Review of Economic Studies (2008) 75, 579–596 0034-6527/08/00240579$02.00c© 2008 The Review of Economic Studies Limited

Statistical Discrimination with PeerEffects: Can Integration Eliminate

Negative Stereotypes?SHUBHAM CHAUDHURI

The World Bank

and

RAJIV SETHIBarnard College, Columbia University

First version received June 2004; final version accepted July 2007 (Eds.)

We introduce peer effects in the costs of human capital acquisition into a model of statistical dis-crimination in labour markets. This creates a link between the level of segregation in social networks andracial disparities in job assignment and wages. We show that this relationship is characterized by disconti-nuities: there is a threshold level of segregation below which negative stereotypes become unsustainable,and steady-state skill levels can change dramatically. This change can work in either direction: skill levelsmay either rise or fall in both groups. Which of these outcomes arises depends on the population shareof the disadvantaged group and on the distribution of the costs of human capital investments. We alsoexamine the effects of affirmative action policies in the presence of peer effects and provide conditionsunder which such policies eliminate negative stereotypes.

1. INTRODUCTION

Stereotypes—beliefs about some unobservable (or yet to be observed) trait of an individual basedupon his membership in an identifiable group—are ubiquitous in economic and social interac-tions. Ever since the pioneering work of Phelps (1972) and Arrow (1973) it has been recognizedthat such stereotypes need not depend on an active taste for discrimination and can be self-perpetuating even in the absence of intrinsic group differences. In these early papers, and in thesubsequent literature on what has come to be termed statistical discrimination, the stability ofstereotypes arises because relevant traits are only imperfectly observable at the individual level.This allows unequal treatment based on ex-ante irrelevant group identities to emerge as an equi-librium outcome.

The contributions of Arrow and Phelps have spawned a considerable theoretical literaturethat explores whether the disadvantages in labour market outcomes experienced by individualsbelonging to certain groups (such as racial and ethnic minorities or women) might be explained bythe prevalence of negative stereotypes amongst employers. There are two strands in this literature.One follows Phelps (1972) in assuming that individual measures of productivity are noisier formembers of disadvantaged groups.1 The other, derived from Arrow (1973), does not assume sucha difference. Instead, the key insight is that a stereotype can influence the behaviour of thosesubject to it in ways that cause the belief to become self-fulfilling. So, for instance, if individuals

1. Aigner and Cain (1977), Borjas and Goldberg (1979), and Lundberg and Startz (1983) all present models ofstatistical discrimination in labour market settings that share this feature; see also Antonovics (2002).

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from certain groups perceive that employers hold negative stereotypes about their capabilities,and are hence less likely to treat them favourably, their incentives to acquire human capital arediminished, thereby reinforcing and perpetuating the employer beliefs.2

The view of human capital acquisition that is implicit in these latter papers is one of a largelyindividual and autonomous process. The treatment that individuals anticipate in the labour marketdoes influence their perceived benefits of acquiring human capital, but the costs of human capitalacquisition are assumed to be exogenously given at the individual level. This assumption goesagainst the view, evident in both a range of theoretical papers as well as a number of empiricalstudies, that human capital investment decisions are subject to a variety of interpersonal external-ities.3 For instance, it is widely held that both the pecuniary and non-pecuniary costs of acquiringhuman capital vary with the social and economic milieu in which an individual operates, and inparticular, with the human capital of others with whom the individual associates, whether in thefamily, the neighbourhood, at school, or in the workplace. And the usual presumption is that theseinterpersonal externalities take the form of positive complementarities—the higher the levels ofhuman capital of others in a group, the lower the costs of human capital acquisition are for eachindividual within it.

If peer effects are important, then the costs of human capital acquisition at the individuallevel can clearly no longer be taken to be exogenous. Instead, these costs will be determined en-dogenously in equilibrium along with labour market outcomes and human capital levels. This hastwo important implications. First, the assumption, typical in the existing statistical discriminationliterature, that the within-group distribution of human capital acquisition costs is the same for allgroups can no longer be sustained, and the hypothesis that groups are ex-ante identical needs tobe recast at a more primitive level.4 Second, in the presence of local human capital spillovers, thelevel of segregation in the economy becomes important as a determinant of human capital invest-ments. This is because the extent to which individuals from different groups interact determineswhether local human capital spillovers are confined within the boundaries of a group or extendacross groups, and that in turn potentially influences whether group-level asymmetries in humancapital investment levels and labour market outcomes can be sustained in equilibrium. Neither ofthese implications has been addressed in the existing literature on statistical discrimination.

This paper explores the link between the level of segregation in the economy (which we taketo be a primitive) and the sustainability of negative stereotypes. In the presence of peer effects,there are potential human capital spillovers from the advantaged group to the disadvantaged, andthe extent to which such spillovers arise depends on the degree of intergroup contact. Increasingintegration therefore tends to lower the costs of human capital acquisition in the disadvantagedgroup while raising those in the advantaged group. If integration proceeds far enough, however,it may result in a qualitative change in the set of attainable equilibria, making it impossible for

2. Coate and Loury (1993), and more recently, Moro and Norman (2003, 2004) are examples of papers in this vein.3. Consider, for example, the following statement in Lucas (1988): “Human capital accumulation is a social ac-

tivity, involving groups of people in a way that has no counterpart in the accumulation of physical capital”. Brock andDurlauf (2001) note that “individual human capital acquisition depends on the behaviours and/or characteristics of com-munity members” for a variety of reasons, including “peer group effects, in which the costs to one person of investing ef-fort in education are decreasing in the effort levels of others”. Such effects are prominent in several models of endogenouscommunity formation (see, among others, Arnott and Rowse, 1987; De Bartolome, 1990; and especially Benabou, 1993,1996a,b). On the empirical front, numerous studies provide evidence suggestive of a variety of peer group and neigh-bourhood effects, among them, Coleman (1966), Summers and Wolfe (1977), Henderson, Mieszkowski and Sauvageau(1978), Borjas (1995), and Ioannides (2001).

4. The hypothesis that group differences are not innate, but result instead from disparate treatment in equilibrium,has been termed the axiom of anti-essentialism by Loury (2002). This is reflected in the ex-ante equality of cost functionsacross groups in standard statistical discrimination models. While we allow cost functions to differ across groups inequilibrium as a result of endogenous differences in peer group human capital, the mapping from peer group humancapital to the distribution of costs is held to be the same for each group. Thus the anti-essentialist hypothesis is maintainedthroughout this paper.

