question:Journal of Economic Literature 2016, 54(1), 3–51 http://dx.doi.org/10.1257/jel.54.1.3 Affirmative Action and the Quality–Fit Trade-off † Peter Arcidiacono and Michael Lovenheim* This paper reviews the literature on affirmative action in undergraduate education and law schools, focusing especially on the trade-off between institutional quality and the fit between a school and a student. We discuss the conditions under which affirmative action for underrepresented minorities (URM) could help or harm their educational outcomes. We provide descriptive evidence on the extent of affirmative action in law schools, as well as a critical review of the contentious literature on how affirmative action affects URM law-school student performance. Our review then discusses affirmative action in undergraduate admissions, focusing on the effects such admissions preferences have on college quality, graduation rates, college major, and earnings. We conclude by examining the evidence on “percent plans” as a replacement for affirmative action. ( JEL I23, I26, I28, J15, J31, J44, K10) 1. Introduction Race-based affirmative-action policies, which seek to provide “affirmative” help to racial minorities, are among the most contentious policies that exist in US labor and education markets. Proponents argue that affirmative action helps level the playing field between majority and minority workers or students. Indeed, the original motivation for affirmative action is the leg- acy of discrimination in the United States, particularly against African Americans. * Arcidiacono: Duke University, NBER, and IZA. Lovenheim: Cornell University and NBER. We thank Kate Antonovics, Jivesh D’Sousa, Peter Hinrichs, and Richard Sander for helpful comments. †Go to http://dx.doi.org/10.1257/jel.54.1.3 to visit the article page and view author disclosure statement(s). Lyndon Johnson summed it up well in his 1965 commencement address at Howard University, where he states You do not take a person who, for years, has been hobbled by chains and liberate him, bring him up to the starting line of a race and then say, “You are free to compete with all the oth- ers,” and still justly believe that you have been completely fair. In this paper, we focus on how affirmative action in higher education—and in particular how racial preferences in admissions—affect the outcomes of minority students. While our focus is on higher education, affirmative action began in the United States with Lyndon Johnson’s 1965 executive order that affected the hiring practices of all fed- eral contractors. These regulations have since expanded to many states, and a large number of private employers have voluntarily enacted 3 4 Journal of Economic Literature, Vol. LIV (March 2016) affirmative action hiring practices. Despite the fact that the focus of Johnson’s original executive order was on hiring practices of federal contractors, colleges and universities throughout the United States quickly adopted these practices as well. The decision to engage in affirmative action in admissions is not federally mandated; instead, both the deci- sion to enact race-based affirmative-action policies and the manner in which minority candidates are given admission preference are decisions made by each postsecondary institution. As a result of the rapid spread of affirmative action in to the higher education sector, race-based admissions preferences are oneofthemostprevalentandcontentiousfix- tures of American postsecondary education. As in the labor market, race-based prefer- ences in college admissions evoke fierce and divided opinions on their merits, divisions that also hold among the Supreme Court jus- tices. Of the three landmark court decisions on affirmative action that examined the use of applicant race in the admissions process at public universities, two were 5–4 decisions. While the use of explicit quota systems and point systems for minority applicants were ruled out in Bakke v. California Board of Regents and Gratz v. Bollinger, respectively, Grutter v. Bollinger permitted the consider- ation of race as one of many factors in uni- versity admissions.1 Writing for the majority that Michigan’s law school admission process was constitutional, Justice O’Conner states that the constitution “does not prohibit the law school’s narrowly tailored use of race in admissions decisions to further a compelling interest in obtaining the educational benefits that flow from a diverse student body.”2 More recently, the Supreme Court ruled in Schuette v. Coalition to Defend Affirmative 1In Fisher v. Texas, the Supreme Court ruled 7–1 to remand the case back to the lower court, stating that the lower court had not applied strict scrutiny. 2 Grutter v. Bollinger, 539 U.S. 306 (2003). Action that a state ban on universities consid- ering race as part of their admissions process does not violate the Equal Protection Clause of the US Constitution. Thus, while univer- sities can use race as a factor in admissions as long as it is “narrowly tailored,” states can prohibit race-based admissions as well, at least among public universities. Whether many university admissions pro- cesses are narrowly tailored in their consider- ation of race—particularly at law schools—is debatable. The extent of race-based prefer- ences varies by ethnic group, with affirma- tive action appearing to be largest for African Americans, relative to Hispanics and other underrepresented groups.3 For example, the average SAT score (math+verbal) for African Americans entering Duke University in 2001 or 2002 was 140 points lower than the average for white students, or about 1.4 stan- dard deviations below the white mean. For Hispanics, the gap was half that (Arcidiacono et al. 2011).4 3There is a myth that the primary beneficiaries of affir- mative action in admissions are women. This is true only to the extent that women are members of particular racial or ethnic groups. There is now a substantial gender gap in educational attainment between men and women. The gap is particularly large for African Americans, where there are two African American women enrolled in a four-year col- lege for every African American man. The gender gap has become so substantial that men are now more likely to receive preferential treatment in admissions than women (Green 2011). Preferences do, however, exist for athletes and legacies. For example, Espenshade, Chung, and Walling (2004) find preferences for athletes to be smaller than that for African Americans, but similar to that for Hispanics, with lower (but still significant) preferences for legacies. 4Because the distribution of SAT scores for African Americans lies to the left of that of whites, even without race-based preferences African Americans would have lower SAT scores than whites within a particular school because they would more likely be on the margin of being admitted. But, it is certain that the gap would be much smaller absent race-based admission preferences. Arcidiacono, Khan, and Vigdor (2011) show that legacies have similar SAT scores to nonlegacies. This is actually indicative of legacy preferences following similar argu- ments: legacies likely come from advantaged backgrounds so their test score distribution should be to the right of that of nonlegacies. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 5 While Grutter v. Bollinger established that the legality of affirmative action policies is rooted in institutions’ preferences for hav- ing a more diverse student body,5 President Johnson’s quote and much of the rhetoric surrounding affirmative action focuses on the desire to support the educational attain- ment of underrepresented minority students. Certainly, the initial impetus for these policies was the hope that affirmative action could play some role in reducing inequalities in life out- comes between minority and majority groups. While racial preferences in admissions can be argued for on the basis of remedy- ing past discrimination or other equity-based reasons, there may be efficiency arguments in its support as well. Durlauf (2008) states that it is not obvious a priori that racial pref- erences are inefficient for two reasons. First, he highlights the fact that racial preferences will be inefficient when there are comple- mentarities between school quality and student preparation. Second, even if aggre- gate human capital would be higher without racial preferences, such admission prefer- ences could increase the allocative efficiency of human capital across communities. That is, the social returns to raising human capi- tal among individuals from heavily minority communities may be particularly high. In such a case, sacrificing productive efficiency in terms of the aggregate amount of human capital for allocative efficiency in terms of the distribution of human capital across dif- ferent groups in the economy may be effi- ciency enhancing. While we are able to make progress on the first issue, the second is beyond the scope of this paper and provides an important caveat to our findings. In this paper, we critically examine the evidence on how affirmative action in 5There is a small literature on the educational and social benefits to majority students of diversity that comes to mixed conclusions. This evidence is summarized in Arcidiacono, Lovenheim, and Zhu (2015). undergraduate and law school admissions affects both education and labor market outcomes.6 Although the effect of affirma- tive action on minority student outcomes is somewhat unrelated to the success of these policies in meeting institutions’ preferences for ethnic diversity, these are important questions to consider because they highlight the potential costs or alternative benefits to these diversity goals. If, indeed, affirmative action harms long-run outcomes of minority students, it suggests that returns to diver- sity within an institution would need to be quite large in order to justify these programs. Conversely, if affirmative action acts to level the playing field, as originally intended, these policies may be particularly desirable because they reduce inequality and allow schools to act on their preferences for diversity. Revealed preference suggests that more selective and higher-resource colleges are preferred by students to their less selective counterparts.7 One then would suspect that more selective colleges also would lead to bet- ter student outcomes. In addition, resources such as expenditures per student are higher at selective colleges, and there is ample evi- dence that these expenditures translate into, for example, higher graduation rates. Thus, policies that sort minority students into more selective schools, which is what affirmative action aims to do, may lead to better edu- cational outcomes for these students. But, is increasing college quality always beneficial? As a result of affirmative action, underrepre- sented minorities, and in particular African 6 We focus on undergraduate and law school admissions due to data availability and the corresponding focus of the literature. There are some papers in the medical literature examining the importance of matching between the race of the doctor and the race of the patient, but these papers suf- fer from concerns about identification. See Durlauf (2008) for a discussion of this work. 7Long (2004) also shows evidence from conditional logit choice models that students place significant and increasing value on college quality when making admis- sions decisions. 6 Journal of Economic Literature, Vol. LIV (March 2016) Americans, are placed in schools where their academic preparation is significantly below that of their white counterparts. The mis- match hypothesis argues that many of the beneficiaries of preferences are so misplaced academically that they would actually be bet- ter off in the absence of affirmative action. Although its antecedents arose much earlier, this hypothesis was popularized in academia by Sander (2004) and in the mainstream by Sander and Taylor’s 2012 book Mismatch: How Affirmative Action Hurts Students It’s Intended to Help, and Why Universities Won’t Admit It. While Sander’s original arti- cle focuses on law school students, his work with Taylor examines the evidence for the mismatch hypothesis for undergraduates as well. At first blush, economists should be very skeptical of the mismatch hypothesis. After all, affirmative action just expands the choice set. If an individual believes a particular school would be too difficult, then the indi- vidual could simply attend another school. By revealed preference, the individual must be better off. Yet, even in the context of rational expectations, where student beliefs are right on average given their informa- tion sets, there are cases where underrepre- sented groups could be made better off in the absence of affirmative action. This comes about because schools may have private information about match quality but are let- ting in lower match-quality students in order to satisfy diversity goals. Thus, the informa- tion sets of both schools and students are the critical components in driving mismatch. Ultimately, the effect of affirmative action on student outcomes is an empirical ques- tion, and we evaluate the evidence to date on whether and how affirmative action in admissions helps its purported beneficiaries.8 8See Holzer and Neumark (2000) for a review of the effects of affirmative action policies both in education and the labor market. Our first goal in what follows is to address the following questions: 1. Under what conditions can affirmative action actually be bad for its beneficia- ries? What are the relevant outcome measures in this case? 2. How can we make the empirical spec- ification flexible enough to allow for the possibility that increasing school quality could be welfare-enhancing for some individuals but not for others? While the second question may seem obvious given the first, it is surprising that many papers—even when explicitly testing for mismatch—estimate models that imply that attending a more selective program is good for everyone or good for no one. Given the answers to the first two sets of questions, we then turn to what the empirical evidence has to say about affirmative action more broadly and the quality–fit trade-off in particular. Focusing on both undergradu- ates and law school students, we discuss the research findings on the following questions: 3. How extensive are race-based preferences? 4. What can we learn from the debate over mismatch in law schools? Does it matter for law school grades, bar pas- sage, future earnings, or some combi- nation thereof? 5. For undergraduates, on what margins does affirmative action matter? The extensive margin (whether individuals attend college at all) or the intensive margin (where individuals enroll)? 6. How does affirmative action affect graduation rates, choice of college major, and earnings? Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 7 7. Are “percent plans,” which provide automatic admission to public universi- ties for students above a given percentile in class or state rank, a viable alternative to affirmative action? What effects do such admissions rules have on student outcomes and how do they compare to the estimated effects of affirmative action on those same outcomes? As we will see, there are many ways that the quality–fit trade-off can manifest itself, but it will be difficult, if not impossible, to say whether the purported beneficiaries are actually made worse off under affirma- tive action. At the same time, there is con- sistent evidence that the fit between the student and the university matters, at least across some dimensions. We highlight areas for future research that can help shed some much-needed light on these important and unresolved questions. The rest of the paper proceeds as fol- lows. In section 2, we discuss the mismatch hypothesis and the conditions for it to hold given that affirmative action expands the choice sets underrepresented groups face. Important to this discussion will be the qual- ity of information individuals have about their prospects for success. In section 3 we discuss ways to empirically test for hetero- geneity in the fit between the school and the student. Section 4 examines mismatch in law school, considering the initial article by Sander as well as the multiple responses it has generated in light of the issues raised in sections 2 and 3. Section 5 focuses on mis- match at the undergraduate level, including the effects of statewide affirmative-action bans. Section 6 discusses percent plans, and section 7 concludes. 2. The Quality–Fit Trade-off The scope for affirmative action policies to benefit minority students hinges, in part, on the extent to which more selective colleges produce better outcomes for all students than less selective colleges. This could be the case because the more selective colleges provide higher monetary inputs and have higher-quality students who in turn gener- ate positive peer effects. We refer to these as “quality effects.” But the match between the college and the student also could be important. More selective colleges may pitch their mate- rial at a higher level, move at a faster pace, and assume that certain concepts and skills are already known. It then may be optimal for some students to attend a less selective college even if a more selective college is in their choice set. In other words, there may be an optimal match between the precollegiate skills of students and schools of different types that varies across the student skill distribution. We refer to these as “match effects.” The extent to which match and quality effects are important may depend on how the benefits to the student are measured. For example, science courses may build on past knowledge in ways that other courses do not. Hence, match effects may be more relevant when considering graduating in the sciences, rather than graduating in the humanities. Whether match or quality effects dominate may also depend on what part of the college quality distribution is being examined. For example, among selective colleges match effects may dominate. However, the sig- nificantly lower resource levels that prevail among nonselective colleges may lead quality effects to dominate in this part of the school quality distribution. Assessing the impor- tance of the fit between the student and the school thus requires flexibility in how the outcome equation is specified. Namely, the returns to academic background need to vary in some way with the quality of the college. Figure 1 shows one way in which heteroge- neity in fit could be modeled. Panel A shows 8 Journal of Economic Literature, Vol. LIV (March 2016) Panel A. College A dominates College B for all levels of academic preparation College A College B Academic preparation Panel B. College A dominates College B for the most academically prepared, College B dominates College A for the least academically prepared a case where College A produces better out- comes for all individuals. In contrast, panel B shows a case where the match effects trump quality effects for a part of the skill distribu- tion: higher levels of academic background are associated with better outcomes at both colleges, but now the college that maximizes the student’s expected outcome depends on the student’s academic background. College A is a particularly good fit for some- one with a strong academic background. College B, however, will not be as stimu- lating for those who are more academically prepared for college, as College B is geared more towards those with relatively weaker backgrounds. In this example, College B will result in better outcomes for those who are less academically prepared for college. As we discuss below, the prior literature on school quality often estimates an average effect that assumes the relationship between quality and outcomes is monotonic for all students. This assumption usually is necessi- tated by data and sample size limitations, but with enough data we could see whether the returns to academic background vary across different schools types. 3. The Role of Information When students know their academic backgrounds and how their academic back- grounds translate into outcomes at different colleges, then students can optimally sort themselves among the colleges to which they have been admitted. Since racial pref- erences expand the choice set for under- represented minorities, when students have all the necessary information—or at least as much information as those of the colleges themselves—there would seem to be little scope for racial preferences to be harmful for minorities.9 9One place where negative impacts could arise would be through stereotyping, which may negatively affect minority students who would have been admitted without racial preferences. However, those admitted under the preferences would still be advantaged as they would be pooled with stronger minority students. Figure 1. Heterogeneity in College Benefits Academic preparation College A College B Outcome Outcome Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 9 As shown in Arcidiacono et al. (2011), when students are lacking information it is possible for minority students to be made worse off as a result of racial preferences, even when these students have rational expectations. The idea hinges on students not having full information regarding their abilities to succeed in school or about the quality of their matches with particular schools. If the university has private infor- mation about the quality of the match but does not reveal this information to the stu- dent, then it is possible that students admit- ted under affirmative action would have been better off attending a different school. This could be the case if schools admitted all students who were a good match, as well as some minority students for whom the match quality is lower. Schools may have incentives to admit this latter group anyway in order to meet diversity goals. Suppose this latter group of students is the group that is admitted under affirmative action. Admitted minority students form expecta- tions of their match quality based in part on observed average outcomes for all minority students admitted to the school and then make enrollment decisions based on the school that maximizes their expected util- ity. However, had the school revealed its private information, the student may have made a different decision. Arcidiacono et al. (2011) distinguish between local mismatch and global mis- match. Under local mismatch, the marginal admitted minority student is worse off under racial preferences but the average benefit for the group is still positive. Under global mismatch, the average benefit for the group turns negative. In their model, as long as schools face binding capacity constraints, some racial preferences will be advantageous for minority students. The model permits an interior optimal amount of racial prefer- ences from the perspective of the minority group. Going past this point results in local mismatch, with further preferences poten- tially resulting in global mismatch. That schools have private information about the quality of the match with the student is a necessary condition for mismatch when stu- dents have, on average, correct information. Information issues become even more salient when the rational expectations assumption is relaxed. Indeed, there is evidence suggest- ing that students—at least those from dis- advantaged backgrounds—have surprisingly poor information about their educational prospects. Hoxby and Avery (2013) show that high-achieving, low-income high-school students apply to schools that are very dif- ferent from their high-income counterparts, often not applying to a single selective insti- tution despite the expected net costs of at least a set of these institutions being lower than the costs of the nonselective institutions they attend. Hoxby and Turner (2013) show that simply providing these high-achieving, low-income students information about their probabilities of admission to different tiers of schools, as well as information about expected costs, has significant effects on the types of colleges and universities these stu- dents attended. These students often have no desire to go to the local school, but are simply unaware of the available options. This research demonstrates that consider- able search frictions exist among students (particularlylow-incomestudents)inhigher education markets. Although their focus is on students who match to lower-quality schools than they are able to attend, it demonstrates that students often do not have sufficient information to make informed decisions about where to attend college. Racial preferences add another layer of uncertainty for students, because affected minority students may not be aware they are being admitted with worse academic credentials than their peers as a result of these preferences. Universities generally state that race is one of many factors that 10 Journal of Economic Literature, Vol. LIV (March 2016) are considered, so students may face signif- icant uncertainty about the degree to which their race affected whether or not they were admitted. For example, Wightman (1998, p. 70) shows that African American and white students had similar beliefs regarding their law-school class rank.10 But, as we will see in section 4, African American students have significantly lower grades in law school than their white counterparts. Measuring whether affirmative action is harmful to its beneficiaries in a utility sense is a difficult task. Namely, it must be the case that by providing additional information about the prospects of success at a particu- lar school, students would make different enrollment decisions. Interventions like the one used by Hoxby and Turner (2013) are geared towards the very top of the distribu- tion. There are no interventions of which we are aware that give underrepresented groups information about their prospects of success at different tiers of institutions. In the absence of such interventions, researchers are forced to use correlates of match quality to estimate the extent of mismatch. If students have poor informa- tion about their match quality, once enrolled they may learn about the quality of the match and make adjustments if they deter- mine that they enrolled in the wrong school. Indeed, using repeated surveys of students at Berea College that are designed to elicit beliefs about academic ability, Stinebrickner and Stinebrickner (2012) show that most low-income students11 are overly optimistic 10 Thirty-eight percent of African American and 34 per- cent of white law students expected to be in the top 10 per- cent of their class. The similar percentages for finishing in the top 25 percent of the class were 77 percent and 79 per- cent for African Americans and whites, respectively. The survey was conducted in the fall of 1991, and the sample was composed of law students in their first semester of law school. 11Berea College is a small liberal arts school in Kentucky that contains a student body composed almost solely of students from low-income backgrounds. about their grades when they enter college and quickly adjust their beliefs when they receive new information in the form of grades. Zafar (2011) finds similar learning dynamics among students at Northwestern University: students tend to be overly opti- mistic about their major-specific outcomes at first and quickly update their beliefs once they begin to receive grades. The main adjustments students can make once they learn about a poor match is to switch to a less demanding major, trans- fer schools, or drop out. Stinebrickner and Stinebrickner (2012) show that learning about one’s ability in the first year of college is strongly related to the decision to drop out, while Zafar (2011) finds that learning is associated with switching majors. A cen- tral reason for the differences in their find- ings is that dropout rates are extremely low at Northwestern (7 percent), while they are much larger at Berea (17 percent). Thus, these students adjust along different margins, which highlights the fact that any mismatch from affirmative action likely gen- erates different responses depending on the characteristics of the school. This prediction has received little attention in the literature. In section 5.4, we examine the evidence on college majors, while we discuss the rela- tionship between affirmative action and col- lege completion in section 5.3. The effect of affirmative action on student transferring has not been examined in prior work, and we view this as an important area for future research.12 12Andrews, Li, and Lovenheim (2014) provide a descriptive analysis of transferring behavior in Texas and show transferring prevalence is very high overall, and especially so at low levels of the college quality distribu- tion. These results suggest that transferring is a margin on which students might adjust to learning about match qual- ity, but no research currently exists that assesses whether this is the case. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 11 Although examining the relationship between affirmative action and these outcomes is of independent interest, it is important to highlight that these measures will be insufficient for establishing whether minorities are actually worse off from affirmative action. For example, suppose we consider college graduation and could show that students admitted to a very selec- tive school because of affirmative action would be more likely to graduate from a lower-tier school. The student may still have preferred to attend the more selective school, even with full knowledge of the lower probability of graduating. This can occur because the individual may receive additional utility from graduating from the more selective school. In discussing the evi- dence on these outcomes, in addition to the varied identification concerns we highlight, it is important to keep in mind that the out- comes being measured may not capture all of the relevant aspects of how schooling choices affect utility, particularly in the long run. 4. The Law-School Debate The discussion over whether affirmative action benefits minority students has been particularly contentious in the realm of law- school admissions. Estimating the effect of affirmative action in law schools differs from examining this question in the con- text of undergraduate admissions for several reasons. First, in contrast to what is seen for undergraduate institutions, affirmative action affects how many minority students attend law school as virtually all laws schools are selective. Hence, for law schools, it is not a matter of just where students attend but whether they attend at all. Second, the data on law schools are not as rich as the data for undergraduates in terms of containing spe- cific information on schools attended and family background characteristics. Weighed against these disadvantages is that, to practice law, one must take a bar exam. The bar exam gives a measure of learning that can be compared across law schools that is typically lacking in the undergraduate literature.13 While bar pas- sage is not a perfect measure of learning, especially if lower-ranked law schools teach more to the test than do higher-quality schools, the importance of passing the bar for one’s legal career makes it a highly rel- evant measure of legal knowledge that also relates to the return one can expect on investment in law school. Furthermore, affirmative action is very aggressive in law- school admissions, which makes this field of particular interest in understanding the effects of racial admissions preferences on student outcomes. To illustrate the last point, we present results analyzing data on admissions deci- sions for Michigan’s law school in 2002, the same year when the Supreme Court took up the Grutter v. Bollinger case.14 The data contain the applicant’s race/ethnicity, LSAT scores, and undergraduate grades, as well as whether the applicant was admitted. We cre- ate an “academic index” to aggregate LSAT scores and undergraduate grades into one measure15 and then standardize the measure so that the mean is zero and has standard deviation one for the applicant pool. Figure 2 shows the distribution of the renormed academic index separately for blacks and whites based on whether or not 13Earnings could be another measure, but affirmative action in the labor market distorts comparisons across racial/ethnic groups. One caveat with using bar passage is that the difficulty of the exam varies across location. 14 These data, along with data in 2003, were obtained by Richard Sander via a Freedom of Information Act request. 15The formula follows Sander and Bambauer (2012, p. 896) and is given by (LSAT − 120) × 10) + (100 × UGPA). One concern may be that our results are driven by lower black undergraduate grades due to affir- mative action in undergraduate admissions. Using just the LSAT produces the same patterns. 