American Economic Association
On the Demographic Composition of Colleges and Universities in Market Equilibrium Author(s): Dennis Epple, Richard Romano, Holger Sieg Source: The American Economic Review, Vol. 92, No. 2, Papers and Proceedings of the One Hundred Fourteenth Annual Meeting of the American Economic Association (May, 2002), pp. 310-314 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/3083422 . Accessed: 01/02/2011 12:44 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=aea. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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On the Demographic Composition of Colleges and Universities in MarketEquilibrium By DENNIS EPPLE, RICHARDROMANO,AND HOLGERSIEG* Achieving diversity in the racial and ethnic makeupof the studentbody is one of the objectives that a college pursuesin making decisions about admission and financial aid.' There has been a tremendousamountof debate aboutpolicies that colleges should adopt, or be proscribedfrom adopting,with respectto diversity. Ratherthan join this largely normativedebate, we adopta positive focus, investigatinghow colleges' questsfor ethnicand racialdiversityaffect admissionand financialaid policies, the college quality hierarchy,and the distributionof whites andnonwhitestudentsacrosscolleges of differing qualities.We providea theoreticalframeworkfor investigatingthe effects of preferencesfor diversity augmentedwith resultsfrom a parallelcomputationalmodel calibratedto U.S. data. The effects of efforts to achieve diversity in colleges depend not only on the weight that colleges give to diversity, but also on the value that prospectivestudentsplace on diversity. Diversity may be of concern to no one, only to colleges, to both colleges and students, or to colleges and a subset of students. We first describe a baseline model without preferencesfor diversity and then examine three alternatives
in which preferences for racial diversity are present. I. Theoretical Background
Here we briefly summarize a theoretical model and results which are developed in detail in Epple et al. (2002a, b). A potential student is characterizedby three variables: race (r), household income (y), and score (s) on a standardizedcollege entranceexam. Race is dichotomous, with individuals either white (w) or nonwhite (nw). Let Fr denote the proportionof race r in the population.Race-conditioneddistributionson (s, y) are assumed to be continuous on 2+. We also assume that (r, s, y) is observablefor all potential studentswho attend college in equilibrium.We assume that all who go to college take the entranceexam. The potentialstudent(or his household)maximizes an increasing utility function of numeraire (x) and educational attainment (a): U = U(x, a). The numeraireis given by x = y
-
Pi if college i is attended, i = 1, 2, ...1,
where pi is tuition at college i. Educational attainmentis an increasingfunctionof score and the perceived quality (Qr) of the college attended: a = a(Qr, s). We examine several college quality measuresdiscussed below. If no college is attended,then Qr = Qr, which can then be interpretedas quality of high school. We assume a positive income elasticity of demandfor school qualityand a nonnegativescore elasticity of demand.Taking as given colleges' tuition and admission policies, students choose among their options to maximize utility. There aren colleges, differentiatedex ante by their "endowmentearnings,"Ri, which consists of all nontuitionearnings.Number the colleges < R,. Subject to their so that RI < R2 < budget constraint,each college chooses tuitionadmission policy and expenditure on educational inputs to maximize its perception of
* Epple and Sieg: CarnegieMellon University,Graduate School of IndustrialAdministration,Pittsburgh,PA 15213; Romano: University of Florida, Departmentof Economics, Gainesville, FL 32611. We thank the National Science Foundation and the MacArthur Foundation for research support.We appreciatethe comments of Stephen Coate on an earlier draft. ' In their pioneering analysis, Charles F. Manski and David Wise (1983) found evidence of significant tuition discounting to nonwhite students. This has been reaffirmed in subsequent research by John Kane and Lawrence M. Spizman (1994) and Epple et al. (2002a). William G. Bowen and Derek Bok (1998) provide a comprehensive analysis of affirmative-action policies in top institutions of higher education. Thomas J. Kane (1998) finds that probabilities of admission for nonwhites are significantly higher in the top quintile (by mean SAT quality, QC, taking as given its endowment score in the student body) of institutions of higher education. and students' equilibrium alternatives. Letting 310
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MARKETIMPLICATIONSOF PEER AND NEIGHBORHOODEFFECTS
ki denote the number of students who attend college i and Ii educationalinputs per student, college costs are assumed to be: (1)
C(ki, Ii)
= F' + V(ki) + kiIi
V'1,V"> O.
