Discrimination and assimilation at school

Journal of Public Economics 156 (2017) 48–58

Contents lists available at ScienceDirect

Journal of Public Economics journal homepage: www.elsevier.com/locate/jpube

Discrimination and assimilation at school夽 Jon X. Eguia1 Michigan State University, United States

A R T I C L E

I N F O

Article history: Received 19 November 2015 Received in revised form 15 August 2017 Accepted 16 August 2017 Available online 24 August 2017 JEL classification: J15 D71 Z13 D62 I24

A B S T R A C T I present a theory of assimilation among students of two different backgrounds with unequal status. Students with a disadvantaged background face an incentive to assimilate into the more advantaged group. The advantaged group chooses the cost of assimilation strategically to screen those who seek to assimilate. In equilibrium, only the most skilled disadvantaged students assimilate. This theory provides a novel explanation of the “acting white” problem. “Acting white” refers to a social phenomenon in which students from disadvantaged ethnicities suffer social sanctions if they obtain good grades. These sanctions raise the cost of acquiring the skills that make it easier to assimilate. Disadvantaged students with low ability impose these sanctions in order to induce their more able co-ethnics to stay in the disadvantaged group. © 2017 Elsevier B.V. All rights reserved.

Keywords: Discrimination Assimilation Acting white Peer effects

Consider a student population divided along ethnic cleavages, and such that one ethnic background enjoys a greater status or privilege than others. Students with a disadvantaged background face an incentive to assimilate into the more advantaged group, adopting its social norms and culture. Because high-skilled students exert better peer effects, students with an advantaged background prefer to screen those who seek to assimilate. They screen optimally by choosing a difficulty of assimilation such that only disadvantaged students with high skills assimilate.

夽 I am grateful to Dan Bernhardt, Jeff Biddle, Alberto Bisin, Renee Bowen, Drew Conway, Matt Dickson, Oeindrila Dube, Raquel Fernandez, Ken Frank, Francesco Giovannoni, Paola Giuliano, Stephan Heblich, Loukas Karabarbounis, Rachel Kranton, Ashley Lyons, Dan Menchik, Maggie Penn, Carlo Prato, Larry Samuelson, Dan Silverman, Jakub Steiner, Will Terry, Leonard Wantchekon, the participants at the 2012 NBER Political Economy Summer Institute and audiences at talks at MPSA 2010, LSE, Northwestern-Kellogg, Stanford-GSB, Chicago, Harvard/MIT, NYU, Oslo, USC, UPF, Bristol, Leicester, MSU, ITAM and CORE for their contribution, and to the Ford Center for Global Citizenship and the Center for Mathematical Studies at the Kellogg School of Management for financial support during the academic year 2010–11. E-mail address: [email protected]. 1 486 West Circle Dr, 110 Marshall-Adams Hall, Dept. Economics, Michigan State U., East Lansing, MI 48824, United States.

https://doi.org/10.1016/j.jpubeco.2017.08.009 0047-2727/© 2017 Elsevier B.V. All rights reserved.

This screening theory provides a novel explanation for the “acting white” phenomenon. “Acting white” refers to the behavior of African-American and Hispanic students who pressure their coethnics not to achieve academic excellence. African-American and Hispanic students who get top grades are less popular at school than those who get lower grades. In contrast, white students with good grades are more popular (Fryer and Torelli, 2010). Social sanctions against skill acquisition emerge from the combination of two reasons: 1) assimilation is too costly for low-skilled disadvantaged students, and 2) skilled individuals generate positive peer effects. It follows that low-skilled disadvantaged students want their skilled co-ethnics to stay in the disadvantaged group. But any disadvantaged student who becomes too skilled ends up exiting the group and assimilating. So low-skilled disadvantaged students use sanctions to dissuade their co-ethnics from becoming too skilled at school. These sanctions deter some students from acquiring the high skills that make it easier to assimilate. These students acquire instead an intermediate level of skill and remain in the disadvantaged group. Because this intermediate level is higher than the group’s average skill, this average increases. Crucially, those who impose the sanctions become better off. This incentive to deter skill acquisition affects only disadvantaged groups because no one leaves the

J. Eguia / Journal of Public Economics 156 (2017) 48–58

advantaged group. In the advantaged group, the incentive is to encourage everyone to acquire more skills. This explanation fits the empirical findings of Fryer and Torelli (2010) on acting white. It explains why popularity and high grades are negatively correlated among African-American and Hispanic students, but positively correlated among whites. The traditional explanation of acting white (Fordham and Ogbu, 1986; Fordham, 1996) is cultural: African-Americans embrace academic failure as part of their identity, and thus they shun those who defy this identity by studying. According to McWhorter (2000), African-Americans engage in self-sabotage: they convince themselves that effort is not rewarded, and thus they do not exert effort. Neither of these accounts fits well with recent empirical findings (Fryer and Torelli, 2010). Austen-Smith and Fryer (2005) propose an alternative theory based on the opportunity cost of studying. They argue that students are distinguished by their social type (low or high) and their economic type. Students with low social type are socially inept: other agents derive more utility from ostracizing them than from socializing with them. In equilibrium, socially inept agents choose to study and are shunned by their peers. Other students differentiate themselves from the socially inept by choosing not to study. While compelling, this reasoning applies to all ethnicities. Thus, it cannot explain the asymmetry across ethnic groups, which is the essence of the acting white phenomenon. In a broader related literature, research on social identity formation shows that agents care about the status of their group (Tajfel and Turner, 1979; Shayo, 2009). Economic theories of identity and culture assume that members of a low status group can pay a cost to assimilate into another group with higher status. This cost is typically assumed to be exogenously given. If the cost is too high, then minorities adopt and pass on to their descendents identities that are anti-achievement (Akerlof and Kranton, 2000), dysfunctional (Fang and Loury, 2005), traditional (Bénabou and Tirole, 2011), ethnic (Bisin and Verdier, 2000, 2001; Carvalho and Koyama, 2013) or less productive (Kim and Loury, 2012).2 I propose a theory that recognizes that the cost of assimilation is endogenous: it depends on the actions of the agents with an advantaged background. The nature of an agent’s social interactions depends on how others treat her. Identity theories do not ask why agents with an advantaged background discriminate against those who assimilate. I show that discrimination arises in equilibrium as agents pursue their own selfish interests. A closer reference is Fryer’s (2007) theory of endogenous group choice. Agents face an infinitely repeated choice to invest in skills that are useful only to a narrow group, or in skills that are valued by society at large. Members of the narrow group reward the accumulation of group specific skills by greater cooperation with the agent. Fryer’s theory features multiple equilibria under standard folk theorem arguments. He describes one equilibrium in which agents invest in group-specific skills. Other equilibria yield different empirical implications. Thus, the model lacks predictive power. Whereas, I show that disadvantaged agents suffer pressure from their peers to acquire a lower level of human capital. This prediction holds in all equilibria. My theory generates unambiguous empirical implications that are consistent with the findings by Fryer and Torelli (2010) on the acting white phenomenon. 1. The model

49

1.1. Players Consider a society with a continuum of agents. Each agent is identified by her background and her ability or talent. Assume that the set of possible backgrounds is {A, D }, where A represents an advantaged background, and D a disadvantaged one. Assume that the set of possible abilities is [0, 1]. Let N = {A, D } × [0, 1] denote the set of players. For each background J ∈ {A, D }, let NJ = J × [0, 1] denote the subset of agents with background J . Assume that the size (or measure) of N, denoted |N|, is equal to one, and that the distribution of agents is uniform over N. Then   for each J ∈ {A, D }, NJ  = 0.5 and the distribution of ability conditional on background J is uniform over [0, 1]. For any i ∈ N, let hi ∈ [0, 1] denote the ability of agent i. Individual ability is private information. I next define players’s strategies. Agents choose their skill and their social group. In addition, some agents choose discrimination levels, or social sanctions. 1.2. Strategies: choice of skill Every agent i ∈ N chooses her academic skill level, denoted si . This choice is subject to constraints given by i s background and ability. Agent i s ability is an upper bound on how skilled agent i can become. For any i ∈ NA , her innate ability is the only bound, and she chooses skill si ∈ [0, hi ]. Agents with a disadvantaged background face some exogenously given handicap (less-educated parents, poverty, a longer bus commute to school, etc.) that limit their acquisition of skills. An agent i ∈ ND chooses to acquire skill si ∈ [0, ghi ] at school,  where g ∈