12 Journal of Economic Literature, Vol. LIV (March 2016) 0.8 0.6 0.4 0.2 0 Notes: Academic index is a weighted average of the applicant’s LSAT score and undergraduate GPA and is normalized to be mean zero and standard deviation one for the population of applicants. Black admit Black reject White admit White reject −6 −4 −2 0 2 X Figure 2. Distribution of Academic Indexes for Applicants to Michigan Law by Race and Accept/Reject Status Source: Data from 2002. they were admitted. For both racial groups, those admitted had significantly higher index values. However, white students who were rejected generally look academically stronger than the black students who were admitted. Indeed, black applicants in 2002 were admitted to Michigan’s law school at a slightly higher rate than white students, 26.9percent to 23.3 percent, despite the average academic index of black applicants being around 1.5 standard deviations lower than the mean white applicant. More striking is where the black admit distribution lies relative to the white admit distribution. The median black admit had an academic index at the second percentile of the white distribution, and the seventy-fifth per- centile of the black admit distribution was at the eighth percentile of the white distribu- tion.16 The difference between the black and white admit distributions is not all due to affir- mative action: if the African American aca- demic index distribution is below the white distribution, this would produce a difference 16If we instead focus on enrollees, the median black enrollee has an academic index less than the first percen- tile of the white enrollee distribution. Indeed, no white enrollees had academic indexes that were lower than the seventy-fifth percentile of the black distribution. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 13 in the incoming qualifications of black versus white students, even in the absence of affir- mative action. However, the fact that these distributions are almost nonoverlapping is suggestive of a large amount of race-based preferences in admissions being given to African American students. The University of Michigan is by no means an outlier either. The data show that the extent of preferential admissions for black students is even more pronounced at other elite public law schools, such as the University of Virginia and the University of Wisconsin. In both of these cases, the median black admit had an aca- demic index that would place him below the first percentile of the white admit at the same school.17 The data also reveal that affirmative action works differently for blacks and Hispanics. While affirmative action is very much pres- ent for Hispanics (the median Hispanic admit at Michigan is at the ninth percentile of the white admit distribution), the median Hispanic admit is at the seventy-eighth percentile of the black admit distribution. Hispanic admission rates were also lower than those for blacks, despite having, on average, better test scores and undergradu- ate grades.18 These tabulations indicate that African American students receive a significant amount of preferential treatment in law school admissions. The extent of affirmative action in law schools is sufficiently large and 17The median black admit at the University of Texas and the University of North Carolina was below the sec- ond and seventh percentiles, respectively, of the white admit distribution. These results refer to two admissions cycles for each school: 2006 and 2007 for Michigan and Wisconsin, and 2005 and 2006 for North Carolina, Texas, and Virginia. These data were obtained by Richard Sander via a Freedom of Information Act request. 18Estimates from a logit model of admission with the right-hand-side variables being LSAT score, undergrad- uate grades, and indicators for race/ethnicity categories reveal that the admission advantage for blacks is roughly twice the advantage for Hispanics. the sorting on prior academic credentials is so prevalent that if mismatch is not present, law schools should be almost perfectly dif- ferentiated by average quality differences (as in panel A of figure 1). In other words, either the large differences in white and black pre-law-school academic characteristics driven by affirmative action policies lead to mismatch or the effect of law school quality is ubiquitous. In the former case, affirma- tive action could lead to negative effects on African American law students, whereas in the latter case, black law students would be unambiguously better off due to affirmative action because of the positive effects of law school quality on outcomes. We now turn to a discussion of the often contentious debate about the empirical support for both of these hypotheses. 4.1 Graduation and Bar Passage 4.1.1 A Framework for Analyzing the Quality–fit Trade-off in Law Schools As we discuss in detail below, the empir- ical literature examining the returns to law- school quality and the existence or extent of mismatch is extremely contentious. A core source of the disagreement among studies examining affirmative action in law schools is the difficulty in producing causal estimates of the effect of attending different-quality law schools on measured outcomes for black and white students. This difficulty is due to the fact that students select schools and schools select students based on factors that are both observable and unobservable to researchers. As a result, there is much concern that the estimates of law-school quality are biased by the existence of unobserved differences in underlying student ability that exist across the school quality distribution. Studies of the effect of affirmative action-driven mismatch on student out- comes seek to understand whether any overall positive effects of school quality are 14 Journal of Economic Literature, Vol. LIV (March 2016) outweighed by the effect of lower academic performance relative to one’s peers. Put dif- ferently, the empirical challenge is to under- stand whether being lower in the academic ability distribution in a school has indepen- dent negative effects on outcomes, and if so how large these effects are in relationship to the gains from attending a higher-quality law school. To fix ideas, denote Yi, Gi, Li, and Xi as individual i’s outcome (such as bar passage or earnings), normalized law-school grades,19 law-school quality tier, and other observable characteristics such as pre-law- school academic credentials. Assume that Yi and Gi can be decomposed as follows: (1) Yi = α1 +Giα2 +Liα3 +Xiα4 +εi (2) Gi = γ1 +Liγ2 +Xiγ3 +ηi, which can also be expressed as the following reduced form equation: (3) Yi = δ1 +Liδ2 +Xiδ3 +μi. If mismatch negatively influences postgraduation outcomes, then 0 > δ2, where δ2 = α3 + α2 γ2. Taken together, equations (1)–(3) show that we can break out the effect of law-school tier into two com- ponents: the direct effect and the indirect effect through law-school grades,20 where the first effect is expected to be positive and the second negative. Thus, if we can identify 19By normalized grades, we mean grades have been normalized to have zero mean within each school. This can alternatively be interpreted as a measure of student rela- tive rank within each school. 20Some researchers, (e.g., Ho 2005b) have raised concerns about controlling for law school grades in the outcome equations because they are themselves affected by law school quality. Rosenbaum (1984) shows that con- trolling for a variable that is affected by the treatment can produced biased treatment effect estimates. However, he also shows that in a selection-on-observables setting, both the treatment and the mediating variable coefficients are identified as long as the observable characteristics are suf- ficient to account for selection into the treatment. δ2 and test whether it is greater than zero, it amounts to a test that matching effects are larger than college quality effects. Virtually all models that researchers have used to test for mismatch in law schools are variants of equations (1)–(2) or (3), with different assumptions regarding the distribution of the errors. There are two issues with this approach. First, it is restric- tive in such a way that precludes the bottom panel of figure 1. Namely, everyone is either hurt or helped by attending an elite school: the test for mismatch does not depend on the match between the school and the student. Hence, the tests in the literature are gener- ally attempts to measure average effects of law-school quality, with the average effects possibly varying by race. This important caveat aside, the second issue is overcoming the selection problem. We now consider the barriers to recovering unbiased estimates of δ2, or alternatively α3 and α2 γ2. For simplic- ity, we focus the discussion on identifying δ2, but the mechanical relationship between δ2, α2, and γ3 means that this discussion gener- alizes to these other parameters as well. The underlying difficulty with identifying δ2 is that law schools have access to informa- tion about students that are unobservable to researchers and that also are likely to affect student outcomes. Thus, students who are similar in terms of observables but who go to law schools of differing quality likely also differ in terms of these unobservable factors: the students attending more elite law schools have higher unobserved ability, conditional on observed ability measures. In this set- ting with selection-on-unobservables across law school quality tiers, researchers have used several methods to identify net effect of quality and matching. One way to pro- ceed is to use what we term a “within-race estimator,” comparing outcomes among underrepresented minority (URM) students at different law-school quality tiers after adjusting for observed ability measures. To Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 15 the extent there is overlap in observed ability across quality tiers, however, it is likely the case that unobserved ability is higher among students in the higher quality tier, biasing δ2 upward. Hence, the within-race estimator likely provides an overestimate of the gains to attending a more prestigious law school. Another way to proceed is to combine within-race estimators in a difference-in- difference setting. This method ostensibly uses whites as a control group for selection amongst URM students. Whether such a comparison is valid and the direction of any bias rests heavily on assumptions about how the joint distribution of observed and unobserved ability differ across groups. This difference-in-difference method can be used to see whether matching effects are important when the selection on unob- servable ability is the same across races for each quality tier. Then, whites can be used as a control group to purge the bias of the within-URM estimates because the bias in δ2 among whites will be identical to the bias for nonwhites. This is, to be sure, a very strong assumption, and if it is violated, estimates of mismatch from equations (1)–(2) or (3) will be biased as well. Here, the bias can go in either direction. First, imagine that URM students have higher unobserved ability than whites (conditional on observed ability). Then, the tier effects will be biased upward more for minority students than for white students, which will lead one to understate mismatch when comparing white versus minority stu- dents. Conversely, if unobservables among URM students are worse than for whites conditional on observed ability, then the difference-in-difference estimator will over- state the extent of mismatch. Note that this is also only a test of whether matching effects matter as the coefficient on tier itself is biased upward for the reasons described above. A third method of identifying the net effect of quality and matching is what we term the “cross-race estimator.” This method compares outcomes of black students to white students unconditional on quality tier. More formally, the cross-race estimator can be written as: (4) Yi = β1 +Blackiβ2 +Xiβ3 +ωi, where Black is an indicator for whether the student is African American. Conditional on observed ability, the mismatch hypothesis predicts that black students will have worse outcomes than whites (i.e., β2 < 0). This model identifies mismatch as long as the dis- tribution of unobservables among African American students is the same as the distri- bution of unobservables among white stu- dents. As Rothstein and Yoon (2009) argue, it is likely that black students have lower unob- served ability than white students based on the fact that they tend to perform worse than white students at the same school and with the same admissions credentials. In this case, β2 will be biased downward, which will over- state the importance of matching. As this discussion highlights, the validity of the different approaches to measuring quality and matching effects in law school rely strongly on the underlying assumptions about how student unobserved ability is dis- tributed across college quality tiers, across racial groups within each tier, and across racial groups independent of tier. Due to these identification concerns, no estimate of the extent of mismatch is perfect. However, we can use this framework to guide our inter- pretation of the differences in results across the methods used, which will help us synthe- size this contentious literature. We now turn to a discussion of this literature, beginning with the seminal work of Sander (2004). 4.1.2 Sander (2004) The law-school debate began with Sander (2004), who argued that large admission pref- erences resulted in blacks learning less in law school and consequently failing the bar at a 16 Journal of Economic Literature, Vol. LIV (March 2016) higher rate.21 Sander speculated that these learning effects were sufficiently large to actu- ally result in fewer black lawyers than if racial preferences were not in place. He also argued that the loss of black lawyers occurs despite racial preferences increasing the number of black students who enroll in law schools as (i) those students who are at the margin of being admitted have a relatively low probability of passing the bar exam and (ii) being severely mismatched results in low bar passage rates. Using data from the Law School Admission Council Bar Passage Study (BPS),22 Sander (2004) shows that, aside from those attend- ing historically black colleges and universi- ties (HBCUs), black students perform very poorly in law school in terms of grade point average. In particular, the bottom decile of the grade distribution contains between 44 and 52 percent of black students, depend- ing on law school tier.23 Since grades are a relative measure of performance and since there are large differences in entering cre- dentials between blacks and whites, it is per- haps not surprising that black students are more likely to be at the bottom of the GPA distribution. Indeed, using 1995 National Survey data that includes school identifiers, Sander (2004) shows that the relative rank of one’s undergraduate GPA and LSAT score within a school are strongly predictive of 21Most of this literature focuses on black-white differ- ences due to affirmative action. To our knowledge, there has been little attention paid to mismatch effects among Hispanic students in law school. 22This dataset follows 1991 law school matriculants. Participation in the study was high for both law schools and their students: 95 percent of law schools participated and 80 percent of the students signed consent forms. The primary disadvantage of the data is that school quality was aggregated into six broad tiers. 23Recall that black students had similar expectations to white students with regard to their expected law school class rank. To the extent that these subjective expectations reflect actual beliefs, the results suggest black students were either unaware of the extent of affirmative action or unaware of how credentials translate into law-school performance. law-school GPA. This finding suggests that policies designed to bring lower-ranked stu- dents into more elite schools could lead to them to perform substantially worse, relative to their peers. While Sander (2004) does not explicitly estimate equation (2), this evi- dence forms the basis of his argument that γ2 is both negative and sizable in magnitude. Sander estimates equation (1) using the BPS data. While he estimates versions of equation (1) for both law-school graduation and first-time bar passage, he focuses on the bar passage results. The resulting literature also focuses on this outcome, so we focus our attention on bar passage. The estimated coef- ficients on (normalized) law school grades, α2, and tier, α3, are positive. Furthermore, conditioning on law school grades and tier leads to no effects of race on bar passage. This drives the remainder of Sander’s anal- ysis that race is not relevant for bar passage. Sander’s approach here is a mixture between the within-race and cross-race estimators. The estimates of α2 and α3 are mostly iden- tified off of white students, because they comprise the majority of his sample. For the reasons described in the previous section, these coefficients are likely biased upward. Including an indicator for Black is in the spirit of the cross-race estimator, but how the biases on α2 and α3 combine with bias on the Black indicator due to differences in unobservables across racial groups is unclear. As a result, signing the bias in this regression on the Black coefficient is very difficult. Sander then breaks out bar-passage rates by LSAT score bins for both blacks and whites. Within each bin, white students are more likely to pass the bar. Since Sander finds no race effects on bar passage, he uses this to justify the assumption that black stu- dents would pass the bar within each of these bins at the same rate as whites had affirma- tive action not been present. He then con- ducts a decomposition analysis that estimates what first-time bar passage rates for blacks Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 17 would have looked like absent racial pref- erences. This is done by assuming that all within-LSAT bin differences between blacks and whites are due to mismatch effects. The results of this exercise suggest that over 45 percent of the black bar failure rate (or about 57 percent of the difference in the black–white failure rate) is due to mis- match.24 This finding implies that attending a higher-tiered law school lowers the probabil- ity of passing the bar on the first try, particu- larly for less-academically-qualified students. The decomposition estimates thus point to a substantial amount of mismatch from affir- mative action that worsens the long-run out- comes of many black law students. Sander argues that the reason for this find- ing is that affirmative action results in black students attending schools where they lack the necessary academic preparation, which in turn lowers the quality of their education. For several reasons, however, his estimates likely overstate the amount of mismatch. First, because black students have lower aca- demic index values than whites overall, this is likely the case within each bin as well. As a result, even within LSAT bins, the black and white students are unlikely to have the same academic ability levels. Second, the assumptions that underlie this approach are identical to those that support the cross-race estimator, as Sander is assuming that there are no other factors (such as unobserved ability) that would drive differences in bar passage rates within each LSAT bin. Similar to the cross-race estimator, this leads to an upwardly-biased estimate of mismatch. Note that Sander does not estimate (2), but he does do so in his response to Ho (2005a). In Sander (2005a), the estimated 24The other reason African American students fail the bar less is because they are more likely to have lower academic index levels. As Sander (2004) highlights, even taking his calculations at face value, eliminating affirmative action could not close the black–white bar passage gap. coefficient on tier in equation (2) for first-time bar passage, γ2, is sufficiently negative that 0 > α2 + α3 γ2. This is sur- prising for two reasons. First, because the estimates are identified predominantly off of whites who comprise the majority of the sample, they are within-race estimates. As discussed above, such estimates typically are biased away from finding mismatch effects. Second, because whites are the majority of the sample, the results imply that the neg- ative net-tier effects are also present for whites. Taken at face value, the estimates in Sander (2004) suggest that attending a more elite law school lowers one’s chances of passing the bar regardless of one’s entering credentials or race. The specification thus only allows for two possibilities: everyone benefits from attending a more elite school or no one benefits. A problematic conclu- sion one could draw from Sander’s results is that everyone is harmed by going to a more elite law school, as the negative effect on GPA swamps the positive direct effect of school quality. Furthermore, if there are cross-race differences in mismatch effects, generalizing these estimates to a sample of African American students could yield misleading conclusions about the extent of mismatch. All subsequent law-school mismatch studies are subject to this same criticism that they imply everyone is either helped or harmed by attending a more elite school. The exception is one part of the analysis in Sander (2005b). Here, Sander allows curva- ture in the relationship between grades and outcomes. He bins percentile class rank and, within each bin and law school tier combination, estimates the joint probabil- ity of graduating and passing the bar on the first attempt. The results reveal a substan- tial penalty for finishing at the bottom of the class across law-school tiers. This suggests the possibility of increasing eliteness being beneficial up to a point. As long as a student 18 Journal of Economic Literature, Vol. LIV (March 2016) is reasonably competitive, the student can reap the benefits of the elite education. Hence, for the objectives of maximizing black bar passage, there may be an opti- mal affirmative-action policy that falls in between the current policy and race-blind admissions.25 4.1.3 The Ensuing Debate The difficulties posed by selection-on- unobservables in identifying mismatch effects in law school has led to a vigorous and often contentious debate about the validity of Sander’s conclusions. Much of the ensu- ing work on law-school mismatch focuses on identifying the effect of law school tier net of the grade effect (i.e., δ2 from equation (2)). A negative estimate of the effect of quality tier in such a model provides evidence of mis- match. While examining the net effect of tier sidesteps the difficulties associated with the endogeneity of relative law-school grades, these analyses face similar identification con- cerns related to selection-on-unobservables that form the basis for many of the objec- tions to the findings in Sander (2004). Rothstein and Yoon (2009) is the only other paper to pursue black–white compar- isons in order to assess mismatch effects. They estimate probit models of bar passage on observed students credentials as well as an indicator for whether the student is black, akin to equation (4). Estimates of this model show that blacks are much less likely to grad- uate and to pass the bar even conditional on the observables, which is consistent with the mismatch hypothesis. However, at the bottom of the credential distribution there is very little overlap in credentials between black and white students due to racial pref- erences affecting the extensive margin. Those white students with poor credentials 25 This point is not explored further in Sander (2005b), as this piece was primarily a reply to criticisms levied against Sander (2004). are likely to have stronger unobservables than similarly-credentialed black students, else they would not have been accepted to any law school. To address this concern, Rothstein and Yoon (2009) report estimates using both white and minority students that focus only on the top four quintiles of the aca- demic index, which is where the white and black academic index distributions have the most overlap. Examining the top four quin- tiles only removes the negative effect of tier on law school graduation and cuts the effect on bar passage by more than half, rendering it insignificant.26 The problem with focusing on the top four quintiles, however, is that three-quarters of the black students fall in this bottom quintile (Rothstein and Yoon 2009, p. 18), and it is these students who are most at risk of being mismatched. Rothstein and Yoon (2009) also estimate equation (3) separately by race, which is the within-race model. This is also the strat- egy pursued by Ho (2005b) and one of the strategies in Ayres and Brooks (2005) and Williams (2013). Rothstein and Yoon (2009) argue that this model provides a lower bound of the extent of mismatch, similar to the arguments described above. Ho (2005b) and Rothstein and Yoon (2009) both find estimates that suggest law-school tier has no effect on eventual bar passage using the within-race analysis. Because these find- ings are likely to be heavily influenced by selection-on-unobservables, they are not necessarily inconsistent with those in Sander (2004). Indeed, he finds evidence of mis- match in models that are biased towards finding mismatch, and the within-race esti- mates find no effect of law-school quality tier in models that are biased against finding mismatch effects. A sensible reading of this 26However, their estimate is still negative and sizable in magnitude, on the order of three percentage points. Furthermore, the 95 percent confidence interval cannot rule out large mismatch effects. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 19 evidence suggests that the correct answer is somewhere in between, which would indi- cate that modest mismatch effects exist. Note that Rothstein and Yoon (2009) do not find quality-tier effects for black stu- dents, while Sander (2005a) does. Part of this is due to the differences in their depen- dent variables. Rothstein and Yoon (2009) examine eventual bar passage, while Sander (2004) examines bar passage on the first try. There is evidence that the black–white gap in first-time bar passage is larger than the gap in eventual passage (Williams 2013). This is an important distinction, however, as failing the bar on the first try has substan- tial costs in terms of increasing study time, reducing earnings, and putting lawyers at a competitive disadvantage in the labor mar- ket. It is thus not surprising that Rothstein and Yoon find smaller mismatch effects than Sander, given that they are examining an out- come that has a smaller racial gap associated with it. Another strategy pursued by Rothstein and Yoon (2009) is the across-race difference- in-difference approach discussed above. They estimate equation (3) separately for white and black students, and then they test for equality of the δ2 coefficients across groups. What is striking is that positive effects of law-school quality emerge for whites, but not for blacks. This suggests either that (i) selection on unobservables is present for whites but not for blacks, or (ii) matching effects are effectively canceling out quality effects for blacks. Under the assumption that selection on unobservables is similar for blacks and whites, however, these esti- mates are consistent with mismatch: African American students at selective law schools are about 2.5 percentage points less likely to pass the bar than white students at selective law schools. Williams (2013) argues that their results further understate mismatch due to the nois- iness of the law-school tier measure, as well as their focus on eventual, rather than ear- lier, bar passage. Williams (2013) finds neg- ative effects of selectivity on both first-time and eventual bar passage, although only the former are statistically different from zero. Additionally, the estimated effects increase when the top two tiers are compared to the bottom two tiers as opposed to comparing the top two tiers to the bottom four, as in Rothstein and Yoon (2009). One caveat is that historically black colleges and univer- sities are in this bottom tier and may oper- ate differently from traditional law school environments.27 As a way of mitigating differences in unob- servables among black students attending different law school tiers, Ayres and Brooks (2005) and Williams (2013) pursue an identification strategy in the spirit of Dale and Krueger (2002).28 Namely, compar- isons are made between black students who attended their first-choice law school and those who reported being admitted to their first-choice law school but attended their second-choice (or lower-quality) law school,29 either due to cost or distance considerations. The descriptive statistics reveal that black students who attend their first-choice school have similar credentials and background characteristics to those who attend their second choice. However, first- choice students placed more importance on academic reputation and less impor- tance on cost than second-choice students. Second-choice students then attended schools that were relatively less selective. A 27Ayres and Brooks (2005) remove these schools in their analysis of eventual bar passage and positive effects of selectivity for blacks when tier is included linearly. Sander (2004) also includes tier in this way. In both cases, it is more appropriate to have only two tiers or dummy out the effects of each tier. 28See section 5.5 for a more complete description of this methodology. 29For ease of exposition, we refer to these students as attending their second-choice school even though the school they attend could be even further down their list. 20 Journal of Economic Literature, Vol. LIV (March 2016) core assumption underlying this method is that students who were admitted to the same types of schools but who attended schools of differing quality have the same unobservable characteristics. Results using this identification strategy point towards small positive effects of attend- ing a more selective school on graduation that often are not statistically significant. But, both Ayres and Brooks (2005) and Williams (2013) find that black students who attend their sec- ond choice (and thus lower-quality) school are more likely to pass the bar the first time they take it, which is evidence consistent with mis- match. Ayres and Brooks (2005) find insignifi- cant effects for passing the bar at a later date, however. Consistent with earlier bar passage results, estimated mismatch effects on the bar exam are strongest earlier on in the career. One interpretation of this finding is that the negative effects of mismatch will be undone over time, but the time pattern of mismatch is important because of the costs associated with delaying bar passage.30 The law-school debate over Sander (2004) has been especially contentious. We find the evidence suggesting that shifting African Americans to less-selective schools would increase bar passage rates, particularly for first-time passage, to be fairly convincing. This is especially the case since the low qual- ity of the data would tend to bias estimates away from finding mismatch. On the other hand, an argument could be made that the data are too noisy and provide sufficiently imprecise information on actual law-school quality that they preclude one from draw- ing any concrete conclusions regarding mis- match. Regardless, the law-school debate makes clear that this is a question that merits 30Williams (2013) also argues that the measurement error in the law-school tier variable results in attenuation bias. Hence he instrumented for law-school tier with the second-choice variable, resulting in massive mismatch effects. However, the estimated effects are so large as to not be plausible. further attention, where more definitive answers could be answered with better data. Our hope is that better datasets soon will become available.31 4.2 Labor Market Outcomes Sander (2004) also argued that affirmative action lowers the earnings of black lawyers, using data from After the JD (AJD). In 2002, the AJD surveyed those who became lawyers (i.e., passed the bar) in 2000.32 As with bar passage and graduation, the mechanism for the earnings result is that attending a more elite school has a positive direct effect on earnings, but also a negative indirect effect through law-school grades. Sander argues that the negative indirect effect is stronger than the positive direct effect, again assum- ing that differences in normalized law-school grades will disappear after affirmative action is eliminated. That this impact is still present even after selecting on bar passage, which itself appears to be affected by mismatch, is notable. Rothstein and Yoon (2009) is one of the few studies that also addresses the earnings question, using data from the BPS data- set. The BPS followed a subsample of the initial participants four to six months after they graduated from law school. Using the same identification strategies they used for bar passage, Rothstein and Yoon find higher 31In Sander v. State Bar of California, the California Supreme Court ruled unanimously in favor of Sander’s petition for access to a deidentified version of public law- school admission data, stating “We conclude that under the common law right of public access, there is a suffi- cient public interest in the information contained in the admissions database such that the State Bar is required to provide access to it if the information can be provided in a form that protects the privacy of applicants and if no countervailing interest outweighs the public’s interest in disclosure.” 