Hence, there are minimalor "custodialcosts" of F + V(ki), and costs thatincrease linearly with educationalinputs in the numberof students. We examine four versions of the model which vary with the assumed quality measures. Let q( 0, I) denote race-independent college quality, with q increasing in (0, I) and 0 the mean entrance-examscore in the college. The baseline model or model I is that developed in Epple et al. (2002a), which assumes thatquality is diversity-independent: QC
QW
underrepresentedrace share a diversity preference [QC _ Q(q(Qi, Ii), FD, [" < Fr] but the overrepresentedrace is diversity-neutral IQ = q(Oi, Ii), Fi > rF] Market equilibrium satisfies utility maximization by all students,quality maximizationby all n colleges, and a market-clearancecondition. To describe the solution to a college's quality-maximization problem, let
tors have a preference for diversity [QC Q(q(0i, Ii), Fr), Fr< Fri. but both races are diversity-neutral [Qw Q= W q(0i Ii)].
Model 3 then has a paternalisticelement. Model 4 assumes that college administratorsand the
2 In additionto learning from one's peers, it is plausible that faculty prefer to teach more acadernically inclined students, helping colleges with better peer groups to hire more effective faculty. The positive model is also consistent with students' reputations (e.g., in the job market) being enhancedby going to college with higher-scoringstudents. There is a large, growing, and controversial literatureon peer effects by social scientists. Here we mentionjust some of the empiricalstudies on peer effects in higher education. Bruce Sacerdote (2001) finds peer effects between roommates on grade point averages. Julian R. Betts and Darlene Morell (1999) find that high-school peer groups affect college grade point average. Peter Arcidiacono and Sean Nicholson (2000) find no peer effects among medical students.
a Z4 > Z3 > 0.
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We now present a propositionthat holds for all four models. PROPOSITION1: Qc < Qc2 < < Qn.The ... college qualityhierarchyfollows the endowment hierarchy. In the next proposition,we refer to boundary loci, which are (generally race-conditioned) two-dimensional loci in the (s, y) plane that partitionstudentsinto colleges andno college in equilibrium.Regardingpricing and the allocation of types we have the following:
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given ability and race, if student I attends college i and student 2 attends college j with Qr > QJ., then Yi > Y2. (b) Within-racescore stratification across the quality hierarchy as perceived by students arises in equilibrium if (qIlq1)i weakly ascends along that quality hierarchy. Under the latter condition and for given income and race, if student I attends college i and student 2 attends college j with then s1 Qr > QJr,
PROPOSITION 2: (a) Tuition in college i equals EMCralong boundaryloci of college i. Along boundary loci, tuition decreases with score, is linkedto race wheneverQcis diversity-
It bears emphasis that, given differences in the distributionof types (s, y) across the races, attendance sets will differ whenever diversity mattersto just colleges or to both colleges and (some) students.The natureand implicationsof these differences is a primary concern, which we examine computationally.
dependent and Fr * Fr, and is independent of income. (b) In the interior of college i's atten-
II. Computational Model and Results
dance set, tuitionexceeds EMC'. Here tuitionis also linked to income. (c) The equilibriumattendancesets are the same as would result if all colleges i instead set tuition to every student equal to the EMCr's that arise in the actual equilibrium. The discounting to higher scores that arises for given race along a college's boundaryloci reflects, of course, the value of higher score as an input to quality (as perceived by colleges). Analogously, when colleges value diversity, a student from the underrepresentedrace on a college's (race-conditioned) boundary locus pays lower tuition than a student of the same score from the overrepresentedrace also on a (race-conditioned)boundary locus of the college. While students in the interior of a college's attendancespace in the (s, y) plane (for given r) pay somewhat higher tuition than their EMCri, the discounting to score and to the underrepresentedrace (the latterin models 2, 3, and 4) persists if the degree of competition is moderate. The positive income elasticity of demandfor college quality and the nonnegative score elasticity, along with Proposition2c, imply that the equilibriumattendancesets satisfy some stratification properties. PROPOSITION3: (a) Within-raceincomestratification across the quality hierarchy as perceived by students arises in equilibrium. For