1 2,1

is a productivity parameter such that 1 − g captures

the inequality of opportunity in skill acquisition. Let s : N−→ [0, 1] denote the skill choice of every agent. I compare two games: a benchmark game without social sanctions, and a game with social sanctions against excessive skill acquisition at school. In the game with social sanctions, each agent i ∈ N is susceptible to peer pressure by other agents from her own background. I model social sanctions  as follows: for each background J ∈ {A, D }, let lJ ∈ J × 0,

1 2

be a representative agent with

background J and low ability. Assume lA chooses a skill threshold sPA ∈ (0, 1] and lD chooses a skill threshold sPD ∈ (0, g].3 Threshold sPJ is observed only by every i ∈ NJ . Assume that every i ∈ NJ who chooses si > sPJ incurs a fixed cost j ∈ R + . This j captures the social cost of overachieving at school, which takes the form of relative social disaffection and lack of popularity. The case j = 0 captures the game without social sanctions, and j > 0 a game with social sanctionsagainst  skill acquisition at school. For any g ∈

1 2,1

and j ∈ R + , let Cg,j denote the game in which

the parameter pair takes the values (g, j). 1.3. Strategies: choice of social group Assume that there are two social groups A and D, characterized by two competing sets of social norms and actions expected from their members.4 Members of the advantaged social group A speak in a certain language, with a certain accent. They adhere to a dress code and a pattern of behavior in social situations, they eat certain foods and not others, and they spend their leisure time on specific

I define a class of games of discrimination and assimilation. 3

Agent lJ acts as a representative agent in the sense that any other agent in   1 would choose the same cutoff. 2

J × 0, 2 See as well network theories (Currarini et al., 2009; de Marti and Zenou, 2017) and empirical findings (Fong and Isajiw, 2000; Echenique et al., 2006; or Patacchini and Zenou, 2016, among others) on inter-ethnic school friendships.

4 Notice that I use calligraphic letters J ∈ {A, D } to refer to backgrounds, while the standard letters A and D refer to each of the two elements of the partition of agents into social groups.

50

J. Eguia / Journal of Public Economics 156 (2017) 48–58

activities. Assume that every agent with an advantaged background immediately belongs to the advantaged social group, that is, NA ⊆ A. An alternative set of norms, behaviors and actions is characteristic of members of the second, disadvantaged social group D. I assume that there is nothing intrinsically better or worse about either set of actions and norms; their only relevant feature is that agents with background A grow up embracing the advantaged norms as their own, whereas, agents with background D are brought up according to the disadvantaged social norms. Assume that any agent with background D can choose to belong to social group D at no cost, or she can embrace the cultural norms of group A to then join A. Let ai ∈ {0, 1} be the choice of agent i ∈ ND . Let ai = 0 denote that i ∈ ND chooses to be part of group D and not to assimilate, and let ai = 1 denote that agent i ∈ ND chooses to adopt the advantaged cultural norms and to become a member of the advantaged group A. If ai = 1, I say that i ∈ ND “assimilates.” Note that the social cost j tied to overachieving at school is unrelated to the assimilation decisions: Any agent i with background D who chooses skill si > sPD suffers the social cost j regardless of her choice of ai ∈ {0, 1}, and any agent who chooses skill si ≤ sPD suffers no such social sanction, regardless of her assimilation decision.5 Let a denote the decisions to assimilate by all agents in ND . Formally, a : [0, 1] → {0, 1} is a mapping from ability to the assimilation decision. Given a, the composition of the social groups is A = NA ∪ {i ∈ ND : ai = 1} and D = {i ∈ ND : ai = 0}. 1.4. Strategies: the cost of assimilation The cost of assimilating is d(1 − si ), where d ∈ R + is the difficulty of learning and embracing the patterns of behavior consistent with membership in A. This difficulty of assimilation d is an endogenous, strategic variable. It can be interpreted as the level of discrimination: if agents with background A are welcoming to those who assimilate, d is small. If agents with background A are hostile or if they give a cold shoulder to those who are trying to assimilate, then d is high. The level of d is a social norm that emerges endogenously through the aggregation of behavior among agents with background A. As I will show below, in the benchmark model all agents with background A agree on the optimal choice of level of discrimination d, and thus I can simplify the exposition by assuming that an arbitrary agent h ∈ NA acts as a representative agent to choose d for the set of agents NA .6 1.5. Timing and strategy sets I model the interaction of the agents as a game with three stages. 1. Agent h ∈ NA chooses the difficulty of assimilation d. Simultaneously, for each J ∈ {A, D }, agent lJ chooses the social sanctions threshold sPJ . Every i ∈ ND observes d and sPD , but not sPA , and every i ∈ NA observes sPA , but neither d nor sPD .7

5 As I prove below, in equilibrium, agents who choose si > sPD also choose ai = 1. Thus, along the equilibrium path, within the set of agents with background D , the subset who suffer social sanctions and incur cost j, the subset who acquire high skills (si > sPD ), and the subset who assimilate (ai = 1) all coincide. However, off-path, if an agent i deviates and acquires high skills but does not assimilate, she incurs cost j; and if another agent deviates and acquires low skills but assimilates, he does not incur cost j. 6 In extensions I introduce richer preferences such that agents with background A disagree on the optimal level of discrimination. In such extension the level of d is a collective choice that aggregates the individual choices of different players with background A. 7 The proofs of the results hold if agents with background A observe sPD or d.

2. Each agent i ∈ N chooses her skill si . This choice is private information.8 3. Agents with background D choose whether to assimilate or not. Payoffs accrue. We can now complete the formal definition   of the strategies for 1 , 1 × R+ . For each i ∈ N, 2

game Cg,j , for any parameter pair (g, j) ∈

let Zi denote the set of feasible pure strategies for agent i. The strategy set for agent lA , who chooses the sanctions threshold sPA ∈ (0, 1], is ZlA = (0, 1] × [0, hlA ], where the first component is the choice of skill threshold that triggers social sanctions, and the second is agent lA ’s choice of her own skill. For agent h ∈ NA , who chooses the discrimination level d ∈ R + , it is Zh = R + × [0, hh ](0,1] . The first component is the choice of d, and the second is the choice of sh as a function of the observed sPA . (0,1]

For every other agent i ∈ NA \{h, lA }, it is Zi = [0, hh ] . For agent lD , who chooses the sanctions threshold sPD ∈ (0, g], it is ZlD = (0, g] × [0, hlD ]R+ × {0, 1}R+ . The first component is the choice of threshold that triggers social sanctions, the second is agent lD ’s choice of her own skill as a function of the observed discrimination level, and the third is her assimilation choice as a function of the observed discrimination level. For any other agent i ∈ ND \{lD }, it is Zi = [0, hi ]R + ×(0,g] × {0, 1}R + ×(0,g] . The first component is the choice of skill si as a function of the observed discrimination and sanctions threshold, and the second component is the assimilation choice ai as a function of the observed discrimination and sanctions threshold. Let Z denote the set of functions with domain N and such that for any i ∈ N, the image of i is an element of Zi . Then Z is the set of pure strategy profiles. 1.6. Utility function Agents derive utility from their own skill, and from the skill of those in their same social group. Agents with greater skills deliver more positive peer effects. As a result, agents are positively affected by the skill of those in their social group. I assume that interactions are more fruitful if all participants share a common skill, i.e. an agent’s own skill and the skill of those in her group are complementary (having more of one makes the other more valuable). Formally, for each social group J ∈ {A, D}, let sJ be the average skill of agents in J.9 Assume that an agent i with skill si in social group J ∈ {A, D} with average skill sJ derives a utility si sJ .10 In addition, agent i may experience costs of assimilation and/or social sanctions. Let Ui (d, sPA , sPD , s, a) denote the utility function of agent i as a function of the discrimination level d, the thresholds that trigger social sanctions, the skill of every agent, and all the assimilation decisions. If we let ai be exogenously fixed at 0 for any i ∈ NA , and we let 1(sP ,1] be the indicator function such that 1(sP ,1] (si ) = 1 if si > sPJ J

J

8 For expositional simplicity, I blackbox how exactly the social sanctions are carried out. If we wished to model how agent lJ sanctions any player i ∈ NJ who chooses si > sPJ , then we would assume that lJ observes si . The relevant assumption is that agents with background A do not observe si for i ∈ ND . If si were publicly observed, screening would be based on skill-contingent discrimination. 9 See the online appendix for a technical exposition on the formal definition of these averages. 10 The theory hinges on peer effects, but not on this particular functional form. The empirical evidence of positive peer effects in schooling is overwhelming (see surveys by Sacerdote, 2011; Epple and Romano, 2011). I present an extension with a more general form of the utility function and I discuss the evidence on peer effects in the online appendix.

