Why Retaining Racial Preferences in College Admissions
May Yield a More-Able Student Body
By
Dong Li and Dennis L. Weisman *
RRH: LI & WEISMAN: RACIAL PREFERENCES
JEL classification: I20; I28
Why Retaining Racial Preferences in College Admissions
May Yield a More-Able Student Body
Abstract
Critics of affirmative action policies contend that the elimination of racial preferences in
college admissions would lead to a “more-able” student body. We develop a simple
model comprised of three classes of college admissions—merit, race and legacy—to
show that a change in admissions policy that reduces racial preferences can lead to a
“less-able” student body. In fact, the change in admissions policy may serve only to
ensure that more admissions are available for “sale” to wealthy alumni through legacy
preferences. The numerical simulations confirm the theoretical results. When there are
multi-dimensional preferences, reducing or eliminating one dimension of preferences
may result in a “less-able” student body.
Keywords: Education, Discrimination, Racial Preferences, Legacy Preferences
Dong Li and Dennis L. Weisman
Department of Economics
Kansas State University
Manhattan, KS 66506-4001
1
I. Introduction
In two recent landmark cases involving the University of Michigan, the U.S. Supreme
Court affirmed that colleges may employ narrowly-tailored, race-based admissions
policies if they contribute to the diversity of the student body. The Court indicated that it
was primarily concerned with diversity of experience rather than diversity of color.1 The
Court’s decisions satisfied neither side of the political spectrum—liberals contending that
colleges should have broad discretion over admissions in order to enhance the overall
educational experience of their students, conservatives contending that such preferences
necessarily produce a “less-able” student body.
What is frequently overlooked in this debate is the fact that racial preferences
represent only one of several types of preferences that prestigious colleges regularly
employ in selecting their student bodies. Other types of preferences include those for
talented athletes and the offspring of wealthy alumni, so-called legacy preferences.2 To
date, there has been no serious challenge to these other types of preferences.3 This is
curious because eliminating one type of preference—say racial preferences—while
retaining other types of preferences could well produce a “less-able” student body.
This
can occur, for example, if (i) the average ability of the students admitted under racial
preferences exceeds that of the new group of students admitted under some other
preference; or (ii) the average ability within the admissions subset decreases rapidly as
the proportion of admissions drawn from that subset increases.
Espenshade et al. (2004) and Espenshade and Chung (2005) find evidence that the
athletes and legacy preferences are larger in magnitude than the racial preference while
the racial preference is much more controversial. Fu (2006) shows that handicapping the
2
stronger contestants may increase the competition, and that therefore racial preferences
may help academic quality-oriented colleges increase the academic quality of their
students.
The format for the remainder of this paper is as follows. Section II formally derives
sufficient conditions for the elimination of racial preferences to produce a “less-able”
student body. Simulations are provided in Section III. The conclusions and policy
implications are discussed in Section IV.
II. The Model
In this section, we formally derive sufficient conditions for the elimination of racial
preferences and the retention of legacy preferences to yield a “less-able” student body.
We begin with a simple numerical example to illustrate the nature of the problem
and to motivate the formal analysis. Consider a pool of 10 applicants with academic
abilities in descending rank order {10, 9, 8, … 2, 1}. Suppose that the racial applicant
pool is {3,5} and that the legacy pool is {2,4}.
4
Suppose that five applicants will be
admitted and that there will be one racial admission and one legacy admission. The
admission pool is {10,9,8,5,4} under both racial and legacy preferences. Suppose that we
eliminate the racial admissions and expand the legacy admissions (i.e., three merit
admissions and two legacy admissions), so that the admission pool is now {10,9,8,4,2}.
In this case, the average ability declines after the elimination of racial preferences,
contrary to the claims of the opponents of the affirmative action policy. The intuition is
straightforward: you will go further down the distribution on one dimension when you
remove another dimension of preferences.
3
A. Assumptions and Definitions
Suppose that there are Z applicants for N available admissions, where N < Z. The
ability level of each applicant q is denoted by Sq, where q = 1, . . . , Z.
The set of
students applying for admission,{S}, can be partitioned into subsets {Sr}, {Sl} and {Sn},
where the superscripts r, l and n indicate, respectively, that the student is an underrepresented, race-based minority,5 alumni offspring, or neither.
Assumption 1. (Well-Ordering) The applicants for admission can be rank-ordered in
descending order of ability level, {S1, . . . , SN, . . . , SZ}.
Assumption 2. The set of rank-ordered admissions, {A}, can be partitioned into subsets of
merit admissions, {Am}, racial-preference admissions, {Ar}, and legacy-preference
admissions, {Al}.
Definition 1. A preference-free admissions set is given by {A} = {S1, . . . , SN}.
Definition 2. A merit admission is the admission Aim of any St  S N .
Definition 3. A racial-preference admission is the admission Arj of any S tr  S N .
