlenberg's (laudable)call for interdistrict income-integration policiesnotwithstanding (Kahlenberg, 2001).

2. Determining the Effects of IncomeIntegration on Racial Integration

What level of racial segregation is likely tooccur under a race-neutral socioeconomic inte-


gration regime? The answer will depend on sev-eral factors: (1) the strength of the associationbetween race and income; (2) the particulars ofthe policies defining socioeconomic integration(e.g., How is income measured? How much in-come balance across schools is required? Whatother socioeconomic factors are considered?);(3) the relationship between racial and incomeresidential segregation in a school district; (4) thefactors determining school assignment (e.g., dis-tance to school, transportation costs, parentalpreferences or choice); and (5) the effect of theincome-integration policy on families' decisionswhether to move into or out of a school districtand whether to enroll their children in public orprivate schools.

We begin by addressing a simpler question-what level of racial segregation is possible undera socioeconomic integration regime?-becausethe answer to this question places upper andlower bounds on the primary question. In addi-tion, our analysis simplifies the issue by focusingon a stylized version of a socioeconomic inte-gration policy that relies solely on income, ratherthan a broader range of socioeconomic factors, inschool assignment. Following this, we use twodifferent strategies to address the question ofwhat level of racial segregation is likely under anincome-integration regime, and then discuss theimplications of our results for school assignmentpolicies.

For the purposes of this article we define per-fect income integration (or desegregation) as thecondition in which each school in a district hasthe same income distribution as the district over-all. This definition of income desegregation cor-responds to the "evenness" definition of racial in-tegration (Massey & Denton, 1988).9 In practice,however, it would be administratively and prac-tically unrealistic for school districts to requireall schools to have exactly identical income dis-tributions, both because districts typically do nothave access to data on families' exact incomes,and because of the administrative and logisticaldifficulties of such a school assignment policy. Amore practical approach, for example, would befor districts to use data on families' poverty sta-tus or free and reduced-price lunch eligibility(rather than exact income data) and to requirethat all schools have poverty or free and reduced-price lunch eligibility rates that fall within somerelatively small range-say, ±10% or ±20% of

55


the overall rate of total district enrollment (as is thecase in Cambridge and in La Crosse, for example).In the analyses that follow we consider the impli-cations of both the "ideal" (complete income in-tegration) and the more practical (approximateintegration) types of income integration.

2.1. Measures of Segregation

Throughout this article we limit our analysesto the case of segregation between two groups(e.g., between Black and White students). Theextension to the multigroup case is more mathe-matically complex, but not likely to lead to sub-stantively different conclusions. 10 We measuresegregation in this article using an easily inter-preted measure of two-group segregation, thenormalized exposure index (denoted V). Thenormalized exposure index measures the gap be-tween the observed exposure level of one groupto another and the exposure level that wouldoccur in a situation of perfect integration, where"exposure" is defined as the average proportionof the second group in the schools attended bystudents of the first group. Formally, V betweengroups m and n is defined as

V=l P,(1)itK.

where 7c, is the proportion of group n in the dis-trict and ,P, is the exposure of group m to groupn, defined by:

R= -- l,p (2)

where ] indexes schools, 7uj is the proportion ofgroup n in schoolj, and tmj and Tm are the numberof students of group m in schoolj and the district,respectively.

The normalized exposure index V has a min-imum of 0, obtained if and only if ,Pn = 71n,which can occur only in the case of perfect racialintegration-if each school has the same propor-tions of group n as in the total district enrollment.The index has a maximum of 1, obtained only inthe case of complete segregation-when no stu-dent of group m attends a school with any stu-dents of group n.I This index has been widelyused in the racial school segregation literature,under a variety of names (Clotfelter, 1999, 2001;Clotfelter, Ladd, & Vigdor, 2003; Coleman,

56

Hoffer, & Kilgore, 1982; see also Farley, 1984;James & Taeuber, 1985; Steams & Logan, 1986).We will refer to it here as the normalized expo-sure index (denoted 1) to suggest its relationshipto the exposure index.

Although a number of other measures of seg-regation are available in the literature (James &Taeuber, 1985; Reardon & Firebaugh, 2002), weuse the normalized exposure index both becauseit is easily interpreted and because the mathemat-ics of our analysis turn out to be far simpler whenwe use this index rather than other measures, suchas the dissimilarity and information theory in-dices. As a measure of the evenness of the distri-bution of two racial groups among schools, V ishighly correlated with the dissimilarity index D(r10.96) and with the information theory index H(1-0.98) in school and residential segregationstudies (Massey & Denton, 1988; Reardon, 1998;Reardon & Firebaugh, 2002), and is a moremathematically appropriate index than the morecommonly used dissimilarity index (because it re-sponds more appropriately to changes in the dis-tribution of students among schools than does D)(Reardon & Firebaugh, 2002). Given the almostexact correspondence between V and H, we usethe guidelines similar to those given by Reardonand Yun (2001; 2003; 2005) to interpret the lev-els of V. A value of Vabove 0.50 is considered ex-tremely segregated (this corresponds roughly tolevels of.D above 0.70, close to what Massey andDenton (1989) require as part of their definitionof "hypersegregation"). Such a value means that,on average, Black students attend schools withonly half the percentage of Whites we would ex-pect given the composition of a school district(and vice versa).12

2.2. The Range of Racial Segregation PossibleUnder Income-Integration Regimes

The range of racial segregation possible underan income-integration regime depends primarilyon the strength of the association between raceand income. For example, if poverty status wereperfectly correlated with race (i.e., if all Blackswere poor, and no Whites were poor), then com-plete income desegregation would necessarilyentail complete racial desegregation-if eachschool had the same proportion of poor students,each would have the same racial proportions aswell. Conversely, if income or poverty statuswere uncorrelated with race (i.e., if the Black and

White income distributions were identical), thencomplete income desegregation could (in princi-ple) be achieved even while maintaining com-plete racial segregation-all schools could bemonoracial yet have identical income distribu-tions. In the intermediate (and more realistic)case, where income distributions differ by race,but overlap considerably, some racial segregation(but perhaps not complete racial segregation)would be possible even under complete incomeintegration.

In addition, the range of racial segregationpossible under an income-integration regime willdepend on how income desegregation is definedand operationalized. First we consider the (unre-alistic) case in which a school district has infor-mation on each family's actual household in-come and where an income-desegregation policyrequires all schools to have identical income dis-tributions. This scenario, while unrealistic, yieldsthe lowest possible upper bound on the range ofpossible racial segregation levels, because it usescomplete information on incomes and requiresperfect balancing of racial distributions acrossschools.

Given that school districts are unlikely to ob-tain actual income data from families, we nextconsider the range of racial segregation levelsthat are possible if a school district uses a di-chotomous measure of income-such as povertystatus or free and reduced-price lunch eligibility-in an income-desegregation plan. If the povertythreshold is set at a very low or very high in-come level, then almost all students will fallinto the same income category, and an income-desegregation policy will put virtually no con-straint on school enrollment patterns. In such acase, any level of racial segregation is possible,since the poverty status indicator will be uncor-related with race. If, however, the poverty thresh-old is set at some point in the middle of the in-come distribution, poverty status may be more orless correlated with race (depending on where thepoverty threshold is set relative to the racial in-come distributions), and so income desegregationwill necessitate some level of racial desegrega-tion. We show in this article how to determinethe location of the optimal poverty threshold-the threshold that minimizes the maximum pos-sible racial segregation.

When using a dichotomous measure of incomesuch as poverty status, administrative and logisti-


cal concerns make attaining perfect income bal-ance across schools impractical. In practice,school districts seeking to achieve income inte-gration are likely to require some approximatelevel of income integration, typically by requiringthat each school have a poverty rate within some(relatively narrow) range of the overall districtpoverty rate. Given this, we next consider the levelof racial segregation possible under a policy thatrequires only approximate income integration.

These analyses quantify the extent of racialsegregation that is possible, in principle, underseveral types of income-desegregation regimesand a range of racial income-distribution patterns.Next we address the issue of what level of racialsegregation is probable under such conditions.

2.2.1. Computing upper and lower bounds on

possible racial segregation

Suppose students are assigned to schools in thefollowing way: first, White students are assignedto schools in such a way that (1) the number ofWhite students in each school is the same; and(2) the White income distribution within eachschool is the same across all schools. Next, thesame procedure is followed for each racial group.Now each school will have the same racial com-position and the same income distribution; hencethe district will show perfect racial and income in-tegration. Because this procedure is possible re-gardless of the racial composition or income dis-tribution within a district (see a formal proof inAppendix A), the lower bound on the level ofracial segregation possible under complete in-come integration will always be 0. But what isthe upper bound? How much racial integration isrequired in order to obtain income integration?