J. Eguia / Journal of Public Economics 156 (2017) 48–58

and 1(sP ,1] (si ) = 0 otherwise, then the utility of an agent i with J background J ∈ {A, D } in social group J ∈ {A, D} can be written as:   Ui d, sPA , sPD , s, a = si sJ − 1(sP ,1] (si )j − ai d(1 − si ).  

 J  Assim. cost Social

(1)

1.7. Solution concept Agents face imperfect information. I assume that agents are sequentially rational, update their beliefs according to Bayes rule and do not play weakly dominated strategies; these three assumptions are common knowledge among agents. I solve by backward induction to find pure Perfect Bayesian equilibria that survive the elimination of weakly dominated strategies. For technical convenience, I also assume that if indifferent about assimilation, agents do not assimilate. Any subsequent mention of an “equilibrium” implicitly means an equilibrium that satisfies these properties. 2. Results As a preliminary observation, note that the first best for the agents with background A is to screen agents with background D , so that only those with a sufficiently high skill level assimilate. I refer to the tuple (d, sPA , sPD , s, a) as an action profile of game Cg,j . Let X denote the set of all possible action profiles. Definition 1. For any agent i ∈ N, an action profile v1 ∈ X is first best for i if v1 ∈ arg max Ui (v). v∈X

An action profile that is first best for every agent with background A maximizes the average skill level of those who end up in group A.  3−

1 g

. An action profile (d, sPA , sPD , s, a) is

first best for all agents with background A if and only if i) ii) iii) iv)

sufficiently small. It follows that for any d, there exists a cutoff correspondence s(d) from the discrimination level to the level of skill, and a cutoff sˆ ∈ s(d), such that each agent with background D ˆ assimilates if and only if her skill is above s. The cutoff that is the first best for agents with background A is

Sanctions

Each agent derives utility from her own skill and from the average skill of the social group she joins, and incurs a disutility j if her skill is above her background’s threshold for social sanctions. Further, agents with background D who assimilate (i ∈ ND such that ai = 1) incur the cost of assimilation d(1 − si ). I assume that the cost function is decreasing in si to capture the intuition that more skilled agents adapt more easily. This completes the definition of game Cg,j = (N, Z, U), where U is the collection of utility functions, one for each agent i ∈ N.

Lemma 1. Define h1 ≡ 2 −

51

si = hi for any i ∈ NA ; si = ghi and ai = 1 for any i ∈ ND such that hi ∈ (h1 , 1]; ai = 0 for any i ∈ ND such that hi ∈ [0, h1 ); and j > 0 ⟹ sPA = 1.

The four conditions state that (i) all agents with background A choose maximal skills; (ii) agents with background D and ability above cutoff h1 choose maximal skills and assimilate; (iii) agents with background D and ability below cutoff h1 do not assimilate (their skill choice is irrelevant); and (iv) agents with background A do not suffer social sanctions. In the game without social sanctions against skill acquisition, agents with background A are able to attain a first best for them as an equilibrium outcome by choosing a particular level of discrimination. Given d, and given any strategy profile by all agents with background D , an agent i with background D prefers to assimilate only if her skill si is high enough so that her cost of assimilating d(1 − si ) is

sˆ ∗ = arg max sA (x) {x}

s.t.

sA (x) =

g + (g − x)(g + x) , 2(2g − x)

where sA (x) is the average skill of the agents in group A as a function of x given that agents with background D assimilate if and only if their skill is above x. I show in the appendix that there is a unique d∗ such that if agent h ∈ A chooses d∗ , the cutoff for assimilation becomes sˆ ∗ . In this unique equilibrium, agents with background A attain their first best: the most skilled among the agents with background D assimilate. Formally: Proposition 2. Assume j = 0. For any productivity parameter g ∈   1 , 1 , there is a unique equilibrium. The unique equilibrium action 2 profile is first best for each agent with background A . The intuition is that agents with background A are able to optimally screen those who assimilate and join their group, by setting a positive but not too large difficulty of assimilation so that only agents with high ability (who in equilibrium are highly skilled) assimilate. As noted in Lemma 1, the assimilation cutoff that is first best for agents with background A is at the skill level that maximizes the average skill sA (x) in the advantaged group (bold solid curve in Fig. 1). If the cutoff for assimilation is at x, the benefit of assimilating for the agent with background D and skill x (thin solid line in Fig. 1) is equal to the difference x(sA (x) − sD (x)), which is a strictly concave function of the assimilation cutoff x; whereas, the cost function is linear with slope −d (dashed line in Fig. 1) and thus it is possible to fix d = d∗ such that the cutoff occurs at exactly the skill level x∗ that maximizes the average skill level of group A. This is the first-best for agents with background A: the difficulty of assimilation is low enough so that every agent whose assimilation benefits the advantaged group chooses to assimilate, while at the same time it is high enough that agents whose assimilation would be detrimental to the advantaged group choose not to assimilate.11 Now consider a game with social sanctions against skill acquisition, i.e. j > 0. I solve by backward induction. First I explain the intuition, then I state the result. At step 2, any agent i with background A and ability hi > sPA chooses skill si ∈ {sPA , hi } and any i with background D and ability hi > 1g sPD chooses si ∈ {sPD , ghi }. At step 1, agent lA who chooses the threshold sPA for agents with background A has an incentive to not sanction at any skill level, because given the complementarity between own individual skill and own group’s average skill, a higher skill level for any i ∈ NA generates a higher utility to all members of NA . Hence, in equilibrium, sPA = 1 and any agent i with background A chooses skill si = hi . Whereas, agent lD ∈ ND who chooses sPD has an incentive to lower the skill level of some agents to prevent them from assimilating. For any discrimination level d and sanctions threshold sPD , there is a cutoff that depends on d and sPD such that agents with background D choose to assimilate if and only if their skill is above this cutoff (Lemma 5 in the Appendix). In equilibrium, low ability agents with background D are hurt by this assimilation process: they are left behind. Setting sPD at the cutoff of assimilation deters some agents

11 I calculate this exact goldilocks level d∗ as a function of the inequality parameter g in the proof of Proposition 2 in the Appendix.