In our calibration,we seek to capturebroad regularitieswith respect to the distributionsof incomes, SAT scores, and college attendance within the United States, and also the broad patterns of variation in SAT scores, endowments, tuitions, and financialaid in U.S. private institutions offering four-year undergraduate degrees. We use data from the U.S. Census and data from the National Center for Education Statistics (NCES). We calibrateusing model 1, assuming six colleges and two groups (whites andnonwhites).Data for nonwhitesare obtained by combining data for the African-American and Hispanic populations. Here we summarizeour calibration,referring readersto Epple et al. (2002b) for details. Nonwhites comprise 20 percent of the population. The distributionsof income are lognormal,with means and medians calibrated to year-2000 U.S. data, implying for nonwhites, ln y JV(10.36, 0.746) and, for whites, ln y JV(10.70, 0.764). We assume that SAT distributions would be normal if all individuals of college age were to take the exam, and we choose parametersof those distributionsso that for the subset of students attendingcollege the moments match the observed values (means of 1,049 and 919 for whites and nonwhites,respectively, and standarddeviations of 192 and 214). The correlationof ln y and SAT is set equal to 0.25 for both races. Per-studentendowment income, assuming a 2-percentreal draw per year,
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is $150, $200, $300, $500, $900, and $4,000, respectively, in our six colleges. We assumea Cobb-Douglasutility-achievement function,with exponentson numeraireconsumption, college inputs,and mean scores of 1, 0.06, and 0.03, respectively.3The exponenton percentage nonwhite,4,,is set equalto 0 in model 1. Our cost functionfor colleges is cubic:22.7 + 200k + 7,043k2 + 925,000(k - k*)3 + kI, where k is the fraction of the population served by the college and k* is the cost-minimizing scale, which we set equal to 0.06. The outside option is assumed to be the same across races and chosen such that the equilibriumproportionof the population attending four-year colleges is the fraction observed in the data, 28.2 percent. A summaryof equilibriumin our calibrated model is provided in the first column of Table 1. Note in Table 1 that the equilibriumproportions of whites and nonwhites are 31.4 percent and 15.4 percent, respectively. The proportions we estimate from the data (but do not use in calibration) are 31 percent and 17 percent. Given that preferences for racial diversity are not present in model 1, the equilibriumvalues differ from the actual values in the "correct" direction. Looking down the column for model 1 in Table 1, we see that the proportion of minority students is smaller in higher-ranked colleges, this due to the differences across races in the calibrated distributions on (s, y). The model is calibratedto the empirical mean SAT scores, averageeducationalcosts, and mean discount to score, so the correspondencebetween the equilibrium values of these variables for model 1 and the empirical values is imposed. The main predictive weakness of the model is thatthe equilibriumvalues of averageincome of college attendeesaremore thanone-thirdhigher than the values we find in our NCES data. We are presently extending our model to permit considerationof preferences for income diversity as well as racial diversity, an extension that we expect will increase income diversity within schools, reducing mean incomes. To provide a meaningful scaling of achievement, we set normed achievement equal to income, calibrating so that mean normed achievementfor whites equals mean income for
3The exponent on score is irrelevantdue to the CobbDouglas specification.