To answer this, we undertake the followingthought exercise. For simplicity, we consider apopulation made up of only Black and White stu-dents, though the method can be generalized tomultiple groups. Suppose we have a populationwhose racial composition and race-specific in-come distributions are illustrated by Figure 1. Inthis figure, the White income distribution is de-scribed by the area shaded light gray; the Black in-come distribution is described by the area shadedin dark gray. The sum of these distributions-thetotal income distribution is described by theheavy black line (note also that the Black incomedistribution is also described by the area betweenthe heavy black total income distribution line and

57


W

LIL

Ez

0

* BlackrE- White

-- Total

50,000 100,000 150,000

Income (dollars)

FIGURE 1. Stylized income distribution, by race.

the light gray White income distribution area). Inthis figure-which corresponds roughly to the ac-tual Black and White income distributions in theUnited States, although our discussion is indepen-dent of the distributional parameters used here-Black incomes are, on average, considerablylower than White incomes.

Now suppose we attempt to assign students toschools in such a way as to produce maximumracial segregation, subject to the constraint thateach school has the same income distribution. Forillustration, suppose that the school district con-sists of four identically sized schools. To maxi-mize racial segregation, we attempt to assign onlyWhite students to the first school, except wherethere are not enough White students in some partof the income distribution (see Figure 2). In thisway, we produce a school (school 1 in Figure 2)that enrolls one-fourth of the student population,has an income distribution that mirrors the popu-lation distribution, and has the maximum numberof White students possible. We repeat this processin the next school, drawing students from the re-maining student population; this school (school 2in Figure 2) also enrolls one-fourth of the studentpopulation, has an income distribution that mir-rors the population distribution, and has the max-imum number of White students possible (amongthe students remaining). We repeat this processuntil all schools are filled. This assignment processwill produce a set of schools with perfect incomeintegration, but maximal racial segregation (For amore formal treatment of the mathematics of com-

puting the maximum possible racial segregation,see Appendix B).

The assignment mechanism described in Fig-ure 2 is, by definition, a race-conscious mecha-nism. It is possible, however, that a similar pat-tern of school assignment could result from arace-blind assignment mechanism. Suppose wehave a city in which the Black and White publicschool student populations are completely resi-dentially segregated from one another-all Blackstudents live nearer to one another than to anyWhite student. Moreover, suppose that the Blackincome distribution is lower than the Whiteincome distribution, but that within each racialgroup there is little or no residential incomesegregation--each Black and White neighbor-hood has an income distribution that matches theoverall Black or White income distribution, re-spectively. Finally, to minimize transportationcosts, suppose we will try to assign each studentto the nearest possible school to his or her home,subject to the constraints of the income-desegre-gation requirements.

Such a situation will result in the maximumpossible racial segregation consistent with theincome-desegregation policy, because most Blackand most White students will attend schools intheir own (racially isolated) neighborhoods. Tosee this, consider the following recursive schoolassignment process: we begin with one school-say, in a White neighborhood-and assign to itstudents from local neighborhoods, attempting toconstruct a student body with the same income

58


Income (dollars)

FIGURE 2. Stylized school enrollments, by race and income.

Black

Wl hite

SBlack, School 1

El White, School 1

E

z

B

E-

Bý

E

100,000

Income (dollars)

=Ez

150,000 050,000 100,000

Income (dollars)

distribution as the whole population. If there arenot enough students from the local neighborhoodin some part of the income distribution, we as-sign additional students from a nearby neighbor-hood, always subject to the constraints of theincome-desegregation policy. To the extent thatonly White students live nearby, this school willbe entirely White, unless there are not enoughWhite students at the lower end of the incomedistribution to create income balance within theschool, in which case, we will assign Black low-income students from other neighborhoods to theschool to fill out the income distribution withinthe school. We repeat this process for each schoolin turn, again attempting to fill the income distri-bution of the school with local students, unlessthere are not enough local students in some partof the income distribution, in which case we as-sign students from other neighborhoods to fill outthe income distribution. We continue this processuntil all students are assigned to schools. Becauseof the strong correlation between race and loca-tion, coupled with the assignment mechanism fa-voring short transportation times, the resultingsegregation patterns will be similar to those de-

scribed in Figure 2, despite the fact that race is notused in school assignment. Thus, the maximumpossible racial segregation is, in principal, ob-tainable even in the absence of a race-consciousassignment mechanism, though the likelihood ofobtaining such patterns will depend in practice onthe extent of residential racial and economic seg-regation in a region. We return to this point below.

2.2.2. Computing the maximum possiblesegregation when income is measureddichotomously

In the above example, we have assumed a con-tinuous income variable (and hence a smoothincome distribution density function), but theanalysis applies equally well to the case in whichincome is measured dichotomously (above orbelow poverty threshold, for example). In thiscase, the income distribution density functionwill be given by a step function rather than asmooth curve. The case where income is mea-sured dichotomously is of particular interest, be-cause school systems generally do not have accessto exact-or even approximate-family incomedata for their students. However, they generally

59

F:BlackIJ White

U Black, School

i White, School 2

B k 1o 12EZ

0

I

50,000 50,000100,000

Income (dollars)

150,000

150,000

m


have free and reduced-price lunch eligibility sta-tus data, and may use this dichotomous indicatorof income to categorize students in an income in-tegration plan.13

When income is measured dichotomously, themaximum possible segregation (as measured bythe normalized exposure index, V) is given by thefollowing simple formula (see Appendix C):

v.= = - 1C , - K-I-(3

So the maximum possible segregation, as mea-sured by the normalized exposure index, dependssimply on the difference in poverty rates betweenBlacks and Whites. It does not depend on theracial composition of the school enrollment. Thisprovides a simple, easily computed measure ofthe extent of maximum possible segregation.14

Moreover, Equation 3 implies that segregationlevels can be quite high unless there are dramat-ically different poverty rates between the racialgroups in question. For example, consider a dis-trict with a White poverty rate of 0.20 and aBlack rate of 0.50-a sizeable poverty gap. Evengiven this substantial race difference in povertyrates, Equation 3 shows that a district could as-sign students to schools in such a way that allschools had equal proportions of poor students,yet the district would still have an extremely highracial segregation level of V= 0.70.

2.2.3. Computing the optimal povertythreshold

Given that it may be administratively muchsimpler for school districts to base an income-desegregation plan on a simple dichotomous in-come indicator rather than on a continuous in-come variable, does it matter what income cutoffpoint is used? And, if so, what income cutoff isoptimal-what income/poverty threshold yieldsthe minimum value of the maximum possiblesegregation described in Equation 3?

Intuitively, a poverty threshold that producesthe strongest association between race and di-chotomous income will be optimal. Such a pointwill be near the middle of the income distribution,since a threshold at the high or low end will resultin both Blacks and Whites falling primarily intothe same income category. Figure 3 illustrates therelationship among the Black and White incomedensities and the optimal poverty threshold."5

Note that the optimal poverty threshold given

these typical (gamma-shaped) Black and Whiteincome distributions falls roughly midway be-tween the Black and White mean incomes (seeAppendix D). Moreover, note that the maximumpossible racial segregation is relatively insensi-tive to the choice of a poverty threshold as long asthe threshold is between the mean incomes of thegroups. This finding implies that, in choosingwhere to set a poverty threshold for an income-integration policy, districts can optimize the racialbalancing effects of the policy by selecting apoverty threshold in the broad range between theWhite and Black mean income for their district.

2.3. Computing Racial Segregation UnderImperfect Income Desegregation

Because it may be administratively or practi-cally unreasonable to require that all schoolshave exactly the same proportion of studentsbelow some defined poverty threshold, we nextconsider the case where an income-integrationpolicy requires only imperfect income balanceacross schools. In general, relaxing the constraintof perfect income balance will-all else beingequal-allow for higher levels of racial segrega-tion. In particular, we consider the case where theproportion of the total student enrollment belowthe poverty threshold is it, and where an income-desegregation policy requires each school to havea poverty rate within ±d of 7rj, so that for eachschool j, ;it- di <rpj:< 7,t + d. Appendix C indicatesthat the maximum possible racial segregationobtainable under such an income-desegregationpolicy is given by

I-= V2d 1-2d.-pI"l -2d 1-2d (4)

where V,,, is the maximum possible racial segre-gation if the income-desegregation policy requireseach school to have exactly the same poverty rate(see Equation 3).16 Note that Equation 4 showsthat even a modest value of d may substantially in-crease the maximum possible segregation. If dis only 10%, for example, V.a is 25% greaterthan Vmx.-

2.4. Results: The Maximum Possible RacialSegregation Consistent With

Income Integration

As we have shown above, the maximum possi-ble racial segregation consistent with an income-integration policy depends in part on the race-

60


_._ 1.0-

U 0.8-0)

a)a)ci)

FU 0.6-

a)"• 0.4-

00-E 0.2 -

E

0.0-

I I

I I

VAI II I

I II I

a)0a)E0

Black Mean Optimal White MeanThreshold

0 25,000 34,700 50,000 75,000 100,000

Poverty Threshold (dollars)FIGURE 3. Maximum possible racial segregation, by poverty threshold (based on ECLS-K estimates of 1998Black and White income distributions in urban districts).

specific income distributions, as well as thespecifics of the income-integration policy. Forinformation on actual race-specific income dis-tributions among the families of public schoolstudents in large school districts in the UnitedStates, we rely on several sources of data. First,we examine data from the Early Childhood Longi-tudinal Study-Kindergarten Cohort (ECLS-K)-a nationally representative sample of kindergartenstudents in the 1998-1999 school year (NationalCenter for Education Statistics, 2001)-to de-scribe the income distributions among White andBlack public school students' families in bothurban and suburban school districts. Second, weexamine 2000 Census data to determine therange of White-Black income ratios across com-munities in the United States, because theECLS-K distribution estimates are national av-erages, but do not provide good estimates of thevariation across districts.