52

J. Eguia / Journal of Public Economics 156 (2017) 48–58

0.6

Avg.skillsA(x) 0.5

0.4

Cost d(1 x)

0.3

0.2

Benefit x(sA(x) sD(x))

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

x*

0.6

0.7

Skill cut off x to assimilate

Fig. 1. Equilibrium assimilation given g = 0.75.

with background D from acquiring a skill level above the cutoff and thus from assimilating. The social sanction that is optimal for agents with background D maximizes sD by inducing as many high ability agents as possible to stay in group D, while lowering their skill level only just as much as needed to deter them from assimilating. Hence in every equilibrium, sPD < g and some agents with background D are deterred from overachieving. The following proposition shows that in equilibrium, highly able students with background D incur social sanctions if they acquire a ∗ high level of skills (any level between sPD and g), while students with background A never incur social sanctions for acquiring skills. Let s∗A denote the equilibrium value of the average skill in group A (sA ).   Proposition 3. Assume j > 0. For any g ∈ 12 , 1 , in any equilibrium, ∗ ∗ social sanctions thresholds are sPD < g and sPA = 1 . Furthermore, ∗ P ∗ there exists an equilibrium in which sD = s∗A and sPA = 1 . Proposition 2 had shown that the equilibrium without social sanctions leads to assimilation. Low-skilled agents with background D become worse off after their better co-ethnics assimilate. Compared to the benchmark with no assimilation, agents with background A and the most able among those who assimilate benefit from assimilation. Whereas, agents with background D who do not assimilate are worse off. Proposition 3 shows that low ability agents with background D react by raising the cost of exiting the disadvantaged social group, which induces some high ability agents to stay in group D. Social sanctions against skill acquisition are strategic barriers to exit. In every equilibrium, highly able agents with background D are pressured to underperform; whereas, agents with background A are not. This is the acting white phenomenon. 3. Empirical implications This screening theory of assimilation and discrimination yields several testable predictions about social sanctions against skill acquisition at school. In this section I compare these predictions with those of the most prominent alternative explanations of acting white in the literature. Definition 2. (Fryer, 2006) “Acting white” is a set of social interactions in which minority adolescents who get good grades in

school enjoy less social popularity than white students who do well academically.12 Fryer (2006) shows that “the popularity of white students increases as their grades increase. For black and Hispanic students, there is a dropoff in popularity for those with higher GPAs.” This peer pressure against academic achievement leads minority adolescents to underperform, and contributes to the achievement gap of African-American and Hispanic students relative to white students.13 Proposition 3 provides an explanation of the acting white phenomenon: students in under-privileged communities dissuade their co-ethnics from acquiring skills in order to increase the cost of assimilation and deter exit from the community. This explanation yields different empirical implications from those of alternative explanations in the literature (see the survey by Sohn, 2011). The “oppositional culture” theory of Fordham and Ogbu (1986) and Fordham (1996) posits that academic failure is an integral part of African-American group identity. Whites embrace studiousness and hard work; whereas, minorities reject these values, embracing instead a counterculture defined in opposition to mainstream values. In particular, they oppose the pursuit of success at school. A second, now traditional explanation is the “self-sabotage” argument: African-Americans engage in willful victimism, persuading themselves that discrimination in the job market makes costly accumulation of human capital not worthwhile (McWhorter, 2000).14 According to theories based on oppositional culture or sabotage, the acting white problem ought to be more severe in schools that are

12 This definition ties the social cost exclusively to good grades (a high si ), and not to any other behavior that may correlate with assimilation. I adhere to this narrow definition. Alternative definitions link social sanctions exclusively to behaviors unrelated to academic achievement (Carter, 2005), or to attaining good grades and to other non-academic behaviors (Fordham and Ogbu, 1986). Further, I use “acting white” only to denote the aggregate social phenomenon characterized by the racial differences in the social effects of achievement. I do not use the terms “acting white” or “acts white” to describe any particular action by an individual agent. 13 Alternative explanations of the achievement gap suggest that education is riskier for the working class (Breen and Goldthorpe, 1997), or that minorities suffer from “stereotype threat” (Steele and Aronson, 1995). These factors may contribute to the achievement gap, but they do not explain the social cost of studying that defines the acting white phenomenon. 14 The term “self-sabotage” is misleading because it improperly anthropomorphizes the African-American minority. No individual African-American engages in self-sabotage; rather, less able students sabotage their more talented peers.

J. Eguia / Journal of Public Economics 156 (2017) 48–58

more segregated. Students in such schools are not exposed to alternative cultures or views. The screening theory in this paper has the opposite empirical implication: the explanation of social sanctions as a strategic exit barrier applies only where there exists an alternative to the disadvantaged culture and exit is an option. Students in mixed schools have opportunities to join a predominantly white social network and abandon the black community. A top student in a fully segregated school has no alternative community to join, so social sanctions are unnecessary and do not arise. Testable Hypothesis 1. The acting white phenomenon occurs in ethnically diverse schools, and not in homogeneously African-American or homogeneously Hispanic ones. Something close to this hypothesis is tested by Mirón and Lauria (1998), Tyson (2006), Fryer (2006) and Fryer and Torelli (2010). They all find that the acting white problem is more severe in more racially integrated schools: in predominantly African-American schools, “there is no evidence at all that getting good grades adversely affects students’ popularity ” (Fryer, 2006). Fryer and Torelli (2010) find this “surprising,” but it fits with the explanation of social sanctions as exit barriers: in a fully segregated school there is no opportunity to assimilate into any other community, no threat of exit, and thus we observe no social sanctions. Fryer (2006) conjectures that perhaps the problem is attenuated if school desegregation leads to cross-ethnic friendships. However, the screening theory implies the opposite: the greater the influence of white culture over African-American students, the greater the risk that the best African-American students assimilate. Fryer (2006) reports that indeed, greater inter-ethnic integration leads to a more severe acting white problem. Fryer and Torelli (2010) conclude that the oppositional culture theory “does quite poorly” in explaining why acting white is less prevalent in majority black schools. Austen-Smith and Fryer (2005) propose an alternative explanation: high-school students shun studious colleagues because studiousness signals social ineptitude. They model a set of students and a black-boxed “peer group.” Each student’s type has two components: an “economic type” and a binary (high or low) “social type.” The economic type determines the marginal utility of time spent studying. The social type determines the marginal utility of leisure time, and it also determines the utility that the peer group derives if it accepts a student (positive if the social type of the accepted student is high, but negative if it is low). Students allocate a unit of time between studying and leisure. In equilibrium, those with a low social type, who face a lower opportunity cost of studying because their marginal utility from leisure is low, study more. The peer group shuns those who study, and accepts those who signal their high social type by not studying. Some high social types thus choose not to study. While this argument is compelling, it applies to all races and social groups: it can explain why students do not want studious friends, but it cannot explain why only African-American and Hispanic students, and not non-Hispanic white students, exhibit this preference. This unexplained asymmetry across ethnic groups is the essence of the acting white phenomenon. In the screening theory I have developed, this asymmetry is obtained as a main result (Proposition 3), derived from primitives (utility functions, distribution of ability and technology for peer pressure) that are symmetric across groups, with the exception of a parameter that measures the inequality of opportunity to acquire skills. Solely from this unequal technology factor, it follows that agents with a disadvantaged background discourage their peers’s acquisition of skills, while agents with an advantaged background never discourage skill acquisition. The signaling theory by Austen-Smith and Fryer (2005) and the screening theory in this paper disagree on one testable empirical

53

implication. If students who obtain good grades are shunned because good grades signal social ineptitude, the popularity of a given student among students of any ethnicity must decrease with the student’s grades. In particular, according to the signalling argument, the popularity of African-American and Hispanic students among students of other ethnicities must decrease. According to my screening theory, a minority student who obtains high grades is moving away from her community and toward assimilation. Therefore, this student is less popular among her co-ethnics (who will be left behind) and more popular among the non-Hispanic white students she is trying to join. We have a test to discriminate between the two theories. Testable Hypothesis 2. The relation between the grades of AfricanAmerican and Hispanic students and their popularity among nonHispanic white students is strictly positive. Austen-Smith and Fryer’s (2005) theory predicts the opposite: a strictly negative relation between grades and number of friends of any race. Fryer and Torelli (2010, Table 5) report that African-American or Hispanic students’ number of friends of other races, increases in grades. Patacchini and Zenou (2016) find that for black teenagers, higher test scores are associated with a higher percentage of out-race friends. In summary, the screening theory of acting white fits well with the following findings: a) The greater prevalence of the acting white phenomenon in more integrated schools, and b) The positive correlation between grades and out-of-race popularity of underprivileged minority students. These findings clash with the predictions of the oppositional identity (Fordham and Ogbu, 1986), self-sabotage (McWhorter, 2000) and signaling theories (Austen-Smith and Fryer, 2005).15 The social sanctions against high achieving African-American and Hispanic students are only an instance of a broader social phenomenon. In groups as diverse as the Buraku outcasts in Japan, Italian immigrants in Boston, the Maori in New Zealand and the working class in Britain, high-achievers have suffered a negative externality from their peer group (see discussions by Fryer (2007) or Sohn (2011)). Jensen and Miller (2017) find evidence of this phenomenon in rural India: parents strategically underinvest in education for their children to prevent them from migrating to the cities. I interpret the acting white phenomenon as yet another exit barrier. The screening theory’s external validity in applications beyond acting white is testable. The theory claims that underprivileged communities deter exit by making skill acquisition costly. If this is correct, acting white is only one instance of a broader phenomenon. An analogous strategic environment is faced by students in rural schools: academic success leads to migration to the city. Therefore, we can test if an “acting urban” phenomenon also arises in a setting with no ethnic divisions. Testable Hypothesis 3. Rural students who obtain top grades are less popular, regardless of their race, than their classmates with lower grades.