32The AJD has the actual school that the lawyer attended. However, law-school grades are self-reported in bins, so adjustments need to be made to renormalize them. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 21 earnings for blacks from attending more selective schools.33 Beyond the differences in identification strategies, a potential way to reconcile the two sets of results comes from Lehmann (2013). Lehmann develops a model of hir- ing and promotion when firms face diver- sity pressures. Using the AJD, Lehmann shows that black lawyers have better early labor-market outcomes, consistent with a model where the cheapest way to satisfy diversity pressures is with newly minted law- yers. After hiring, however, black lawyers receive worse tasks and are less likely to be promoted. Lehmann’s results point to the difficulty of comparing black and white out- comes when affirmative action is present in the labor market, as well as pointing out that these sorts of comparisons may change with the age of the cohort. 4.3 The Extensive Margin In contrast to undergraduate institutions, almost all law schools are selective. Hence, an affirmative-action ban may result in some minority law school applicants, who would have been admitted prior to the ban, being unable to obtain admission to any law school. Assuming that the number of black applicants would remain unchanged and that their admissions and application behavior would now match those of whites with similar LSAT scores and grades after an affirmative-action ban, Wightman (2003) estimates a 14.1 percent drop in the number of black applicants who would be admitted to law school if an affirmative-action ban were passed that was based on applications for law-school admissions in 2001. This is the number Sander (2004) uses in his calculation of the extensive margin. 33They also find that black students at elite schools were more satisfied with their jobs than matched students of either race at less elite institutions. Both Chambers et al. (2005) and Rothstein and Yoon (2008) argue that this number is too low. Rothstein and Yoon (2008) use a grid model for 1990–1991 and show there would be a 60 percent drop in law school enrollees, with a 90 percent drop at the most selective schools. Chambers et al. (2005) extend the grid model in Wightman (2003) to other years, showing that in 2003 and 2004—the most recent years available at the time—the decline would be much higher, at 32.5 per- cent in 2004.34 Chambers et al. (2005) also argue that the drop in admissions may trans- late into lower take-up rates, but there are no data to provide support for or against this claim. There are at least two reasons, however, why estimates of the extensive margin may be overstated. First, the grid model assumes after a ban that black applications will look like white applications. But, the least selec- tive law schools are historically black insti- tutions, which are likely more appealing to black applicants than white applicants.35 How historically black institutions would respond to a surge in black applications following36an affirmative ban is an open question. Second, universities may respond to affirmative-action bans by looking at other ways of increasing diversity by, for example, putting more weight on characteristics that 34 Sander (2005b) argues that those two years are outli- ers driven by a surge of white applications; looking over the previous ten years suggests that the 14.1 percent number is representative. The magnitude of the extensive margin may depend on whether what was happening in 2003 and 2004 reflects permanent or temporary change in applica- tion behavior. 35Rothstein and Yoon (2008) predict an 84 percent drop in minority enrollees at HBCUs, which they acknowl- edge is too high. However, they argue that the movement towards HBCUs would come in part from other law schools, resulting in those schools being even less diverse. 36Of course, having the extensive margin being some- what removed due to historically black institutions may be unsatisfying, harkening back to “separate but equal” education. 22 Journal of Economic Literature, Vol. LIV (March 2016) are correlated with particular races.37 Yagan (2014) provides evidence of this, looking at law-school applications from an elite college to University of California (UC) Berkeley and UCLA before and after Proposition 209, which banned the use of racial preferences in admissions. Yagan shows that although admissions probabilities fell substantially for black applicants after Proposition 209, black applicants still had an advantage in admis- sions over white applicants after controlling for a set of observed credentials. Long and Tienda (2008) show qualitatively similar results in Texas. 5. Undergraduate Decisions and Outcomes 5.1 The Extent of Racial Preferences Unlike in the law-school setting, affirma- tive action is present only at a subset of all undergraduate institutions. In The Shape of the River, Bowen and Bok (1998) argue that only 20 to 30 percent of four-year colleges practice racial preferences, as most schools simply are not selective. These results are consistent with Kane (1998) and Arcidiacono (2005), who both estimate models of admis- sions using nationally representative data in which students self-report where they have been admitted. Both find that racial prefer- ences are large at the top schools, with little scope for racial preferences at the bottom schools. At these top schools, however, racial pref- erences result in substantial differences in academic backgrounds between majority and minority enrollees.38 Using the College 37See Chan and Eyster (2003) and Fryer, Loury, and Yuret (2008) for models of how these adjustments could occur in equilibrium. 38Kane (1998) points out that even in the absences of racial preferences minorities would, on average, have lower test scores within each school given that lower minority test scores would place them more on the margin and Beyond dataset, which focuses on a set of highly selective colleges,39 Arcidiacono, Khan, and Vigdor (2011) show within-school SAT score gaps of at least 140 points.40 It is typically difficult to get information on the accept–reject decisions of universities. But there are data from the University of California system, acquired under a Freedom of Information Act request, that can be used to open the black box of university admis- sions. This individual-level dataset covers applicants from the years 1992 to 2006 and follows these individuals through to gradu- ation.41 While missing many covariates that are likely of importance (such as gender), the data contain a continuous measure of aca- demic preparation. Namely, they include an academic index, which is a weighted average of the student’s SAT score and high-school grades.42 We standardize the academic index so that it has zero mean and standard devia- tion one for the set of individuals that applied to a UC school during this period. The data show that half of the minority applicants who applied to UC Berkeley were admitted compared to 35 percent of white applicants. Minority applicants to UC Berkeley had significantly higher probabil- ities of being admitted despite academic indices that were almost one standard devi- ation lower than white applicants. Figure 3 of being admitted. Given the magnitude of the differences in the entering credential of enrollees, however, it is clear that racial preferences at top schools are large. 39This is the same dataset as used in The Shape of the River. 40The SAT score here includes only the verbal and math sections. 41The year the individual applied is given only by its three-year interval. In 1998, Proposition 209 was imple- mented, which banned the use of racial preferences in California. Hence, two (three) three-year periods are offered before (after) the implementation of Proposition 209. See Antonovics and Sander (2013) for a fuller descrip- tion of the data. 42The SAT score is rescaled on a range from 0 to 600 and high school grades are rescaled from 0 to 400. The two numbers are then added together. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 23 0.8 0.6 0.4 0.2 0 −6 −4 −2 0 2 Academic index Figure 3. Distribution of Academic Indexes for Applicants to UC Berkeley by Minority and Accept/Reject Status Notes: Academic index is a weighted average of the student’s SAT score and high-school GPA and is normal- ized to be mean zero and standard deviation one for the population of applicants. Minority, admit Minority, reject White, admit White, reject Data source: UCOP, years 1995–1997. shows the academic index distribution for those admitted and rejected by UC Berkeley for both minority and white applicants. For both minority and white students, admits have academic indices, on average, around one standard deviation higher than rejects. But the median white reject has an academic index higher than the median minority admit. In fact, the median minority admit’s academic index is at the seventh percentile of the white admit distribution. To further demonstrate the extent of racial preferences at UC Berkeley during this period, Figure 4 compares the academic index distribution for minority admits at UC Berkeley to the academic index distribution for whites who applied to any UC school. The distributions look remarkably similar: randomly drawing from the pool of white students who applied to any UC school would produce an academic index distribu- tion similar to that of minority admits at UC Berkeley.43 As shown in Antonovics and Backes (2014a), UC Berkeley had the largest racial preferences in the UC system. After condi- tioning on a set of observables, minorities had increased chances of admission at the top 43 Looking at enrollees instead of admits shifts the dis- tribution of academic indexes to the left, with the median minority index falling by 0.24 standard deviations. This would place the median minority enrollee at Berkeley below the ninth percentile of white enrollees at Berkeley. 24 Journal of Economic Literature, Vol. LIV (March 2016) 0.5 0.4 0.3 0.2 0.1 0 Figure 4. Distribution of Academic Indexes for Minority Berkeley Admits and White UC Applicants Notes: Academic index is a weighted average of the student’s SAT score and high school GPA and is normal- ized to be mean zero and standard deviation one for the population of applicants. Data source: UCOP, years 1995–1997. Minority, admit Berkeley White, apply UC −4 −2 0 2 4 Academic index four UC schools, ranging from 40 percent at UC Berkeley to 21 percent at UC San Diego. Minority advantages were smaller at the next set of schools, likely for two reasons. First, there is less scope for racial preferences at the bottom set of schools. For example, for those who applied to UC Santa Cruz in this period, both whites and minorities had an over 80 percent chance of being admitted. Second, selection on unobservables is likely stronger at more elite schools as a result of minorities being aware of affirmative action. Those applying to the bottom schools prob- ably were aware that their chances of admis- sion were lower at UC Berkeley, despite racial preferences. Dillon and Smith (2013) provide some additional information on how affirmative action affects sorting using the National Longitudinal Study of Youth of 1997 (NLSY97) data. Their paper is focused on characteriz- ing the extent of academic mismatch more generally, which they define as differences between student rank on the Armed Services Vocational Aptitude Battery (ASVAB) test given to NLSY97 respondents and college rank on an index of quality constructed from several measures of institutional character- istics. This analysis has the value of being nationally representative, although their small sample precludes them from examining mis- match along the college quality spectrum. While the aim of Dillon and Smith (2013) is measuring and characterizing overall mismatch in higher education, they show some evidence that speaks directly Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 25 to mismatch that might be caused by affir- mative action. In particular, they show that despite the clustering of black students in lower-quality schools, they are much more likely to be “over-matched” and less likely to be “under-matched” than whites.44 This is somewhat mechanical, as blacks have lower test scores and hence are naturally more likely to be overmatched than under- matched, relative to whites. Surprisingly, Dillon and Smith (2013) also show that black students are no more likely than whites to be overmatched or undermatched given the same test scores and high-school grades. However, this finding only holds when focusing on students at four-year schools. When two-year students are included, blacks are more likely to be overmatched than whites conditional on credentials. It is difficult to fully explain this result with the evidence given in their paper, but one likely explanation is that more black students are overmatched at two-year schools and non- selective four-year schools, relative to white students.45 The findings of Dillon and Smith of no overmatch among black students stands in contrast to much of the prior litera- ture. In particular, Bowen and Bok (1998) provide extensive evidence of overmatch among African American students. It thus is instructive to understand why these stud- ies differ in their conclusions. One core difference between them is that Dillon and Smith consider the entire distribution of colleges, while Bowen and Box examine just elite schools. Arcidiacono, Khan, and Vigdor (2011) show how the share of black 44“Over-matched” (“under-matched”) is defined as the ASVAB rank being more than 20 points below (above) the college rank. 45Because Dillon and Smith categorize a student as overmatched if he or she has an ASVAB score that is twenty percentile points below the mean of the school he or she is attending, it is entirely possible that black students are overmatched at community colleges. students varies with the average SAT score of the school using data from US News & World Report for 1991. Figure 5 is repro- duced from their paper. The figure shows a U-shaped pattern, where the lowest and highest quality schools, as measured by the average SAT scores of their student body, have the highest share of black students. Because of this feature of how African Americans sort into different-quality post- secondary schools, it is possible for there to be little overmatch among blacks across the whole four-year quality distribution while still exhibiting overmatch among the more- elite schools. It also is important to high- light that Bowen and Bok use older data (their most recent cohorts matriculated in 1989), and the extent of mismatch may have changed over time. This discussion makes clear that much more work needs to be done to understand the extent of overmatch among African Americans across the post- secondary quality distribution. The U-shaped curve shown in figure 5 is of interest in its own right, and it is useful to consider what factors might cause this pattern. First, affirmative action at the top schools can lower the share of black stu- dents in the schools just below them. To the extent that the fall in the black share at this next set of schools makes them less attractive to black students (for example, they are perceived as not having a criti- cal mass), these students may choose to attend lower-quality schools where, condi- tional on their credentials, they are under- matched. Another possibility is that the middle-tier schools practice less affirmative action. Third, black students may be less informed, on average, about their college possibilities. As illustrated by Hoxby and Avery (2013), those who are economically disadvantaged seem to make poorer college choices than their wealthier counterparts. Economically disadvantaged blacks may then choose colleges at the bottom of the 26 Journal of Economic Literature, Vol. LIV (March 2016) 0.12 0.1 0.08 0.06 0.04 600 800 1,000 1,200 Average SAT score 1,400 Figure 5. College Percent Black as a Function of Average SAT Score Source: America’s Best Colleges produced by US News & World Report for 1991. quality distribution even though they would have been a better match for a school in the middle of the quality distribution. Regardless of the reasons for the U-shaped pattern, the removal of affirmative action would affect the sorting of black students into selective colleges. By altering minority college enrollment behavior, removing or restricting racial preferences could impact minority graduation rates, major choices, and the returns to college that they experience in the labor market. Below, we discuss what is known about how affirmative-action policies and bans affect each of these outcomes, as well as highlight questions that remain for future research. 5.2 Enrollments Before discussing the effects of affirma- tive-action policies on longer-run outcomes such as college completion and wages, we provide a summary of the evidence on how affirmative action affects minority enroll- ment behavior, both overall and across the higher-education quality distribution. Many of the effects of affirmative action on stu- dent outcomes flow through changes in whether and where students enroll, thus understanding the effect of racial admissions preferences on minority enrollment is an important first step in thinking about these broader outcomes. Share black Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 27 Figures 3 and 4 suggest that affirmative- action policies act to sort minority students to more selective schools, which implies that repealing them will lower minority representation at these schools. Structural models of how students sort into schools by Arcidiacono (2005) and Howell (2010) have been used to predict how the removal of affirmativeactionaffectsboththeextensive margin (whether students enroll at all) and the intensive margin (where they enroll). Both of these papers have individuals make their application decisions based on expec- tations regarding their probabilities of admittance. These models predict substan- tial reductions in the share of minority stu- dents attending the most selective schools following a ban but, because many schools are nonselective, they predict little effect on the share of minorities enrolled in four-year colleges overall.46 Recent affirmative-action bans allow us to address how affirmative action affects enrollment decisions directly. The enact- ment of these bans provides quasi-exper- imental variation in affirmative-action policies, and a large portion of studies on which we will focus in this and subsequent sections examines these policy changes. Table 1 shows the timing of the bans in each state, as well as the source of each ban’s enactment. The bans come from a mix of sources, ranging from voter initiatives to executive orders and court rulings. The variation in time and location of these bans provides useful variation with which to identify the effect of affirmative-action 46Epple, Romano, and Sieg (2008) calibrate a general equilibrium model of how students sort into colleges when colleges are differentiated only in terms of quality (as in panel A of figure 1). In their model, schools adjust their admissions decisions in reaction to an affirmative-action ban so as to attain more diversity by weighting character- istics correlated with race more heavily. They too show substantial predicted reductions in enrollments at top-tier colleges. policies on college outcomes. Importantly, these bans only affect public universities, so private schools in these states can continue to practice race-based affirmative action. Consistent with the structural predic- tions, the general consensus from empir- ical analyses of affirmative-action bans is that elimination of race-based preferences reduced underrepresented minority stu- dent enrollment at flagship and elite public schools but had little impact on the overall likelihood of four-year college enrollment. Kain, O’Brien, and Jargowsky (2005) exam- ined enrollment behavior among Texas pub- lic college attendees between 1991–2002. They show evidence that when affirmative action was banned in 1997, minority enroll- ment dropped substantially at the state flag- ship schools—University of Texas at Austin and Texas A&M–College Station. In the two years post-Hopwood,47 the number of black students enrolling at UT–Austin dropped to 68 percent of the pre-Hopwood level and dropped to 72 percent of the pre- ban level at Texas A&M. Hispanic enroll- ment also declined by 6 and 25 percent of the preban levels at UT–Austin and Texas A&M, respectively. However, white enroll- ment jumped by 7 percent at these schools, which is consistent with affirmative-action bans reducing enrollment levels among underrepresented minorities at elite public schools when race-based admission policies are eliminated. Long and Tienda (2008) complement these results using administra- tive data from the pre- and postban eras at Texas A&M and UT–Austin. They estimate probit regressions of admission at these schools and show that post-Hopwood, these schools no longer gave independent weight to race and ethnicity in admissions. Hinrichs (2012) and Backes (2012) exploit the timing of affirmative-action bans across 47Hopwood v. Texas is the court case that led to the affirmative-action ban in Texas. 28 Journal of Economic Literature, Vol. LIV (March 2016) TABLE 1 Affirmative Action Bans in Undergraduate Admissions by State State Texas California Washington Florida Georgia** Michigan*** Arizona New Hampshire Oklahoma Year of Enactment 1997* 1998 1999 2001 2002 2006 2010 2012 2013 Source of Ban Court Ruling: Hopwood v. Texas Voter Initiative (Prop. 209) Voter Initiative (Initiative 200) Executive Order 99-281 Court Ruling: Johnson v. University of Georgia Voter Initiative (Proposition 2) Voter Initiative (Proposition 107) Legislative Act (HB 623) Voter Initiative (State Question 759) * The affirmative-action ban in Texas was overturned in 2003 by the 5th Circuit Court of Appeals. ** This ruling only affects the University of Georgia. The race-based admission system used was ruled unconstitu- tional, but other public institutions in Georgia still consider race in admissions. *** In July 2011, the Michigan affirmative action ban was ruled unconstitutional by the 6th Circuit Court of Appeals. The case was argued in front of the Supreme Court in October 2013, but no decision has been made on the constitutionality of this law. several of the states shown in table 1 to estimate difference-in-difference models of college enrollment. Due to their use of plau- sibly exogenous affirmative-action variation, these are the most informative estimates of how affirmative action affects flagship enroll- ment among underrepresented minority students. Using data from the American Community Survey and the Current Population Survey, Hinrichs (2012) shows that affirmative-action bans had no overall effect on the likelihood that minority students attended any college or attended a four-year college. However, with IPEDS data on col- lege enrollment by race and institution, he shows bans led to large declines in black and Hispanic enrollment shares at more elite uni- versities. In particular, enrollment by black students at US News top-fifty-ranked schools declined by 1.6 percent and Hispanic enroll- ment at these schools dropped by 1.8 per- cent. Because these declines include private schools that were not subject to these bans, these results suggest the affected minority students did not just shift enrollment to other schools of similar quality that were still practicing affirmative action. Furthermore, he shows that the effects on enrollment were larger at schools that had lower admissions rates and higher SAT scores. Overall, these estimates indicate that banning affirmative action had large impacts on the quality of schools in which underrepresented minority students enroll. Backes (2012) shows very similar results using an expanded time frame and a more detailed set of institutional qual- ity measures. Research on enrollment responses to affir- mative-action bans also shows that most of the enrollment effect came from changes in admissions behavior: application pat- terns among minority students changed little following affirmative-action bans (Card and Krueger 2005; Antonovics and Backes Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 29 2013).48 Thus, affirmative-action bans alter admissions rules such that minority students are less likely to gain admittance to more elite in-state, public schools. This reduces the quality of schools these students attend, although it does not affect the likelihood underrepresented minority students enroll in a four-year college. In what follows, we discuss the potential effects of this type of re-sorting on college graduation, college major, and labor-market returns to college investment. An important caveat to all of these studies is that the measured effects are in an envi- ronment where racial preferences are still in place at all private schools and at public schools in states that do not have bans. A uni- versal ban on racial preferences would likely have very different effects, for example, on the share of minorities at flagship universities than a local ban (a ban on racial preferences at state institutions). Minorities admitted to the flagship school under a universal ban will have fewer options than those admit- ted under a local ban, suggesting minority students admitted to the flagship would be more likely to attend under a universal ban than a local ban. 5.3 Graduation Affirmative-action policies can influence graduation rates through two conflicting mechanisms. As discussed in section 5.2, race-based admissions policies can have a siz- able influence on the distribution of minority students across schools of differing quality. The Texas and California experience with banning affirmative action had a large impact 48 Long (2004) compares changes in SAT score-sending rates among white and minority students in the late 1990s and finds some evidence of a relative decline in minority applications to the top in-state institutions, especially in California. However, he also shows similar patterns in applications to private and out-of-state colleges, which is inconsistent with these patterns being driven by affirma- tive-action changes. on the representation of minority students at the more selective, in-state universities. The mismatch hypothesis predicts that banning race-based admissions could increase the rate at which they graduate from college. This would occur because students are now better matched, academically, for the schools in which they enroll. However, there also is a growing litera- ture (discussed below) suggesting that col- lege quality and collegiate resources that are strongly correlated with measures of college quality exert an independent influence on students’ abilities to obtain a college degree. If such college quality effects are large for students affected by race-based admissions policies, then affirmative action policies could raise collegiate attainment among underrepresented minority students. 5.3.1 Graduation and College Quality We begin this section with an overview of the literature on the effect of college qual- ity on BA completion. Table 2 shows six-year graduation rates across different higher edu- cation sectors in the United States for the 2011–2012 school year. We split the four-year higher-education sector into four mutually exclusive and exhaustive groups: flagship public universities,49 nonflagship public uni- versities, highly selective private schools, and less-selective private schools. The highly selective private schools are the top-sixty- five private universities and colleges accord- ing to the 2012 US News and World Report rankings, as well as the top-fifty liberal arts schools. We have chosen to categorize the higher education market in this way because these sectors represent the broad choices students face when deciding what type of college to attend. The differences across the tiers are substantial: public flagships and 49 The flagship public schools are straightforward to identify. See Lovenheim and Reynolds (2013) for a description of this classification. 30 Journal of Economic Literature, Vol. LIV (March 2016) TABLE 2 Median Institutional Characteristics by College Quality Tier College characteristic Six-year graduation rate Black six-year graduation rate Hispanic six-year graduation rate White six-year graduation rate Student–faculty ratio Per-student expenditure Per-student instructional expenditure Faculty salaries Flagship Public 0.702 0.598 0.670 0.715 11.59 39,685 12,081 91,177 Nonflagship Public 0.444 0.333 0.393 0.460 13.84 15,741 6,054 65,961 Highly Selective Less-Selective Private Private 0.868 0.484 0.810 0.333 0.818 0.417 0.869 0.514 6.36 13.48 Notes: The per-student expenditures and student–faculty ratios use all student enrollments, not just undergraduates. Faculty salaries and student–faculty ratios use all instructional staff, which includes nontenure-track lecturers. Source: 2011–2012 IPEDS data. elite privates have six-year graduation rates of 70.2 percent and 86.8 percent, respec- tively, compared to between 44–48 percent in the other two sectors. These differences are particularly pronounced for Hispanic and black graduation rates, which highlights the potential importance of school quality for these groups. In addition, the table shows resource differences across sectors in terms of student–faculty ratios, per-student expen- ditures, and faculty salaries. For all measures, the highly selective private schools have much higher resource levels than any other sector, with public flagships also having substantially more resources than lower-ranked public and private schools. Thus, affirmative-action bans that re-sort minority students from flagship universities to nonflagship public schools will, in effect, reduce the amount of postsecond- ary resources to which these students have access, which in turn could lead to worse postsecondary outcomes. The graduation-rate differences across sectors shown in table 2 also reflect factors that make it difficult to identify the causal effect of college quality on graduation. Due to the nature of how students select into different quality tiers as a function of precollegiate academic training, the most academically qualified students sort into the highest-quality schools. The outcome differences across school types thus could reflect prior academic training, and it there- fore would be incorrect to extrapolate these graduation rates to students who were admitted under an affirmative-action pol- icy. Additionally, it is possible that collegiate resources have heterogeneous effects on stu- dents with different precollegiate academic abilities, which forms the basis of the mis- match hypothesis. The literature on the effect of college qual- ity on graduation rates attempts to control for differences in the preexisting academic capabilities of students in order to isolate the role of college quality.50 Although not with- out its difficulties in terms of causal identifi- cation due to the college selection process, 50 In this literature, the terms “college quality,” “college resources,” and “college selectivity” are used interchange- ably. We follow that convention here. 54,028 20,478 96,101 17,232 5,433 54,520 Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 31 prior work shows consistent evidence that college quality has a large and positive effect on the likelihood a student graduates from a four-year school, on average.51 Bound, Lovenheim, and Turner (2010) show evidence from the National Longitu- dinal Survey of the High School Class of 1972 (NLS72) and the National Educational Longitudinal Study of 1988 (NELS:88) that once one controls for an extensive set of stu- dent background characteristics, including high school test scores, large completion differences across college quality tiers still remain. The NLS72 and NELS:88 datasets are nationally representative and follow the high school classes of 1972 and 1992, respec- tively, as they transition to college and then into the workforce. Their categorization of schools follows closely with the one pre- sented in table 2, and their estimates indicate that students beginning college at a top-fifty public school or at an elite private school are significantly more likely to obtain a BA within eight years than are students who begin col- lege at less-selective four-year public schools. Furthermore, this gap grew substantially across cohorts, suggesting that college quality is becoming more important over time. Bound and Turner (2007) use a different and novel source of variation in school quality derived from population variation to identify the role of collegiate resources on graduation 51Prior research also has addressed the question of whether starting college in a two-year school versus in a four-year school affects the likelihood of BA receipt. Rouse (1995) shows evidence from community college openings that some students are diverted from a four-year school and are less likely to complete a BA as a result. Using matching methods to help account for the nonrandom selection of students across school sectors and the non- overlap in observable characteristics between two- and four-year students, Long and Kurlaender (2009), Reynolds (2012), and Doyle (2009) all report large negative effects of two-year attendance on eventual BA receipt. Because most affirmative action policies affect admission and enrollment at four-year schools due to the open-enrollment nature of community colleges, we restrict our attention to the litera- ture that focuses on that sector. that is less prone to the biases associated with the selection-on-observables approach. Their insight is that relatively large cohorts of stu- dents represent a demand shock for college. Since the supply of spaces at higher-resource schools tends to be inelastic, and since state funding responds to demand increases with long lags, students born in larger cohorts are less likely to attend a selective school and probably experience lower resources when they are there. They term this phenome- non “cohort crowding.” They show that the elasticity of college graduation with respect to birth cohort size is smaller than the elas- ticity of college enrollment with respect to cohort birth cohort size. This finding is driven by some combination of students in larger cohorts attending less-selective col- leges and with per-student resources declin- ing at less-selective schools when there is a positive demand shock. Regardless of which effect dominates, both factors point towards resources mattering for degree attainment.52 Another source of variation research- ers have used to inform the question of how college quality affects college gradu- ation rates is state merit aid systems. State merit aid programs are prevalent and grow- ing in the United States, and they offer higher-performing students scholarships to attend college in-state. Cohodes and Goodman (2014) study the Massachusetts Adams Scholarship that provides free tuition (but not fees) for high-performing students in Massachusetts. Using a regression discon- tinuity design surrounding the state test score eligibility cutoffs, they show that becoming eligible for this scholarship reduced college 52 One objection to their approach is that students from larger cohorts may experience crowding in elementary and secondary schools that makes them less academically pre- pared for college. Bound, Lovenheim, and Turner (2010) conduct a similar analysis in which they control for student precollegiate test scores. Their findings are very similar, which suggests reduced precollegiate training is not driv- ing the cohort crowding result. 