ECLS-K data show that in cities and suburbs,the national mean White income is approxi-mately 2.1 times the mean Black income (mean1998 White and Black incomes were $52,100and $24,600 in urban districts and $68,100 and$31,700 in suburban districts, respectively).17 In

addition, Census 2000 data show that White-Black household income ratios in central citiesand suburbs range from roughly 1.0 to 2.0.18 Thesefigures suggest the extent of variation across com-munities in White-Black income ratios, althoughthey are not exactly comparable to the ECLS-Kfigures because they are based on the averageincomes of all White and Black households (in-cluding those without children), whereas theECLS-K estimates are based on the average fam-ily incomes of White and Black public school stu-dents. Based on these figures, in our results belowwe report estimates of possible racial segregationfor places with White-Black income ratios rang-ing from 1.0 to 3.0, although there appear to befew places where the White-Black income ratioamong public school students' families is greaterthan 2.0.

Figure 4 illustrates the relationship betweenthe White-Black income ratio and the maximumpossible racial segregation (as measured by V)under four types of income-integration regimes:perfect income integration using complete in-come information; perfect income integrationusing an optimal poverty threshold; and approxi-mate integration using an optimal poverty thresh-

61

- Maximum V

- White Income Density

- Black Income Density


old and poverty rates of ±10% and of ±20%. Notethat under each type of income-integration policy,the maximum possible value of V is insensitiveto the racial composition of the district.

First we report the maximum possible segrega-tion under income-based school assignment poli-cies where income is treated continuously. As weexpect, at White-Black income ratios close to 1.0,racial segregation may be quite high. At an in-come ratio of 2.1, roughly the national White-Black income ratio observed in the ECLS-K pub-lic school data, V has a maximum possible valueof 0.56, meaning that complete income balanceacross schools could be achieved even whileBlack and White students attended schools withonly an average of 44% as many members of theother group as in the overall district enrollment.This is a nontrivial level of segregation; in fact itis higher than the segregation levels in most urbanschool districts in the United States.

Under a policy that uses a dichotomous in-come (poverty) measure, the income thresholdthat produces the lowest possible maximum levelof segregation (given the Black and White in-come distributions derived from the ECLS-Kdata above) is roughly $35,000 (see Figure 3).

1 .- l - -- --ý- -

0.8-

-)-0.6-

0)

2 0.4-0)

Conveniently, this corresponds roughly to the in-come cutoff for reduced-price lunch eligibilitystatus for a family of four in 2000.19 At this opti-mal dichotomization point, an income-integrationpolicy guarantees roughly 25% less racial inte-gration than would a policy using continuousmeasure of income-at a White-Black incomeratio of 2.1, the maximum possible racial segre-gation under a dichotomous income measurewould be 0.66, compared to 0.56 using a contin-uous income measure.

Figure 4 also illustrates the loss of efficiencythat comes from allowing schools to vary some-what in their poverty rates. Allowing schoolpoverty rates to vary by ±10% of the districtmean guarantees very little racial integration,while allowing them to vary by ±20% guaran-tees no racial integration, except in districtswith extremely high racial income disparities.Finally, we note that the loss of efficiency thatcomes from allowing approximate income inte-gration is greater than the loss that comes fromusing a dichotomous rather than a continuousincome measure.

The results shown in Figure 4 suggest that it ispossible, in principle, for racial segregation to re-

Optimal Dichotomous Income Measure, +/-20% Balance

Optimal Dichotomous Income Measure, +/-10% Balance

Optimal Dichotomous Income Measure, Perfect Balance

Continuous Income Measure, Perfect Balance

1.0 1.5 2.0

White-Black Income Ratio

FIGURE 4. Maximum possible racial segregation, by White-Black income ratio and incomeintegration policy.

62

0.2-

0.0-

2.5 3.0

-------------


main unchanged (or even increase, though thiswould be likely only in districts that currently haveracial integration policies in place that would bediscontinued under an income-integration policy)from current levels in districts that implementincome-based integration plans. Moreover, Fig-ure 4 shows that, even at White-Black incomeratios much higher than those present in U.S.cities, income-integration policies would not guar-antee low levels of racial segregation.

In sum, income-integration policies providelittle or no guarantee of reductions in racial seg-regation. Under any practical income-integrationregime, high levels of racial segregation are, inprinciple, possible, unless race and income werefar more highly correlated than they are in fact.Nonetheless, the results illustrated in Figure 4are upper bounds on the level of possible racialsegregation; obtaining these levels of segrega-tion would certainly require some level of race-conscious school assignment policy. In the fol-lowing section, we compute lower bounds on thelevel of racial segregation that is probable underan income-integration regime.

2.5. Estimating Probable Levelsof Racial Segregation Under

an Income-Integration Regime

The preceding analyses determined the maxi-mum possible racial school segregation under var-ious conditions. In general, the results indicate that,under conditions typical of large school districtsin the United States and under practical income-desegregation policies, achieving income desegre-gation guarantees little to no racial integration-inother words, income-integration policies wouldplace only a very high upper bound on the level ofpossible racial segregation. This is not to say,

however, that no racial integration would resultfrom an actual income-desegregation policy. Weemploy two strategies to provide plausible esti-mates of the lower bound of probable levels ofracial integration that would be obtained underrace-blind income-integration district policies.First, we examine patterns of racial and socio-economic residential segregation in large U.S.cities. Second, we conduct simulations, startingwith patterns of racial and socioeconomic schoolcomposition in large U.S. districts, to determinethe effect that income integration would have onpatterns of racial school segregation. While nei-

ther approach is definitive, both approaches areinformative regarding the probable effects of in-come-integration policies on racial segregationpatterns.

2.5.1. The relationship betvveen racial andincome residential segregation

The actual level of racial segregation likelyunder an income-desegregation policy will de-pend not only on the particulars of the policy butalso on the degree of residential racial and in-come segregation within a school district, be-cause racial school desegregation will be morelikely under an income-integration regime in adistrict where income segregation is high butracial segregation is low. Conversely, in a dis-trict where residential racial segregation is highbut where the level of income segregation withineach racial group is low, income integrationamong schools could be achieved largely by hav-ing students attend schools in their own (raciallysegregated, but economically diverse) neighbor-hoods, and so would be unlikely to produce sig-nificant racial desegregation as a by-product.

To see this, imagine a district composed of fourstylized types of households-poor White house-holds, poor Black households, nonpoor Whitehouseholds, and nonpoor Black households. If thepoor White and Black households are near eachother and far from nonpoor households, then anumber of students will have to travel to rela-tively distant neighborhoods (some of which arepredominantly White and some of which are pre-dominantly Black) to achieve income balanceacross schools. Under a race-blind policy, thiswill likely lead to substantial racial integration,since poor Blacks will be equally likely to attendschool with nonpoor White students as with non-poor Black students, and vice versa. If, however,poor and nonpoor Black households are near eachother, and far from both poor and nonpoor Whitehouseholds, then an efficient transportation sys-tem that attempted to minimize the distance thatstudents travel to school would likely result inmany poor and nonpoor Blacks attending schooltogether and many poor and nonpoor Whites at-tending school together. In this case, income in-tegration might be achieved without producingsubstantial racial desegregation.

A more formal consideration of these spatialdynamics and their impact on school socioeco-nomic and racial integration patterns is beyond

63


the scope of our analyses here. We can, however,get a rough sense of how the relationship betweenracial and income residential segregation mightimpact the probable level of racial segregationunder a race-neutral income-desegregation policyby examining existing patterns of racial and in-come segregation in U.S. cities. Research con-sistently shows that income disparities betweenracial groups do not fully explain patterns ofracial segregation; racial segregation is quite higheven among households with similar income lev-els (see, for example, Clark & Ware, 1997; Dar-den & Kamel, 2000; Farley, 1995, 2003). As are-sult, economic segregation levels are typicallysubstantially lower than racial segregation levels,both within racial groups and in the total popula-tion (Massey & Fischer, 2003). An examinationof data from the 2000 Census illustrates this pat-tern in U.S. urban areas (Figure 5).