15 However, identifying whether social sanctions are triggered by achievement or by other behaviors unrelated to achievement is difficult, and the evidence on racial differences in the causal effect of achievement on popularity is disputed. Wildhagen (2011) and others find that there is no difference across races in social sanctioning against high achievement. Carter (2005) argues that social sanctions follow from the adoption of certain cultural norms relating to clothes, music or speech pattern, and not from achievement.

54

J. Eguia / Journal of Public Economics 156 (2017) 48–58

In the United States, this can be tested by using the longitudinal National Study of Adolescent to Adult Health (Add Health) dataset. An analogous prediction applies to other countries and contexts; in the words of Fryer and Torelli (2010): “any group presented with the same set of payoffs, strategies and so on, would behave identically.” 4. Extensions I consider several extensions and generalizations to the theory. 4.1. Homophily Assume that agents with background A have in-group preferences captured by an homophily parameter c ∈ R + so that their utility decreases proportionally in the size of the set of agents who assimilate. Dropping the terms on social sanctions and cost of assimilation (which do not apply to background A), assume that the utility function of each i ∈ NA is: 1 Ui (s, a) = si sA − c 

ai dhi . 0



(2)



Homophily

Recall that agents with background D assimilate if and only if their skill is above a cutoff x, and the average skill in the advantaged group as a function of this cutoff is maximized where sA (x) = 0. With in-group preferences and cutoff x for assimilation, the homophily term in the utility function is −c(1 − gx ). Assuming c is small enough to guarantee an interior solution, the utility of agents with background A is maximized where sA (x) + cg = 0. The solution is strictly increasing in c. The proof is in the online appendix. Therefore, fewer agents assimilate if those with an advantaged background have in-group preferences. 4.2. Decentralized choice of discrimination In the benchmark model, a representative agent h chooses d on behalf of all agents with background A. Here I suggest an alternative micro-foundation for a decentralized choice of discrimination d. ˆ A ⊂ NA of arbitrary Suppose d is chosen collectively by a subset N size N ∈ N.16 Label these agents according to their ability, so that ˆ A strategically chooses h1 ≤ h2 ≤ . . . ≤ hN . Assume that each i ∈ N ¯ ¯ di ∈ [0, d] for some sufficiently high d ∈ R++ , and that the vector (d1 , . . . , dN ) aggregates into a difficulty of assimilation d ∈ R ++ . Assume that d is set at the m − th largest value among the set of individual choices {d1 , d2 , . . . , dN } for some integer m ∈ {1, . . . , N}; for instance d could be the minimum, or the median, or the maximum. The intuition is that at least m agents must support erecting a given barrier to assimilation in order for this barrier to materialize. With the benchmark utility function given by expression (1) and no in-group preferences, all agents with background A have comˆ A and it mon preferences to maximize sA , so di = dj = d ∀i, j ∈ N was without substantive loss to assume instead that a representative agent h chooses d for the group. With in-group preferences given by utility function (2), agents with background A have conflicting interests about assimilation: they all want agents with skills above a cutoff to assimilate, but they disagree on the ideal cutoff. Low-skilled agents benefit less from the increase in the average skill sA caused by assimilating highskilled agents, but they incur the same homophily cost. As a result,

16 If we let all agents with background A choose d, given that they are infinitely many, the strategic incentives to choose optimally vanish. Keeping the number finite generates strict incentives to choose optimally.

among all agents with background A, those with low skills prefer more discrimination and less assimilation than those with high skills (Corollary OA3 and Proposition OA5 in the online appendix).  ˆ If we assume instead that d = N1 N i=1 di , then members of NA ¯ because with skill below some level discriminate maximally (d = d) i

they want their group to discriminate more; whereas, those with skill above this level do not discriminate at all ( di = 0) because they find that their group already discriminates too much. Under either assumption on how discrimination aggregates, the implication is that among agents with background A, those with low skills want more discrimination and less assimilation than those with high skills. This prediction is testable. Testable Hypothesis 4. Education is positively correlated with more favorable attitudes toward immigrants and ethnic minorities. Related evidence on the relationship between education and attitudes toward immigration is consistent with this prediction. Hainmueller and Hiscox (2007) find that in Europe “people with higher levels of education and occupational skills are more likely to favor immigration” .This effect is still positive, though smaller, in poorer countries (see Fig. 3 in Mayda, 2006). In the online appendix I provide additional extensions that incorporate the following features into the model: c) Generalized functional forms, in which the complementarity between own skill and group skill is given by a function vi (si , sJ ) that is increasing in both arguments. d) A majority of size k > 12 with an advantaged background, and a minority of size 1 − k with a disadvantaged background. e) An endogenous cost j of social sanctions, and a decentralized choice of the threshold for social sanctions sPD . f) Partial assimilation. 5. Summary and policy implications I have presented a rational choice theory of assimilation and discrimination at school with two main results. First, students with an advantaged background use discrimination as a screening device. Highly skilled outsiders assimilate, and the rest do not (Proposition 2). This screening equilibrium is optimal for the advantaged group because the students who assimilate into it generate positive peer effects. This assimilation process splits disadvantaged communities into two subgroups: the most skilled students assimilate, and all others are left behind. Policies that target only the top of the skill distribution, such as affirmative action in college admissions, sharpen the divide between the most and least successful within the disadvantaged community: “enforcement of affirmative action guidelines was beneficial, but only to more qualified blacks” (Son et al., 1989). More generally, “a variety of (...) shifts have brought great progress to middle-class blacks, (...)pushing a less qualified black population into an increasingly isolated and alienated underclass” (Son et al., 1989). This downside can be attenuated by pairing any policy that targets the top of the distribution with measures that compensate those left behind. The second finding is that the acting white phenomenon is an strategic exit barrier erected by students with a disadvantaged background and low ability (Proposition 3). Social sanctions on excessive skill acquisition induce some talented disadvantaged students to choose only a moderate level of skills and to stay in the disadvantaged group. Because the level of skills they choose is higher than the average skill in the disadvantaged group, the group benefits from retaining these talented students. The social sanctions in the acting white phenomenon reduce aggregate social welfare, but they increase the aggregate skill levels of those left in the disadvantaged community. Crucially, they make the students who impose the sanctions better off.

J. Eguia / Journal of Public Economics 156 (2017) 48–58

This screening theory yields some testable empirical implications that fit well with previous findings on social sanctions against skill acquisition at school (Hypotheses 1 and 2), and it proposes other implications that have not been systematically tested yet (Hypotheses 3 and 4). Implications for education policy follow. A suggestion to address the acting white problem is to create incentives so that all students become stakeholders in the success of their most able classmates. Coleman (1961) argued that academic high achievers are not popular because studying only produces an individual gain, with no spillovers to the rest of the school. If students perceive that it is in their own best interest for others to obtain top grades, then their attitude toward students with top grades will be more favorable. Policy interventions can change individual incentives in the classroom by providing contingent rewards based on observed behavior. Financial incentive schemes typically reward individuals for their own behavior or achievement, without attention to peer effects (see a survey by Slavin (2010). These incentives reinforce the perception that educational achievement is a purely individualistic good. I suggest instead to distribute the conditional rewards to a group of peers. A program that rewards every classmate or peer of a good student changes educational achievement from an individualistic good that only benefits the student, into a public production good that immediately benefits every member of the community through the contingent collective reward. With contingent collective rewards, the most able students produce a public good enjoyed by all their classmates. I conjecture that there would be no social sanctions against achieving the high grades that generate this public good. Appendix A Proof of Lemma 1. For each social group J ∈ {A, D}, let sJ (x) be the average skill in group J as a function of x assuming that agents in ND acquire maximum skill and assimilate if and only if their skill is strictly above x. Then  sA (x) =

g+x g−x 1 + 2 2 g



1 1+

g−x g

=

g + (g + x)(g − x) , 2(2g − x)

g+x where g−x g is the measure of agents who assimilate and 2 is their average skill. The first order condition for maximization of sA (x) is



1 2(2g − x)

2

 g2 − 4gx + g + x2 = 0,

or equivalently g2 − 4gx + g + x2 = 0.   The only solution in the interval 12 , 1 for this condition is x =  2g − g (3g − 1). Because the first derivative of sA (x) is continuous, positive for x = 12 and negative for x = 1 and zero only for    x = 2g − g (3g − 1) within interval 12 , 1 , it follows that the second order condition is satisfied as well and x∗ (g) = 2g −

 g (3g − 1).