32 Journal of Economic Literature, Vol. LIV (March 2016) quality by shifting students from high- er-quality private schools to lower-quality public universities. They also show that BA attainment rates within four years declined by 1.7 percentage points, which is consis- tent with declining school quality on average worsening college outcomes.53 While the estimates in Cohodes and Goodman (2014) also could be driven by changes in family budgets and the fact that treated students are more likely to go to school closer to home and maintain their high-school peer groups, they show evidence that it is the highest-scoring students whose BA attainment is most affected. Since these are the students who are most likely to experience a decline in college quality due to the scholarship, their results are consistent with an important role for college quality in driving college comple- tion. Cohodes and Goodman (2014) also may be picking up matching effects, as students who are qualified to receive the merit aid pro- gram may be the most likely to benefit from attending a higher-quality college. 5.3.2 Graduation and Fit This research on the graduation effects of college quality examines average effects, but average effects may not be relevant for all students if there is significant mis- match. Several papers in the literature have examined the empirical relevance of “mis- match,” particularly along racial/ethnic lines. Loury and Garman (1995) provide one of the first analyses of the effect of mismatch among whites and blacks using data from 53The finding that state merit aid reduces BA attain- ment is far from universal in this literature. Fitzpatrick and Jones (2012) and Sjoquist and Winters (2012) find that the implementation of a state merit aid program does not affect graduation rates, while Scott-Clayton (2011) shows positive effects of state merit aid on collegiate attainment. A major difference between these studies is in the quality of schools students would attend in the absence of merit aid. These differences likely drive much of the heteroge- neity in findings in this literature, although no work has directly tested this hypothesis. the NLS72. Using selection-on-observables models that attempt to control for selection using SAT scores, parental income and educa- tion, and student demographic variables, they examine whether black and white students are more or less likely to graduate if they are “bet- ter matched” to their college in terms of their own SAT score being closer to the college median. Consistent with Bound, Lovenheim, and Turner (2010), they find college quality has a sizable average impact on the likeli- hood of obtaining a BA for both blacks and whites, but lower-SAT whites and blacks both are less likely to graduate when they attend a higher-SAT school. This penalty is particularly large for black students. The authors interpret these results as evidence of mismatch, as less academically prepared students are less likely to get a four year degree when they attend a higher-resource school, despite the higher level of resources they experience there. Light and Strayer (2000, 2002) provide some additional evidence on the extent of mismatch using data from the NLSY79. They model college graduation as a two-stage process: in the first stage, students decide whether to go to college and which college type to attend, and in the second stage they make a graduation decision. They estimate these two stages simultaneously and allow for correlations of the error terms across equations. This method explicitly allows for unobservables to influence both enroll- ment and completion. The results from Light and Strayer (2000) show that the pre- dicted likelihood of graduation among the lowest-quartile Armed Forces Qualification Test (AFQT) students declines monotoni- cally as college quality increases. As student academic ability rises, this pattern largely reverses.54 These results again suggest that 54The reversal is not perfect. A surprising finding of Light and Strayer (2000, 2002) is that schools in the second lowest quartile are associated with the highest graduation rates, though the standard errors are large. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 33 policies that induce less academically pre- pared students to attend higher-quality schools might be counter-productive in terms of graduation. In a related paper using similar methods, Light and Strayer (2002) find similar patterns. Here, minority status has a positive effect on graduation rates, a positive intercept regardless on the what college the student attends. Light and Strayer (2002) argue that this positive effect may be result of affirmative action programs being successful. However, this ignores the effect of minority status on college quality, the primary channel through which racial preferences affect outcomes. And here, the results indicate match effects are important and generally dominate quality effects. That the minority intercept is positive may instead reflect higher returns to a college degree for African Americans.55 5.3.3 Evidence from Affirmative-Action Bans The recent affirmative-action bans make it possible to directly test how affirmative action affects graduation rates. Hinrichs (2012) examines the effect of affirmative-ac- tion bans for public postsecondary schools that were enacted in California (1998), Florida (2001), Texas (1997), and Washington (1999). Using American Community Survey data from 2005–2007 that links each respon- dent to whether an affirmative-action ban was in place in his state of birth when he was eighteen, Hinrichs (2012) estimates difference-in-difference models that exam- ine how ban enactments change the BA attainment rate of minority students. He finds no effect: his preferred specification produces a coefficient on the ban that is small in absolute value and is not statistically significant. His estimates are precise enough that he can rule out effects larger than 0.8 of 55 Both Arcidiacono (2005) and Arcidiacono, Bayer, and Hizmo (2010) find that African Americans receive a larger wage bump from college graduation than whites. a percentage point (or 7 percent relative to thesamplemean)declineinBAattainment rates.Infollowuppapers,Backes(2012)and Hinrichs (2014) examine the question of how affirmative-action bans affect completion using institutional data from the Integrated Postsecondary Education Data System (IPEDS).56 Backes (2012) examines two addi- tional states, Georgia (2002)57 and Michigan (2004), focusing on six-year graduation rates at public schools. He finds a significant neg- ative graduation effect on the share of public school graduates who are black, but the mag- nitude is very small.58 Hinrichs (2014) exam- ines both four- and six-year graduation rates, where the dependent variable is the share of blacks (or Hispanics) who graduated from a particular type of institution. Considering all institutions, no negative effects were found for either blacks or Hispanics. Backes (2012) and Hinrichs (2014) also look at the share of graduates of particular races at selective colleges. Consistent with the findings on enrollment, the share of graduates who are underrepresented minorities at selective col- leges falls. These estimates thus point to little impact of affirmative-action bans on minority graduation rates. Cortes (2010) analyzes the effect of the Texas affirmative-action ban using data from six public universities of differing quality in Texas. In particular, her data contain the two flagship universities in Texas, as well as a set of less selective public schools and two private schools. The identification strategy employed is to compare changes in six-year graduation rates among those in 56A potential issue with using IPEDS for this ques- tion is that some minority students may react to a ban by attending college out of state. 57 There is some ambiguity about how to treat Georgia’s affirmative action policy change, as only the University of Georgia eliminated racial admissions preferences. 58Note that by focusing on public schools a possible substitution from public to private schools that can still use affirmative action is ignored. 34 Journal of Economic Literature, Vol. LIV (March 2016) the top decile of their high school class to changes among lower-ranked students in a difference-in-difference framework. The argument for this methodology is based on the fact that the Texas Top 10 Percent Rule guaranteed admissions to any public Texas university for students in the top 10 per- cent of his or her high-school class,59 and students in the top 10 percent were highly likely to be admitted to Texas A&M or UT Austin prior to the affirmative-action ban. The results indicate that, after the ban, minority students in Texas were between 2.7 and 4.0 percentage points less likely to earn a BA within six years. However, these effects are due predominantly to an increase in graduation rates among those in the top decile (who are assumed to be untreated by the ban), rather than a decline among low- er-ranked students. This pattern is more consistent with the Top 10 Percent Rule positively impacting top-decile students than with the affirmative-action ban driv- ing a decline in completion rates for low- er-ranked students. Arcidiacono et al. (2014) examine how the affirmative-action ban in California affected five-year graduation rates, paying particular attention to issues of the match between the student and the school. They show gradu- ation rates for underrepresented minori- ties in the UC system increased by over 4 percentage points after the ban. They then decompose this change into three parts: improvement in the fit between the stu- dent and the school, selection into the UC system as minorities were a smaller share of the student body after the ban, and gains due to universities responding to the ban by changing how academic preparation trans- lates into graduation. Their model allows the returns to academic preparation to vary 59We discuss the Texas top 10 percent rule as well as other related “percent plans” as a replacement for affirma- tive action in section 6. across campuses, consistent with figure 1. The results show patterns consistent with the bottom panel of figure 1: the most selec- tive schools have the highest graduation rates for the most prepared students, while the least selective schools in the system had the highest graduation rates for the least prepared students. Reshuffling minority stu- dents within the system according to their sorting patterns after the ban results in grad- uation gains for minority students between 0.1 and 1.2 percentage points, depending on the controls for selection.60 Substantially higher gains are found for those in the bot- tom quartile of the minority preparation dis- tribution, where the resulting improvements in fit improve graduation rates by between 0.8 and 2.2 percentage points. At the same time, these students are also the ones most likely not to be admitted to any school in the system in the presence of a ban. The results of Arcidiacono et al. (2014) also suggest that universities may respond to affirmative-action bans by focusing more resources on ensuring that the minority stu- dents they do have actually graduate. Their lower-bound estimates suggest that, con- ditional on both observed and unobserved ability, students had significantly higher probabilities of graduating after the ban. Antonovics and Backes (2014a) show fur- ther evidence of endogenous institutional responses to affirmative-action bans. In the UC system, one of the responses to the ban was to adjust the admissions weights placed on different background characteristics in diversity-enhancing ways. Universities may also adjust how they recruit students and how they invest in students conditional on arrival, as occurred in Texas with the Century 60The smallest results are for the baseline case that only controls for selection on observables. Larger results are found when controls similar to those used in Dale and Krueger (2002) are used. See section 5.5 for a discussion of Dale and Krueger (2002). Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 35 and Longhorn scholarship programs that were implemented post-Hopwood.61 Such changes may provide additional support for minority students at more elite schools after affirmative-action bans are implemented. Fit seems to be more important as the graduation window shortens. Arcidiacono, Aucejo, and Hotz (2016) show that minority four-year graduation rates in the UC sys- tem are highest at UC Santa Cruz, despite minorities at UC Santa Cruz having sub- stantially worse academic backgrounds than those at places like UC Berkeley and UCLA. This pattern does not show up in the raw data for five-year graduation rates, suggesting that match effects may affect time to degree, as well as completion rates. Arcidiacono and Koedel (2014) examine eight-year graduation rates in the Missouri system. While there is some evidence that fit matters when compar- ing the most selective schools in the system to moderately selective schools, there are a set of schools at the bottom of the quality distribution that appear to be worse for all students. This may be the result of quality effects dominating any match effects at the bottom of the school-quality distribution or because of the wide graduation-rate window considered in this study. In sum, the evidence suggests that (1) increasing college quality positively affects graduation rates on average, (2) these positive effects turn negative for those with relatively low levels of academic preparation, and (3) affirmative-action bans have had little effect on the overall grad- uation rates of minority students. These results are consistent with offsetting influ- ences of quality effects and match effects. Given that more selective schools have an overall positive effect on graduation, racial preferences that are milder than those 61See Andrews, Ranchhod, and Sathy (2010) for a description of these programs and their effect on minority enrollment. currently in place, at least at the very best schools, could result in higher graduation rates for minority students, taking advan- tage of the higher resources associated with more selective schools without being so aggressive such that match effects degrade educational outcomes. However, the esti- mated graduation gains from milder pref- erences at the best schools are small. As Arcidiacono and Koedel (2014) note, sub- stantially higher gains could be achieved from policies that lead to college quality increases among minority students attend- ing the very bottom schools. 5.4 Choice of Major Many studies have documented substan- tial differences in the monetary returns to different majors.62 Relatively few studies have analyzed how the fit between the stu- dent and the school affects one’s choice of major, however, but it is here where fit may be the most important. This is in part due to the very different demands placed on students in different majors. Majors in sci- ence, technology, engineering, and math (STEM), as well as in economics, have very different grading distributions (Sabot and Wakeman-Linn 1991; Johnson 2003) and are associated with more study time than other majors (Brint, Cantwell, and Saxena 2012; Stinebrickner and Stinebrickner 2011). Differences in grading standards and study time have disproportionate effects on those who are at the bottom of the prepa- ration distribution. Within colleges, there is massive sorting surrounding who persists in majors in the sciences, with those who have higher SAT scores (and in particular SAT math scores) being much more likely to persist in the sciences (Arcidiacono 2004; Arcidiacono, Aucejo, and Hotz 2016). 62See Altonji, Blom, and Meghir (2012) for a recent review. 36 Journal of Economic Literature, Vol. LIV (March 2016) Since affirmative action results in minority students being relatively less prepared than their nonminority counterparts within a given school, differences in expectations across fields can result in minorities shift- ing away from the sciences over the course of their college careers. Using data on Duke students, Arcidiacono, Aucejo, and Spenner (2012) showed that, conditional on gender, African Americans were more likely than white students to have an initial major in the sciences. However, their probability of finishing a major in the sciences was much lower. For example, of those who expressed an initial interest in the sciences, 54 per- cent of African American males finished in the humanities or social sciences compared to 8 percent of white males. Arcidiacono, Aucejo, and Spenner show that this entire racial gap can be accounted for by controlling for academic background: there are no dif- ferences in major switching patterns across races once one accounts for differences in academic background. What the Arcidiacono, Aucejo, and Spenner study cannot do is say whether the African American students at Duke would have been more likely to persist in science majors had they attended a less selective school. That is, is it relative preparation or absolute preparation that matters for obtain- ing a science degree? Smyth and McArdle (2004) use the College and Beyond data63 and estimate models of graduating in a sci- ence field, controlling both for the differ- ence between the individual’s SAT math score and the average SAT score of their school as well as the average SAT score of the school itself.64 Smyth and McArdle (2004) 63 This is the same data used in Bowen and Bok’s semi- nal book The Shape of the River and in Dale and Krueger (2002). 64 Smyth and McArdle’s study was motivated by Elliott et al. (1996), who examine data from eleven colleges of varying selectivity. At each of the schools in their data, individuals who graduate are divided by their tercile of the find support for relative preparation matter- ing for attaining a STEM degree: the total effect of college quality (the direct effect and the effect through relative preparation) on STEM attainment is negative. Arcidiacono, Aucejo, and Hotz (2016) also explore the issue of institutional fit but allow sufficient flexibility such that better students may see higher probabilities of graduating in the sciences at better schools, consistent with the bottom panel of figure 1. They use data from the cohorts of students applying to the University of California sys- tem between 1995 and 1997. These data show substantial within-school gaps in SAT scores between minority students who start in the sciences and finish in the sciences and minority students who start in the sciences but finish outside of the sciences—gaps that are much smaller for nonminority stu- dents.65 For example, at UC Berkeley the gap is 105 points for minority students and only 21 points for nonminority students. The probability of graduating with a science degree conditional on starting in the sciences at UC Berkeley is 61 percent for nonminority students, but just over half that (31 percent) for minority students. Graduating in the sciences with a low SAT score is difficult at UC Berkeley, but would these same students have a higher probabil- ity of graduating in the sciences from a school like UC Riverside? The answer appears to be yes. Denote the median SAT score for minorities at Berkeley as A. The minority school’s SAT math score distribution. They find that the share of science degrees awarded to each tercile was very similar across institutions despite large differences in the average SAT math scores of the schools, with the top tercile generally producing over half the science graduates. While more selective schools also produced more science gradu- ates overall, the net effect is still that attending a less selec- tive institution makes persisting in the sciences more likely. A caveat with this study is that it focuses on college grad- uates, and dropout rates may differ across the institutions. 65 Finishing in the sciences and graduation probabilities both refer to five-year graduation rates. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 37 students below this score who are interested in the sciences have a five-year graduation rate in the sciences of 13 percent. Now con- sider minority students at Riverside, again with an initial interest in the sciences, who have SAT scores below A.66 Their five-year graduation rate in the sciences is 20 percent, despite likely having lower SAT scores overall and likely having a worse academic record.67 Arcidiacono, Aucejo, and Hotz estimate a model of major choice, controlling for some observed preparation measures as well as Dale and Krueger (2002) controls, where schools differ in their major-specific returns to academic preparation. Hence, as in fig- ure 1, it is possible for the most prepared stu- dents to be best matched at the most selective schools and less prepared students to be best matched at less selective schools. Estimates of the model reveal exactly this pattern. After accounting for selection, the vast majority of minority students, who are coming in with significantly worse preparation than their nonminority counterparts, would have higher graduation probabilities in the sciences had they attended a less selective school in the UC system. On the other hand, nonminority students are generally well-matched: a much smaller share would see higher gradua- tion probabilities in the sciences had they attended a less selective school.68 Note that the studies discussed above have focused on selective schools. Arcidiacono (2004, 2005) both analyze the choice of col- lege and major using the NLS72 dataset and come to a different conclusion on how col- lege quality and the choice of major interact. 66Eighty-seven percent of entering minority science majors at Riverside have SAT scores below A. 67 Note that the overall graduation rate for this group is higher at Berkeley. Berkeley does an excellent job of grad- uating minority students, just not in the sciences. 68 Indeed, Arcidiacono, Aucejo, and Hotz find that non- minority students in the UC system would have had lower graduation rates in the sciences had they been assigned to schools like their minority counterparts. Namely, increasing college quality makes majoring in the sciences more attractive. There are two reasons why we do not put as much weight on these studies in terms of how college quality interacts with major choice. First, they placed considerable structure on how college quality and major choice inter- acted by constraining college quality to make particular majors more or less attractive for all students, as opposed to allowing for the complementarities discussed in section 2. As the discussion of Arcidiacono, Aucejo, and Hotz (2016) suggests, mismatch only becomes an issue when one’s preparation is severely behind one’s peers.69 Second, there are no controls for intended major, and those who intend to major in the sciences are, on average, stronger students and therefore more likely to attend higher-quality colleges. But it is possible that the difference in results is partly driven by the national data used in Arcidiacono (2004, 2005). Arcidiacono and Koedel (2014) find that the lowest-quality four-year institutions in the Missouri system have very few if any STEM majors. It may be the case that col- leges’ quality effects dominate at the bottom of the distribution, whereby better schools are better for everyone, while matching effects are more important at the top of the distribution. The affirmative-action debate has typically focused on racial preferences at the very best institutions, but preferences may be most beneficial further down the col- lege quality distribution. 5.5 Labor Market Outcomes Ultimately, when assessing the impact of affirmative action on minority students, we care about how their long-run life outcomes are affected. An important component of such outcomes is labor-market performance, 69 Note then that it is not just a matter of being a small fish in a big pond as Gladwell (2013) suggests, but how well one is prepared for the material being taught. 38 Journal of Economic Literature, Vol. LIV (March 2016) such as earnings, wages, and the likelihood of employment. Affirmative action will potentially affect labor-market outcomes through its influence on the myriad of col- legiate outcomes considered above: quality of schools attended, the likelihood of grad- uation, and choice of major. Affirmative- action policies could also have direct effects on labor-market outcomes that work through employer beliefs. As shown by Coate and Loury (1993), affirmative action can lead to employers developing “negative stereotypes” about minority workers that lower their returns to a given educational investment. In equilibrium, employers will believe minority workers to be less productive, will pay them less, and thus will lower the incentive for these workers to invest in labor-market skills (including education). If employers develop beliefs about worker productivity that are a function of school attended and race, then such an equilibrium could arise due to affir- mative action in the higher-education market. Indeed, the concern that employers will treat underrepresented minority workers from elite schools differently because of affirma- tive action is a core argument opponents of affirmative action adduce against these pol- icies. Furthermore, affirmative action could have important distributional consequences for minority workers, with affirmative-action bans potentially influencing wages among very high-skilled minorities. The results from affirmative-action bans discussed in section 5.2 highlight that a pri- mary effect of affirmative action on minority students is to induce higher rates of atten- dance at more elite schools. With little effect on overall graduation rates, the effect of affirmative action in admissions on earnings will flow predominantly through the choice of major and college quality. We begin our discussion by considering the returns to college quality in general. This is a growing area of research that has made many recent advances due to growing data quality that allow one to link students’ K–12 edu- cation records to higher-education records andfutureearnings.Asintheestimationof college quality effects on higher BA attain- ment rates, the struggle in all studies on the returns to college quality is to deal with the fact that students with higher precollegiate academic ability that is independently valued in the labor market are more likely to sort into higher-quality schools. Put differently, student sorting across the quality distribu- tion of the higher-education market is highly nonrandom, and even when the characteris- tics of students at different types of schools overlap we must still be concerned about why they wound up in different institutions and whether these differences are correlated with unobserved attributes that will drive labor-market outcomes. The literature to date has used three methods to address this endogeneity concern: selection on observ- ables, regression discontinuity using admis- sions rules, and matching based on students with similar application and admissions histo- ries. We discuss each of these in turn below, and we also point interested readers to the thoughtful review provided in Hoxby (2009). The largest group of studies uses student observable characteristics to control for the differences across students in underlying earnings potential across the higher education quality distribution. In early work on this ques- tion, Brewer, Eide, and Ehrenberg (1999) use the rich demographic and precollegiate test- score data in the NLS72 and High School and Beyond (HSB) datasets to estimate wage and earnings premiums for those who attend schools from different higher-education sec- tors.70 Their results indicate that students attending a high-quality public university (defined as being most or highly compet- itive in the Barron’s ratings) earn as much as 25 percent more than those attending a 70 Note that the student need not graduate from college to be included in the sample. Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 39 low-quality one (defined as being less or non- competitive). In addition to the concern that there is selection on unobservable character- istics, such as student motivation, there is a concern that OLS models will produce mis- leading estimates because of limited overlap in the observable characteristics of students attending schools of differing quality. If there was full sorting on background characteristics, then the only way to adjust for differences in background characteristics across schools would be to impose a poten- tially restrictive functional form assump- tion. This issue is taken up directly in Black and Smith (2004). They use the National Longitudinal Study of 1979, which simi- lar to NLS72 and HSB, contains detailed background and precollegiate cognitive skill measures, and they estimate matching models that allow them to both assess the degree of common support across the dis- tribution of college quality and to examine how sensitive the results are to excluding the nonoverlapping set.71 Their results demon- strate a surprising amount of overlap in the propensity score distributions of the likeli- hood of being in the top versus the bottom quartile of college quality as a function of the observables. They find a sizable effect of college quality: going from the bottom to the top quartile of college quality increases wages by 14 percent for men and 8 per- cent for women.72 Although these estimates are difficult to compare with Brewer, Eide, and Ehrenberg (1999) due to differences in 71 Black and Smith (2004) measure college quality using an index based on average faculty salaries in 1997, average SAT score of the entering class in 1990, and the average first-year retention rate in 1990. Black and Smith (2006) use a more extensive set of institutional characteristics to measure college quality, again showing significant col- lege-quality effects. 