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TABLE 1-SUMMARY OF EQUILIBRIA IN THE CALIBRATED MODELS
Model Variable
1
2
3
4
Percentagein college White Nonwhite
31.4 15.4
30.2 20.1
30.7 18.3
30.4 19.3
Percentagenonwhite Outside college College 1 College 2 College 3 College 4 College 5 College 6
23.6 13.0 13.1 11.9 10.6 8.8 5.8
22.3 14.1 14.2 14.3 14.3 14.3 13.7
22.8 16.3 15.2 13.8 12.0 9.3 8.4
22.5 15.1 14.7 14.3 13.5 11.9 11.1
1,045 917
1,046 910
1,045 913
1,046 912
589 13,007
2,067 584 12,870
1,284 593 13,065
1,903 592 13,043
-1.8 6.6
-0.8 4.2
-1.2 5.7
Average SAT White Nonwhite ' Nonwhite aida Aid/l00 SATb Expenditurec Effects on future incomed
White Nonwhite
Average additionalaid to nonwhites,relativeto whites. Average aid per 100 points on SAT. 'Average college expenditureper student. d Percentagechange, based on changes in mean normed achievement. a b
whites and mean normed achievementfor nonwhites equals mean income of nonwhites. Changes in normed achievement across the models can thenbe interpretedloosely as effects on futureincome. These are presentedin the last two lines of Table 1 again using Model 1 as the baseline. The second column of Table 1 is the equilibrium for model 2 when the exponent on diversity in the college-quality function, 4',equals 0.0015. This value of 4,yields additionalfinancial aid to nonwhite relative to white students that is on average approximatelythe value estimated by Epple et al. (2000a) for AfricanAmerican students. With this value of 4i, the proportionof nonwhites attendingcollege rises to 20.1 percent while the proportionof whites declines to 30.2 percent. Thus, this value of 4i implies an increase in the proportionof nonwhites to a level that exceeds the observed proportion of 17 percent. Looking down the
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column for model 2, we see thatpreferencesfor racial diversity in model 2 increase the proportion of minority students in all schools, but with proportionallygreatereffect in the higherrankedschools. Models 3 and 4 in Table 1 assume the same value of qrin the college quality function as in model 2 for those who value diversity. Intuition suggests that model 2 would yield the largest discount to nonwhite students and the largest effect on their attendancein college; model 4 should have the next-highesteffects, and model 3 should have the lowest effects. Looking across the columns for models 2-4, we see that the aid amounts for nonwhite students and the proportionsof nonwhites in the different colleges confirm these expectations. We also see that model 2 not only has the highest average proportion of nonwhites, but also the most nearlyequal proportionsacross colleges. Model 3 yields the lowest proportionof nonwhites and the most variationin this proportionacross the colleges. The differing preferencesfor diversity have very modest effects on mean SAT scores and discounts for high SAT scores, but quite substantial effects on participation of nonwhites. This is due to there being many nonwhite students on the margin of attending college.
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are an exaggeration. State-affiliated colleges may pursue substantially different objectives. Student preferences are likely to vary in more ways than we have allowed. Nevertheless, we believe that the modeling here helps to clarify the equilibrium promotion of diversity in higher education and provides a useful building block. REFERENCES Arcidiacono, Peter and Nicholson, Sean. "The
Mirage of Peer Effects: Evidence from U.S. Medical Schools." Mimeo, Duke University, 2000. Betts, Julian R. and Morell, Darlene. "The De-
terminantsof UndergraduateGPA: The Relative Importance of Family Background, High School Resources, and Peer Group Effects." Journal of Human Resources, Spring
1999, 34(2), pp. 268-93. Bowen, William G. and Bok, Derek. The shape of the river: Long term consequences of considering race in college and university admis-
sions. Princeton, NJ: Princeton University Press, 1998. Epple, Dennis; Romano, Richard and Sieg,
Holger."PeerEffects, FinancialAid, and Selection of Studentsinto Colleges and Universities." Journal of Applied Economics, 2002a
III. Conclusion
(forthcoming). "On Affirmative Action in Higher Education." Mimeo, Carnegie Mellon University, 2002b. 0
We find that an objective of colleges that places weight on the racial mix of their student body can have predictionsthat are broadlyconsistent with what is observed. Not surprisingly, colleges will be more diverse if households also value diversity. What may be more useful in choosing among the specifications is that predicted variation in diversity along the college quality hierarchy is substantially lower when households value diversity than when they do not. This paperis partof a largerprojectaimed at empirically identifying objectives and interactions of institutionsof higher education.We are the firstto admitthatthe spartanmodel here can only go so far in predictingthe characteristicsof higher education in the United States. The model fails to predict the substantial needbased aid that is observed in the data. The neat stratificationproperties that are predicted
Kane, John and Spizman, Lawrence M. "Race,
Financial Aid Awards and College Attendance: Parents and Geography Matter." American Journal of Economics and Sociology, January 1994, 53(1), pp. 85-97.
Kane, Thomas J. "Do Test Scores Matter? Racial and Ethnic Preferences in College Admissions," in Christopher Jencks and Meredith Phillips, eds., The black-white test
score gap. Washington,DC: Brookings Institution Press, 1998, pp. 431-56. Manski, Charles F. and Wise, David. College choice in America. Cambridge, MA: Harvard
University Press, 1983. Sacerdote, Bruce. "Peer Effects with Random
Assignment: Results for Dartmouth Roommates." Quarterly Journal of Economics,
May 2001, 116(2), pp. 681-704.