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negative stereotypes to be sustained. In this case a shift to symmetric treatment of groups istriggered, resulting in discontinuous changes in skill levels. While this tends to equalize labourmarket outcomes across groups, it can do so in one of two quite distinct ways. Depending onthe population share of the disadvantaged group and the distribution of human capital investmentcosts, the resulting shifts can leave both groups better off or both groups worse off than under thestatus quo. This suggests that under certain circumstances, both the advantaged and the disadvan-taged group may be supportive of vigorously pursued integrationist policies, while under othercircumstances, both may be opposed. More generally, the pursuit of integration, whether in theschools or in housing markets, may have significant and identifiable equity and efficiency effectsthat extend well beyond the specific arenas in which integrationist policies are implemented.

A further consequence of peer effects is that group inequality in human capital investmentscan persist even in the face of a colour-blind policy that requires employers to provide identicaltreatment to workers with identical signals. This cannot happen when the costs of human capitalaccumulation are exogenously given and identical across groups. We explore the effects of suchpolicies on equilibrium stereotypes below and show that when peer effects are sufficiently weak,a policy of equal treatment does in fact result in equal outcomes.

Our work is related to and complements a number of papers in several areas. First and mostobviously, it is related to the contributions of Coate and Loury (1993) and Moro and Norman(2003). These papers examine whether concerted public action in the form of temporary affirma-tive action programmes can eliminate self-confirming discriminatory stereotypes. We investigatethe related and as yet unexplored question of whether, starting from a situation where statisticaldiscrimination is prevalent, public efforts towards greater integration of previously segregatedgroups can eliminate negative stereotypes in labour markets.

A related literature addresses the efficiency implications of statistical discrimination. Schwab(1986) is an early example while Norman (2003) provides the most recent and thorough analysis.The central trade-off emphasized in Norman (2003) is that between the cost inefficiencies impliedby statistical discrimination—individuals in the disadvantaged group who have low costs of hu-man capital acquisition choose not to invest, whereas higher cost individuals in the advantagedgroup do—and the gains from better matching (of qualified individuals to jobs that require highlevels of human capital and unqualified individuals to jobs that do not) possible under statisticaldiscrimination. We introduce an additional channel through which the elimination of negativestereotypes may affect efficiency, namely local human capital spillovers. Our findings are alsorelated to the literature on endogenous community structure and stratification, where peer effectsare shown to matter for the extent and efficiency implications of stratification (Benabou, 1993,1996a,b).

Finally, Moro and Norman (2004) have criticized standard statistical discrimination modelson the grounds that these models assume that human capital investment decisions of individualsfrom different groups are separable. The assumed separability has the somewhat implausible—from a positive political economy perspective—implication that the privileged group has no in-centive to resist public efforts to eliminate negative stereotypes. To address this shortcoming,Moro and Norman (2004) adopt a more general production technology under which they obtaincross-group effects operating through wages and job assignments in general equilibrium. In ourcase cross-group effects arise through human capital spillovers. As a consequence, the advan-taged group may have an incentive to resist integration.5

5. Benabou (1993) combines the local spillovers that we focus on in our paper, with the global interactions result-ing from production complementarities that are the focus of Moro and Norman (2004), but does so in a framework whereindividual human capital is perfectly observable. Bowles, Loury and Sethi (2007), building on Loury (1977), explore thelink between segregation and persistent inequality and find threshold effects similar to those identified in this paper, butagain in a model with observable human capital.

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The remainder of the paper is organized as follows. In the next section, we incorporate peereffects into an otherwise standard model of statistical discrimination. Section 3 contains numer-ical examples that illustrate how integration, if it proceeds beyond some threshold, can result ina qualitative change in the set of attainable equilibria. In Section 4 we establish conditions underwhich integration eliminates negative stereotypes and results in higher skill levels in both groups,as well as conditions under which integration is uniformly skill reducing. The effects of an affir-mative action policy (requiring equal treatment of workers with identical signals) are explored inSection 5, and Section 6 concludes with a discussion of empirical and policy implications.

2. STATISTICAL DISCRIMINATION WITH PEER EFFECTS

2.1. Wages and job assignment

Consider the following variant of the Coate and Loury (1993) model. There are two groups ofworkers, 1 and 2, and group membership is costlessly observable. Workers may either pay acost to become qualified or may remain unqualified. This cost should be interpreted as beinga net cost, taking into account any intrinsic benefits that may be derived from education. Thedistribution of costs within each group is determined endogenously in equilibrium, in a mannerto be specified below.

Firms may assign workers to one of two jobs. One of these may be done equally well byqualified and unqualified workers and results in zero pay-offs to both firms and workers. Theother yields a positive aggregate pay-off xq if a qualified worker is assigned and a negative pay-off xu otherwise. Nothing essential is lost by setting xq = 1 and xu = −1.

Let si be the proportion of workers in group i who choose to become qualified. Firms canobserve these population proportions, but cannot observe individual characteristics of workersprior to assignment. They observe a noisy signal, which can take one of two values: P (positive)or N (negative). The probability that this signal is P when the worker is qualified is p, and theprobability that this signal is P when the worker is unqualified is q < p.

When a firm observes a P worker belonging to group i , the posterior probability that theworker is qualified is

θ(si ) = psi

psi +q(1− si ). (1)

Similarly, when a firm observes an N worker belonging to group i , the posterior probabilitythat the worker is qualified is

ϕ(si ) = (1− p)si

(1− p)si + (1−q)(1− si ). (2)

Since p > q, θ(si ) > ϕ(si ) for all si ∈ (0,1). Firms will assign workers to the skilled task ifthe expected pay-off is positive. For a P worker, this requires 2θ(si ) − 1 > 0. Similarly, forN workers to be allocated to the skilled tasks requires 2ϕ(si )−1 > 0.

We assume that competition among firms for workers results in zero profits, with wagesequal to expected productivity (conditional on the signal) for workers.6 Hence wages are

wp(si ) = max{2θ(si )−1,0},wn(si ) = max{2ϕ(si )−1,0}.

6. The assumption of zero profit for firms and wages equal to expected productivity may be obtained endogenouslyas the subgame-perfect equilibrium of a game in which two or more firms make simultaneous wage offers contingent onsignals.

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Each worker is assigned to a job on the basis of the inference that firms draw from her signal,which in turn depends on the skill share in the group to which the worker belongs.7

A worker will wish to become skilled if doing so would yield an expected increase in thewage that exceeds the cost of becoming skilled. Hence a worker with cost c in group i will derivea positive net benefit from investing if and only if

pwp(si )+ (1− p)wn(si )− c > qwp(si )+ (1−q)wn(si ),

which simplifies to yield(p −q)(wp(si )−wn(si )) > c.

Let b(si ) = (p − q)(wp(si ) − wn(si )) denote the expected benefits of investing when theskill share in group i is si .