72These are similar to the OLS estimates, and for both men and women the effects grow substantially, to 25 percent and 16 percent, when the estimation sample is restricted to the “thick support region” where students are most similar in terms of observables across school types. the way they measure school quality and not separating public and private schools, they both indicate substantial returns to attending a higher-quality college. Arcidiacono (2005) is the only study that links racial preferences in admissions to future earnings. Using data from the NLS72, he estimates a structural model where stu- dents first decide the set of colleges to which they will apply given expectations about their probabilities of admission, financial aid conditional on admission, enrollment and major choice decisions, and future earnings. Arcidiacono’s results on earnings show small returns to college quality as measured by the average SAT math score of the student body, particularly after allowing for the errors in the various parts of the model (applications, admissions, earnings, etc.) to be correlated.73 However, as pointed out by Black and Smith (2004, 2006), using only one noisy measure of college quality may bias downward the estimated returns. Arcidiacono’s model is estimated in a regime where racial preferences are pres- ent. The structural parameter estimates are then used to forecast how black student decisions over applications, enrollment, and major choice would change if they faced the admission and/or financial-aid rules of whites. The effects of removing racial preferences in admissions on expected black earnings, even among those most likely to attend a selective college, were negative but quite small. This is due to racial preferences in admissions hav- ing little effect on enrollment and the low estimated returns to college quality.74 73 Similar low returns, albeit with the same dataset, are found in Arcidiacono (2004). 74Arcidiacono (2005) finds slightly larger negative effects from removing racial preferences in financial aid as racial preferences in financial aid affect whether black students attend college at all. He estimates that the returns to college are significantly higher for blacks than whites, given the same background characteristics. Racial prefer- ences in financial aid have the added advantage that mis- match effects are unlikely to result. 40 Journal of Economic Literature, Vol. LIV (March 2016) These results are based on older cohorts who graduated from college in the 1970s and 1980s. It thus is important to consider some more recent evidence. Long (2010b) shows suggestive evidence that the returns to college quality has increased over time. He estimates the effect of college quality in the NLS72, HSB, and NELS:88 surveys, and he shows that the effect on earnings of a one standard deviation increase in quality has grown from 2.6 percent to 4.8 percent across cohorts. However, this change was not sta- tistically different from zero. Interestingly, the return to college quality for African American students in NELS:88 dropped to zero from the 7.5 percent estimate in HSB. Thus, the increasing returns do not appear to be uniformly distributed across students, and if the returns to quality have dropped for black students this has important impli- cations for how affirmative action changes have affected the returns to education for this group. Evidence from the most recent cohorts comes from Texas, where it is possible to link students’ K–12 records to higher-education outcomes and labor market earnings starting from the 1994 high school graduating cohort (Andrews, Li, and Lovenheim forthcoming). Andrews, Li, and Lovenheim (forthcoming) is distinguished by the large volume of stu- dent high school test score controls that the longitudinal datasets used in other work do not have and by the large sample sizes that provide more statistical power. They estimate OLS models of the effect of UT Austin and Texas A&M graduation on earnings relative to earnings from graduates at other four-year schools in Texas. These estimates are similar to those from prior work, suggesting an earn- ings premium from UT Austin of 11.5 per- cent and from Texas A&M of 21 percent. For black and Hispanic students, however, the returns to UT Austin are small, while the returns to Texas A&M are about 20 per- cent. Furthermore, they estimate quantile treatment effects that show how college quality shifts the earnings distribution. At UT Austin, there is a wide distribution, with earnings at the top of the distribution shift- ing out much more than at the bottom. The effect of Texas A&M on the earnings distri- bution is much more constant. These results may be consistent with the bottom panel of figure 1 if the quantiles correlate strongly with academic preparation, which would explain the lower returns at UT Austin for blacks and Hispanics. Alternatively, the wide distribution at UT Austin may simply reflect UT Austin being a risky proposition regard- less of where one lies in the preparation distribution. Overall, the evidence from studies that rely on student-observable characteristics to account for selection into schools of differing quality points towards significant effects of college quality on earnings. However, there is much concern that despite the large volume of information contained in the conditioning sets used, there is still residual selection on unobserved characteristics that could exert an independent influence on labor-market outcomes. Hoekstra (2009) takes a unique approach to solving this identification prob- lem by using an admission cutoff rule at a large flagship university based on SAT scores and GPA. Students are assigned a score based on a combination of SAT and GPA, and admission rules use a cutoff for each SAT–GPA combination that form the basis of a regression discontinuity (RD) design. He shows that students attending the flag- ship because they score just above the cutoff earn 25 percent more than those who do not attend because they are just below. A com- plication with this study is that he does not observe what happened to students who do not attend the flagship, so the counterfactual is difficult to assess. However, in terms of affirmative action, this paper is relevant for two reasons. First, the effect is focused on the students at the margin of being academically Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 41 eligible for admission at a flagship university, and underrepresented minority students who are impacted by affirmative-action rules are more likely to be in this group. Second, the correct counterfactual when thinking about imposing stronger or weaker admis- sions requirements is what students would do if not admitted. Thus, Hoekstra’s results are applicable when attempting to pre- dict the policy effect on earnings of impos- ing higher admissions requirements for a marginally admitted group.75 However, an important caveat to his findings with respect to affirmative action is that he focuses only on white students; it is unclear whether one can generalize his findings to other racial and ethnic groups. The final method researchers have used to identify the effect of college quality on earnings is through a matching technique that compares earnings across students who attend different-quality universities but who applied to and were admitted to the same set of schools. Dale and Krueger (2002) were the first to implement this method using College and Beyond and NELS:88 data, and the motivation is that the student unobserv- able qualities one might be concerned about in the selection-on-observable studies are contained in the application and admission set. They show that once one conditions on the application and admissions sets, there is no evidence of an earnings return to col- lege quality. This finding is reconfirmed in a follow-up study (Dale and Krueger 2014), however both papers find sizable college quality effects for students from low-income families. To the extent underrepresented 75Zimmerman (2014) shows a similarly sized effect on earnings using an admission cutoff rule for a low-se- lectivity school in Florida. This finding suggests that earn- ings returns to college are not limited to the high-quality schools, but we focus this discussion on the flagship and selective schools because these are the schools whose minority enrollments are most affected by affirmative action. minority students are more likely to be from lower-incomefamilies,theseresultssuggest college quality may still be an important com- ponent of future earnings for this group. This methodology has several appealing attributes, most notably that it is comparing students who are very similar in terms of the types of schools they express a desire to attend and the types of schools that they are academi- cally eligible to attend.76 But this method also has generated controversy because of the concern over what is driving the residual selection. That is, why do two different stu- dents who have identical choice sets choose schools of different quality? Some of the potential explanations could be the types of unobservables this model is designed to over- come (e.g., differences in student motiva- tion). Or, there could be differences in match quality that are highly important in this con- text, driven by differences in schools’ focus in terms of academic area or other attributes students value. Finally, the differences in col- lege quality within matched sets, particularly at the top of the quality distribution, tend to be small. It could be the case that the effects of college quality are nonlinear in means, which could complicate comparisons across very similar schools. Our conclusion from this literature is that, on the whole, the evidence supports the claim that there are significant returns to college quality. Even Dale and Krueger (2002) show sizable effects for low-income students. The estimates are particularly large when examining the elite public schools, which typically are state flagship universities. Table 2 shows that these schools have sub- stantially higher resources and higher grad- uation rates, both overall and for minority students. It is reasonable to conclude that these higher resources allow these schools to 76Indeed, Dale and Krueger (2014) shows positive returns for minorities for the most recent cohort. However, similar positive findings are not found for the older cohort. 42 Journal of Economic Literature, Vol. LIV (March 2016) have a larger effect on human capital than lower-resource schools. A prediction that stems directly from this evidence and from the enrollment effects discussed in sec- tion 5.2 is that affirmative action likely raises the earnings of minority students due to its importance in driving minority enrollment in selective universities and the substantial earnings returns that tend to follow. At the same time, many of the results in this literature do not speak that closely to the question of how admitting lower-abil- ity minority applicants through affirmative action affects their future earnings. More broadly, this literature has not explored how the match between the school and the stu- dent affects the returns to college quality. This is an important hole in the returns-to- college-quality research and is thus a ripe area for future research. The methods used by some of the more prominent papers in this literature, namely Dale and Krueger (2002, 2014) and Hoekstra (2009), are par- ticularly ill-suited for this task. In the case of Dale and Krueger, due to the noisiness of college quality measures and the fact the students’ first and second choices do not differ in terms of measured quality by that much, any mismatch differences across first- and second-choice schools are likely to be small. With respect to Hoekstra (2009), his estimates are identified off of academically marginal students whose outside options are worse in terms of quality. While minority students are more likely to be academically marginal, the existence of affirmative-action policies makes their outside options far bet- ter than an observationally equivalent white student.77 These studies also generally focus on earnings not conditional on graduation.78 77 Hoekstra focuses only on white students in his analy- sis, so he cannot test this hypothesis. 78 An exception is Andrews, Li, and Lovenheim (forth- coming), which examines college quality earnings premi- ums among graduates. Since college quality overall has a positive effect on graduation rates, this may be the channel through which it influences earn- ings. But as we saw in section 5.3.1, higher graduation probabilities from attending a higher quality college may not result if the student is significantly underprepared, rela- tive to his peers. 5.6 Summary of the Evidence This section reviewed the large literature on the varied returns to college quality, as well as the evidence on how affirmative-ac- tion bans affect student outcomes. Given the quantity of research discussed, we briefly summarize the lessons learned from this research and highlight some remain- ing research questions. First, we find the research rather clear in showing that, on average, there is a positive return to college quality in terms of the likelihood of gradua- tion as well as on subsequent earnings. The identification strategies supporting these conclusions are all imperfect, but sizable effects have been found using a variety of methods, time periods, and data sources. The literature is much less clear about how the match between the school and stu- dents affects these outcomes more generally and how affirmative action impacts these outcomes specifically. The research using affirmative-action bans is certainly the most straightforward to interpret, and this work shows no effect of these bans on the like- lihood of graduation. This is a surprising result, because these studies demonstrate that affirmative-action bans significantly reduce college quality for African American and Hispanic students. These results can be reconciled by the changes in matching effects from the bans canceling out the col- lege-quality effects, though more evidence is needed. There is also a lack of understand- ing of how universities might respond to changes in affirmative-action policies. If uni- versities shift resources to underrepresented Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 43 minority students in order to compensate for externally imposed admission constraints, this could have sizable impacts on student outcomes. No research exists that examines institutional responses to affirmative-action bans, which is a notable gap in the literature. In addition, there is no work estimating how affirmative-action bans influence earnings. Understanding how these bans influence earnings may yield some insight into how affirmative action policies affect long-run outcomes, and this line of inquiry thus is ripe for future study. As noted above, graduation is an imper- fect measure of how affirmative action impacts students because they can adjust along other margins. Changing majors is one of the central ways in which students can compensate for a poor academic match. The evidence suggests that relatively less prepared students are likely to switch into less demanding majors. Furthermore, if these students had gone to slightly lower-quality schools, they would have had a better chance of completing the more dif- ficult major. The central question that arises from this research is what are the returns to majors versus school quality, and how do the returns to major and school combinations differ across different student types. It is extremely difficult to identify the returns to college majors and to major– school choice combinations.79 Nonetheless, Gemici and Wiswall (2014) suggest that choice of major has become more import- ant over time; this is an important area for future research. 79This is not the case in other countries where admis- sions to school–major combinations are sometimes a reflection of only one’s test score, implying regression dis- continuity methods may be employed. See, for example, Hastings, Neilson, and Zimmerman (2013) and Kirkeboen, Leuven, and Mogstad (2014). The usefulness of disconti- nuities is limited to local effects when matching effects are present, though. 6. Percent Plans as an Alternative to Affirmative Action In the wake of court decisions and voter referendums eliminating affirmative action in higher education (see table 1), states have been increasingly turning to “percent plans” in order to achieve diversity goals.80 Percent plans guarantee admission to a set of schools to students who are in the top X percent of their high school class. The idea behind such rules is that high schools tend to be ethnically and racially segregated, and so a percent plan will necessarily expand access to low-income minority students who are in heavily nonwhite schools. The most prominent percent plan is in Texas. The Texas Top 10 Percent Rule went in to effect in 1998, and it guaranteed all students in the top 10 percent of their high school class automatic admission to any public college or university in Texas. California also has an admission policy in which any student in the top 9 percent of their class or in the top 9 percent of students overall in the state are guaranteed admission to some University of California campus.81 80 This section examines percent plans as a method for states to achieve diversity goals in lieu of race-based affir- mative action. If the goal is instead to better the schooling outcomes of minority students, they have a much larger set of policies with which to work, including financial aid, income supports for low-income families, college coun- seling, information dissemination, and investments in minority students’ education at younger ages. There are large literatures examining the effects of each of these pol- icies that are beyond the scope of this review. 81The percent plan in California originally considered only those in the top 4 percent of their graduating class or in the top 12.5 percent of students in the state. These admission rules were changed for the incoming class of 2012. The within-school ranking in California is based on GPA, while the statewide rankings are based on an aca- demic index, which is a weighted average of SAT/ACT score and high-school GPA. Florida has a percent plan as well, called the Talented 20. Under this plan, which was enacted in 2000, the top 20 percent of students in each high school class who complete a college preparatory cur- riculum are guaranteed admission to at least one of the public universities in Florida, but in practice this program does not bind. 44 Journal of Economic Literature, Vol. LIV (March 2016) In this section, we provide a brief over- view of the evidence on how these percent plans affect student outcomes in terms of enrollment and graduation, and we discuss evidence on some of the unintended con- sequences of these rules that policymakers need to consider when deciding on whether to replace existing affirmative-action poli- cies with a percent plan. Most of the evi- dence comes from studying the Texas Top 10 Percent Rule, so we focus our discussion there. Kain, O’Brien, and Jargowsky (2005) examine eight cohorts of college students in Texas that span the preban period, the two years the ban was in effect, and the Top 10 Percent Rule implementation period. They show that the enrollment declines among African Americans at UT Austin and Texas A&M post-Hopwood discussed in section 5.2 were partially reversed after the Top 10 Percent Rule was implemented. However, black enrollment at the flagship schools still remained 14 percent below the preban levels. Among Hispanic students, enrollments at UT Austin fully recovered, relative to preban levels, and at Texas A&M it remained at about 20 percent below the preban level. Overall, the evidence sug- gests that the Texas Top Ten Percent Rule went some way to undoing the declines in minority enrollment at Texas flagship uni- versities after affirmative action was banned, but the recovery was not full, especially for African Americans. The enrollment pat- terns also show a large increase in white enrollment at these three schools, jumping 27 percent after the implementation of the Top 10 Percent Rule. Long (2004b) argues that enrollment trends are misleading because they are con- founded by growing minority populations within the state. He performs simulations of enrollment probabilities using the NELS:88 survey and shows that percent plans are unlikely to affect minority enrollment for two reasons. First, most minority students in the top decile of their class are admitted to the state flagship anyway. Second, under- represented minority students are not con- centrated enough in most areas and they are unlikely to be in the top portion of their high schools for this policy to have large effects on their enrollment patterns. A source of variation that is increasingly being exploited to identify the effects of per- cent plans on students is the discontinuity in access to flagship universities that occurs just above the rank cutoff. This methodology is not subject to the criticisms outlined in Long (2004b), but they do have the drawback that the estimates are only local to the specific cutoff. Thus, these studies do not tell us the overall effect of percent plans on the com- position of the student body; they can only inform how students respond when given admission guarantees to flagship schools. They are informative about how students respond to admission guarantees, but the estimates are too local to tell us much about the overall impacts of percent plans on the state postsecondary system (or on flagship schools, per se). There are two studies that have used regression discontinuity methods to study the Texas Top 10 Percent Rule, and they come to somewhat different conclusions. Niu and Tienda (2010) use data from the Texas Higher Education Opportunity Project (THEOP), a state-representative, longitudinal survey of Texas public high-school students that began in 2002. Class ranks were verified using tran- scripts among those who persist in the survey to the second follow-up. The main finding is a 12–14 percentage point increase in the likelihood of Hispanic students who are just over the 10 percent rank cutoff enrolling in one of the two Texas flagships. Students at predominantly minority schools also experi- ence a jump of 14 percentage points in the likelihood of enrolling in a flagship at the cut- off. White students and students in majority white schools are unaffected, as are African Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 45 American students. This evidence is consis- tent with the tabulations in Kain, O’Brien, and Jargowsky (2005), which shows Hispanic enrollment at Texas flagships increased sub- stantially after the Top Ten Percent rule was implemented, relative to enrollment in the ban years. Daugherty, Martorell, and McFarlin (2014) examine how the cutoff not only affects whether students attend the flagship school, but also how it affects attendance patterns at other schools. They match six cohorts of students to a large, unnamed dis- trict in Texas and estimate regression dis- continuity models using administrative data on class rank that is linked to college enroll- ment outcomes both within and outside of Texas. They show that, while being in the top 10 percent increases the likelihood students enroll in a state flagship, it displaces enroll- ment in private and out-of-state universities. As a result, students just above and below the top 10 percent cutoff experience the same level of college quality.82 Interestingly, they also find no effect on the likelihood of enrollment, even though the baseline enroll- ment rate in college is only 30 percent in the study’s district (and is only 58 percent among those in the top decile of class rank). This finding highlights the fact that percent plans influence where students attend, not whether they attend at all, similar to affirma- tive-action policies more generally. The findings of this paper are relevant for affirmative-action policy, as they suggest that expanding admission preferences at any one school (or among a small number of schools) may simply re-sort students among similar-quality institutions. This is perhaps an explanation for the lack of graduation 82 Daugherty, Martorell, and McFarlin (2014) also high- light that they use class rank at the end of eleventh grade, which is the relevant rank for college applications, while Niu and Tienda (2010) use the rank at the end of twelfth grade. They show evidence that this difference can account for the different findings across the two studies. effects found in the studies examining affir- mative-action bans discussed in section 5.3.3. Furthermore, this study seemingly con- flicts with the findings in Hoekstra (2009). Hoekstra finds large effects of attending the state flagship on earnings using an admission discontinuity. While a full understanding of why the findings differ is difficult (and is a ripe area for further research), the expla- nation likely is related to the fact that the students in the Daugherty, Martorell, and McFarlin (2014) study are not marginal for admission to the state flagship in terms of academic quality. Indeed, it was highly likely they would have been admitted prior to the Top 10 Percent Rule. In the Hoekstra study, the students who are barely admitted are marginal for enrollment by design, so their alternatives are very likely to be low- er-quality institutions. This difference high- lights the limited applicability of the results in Hoekstra (2009) for affirmative action, as most underrepresented minority students admitted under affirmative action will have access to an array of similarly competitive schools. In addition to the direct effects on which types of schools students attend, percent plans can have general equilibrium effects that often are not intended by policymakers. A prime example is found in Cullen, Long, and Reback (2013), who show that a small but nontrivial proportion of student transfers across high schools after the Top 10 Percent Rule was implemented were to improve their chances of being in the top 10 percent. Not only does this strategic behavior reduce the high school quality students experience, it also can undo the effectiveness of the per- cent plan by displacing minority students with nonminority transfer students. Long, Saenz, and Tienda (2010) show that the Top 10 Percent Rule expanded access to flagships geographically, with a large drop in the pro- portion of minority students from “feeder” schools and a higher percentage coming from 46 Journal of Economic Literature, Vol. LIV (March 2016) rural towns and small cities. Thus, the per- cent plan in Texas altered the composition of minoritystudents,theeffectsofwhichonthe academic attainment of these students or on their peers at flagship universities currently is not known. In addition, percent plans can alter grading incentives placed on schools and/or specific teachers, can alter the distri- bution of resources in schools to students of different ability levels, and can affect student effort incentives for those close to the per- centile cutoff. The existence and extent of these effects on students and schools has not been studied previously, and we view these as important questions for future research. 7. Conclusion Intense debate continues in the United States over the role of affirmative action in college and graduate-school admissions. Although the Supreme Court has ruled that the constitutional basis for affirmative action must be rooted in schools’ prefer- ences for a diverse student body, the origi- nal impetus for these policies and much of the resulting focus among policymakers and researchers alike is on whether and how affirmative action “levels the playing field” between underrepresented minority and white students. Of central importance, then, is how racial preferences in college and graduate school admissions influence stu- dents’ long-run educational and labor-mar- ket outcomes, relative to a counterfactual of race-neutral admissions. From a policy perspective, understanding the extent and effect of affirmative-action policies is of primary importance. The liter- ature clearly shows positive average effects of college quality on a host of outcomes. This suggests that mild racial preferences will have a positive impact on minority out- comes. The issue is whether racial prefer- ences in their current form are so strong that mismatch effects may arise. At more selective undergraduate schools and among virtually all law schools, affirmative action is practiced extensively, particularly for African Americans, and leads to substantial differences in the academic backgrounds of minority and majority students. These differ- ences are particularly striking in law schools, where the median black admit may have cre- dentials that are below that of the first per- centile of white admits. As we argue throughout this paper, the net impact of racial preferences comes down to whether the strength of the overall college quality effect is larger or smaller than any match effects. If matching effects are dom- inant given the current levels of racial pref- erences, then the dual goals of affirmative action of generating a diverse student body and supporting the educational attainment of minority students may be at odds with one another. Large matching effects would sup- port the need for more policy consideration of what the optimal amount of affirmative action is at different school types in order to balance the negative matching effects with institutional diversity goals. Conversely, absent matching effects, these policy goals can be complementary. The evidence suggests that racial prefer- ences are so aggressive that reshuffling some African American students to less selective schools would improve some outcomes due to match effects dominating quality effects. The existing evidence indicates that such match effects may be particularly rel- evant for first-time bar passage and among undergraduates majoring in STEM fields. However, shifting minority undergradu- ates to low-resource, nonselective schools ultimately may undo any gains from higher match quality, and shifting minorities out of law schools altogether could lead to worse labor market outcomes among these stu- dents than had they been admitted to some law school. Alternatively, schools that wish to practice extensive affirmative action could Arcidianco and Lovenheim: Affirmative Action and the Quality–Fit Trade-Off 47 provide targeted services to these students in order to overcome any mismatch induced by their admission policies, such as offering tutoring and remedial classes. While the evi- dence on targeted college services is scant,83 it is plausible such interventions could be successful in mitigating any negative match effects. The extent to which schools can suc- cessfully target services to less academically prepared students is an important question for future research. The past several decades have witnessed a revolution in data availability, with many states, as well as the federal government, making data available to researchers that per- mit one to follow students from their K–12 education through college and into the work- force. These datasets have immense poten- tial to inform most education policy debates, and using these data to study the effects of affirmative action on long-run outcomes would be quite fruitful. This paper outlines many of the gaps in our current knowledge and the difficulty in credibly disentangling the relevance of fit versus mismatch as they relate to affirmative action. We are hopeful that these longitudinal administrative data- sets will allow us to gain new insights into the effects of affirmative action on minority educational and labor market outcomes in the near future. Administrative data would also permit more exploration of how racial preferences in college admissions affect other margins, such as precollege investment decisions. If, in equilibrium, affirmative action leads employ- ers to believe minority students graduating 83The existing research has suffered from identifica- tion issues surrounding student selection into the col- lege services being offered. Abrams and Jernigan (1984) and Chaney et al. (1998) find positive effects of targeted services to lower-performing students. However, neither study can credibly separate student selection into the pro- gram from the effects of the program on student outcomes themselves, and neither focuses on students impacted by affirmative action. from more elite schools are lower produc- tivity, affirmative action may induce lower minority effort (Coate and Loury 1993). Conversely, affirmative action bans could lower minority human capital investment by making it less likely they will be admitted to higher-quality colleges that may have higher returns. 