Figure 5 shows the relationship between racial(Black-White) segregation and within-race incomesegregation in the 228 metropolitan area centralcities with at least 5% Black population in 2000.Although racial segregation is high in many ofthese cities, with an average of 0.31 (recall that0.50 is considered extremely segregated), residen-tial income segregation (here measured as the seg-regation between those with household incomes

1.0-

c 0.8-

0)

0.

CLC0)

c-~

0zSL0.3-00

0.0

0.00

above and below $35,000)20 is quite low in com-parison, averaging 0.12 for Black households and0.09 for White households. In all U.S. cities withhigh levels of Black-White segregation, incomesegregation within racial groups is low. This sug-gests that a race-neutral income-desegregationpolicy would be able to achieve income balanceacross schools without substantially reducingracial segregation. The evidence of high racial seg-regation among households with similar incomeslikewise limits the potential of income-integrationplans to reduce racial school segregation. Whilethe theoretical range of possible racial segrega-tion levels that could obtain under a race-neutralincome-desegregation regime is, in general, quitebroad, the most probable result in districts withhigh levels of racial residential segregation and lowlevels of residential income segregation would bepatterns of school enrollment that mirrored theseresidential patterns.

2.5.2. Simulated racial segregation underincome-integration regimes

Our second approach to estimating the likelylevel of racial segregation that would result froman income-integration policy relies on actualschool enrollment data as a basis for simulatingthe patterns of enrollment that might result from

* Blacko White

* 100 049 % (O %

k,80 0 0 0

0.251

0.501

0.75I

1.00

Black-White Segregation (V)FIGURE5. Patterns of racial and within-race income segregation, U.S. central cities at least 5% Black,2000.

64

income-integration policies. While these esti-mates are useful, it is important to note that theseare lower-bound estimates (and probably im-plausibly low), since they rely on race-neutral re-assignment of poor and nonpoor students amongschools, without regard for patterns of residentialsegregation and home-school distances and trans-portation costs.

We adopt a simulation approach similar tothat used by Chaplin (2002). Like Chaplin, weexamine racial and economic enrollment patternsamong schools and simulate the race-neutral re-assignment of students among schools under anincome-integration constraint, and then computethe resulting racial segregation obtained. Our ap-proach differs in several key ways from Chap-lin' s, however. First, Chaplin's definition of eco-nomic integration required that no school havemore than 50% of students eligible for free or re-duced-price lunch, whereas our "evenness" def-inition (Section 2.1) requires that all schoolshave poverty rates identical (or at least close) tothose of their district as a whole. Ours is thus amuch more stringent income-integration require-ment. Second, because his definition of inte-gration was impossible to meet in districts withgreater than a 50% poverty rate, Chaplin in thesecases simulated economic integration amongschools over larger geographic areas than dis-tricts, such as counties or entire metropolitanareas. Because both racial segregation and racialincome disparities are generally greater oversuch large areas than within individual districts,Chaplin's approach will tend to yield estimatessuggesting more substantial racial desegregationeffects of economic integration than will ours.For both practical and political reasons, however,we consider it unlikely that economic integrationwould be implemented on a scale larger than anindividual school district, and so would arguethat our estimates more plausibly represent theextreme of what is possible in any foreseeableimmediate future. Finally, we report resulting lev-els of within-district racial segregation, whereasChaplin reports resulting levels of overall racialsegregation across the United States.

We use data from the Common Core of Data(CCD) (National Center for Education Statistics,2003) from 89 of the 1100 largest school districtsin the United States in the 2001-2002 schoolyear (11 of the largest districts were missingreduced-price lunch eligibility data) to estimate


a lower bound on the level of racial segregation.Using data on the poverty (measured by free andreduced-price lunch eligibility status) rate andracial composition of each school, we estimatethe White and Black poverty rates for each dis-trict and compute upper and lower bounds on thelevel of racial segregation possible (1) under in-come integration requiring perfect poverty ratebalance, and (2) under income integration re-quiring approximate (±10% of district povertyrate) balance.

We make the simplifying assumption that theWhite and Black poverty rates in each school arethe same as the overall school poverty rate-ineffect, we assume that race and poverty status arenot correlated within each school (although theymay be in the district).21 This enables us to esti-mate the White and Black poverty rates in eachdistrict, which yields estimates of the maximumpossible racial segregation under an income-segregation regime (from Equations 3 and 4).We then determine whether the poverty ratewithin a given school is above or below the dis-trict poverty rate and compute the number ofpoor or nonpoor students in each school whowould need to exchange places with (nonpoor orpoor, respectively) students in another school toproduce a level of economic integration com-mensurate with the income-integration regime(perfect or approximate balance). We assume thatany poor and nonpoor students transferring out ofa school will be proportionately divided betweenracial groups within that school, and that studentstransferring in will be racially mixed in propor-tion to the pool of poor and nonpoor studentsavailable in the district to transfer in (the poolconsists of those who have transferred out ofother schools). This approach simulates a raciallyproportionate reassignment of students amongschools to achieve income balance. Details of thesimulation method appear in Appendix E.

Following this simulated reassignment, wecompute the racial segregation level in the dis-trict. This will be a lower bound on the level ofracial segregation likely following income inte-gration because the racially proportionate reas-signment does not take into account residentialsegregation and transportation efficiency. In adistrict with substantial racial residential segre-gation but little within-race income segregation,the reassignment of students among schools willmirror residential patterns (to the extent possible

65


under the income-segregation constraints): Blackstudents will be more likely to be assigned to pre-dominantly Black schools, and vice versa.

When we compare the existing White-Blacksegregation levels in these 89 districts with theWhite-Black segregation levels obtained from asimulated racially proportionate reassignment ofpoor and nonpoor students among schools, it ap-pears that racial segregation could be reducedsubstantially under an income-integration policy(Figure 6). However, just as the upper-bound es-timates of possible segregation are likely muchhigher than would be achieved in practice, thesesimulation estimates are likely much lower thanwould result in practice, for several reasons.First, the racial segregation levels illustratedin Figure 6 represent what we would expect if:(1) the minimum possible number of studentswere reassigned among schools, and (2) if schoolreassignment patterns were proportionate to theracial composition of the available reassignmentpool. The latter is not likely, particularly in theschool districts with high levels of racial segre-gation, because these high levels of racial segre-gation result, in large part, from high levels ofracial residential segregation. As evident in Fig-ure 5, however, districts with high levels of resi-dential racial segregation uniformly have lowlevels of within-race income segregation; in these

1.0-

- 0.8-

0.6-

CO)

C 0.4-

U) 0.2-09a- .2

0.0-

districts, any reassignment plan that attempts tominimize distances traveled will reassign manyBlack and White students to schools relativelynearby. As a result, the racial integrating effect ofthe income-desegregation policy is likely to be farless than that illustrated in Figure 6.

Our simulation also certainly underestimatesresulting racial segregation in districts with exist-ing racial desegregation plans. In these districts,if the income-integration policy replaces the racialdesegregation policy, then the "baseline" patternsof school enrollments used in the simulation areinappropriate, as they are only the baseline underthe racial desegregation plan. A more realisticbaseline would be school assignments that mir-rored neighborhood patterns. If these are highlysegregated (as they would likely be in districtsthat had school desegregation court orders), thenthe effects of an income-integration policy wouldbe weaker, for the reasons described above. Infact, the only type of district for whom the lower-bound estimates shown in Figure 6 might occur inpractice would be districts with low levels of res-idential racial segregation and no racial desegre-gation plan. But such districts tend to have lowlevels of school racial segregation in the firstplace, so the ancillary racial integration benefitsof an income-integration policy are likely to ac-crue primarily where least needed.

Maximum, +/-10%-- Poverty Balance .... "-- Maximum, PerfectPoverty Balance

Current SegregationMinimum, +/-10%Poverty BalanceMinimum, PerfectPoverty Balance

*

0 0

66

0.0 0.2 0.4 0.6

Current Racial Segregation (V)FIGURE 6. Simulated maximum and minimum possible racial segregation, 89 large districts.

<>0

3. Conclusion

Our primary goal in this article was to ex-amine the extent to which-and under whatconditions-income-based school assignmentpolicies might guarantee racial school integra-tion. Our analysis shows that even under the moststringent form of income-based integration-school assignment based on exact family incomelevels-and assuming that income balance isachieved completely, income integration doesnot guarantee even a modest level of racial de-segregation. The only situation in which incomeintegration would necessarily produce racial de-segregation is if the White and Black incomedistributions were far more different than theyare in fact.