Definition 4. A legacy-preference admission is the admission Akl of any Stl  S N .
Assumption 3. (Continuity) d ( Ats , Ats1)  Ats  Ats1  0 s  m, r , l ; t  i, j, k . 6
Assumption 4. The order of admissions within any subset, { Am },{ Ar },{ Al }, proceeds
sequentially in decreasing rank-order from “most-able” to “least-able.”7
In the benchmark case, the average ability of the student body is given by
Nm
Nr
Nl
i 1
j 1
k 1
m
r
l
(1) A  [  Aim   Arj   Akl ] / N  sm A  sr A  sl A ,
4
where N m  N r  N l  N , and ss  N s / N , s  m, r , l is the share of each of the three
s
Ns
groups represented in the set of N admissions and A   Ats / N s , s  m, r , l ; t  i, j, k is
t 1
m
r
the average ability level within the admissions subset. By Definitions 2-4, A  A and
m
l
A  A , respectively.
Definition 5.
An admissions policy consists of a three-tuple of admission shares
{sm , sr , sl } , with 0  ss  1 and  ss  1, for s = m, r and l.
s
 A ss
 0, s = m, r and l define the average-ability elasticities.
Definition 6. Let  s 
ss As
B. Key Findings
Consider now a “small” change in the admissions policy that allows for a reduction
in the share of racial preference admissions ( dsr  0 ) and an offsetting increase in the
shares of merit and legacy preference admissions ( dsm  dsl  0 ). This implies that
(2) dsm  dsr  dsl  0.
Proposition 1. The change in admissions policy leads to a “less-able” student body
when
m
(3)
l
A (1   m )dsm  A (1   l )dsl
(dsm  dsl )(1   r )
r
A.
Proof: Taking the total differential of (1) implies that
m
r
l
r
l
A
A
A
)dsm  ( A  sr
)dsr  ( A  sl
)dsl .
(4) d A  ( A  sm
sm
sr
sl
m
dsr  (dsm  dsl ) follows from (2). Substitution yields
5
m
m
(5) d A  ( A  sm
r
l
r
r
A
A
A
)dsm  ( A  sr
)( dsm  dsl )  ( A  sl
)dsl .
sm
sr
sl
Setting d A  0 in (5) and appealing to Definition 6 yields
m
r
l
(6) A (1   m )dsm  A (1   r )(dsm  dsl )  A (1   l )dsl  0.
r
Solving for A in (6) yields the result in (3). ■
r
Proposition 1 establishes a lower-bound for A sufficient for the change in education
policy to lead to a “less-able” student body.
Lemma 1. The change in admissions policy leads to a “more-able” student body when
dsl  0.
Proof: When dsl  0 , (4) reduces to
m
r
r
A
A
)  ( A  sr
).
(7) d A  ( A  sm
sm
sr
m
Let sm 1  sr  sl , then
m
r
r
A
A
 A  sr
,
(8) d A  A  (1  sr  sl )
sm
sr
m
or
m
m
r
r
A
A
A
 A )  ( sr  sl )
 sr
.
(9) d A  ( A 
sm
sm
sr
m
The second and third terms on the right-hand side of (9) are non-negative. Also, since
m
r
A
 A , so the first term on the right-hand
A  A and A  A for all i and j, A 
sm
m
r
m
i
r
j
m
side of (9) is positive. It follows that d A  0. ■
6
Because the ability of each additional merit admission exceeds the ability of each
displaced racial-preference admission, the change in admissions policy necessarily leads
to a “more-able” student body.
Lemma 2. The change in admissions policy when dsm  0 leads to a “less-able” student
body whenever:
l
r
a) A  A and  l   r ; or
l
r
b) A  A and  l   r .
Proof: Under the conditions stated in the proposition, (3) reduces to
(10) Al (1   l )  Ar (1   r ).
Equation (10) is satisfied if either condition a) or condition b) holds. ■
The change in admissions policy leads to a “less-able” student body if (i) the average
ability of legacy-preference admissions is lower than that of racial-preference admissions
and the decrease in ability is comparable as one moves further down the respective
applicant pools; or (ii) the average ability levels are comparable across the two
admissions subsets, but the average ability level of legacy-preference admissions
decreases at a faster rate than that of racial-preference admissions.8
Proposition 2. There exists a change in admissions policy of the form
dsr  0, dsm  0, dsl  0 and dsm  dsr  dsl  0 that leads to a “less-able” student body
whenever:
l
r
a) A  A and  l   r ; or
l
r
b) A  A and  l   r .