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question:The Efficiency of Race-Neutral Alternatives to Race-Based Affirmative Action: Evidence from Chicago’s Exam Schools† By Glenn Ellison and Parag A. Pathak* Several K-12 and university systems have adopted race-neutral affirmative action in place of race-based alternatives. This paper explores whether these plans are effective substitutes for racial quotas in Chicago Public Schools (CPS), which now employs a race-neutral, place-based affirmative action system at its selective exam high schools. The CPS plan is ineffective compared to plans that explicitly consider race: about three-quarters of the reduction in average entrance scores at the top schools could have been avoided with the same level of racial diversity. Moreover, the CPS plan is less effective at adding low-income students than was the previous system of racial quotas. We develop a theoretical framework that motivates quantifying the inefficiency of race-neutral policies based on the distortion in student preparedness they create for a given level of diversity and use it to evaluate several alternatives. The CPS plan can be improved in several ways, but no race-neutral policy restores minority representation to prior levels without substantially greater distortions, implying significant efficiency costs from prohibitions on the explicit use of race. (JEL H75, I21, I28, J15) Affirmative action is one of the most contentious issues in American public policy, particularly in school admissions. In the 1970s, race-based plans were widespread in K-12 and university admissions. Since then, such plans have been challenged on multiple fronts. Some states have banned race-based affirmative action.1 In 2003, the US Supreme Court established a standard of “strict scrutiny” for race-based plans at public institutions, requiring that they must serve a compelling government * Ellison: Department of Economics, Massachusetts Institute of Technology, and NBER (email: gellison@mit. edu); Pathak: Department of Economics, Massachusetts Institute of Technology, and NBER (email: ppathak@mit. edu). Debraj Ray was the coeditor for this article. We thank Katie Ellis, Susan Ryan, and the staff at Chicago Public Schools for their expertise and help with the data. Vivek Bhattacharya, Adrian Blattner, Jackie Bredenberg, Brandon Enriquez, Raymond Han, Alex Olssen, and Rahul Singh provided excellent research assistance. We’re grateful to seminar participants at CESifo, Harvard Law, MIT, and Tufts for comments. Ellison acknowledges support from the Toulouse Network for Information Technology. Pathak acknowledges support from the William T. Grant Foundation and National Science Foundation grants SES-1056325 and SES-1426566. This study was approved by MIT’s committee on human subjects as IRB 1506128499 on July 15, 2015. †Go to https://doi.org/10.1257/aer.20161290 to visit the article page for additional materials and author disclosure statements. 1 California and Michigan passed constitutional amendments by ballot in 1996 and 2006, respectively. 943 944 THE AMERICAN ECONOMIC REVIEW MARCH 2021 interest that cannot be effectively achieved in a race-neutral manner.2 Many public institutions have adopted alternative plans that avoid explicitly using race. We explore the consequences of the shift from race-based to race-neutral affir- mative action at Chicago Public Schools (CPS). From 1980–2009, the assignment process for Chicago’s exam high schools involved racial quotas that restricted White enrollment. In 2010, CPS shifted to a race-neutral system that targets socioeco- nomic integration via a neighborhood-based approach. In the system, admissions preferences are based on the socioeconomic characteristics of the census tract in which a student resides. Each tract is placed into one of four tiers. Within each tier, the tie-breaker is based on a composite score which combines results of a special- ized entrance exam, prior standardized test scores, and grades in prior coursework. Students rank schools and a deferred acceptance algorithm generates placements, factoring in both tier and tie-breaker.3 We focus on how this change has affected the minority and low-income composition, and curriculum matching for the incom- ing class at the two most selective exam schools, Walter Payton College Prep and Northside College Prep. Chicago’s race-neutral plan has substantially increased the spread of composite scores at Payton and Northside. This is driven by two forces: the CPS plan admits students who have much lower composite scores than any student who would have been admitted under a race-based plan, and the CPS plan fails to admit many high achieving low-income and minority students. Even though the plan aspires to broaden the definition of disadvantaged students, the CPS plan admits fewer low-income students than would a pure racial quota with a comparable effect on the distribution of scores of admitted students. Moreover, within-school achievement gaps between majority and minority students widen compared to racial quotas. A race-neutral policy remains far away from the optimal admissions plan for two reasons. First, the policy may not have been chosen well. We show that CPS’s plan can be substantially improved with some relatively minor changes. By reweighting and transforming the CPS tract-level socioeconomic measures, we can improve on the performance of the plan. Further improvements are possible if we use appli- cant’s free or reduced price lunch (FRPL) status.4 Second, the policy is bounded away from the optimal frontier because of the con- straint that it is race-neutral. Our most important conclusion is that a substantial portion of the minority inefficiency of the CPS plan appears to be an inevitable consequence of a restriction on using race. We draw this conclusion from an exploration of more complex plans constructed using machine learning techniques. Admissions policies which may be beyond a legally permissible use of race cannot achieve much more than what our simple improvements to CPS’s tract-level variables achieve. Possibilities for race-neutral plans differ by school and are even more limited at Northside. We anchor our empirical investigation with a model of elite schools. In the model, students are assumed to benefit from a curriculum tailored to their needs and from learning within a diverse student body. Affirmative action plans can 2 A June 2007 Supreme Court decision applied the earlier decision to strike down race-based admissions plans for public schools in Seattle and Jefferson County, Kentucky. 3 Dur, Pathak, and Sönmez (2020) describe the assignment mechanism in more detail. 4 CPS chose not to use individual-level data in part due to concerns about whether the data would be reliable if used. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 945 benefit students by increasing diversity, although this comes at a cost of reducing curriculum-matching benefits. We propose that the efficiency of plans be assessed by measuring how close a plan is to minimizing the adverse effects on the dis- tribution of student-preparedness conditional on the level of diversity. Our over- all efficiency measure compares a given plan with other plans that simultaneously achieve the same level of minority and low-income representation. Our minority and low-income efficiency measures reflect how efficiently a plan accomplishes diversity on each dimension separately. Our efficiency measures involve subjective decisions about which adverse effects to consider and how to cardinalize them. We have tried to be conservative in labeling plans as inefficient, both relative to a utility-based model that motivates the measure and in ignoring additional disadvantages of the CPS plan, e.g., its contribution to stereotype reinforcement. Under purely score-based admissions, Payton and Northside would have few low-income and minority students. Race-based or free or reduced price lunch (FRPL)-based affirmative action plans can substantially increase minority and/or low-income representation with a modest effect on the average composite score of admitted students. However, according to our metric, the CPS race-neutral plan is 28 percent and 22 percent efficient at Payton and Northside compared to the race- and FRPL-based benchmark. Focusing more narrowly on the CPS plan as a means to boost minority enrollment, the CPS plan is only about 26 percent and 19 percent efficient at Payton and Northside. Focusing narrowly on the plan as a means to boost low-income enrollment, it is even more inefficient. By making fairly straightforward modifications to how CPS measures socioeconomic status and how SES is used in the assignment mechanism, it is possible to improve overall efficiency to 44 percent and 39 percent at Payton and Northside. But this is about all that can be done: a more involved change (that may not survive legal challenges) results in a modest additional efficiency improvement. The model we use to motivate our approach builds on several ideas in the previ- ous literature. In particular, our model of curriculum-matching and school system design generates a trade-off between score maximization and diversity studied in other papers. The effect of CPS’s plan on the distribution of admitted students ver- ifies the practical relevance of Chan and Eyster’s (2003) observation that optimal race-neutral plans may expand the lower-tail of the admitted student distribution by rejecting some very highly qualified minorities and admitting some majority candidates who would have not been admitted under a race-based plan. Our find- ing that within-school achievement gaps are larger under the CPS plan than under a race-based plan empirically confirms a phenomenon that Ray and Sethi (2010) noted was a necessary feature of the constrained optimal plan, but which might have been regarded as counterintuitive. Finally, our approach to developing an upper bound on what may be achieved, exploring what can be done with plans that probably go beyond what would be legally permissible, follows Fryer and Loury’s (2005) admonition that too-efficient forecasts of race are not race-blind in a meaningful sense. The most closely related empirical antecedent to our paper is Fryer, Loury, and Yuret (2008). We follow their main approach of comparing the effects that race-based and race-blind affirmative action procedures would have on class com- position via counterfactual simulations. Importantly, however, we overcome the 946 THE AMERICAN ECONOMIC REVIEW MARCH 2021 limitations of their dataset, which only contains information on admitted students and not unsuccessful applicants, making it challenging to assess how policies would work in practice. Cestau, Epple, and Sieg (2017) develop an econometric model of the referral process for taking the admissions tests for selective elementary schools. They report that profiling by race and income together with affirmative action based on FRPL status can achieve 80 percent of level of Black enrollment as a race-based affirmative action plan. Corcoran and Baker-Smith (2018) study admissions poli- cies at New York’s exam schools, focusing on a descriptive account of application decisions. Though their main interest is not in affirmative action, they simulate top 10 percent rules based on seventh grade scores and find that such policies lead to an increase in Black and Hispanic representation at the schools examined. Epple, Romano, and Sieg (2008) develop an equilibrium model of affirmative action and tuition policies to show that a ban on affirmative action leads to a decline in minority students at top-tier colleges. The rest of this paper is organized as follows. Section I describes Chicago’s exam schools and the new race-neutral assignment policy. Section II computes the trade-off between diversity and selectivity at Payton and Northside. It also reports on racial, income, and achievement gaps under the CPS policy compared to a race and FRPL-based benchmark. Section III develops a model of optimal admissions with affirmative action to motivate our measure of the relative efficiency admissions plans. Section IV computes the relative efficiency of Chicago’s plan. Section V stud- ies the extent that simple or larger-scale changes to Chicago’s plan could increase its efficiency. Section VI concludes. I. Chicago’s Exam Schools A. Schools and Admissions Process In 2014, Chicago Public Schools operated ten “selective enrollment” or exam high schools, listed in Table 1.5 Each school offers students advanced curricula and a high-achieving peer group compared to most CPS schools. Payton and Northside are the most selective and highest-performing. Table 1 shows that Payton and Northside are also fairly small: 233 and 265 students enroll respectively, compared to total eighth grade CPS enrollment of about 28,000. Jones and Whitney Young are the next most selective; their average ACT scores are at the eighty-sixth and eighty-ninth per- centile, respectively. Performance at the other exam schools is substantially lower. To obtain an exam school seat, applicants must take an admissions test and sub- mit a ranking of up to six schools. Students who are competitive for Northside or Payton will be sufficiently far above the cutoff for the bottom half of the schools so that they can be certain that they will be admitted to any of them, and should there- fore list their top six choices in their true preference order.6 This fact motivates our 5 We omit South Shore College Prep because it only became a selective enrollment school in 2013–2014 so statistics on 2014 graduates would not reflect the current admissions procedure and curriculum. 6 Haeringer and Klijn (2009) and Pathak and Sönmez (2013) also show that for students who prefer less than six schools, it is weakly dominant strategy to rank schools truthfully. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 947 Table 1—Chicago’s Exam High Schools Northside College Prep Walter Payton College Prep Whitney Young Magnet Jones College Prep Lane Technical Brooks College Prep Lindblom Math and Sci Acad Westinghouse King College Prep Mean ACT score (percentile) (1) 30.1 (94) 30.2 (94) 28.3 (89) 26.8 (86) 24.7 (78) 23.5 (74) 22.8 (69) 21.4 (57) 20.5 (57) Average number of AP exams passed (2) 4.68 5.20 3.04 2.25 2.13 0.81 1.34 0.79 0.25 Average AP score (3) 3.83 3.74 3.20 3.18 2.91 1.97 2.42 2.01 1.80 Enrolled students (4) 265 233 468 427 925 199 228 278 148 Notes: ACT Percentiles are for rounded average ACT scores by school, and based on national distributions for high school graduates from 2015, 2016, and 2017 (ACT 2020). ACT data was received from the Prairie State Achievement Examination (ISBE 2014). Average number of AP exams passed refers to the average number of AP exams ever passed (score 3 or higher) for the cohort of seniors in 2014. Average AP score refers to the average of the mean score achieved by the same cohort. Enrolled students refers to the number of ninth grade students in 2014. assumption that rankings for these schools for students correspond to true prefer- ences and would not change under the alternate admissions policies we consider.7 The CPS assignment procedure starts by assigning each student a “composite score,” which we rescale from 0 to 100. The composite score places equal weight on three factors: the entrance exam score, the student’s seventh grade score on Illinois’ statewide standardized test, and the student’s seventh-grade grade-point average. The composite score can roughly be thought of as corresponding to a student’s national percentile. For example, an applicant would receive a score of 98.9 if she achieved a score CPS deemed to be at the national ninety-eighth percentile on their admissions test, had ninety-seventh percentile scores in both English and Math on the ISAT, and had a perfect middle school GPA.8 Each student is also assigned to one of four SES tiers. The SES tiers are determined by a place-based affirmative action scheme. Student addresses are mapped to each of about 800 census tracts. Six characteristics of each census tract are then used to construct the SES score for the tract: (i) median family income, (ii) percentage of single-parent households, (iii) percentage of house- holds where English is not the first language, (iv) percentage of homes occupied by the homeowner, (v) adult educational attainment, and (vi) average Illinois Standards Achievement Test (ISAT) scores for attendance-area schools.9 Tracts are ranked by these scores, and then divided into four tiers, each with approximately the same num- ber of school-age children. Tier 1 tracts are the most disadvantaged, while Tier 4 tracts are the most advantaged. Panel A of Figure 1 illustrates the distribution of tiers 7 Student preferences over schools could change if changes to the admissions procedure made large enough changes to the achievement levels or demographics of the admitted students. There is also evidence from lab exper- iments that participants do not submit their preferences truthfully in strategy-proof mechanisms (see, e.g., Chen and Sönmez 2006 and Li 2017). Students might also move in response to a change in the admissions process. Cullen, Long, and Reback (2013) document that this has occurred in Texas after Texas adopted a school-based admissions procedure for college. The online Appendix shows that such behavior seems to be quite rare in our context. 8 The percentiles are not calibrated to the same population, and a full 100 GPA points are given to any student with straight As in seventh grade so the interpretation of scores as percentiles is not precise. 9 The SES score is the average of a tract’s percentile rank within the city of Chicago on the six dimensions. 948 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Panel A. Tiers Panel B. Payton admittees Panel C. Northside admittees Figure 1. Chicago Tiers and Admitted Students at Payton and Northside Notes: Panel A shows the geographical distribution of tiers across tracts. Panels B and C show the number of appli- cants admitted to Payton and Northside, respectively, by tract. Tier 1 Tier 2 Tier 3 Tier 4 1 Admittee 2 Admittees 3 Admittees 4 Admittees None throughout the city. Many of the highest-tier census tracts are at the northern or west- ern edges of the city or along Lake Michigan north of downtown. The CPS assignment mechanism is based on deferred acceptance (DA). In the system, each school is divided into five sub-schools. Merit seats, representing 30 per- cent of the seats, are open to all students regardless of their SES tier. Tier seats, each with 17.5 percent of the available seats, prioritize applicants from the corresponding tier, using the composite score to break ties. In the algorithm, applicants who rank a school are first considered for merit seats and then for the seats reserved for their tier. Tier seats are almost exclusively assigned to applicants from the corresponding tier.10 An unmatched student is defaulted to their neighborhood school.11 An unmatched student also could matriculate at other choice options including career and technical academies, magnet schools, military academies, and charter schools. One feature of Chicago that makes it attractive for studying affirmative action is that the two most selective exam schools are differently situated, so we can study how race-neutral affirmative action plans fare in different environments. Figure 1 shows the schools’ locations and where their students live. Payton is closer to the center of Chicago and enrolls students from across the city. Northside is a relatively attractive location for students from the northern parts of the city which consist of primarily tier 3 and tier 4 neighborhoods. For this reason, there may be greater 10 Dur, Pathak, and Sönmez (2020) provide more details on Chicago’s DA implementation with merit and tier seats. 11 A small number of students may also be admitted to each school under special education and “principal’s dis- cretion” programs. We have no way of knowing if any students would have been admitted via these programs under alternative admissions policies, so in all of our comparisons we assume that under the current policies the school consists only of those students selected by the main deferred acceptance algorithm. All statistics on the admitted classes at Payton and Northside refer to students admitted through the main admissions process, and therefore will not exactly match other measures of the schools’ demographics computed directly from the set of enrolled students. 1 Admittee 2 Admittees 3 Admittees 4 Admittees 6 Admittees 7 Admittees 8 Admittees None VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 949 efficiency costs at Northside if an affirmative action plan fails to offer admission to the low-income and/or minority students living in these neighborhoods. B. Data Our primary data consist of application files from the Chicago Public Schools for the 2013–2014 year and a separate file containing the factors underlying the tier formula for each census tract (CPS 2014). Each record includes students’ composite scores and underlying components, school choices, tier, and final assignment.12 We augment this file with data on the census tract factors used to construct tiers from Eder and Gregg (2014) and additional information on tract-level characteristics from IPUMS (Manson et al. 2010). Our analysis sample has 16,818 students and 77,051 student choices. The average applicant ranks about 4.6 schools. Applicants who rank all six choices are more likely to be Black, to qualify for a subsidized lunch, and have lower average composite scores.13 Applicants to exam schools are positively selected compared to CPS eighth grad- ers. Table 2 shows that the standardized test scores of applicants (measured as per- centiles) are well above average. Exam applicants are also less likely to qualify for a subsidized lunch than a typical CPS eighth grader. Compared to the broader pool of exam applicants, applicants who rank Payton and Northside are somewhat more positively selected as shown in columns 3 and 6 of Table 2. Northside has fewer Black and more Hispanic applicants compared to Payton. Race-neutral admissions plans will by necessity either admit fewer minority stu- dents or reduce average composite scores at the most selective schools compared to race-based plans. Column 4 reports characteristics of students admitted under the current CPS procedure, while column 5 reports on students who would have been admitted had CPS used racial quotas to maintain the same level of Black and Hispanic representation at Payton and Northside as under the previous race-based process.14 At Payton, the shift to a race-neutral plan has cut the fraction of Black admits nearly in half, from 28 percent to 15 percent. Average composite scores are unchanged, and there have been only small reductions in the fraction of Hispanic students and the fraction of students qualifying for free or reduced price lunch (FRPL). At Northside, the CPS plan increased minority and FRPL enrollment and decreased average composite scores.15 Across both schools, 124 of the students 12 The application files also indicate whether a student qualifies for special education programming; because those applicants are assigned through a different process, we exclude them from our analysis sample. 13 The fact that an applicant does not rank all schools does not necessarily imply that she has not revealed her preferences truthfully. Rather it could indicate that any unlisted choice is less preferred to the applicant’s outside option. For high-scoring applicants, the fraction of applicants who do not rank all six choices increases with the composite score. Many such high-scoring applicants would likely opt for schools outside of the public district if not admitted to the most selective exam schools. 14 We implemented racial quotas at Payton and Northside by setting separate Black, Hispanic, and White/Asian quotas to be equal to the average share of Black, Hispanic, and White/Asian students from the three years prior to the adoption of race-neutral admissions. The number of offered seats is not exactly the same as the number of students who enroll in Table 1 because of new offers made after the initial round. 15 The share of Black and Hispanic students was much lower at Northside than at Payton under the former race-based policy. This was a side effect of the fact that the 1980 consent decree required that no more than 35 percent of a school be White. Northside had a substantially larger Asian enrollment than Payton and by 2008, Asian students took up many of the non-White seats at Northside. 950 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Table 2—Descriptive Statistics on Applicants Offered Payton Racial Quota (5) 0.28 0.24 0.67 0.26 98.0 99.3 97.5 97.1 Offered Northside Black Hispanic Female FRPL Composite score GPA Standard tests Admissions test Observations (2) (3) (4) 0.39 0.37 0.15 (7) (8) 0.06 0.05 CPS 8th graders (1) 0.42 0.45 0.50 0.86 27,944 Exam Applicants Current applicants Payton CPS Applicants Northside (6) 0.26 0.50 0.57 0.72 66.87 67.39 78.68 54.56 8,274 Current CPS Racial quota 0.42 0.44 0.22 0.55 0.55 0.65 0.74 0.71 0.25 62.17 65.89 97.99 61.82 65.67 99.36 75.86 78.30 97.61 48.82 53.70 97.01 16,818 10,549 220 0.29 0.23 0.65 0.65 0.34 0.24 97.75 99.16 97.49 96.58 98.93 99.90 98.24 98.65 259 259 Notes: This table shows descriptive statistics for the CPS students enrolled in eighth grade and applicants to exam schools in our analysis sample. Current CPS refers to the current tier-based admission scheme and GPA to seventh grade average grades. admitted under the new policy, or roughly one-quarter of admitted students, would not have been admitted to the same school under the racial quota policy we simulate. C. The CPS Composite Score The CPS composite score is closely related to curricular offerings and educa- tional outcomes at schools. The relationship between an applicant’s composite score and an exam school’s distribution therefore provides a measure of whether the stu- dent is suited for the exam school’s curriculum. Advanced Placement (AP) courses are advanced college-level curricula offered to high school students who may obtain placement or credit for high enough scores. Exam schools offer AP courses, and more selective schools are able to offer more such courses and teach them at a more advanced level. Panel A of Figure 2 examines the relationship between entering students’ composite scores and AP test performance, where AP test performance data is from CPS (2014). To measure performance, we define AP Performance as the number of points in excess of two that students achieve on all of the AP tests that they take in 11th or 12th grade.16 We then plot average AP Performance against student composite score, for students with scores greater than 85. Recall that we have scaled the CPS composite score to run from 0 to 100, so this represents roughly the top 15 percent of students. AP Performance measures both the relationship between student ability and AP test performance and course offerings at any given exam school. At less selective exam schools, students have both lower baseline scores and tend to have less access to AP courses. Students with composite scores below 90 typically pass few AP tests. Expected AP performance increases sharply among students with composite scores greater than 95. In particular, students 16 Specifically, we give them no credit for tests on which they score 1 or 2, one point for each test on which they score 3, two points for each score of 4, and three points for each score of 5. The motivation for this transformation rather than simply adding all AP scores is that scores of 1 or 2 are generally regarded as failing scores. We do not want results to be affected by differences in whether students did or did not take tests on which they would receive such scores. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 951 Composite score and AP results 15 10 5 0 0.5 0.4 0.3 0.2 0.1 0 Figure 2. Relationship between CPS Composite Scores and AP Performance and Scores and Admission to the Most Competitive Colleges Notes: The sum of AP Scores is the sum of the AP scores ever achieved by a student, where only scores in the range of 3–5 are considered, and rescaled to 1–3. Most competitive colleges are defined as in the 2009 Barron’s Profile of American Colleges. with the highest composite scores have expected AP Performance of 14, those with composite scores of 99 have expected AP Performance of about 11, and those at 98 have about 8. Therefore, if a student with a composite score in the low 90s or below replaces one with a composite score in very high 90s, we anticipate a substantial efficiency loss due to curriculum matching. 85 90 95 100 Composite score Composite score and enrollment in “most competitive” college 85 90 95 100 Composite score Proportion enrolled Average sum of AP scores 952 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Suggestive evidence of curriculum mismatch from assigning lower scoring stu- dents to Payton or Northside is also present when we compare composite scores to college-going. Panel B of Figure 2 plots the relationship between the compos- ite scores and whether students go on to attend a “most competitive” college after graduation.17 As with AP Performance, the predictive power of composite score in the upper tail is striking: more than 40 percent of students with the highest possible composite score attend a most competitive college, while only about 5 percent of students with composite scores between 85 and 90 do. The strength of the relationship between composite score and AP Performance and college-going motivates treating the composite score as a proxy for the curricu- lum to which students are matched. When we examine how affirmative action plans affect within-school standard deviation in the composite score, Figure 2 can serve as a reference to translate the cardinal scale of composite scores into units of AP and college outcomes. II. Trade-Offs between Diversity and Selectivity A. Chicago’s Frontiers We begin our analysis by computing the trade-off between diversity and selectiv- ity at Payton and Northside. Figure 3 plots the frontier at both schools, as well as a single point (marked with a triangle) corresponding to the class composition if CPS were to adopt a purely score-based admissions procedure. Under this policy, the average composite score of admitted students would be 99.1 at Payton and 99.0 at Northside. Since these means are near the upper bound score of 100, there is obvi- ously little within-school variation in the composite scores. The majority of students would have composite scores in the 99 to 100 range and all students would have scores of at least 97. The position of these points relative to the x- and y-axes show that the schools would also not be very diverse under purely score-based admis- sions. The x-axis position reflects that only 21 percent of the students at Payton and 19 percent of the students at Northside would be Black or Hispanic, hereafter underrepresented minorities. The y-axis positions reflect that only 15 percent of the students at Payton and 23 percent of the students at Northside would be sufficiently low income to qualify for FRPL. The three curves in the figure correspond to admissions policies where some admitted under score-based admissions are replaced with lower-scoring minority and/or low-income students.18 The innermost curve contemplates a small change in class composition, where the average composite score is 98.8. Such plans can have a nontrivial effect on minority and low-income representation even though only a limited number of offers change. The other two curves illustrate admis- sions policies which result in larger changes in the composite score, reducing 17 We define “most competitive” using the 2009 Barron’s Profile of American Colleges, provided to us by David Deming in electronic form. 18 To compute these curves, we add bonus points to an applicant’s score if she is minority or low-income. With these adjusted scores, we re-compute the outcome of the deferred acceptance algorithm for a grid of bonus points. The plotted curves correspond to minority and low-income percentages which correspond to the given average composite score. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 953 0.55 0.45 0.35 0.25 0.15 0.15 0.55 99.10 98.79 98.80 98.43 98.40 98.38 0.45 Payton Northside 98.02 98.04 98.07 98.11 98.41 98.40 98.12 98.81 98.81 98.79 98.80 97.96 98.80 0.25 0.35 Minority share 0.15 0.25 Minority share 97.97 0.45 98.40 98.41 98.42 98.40 98.10 98.36 98.39 97.98 97.96 98.79 98.80 97.99 98.82 98.96 98.01 97.97 97.97 98.42 98.42 98.39 0.35 0.25 0.15 98.80 98.80 98.80 98.79 97.75 98.00 0.55 0.45 0.55 0.35 Figure 3. Levels of Minority and Low-Income Representation That Can Be Simultaneously Achieved at Various Score Levels Notes: This figure plots the frontiers for Payton and Northside for several levels of the composite score. The trian- gle corresponds to a purely score-based admission scheme. The black diamond corresponds to the CPS scheme. The green, red and blue curves represent levels of minority and FRPL representation that can be achieved, while holding the composite score fixed at 98.8, 98.4, and 98, respectively. the average to 98.4 and 98.0, respectively. These policies make larger changes to student demographics feasible. The shape of the frontiers demonstrates three main findings. First, absent restric- tions on the form of affirmative action, it is possible to make substantial changes in school demographics with only modest changes in the distribution of admitted students’ scores. Looking at the endpoints of the innermost curve for Payton, we see that changes that keep the average score at least 98.8 can increase minority representation from 29 percent to 37 percent or increase the share of FRPL students from 20 percent to 30 percent. The pattern is similar at Northside: it is possible to increase minority representation from 23 percent to 33 percent or increase the FRPL share from 26 percent to 37 percent, while maintaining average scores of at least 98.8. Unrestricted affirmative action would therefore be a powerful tool in Chicago. Second, under admissions policies that change composite scores as much as Chicago’s race-neutral system, there is substantially greater scope to alter school demographics. The outermost curves in the figure describe assignments which maintain an average composite score of 98.0, and hence can be thought of as mak- ing changes to the composite score distribution which are about as large as CPS has made. (Recall that the average composite score under the CPS policy is 98.0 at Payton and 97.7 at Northside.) With changes of this magnitude, the minority share at both Payton and Northside could be increased to over 50 percent. Moreover, the FRPL shares could be increased to about 43 percent at Payton and 55 percent at Northside. Free or reduced price lunch share Free or reduced price lunch share 954 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Third, all of the frontiers are highly concave: they are quite flat near their left endpoints and quite steep near the right endpoints. The flatness means that a modifi- cation of a pure income-preference that replaces a portion of the income preference with explicit consideration of race can substantially increase minority representation while having negligible effect on average scores and almost no effect on low-income representation. For example, the leftmost and third from left points on the outmost Payton frontier indicate that one could raise minority representation at Payton from 37 percent to 48 percent while holding the average score at Payton fixed at 98.0 and reducing subsidized lunch representation just from 43 percent to 40 percent. Similarly, the steepness of the right portions of the curve means that a modification of a pure racial-preference plan that bases a portion of the preference on income rather than race can substantially increase low-income representation while only slightly reducing minority representation. For instance, the near verticality of the outermost curve for Payton near its right endpoint means that at average composite scores of 98.0 it is possible to increase low-income representation from 27 percent to 38 percent while only reducing minority representation from 53 percent to 50 per- cent. Together these two features suggest that policymakers who value both racial and income diversity would presumably have a preference for policies that explicitly considered both race and income. Implementing a policy that achieves an interior point on a curve requires explicit use of both FRPL eligibility and minority status. Assuming this is possible, policies can be implemented using simple bonus schemes. Students have x bonus points added to their score if they qualify for FRPL and y bonus points added if they are a minority. Assignments are then made simply by accepting students in the order of their bonus-adjusted scores. The ratio x/y determines where we end up between the fewer minorities/more low-income or the more minorities/fewer low-income end of each curve. The magnitudes of the bonuses will determine whether we reach a curve with a higher or lower average composite score. For example, in the case of Payton, the rightmost (53 percent minority, 28 percent subsidized lunch) point on the average-score-98 curve is obtained by giving 6.4 points for minority status and no points for being low-income. The leftmost (37 percent minority, 43 per- cent subsidized lunch) point on the same curve is obtained by giving 6.9 points for low-income status and no points for being a minority. The allocation two points to the right on the same curve where minority representation is substantially higher (48 percent) and subsidized lunch representation is only a little lower (40 percent) are obtained by giving 2.6 points for minority status and 5 points for subsidized lunch eligibility, respectively. B. Within-School Heterogeneity in Race, Income, and Composite Scores A separate concern about affirmative action policies is that they may contribute to within-school achievement gaps that adversely affect minority students for rea- sons, e.g., stereotype formation, that we did not try to incorporate into our mod- el.19 This might also be thought to be an advantage of race-neutral plans: when 19 See, e.g., Steele and Aronson (1995) and Austen-Smith and Fryer (2005) for further discussion. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 955 preferences are given only to minority students, it is clear that the lowest scor- ing students in a school will be minorities. But, Ray and Sethi (2010) note that race-neutral plans do not lower achievement gaps when optimally designed. When the distribution of scores for minorities and non-minorities differ, an optimal plan admits students from a region of score distribution heavily populated by minori- ties, but below the region for the remaining students, which increases the mean achievement gap. In this section, we illustrate how the gap in composite scores is affected as more seats are reserved for minorities under a quota system and with CPS’s shift to a race-neutral policy. We find that the Ray and Sethi (2010) concern is not purely academic. At any given level of minority representation the majority-minority achievement gap is wider under CPS’s tier-based plan than it would be under a race-based plan. CPS’s race-neutral admissions policy has created substantially more heteroge- neous student bodies, measured by race or income, at Payton and Northside than a pure score-based admissions process. The histograms in panel A of Figure 4 show what the composite score distribution would be at Payton and Northside under score-based admissions with no set-aside. Each histogram bar is divided into two portions: the lower dark gray and red portion reflects the number of underrepre- sented minority students with composite scores in the band and the upper light gray and red portion reflects the number of White or Asian American students. The upper histograms show that within-school racial achievement gaps are small under a purely score-based admissions policy. Without affirmative action, there would be little within-school variation at either school: about 60 percent of students at each school would have composite scores of at least 99 and all students would have scored at least 97. The average score gap between majority and minority students is 0.4 at Payton and 0.2 at Northside. The average FRPL score gap is 0.4 at both schools. The second row of histograms in the figure illustrate the class composition at Payton and Northside under benchmark affirmative action policies that use race and FRPL data to make Payton and Northside as diverse as they are under the cur- rent CPS policy on both dimensions. Note that a bonus scheme in which students are assigned a school-independent number of bonus points for minority and FRPL status cannot simultaneously match the current demographic composition of both Payton and Northside. However, it seems unlikely that CPS would choose to imple- ment a scheme with school-based bonuses. Hence, we have chosen throughout to report on two separate race and FRPL-based benchmarks. The Payton benchmark gives 2.9 points for minority status and 1.3 points for FRPL, while the Northside benchmark uses bonuses of 2.8 and 1.5 points, respectively. There is relatively little increase in within-school heterogeneity under these benchmarks. The histograms in the second row show that achievement gaps would be only moderately larger if CPS had used these plans to bring minority and FRPL representation up to their current levels. For example, the left histogram in Panel B corresponds to a policy in which 36.8 percent of the students assigned to Payton are underrepresented minorities and 24.5 percent are eligible for FRPL. These bench- mark plans replace many of the students with scores between 97 and 98.5 with minority and/or FRPL eligible students with scores between 95 and 96, but admit few students with scores under 95. The average majority-minority score gap is 1.8 at both schools, and the average FRPL gap is 1.5 at Payton and 1.4 at Northside. 956 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Panel A. No set aside 150 100 50 Payton Minority share: 0.214 FRPL share: 0.150 Mean score: 99.1 Min. score: 97.1 Mean majority−minority gap 0.4 Mean FRPL gap 0.4 150 100 50 Northside Minority share: 0.189 FRPL share: 0.228 Mean score: 99.0 Min. score: 97.0 Mean majority−minority gap 0.2 Mean FRPL gap 0.4 90 95 100 Northside Minority share: 0.355 FRPL share: 0.340 Mean score: 98.7 Min. score: 94.2 Mean majority−minority gap 1.8 Mean FRPL gap 1.4 90 95 100 Northside Minority share: 0.355 FRPL share: 0.340 Mean score: 97.7 Min. score: 87.3 Mean majority−minority gap 3.8 Mean FRPL gap 2.9 90 95 100 150 100 50 Payton Minority share: 0.368 FRPL share: 0.245 Mean score: 98.0 Min. score: 89.0 Mean majority−minority gap 3.2 Mean FRPL gap 2.5 150 100 50 00 85 90 95 100 85 Panel B. Race- and FRPL-based benchmark 150 100 50 Payton FRPL share: 0.245 Mean score: 98.8 Min. score: 94.6 Mean majority−minority gap 1.8 Mean FRPL gap 1.5 150 100 50 Minority share: 0.368 00 85 90 95 100 85 Panel C. Current tier system 00 85 90 95 100 85 Figure 4. Within-School Score Distributions under Alternate Admissions Policies Notes: Each histogram bar is divided into four portions: green and light gray reflect the number of non-minority students and magenta and dark gray reflect the number of minority students. The subportions in green and magenta represent FRPL students in the respective subportion. Score refers to the composite score. Switching to CPS’s tier-based affirmative action affects within-school achieve- ment gaps in two opposing ways. One effect that works to reduce racial gaps is that the policy admits some relatively low-scoring White and Asian American students who live in low SES neighborhoods. Working in the opposite direction, however, is the fact that a number of high-scoring minority students living in medium to high SES neighborhoods are now being denied admission and that almost all of the lowest-scoring admits (all of whom come from tier 1) are minorities. Given the opposing effects, it is not a priori clear how the shift to a race-neutral policy will affect within-school gaps. Panel C shows that the tier-based plan increases the within-school racial gaps. Under the race- and FRPL-based benchmark, the majority-minority score gap in average scores is 1.8 points at both Payton and Northside. Under the CPS plan, it is 3.2 points at Payton and 3.8 points at Northside. (As before, the larger gap Counts of applicants Counts of applicants Counts of applicants Counts of applicants Counts of applicants Counts of applicants Majority − FRPL Majority − No FRPL Minority − FRPL Minority − No FRPL VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 957 at Northside reflects that the CPS plan is less efficient at Northside.) A compari- son of the histograms in the second and third rows of Figure 4 brings this out in more detail. At both schools, the shift to the race-neutral plan reduces the number of majority students with scores in the 98 to 99 range. The CPS plan does add a large number of majority students with scores between 95 and 97, but almost all of the added students with scores below 95 are underrepresented minorities. The most striking difference is that the left tail of low-scoring students now extends down to 89 at Payton and 87 at Northside. This means that teaching and curricula in the two schools must accommodate a substantially more heterogeneous student population relative to an affirmative action policy which uses race and FRPL status. The histogram also shows that the CPS policy has displaced most students with composite scores of between 98 and 99. As a result, the classes at Payton and Northside have an unusual composition: the majority of students have scores of at least 99, but the rest are spread out over a wide range. Since the policies in panel B and C result in the same level of diversity, we conclude that most of the increase in within-school heterogeneity could have been avoided if there were no restrictions on the type of affirmative action which could be practiced. Within-school majority-minority achievement gaps are larger under the tier-based policy for any fixed level of minority representation. This fact can be seen in Figure 5, which plots the difference between the average composite score of admitted majority and minority students under CPS-like and pure racial-preference policies for each level of minority representation. The solid lines illustrate racial quota pol- icies varying the numbers of reserved slots, with blue circles for Payton and red triangles for Northside. At the left endpoint, the curves have a value of 0.2, which reflects the small difference in average scores of admitted majority and minority students under a purely score-based admissions process. Achievement gaps grow by about 1 point for every 10 points of minority share: they are slightly above 1 point when the school is 30 percent minority, around 2 points when the schools are 40 percent minority, and around 3 points at 50 percent minority. The dashed lines give corresponding numbers for policies based on the CPS tiers: for each desired level of minority representation, we find the CPS tier reservation percentage that produces that level of minority representation and report the majority-minority score gap under that policy. At Payton, the majority-minority gaps are small at first, but grow once enough seats are allocated by tier to increase minority representation beyond 30 percent. The gaps grow even more quickly at Northside, again illustrating that the lower efficiency of the CPS plan in the Northside environment is associated with a larger majority-minority score gap. Compared to a racial quota, the CPS policy roughly doubles the magnitude of the majority-minority score gap at Northside.20 Figure 5 also reports the majority-minority gap in scores for the twenty-fifth percen- tile of the distribution. The gap at this quantile are wider than the gap at the mean at both Payton and Northside. 20 While we focus on the effect of CPS’s race-neutral affirmation action system on Payton and Northside, CPS’s tier-based plan produces a higher SD at every school relative to the benchmark. The effects are largest at Jones and Young, which are the next two schools in the selectivity hierarchy following Payton and Northside. Online Appendix Figure A.1 presents histograms similar to those in Figure 4 illustrating the impact of the CPS plan on the other schools. 958 THE AMERICAN ECONOMIC REVIEW MARCH 2021 10 Mean gap 25th percentile gap 10 88 66 44 22 00 0.2 0.3 0.4 0.5 0.6 0.7 0.2 Minority share Figure 5. Within School Gaps in Average Composite Scores: Quotas versus CPS Tier-Based Plan Notes: Each point plots the composite score gap between majority and minority students for a given admissions policy. In the left panel, the score gap is measured as the difference in mean majority and mean minority scores. The right panel considers the gap at the twenty-fifth percentile of each distribution of scores. The solid line rep- resents benchmark policies, varying the magnitude of the weight for minority status. The dashed line represents CPS tier policies, increasing the percentage of reserved seats. Blue circles indicate Payton and red triangles indi- cate Northside. C. Targeting Disadvantaged Applicants Compared to the benchmark, the CPS plan must be rejecting some students with high composite scores and instead accepting lower-scoring students with the same minority-FRPL status. In this section, we investigate further to what extent admis- sions probability depends on student characteristics and examine the attributes of students who are displaced. Figure 6 reports the probability of admissions for students stratified by FRPL and minority status at Payton and Northside under the CPS policy. The vertical line in each panel corresponds to the lowest composite score of an admitted student under the race- and FRPL-based benchmark plan. That is, at Payton, any minority and FRPL student with a composite score of at least 94.5 would have been guaranteed admission under this alternative. In contrast, under the CPS policy, students with scores of 89 have a positive probability of being admitted. Their positive probability comes at the expense of higher scoring minority/FRPL students who do not obtain admissions for certain. At Northside, minority/FRPL students with scores of 87 have a positive admissions probability, when the lowest score to gain admittance under the benchmark would be 94.2, and the admissions probability for students with scores 90–95 is less than 0.5. The positive probability to the left of the vertical line, and the fact that points to the right of the vertical are not equal to 1 provide another visualization of the inefficiency of the CPS plan. 0.3 0.4 Minority share 0.7 0.5 0.6 Score gap Score gap Payton benchmark Northside benchmark Payton tier Northside tier VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 959 Payton Minority - FRPL 11 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 Minority - No FRPL 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 056000 80 85 90 95 100 Composite score Majority - FRPL 11 80859095100 Composite score Majority - No FRPL 80859095100 Composite score Northside 80859095100 Composite score Minority - No FRPL Minority - FRPL 11 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 97.67 0.2 067 022 078 80859095100 Composite score Majority - FRPL 11 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 80859095100 Composite score Majority - No FRPL 011056 80 85 90 95 100 80 85 90 95 100 Composite score Composite score Figure 6. Probability of Admission and Composite Score Notes: This graph shows the probability of admission under the current CPS tier plan by composite score bucket, separately for four groups of minority and FRPL combinations. The vertical line represents the lowest composite score among students who would have been admitted under the race- and FRPL-based benchmark plan, subtracting the bonus points in the respective group. The figure shows that the CPS’s plan does not simply admit the wrong minority-FRPL students. The other three panels illustrate that the CPS plans admits students with lower composite scores than the benchmark plan, and it also does not admit all pupils with higher scores. For example, even majority/non-FRPL students 94. 96. 98. 94. 95. 97. 98. Admission probability Admission probability Admission probability Admission probability Admission probability Admission probability Admission probability Admission probability 960 THE AMERICAN ECONOMIC REVIEW MARCH 2021 with scores of 94 are admitted with positive probability at Payton and Northside, even though they do not increase diversity and have substantially lower composite scores than the composite score these types of applicants would have faced under the benchmark plan.21 III. Measuring the Efficiency of Affirmative Action Plans A. Modeling Affirmative Action in Admissions Several models show how race-based affirmative action can enhance welfare.22 In these models, restrictions on the use of race will always lead to welfare losses. Chan and Eyster (2003) first formulated this finding in a model in which an elite school values both diversity and the average quality of admitted students. They also noted that welfare losses can be severe and take on striking forms: constrained opti- mal plans can deny admission to the some of the most-qualified students of every race; they can reject some minority students who are more qualified than some admitted majority students; and they can reduce average student quality as well as minority representation. Ray and Sethi (2010) note that absent the monotonicity restriction imposed by Chan and Eyster (2003) optimal policies will generically be non-monotone and accept some less-qualified students while simultaneously deny- ing admission to some who are more qualified. Consider a school system that serves a heterogeneous set of students. Each student has a type vector (θ, 𝐱, z), where θ describes the best curriculum for the student, 𝐱 is a vector of student characteristics (such as race or socio-economic status), and 𝐳 is “proxy” that may be used in assigning students to schools. Suppose that the expected educational outcome of a student of type (θ, 𝐱, 𝐳) when assigned to school s is (1) Vs(θ,𝐱,𝐳) = h(θ,𝐱,𝐳) − k(θ − cs)2 − d∥𝐱–s − 𝐱∗∥, where cs is the curriculum at school s, 𝐱–s is the mean of the vector of characteristics of students in school s, and 𝐱∗ is the composition of an optimally diverse school. The function h gives the component of a student’s expected outcome which does not depend on school s. The parameter k indexes the importance of providing stu- dents with a curriculum that is matched to their type θ. The parameter d indexes the importance of losses from schools having demographics that differ from x∗. This loss term might reflect the value of discussions in diverse classrooms. An optimal affirmative action plan balances curriculum matching against concerns for diversity. An important assumption in this model is that the education benefits to diversity and optimal curriculum matching do not depend on whether the student is in the major- ity or minority group. 21 Figure A.2 in the online Appendix reports additional information on the tract characteristics of students dis- placed under the CPS plan. Almost all of the displaced students come from tracts that are above the median in the CPS SES index in its income and education components. 22 Coate and Loury (1993) develop a model based on statistical discrimination and skill investment, Athey, Avery, and Zemsky (2000) and Chung (2000) develop frameworks with mentoring and role models, and Fu (2006) considers investment incentives with exogenous asymmetries. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 961 Suppose that the school system operates schools indexed by s = 1,2,...,S. Assume that the school system chooses both a student assignment function A : Θ × X × Z → S and the curricula at each school 𝐜 = (c1, c2, ..., cs).23 When affirmative action is unrestricted, we assume that the school system knows the distribution of student types and can choose any function A(θ, 𝐱, 𝐳) and any curricula 𝐜. Suppose the social welfare function aggregates student outcomes as follows: (2) WA,c ≡ ∑s ∫{θ,𝐱,𝐳|A(θ,𝐱,𝐳)=s}(h(θ,𝐱,𝐳) − k(θ − cs)2 − d∥𝐱–s − 𝐱∗ ∥)dμ(θ,𝐱,𝐳), where μ is the distribution over types. The unrestricted optimal assignment policy thus maximizes: {A∗(θ,𝐱,𝐳),𝐜∗} = argmaxWA,c. A(θ,𝐱,𝐳),𝐜 Our utilitarian welfare function implicitly assumes that the motivation for maintain- ing diversity is that it affects educational outcomes of students in the system. But the same social welfare function could also capture situations where diversity benefits accrue to others, e.g., they could be realized when the current generation of students serves as role model for future students as in Chung (2000), or they could reflect the preferences of voters or politicians for diverse schools. The online Appendix devel- ops some properties of the optimal assignment policy. The most important result is that when each school sets its curriculum optimally, school-specific welfare simpli- fies to WAs,c =−kVar(θ|A(θ,𝐱,𝐳)=s)−d∥𝐱–s−𝐱∗∥. That is, welfare is determined by the combination of within-school variance in the curricula to which students are best matched and demographic diversity. B. Measuring Relative Efficiency The optimal admissions plan will often be infeasible for two reasons: (i) schools may be legally prohibited from basing admissions decisions on some dimension 𝐱 and (ii) schools may not observe some dimensions of 𝐱. When this happens, school systems can only implement rules A(θ, 𝐳) that involve proxy variables 𝐳 imperfectly correlated with 𝐱. { } The difference in welfare from adopting a given race-neutral plan Aˆ(θ,𝐳),𝐜ˆ rather than {A∗(θ, 𝐱, 𝐳), 𝐜∗} is WA∗,c∗ − WAˆ ,cˆ. Several factors make estimating this quantity infeasible. Computing the optimal policy and evaluating the welfare func- tion requires knowing what dimensions of diversity matter (to student achieve- ment or school-board preferences) and estimates of the importance of curriculum mismatch and diversity for student achievement. Therefore, we take an alternative 23 Chan and Eyster (2003) and several subsequent papers consider a more general class of plans which may involve random assignment. The CPS plan was deterministic and we simplify the discussion that follows by only considering such plans. 962 THE AMERICAN ECONOMIC REVIEW MARCH 2021 approach that compares a given race-neutral policy to a well-defined race-based benchmark policy. One school assignment plan, available regardless of whether it is legal to condi- tion on x, ignores both x and z and chooses the assignment plan that minimizes the average within-school variance in θ. Other plans involving affirmative action can improve diversity, but improvements come at the expense of increasing within-school variance in θ. The relative efficiency measure we define below focuses on how well plans make this trade-off, i.e., how much of a loss in curriculum matching is incurred for each unit of improvement in diversity. Let sˆ be a school of interest. Let {Aˆ (θ, x, z), cˆ} be some assignment plan of inter- est in which the curriculum is optimal with respect to the school assignments.24 Suppose that Aˆ assigns ns students to each school s and write xˆsˆ for the average demographics of school sˆ under this plan. To define our relative efficiency mea- sure for plan Aˆ as school sˆ, we first construct two benchmark assignment plans that serve as points of comparison. First, let ASB be the “score-based” admis- n ,...,n 1S sion plan in which students are prioritized in order of θ and assigned in order to their most-preferred school provided it is not full given the capacities n1,...,ns. Under the score-based admissions plan, a student is assigned her most preferred school among those with available capacity when it is her turn to choose. Next, let ARB be the “race-based’ affirmative action plan which prioritizes students sˆ,n ,...,n ,xˆ 1 ssˆ on the basis of θ plus an underrepresentation bonus and provides members of groups which would be less represented at school sˆ under plan ASB with the smallest num- ber of bonus points that are sufficient to make the demographics of school sˆ match the demographics, xˆsˆ, that it has under plan Aˆ . We then define the relative efficiency of school assignment plan Aˆ at school sˆ by (3) Relative Efficiency(Aˆ , sˆ) RB (θ,x,z) = sˆ ) SD(θ|Asˆ,n1,...,ns,xˆsˆ(θ,x,z) = sˆ − SD θ|A ) ( SB n1,...,ns = ___________________________________________ . SB SD(θ|Aˆ(θ,x,z) = sˆ)−SD(θ|An ,...,n(θ,x,z) = sˆ) Relative efficiency can be thought of as measuring the fraction of the welfare losses from inferior curriculum-matching at school sˆ, which were necessary to achieve the level of diversity which plan Aˆ achieves. The measure compares Aˆ to another plan which uses bonuses based on x to achieve the same level of diversity. For example, if plan Aˆ increases the standard deviation of θ within school sˆ by 4 units for every 0.01 increase in the representation of some group and the alternate plan ARB increases the standard deviation of θ within school sˆ by only 1 unit for every 0.01 increase in the representation of that group, then we would say that plan Aˆ is only 25 percent effi- cient at school sˆ. That is, 25 percent of the loss in curriculum-matching benefits that plan Aˆ entails were necessary to achieve the diversity benefits that plan Aˆ achieves while the remaining 75 percent were due to the restriction that prevents using x in the admissions process and/or to plan Aˆ having been suboptimally designed relative to the constraints. 24 We will make this assumption about curricula throughout this section and henceforth describe assignment plans solely by their assignment functions. 1s VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 963 School-specific welfare depends on within-school variance in curricula and demographic diversity. The score-based benchmark is a natural point of comparison because the equivalence of minimizing within-school standard deviation and max- imizing across school standard deviation in mean scores means that a plan which considers only θ in admissions can be thought of as a plan designed to roughly maximize one dimension of welfare. The race-based benchmark, which focuses on maximizing average scores at school sˆ subject to a diversity constraint, is another natural point of comparison if we think of trying to reduce the system-wide standard deviation given a diversity constraint.25 Cardinalization of the welfare losses in curriculum matching that have been incurred in the course of increasing diversity involves taking a position on the appropriate scale. While our model suggests using the within-school variance in composite scores as a measure of welfare losses, we use the within-school standard deviations, SD(θ | Aˆ (θ, x, z) = sˆ), instead. We make this choice to be more conser- vative in labeling plans as inefficient, cognizant of the fact that the model-based argument for using variance instead would be relying heavily on the assumption that welfare losses are quadratic in the mismatch in composite scores. We are also trying to be conservative in omitting other potential welfare losses from our calculation. For example, we noted that the CPS plan has much larger majority-minority score gaps than our benchmark plans, and the impact that this may have on the formation of stereotypes is another welfare concern that might be regarded as making the CPS plan even less efficient than it appears in our measures. We also report a measure of efficiency using the mean composite score in the online Appendix. For a given admissions plan, this measure takes the ratio of the difference in the average score under the optimal plan for corresponding level of diversity compared to purely score-based admissions to the difference in the aver- age score under the plan minus the average under purely score-based admissions. That is, this metric simply replaces the standard deviation in equation (3) with the mean. In many cases, the standard deviation measure and the mean produce similar efficiency measures, so we opt to emphasize the standard deviation measure. The similarity is not surprising. For example, if all students have a score of 100 under the score-based plan, plan A has a fraction 1 − p of students having a score of 100 and fraction p having a score of 100 − y, and plan A’s benchmark has a fraction 1 − p at 100 and fraction p at 100 − x, then the mean and standard-deviation based defini- tions both measure the efficiency as x/y. ˆ In the discussion above, we have defined the relative efficiency of a plan A con- sidering all demographic changes it entails. In practice, affirmative action plans change the demographics of schools in multiple dimensions: they affect the fraction of minority students, the fraction of low-income students, etc. Our primary mea- sure, which we will refer to as overall efficiency, considers benchmark plans ARB that achieve exactly what plan Aˆ achieves on both of these dimensions. To better 25 The plan that minimizes within-school standard deviation in θ just at school sˆ, in contrast, would not be a sensible benchmark: one can reduce within-school standard deviation at any school almost to zero simply by deny- ing admission to all of the highest-achieving students and instead accepting students from some point in the middle of the distribution where there are many students from every demographic group. Our benchmarks are designed to avoid the concern that one is achieving a favorable result at school sˆ at the expense of increasing within-school variance at other schools. 