 The average skill level sA = 2g − g (3g − 1) is achieved if and only if (up to any deviation by a mass zero of agents, who have no effect on averages) the following four conditions hold: i) every i ∈ NA chooses maximal skills according to her potential (si = hi ); ii) any j ∈ D ×     2 − 3 − 1/g, 1 chooses skill sj = ghj (so that sj > 2g− g (3g − 1))    and assimilates; and iii) any j ∈ D × 0, 2 − 3 − 1/g chooses not to assimilate. Further, a first best for agents with background A requires that none of them suffer sanctions for skill acquisition, so it requires sPA = 1 for any j > 0. 

55

I establish three lemmas used to prove Propositions 2 and 3. Let ZWU ⊂ Z be the set of strategy profiles in which no agent plays a weakly dominated strategy. Lemma 4. Assume agents play an arbitrary z ∈ ZWU . Then for any i ∈ NA and any j ∈ ND , si ∈ {hi , sPA } and sj ∈ {ghj , sPD } . Proof. For any i ∈ N, since si is private information, it has no effect on the actions taken by any other agent at stage 3. For each social group J ∈ {A, D}, expression si sJ − ai d(1 − si ) is increasing in si (strictly if sJ > 0); it follows that for each J ∈ {A, D }, the utility of agent i ∈ NJ is increasing in si for si ∈ [0, sPJ ) ∪ (sPJ , 1] (strictly if sJ > 0). Hence choosing any si ∈ / {hi , sPA } is dominated either by si = hi or by P si = sA for any i ∈ NA , and choosing any sj ∈ / {ghj , sPD } is dominated P either by sj = ghj or by sj = sD for any j ∈ ND .  Lemma 4 together with sPJ > 0 (true by assumption) imply that sA > 0 and it also implies that if D contains more than one agent, then sD > 0. Because assimilating is less costly for high-skilled agents, in equilibrium there is a cutoff such that agents with background D assimilate if and only if their skill is above the cutoff. Let z ∈ Z be a strategy profile, and for any i ∈ N, let z −i be the strategy profile that contains a strategy for every agent in N\{i}. Lemma 5. Assume z ∈ Z is an equilibrium strategy profile. For any (d, sPD ) played at the first stage, there exists a skill cutoff s(d, sPD ) ∈ [−1, g] such that according to profile z, any agent i ∈ ND chooses ai = 1 if si > s(d, sPD ) and chooses ai = 0 if si ≤ s(d, sPD ) . Proof. Consider any (d, sPD ) such that if all players play according to z at stages two and three and every i ∈ NA \{h} plays z throughout the game, then sA ≤ sD . By sequential rationality, in this case no agent assimilates, so the statement holds trivially with s(d, sPD ) = g. Consider instead any (d, sPD ) such that sA > sD (under the same conditions). Given (d, sPD ) and given the anticipated play according to z −i following (d, sPD ), the benefit of assimilating si (sA − sD ) is increasing in si and the cost of assimilating d(1 − si ) is decreasing in si . Therefore, agent i chooses ai = 1 if and only if si is above some cutoff. Further, for any i, j ∈ ND such that si > sj , given (d, sPD ) and given z −i and z −j , the benefit of assimilating is greater and the cost is lower for i than for j. Therefore, it must be that according to z, {aj = 1 ⟹ ai = 1}. Hence for any (d, sPD ) ∈ R+ , there is a cutoff s(d, sPD ) ∈ [−1, g], such that any i ∈ ND strictly prefers to choose ai = 1 if and only if si is strictly above s(d, sPD ).  Lemma 6. There exist cutoffs h(j, sPA ) ∈ [sPA , 1] and h(j, sPD , d) ∈ [ 1g sPD , 1] such that in equilibrium: i) si = hi ∀i ∈ A × [0, sPA ] ∪ (h(j, sPA ), 1] and si = sPA ∀i ∈ A × P

sA , h(j, sPA ) ; and ii si = ghi ∀i ∈ D × [0, 1g sPD ] ∪ (h(j, sPD , d), 1] and si = sPD ∀i ∈ D ×   1 P P g sD , h(j, sD , d) . Proof. At the second stage, given that sA > 0 (established above), any agent i ∈ A × [0, sPA ] uniquely maximizes her utility by choosing si = hi . Any agent i ∈ A × (sPA , 1] faces a trade-off: choosing si = hi she incurs a cost j ≥ 0, but compared to choosing si = sPA < hi , she derives a benefit (hi − sPA )sA > 0. This benefit is strictly increasing in hi , while the cost is j ∀hi . And for any i, j ∈ NA such that hi > hj , for any fixed strategies by any agent in N\{i, j}, the benefit for i of

56

J. Eguia / Journal of Public Economics 156 (2017) 48–58

choosing si = hi is strictly greater than the benefit for j of choosing sj = hj . Thus, there exists a cutoff h(j, sPA ) ∈ [sPA , 1] and such that any i ∈ A × (sPA , h(j, sPA )) chooses si = sPA and any i ∈ A × (h(j, sPA ), 1] chooses si = hi .   Similarly, any agent i ∈ D × 0, 1g sPD uniquely optimizes at si =   ghi . Any agent i ∈ D × 1g sPD , 1 faces a trade-off: choosing si = ghi > sPD she incurs a cost j, but she derives a benefit (ghi − sPD )sD > 0 if she does not assimilate, and a reduced cost of assimilation plus (ghi − sPD )sA if she assimilates. For any i ∈ ND , the benefit of choosing si = ghi is increasing in hi , while the cost is fixed at j. In addition, for any i, j ∈ ND such that hi > hj , for any fixed strategies by any agent in N\{i, j}, the benefit for i of choosing si = hi is strictly greater than the benefit for j of choosing sj = hj . Thus, there exists     a cutoff h(j, sPD , d) ∈ 1g sPD , 1 such that any i ∈ D × 1g sPD , h(j, sPD , d)

so in order for agent (D , x∗ (g)) to be indifferent about assimilation, it must be ∗ x∗ (g)(sA (x∗ (g)) − s = d(1 − x∗ (g)), D (x   (g)))  or   1 2g − g (3g − 1) − g − g (3g − 1) 2g − g(3g − 1) 2     = d 1 − 2g − g (3g − 1) ,

so we can express the equilibrium discrimination as a function of g thus:     2g − g (3g − 1) 1   g− g (3g − 1)  2 1 − 2g − g (3g − 1)  2  2 g − 12 g (3g − 1)  . =  1 − 2g − g (3g − 1)

d∗ (g) =

chooses si = sPD and any i ∈ D × (h(j, sPD , d), 1] chooses si = ghi . 