If the level of skill acquisition in a group is sufficiently small, workers drawn from thatgroup will be assigned to unskilled jobs even if they are associated with positive signals. Definesx as the level of skill acquisition in a group that induces firms to be indifferent between placinga worker with a positive signal in a skilled job versus an unskilled one. Specifically sx is definedby the solution to 2θ(s)−1 = 0, which yields

sx = q

p +q. (3)

Clearly, for any group with skill share si < sx , all workers are assigned to the low-skill task,regardless of signal. Similarly, if si is sufficiently high, even workers with negative signals will beassigned to high-skill jobs. For intermediate values of the skill share si , workers will be assignedto the high-skill job if and only if they have a positive signal. These considerations imply thatb(si ) is single peaked, takes its maximum at some interior value of si , and b(si ) = 0 for si < sx

and si = 1. The point at which b(si ) attains its maximum is precisely the level of si at which firmsbegin to assign N workers to skilled jobs.

2.2. Human capital acquisition

A standard assumption in the statistical discrimination literature is that the ex-ante costs of ac-quiring human capital are exogenously given and are the same for both groups. We relax thisassumption and allow for the possibility that the costs of becoming skilled depend on the extentto which one’s peers invest in human capital. This in turn will depend on the human capital levelsin the two groups, as well as the extent of integration or intergroup contact.

A key component of our analysis is the extent to which one’s peers are drawn from the sameracial or ethnic group as oneself. Under perfect integration, the composition of one’s peer groupwould be independent of the group to which one belongs. In other words, each individual’s peergroup would have a racial composition that mirrors that of the population as a whole. Undercomplete segregation, on the other hand, peer groups would be homogeneous. We use a measureof segregation that captures the extent to which the composition of one’s peer group lies betweenthese two extremes.

Specifically, suppose that an individual belonging to group i has a proportion η of peerswho are drawn from the same group i , while the remaining (1 − η) are drawn at random from

7. While there are no legal impediments to conditioning wages on performance-related signals (as opposed toconditioning directly on race or gender, for example), many of the arguments presented below hold also if worker com-pensation is based only on task assignment and not on the signal. In fact, wages are identical in the two models except inthe case where si is so high that all workers are assigned to the skilled job regardless of signal. But no si in this range canbe an equilibrium when wages are independent of signals since no worker would pay a positive cost to become skilled inthis case.

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the population at large. Additionally, let β denote the share of the overall population belongingto group 1. Then an individual belonging to group 1 will have a peer group in which a propor-tion η+ (1−η)β individuals also belong to group 1, while a proportion (1−η)(1−β) belong togroup 2. These two proportions are referred to in the empirical segregation literature as theisolation index and the exposure index respectively. Similarly for group 2, the isolation index isη + (1 −η)(1 −β), and the exposure index is (1 −η)β. Note that the exposure of a minority toa majority group is greater than the exposure of a majority to a minority group, as one wouldexpect. The parameter η is sometimes refereed to as the correlation ratio (Massey and Denton,1988). It is a normalized index of isolation, ranging between between 0 and 1, and will be themeasure of segregation used in the remainder of this paper.

The mean skill share in the population is simply s = βs1 + (1 −β)s2. Let σi be the meanpeer group skill level for individuals in group i (the share of their peers who become skilled).When the groups are completely segregated and have no contact with each other, the meanskill share experienced by each group will simply be given by σi = si . On the other hand,under complete integration, each group would experience the same peer group skill share,which in turn would equal that in the population as a whole: σ1 = σ2 = s. These are the twoextremes within which peer group skill shares will lie for intermediate levels of segregation.Specifically,

σi (s) = ηsi + (1−η)s.

Lower values of the segregation index η correspond to greater integration and hence asmaller distance between the peer group skill shares experienced by members of the two groups.8

We now specify the manner in which the costs of skill acquisition are distributed within andacross groups. Let G(c,σ ) denote the proportion of individuals with costs below c in a group withmean peer group skill level σ. Positive spillovers from human capital accumulation within peergroups are reflected in the assumption that G is increasing in σ. Although the costs of humancapital accumulation may now differ across groups, which experience different levels of peergroup quality, it is assumed that the function G is identical across groups. Hence groups are notassumed to be innately different in any meaningful sense.

Consider a group with skill share si and peer group skill share σi . Assuming that the workerswho become skilled are those who have the lowest costs of doing so, we may define c(si ,σi ) asthe cost of becoming skilled for the marginal worker. This is the worker whose cost of becomingskilled is such that a proportion si of workers have lower cost. This marginal worker will wishto become skilled if and only if the benefits b(si ) from doing so exceed this cost. A pair of skillshares (s1,s2) is a (Bayes–Nash) equilibrium if each worker chooses optimally, given the choicesof other workers, firms assign workers to jobs optimally given their beliefs, and these beliefs areconsistent with the actual skill shares in each group. This implies that for each i,

b(si ) = c(si ,σi ) = c(si ,ηsi + (1−η)(βs1 + (1−β)s2)). (4)

Note that in the presence of peer effects, the cost distribution in each group depends on theskill acquisition in the other group, an interdependence that does not arise in standard models ofstatistical discrimination. Since the peer group skill shares σi depend on the skill shares withineach group (as well as the level of integration), the two cost functions are determined endoge-nously together with the skill shares (s1,s2) in equilibrium.

8. We thank an anonymous referee for pointing out to us that this specification is also consistent with a hypothesisof aggregate externalities, where a group’s cost distribution depends on a weighted average of own-group skill share andpopulation skill share.

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FIGURE 1

Statistical discrimination under complete segregation

2.3. Complete segregation

In the special case of complete segregation (η = 1) the equilibrium conditions (4) reduce tob(si ) = c(si ,si ) so the the two groups can be analysed independently. An equilibrium (sl ,sh) withstatistical discrimination under complete segregation is shown in Figure 1.9 The (endogenous)functions c(s,sl) and c(s,sh) represent the manner in which the cost to the marginal worker ofbecoming skilled varies with the skill share in the group, holding constant the peer group skillshares at the equilibrium values sl and sh respectively. In group 1, a proportion sl of workers havecosts below b(sl), while in group 2 a proportion sh of workers have costs below b(sh).