Moreover, our analyses show that the extentof racial integration guaranteed by an income-integration policy is very sensitive to the way inwhich income integration is defined in practice.For a given pattern of Black and White incomedistributions, a policy that required perfect in-come distribution balance, with income mea-sured by a continuous variable, would producethe greatest guarantee of some ancillary racialintegration. Such a policy is impractical, bothbecause school districts do not have access toexact family income and because perfect balanceamong schools is an unreasonable goal for aschool district. Under most racial desegregationplans, school districts have typically requiredschools to have racial compositions that fallwithin some moderate range of racial composi-tion. It is likely that districts pursuing income-based desegregation will do the same. Our analy-ses show, however, that the less strictly a planensures perfect income balance, the weaker itslikelihood of ensuring any substantial racialintegration.

Not only are districts unlikely to achieve per-fect income balance, they are unlikely to be ableto use a continuous income measure for schoolassignment. Dichotomous measures, however,are somewhat weaker guarantors of racial segre-gation than continuous measures. If school dis-tricts based school assignment on a simple di-chotomous indicator of socioeconomic status,such as poverty status or free and reduced-pricelunch eligibility status, socioeconomic integra-tion would guarantee even less racial integrationthan would socioeconomic desegregation based


on a continuous income measure. We also demon-strate that the extent to which school assignmentbased on a dichotomous income indicator willguarantee racial desegregation depends on theincome level at which the distribution is di-chotomized. Based on these analyses, we suggestthat if districts use a dichotomous measure, theoptimal choice of a poverty threshold will be apoint on the income distribution somewhere be-tween the mean Black and White income levels-a point that turns out to be conveniently close tothe federal reduced-price lunch eligibility thresh-old, the income measure most readily availableto school district officials. Nonetheless, the pri-mary thrust of our analyses is to show that a race-blind income-based school assignment policy isunlikely to ensure even modest levels of racialdesegregation. Unless the White-Black mean in-come ratio is greater than 3-a ratio that is notfound in any U.S. city-using a dichotomous in-come measure will not guarantee a segregationlevel lower than 0.50, which is an extremely highlevel of segregation. This is not to say that racialand income desegregation are incompatible. Infact, it is always possible-in principle-toachieve both complete racial and socioeconomicintegration under any income-integration regime.But the clear implication of our analyses is thatracial integration is not guaranteed by a race-neutral income-integration policy.

Our analyses here focus on the possible andprobable effects of within-district income-integration policies in large urban districts. Ab-sent income reassignment options that rely oninter-district or metropolitan transfers to achieveregional income balance among schools, the ef-fect on racial school desegregation is likely to beextremely limited. It may be that the adoption ofinter-district transfer policies coupled with in-come integration would lead to greater racialschool desegregation, particularly if income andrace are more highly correlated among districtsthan within districts. Yet, the same set of limit-ing conditions-residential segregation patterns,in particular, and transportation distances andcost-would likely influence the potential racialdesegregation resulting from such a policy. Ad-ditional analyses would be necessary to deter-mine the likely effects of inter-district incomeintegration policies.

One might ask whether a socioeconomic-integration policy that used multiple socioeco-

67


nomic factors-including, perhaps, parentaleducation, neighborhood poverty rates, publichousing assistance-as well as other factors, suchas prior achievement and English proficiency,might produce greater ancillary racial integra-tion than a strictly income-based policy. Althougha full analysis of the implications of a multiplefactor policy is beyond the scope of this article,one can guess at the likely results: multiple fac-tors are better than a single factor to the extentthat the multiple factors are collectively morehighly correlated with race than income alone,although the extent of improvement will likely de-pend strongly on the specifics of the policy. Ingeneral, we would expect that a policy that re-quired exact balancing among schools on multi-ple factors, each of which was partially correlatedwith race (net of the other included factors),would guarantee lower levels of racial segrega-tion than would the use of an income measurealone. However, a multiple factor policy may notguarantee much, if any, additional racial integra-tion if only approximate balance among schoolswere required, since even mildly inexact balancesubstantially diminishes the guarantee of racialintegration.

The second goal of this article was to estimatethe probable level of racial integration that mightresult from a race-neutral income-integration pol-icy. Examination of residential racial and in-come segregation patterns in U.S. cities suggeststhat whereas racial desegregation is a possible re-sult of an income-desegregation policy, it is nota probable outcome, given the low levels of res-idential income segregation relative to levels ofresidential racial segregation in most cities.While our lower-bound estimates of the racial in-tegration effect of income integration suggest thepossibility of substantial ancillary racial integra-tion impacts, these impacts are likely to be sub-stantial only in districts with low levels of racialsegregation relative to income segregation-districts, in other words, where racial segregationis not sufficiently high to be a pressing concernin the first place. In fact, the patterns of racial andincome segregation in urban areas in the UnitedStates strongly suggest that transportation policymay be the strongest determinant of whetherincome-integration policies result in significantracial integration. In a district that guaranteedfree and efficient transportation for all studentsto any school in the district, and that used parental

choice or a lottery in conjunction with anincome-integration constraint, school enrollmentpatterns would be potentially only weakly con-strained by residential segregation patterns. (Theprimary constraint would be due to parental orstudent preference for nearby schools, but thismight be offset by preferences for quality schools,which may not always be nearby;,see, for exam-ple, Hastings, Kane, & Staiger, 2005.) In such acase, we would expect racial segregation patternsunder an income-integration regime to look muchmore like our lower-bound estimates shown inFigure 6. In the absence of strong transportationand choice or lottery mechanisms to counter resi-dential segregation patterns, however, an income-integration regime is likely to result in many stu-dents attending schools relatively near their homesin racially segregated neighborhoods.

Finally, it bears repeating that none of ouranalyses are meant to suggest that income de-segregation among schools might not have sub-stantial positive benefits for students. Socioeco-nomic integration may be quite valuable in itsown right, regardless of its racial impacts. Ourfindings simply indicate that income and racecannot stand as proxies for one another in schoolintegration policies. Absent some substantial de-cline in racial residential segregation, race-neu-tral assignment policies are unlikely to producesignificant racial school desegregation.

Appendix A: Demonstrating That IncomeAnd Racial Integration Are Compatible

Suppose we have J schools, with the enroll-ment of schoolj given by tp, and M distinct race/ethnic groups in the population, indexed by m,each making up proportion 7;m of the total popu-lation. Let income be measured by I and let (Dm(!)be the income density function for group m, andlet (D(/) be the income density function of thepopulation. Then the total income density func-tion is simply the weighted average of those ofthe M groups:

M (1=D (I) (I). (Al)

To verify that it is always possible to achieveboth complete racial and complete income de-segregation in the schools, we enroll tmj = 7tmtj stu-dents of each race m in each school j (so thatiumj = 7cm for all m andj) such that the income dis-

68


tribution of these students is the same as that ofthe total population of students of race m, that is,Dmnj(l) = m(/). Now the total income distributionof schoolj will be identical to the income distri-bution of the population:

M M

(Dj(I)=17u ,(D/()=Iic.o(D.()=(D(I. (A2)

Likewise, the racial composition of each schooljwill be identical to that of the whole population, sowe will have perfect integration by both race andincome. Note that this result does not depend onthe racial composition, the income distribution, orthe relationship between the two. Complete racialand income integration are mathematically com-patible for any population, regardless of the rela-tionship between race and income.

Appendix B: Computing Maximum PossibleRacial Segregation

We next determine the pattern of school en-rollments that yields the maximum possible levelof racial segregation consistent with perfect in-come integration. Let ID(I) and OWb(I) denote theWhite and Black income-density functions, re-spectively. Perfect income integration requiresthe income distribution in each school to be(tIT)O(I). To produce maximal racial segrega-tion, we assign students as follows:2 2 in the firstschool, we assign only White students, except inregions of the income distribution where thereare not enough White students to make the in-come distribution in the school match that of thepopulation. In these regions, we assign as manyWhite students as we can, and then fill the re-maining part of the income distribution withBlack students. We repeat this for each subse-quent school, drawing from the remaining poolof students.

Formally, we let cj denote the cumulative totalenrollment of schools 1 throughj; and we define ijto be the point on the income distribution such that

T

for allj = 1, .... , J - 1 (in addition, we defineio = 0 and ij = +-). Now the above assignmentmechanism will yield enrollments of Black (bj)and White (wj) students in schoolj given by

ij- ibi f tiDId

+ f [cjDP(I)- Tt.D. (I)]dI

ij ((wj f [Thw(Dw (I)-c_,-.(Dl()]dI(B2)

ij

From these racial compositions, we compute thesegregation index V.

Appendix C: Maximum Possible SegregationWhen Income Is Measured Dichotomously

Let 7, and ntb denote the White and Black pop-ulation proportions and let 7E, it, and nbp denotethe total, White, and Black poverty rates, respec-tively. Assume, without loss of generality, thatlTwp < nbp. Further, assume a policy requires allschools to have poverty rates within ±d of ;x.