Proof of Proposition 2. First step. I show that in equilibrium, every i ∈ NA chooses si = hi and every i ∈ ND chooses ghi . Since j = 0, sPJ is not payoff-relevant; assume any value is chosen. Using the notation from Lemma 6, h(0, sPA ) = sPA and h(0, sPD , d) = 1g sPD . Hence by Lemma 6, every i ∈ NA chooses si = hi and every i ∈ ND chooses ghi . Second step. I show that in equilibrium, there is a cutoff for assimilation such that agents with background D assimilate if and only if their skill is above the cutoff (Lemma 5). I show below that the solution for the value of the cutoff is in equilibrium unique. Until I prove uniqueness, I let s : R+ ⇒ [0, g] denote the set of all possible cutoffs for assimilation that constitute an equilibrium of the continuation game after d is chosen and observed by agents with background D , so that s(d) is the set of possible cutoffs for a given d. Third step. I find the cutoff that generates the best outcome for all agents in NA . For any x ∈ [0, g], there exists a unique value of d such that agent i ∈ ND with skill si = x is indifferent between assimilating or not given that other agents with background D assimilate if and only if their skill is above x. Let this value be denoted by s −1 (x). The optimal cutoff of assimilation for agent h ∈ A is x∗ = arg max sh sA (x) x∈[0,g]

s.t. sA (x) =

g+(g−x)(g+x) , 2(2g−x)

g+g2 −x2 . x∈[0,g] 2(2g−x)

which is arg max

This optimiza-

tion problem is the maximization of a continuous function over a compact set, hence it has a solution x∗ . Then there exists an equilibrium in which h ∈ A chooses d = d∗ = s −1 (x∗ ), every i ∈ NA chooses si = hi , every j ∈ ND chooses sj = ghj and every j ∈ ND chooses aj = 1 if and only if sj > x∗ , or, equivalently, if and only if hj > h∗ = 1g x∗ . In particular, from Lemma 1, we obtain the solution x∗ (g) = 2g −

 g (3g − 1)

It follows that h∗ (g) ∈



1 2,1

and 

h∗ (g) = 2 −

for any g ∈



1 2,1

 

3 − 1/g.

.

Fourth step. I establish uniqueness. To prove uniqueness, we calculate the equilibrium level of discrimination d∗ . Note that   g (3g − 1)))(g + (2g − g (3g − 1)))  2(2g − (2g − g (3g − 1)))   g + 2g g (3g − 1)−3g2 = 2g − g (3g − 1), and =  g (3g − 1)  1 g (3g − 1), sD (x∗ (g)) ≡ s∗D (g) = g − 2 sA (x∗ (g)) ≡ s∗A (g) =

g + (g − (2g −

Next define b(x) = x(sA (x) − sD (x)) as the “benefit of assimilation” function, which measures the benefit of assimilating for the agent at the cutoff x, as a function of the cutoff, given that agents with background D assimilate if and only if their type is above the cutoff x. The function b(x) is  x(sA (x) − sD (x)) = x

 g + (g − x)(g + x) x − . 2(2g − x) 2

Its first derivative is  

g + (g − x)(g + x) x ∂ x − 2(2g − x) 2 ∂x

 =

g

(2g − x)

2

  g2 − 4gx + g + x2 .

Its second derivative is 



  ∂ 3g − 1 g = 2g2 < 0, g2 − 4gx + g + x2 ∂ x (2g − x)2 (x − 2g)3 so the function is concave. Given that the benefit function b(x) is concave, and the cost of assimilation function c(d, si ) = d(1 − si ) evaluated at si = x is linearly decreasing in x, it follows that for any d, the net benefit of assimilating b(x) − c(d, x) is either always increasing in x (in which case it can be zero only once), or first increasing and then decreasing (in which case it could be zero once, or twice at two different equilibria). Suppose b(x) − c(d, x) > 0 at x = g and d = d∗ (g); then even if b(g) − c(d∗ (g), g) is first increasing and then decreasing, it can only be zero once, and the continuation game that follows after advantaged agent h chooses d∗ (g) has a unique equilibrium at this value of x such that b(x) = c(d∗ (g), x). Since this is the unique first best for agent h, it follows that h indeed chooses d. Thus we want to show that b(x) − c(d, x) > 0 at d = d∗ (g) and x = g. Note that  g g + (g − g)(g + g) − 2(2g − g) 2  2  1 2 g − 2 g (3g − 1)   (1 − g ) . −  1 − 2g − g (3g − 1)

b(g) − c(d∗ (g), g) =g



A few lines of algebraic manipulations (available from the  author)  show that this expression is strictly positive for any g ∈ 12 , 1 , so   b(x) − c(d, x) > 0 at x = g and d = d∗ (g) for any g ∈ 12 , 1 as desired. Thus, there is only one equilibrium of the continuation game after d∗ is chosen, and d∗ is indeed chosen in the unique equilibrium of the whole game. Therefore, the equilibrium outcome is unique. 

J. Eguia / Journal of Public Economics 156 (2017) 48–58

Proof of Proposition 3. I first prove the existence claim. By Lemma 5, at the assimilation stage there is a cutoff s(d∗ , (sPD )∗ ) ∈ [−1, g] such that any agent i ∈ ND chooses ai = 1 if si > s(d∗ , (sPD )∗ ) and chooses ai = 0 if si ≤ s(d∗ , (sPD )∗ ). Unlike in the proof of Proposition 2, the cutoff may not be unique; if it is not unique, arbitrarily select the solution with the highest cutoff and fewest agents assimilating. By Lemma 6, at the skill-acquisition stage there exist cutoffs h(j, sPA ) ∈ (sPA , 1] and h(j, sPD , d) ∈ ( 1g sPD , g] such that in equilibrium: i) si = hi ∀ i ∈ A × [0, sPA ] ∪ (h(j, sPA ), 1] and si = sPA ∀ i ∈ A × P

sA , h(j, sPA ) ; and ii) si = ghi ∀ i ∈ D × [0, 1g sPD ] ∪ (h(j, sPD , d), 1] and si = sPD ∀ i ∈   D × 1g sPD , h(j, sPD , d) .

At the first stage, note first that in equilibrium (sPA )∗ = 1. Choosing sPA < 1 causes any i ∈ A × (sPA , h(j, sPA )) to choose skill si = sPA < hi , which reduces sA and makes agent lA strictly worse off. Given j ∈ R+ , for any sPD ∈ [0, g] and any d ∈ R+ , let f (sPD , d, j) denote the ability of the marginal agent j ∈ ND who is indifferent between choosing sj = ghj and making the optimal assimilation decision, or choosing sj = sPD and making the   optimal assimilation decision, given that: i) any i ∈ D × 0, 1g sPD chooses si = ghi and   ai = 0; ii) every i ∈ D × 1g sPD , f (sPD , d, j) chooses si = sPD and ai = 0; and iii) every i ∈ D × ( f (sPD , d, j), 1] chooses si = ghi and ai = 1. If no agent is indifferent, let f (sPD , d, j) = 1. Note that because the utility functions are continuous in ability, function f (sPD , d, j) is continuous in sPD . Assume (Assumption A1) that every i ∈ D × [0, 1g sPD ] chooses si =   ghi and ai = 0, every i ∈ D × 1g sPD , f (sPD , d, j) chooses si = sPD and ai = 0, and every i ∈ D × ( f (sPD , d, j), 1] chooses si = ghi and ai = 1. I later verify that in equilibrium, there is no profitable deviation from Assumption A1, given the equilibrium values of sPD and d. Given A1 and for any given d, consider sA : [0, g] −→ [0, 1] as a function of sPD . Then sA (sPD ) − sPD is a continuous function that is strictly positive for sPD = 0 and strictly negative for sPD = g. Thus, by the Intermediate Value Theorem, there exists a value of sPD such that sA (sPD ) = sPD . I construct an equilibrium in which (sPD )∗ = sA ((sPD )∗ ) and d∗ is such that any i ∈ ND with si = (sPD )∗ is indifferent between assimilating or not, given A1. I now check that there is no profitable deviation. By construction, at stage 3, i ∈ ND with si = (sPD )∗ is indifferent about assimilation, so in equilibrium does not assimilate. This also implies that any i ∈ ND with si > (sPD )∗ chooses ai = 1 and any i ∈ ND with si < (sPD )∗ chooses ai = 0 as assumed. At stage 2, every i ∈ NA chooses si = hi , which is optimal given that at stage 1 (sPA )∗ = 1. Every i ∈ D × [0, 1g (sPD )∗ ] chooses si = ghi as assumed, which is opti  mal; every i ∈ D × 1g (sPD )∗ , f ((sPD )∗ , d∗ , j) chooses si = (sPD )∗ , which is again optimal by the definition of f (sPD , d, j); and every i ∈ D × ( f ((sPD )∗ , d∗ , j), 1] chooses si = ghi , which is optimal by the definition of f (sPD , d, j). Hence, the assumptions at the second stage are verified. At stage 1, we have already argued that (sPA )∗ = 1. We now note that choosing sPD < (sPD )∗ is not a profitable deviation: it induces some agents in D × ( 1g sPD , 1g (sPD )∗ ) to choose si = sPD < ghi , which reduces the average skill level sD . Similarly, choosing sPD > (sPD )∗ is not a   profitable deviation: it induces any i ∈ D × 1g (sPD )∗ , 1g sPD to choose si = ghi and ai = 1, which reduces sD .