Under complete segregation, an asymmetric equilibrium exists if and only if there exist atleast two stable symmetric equilibria. Hence in the example shown in Figure 1, both (sl ,sl) and(sh,sh) are equilibria. In fact, it is easily shown that the set of symmetric equilibria is invariantin the level of segregation η: the skill profile (sl ,sh) is an equilibrium with η = 1 if and only if(sl ,sl) and (sh,sh) are both equilibria for all η ∈ [0,1]. We shall assume in the remainder of thepaper that there exists an equilibrium with statistical discrimination under complete segregationand hence that there exist (at least) two symmetric equilibria at all levels of segregation.10

2.4. Stability

In this paper we focus on equilibria that are dynamically stable in the sense that small pertur-bations in skill shares in the neighbourhood of the equilibrium are self-correcting. Specifically,suppose that at any disequilibrium skill profile (s1,s2), the skill share in group i will rise ifand only if the benefits of skill acquisition exceed the costs for the marginal worker. This is thestandard myopic tatonnemont process used as the basis for equilibrium selection in models of

9. The figure is based on the following specification: p = 0·8, q = 0·2, and c(s,σ ) = (1·4−σ 0·5)s −0·1.10. In the example depicted in Figure 1, any state (s1,s2) ∈ {sl ,sm ,sh}×{sl ,sm ,sh} is an equilibrium under com-

plete segregation, so there are three symmetric and six asymmetric equilibria.

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statistical discrimination (Akerlof, 1976) and neighbourhood sorting (Benabou, 1993). Theimplied dynamics of skill shares are

si = f (b(si )− c(si ,σi )), (5)

where f is an arbitrary strictly increasing and continuous function satisfying f (0) = 0. Undercomplete segregation the dynamics of skill shares in the two populations are independent, sinceσi = si . In this case, the stability of any equilibrium skill profile (s1,s2) requires only that for eachsi , the slope of the cost function c(si ,si ) exceeds that of the benefit function at the equilibrium.This is the case for the equilibrium (sl ,sh) depicted in Figure 1. In fact, the symmetric equilibriaat (sl ,sl) and (sh,sh) are also stable. Consider, however, equilibria in which one or both groupshave skill share sm, where sl < sm < sh and b(sm) = c(sm,sm); see Figure 1. All such equilibriaare dynamically unstable. To see this, note that starting from sm , if the skill share in a group wereto rise, the benefits would strictly exceed the costs of skill acquisition for the marginal worker,leading to cumulative divergence away from sm and towards sh . Similarly a small decline in skillshare below sm would lead to further declines under the dynamics, eventually resulting in skillshare sl .

When some degree of integration exists, the dynamics of skill shares are no longer indepen-dent in the two populations, since σi depends on both s1 and s2. In this case we say that a state(s1,s2) is (locally asymptotically) stable if there exists a neighbourhood N of (s1,s2) such that,for any initial state in N , the dynamics (5) converge to (s1,s2). As noted above, any symmet-ric state that is an equilibrium under complete segregation is also an equilibrium at any level ofsegregation. This equivalence extends to stable symmetric states:

Proposition 1. Any symmetric state that is a stable equilibrium under complete segrega-tion is also a stable equilibrium at all levels of segregation.

Proposition 1 implies that changes in the level of segregation do not affect the existence andstability of symmetric equilibria. This is not the case with asymmetric equilibria, as we show inthe next section.

3. EFFECTS OF INTEGRATION

Suppose that the economy is initially at an asymmetric equilibrium (sl ,sh) with sl < sh, and con-sider the effects of a decline in segregation η. The immediate effect of such a decline would be tobring the two groups into greater contact, thus lowering the costs of human capital acquisition inthe initially disadvantaged group and raising them in the initially advantaged group. This in turnwould result in changes over time in skill levels and convergence to a new steady state. Intuitionsuggests that at the new steady state, the skill share in the former group will be higher, and inthe latter group lower, relative to the initial state. This is indeed the case, provided that negativestereotypes continue to prevail in the new steady state. However, as the following examples show,stable asymmetric equilibria may exist only if segregation is sufficiently great, so that increas-ing integration, beyond some threshold, is simply inconsistent with the persistence of statisticaldiscrimination.

Example 1. Suppose β = 0·5, p = 0·8, q = 0·2, and c(s,σ ) = (1·2−σ 0·45)s −0·1. Thenfor η = 1, there is a stable equilibrium at (sl ,sh) = (0·12,0·97). There exists η � 0·69 suchthat a stable asymmetric equilibrium with s1 < s2 exists if η > η, but not if η < η. Trajectoriesoriginating at (sl ,sh) converge to the stable asymmetric equilibrium with s1 < s2 if η > η, butconverge to (sh,sh) if η < η.

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CHAUDHURI & SETHI STATISTICAL DISCRIMINATION WITH PEER EFFECTS 587

FIGURE 2

Effect of integration on steady-state skill shares (Examples 1 and 2)

The left panel of Figure 2 shows how steady-state skill shares vary with integration in this ex-ample, assuming that the economy is initially both segregated and highly unequal. Greater in-tegration reduces the steady-state skill share in the initially advantaged group and raises it inthe initially disadvantaged group, as one might expect. However, if segregation falls below thethreshold η, stable unequal steady states no longer exist, and convergence occurs to a state inwhich both groups have high levels of human capital. In the neighbourhood of η, a small changein integration can result in a large and discontinuous shift in steady-state skill levels, such thateven the initially advantaged group has higher levels of human capital relative to unequal statesin which segregation is above η. In this case a policy of integration can benefit both groups.

Consider, however, the following example, which is identical to the first except for the spec-ification of the cost function c(s,σ ).

Example 2. Suppose β = 0·5, p = 0·8, q = 0·2, and c(s,σ ) = (1·33 − σ)s − 0·1. Thenfor η = 1, there is a stable equilibrium at (sl ,sh) = (0·08,0·92). There exists η � 0·74 suchthat a stable asymmetric equilibrium with s1 < s2 exists if η > η, but not if η < η. Trajectoriesoriginating at (sl ,sh) converge to the stable asymmetric equilibrium with s1 < s2 if η > η, butconverge to (sl ,sl) if η < η.

In this example, if segregation falls below the threshold η, convergence occurs to a state inwhich both groups have lower levels of steady-state human capital relative to unequal states inwhich segregation is above η (see the right panel of Figure 2). Integration eliminates negativestereotypes but results in a shift to the less-efficient symmetric equilibrium. Integration raises thehuman capital of the disadvantaged group only until η is reached, after which the skill shares ofboth groups collapse to levels below those at any asymmetric equilibrium. Equal treatment comesat a welfare cost to both groups, although the loss inflicted on the previously advantaged groupis clearly greater.

These examples illustrate not only that integration can eliminate negative stereotypes, butthat its implications for human capital investment may depend in critical ways on underlying

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parameters. In the next section we identify conditions under which the elimination of negativestereotypes is skill enhancing and those in which it is skill reducing. It turns out that the popula-tion share of the disadvantaged group and the distribution of human capital acquisition costs arecritical in this regard.