964 THE AMERICAN ECONOMIC REVIEW MARCH 2021 understand a plan’s performance, we also separately report plans’ relative effi- ciency as a means to improve single dimensions of diversity. For example, if plan Aˆ increases the representation of minority students at sˆ from 15 percent to 30 per- cent and increases the representation of low-income students from 20 percent to 25 percent, we will measure the “minority-only efficiency” of the plan by a similar formula, but using in place of ARB a benchmark that maximizes scores at school sˆ imposing only the constraint that minority enrollment must be at least 30 percent. We will measure the “income-only efficiency” by using in place of ARB a benchmark that only imposes the constraint that low-income enrollment must be at least 25 per- cent. At Payton, the bonus points for race under the minority-only benchmark are 3.4 points and the bonus points for FRPL under the income-only benchmark are 2.9. At Northside, the bonus points for race under the minority-only benchmark are 2.8 and the bonus points for FRPL under the income-only benchmark are 2.1. IV. How Efficient Are Chicago’s Policies? Our efficiency calculation evaluates an affirmative action policy in terms of the portion of the increase in within-school variation in θ that was necessary to achieve the level of diversity which is being achieved. For our efficiency calculations, we will take θ to be the composite score. In terms of histograms in Figure 4, the actual increase in standard deviation (SD) can be thought of as calculated by computing the difference in SD between panel C and panel A for each school. The necessary increase in SD would be the difference between the SD of rescaled versions of the panel B and panel A for each school. Here, this calculation says that the current CPS policy has an overall efficiency of 28 percent at Payton and 22 percent at Northside. Put differently, the CPS policy has increased within-school SD in composite scores by about four times as much as was necessary to achieve the diversity that is being achieved. We more directly study the CPS’s plan as a tool for racial and socioeco- nomic integration. A. The Policy as a Tool for Racial Integration The CPS policy departs from our race- and FRPL-based benchmark policy because it does not use student-level race data to increase minority representa- tion, nor does it use student-level income data (measured by subsidized lunch eligibility) to increase low-income representation. Figure 7 examines the trade- off between increasing minority representation and reducing within-school SD in student ability/preparation at Payton and Northside. Specifically, the figure describes the classes that would be admitted to each school under various affir- mative action plans on two dimensions. The x-axis is the percent of students who would be underrepresented minorities. The y-axis gives the within-school SD in composite scores. The y-axis is plotted on a flipped scale so that one can regard higher points on either axis as desirable. The left panel corresponds to Payton and the right panel to Northside. The red circles describe the outcomes under a set of benchmark race-based pol- icies which provide varying levels of advantage to underrepresented minority stu- dents. The upper left point on these curves correspond to the purely score-based VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 965 Payton Minority share 0 0.1 0.2 0.3 0.4 0.5 0.6 00 11 22 33 44 55 Northside Minority share 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 7. Feasible Composite Score Spread at Various Levels of Minority Representation Notes: This figure compares the standard deviation of composite scores for several levels of minority representation. The upper left point of each curve corresponds to a purely score-based policy. Red circles correspond to policies that give bonus points for minority status, increasing by 1.5 point increments. Blue squares correspond to the anal- ogous policy which instead rewards FRPL status. Green triangles represent simulations of admission schemes that reserve various fractions of seats to be allocated evenly across CPS tiers, increasing by 10 percent increments. The black square corresponds to the current CPS policy with 70 percent of seats reserved. policy, which produces classes in which 21.4 percent of Payton students and 18.9 percent of Northside students are underrepresented minorities. The points further to the right on these curves correspond to increasing the minority preference: the first dot corresponds to a minority preference of one point, the second to a minority pref- erence of two points, and so on. The flatness of the left portions of the curves is a fur- ther illustration that race-based affirmative action policies can substantially increase minority representation while only making small changes in the distribution of stu- dent preparation. The increasing steepness of the curves as we move to right shows that there is limited scope for making such low-cost changes. Larger increases in minority representation will require more substantial increases in within-school SD. The green triangles describe the incoming classes when one implements CPS-like policies that reserve various fractions of the seats at Payton and Northside to be allo- cated evenly across the CPS tiers. The upper left point on these curves is the pure score-based admissions policy. Each subsequent point corresponds to allocating an additional 10 percent of the seats by tier. The large black square is the current CPS policy which involves a 70 percent reservation for tier seats and produces a class that is 37 percent minority. The blue squares describe the income classes under a set of FRPL-based benchmark policies, where bonus points are given for low-income status. The vertical difference between the two curves provides another illustration of the inefficiency of the CPS policy. Our relative efficiency measure applied to this solely race-based comparison (i.e., minority-only efficiency) would say that the CPS policy is 26 percent efficient at Payton, i.e., the actual within-school SD rela- tive to the score based policy 2.71 − 0.80 = 1.91, is 1/0.258 = 3.9 times as large Benchmar FRPL Tier k Standard deviation of composite score Standard deviation of composite score 966 THE AMERICAN ECONOMIC REVIEW MARCH 2021 as the increase in SD (1.29 − 0.80 = 0.49) sufficient to achieve the same level of minority representation under the benchmark race-based policy. The CPS policy is even less efficient as a means to increase minority represen- tation at Northside. In the right panel of Figure 7, the green and red curves diverge even more quickly than in the left panel. The current CPS policy is only 19 per- cent minority-only efficient at Northside. The minority-only efficiency of the FRPL benchmark is close to that of the CPS policy at Payton, but at Northside the FRPL benchmark is midway between the race-based benchmark and the CPS policy at minority shares above 35 percent. The difference between the policies’ performance at Payton and Northside is driven in part by the demand patterns of applicants to both schools. That is, place-based affirmative action plans may be less effective in environments similar to Northside than in environments similar to Payton, since Payton naturally draws students from nearby minority neighborhoods. Note that the degree to which a social planner should care about relative effi- ciency would vary with the size of the change in school composition that is being contemplated. Near the left endpoint, low relative efficiency would not be a big concern because the increase in within-school SD (which is what enters into wel- fare) is still fairly small in absolute terms even for an inefficient policy. The form of the graphs, however, bolsters the case that relative efficiency is a useful concept in showing that it provides a measure which is somewhat robust to how intensively the policy level is used. For example, if one were to analyze the CPS-like policies which involve reserving 30 percent to 80 percent of the seats at Payton by CPS tier, one would find that they are all 26 percent to 33 percent efficient as means to increase minority representation. B. The Policy as a Tool for Socioeconomic Integration The nature of CPS policy’s inefficiency as a means for admitting low-income stu- dents differs from its inefficiency as a means for admitting minority students since CPS faced a legal prohibition on using race in admissions, whereas it could have used FRPL eligibility if it had chosen to do so. We nonetheless study the CPS policy’s efficiency as a means for admitting low-income students for two reasons. First, many other cities and states have adopted placed-based affirmative action policies, so it is of more general interest to know how efficiently one such policy is aiding low-income students.26 Second, while the “low-income” inefficiency of the CPS policy could be eliminated by incorporating student-level data on FRPL eligibility, socioeconomic disadvantage involves much more than income. CPS does not have the student-level data to measure these other dimensions of disadvantage. For example, CPS does not know if a student’s parent(s) are supportive of their children’s education; how many books they have in their home; whether the student has been affected by domestic or neighborhood violence; whether the student’s teacher encouraged them to pursue high achievement; etc.27 Place-based 26 Texas’s well-known top 10 percent policy for college admissions can be thought of as one example of a much coarser place-based policy which puts all students who attend the same public high school in an equivalence class. 27 The breadth of disadvantage motivates the multifaceted approach to addressing disadvantage in the Harlem Children’s Zone studied in Dobbie and Fryer (2011). Fryer and Levitt (2004) show that the number of books in the VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 967 Payton Free and reduced price lunch share 0 0.1 0.2 0.3 0.4 0.5 0.6 00 11 22 33 44 55 Northside Free and reduced price lunch share 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 8. Feasible Composite Score Spread at Various Levels of Free or Reduced Price Lunch Eligibility Notes: This figure compares the standard deviation of composite scores for several levels of low income represen- tations. The upper left point of each curve corresponds to a purely score-based policy. Red circles correspond to policies that give bonus points for FRPL status, increasing by 1.5 point increments. Blue squares correspond to the analogous policy which instead rewards minority status. Green triangles represent simulations of admission schemes that reserve various fractions of seats to be allocated evenly across CPS tiers, increasing by 10 percent increments. The black square corresponds to the current CPS policy with 70 percent of seats reserved. policies are a possible way to address such disadvantages, and the lessons learned form examining the degree to which the CPS policy is benefiting low-income students may also apply to the effectiveness of place-based policies in aiding students who are disadvantaged on other dimensions on which schools (and we) lack data. Figure 8 reports on the trade-off between increasing low-income representation and reducing within-school composite score SD at Payton and Northside. The figure describes the classes that would be admitted to each school under various affirmative action plans. Figure 8 differs from Figure 7 since the x-axis reports the fraction of students eligible for FRPL instead of the number of minority students. As with racial diversity, bonus schemes can substantially increase low-income representation from its initial level with only small changes in the composite score distribution. The red circles in Figure 8 illustrate the trade-off that CPS faces when it gives bonus points to low-income students to increase their representation. The curves are nearly horizontal close to their left endpoints. This fact shows that it is possible to substantially increase low-income representation with only a slight change in the distribution of composite scores. But, the steeper slopes at higher levels of low-income representation show that more substantial increases in within-school variation are inevitable, even for levels of representation well below the CPS aver- age FRPL eligibility of 86 percent. Comparing the two schools, Northside will have home is a powerful predictor of student performance at the beginning of kindergarten and including this and a host of other controls can account for the racial achievement gap at that time. Benchmark Race Tier Standard deviation of composite score Standard deviation of composite score 968 THE AMERICAN ECONOMIC REVIEW MARCH 2021 substantially more low-income students than does Payton under any given level of low-income preference. The green triangles describe the student bodies produced by variants of the CPS policy with different tier reserve sizes. Despite its focus on socioeconomic disadvantage, the CPS plan is not very efficient as a method for increasing the number of low-income students. At Payton, for example, the current CPS policy increases the proportion of students on FRPL from 15 percent to 25 percent at the cost of increasing the SD in composite scores from under 1 to almost 3. A simple FRPL quota could have achieved the same increase in subsidized lunch representation while only increasing the SD to about 1.1 SD. Our income-only efficiency metric says that the CPS policy is only 17 percent efficient as a method of increasing low-income representation. We take this as a strong cautionary mes- sage. Place-based affirmative action policies will provide a substantial benefit to students living in disadvantaged areas. But it is not at all clear that the benefit will go to students who are themselves disadvantaged. The third series in Figure 8 provides another striking comparison: the CPS policy is worse than using racial quotas for low-income representation. The blue squares describe the classes that would have been admitted under a purely race-based plan that only explicitly favors students who belong to an underrepresented minority group. Strikingly, this graph lies above the CPS plan in the figure. That is, for every level of low-income representation, the CPS policy is actually worse than using racial quotas. The fact that CPS’s SES-based policy is less efficient than racial quotas as a means to increase low-income representation may at first seem paradoxical. CPS’s SES index is based on several variables known to be highly correlated with poverty. How could it possibly be less efficient as a means to admit low-income students than a pure racial preference? One explanation involves the fact that once we learn that a student from a low- SES census tract has an extremely high composite score, it means that the stu- dent is unusual for their census tract. This fact means that the relationship between tract and student characteristics is less reliable exactly for these kids. For example, most Black students in CPS are poor. An even higher percentage of students from very low-SES census tracts are poor. But once we learn that a Black student and a student from a low-SES census tract achieved a very high composite score, it is likely that neither one is poor. The reason racial quotas outperform, however, is that minority status retains more of its power as a predictor of poverty than the tract-level SES score. The fact that the CPS plan is less efficient as a means to admit low-income stu- dents than racial quotas is a strong criticism of place-based plans like Chicago’s. It has been recognized that plans that do not explicitly consider race will be less effective as a means to increase minority representation. It has been generally pre- sumed that this disadvantage should be thought of as part of a trade-off, wherein race-neutral plans will benefit students who are themselves disadvantaged but not necessarily minority. Our analysis of the CPS plan shows that this potential off- setting benefit need not exist: the CPS place-based plan is both less efficient as a means to increase minority representation and less efficient as a means to benefit low-income students than a race-based plan. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 969 V. How Efficient Can Race-Neutral Policies Be? So far we have compared the CPS plan with benchmark race-based policies which are infeasible in the current legal/political environment. In this section, we explore the extent to which CPS might improve on its plan given current restrictions on the use of race. We do not construct the constrained-optimal plan: defining such a plan would require knowledge of how CPS values diversity and within-school variance and an understanding of political constraints. We instead pursue two more limited aims. Section VA discusses the extent to which small and potentially feasible changes to the CPS plan would improve minority-only efficiency. Section VB attempts to provide an upper bound on the efficiency CPS could have achieved by going beyond what might be legally or practically feasible, but still not directly using race. A. Can Simple Changes Make the CPS Plan More Efficient in Increasing Minority Representation? Several aspects of the CPS plan seem crude. The census tract SES index is an unweighted sum of six variables when some variables may be more important than others. The continuous tract-level CPS measure, which takes on 800 different values, is discretized into four bins before being used in the assignment process, suggest- ing it falls short of exploiting the full range of available information. And the plan makes no use of student-level data on FRPL eligibility. The decision not to make use of individual-level data potentially reflects the belief that SES tier is predictive of disadvantage, but it could alternately reflect a feasibility or political constraint.28 We therefore explore whether two simple plans can make the CPS plan more efficient as a means to increase minority representation: one using only data CPS is now using and one also using individual-level FRPL data.29 One class of potentially feasible modifications to the CPS plan constructs the SES index as a weighted average of any subset of the disadvantage indicators CPS is now using, and then gives preferences to students as a linear function of the con- tinuous SES index. A rough intuition for what might make this type of plan work well is if the SES index is highly correlated with minority status among students with high composite scores. To find weights, we regressed an indicator for minority status on the six CPS variables, running the regression on the subsample of appli- cants with composite scores of at least 96. Some variables have the “wrong” sign in this regression, e.g., it suggests favoring students from higher-income census tracts, and it seems debatable whether CPS could use such an index, so we sequentially 28 A reporter asked Richard Kahlenberg, the architect of the CPS plan, directly about this issue, “The new admissions policy uses the socioeconomic status of a student’s census tract, not an individual student. Is it possible that schools will draw, and admit, students who are the exceptions within their neighborhoods?” His response mentions both concerns about whether individual-level data would be reliable and confidence that the “rich” set of variables that go into the SES index will make it a reliable indicator of disadvantage. “There is that possibility. But there are problems with using the individualized data. FRPL data are presented by individual parents. There is usually very little verification that goes on. Using census data, we are able to get a rich set of factors. You could ask all these questions of people, but once it became clear that there was an advantage to being in one category, there might be an incentive for individuals to provide false information.” See Harris (2009). 29 We focus only on minority efficiency because measured the income efficiency can be increased to 100 percent using the student-level data. As mentioned earlier, it is not clear whether this corresponds to 100 percent efficiency in addressing true disadvantage when disadvantage is multi-dimensional. 970 THE AMERICAN ECONOMIC REVIEW MARCH 2021 Table 3—Efficiency of Alternate Race-Neutral Affirmative Actions Plans Within Maj.- Avg. school Min. score SD Gap Share Share minority FRPL (5) 0.245 0.236 0.236 0.286 0.264 0.245 0.340 0.340 0.359 0.452 0.429 0.340 Percent efficiency (SD) Minority Income Overall only only (6) (7) (8) 28.3 25.8 17.1 43.6 40.8 22.5 49.0 45.9 25.3 69.8 49.0 59.6 73.2 56.1 57.1 100.0 91.1 60.4 22.1 18.5 9.7 38.8 32.5 17.0 45.3 34.0 21.6 76.2 36.5 62.8 64.9 36.1 48.4 100.0 83.7 43.7 (4) 98.3 2.01 2.06 0.368 98.3 1.87 1.97 0.368 98.4 1.80 1.93 0.368 98.5 1.68 1.95 0.368 98.8 1.34 1.77 0.368 97.7 3.18 3.79 0.355 98.0 2.21 2.14 0.355 98.0 2.15 2.07 0.355 98.0 2.07 1.85 0.355 98.0 2.08 1.88 0.355 98.7 1.42 1.84 0.355 Panel A. Walter Payton College Prep Current CPS plan Reweighted SES bonus LASSO SES bonus SES Bonus with FRPL LASSO with FRPL Race and FRPL-based benchmark Panel B. Northside College Prep Current CPS Plan Reweighted SES bonus LASSO SES bonus SES Bonus with FRPL LASSO with FRPL Race and FRPL-based benchmark (1) (2) (3) 98.0 2.71 3.17 0.368 Notes: Reweighted SES bonus refers to a plan that uses a subset of the SES disadvantage indicators: CPS’s Adult Education Index, Percentage of Single-Parent Families, and Local Elementary School ISAT Score variables. LASSO SES bonus adds the estimate for the probability of being a minority from a LASSO model as additional SES indicator. The SES bonus with FRPL and LASSO with FRPL refer to plans that additionally take the individ- ual FRPL status into account. The race- and FRPL-based benchmark plan uses weights on individual minority and FRPL status to maximize the average composite score, while matching both the minority and FRPL share achieved in the current CPS plan. drop wrong-sign variables from the regression until all weights are positive. We end up with a tract-level SES index that is a simple weighted average of just three variables: CPS’s Adult Education Index, Percentage of Single-Parent Families, and Local Elementary School ISAT Score variables.30 We do not know whether some political constraint or data quality concern led CPS to conclude that it should ignore student-level FRPL status and only use tract-level variables. Nevertheless, to demonstrate the additional predictive power of using such information, we also constructed a student-level SES index variable which turns out to be a weighted average of four variables: the individual-level FRPL indicator and the same three variables that are involved in our tract-level index. We then considered alternative race-neutral assignment plans in which students’ priorities at exam schools are the sum of their composite scores and a scale factor times of one of our two new SES indexes. We chose the scale factor so that the allo- cations produced would assign exactly as many minorities to Payton and Northside as CPS’s actual plan. Table 3 provides statistics comparing the classes admitted under these plans with the classes admitted under the CPS plan. Panel A examines how the plans perform at Payton. Our simple modification raises minority-only effi- ciency from 26 percent to 41 percent. This increase corresponds to a reduction in within-school SD from 2.7 to 2.0. This reduction is quite far from eliminating the undesired side effect of the shift to race-neutral affirmative action, but it is a sizable enough shift to seem worthwhile. Another appealing feature of the modified plan is 30 Details on these regressions are provided in online Appendix Table A1. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 971 that it reduces the majority-minority gap in average composite scores from 3.2 to 2.1 points. Adding individual FRPL data allows for a further improvement at Payton. Minority-only efficiency increases to 49 percent. The modified plan now also admits more FRPL-eligible students than the CPS plan. Panel B reports comparable statistics for Northside. The version that uses only tract-level variables once again improves minority-only efficiency and simultane- ously reduces the majority-minority gap in average composite scores. But, the fact that minority-only efficiency only reaches 33 percent provides further evidence for our initial conjecture that race-neutral affirmative action would be more difficult to carry out in an environment like Northside. Similar to Payton, the inclusion of individual-level FRPL status results in further improvement and raises minority-only efficiency to 37 percent. It is possible that even these simple plans would run into legal trouble. Opponents of affirmative action could challenge the use of a weighted average, charging that the weights were derived from an attempt to influence racial out- comes of the school assignment plan rather than from their relative importance as components of diversity or disadvantage. We will not take a stance on this discus- sion and report in online Appendix Table A.6 that variants of these plans that use unweighted averages perform nearly as well as plans using weighted averages. The performance of several other plans, including neighborhood-based plans more directly analogous to the Texas top 10 percent plan, are also discussed in our working paper and in the online Appendix. B. More Sophisticated Plans as an Upper Bound We next ask whether substantially higher welfare gains could be reached with even more sophisticated race-neutral plans. We believe that the answer to this question is no. This view derives from an analysis of what could be done under a much more ambitious plan which already seems to go well beyond might be legally permissible. To explore how far one might be able to increase efficiency, we began by man- ually browsing through the many thousands of tract-level variables available from Manson et al. (2010) and identified a subset of 145 that we thought might turn out to be useful predictors of minority status.31 The number of variables is sufficiently large relative to the number of high-scoring minority applicants to make overfit- ting an important concern, and it also seems implausible that CPS could choose an SES index with such a large number components. Accordingly, we use a LASSO regression to pick a parsimonious index that predicts minority status among students with high composite scores.32 Specifically, we estimate a LASSO regression on the subsample of applicants with composite scores of at least 96 with minority status as the dependent variable and both the CPS variables and our added 145 variables as potential explanatory variables.33 The LASSO procedure led to a model that uses nine of the explanatory variables. The variables it chose magnify concerns that an 31 Not all variables are available for all tracts. We impute the mean value for missing values. 32We also investigated the performance of random forest models. They did not fare quite as well in out-of-sample fit. 33 The regularization parameter was chosen by cross validation. Details are in online Appendix Table A5. 972 THE AMERICAN ECONOMIC REVIEW MARCH 2021 approach of this form would not pass legal muster. Two variables it selects are the fractions of the foreign-born population in the census tract who come from Asia and Europe, which while not technically a use or race, run the risk of being seen by courts as having been chosen by the machine learning algorithm to penalize tracts with many Asians and Whites. We then implemented race-neutral affirmative action plans as in the previous sec- tion. We treat the predicted probability of being a minority that comes out of the LASSO model as if it were an SES index and rank students on a weighted average of their composite scores and their predicted minority status. As above, we implemented two versions of each of the above plans. One uses a weight that makes the underrep- resented minority share at Payton exactly match its value under the current CPS plan. The other exactly matches the current underrepresented minority share at Northside. The LASSO model is able to improve on our simple modifications at both Payton and Northside. But the primary result that we would emphasize is the magnitudes: the results indicate that even such a fairly complex model cannot produce significantly larger gains than the plans presented previously. At Payton, the LASSO model achieves 46 percent minority efficiency when constructed without student-level FRPL data and 56 percent efficiency, when using such individual-level data. The improvement from LASSO without free lunch is only about 5 percentage points bet- ter than we were able to do achieve in our simple three variable model and remains far from 100 percent. At Northside, the LASSO model adds essentially nothing to what we were able to do earlier to improve minority efficiency. We conclude that the moderate improvements that we were able to make to the CPS plan in the previous section are probably not far from the upper bound under the given constraints. Constraining affirmative action plans to be race-neutral and place-based will substantially increase the costs that are incurred if one tries to increase minority representation at Chicago’s most elite public schools. Table 3 also reports overall efficiency measures for the plans discussed above. They are as high as 76 percent for plans that use the individual-level FRPL data. When the overall efficiency evaluated using the mean, rather than the standard deviation, the overall efficiency is never greater than 62 percent.34 Even these bounds should be regarded as an overly optimistic view of what can be accomplished in a race-neutral manner because it assumes that the FRPL variable that is now being used perfectly captures the socioeconomic disadvantage that CPS cares about, and gives our modi- fied plan a great deal of credit for capturing such disadvantage accurately. We believe that the exercise of using an income proxy to examine how efficient a plan admits disadvantaged students is better motivated for plans that do not make use of an income proxy variable. It would be interesting to examine the overall efficiency of plans that use the income variable also considering how efficiently they admit students who are disadvantaged on other dimensions, e.g., coming from single-parent families or hav- ing parents who did not attend college, but we lack the data to do so. It is also worth noting that the potential ways to increase efficiency we have examined only have a moderate impact on the number of Asian students. The pro- portion of Asian students is moderately lower under our benchmark plans that use 34 These and other additional results are shown in online Appendix Table A6. VOL. 111 NO. 3 ELLISON AND PATHAK: RACE-NEUTRAL ALTERNATIVES 973 race and FRPL data. At Payton, 12.3 percent of students at Payton are Asian under the CPS plan and 10.0 percent are under the race and FRPL-based benchmark. At Northside, 24.7 percent of students are Asian under CPS plan and 23.6 percent are under the race and FRPL-based benchmark. Some of the alternate race neutral plans we have considered slightly increase Asian representation at one or both schools. See online Appendix Table A6. VI. Recap and Conclusions This paper evaluates affirmative action plans when schools develop curricula tai- lored to students’ ability or preparation levels and a diverse learning environment. Absent any restrictions on its form, we show that affirmative action in Chicago’s exam school sector would be a powerful tool. There are sufficiently many minority and low-income students with composite scores within two or three points of the purely score-based cutoffs, which allows for a substantial increase in low-income and/or minority enrollment without a large impact on the distribution of admitted students’ composite scores. For example, it is possible to roughly double minority and/or low-income enrollment by giving preferences on the order of two or three points. Place-based affirmative action plans expand the definition of disadvantage to include neighborhood characteristics and Chicago’s plan is one of the most ambi- tious and sophisticated of such plans. However, it performs worse than racial quotas in increasing minority or low-income access. This fact shows broadening the defi- nition of disadvantage need not entail a trade-off between minority enrollment and other measures of disadvantage. The CPS plan incurs higher curriculum-matching losses and widens within-school racial achievement gaps. We show that there is room to improve the efficiency of the CPS plan. By altering CPS’s index of socioeconomic status, using continuous bonuses rather than dis- crete SES tiers and exploiting individual-level data on FRPL status, it is possible to achieve the same levels of minority and FRPL representation with significantly fewer distortions. However, a key finding is that there are still significant limits to the efficiency of place-based affirmative action policies. One main reason is that the high-achievers that Payton and Northside admits are by definition outliers for their neighborhood, which complicates the relationship between neighborhood character- istics and true measures of disadvantage. While we have focused on Chicago, contentious debates about the lack of diver- sity at exam schools have taken place in other cities. The effectiveness of any affir- mative action plan depends both on the design of the plan and the city in which it is implemented. Our ability to simulate the effects of different plans in Chicago have given us some scope to provide estimates of the range of efficiencies that are possi- ble under different plans. The differing levels of efficiency at Payton and Northside demonstrate that a substantial range in efficiency exists both across plans and tar- gets; while we achieved more than 50 percent minority-only efficiency at Payton, the most efficient plan reached just 36 percent at Northside. Our framework can be applied to other cities to help quantify the efficiency of affirmative action and eval- uate the performance of alternatives. Our framework to evaluate affirmative action plans is based on the premise that grouping together high ability students allows for schools to offer advanced 974 THE AMERICAN ECONOMIC REVIEW MARCH 2021 curricula that would not be available elsewhere. 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