First, note that if 7t - d < 7rp < 7tbp < it + d, thenwe can obtain complete racial segregation underthe income-integration constraints simply by as-signing all Black students to one school and allWhite students to another. In the case whereTCWp < u - d, < 7c + d < nb, however, some racialintegration will be required under the incomeconstraints. To achieve maximum segregation inthis case, we construct three school populationsas follows.2Y We first assign to school 1 all poorWhites and enough nonpoor Whites to obtain apoverty rate of 7t - d. This school will be 100%White, with total enrollment tj = Tn,7cw,/(rp - d).Second, we assign to school 2 all nonpoor Blacksand enough poor Blacks to obtain a poverty rateof;p + d. This school will be 100% Black, withtotal enrollment t2 = TRb( 1 - nb,)/(I - ,p-- d). As-sign all remaining Blacks (who will all be poor)and Whites (who will all be nonpoor) to the thirdschool. This school will have Thc(7p - d - nýp)/

(7t - d) White students and Tb(n•bp - n, - d)l(I - -g - d) black students.

This arrangement of students will give themaximum possible segregation while maintain-ing income integration, because there is one all-White school with as many Whites as possiblegiven the income constraints, one all-Black schoolwith as many Black students as possible, and oneschool with the remaining students. Defining

69


z.=icp-d-iu,p and zb= 7cbp-7uP-d, and substi-

tuting the racial counts from the three schoolsinto Equation 1 yields

v. (d) =1- býwICw

1 b U

[u Tirb (1-7tp-d) (7c,-d)(c1)(1-K, -d) (7, - d)

FrTTCz, (7U, -d)L+ Thwzý (I - n, -d)]

nbZb (n, -d)+ 7cwzw (I -7u- d)- zbZ,,

Appendix D: Computing Optimal PovertyThreshold

To minimize the maximum possible segrega-tion given by Equation 3, we need to find thepoint Ip on the income distribution where thequantity l7cbp - np I is maximized. The povertyrate 7tcp for a group m is simply the proportion ofthe population of group m with incomes below I,so we have

7 1 P p(,() - o(I)dI

0 0 1

(D1)

Taking the derivative of both sides of this yields

d 1 p• [ 0 , ( 1, ) _ (D . I ) ] .dI,

7ubzb (TCP - d) + n,Zw (1- 7CP - d)

Note that if d = 0 (if the policy requires perfectincome integration), we have

Z =C(C 2)

Z, = 11. (11bp - 7.)

and so

,(0)o- 7cbZb7CP + 7,,,z (i- ,,) - ZbZ.7cbzb7UP + rn.z ý (1 - 7CP) ( 0

Note also that if tc,, =7cb = 0.5, then we have

zý = Zb= z = 0.5(7tbp - 7c, - 2d) (C4)

and

nbzb (7c, - d) + 7c.z. (1- n, - d) - z,z,7cbzb (up- d)+ 7cz,, (I 1P- -d)

0.5z (1 - 2d)- z-

0.5z (1 - 2d)

1- (7t, -- 7c,-)

(1-2d)

-V. (o)(1- 2d)

(D2)

This expression equals 0 (and so Equation 3 isminimized), when the cutoff point Ip satisfies

(D3)

Note that the optimal income cutoff does not de-pend on the proportions of Whites and Blacks inthe population.

Appendix E: Simulating StudentReassignment

Let wi, bi, and tj indicate the number of White,

Black, and total students enrolled in school i, re-

spectively. Let 7;,pi, Tcbj, , and itc, indicate theWhite, Black, and total poverty rate in school i,respectively. Let 7up indicate the overall povertyrate among students in the district. In order thateach school have the same poverty rate as that ofthe district, we transfer xi poor students out ofschool i, and transfer xj nonpoor students intoschool i, where

xj = ti (7c,, - up). (El)

Note that xi will be positive (meaning that poorstudents are transferred out and nonpoor studentsare transferred in) for schools with Icp, > nIp, and

negative (meaning the opposite) for schools withTcpi < Itp. Let XP and X, indicate the total numberof poor and nonpoor students, respectively, thatare transferred among schools, and let X,p, X,ý,Xbp, and Xb,, indicate the total numbers of White

and Black poor and nonpoor students transferred.Assuming that poor and nonpoor students to be

V.(d)

70

0, I,,P) =(,,,.(Ij).

transferred out of a school are selected propor-tionately to their racial composition (as we wouldexpect under a race-neutral policy), and if we as-sume a 0 correlation between race and povertywithin each school (i.e., n,,pi = itbpi = itcp), then

these total counts of transferring students aregiven by

7I.Piwpi i W,

X,= IXi .Xiqn'>.1 7U = 4 x,-, ti

itbPlb• blXpb = I X.nb = Y, X--

11X'>., 7Tp1ti ik'i-p ti ( 2-- ~ ,i~-L(E2)

X _ = Y X (1 - 7c ,pi) w i W,-

S= Ii (1 - Itcp) b, = X- bi

The poor students transferred out of schools withan initial excess of poor students make up thepool of students available to transfer into schoolsthat had a deficit of poor students initially, whilethe nonpoor students transferred out of schoolswith an initial deficit of poor students make upthe pool of students available to transfer intoschools that had an excess of poor students ini-tially. If these pools of students are reassigned toother schools in proportion to the racial compo-sition of the available pool, then the number ofWhite and Black students in each school after thetransfer process is completed will be wi and b,:

}i+ i (1-7Jwf X W itfitp <iteac sch -(1o -w pi)t,n X)Win w h an , income-

ifnt io pol-7CwpAW X.nw3

bicy requI-res o bly ap xim bala ifn thi lgipb(I-7E,i)ti X,P

b i - X i + jx n b if 7Cp i > TCP .7rpi.tj Xý

We compute the racial segregation following re-assignment using the new racial compositions ofeach school (wý and bD).

In the case where an income-integration pol-icy requires only approximate balance, the logic

of the simulation is similar, though with one com-plication. We must exchange enough poor andnonpoor students among schools so that for each


school 1i7c- 7c, k< d. However, in a given districtthe minimum total number of poor students whomust transfer out of schools with an excess ofpoor students will not necessarily equal the min-imum total number of nonpoor students whomust transfer out of schools with an excess ofnonpoor students.24 In this case, we transfer ad-ditional poor or nonpoor students (dependingwhich group requires more transfers out) out ofschools until the total number of poor and non-poor transferring students are equal. Specifically,we find the minimum value of 7t, and the maxi-mum value of 7t. such that it, - d!< n• <i • <t it, <It, + d and that defining xi as

( -- ) if ip, <i7txi= t 0r% if 7c, < Tj<itc.

ttj (7pi -- n) if lcpi >it.(E4)

yields Xp =X, when this value of x, is substitutedinto Equation E3. In effect, if requiring lit7p -

it •< d yields more poor than nonpoor transferees,we are raising the minimum tolerable povertylevel above itp - d until the number of nonpoortransferees equals the number of poor transferees(and vice versa).

Notes

'Brown v Board of Education, 347 U.S. 483 (1954).2Comfort v. Lynn School Committee, 418 F.3d 1 (1st

Cir. 2005); Parents Involved in Community Schools v.Seattle School District No. 1, 426 F.3d 1162 (9th Cir.2005).

3For a more detailed discussion of recent legal de-velopments in public school desegregation policy, seeMa and Kurlaender (2005).

"4Grutter v. Bollinger, 123 S.Ct. 2325 (2003).51n determining whether a policy is narrowly tai-

lored, courts have generally followed the frameworkset forth in United States v. Paradise (480 U.S. 149,107 S.Ct. 1053, 94 L.Ed.2d. 203 (1987)). an employ-ment case dealing with a remedial affirmative actionplan. That framework requires an examination of (1) thenecessity of the policy, (2) the burden of the policy oninnocent third parties, (3) the efficacy of race-neutralalternatives, (4) the duration of the policy, (5) the flex-ibility of the policy, and (6) the relationship of numericgoals to the relevant population. Subsequent caseshave modified this framework to better address the dis-tinct issues raised in the education context (Ma &Kurlaender, 2005).

6La Crosse, WI, 2002-2003 school enrollment infor-mation, obtained at: http://www.lacrosseschools.com/schools.htm.

71


7SFNAACP v. SFUSD 576 F. Supp. 34 (1983).8Brian Ho v. SFUSD, 965 F. Supp. 1316, 1319, n. 2

N.D. Cal 1997.9Some examples of measures of evenness are the

dissimilarity index (D), the information theory index

(H), and the normalized exposure index (V) (see also

James & Taeuber, 1985; Massey & Denton, 1988).IONote that in the two-group case, for mathematical

simplicity and without loss of generality, we computeall racial proportions as if there were no other studentsin the schools. Thus, in each school and in the popula-tion, the proportions of Whites and Blacks sum to 1.