4. MAIN RESULTS

Consider an economy that is initially at a stable asymmetric equilibrium (sl ,sh) under completesegregation (η = 1), and suppose that segregation declines to some level η < 1. We are interestedin conditions under which such a decline in η results in convergence to a symmetric equilibriumif and only if η falls below some threshold η ∈ (0,1). If there exists such a threshold, we say thatintegration eliminates negative stereotypes. As noted above, a transition to a symmetric equilib-rium can result in lower skill shares in both groups, or higher skill shares in both. We say thatintegration is skill enhancing if it results in a transition to (sh,sh), as in Example 1. If, instead,convergence occurs to (sl ,sl), as in Example 2, we say that integration is skill reducing.11 Which(if any) of these phenomena arises depends on two key attributes of the model: the populationcomposition and the distribution of human capital acquisition costs.

In order to identify these effects analytically, we adopt a simple parametric specification forthe cost distribution. Suppose that costs are uniformly distributed with support [−ε,γ − ε], whereε > 0, and γ depends on the level of human capital σ in one’s peer group and on a shift parametera. For given values of σ and a, we therefore have a cost function that is linear in s:

c(s,σ ) = γ (σ,a)s − ε.

Positive spillovers from human capital accumulation imply that γ (σ,a) is strictly decreasingin σ . We adopt the convention that γ is increasing in a, so higher values of a correspond to abroader range of costs for any given σ and hence higher costs for the marginal worker at anygiven s. We assume that γ (σ,a) > ε for all σ and a, which ensures that all equilibria are interior.

In general there may exist values of a such that equilibria with statistical discriminationcannot exist even under complete segregation. This can happen when a is sufficiently small (inwhich case both groups will have high levels of human capital in the unique symmetric equi-librium) or when a is sufficiently large (in which case both groups will have low levels ofhuman capital in the unique symmetric equilibrium). Let amin and amax be defined, respectively,as the lower and upper bounds for a such that stable equilibria with statistical discriminationexist under complete segregation. We assume that the interval [amin,amax] is well defined andnon-empty, otherwise equilibrium statistical discrimination would be impossible for any param-eter values.

By definition, for any a ∈ [amin,amax] there exists an asymmetric equilibrium (sl ,sh) withsl < sh when η = 1. The following results establish conditions under which integration elimi-nates negative stereotypes and characterize the set of parameter values for which skill levels riseor fall.

Proposition 2. For any β ∈ (0,1), there exist am,an ∈ (amin,amax) such that (i) integrationeliminates negative stereotypes and is skill enhancing for all a ∈ (amin,am), and (ii) integrationeliminates negative stereotypes and is skill reducing for all a ∈ (an,amax). Both am and an aredecreasing in β.

11. We are assuming here that there are at most two stable symmetric equilibria (sl ,sl ) and (sh ,sh) in the model,as depicted in Figure 1. As noted above, this is consistent with as many as nine equilibria under complete segregation, ofwhich four are dynamically stable.

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FIGURE 3

Parameter range for which integration eliminates negative stereotypes

Hence integration can eliminate negative stereotypes and raise (lower) skill levels in both groupsfor any population composition provided that the costs of human capital accumulation are suffi-ciently low (high). The threshold values am(β) and an(β) are both decreasing in β.

The result is illustrated for a particular numerical specification of the model in Figure 3.Integration eliminates negative stereotypes and is skill enhancing if the parameters a and β lie inthe lower-left shaded region of the figure. Similarly, integration eliminates negative stereotypesand is skill reducing if the parameters a and β lie in the upper-right shaded region. Note that whena is sufficiently small, integration is skill enhancing if β is also sufficiently small. And when ais sufficiently large, integration is skill reducing if β is also sufficiently large. Specifically, thefollowing is implied by Proposition 2.

Corollary 1. For any a ∈ (am(1),am(0)) there exists βl > 0 such that integration elimi-nates negative stereotypes and is skill enhancing if β ∈ (0,βl), and for any a ∈ (an(1),an(0))there exists βh < 1 such that integration eliminates negative stereotypes and is skill reducing ifβ ∈ (βh,1).

If the disadvantaged group is a sufficiently small minority, both groups can benefit from a policyof integration if negative stereotypes are eliminated as a result. In this case one might expectwidespread popular support for integrationist policies even within the ranks of the advantagedgroup. On the other hand, integration may be skill reducing for both groups if the initially dis-advantaged group is a sufficiently large majority. Note that it is possible for an(1) to be strictlysmaller than am(0), as in Figure 3. In this case there will exist values of a (between the dashedlines in the figure) for which integration eliminates negative stereotypes if the population shareof either group is sufficiently small.

Our assumption that costs are uniformly distributed for any given level of peer group qualityσ and that changes in σ affect the cost distribution by shifting the upper bound of its support is

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clearly restrictive. The logic underlying our results appears to be quite general, however.12 Togain some intuition for our findings, consider a completely segregated economy initially at anasymmetric equilibrium in which one group of individuals is subject to negative stereotypes. Asshown above, since there exists an asymmetric equilibrium under complete segregation, theremust exist multiple stable equilibria under all levels of segregation. Now consider what happensas the economy becomes increasingly integrated. The initially disadvantaged group benefits fromgreater contact with the other group, while the initially advantaged groups loses. If the popula-tion share β of the disadvantaged group is small, the latter effect will tend to be outweighedby the former. This effect will be especially pronounced if a is small. Integration under thesecircumstances will raise the skill levels in the disadvantaged group rapidly without significantlylowering the skill levels in the advantaged group. Once a point is reached at which stable asym-metric equilibria can no longer be sustained, skill levels in the disadvantaged group would haverisen far enough to place the economy in the basin of attraction of the high-skill symmetric equi-librium. The same arguments apply in reverse when both β and a are large, in which case thelow-skill symmetric equilibrium is reached.

Our results have been obtained under the assumption that ε > 0. That is, there exist someindividuals who would prefer to become skilled even if no other individual (in either group)were to become skilled. If, instead, we were to assume that ε < 0, then the low-skill symmetricequilibrium would entail zero skill levels in both groups. In this case, starting from an asymmetricequilibrium under complete segregation in which one group has a skill share of 0, integrationcannot result in a movement to the symmetric high-skill equilibrium. This is because no matterwhat the values of β and a, the costs of skill accumulation would be positive for all membersof the disadvantaged group when the skill share in that group is 0. In other words, when ε < 0,c(0,σ ) = −ε > 0. Since b(0) = 0, no member of the group will have any incentive to accumulatehuman capital at any level of integration. Hence, if integration eliminates negative stereotypes,it can only do so in a manner that is skill reducing. More generally, if integration is to be skillenhancing, there must be at least some subset of individuals who would choose to accumulatehigh levels of human capital even when all others in their group do not.