57

Deviating to d < d∗ causes all i ∈ ND with si = sPD and some i ∈ ND with hi < sPD to choose ai = 1, which reduces sA . Because f ((sPD )∗ , d∗ , j) ≤ f ((sPD )∗ , d, j) for any d = d∗ , deviating to d < d∗ 

also induces any i ∈ D × f ((sPD )∗ , d∗ , j), f ((sPD )∗ , d, j) to choose skill si = (sPD )∗ instead of si = ghi > (sPD )∗ , which reduces sA further. Choosing d > d∗ is not profitable, because it induces any i ∈ D × [ f ((sPD )∗ , d∗ , j), f ((sPD )∗ , d, j)) to choose ai = 0, which has no effect if f ((sPD )∗ , d∗ , j) = f ((sPD )∗ , d, j)), and reduces the average skill sA otherwise. Thus, there is no profitable deviation at any stage, and the strategy profile constitutes an equilibrium. Next I prove the claim that in every equilibrium sPD < g and sPA = 1. First note that as mentioned above, choosing sPA < 1 causes any i ∈ A × (sPA , h(j, sPA )) to choose skill si = sPA < hi , which reduces the average skill level sA and makes lA strictly worse off. Suppose sPD = g. Then, the unique equilibrium with sPD = g and  P sA = 1 is such that any i ∈ ND with hi > h∗ = 2 − 3 − 1/g assimP ilates. Suppose lD (the agent who chooses sD ) deviates to choose

sPD = gh∗ . Then any i ∈ D × h∗ , f ((sPD ), d∗ , j) chooses hi = sPD and ai = 0, which increases the average skill level sD , thus benefit lD . Therefore, the original profile with sPD = g is not an equilibrium and in every equilibrium sPD < g.  Appendix B. Supplementary material Supplementary material to this article can be found online at https://doi.org/10.1016/j.jpubeco.2017.08.009.

References Akerlof, G.G., Kranton, R.E., 2000. Economics and identity. Q. J. Econ. 115 (3), 715–753. Austen-Smith, D., Fryer, R.G., 2005. An economic analysis of acting white. Q. J. Econ. 120 (2), 551–583. Bénabou, R., Tirole, J., 2011. Identity, morals and taboos: beliefs as assets. Q. J. Econ. 126 (2), 805–855. Bisin, A., Verdier, T., 2000. Beyond the melting pot: cultural transmission, marriage and the evolution of ethnic and religious traits. Q. J. Econ. 115 (3), 955–988. Bisin, A., Verdier, T., 2001. The economics of cultural transmission and the dynamics of preferences. J. Econ. Theory 97 (2), 298–319. Breen, R., Goldthorpe, J.H., 1997. Explaining educational differentials. Toward a formal rational action theory. Ration. Soc. 9 (3), 275–305. Carter, P.L., 2005. Keepin’it Real: School Success Beyond Black and White. Oxford University Press. Carvalho, J.P., Koyama, M., 2013. Resisting Education. SSRN wp 2290006. Coleman, J.S., 1961. The Adolescent Society. Free Press New York. Currarini, S., Jackson, M.O., Pin, P., 2009. An economic model of friendship: homophily, minorities and segregation. Econometrica 77 (4), 1003–1045. de Marti, J., Zenou, Y., 2017. Segregation in friendship networks. Scand. J. Econ. 119 (3), 656–708. Echenique, F., Fryer, R.G., Kaufman, A., 2006. Is school segregation good or bad? Am. Econ. Rev. Pap. Proc. 96 (2), 265–269. Epple, D., Romano, R.E., 2011. Peer effects in education: a survey of the theory and evidence. In Handbook of the Economics of Education. vol. 3. pp. 1053–1163. (Ch. 20). Fang, H., Loury, G., 2005. ‘Dysfunctional identities’ can be rational. Am. Econ. Rev. Pap. Proc. 95 (2), 104–111. Fong, E., Isajiw, W.W., 2000. Determinants of friendship choices in multiethnic society. Sociol. Forum 15 (2), 26–32. Fordham, S., 1996. Blacked Out: Dilemmas of Race Identity and the Success at Capital High. University of Chicago Press, Chicago. Fordham, S., Ogbu, J.U., 1986. Black students’ school successes: coping with the burden of ’acting white’. Urban Rev. 18 (3), 176–206. Fryer, R.D., 2006. Acting white. Educ. Next 6 (1), 53–59. Fryer, R.D., 2007. A model of social interactions and endogenous poverty traps. Ration. Soc. 19 (3), 335–366. Fryer, R.D., Torelli, P., 2010. An empirical analysis of ‘acting white’. J. Public Econ. 94 (5–6), 380–396. Hainmueller, J., Hiscox, M.J., 2007. Educated preferences: explaining attitudes toward immigration in Europe. Int. Organ. 61 (02), 399–442. Jensen, R.T., Miller, N.H., 2017. Keepin’ ’em Down on the Farm: Migration and Strategic Investment in Children’s Schooling. NBER w.p. 23122. Kim, Y.C., Loury, G., 2012. Stereotypes and inequality: a ‘signaling’ theory of identity choice. KDI J. Econ. Pol. 34 (2), 4–15.

58

J. Eguia / Journal of Public Economics 156 (2017) 48–58

Mayda, A.M., 2006. Who is against immigration? A cross-country investigation of individual attitudes toward immigrants. Rev. Econ. Stat. 88 (3), 510–530. McWhorter, J.H., 2000. Losing the Race: Self-Sabotage in Black America. Free Press, New York. Mirón, L.F., Lauria, M., 1998. Student voice as agency: resistence and accommodation in inner-city schools. Anthropol. Educ. Q. 29 (2), 189–213. Patacchini, E., Zenou, Y., 2016. Racial identity and education in social networks. Soc. Networks 44, 85–94. Sacerdote, B.I., 2011. Peer effects in education: how might they work, how big are they and how much do we know thus far? Handbook of the Economics of Education. vol. 3. pp. 249–277. (Ch. 4). Shayo, M., 2009. A model of social identity with an application to political economy: nation, class, and redistribution. Am. Polit. Sci. Rev. 103 (2), 147–174. Slavin, R.E., 2010. Can financial incentives enhance educational outcomes? Evidence from international experiments. Educ. Res. Rev. 5, 68–80.

Sohn, K., 2011. Acting white: a critical review. Urban Rev. 43 (2), 217–234. Son, S., Model, S.W., Fisher, G.A., 1989. Polarization and progress in the black community: earnings and status gains for young black males in the era of affirmative action. Sociol. Forum 4 (3), 309–327. Steele, C.M., Aronson, J., 1995. Stereotype threat and the intellectual test performance of African Americans. J. Pers. Soc. Psychol. 69 (5), 797–811. Tajfel, H., Turner, J.C., 1979. An integrative theory of intergroup conflict. In: Austin, W., Worchel, S. (Eds.), The Social Psychology of Intergroup Relations. pp. 33–47. Tyson, K., 2006. The making of a ‘burden’: tracing the development of a ‘burden of acting white’ in schools. In: Hovart, E., O’Connor, C. (Eds.), Beyond Acting White: Reframing the Debate on Black Student Achievement. Rowman and Littlefield., Wildhagen, T., 2011. Testing the ‘Acting White’ hypothesis: a popular explanation runs out of empirical steam. J. Negro Educ. 80 (4), 445–463.