Proof: By Lemma 1, the change in admissions policy leads unequivocally to a “moreable” student body when dsl  0. By Lemma 2, the change in admissions policy leads to
a “less-able” student body when condition a) or condition b) holds. By continuity of
7
ability within the admissions subsets (Assumption 3), there must exist a change in
admissions policy of the form dsr  0, dsm  0, dsl  0 and dsm  dsr  dsl  0 that leads to
a “less-able” student body. ■
Due to the lack of real data, we resort to simulations in the next section to
empirically examine this issue.
III. Simulations
In this section we conduct simulations of the changes in admission policy. Values of the
parameters are chosen to mimic a hypothetical selective college. It should be pointed out
that the results are qualitatively similar when different values of the parameters are
chosen.
Assume that the ability of the three groups of applicants is randomly drawn from a
tri-variate normal distribution N(μ,Σ) where μ=[μm, μr, μl] is the mean ability vector of
the three groups, Σ is a 3 by 3 positively definite variance-covariance matrix.
Assume that there are 10,000 applicants, of which 1,500 are racial applicants, 1,500
are legacy applicants, and 7,000 applicants are neither. Suppose that there are 1,500
admissions available. Under the current policy there are 1050 merit admissions, 225
racial-preference admissions, and 225 legacy-preference admissions.
In the results
presented below, we assume that if we eliminate the racial-preference admissions, there
are 450 admissions for legacy preferences. In other words, the entirety of the racialpreference admission quota goes to the legacy-preference admissions. This corresponds
to the case where dsm  0 as in Lemma 2.9 Under each parameterization of the model,
1000 repetitions are carried out to compare the average student ability 10 under the current
policy (with both racial-preference admissions and legacy-preference admissions) with
8
that under the proposed policy (with only legacy-preference admissions).
In each
simulation, merit admissions are allocated purely based on the order of ability, regardless
of racial or legacy status. Racial admissions and legacy admissions are subsequently
allocated in the order of ability within the respective applicant pool.
Of the 1000
repetitions, we calculate the probability that the average student ability under the current
policy (with both racial and legacy admissions) is higher than that under the proposed
policy (with only legacy admissions).
We vary μ and Σ in our simulations to find the break-even point in terms of average
student ability for the two policies. Different parameterizations of μ and Σ leads to
different conditions analyzed in Section 3. For example, the conditions that Σ is an
identity matrix and μl < μr coincide with condition a) in Lemma 2. Similarly, the
conditions that μl = μr and σl > σr with all covariances equal to zero coincide with
condition b) in Lemma 2. In this case, the ability of the legacy applicants declines faster
than that of the racial applicants. In the interest of economy of presentation, we present
only the results based on Σ as an identity matrix and various values for μ . Figures based
on varying Σ with fixed μ are qualitatively similar and are available upon request.
[Figure 1 is inserted around here]
[Figure 2 is inserted around here]
We fix μm = 0 in all of the simulations. This is arbitrary and trivial for the results.
The results are summarized in Figure 1. For example, when μl = - 3 and μr= - 1 (the top
left point in the graph), abolishing the racial admissions leads to a less-able student body
with 100% probability. When μl = μr the probability is still 100% or close to be 100%.
Only when μl is much larger than μr does abolishing the racial admissions preference lead
9
to a more-able student body. To better understand the graph, in Figure 2 we present a
two dimensional cut of Figure 1 at μl = 0. In Figure 2 we can see that the break-even
point is around μr= -1. This means that if the mean ability of the racial applicants is
within one standard deviation less than the mean ability of the legacy applicants, the
current policy dominates the proposed policy. Only if the mean ability of the racial
applicants is significantly (more than one standard deviation) less than that of the legacy
applicants does the proposed policy lead to a more-able student body.
[Figure 3 is inserted around here]
[Figure 4 is inserted around here]
To closely mimic reality, we also consider the case in which legacy admissions are
not based on the order of ability among the legacy applicants. It is generally believed
that selective colleges allocate racial admissions to the best minority applicants while
they do not necessarily allocate legacy admissions to the best legacy applicants, but rather
to the legacy applicants from more affluent families.11, 12 Here we assume that legacy
admissions are allocated randomly among the legacy applicants who have not been
admitted based on merit. The results are presented in Figures 3 and 4. As one can see
the results are even stronger than those in Figures 1 and 2. For example, Figure 4 shows
that only if the mean ability of the racial applicants is more than two standard deviations
lower than that of the legacy applicants does the proposed policy leads to a more-able
student body.
Our simulation results show that for a large parameter space, abolishing racialpreference admissions leads to a less-able student body, contrary to what opponents of
10
such policies may claim. The numerical simulations confirm the theoretical findings in
Section 2.
It should be pointed out that we maintain the assumption that the policy change
would cause no behavior change in the applicants. For example, we assume in our
simulations that the number of applicants from each group remains the same regardless of
policy changes.
In practice, one might expect the policy change to induce some
behavioral changes. For example, Long (2004a,b) finds that in response to the
elimination of affirmative action in California and Texas minority students sent SAT
reports to lower quality schools. Dickson (2006) reports similar findings, while Card and
Krueger (2005) find no behavioral change in highly qualified black or Hispanic students.