5. AFFIRMATIVE ACTION

We have assumed to this point that firms are free to assign workers to jobs and to set wagesbased only on profit considerations. At asymmetric equilibria, this implies unequal treatment ofgroups in the sense that individuals belonging to a low-skill group face different labour marketprospects even conditional on the value of their signal. If such signals were publicly observableand verifiable, government policy could restrict the actions of firms to require equal treatmentof workers who belong to different groups but are otherwise identical. This policy is processoriented, in the sense that it requires employers to assign workers to jobs and wages based onlyon their signals (independently of group membership).13

Suppose that employers were constrained to make job assignments and pay wages basedonly on a worker’s signal, independently of group membership. The proportion skilled among

12. Our results may be extended (with minor modifications to the proofs) to accommodate the case in which changesin σ shift the entire support of the cost distribution and not just the upper bound. In this case we would require that thelower bound of the support of the cost distribution be negative for sufficiently high values of σ (and not necessarily forall values of σ as currently assumed). Furthermore, it is possible to construct examples such as those shown in Figure 2using non-uniform distributions; we have done so for parametric examples using exponentially distributed costs.

13. Even if individual signals are not publicly observable, such a policy can be implemented using quotas if theaggregate distribution of skills in each group is known. For reasons discussed below, such a policy would eliminatenegative stereotypes in the Coate and Loury (1993) model, although it need not do so here. Coate and Loury focus on asomewhat different affirmative action policy, in which quotas are based not on skill distributions but on population shares.

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those with a positive signal is given by

θ (s1,s2) = βps1 + (1−β)ps2

β(ps1 +q(1− s1))+ (1−β)(ps2 +q(1− s2)).

Similarly, the proportion skilled among those with a negative signal is

ϕ(s1,s2) = β(1− p)s1 + (1−β)(1− p)s2

β((1− p)s1 + (1−q)(1− s1))+ (1−β)((1− p)s2 + (1−q)(1− s2)).

Wages are then given by

wp(s1,s2) = max{2λ(s1,s2)−1,0},wn(s1,s2) = max{2µ(s1,s2)−1,0}.

The benefits of investment are now the same for each group and are given by

b(s1,s2) = (p −q)(wp(s1,s2)− wn(s1,s2)).

The dynamics of skill shares are now

si = f (b(s1,s2)− c(si ,σi )), (6)

for i = 1,2, and equilibria are states at which

b(s1,s2) = c(si ,σi ). (7)

A policy of equal treatment (conditional on signal) implies equality across groups in thebenefits of skill acquisition. In the Coate and Loury (1993) model, this policy necessarily re-sults in group equality in skill distributions, since the costs of human capital accumulation areexogenously given and identically distributed across groups. In the presence of peer effects andsegregation, however, inequality in skill levels can persist even in the face of equal treatment,since the costs distributions are determined in equilibrium together with the skill levels. Thefollowing example demonstrates this possibility.

Example 3. Suppose β = 0·10, η = 1, p = 0·8, q = 0·2, and c(s,σ ) = (1·4−σ)s −0·1.Then there is an asymmetric equilibrium (s1,s2) = (0·08,0·89), in which all workers in the lowskill group are assigned to unskilled jobs, and all others are assigned to skilled jobs with wageswp = 0·94, wn = 0·35. If a policy of equal treatment is imposed, then trajectories initially atthis state converge to (s1,s2) = (0·44,0·96), with all workers assigned to skilled jobs and wageswp = 0·95 and wn = 0·41.

This example illustrates not only the persistence of group inequality in skill levels underequal treatment, but also the somewhat surprising possibility that such a policy can eventuallyraise wages and skill levels in both groups. The initial effect of the policy is to lower expectedproductivity and hence wages in skilled jobs, conditional on either signal. However, productivityconditional on a positive signal declines much less than productivity conditional on a negativesignal, raising the benefits to skill acquisition. As skill shares in both groups rise, so do produc-tivity and wages, but the wage gap between workers with positive signals and those with negativeones narrows. Eventually the benefits of skill acquisition fall to the costs of skill acquisitionfor the marginal worker in each group, with the resulting equilibrium having higher wages andproductivity than the asymmetric equilibrium without affirmative action.

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What can be said more generally about the implications of affirmative action policies? Con-sider an economy which is initially at an asymmetric equilibrium (sl ,sh) under complete segre-gation. We say that affirmative action eliminates negative stereotypes if it results in a transitionto a state with equal skill shares. As in the case of integration, the transition can be (sh,sh) orto (sl ,sl). We say that the policy is skill enhancing if the former occurs and skill reducing if thelatter. In the extreme case of complete segregation we have σi = si , and so the equilibrium condi-tion (7) reduces to b(s1,s2) = c(si ,si ) for each i . Hence if c(s,s) is increasing for all s ∈ [0,1],equal treatment must result in equality of outcomes. Whether the effect is skill enhancing or skillreducing depends on the population composition:

Proposition 3. Suppose that η = 1, and c(s,s) is increasing for all s ∈ [0,1]. Then thereexist βl > 0 and βh < 1 such that that affirmative action eliminates negative stereotypes andis skill enhancing for all β ∈ (0,βl), and affirmative action eliminates negative stereotypes andis skill reducing if β ∈ (βh,1).

This result is very intuitive. When the population share of the initially disadvantaged groupis sufficiently small, the imposition of the policy does not affect the optimal assignment of work-ers in the initially advantaged group. That is, firms comply with the policy by assigning workersin the low-skill group to high-skill jobs. This lowers expected productivity in skilled occupationsand hence the wage rates wp and wn, but to a negligible extent when β is small. More importantly,it raises the expected benefits of human capital accumulation for workers in the low-skill groupfrom 0 to (p −q)(wp −wn), and, provided that c(s,s) is increasing, results in eventual equaliza-tion of skill shares across groups.14 The same logic applies in reverse if β is large. Hence thereis an interesting parallel between integration and affirmative action, in the sense that both caneliminate negative stereotypes and result in greater skill levels (if β is low) or lower skill levels(if β is high). However, the mechanism through which the two policies operate are quite differ-ent. While affirmative action alters the labour market returns to human capital, social integrationaffects the costs of human capital investments.