"INote that Vis symmetric with respect to m and n:it does not matter whether we derive it from the expo-sure of m to n or the exposure of n to m.

120f the 101 large (20,000+ enrollment), predomi-

nantly Black and White public school districts (districts

where Black and White students together make up at

least 90% of district enrollment) in the United Statesin 2001-2002, only four districts had segregation lev-

els of V> 0.50. These were Atlanta, Georgia (0.57);

Mobile County, Alabama (0.54); Calcasieu Parish,Louisiana (0.53); and Huntsville, Alabama (0.52).(Authors' tabulations of Common Core of Data, Na-

tional Center for Education Statistics,.2003.)13Apart from the difficulties in getting families to re-

port actual income to school districts, asking them todo so may create a kind of "moral hazard"--an incen-tive for nonpoor families to report lower than actual in-

comes, since reporting a lower income reduces theprobability that a student (particularly a nonpoor stu-

dent living in a low-poverty neighborhood) would beassigned to a school far from his or her home in order

to achieve income balance across schools.14If we use a different measure of segregation, such

as the dissimilarity index or the information theory

index, the maximum segregation level does depend

on the racial composition of the school enrollment,but only very weakly.

15The Black and White income densities shown hereare based on data from the Early Childhood Longitu-dinal Study-Kindergarten Cohort (National Center for

Education Statistics, 2001); see Section 2.2.5 for moredetail. -

16Strictly speaking, Equation 4 holds exactly only if

the student population is 50% Black and 50% White(see Appendix C). Additional analyses (not shown) in-dicate that, if the racial composition is different than

this, Equation 4 underestimates the true V1, and so is a

conservative (i.e., low) estimate of the maximum pos-

sible racial segregation under this type of income-de-segregation plan.

17Moreover, for all four distributions (White and

Black city and suburban school students' families) the

median income is about 0.8 of the mean income,which corresponds to shape and location parametersfor a gamma distribution in each case of x= 1.56 and

72

S= (0.64) x (mean income), respectively. In all of oursubsequent analyses we assume both the Black andWhite income distributions have this common shape,

as it simplifies computations. Additional sensitivityanalyses to examine the sensitivity of our results to dif-

ferent shape parameters of the gamma distributionsuggest that decreasing the standard deviation of in-

come by 10% (which corresponds to an increase in (Xof roughly 20%) reduces the maximum possible racial.

segregation obtained under complete income integra-tion by 0.05-0.07 points, as measured by V.

18Across central cities, the mean White-Black in-

come ratio is 1.51, with a standard deviation of 0.23;across suburbs, the mean ratio is 1.43 with a standarddeviation of 0.21.

19Students are eligible for free or reduced-price

lunch if their family income is less than 130% or 185%of the poverty threshold, respectively. In 1998, thepoverty threshold for a family of three was $13,133;for a family of four the poverty threshold was $16,588.

These correspond to reduced-price lunch eligibilitythresholds of $24,296 and $30,688. The optimal di-chotomization point in 1998 (which we use in ouranalyses here) was $34,700, roughly twice the povertythreshold for a family of four. Figure 3 indicates thatour results would be relatively insensitive to changesin the dichotomization point as long as the point is be-tween the mean Black and White incomes, so usingthe reduced-price lunch eligibility threshold as a di-

chotomization point is both administratively practicaland produces near-optimal results (relative to otherdichotomization points).

20The results here are insensitive to the choice of a

poverty threshold. We also computed income segre-

gation using a wide range of other poverty cutoffs,from $10,000 to $100,000, and in each case obtainedvery similar results.

21Chaplin (2002) examines the plausibility of the as-

sumption that the within-school correlation betweenrace and poverty is 0. While he finds some discrepancy

between CCD and Census data on poverty rates, hissimulation results (which are similar in method toours) are insensitive to alternative plausible assump-tions about the correlation of race and poverty withinschools. Thus, we assume a 0 race-poverty correlationwithin schools.

22The following derivation relies on several as-sumptions. First, we assume that all schools in the sys-

tem are the same size (tj = t for all j). While this is un-likely to be true, it matters little in general. If schools

have different sizes, then the order in which we fill theschools will matter, and we can get different levels ofsegregation from the above procedure. Simulations

(not shown) indicate that if there are 20 or more

schools of roughly the same order of magnitude insize, the maximum obtainable segregation does not de-

pend on the number or sizes of the schools. In districts

with fewer schools, differences in the sizes of schoolsmay matter, but in general, large urban school districtshave far more than 20 schools.

Second, we assume schools can enroll fractions ofpersons, since bj and wj need not be integers. Nonethe-less, in a district with even a modest number of stu-dents, rounding to integers will make a negligible dif-ference in the computed segregation levels.

Third, we assume the income distributions for eachracial group are relatively simple-in particular, wehave assumed that for each schoolj, a single incomepoint ij is defined by Equation B 1. This would, for ex-ample, be true if (D. and (Db define unimodel distribu-tions with the mode of (D. greater than that of Ob,

which is realistic given patterns of income distributionin the United States.

Fourth, we describe here an approach for comput-ing the maximum possible segregation between tworacial groups. The method could be adapted to applyto multiple groups, however, if groups were orderedfrom highest average incomes to lowest, and we filledschools beginning with the highest income group, fol-lowed by the next highest, and so on. This would, ingeneral, yield the maximum possible racial segrega-tion for a given set of income distributions.

23Because segregation is not changed if a school isdivided into two or more schools, each with identicalcompositions as the original school (James & Taeuber,1985; Reardon & Firebaugh, 2002), this approachgeneralizes to any school system with three or moreschools.

2To see this, consider the case where d = 0.2 and adistrict has an overall poverty rate of 7t = 0.15; in thiscase, there can be no schools with an excess of non-poor students, though there may be schools with an ex-cess of poor students (i.e., schools where 7c, > 0.35).Thus, there will be poor students who must leave high-poverty schools, but there will be no available slots forthem vacated by nonpoor students in low-povertyschools.

References

Anyon, J. (1997). Ghetto schooling: A political econ-omy of urban educational reform. New York:Teachers College Record.

Boozer, M., Krueger, A., Wolkon, S., Haltiwanger, J.,& Loury, G. (1992). Race and school quality sinceBrown v. Board of Education. In M. N. Baily & C.Winston (Eds.), Brookings papers on economic ac-tivity: Microeconomics (pp. 269-338). WashingtonD.C.: The Brookings Institution.

Chaplin, D. (2002). Estimating the impact of eco-nomic integration of schools on racial integration.In The Century Foundation Task Force on the Com-mon School (Ed.), Divided we fail: Coming togetherthrough public school choice (pp. 87-113). NewYork: The Century Foundation Press.


Clark, W. A. V., & Ware, J. (1997). Trends in residen-tial integration by socioeconomic status in SouthernCalifornia. Urban Affairs Review, 32(6), 825-843.

Clotfelter, C. T. (1999). Public school segregation inmetropolitan areas. Land Economics, 75(4), 487-504.

Clotfelter, C. T. (2001). Are whites still fleeing? Racialpatterns and enrollment shifts in urban public schools,1987-1996. Journal of Policy Analysis and Man-agement, 20(2), 199-221.

Clotfelter, C. T., Ladd, H. F., & Vigdor, J. L. (2003).Segregation and resegregation in North Carolina'spublic school classrooms. North Carolina Law Re-view, 81(4), 1463-1511.

Coleman, J. S., Campbell, E. Q., Hobson, C. J., Mc-Partland, J., Mood, A. M., Weinfeld, F. D., et al.(1966). Equality of educational opportunity. Wash-ington, DC: U.S. Department of Health, Education,and Welfare, Office of Education.

Coleman, J. S., Hoffer, T., & Kilgore, S. (1982).Achievement and segregation in secondary schools:A further look at public and private school differ-ences. Sociology of Education, 55(April/July),162-182.

Community Advisory Conmnittee on Student Assign-ment. (2005). Recommendations for student assign-ment in the San Francisco Unified School District.San Francisco: San Francisco Unified School District.

Darden, J. T., & Kamel, S. M. (2000). Black residentialsegregation in the city and suburbs of Detroit: Doessocioeconomic status matter? Journal of UrbanAffairs, 22(1), 1-13.

Dawkins, M. P., & Braddock, J. H. (1994). The contin-uing significance of desegregation: School racialcomposition and African American inclusion inAmerican society. Journal of Negro Education, 63(3),394-405.

Duncan, G., Boisjoly, J., Levy, D., Kremer, M., &Eccles, J. (2003). Empathy or antipathy? The con-sequences of racially and socially diverse peers onattitudes and behaviors (Working Paper): WorkingPaper no. 326, Joint Center for Policy Research.