6. DISCUSSION

Peer effects (or more broadly, local complementarities) have featured prominently in analyses ofhuman capital formation. Somewhat surprisingly, such effects have been ignored in the literatureon statistical discrimination in labour market settings. This is the case even in that strand of theliterature that emphasizes how negative stereotypes might be perpetuated through the endoge-nous human capital acquisition decisions of those subject to such beliefs. We see the statisticaldiscrimination literature and the literature on peer effects in human capital accumulation as beingnaturally complementary and have taken a step towards bridging the gap between the two. Withthe introduction of peer effects, the level of segregation in the economy becomes relevant to thedetermination of human capital investments and employer beliefs. Starting from a situation inwhich statistical discrimination is prevalent, integration on a large-enough scale can eliminatenegative stereotypes by making it impossible for a stable discriminatory equilibrium to be sus-tained. Whether this results in an increase in skill levels or a decline depends on the populationshare of the initially disadvantaged group and the manner in which the costs of human capitalacquisition are distributed.

The model has a number of empirical and policy implications. Integrationist policies, whichsucceed in raising the level of intergroup contact, can have effects on the perpetuation of negative

14. Hence it is the non-monotonicity of c(s,s) in Example 1 that allows negative stereotypes to persist in the faceof affirmative action policies.

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stereotypes. A sufficiently high level of such contact can make equilibrium stereotypes unsus-tainable and result in large, discontinuous changes in wage inequality. Hence policies that fostergreater intergroup contact, such as the integration of schools, neighbourhoods, and workplaces,can serve as an alternative to affirmative action policies in eliminating negative stereotypes. Bothaffirmative action and pro-integration policies have potential pitfalls, although for very differentreasons. As shown by Coate and Loury (1993) affirmative action can further reduce the incen-tives for human capital accumulation in a group initially facing a negative stereotype. Integration,on the other hand can reduce the incentives for human capital accumulation in the initially ad-vantaged group. If this effect is sufficiently strong, the economy can transition to a symmetriclow skill equilibrium in which both groups are left worse off. On the other hand, if this effectis weak relative to the positive incentive effect on the initially disadvantaged group, integrationcan leave both groups better off in the long run as the economy shifts to a symmetric high-skillequilibrium.

Our findings also have implications from a political economy perspective. The interests ofadvantaged and disadvantaged groups may be opposed at certain levels of segregation, but alignedat others. In metropolitan areas in which the stereotyped group is relatively small, vigorous in-tegrationist policies may be uniformly skill enhancing in the long run and hence might enjoywidespread popular support. On the other hand, in regions where the initially advantaged groupis small relative to the disadvantaged, the opposite is likely to be true. Vigorous integrationistpolicies in this case can be uniformly skill reducing and hence face widespread popular opposi-tion. These considerations suggest that high levels of segregation, as well as negative stereotypesand group inequality, will be most persistent in areas where the disadvantaged group is relativelylarge.

We have focused in this paper on the effects of greater integration of the viability of neg-ative stereotypes in labour market interactions. In doing so, we neglected the feedback effectsrunning from changes in racial inequality to equilibrium levels of segregation. Such effects canbe complex and sometimes counterintuitive, with declining racial inequality being consistentunder certain conditions with increased segregation (Sethi and Somanathan, 2004). In practice,inequality and segregation are jointly determined through decentralized choices in markets andsocial interactions. A fully endogenous treatment of both inequality and segregation is beyondthe scope of this paper, but remains an important goal for future research.

APPENDIX

Proof of Proposition 1. Let (sk ,sk ) denote a symmetric stable equilibrium under complete segregation. Thenthere exists a neighbourhood N of (sk ,sk ) such that for all (s1,s2) ∈ N , c(si ,si ) > b(si ) if si > sk and c(si ,si ) < b(si ) ifsi < sk (see Figure 1). Given any state (s1,s2) ∈ N , let si = min{s1,s2} and s j = max{s1,s2}. Then s ≡ βs1 + (1−β)s2satisfies si ≤ s ≤ s j , and so σi ≡ ηsi + (1−η)s ≥ si , and σ j ≡ ηs j + (1−η)s ≤ s j . Hence c(si ,σi ) ≤ c(si ,si ) < b(si )

if si < sk , which implies from (5) that si > 0. Similarly, c(s j ,σ j ) ≥ c(s j ,s j ) > b(s j ) if s j > sk , which implies from (5)that s j < 0. Hence along all trajectories initially in N , min{s1,s2} is strictly increasing whenever it lies below sk , andmax{s1,s2} is strictly decreasing whenever it lies above sk . All trajectories initially in N therefore converge to (sk ,sk ),

which is therefore locally asymptotically stable. ‖

Proof of Proposition 2. Recall that sx = q/(p + q), and define γx = ε/sx (see Figure 4). Let sz > sx denotethe larger solution to sγx − ε = b(s), and sxz = βsx + (1 − β)sz the neighbourhood skill share that would prevailunder complete integration at the state (s1,s2) = (sx ,sz). Define am as the value of a, which solves γ (sxz ,a) = γx .

Then γ (sz ,am ) < γx < γ (sx ,am ) so there exists an asymmetric equilibrium under complete segregation. Hence am ∈(amin,amax). Note that γ (sxz ,am ) = γx implies that c(sx , sxz) = 0 and c(sz , sxz) = b(sz) when a = am , so there is anasymmetric equilibrium (s1,s2) = (sx ,sz) under complete integration (η = 0); see Figure 4. Note that sxz is decreasing inβ, and γ (sxz ,a) is increasing in a, so am is decreasing in β. Now consider any a ∈ (amin,am ). In this case c(sx , sxz) < 0,and c(sz , sxz) < b(sz) so (sx ,sz) cannot be an equilibrium under complete integration (η = 0).

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FIGURE 4

Definitions of sw, sx , sy , sz , γx , and γy

Claim 1. Given any a ∈ (amin,am ), there exist η ∈ (0,1) and s ∈ [sz ,1] such that when η = η, the state (sx , s) isan equilibrium.

Proof of Claim 1. We need to show that for any a ∈ (amin,am ), there exists η ∈ (0,1) and s ∈ [sz ,1] such that

c(sx , ηsx + (1− η)(βsx + (1−β)s)) = 0, (A.1)

c(s, ηs + (1− η)(βsx + (1−β)s)) = b(s). (A.2)

Let S denote the compact set [0,1]× [sz ,1], and define 1 and 2 as follows:

1(η,s) = c(sx ,ηsx + (1−η)(βsx + (1−β)s)),

2(η,s) = c(s,ηs + (1−η)(βsx + (1−β)s))−b(s).

Given any s ∈ [sz ,1], define η′(s) as the unique solution to 1(η,s) = 0. To see that such a solution exists and isunique, note that 1(η,s)is continuous and decreasing in η, and that

1(0,s) = c(sx ,βsx + (1−β)s) ≤ c(sx , sxz) < 0,

1(1,s) = c(sx ,sx ) > 0,

since s ≥ sz and a ∈ (amin,am ). Note that η′(s) ∈ (0,1). Now consider any η ∈ [0,1] and define s′(η) as the largestsolution to 2(η,s) = 0. To see that such a solution exists, note that 2(η,s) is continuous in s and that

2(η,sz) = c(sz ,ηsz + (1−η)sxz)−b(sz) ≤ c(sz , sxz)−b(sz) < 0,

2(η,1) = c(1,η+ (1−η)(βsx +1−β))−b(1) ≥ c(1,1) > 0,

since a ∈ (amin,am ). Let F : S → S denote the function defined by F(η,s) = (η′(s),s′(η)), where η′ and s′ are asdefined above. Since F is a continuous mapping and S is compact, there exists (η, s) ∈ S such that F(η, s) = (η, s) fromBrouwer’s fixed-point theorem. Since 1(η, s) = 2(η, s) = 0, (η, s) satisfies (A.1) and (A.2). This proves the claim.

Since a ∈ (amin,amax), there is an asymmetric equilibrium (sl ,sh) when η = 1. From Claim 1, when η = η, thereis an asymmetric equilibrium at (sx , s). It can be shown that for any η ∈ (η,1), there exists an asymmetric equilibrium

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(s1,s2) ∈ (sl ,sx )× (s,sh). (The equilibrium (s1,s2) varies continuously with η within this range.) For η = η, consider astate (s1,s2) = (sx + ε, s) where ε > 0 but small. Here s1 > 0 and s2 > 0. Also, s1 > 0 for all (s1,s2) ∈ (sx ,sz)× (s,1]and s2 > 0 for all (s1,s2) ∈ (sx ,1]×(sz , s]. So the set (sz ,1]×(sz ,1] is absorbing under the dynamics (5). Since the onlyequilibrium in this set is (sh ,sh), all trajectories initially at (s1,s2) = (sx + ε, s) converge to (sh ,sh) for any ε, howeversmall. For any η < η, therefore, the state (sx , s) is in the basin of attraction of the symmetric equilibrium (sh ,sh). Henceintegration eliminates negative stereotypes and is skill enhancing for all a ∈ (amin,am ).

To prove part (ii) of the result, define sy as the level of skill acquisition at which (b(s)+ ε)/s is maximized, andlet γy = (b(sy) + ε)/sy . Let sw < sx denote the smaller solution to sγy − ε = b(s), as shown in Figure 4. Let swy =βsw +(1−β)sy denote the neighbourhood skill share that would prevail under complete integration at the state (s1,s2) =(sw,sy). Define an as the value of a that solves γ (swy ,a) = γy . Then γ (sw,an) < γx < γ (sy ,an) so there exists anasymmetric equilibrium under complete segregation. Hence an ∈ (amin,amax). Note that γ (swy ,an) = γy implies thatc(sw, swy) = 0 and c(sy , swy) = b(sy) when a = an , so there is an asymmetric equilibrium (s1,s2) = (sw,sy) under com-plete integration; see Figure 4. Note that swy is decreasing in β, and γ (swy ,a) is increasing in a, so an is decreasing in β.

Now consider any a ∈ (an ,amax). The proof of the following claim follows the same logic as that of Claim 1 above,and is therefore omitted:

Claim 2. Given any a ∈ (an ,amax), there exist η ∈ (0,1) and s ∈ [0,sx ] such that when η = η, the state (s,sy) isan equilibrium.

Since a ∈ (amin,amax), there is an asymmetric equilibrium (sl ,sh) when η = 1. From Claim 2, when η = η, thereis an asymmetric equilibrium at (s,sy). It can be shown that for any η ∈ (η,1), there exists an asymmetric equilibrium(s1,s2) ∈ (sl , s)× (sy ,sh). (The equilibrium (s1,s2) varies continuously with η within this range.) For η = η, consider astate (s1,s2) = (s,sy −ε) where ε > 0 but small. Here s1 < 0 and s2 < 0. Also, s1 < 0 for all (s1,s2) ∈ (s,sw)× [0,sy),and s2 < 0 for all (s1,s2) ∈ [0, s) × (sw,sy). The set [0,sw) × [0,sw) is therefore absorbing under the dynamics (5).Since the only equilibrium in this set is (sl ,sl ), all trajectories initially at (s1,s2) = (s,sy − ε) converge to (sl ,sl ) forany ε, however small. For any η < η, therefore, the state (s,sy) is in the basin of attraction of the symmetric equilibrium(sl ,sl ). Hence integration eliminates negative stereotypes and is skill reducing for all a ∈ (an ,amax). ‖

Proof of Proposition 3. At the initial state (sl ,sh), b(sl ) = c(sl ,sl ) and b(sh) = c(sh ,sh). Note thatlimβ→0 θ (s1,s2) = θ(s2), and limβ→0 ϕ(s1,s2) = ϕ(s2). Hence limβ→0 b(s1,s2) = b(s2). Now consider the dynamics(6) after the imposition of the policy, starting from the initial state (sl ,sh). In the limiting case β = 0, s2 = f (b(s2)−c(s2,s2)), and s1 = f (b(s2) − c(s1,s1)). Since c(s,s) is increasing and sl < sh , we have s1 > 0 and s2 = 0 for all(s1,s2) ∈ [0,sh)×{sh}. Hence affirmative action results in a transition to (sh ,sh). Since trajectories are continuous in β

and the rest point (sh ,sh) is isolated, the same is true for any β sufficiently small.Now consider the case of large β, and note that limβ→1 b(s1,s2) = b(s1). In the limiting case β = 1, starting from

the initial state (sl ,sh), the dynamics (6) satisfy s2 = f (b(s1)−c(s2,s2)) and s1 = f (b(s1)−c(s1,s1)) = 0. Since c(s,s)is increasing and sl < sh , we have s1 = 0 and s2 < 0 for all (s1,s2) ∈ {sl }× (sl ,1]. Hence affirmative action results in atransition to (sl ,sl ). Since trajectories are continuous in β and the rest point (sl ,sl ) is isolated, the same is true for any β

sufficiently close to 1. ‖

Acknowledgements. This material is based upon work supported by the National Science Foundation under GrantNo. SES-0133483. The findings, interpretations, and conclusions in the paper are those of the authors and do not representthe view of the World Bank, its Executive Directors, or the countries they represent. We thank Yeon-Koo Che andPeter Norman for helpful comments on an earlier version, as well as two referees and the editor, Maitreesh Ghatak,for suggesting a number of improvements. The analysis in Section 5 (especially Proposition 3) owes a great deal to theefforts of an anonymous referee who urged us to explore the effects of affirmative action in the presence of endogenoushuman capital investment costs and conjectured that standard results may not hold in this setting.

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