Entwisle, D. R., Alexander, K. L., & Olson, L. S.(1997). Children, schools, and inequality. Boulder,CO: Westview Press.

Farley, J. E. (1984). P* segregation indices: What canthey tell us about housing segregation in 1980?Urban Studies, 21(3), 331-336.

Farley, J. E. (1995). Race still matters: The minimalrole of income and housing costs as causes of hous-ing segregation in St. Louis, 1990. Urban AffairsReview, 31, 244-254.

Farley, J. E. (2003). Race not class: Explaining racialhousing segregation in the St. Louis metropolitanarea, 2000. Paper presented at the annual meetingof the Midwest Sociological Society. April 16-19.

Finder, A. (2005, September 25). As test scores jumpRaleigh credits integration by income. The New YorkTimes, Section 1, Column 5, National Desk, p. 1.

73


Fiske, E. B. (2002). Controlled choice in Cambridge,Massachusetts. In The Century Foundation TaskForce on the Common School (Ed.), Divided wefail: Coming together through public school choice(pp. 167-208). New York: Century Foundation Press.

Flinspach, S. L., & Banks, K. E. (2005). Moving be-yond race: Socioeconomic diversity as a race-neutralapproach to desegregation in Wake County Schools.In J. C. Boger & G. Orfield (Eds.), School resegre-gation: Must the South turn back? (pp. 261-280).Chapel Hill: The University of North Carolina Press.

Flinspach, S. L., Banks, K. E., & Khanna, R. (2003).Socioeconomic integration policy as a tool for di-versifying schools: Promise and practice in twolarge school systems. Paper presented at the ColorLines Conference Harvard University, Cambridge,MA.

Gottlieb, A. (2002). The case for economic school in-

tegration. Denver CO: The Piton Foundation.Gurin, P., Dey, E. L., Hurtado, S., & Gurin, G. (2002).

Diversity and higher education: Theory and impacton educational outcomes. Harvard Educational Re-view, 72(3), 330-366.

Hallinan, M. T. (1998). Diversity effects on studentoutcomes: Social science evidence. Ohio State LawJournal, 59(3), 733-754.

Hallinan, M. T., & Williams, R. (1989). Interracialfriendship choices in secondary schools. AmericanSociological Review, 54(1), 67-78.

Hanushek, E. A., Kain, J. F., & Rivkin, S. G. (2002).New evidence about Brown v. Board of Education:The complex effects of school racial composition onachievement. Unpublished manuscript.

Hastings, J. S., Kane, T. J., & Staiger, D. 0. (2005).Parental preferences and school competition: Evi-dence from a public school choice program (Work-ing Paper No. 11805). Cambridge, MA: NationalBureau of Economic Research.

James, D. R., & Taeuber, K. E. (1985). Measures ofsegregation. Sociological Methodology, 14, 1-32.

Jones, R. (2002). Defining diversity. American School

Board Journal, 189(10), 18-23.Kahlenberg, R. (2001). All together now: Creating

middle-class schools through public school choice.Washington D.C.: Brookings Institution.

Kane, T. J. (1998). Misconceptions in the debateover affirmative action in college admissions. InG. Orfield & E. Miller (Eds.), Chilling admissions:The affirmative action crisis and the search for alter-natives. Cambridge: Harvard Education PublishingGroup.

Kurlaender, M., & Yun, J. T. (2003, April). Schoolracial composition and student outcomes in a mul-tiracial society. Paper presented at the annualmeeting of the American Educational ResearchAssociation, Chicago, IL.

74

Ma, J., & Kurlaender, M. (2005). The future of race-conscious policies in K-12 public schools: Supportfrom recent legal opinions and social science re-search. In J. Boger & G. Orfield (Eds.), School reseg-regation: Must the South turn back? (pp. 239-260).Chapel Hill: The University of North Carolina Press.

Massey, D. S., & Denton, N. A. (1988). The dimen-sions of residential segregation. Social Forces, 67(2),281-315.

Massey, D. S., & Denton, N. A. (1989). Hypersegre-gation in U.S. metropolitan areas: Black and His-panic segregation along five dimensions. Demogra-phy, 26(3), 373-39 1.

Massey, D. S., & Fischer, M. J. (2003). The geogra-phy of inequality in the United States, 1950-2000.Brookings-Wharton Papers on Urban Affairs.

Mayer, S. E. (2002). How economic segregation affectschildren's educational attainment. Social Forces, 81,153-176.

National Center for Education Statistics. (2001). Earlychildhood longitudinal study-kindergarten cohortrestricted-use base year: Child file, teacher file, andschool file. From http://www.nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2000097

National Center for Education Statistics. (2003). Com-mon core of data (CCD) public elementary and sec-ondary school universe data. From http://www.nces.ed.gov/ccd/pubschuniv.html

Natriello, G., McDill, E. L., & Pallas, A. M. (Eds.).(1990). Schooling disadvantaged children: Racingagainst catastrophe. New York: Teachers CollegePress.

O'Keefe, 0. (2005, October 27). Personal communi-cation to M. Kurlaender. Education Placement Cen-ter, San Francisco Unified School District.

Reardon, S. F. (1998, March). Measures of racial di-

versity and segregation in multi-group and hierar-chically-structured populations. Paper presented atthe annual meeting of the Eastern Sociological So-ciety, Philadelphia, PA.

Reardon, S. F., & Firebaugh, G. (2002). Measures ofmulti-group segregation. Sociological Methodology,32, 33-67.

Reardon, S. F., & Yun, J. T. (2001). Suburban racialchange and suburban school segregation, 1987-1995.Sociology of Education, 74(2), 79-101.

Reardon, S. F., & Yun, J. T. (2003). Integrating neigh-borhoods, segregating schools: The retreat fromschool desegregation in the South, 1990-2000.North Carolina Law Review, 81(4), 1563-1596.

Reardon, S. F., & Yun, J. T. (2005). Integrating neigh-borhoods, segregating schools: The retreat fromschool desegregation in the South, 1990-2000. InJ. C. Boger & G. Orfield (Eds.), School resegrega-tion: Must the South turn back? (pp. 51-69). ChapelHill, NC: University of North Carolina Press.

Reardon, S. F., Yun, J. T., & Eitle, T. M. (2000). Thechanging structure of school segregation: Measure-ment and evidence of multi-racial metropolitan areaschool segregation, 1989-1995. Demography, 37(3),351-364.

Rivkin, S. G. (2000). School desegregation, academicattainment, and earnings. Journal of Human Re-sources, 35(2), 333-346.

Rumberger, R., & Palardy, G. (2002). Does segrega-tion (still) matter? The impact of student compo-sition on academic achievement in high school.Teachers College Record, 109(9), 1999-2045.

Schofield, J. W. (1995). Review of research on schooldesegregation's impact on elementary and sec-ondary education students. In J. Banks & C. A. M.Banks (Eds.), Handbook on research of multicul-tural education (pp. 597-616). New York: Simonand Schuster MacMillan.

Silberman, T. (2002). Wake County schools: A questionof balance. New York: The Century Foundation.

Steams, L. B., & Logan, J. R. (1986). Measuringtrends in segregation: Three dimensions, three mea-sures. Urban Affairs Quarterly, 22(1), 124-150.

Wells, A. S., & Crain, R. L. (1994). Perpetuationtheory and the long-term effects of school deseg-regation. Review of Educational Research, 64(4),531ff.


Authors

SEAN F. REARDON is an Associate Professor ofEducation, School of Education, Stanford University,CERAS Building, 520 Galvez Mall, #526, Stanford,CA 94305; [email protected]. His areas ofspecialization are education policy, segregation, race,and quantitative methods.

JOHN T. YUN is an Assistant Professor of Educa-tion, Gervitz Graduate School of Education, PhelpsHall, University of California, Santa Barbara, SantaBarbara, CA 93106; [email protected]. Hisareas of specialization are education policy, segrega-tion, high-stakes testing, and using data to improveinstruction.

MICHAL KURLAENDER is an Assistant Profes-sor of Education, University of California, Davis, 2051Academic Surge Building, One Shields Avenue,Davis, CA 95616; [email protected]. Herareas of specialization are social stratification, educa-tion policy, and quantitative methodology.

Manuscript received June 4, 2004Revision received December 1, 2005

Accepted January 27, 2006

75

COPYRIGHT INFORMATION

TITLE: Implications of Income-Based School Assignment Policiesfor Racial School Segregation

SOURCE: Educ Eval Policy Anal 28 no1 Spr 2006WN: 0610503465006

The magazine publisher is the copyright holder of this article and itis reproduced with permission. Further reproduction of this article inviolation of the copyright is prohibited. To contact the publisher:http://www.aera.net/

Copyright 1982-2006 The H.W. Wilson Company. All rights reserved.

implications of income-based school assignment policies for racial

Documents