IV. Conclusions
Thomas Paine observed more than 200 years ago that “virtue is not hereditary” (Paine,
1776, p. 54). It is apparent that at least in some cases ability is not hereditary either. This
has given rise to so-called legacy preferences whereby wealthy alumni are essentially
able to “buy” admissions for their less-able offspring. To date, legacy preferences have
attracted relatively little attention as the focus has been on racial preferences and recent
court decisions circumscribing its use.
Critics of race-based admissions policies contend that eliminating racial preferences
will necessarily lead to a “more-able” student body. As the analysis in the previous
sections attest, this need not be the case. In fact, the elimination of racial preferences in
combination with the retention of legacy preferences could well lead to a “less-able”
student body.13 Hence, if the real intent of the policy change is to produce a “more-able”
student body, then consideration should perhaps be given to eliminating all preferences,
11
not
simply
racial
preferences
(i.e.,
a
preference-free
admissions
set).
Admittedly, such policy prescriptions are unlikely to be popular with college
administrators as they greatly limit their ability to extract contributions from wealthy
alumni.
Finally, it should be recognized that eliminating racial preferences while retaining
legacy preferences may prove to be doubly-damning to minorities—possibly even
reversing the laudable gains in college graduation rates that these groups have achieved
over the last two decades (Bowen and Bok, 1998).
The concern is that past
discrimination in this country, inclusive of its implications for the distribution of
wealth,14 may limit the degree to which minorities can avail themselves of legacy
preferences.
12
References
Bowen, William G. and Derek Bok. The Shape of the River: Long-Term Consequences
of Considering Race in College and University Admissions. Princeton NJ: Princeton
University Press, 1998.
Card, David and Alan B. Krueger. “Would the Elimination of Affirmative Action Affect
Highly Qualified Minority Applicants? Evidence from California and Texas.” Industrial
and Labor Relations Review, Volume 58, 2005, pp. 416-34.
Dickson, Lisa. “Does Ending Affirmative Action in College Admissions Lower the
Percent of Minority Students Applying to College?” Economics of Education Review,
Volume 25, 2006, pp. 109–119.
Espenshade, Thomas J. and C.Y. Chung. “The Opportunity Cost of Admission
Preferences at Elite Universities.” Social Science Quarterly, Volume 86, 2005, pp. 293305.
Espenshade, Thomas J., C.Y. Chung, and J.L. Walling. “Admission Preferences for
Minority Students, Athletes, and Legacies at Elite Universities.” Social Science
Quarterly, Volume 85, 2004, pp. 1422-46.
Fu, Qiang. “A Theory Of Affirmative Action in College Admissions.” Economic Inquiry,
Volume 44, 2006, pp. 420–428.
Golden, Daniel, The Price of Admission: How America’s Ruling Class Buys Its Way Into
Elite Colleges–and Who Gets Left Outside the Gates. New York: Crown Publishers,
2006.
Goldstein, Amy. “Bush Hits ‘Legacy’ College Admissions; President Addresses Minority
Journalists.” The Washington Post, Washingtonpost.com, August 7, 2004, Section A.
Gratz et. al. v. Bollinger et. al., Case No. 02-516, Before The United States Supreme
Court, Decided June 23, 2003.
Grutter v. Bollinger et. al., Case No. 02-241, Before The United States Supreme Court,
Decided June 23, 2003.
Lipsey, R.G. and K. Lancaster. “The General Theory of the Second Best.” Review of
Economic Studies, Volume 24, 1956, pp. 11-32.
Long, Mark C. “College Applications and the Effect of Affirmative Action.” Journal of
Econometrics, Volume 121, 2004a, pp. 319-342.
Long, Mark C. “Race and College Admissions: An Alternative to Affirmative Action?”
Review of Economics and Statistics, Volume 86, no. 4, 2004b, pp. 1020-33
13
Loury, Glenn C. The Anatomy of Racial Inequality. Cambridge MA: Harvard University
Press, 2002.
Paine, Thomas. Common Sense, New York: Barnes and Noble, 1995 (Originally
published in 1776).
14
15
16
*
The authors are grateful to Alfred Kahn for helpful discussions and to Inung Jung and Daigyo Seo for
expert research assistance. The usual caveat applies.
Li: Associate Professor, Department of Economics, Kansas State University, Manhattan, KS 66506-4001,
(785) 532-4572, dongli@ksu.edu.
Weisman: Professor, Department of Economics, Kansas State University, Manhattan, KS 66506-4001,
(785) 532-4588, weisman@ksu.edu.
1
The Court observed that “Enrolling a ‘critical mass’ of minority students simply to assure some specific
percentage of a particular group merely because of its race or ethnic origin would be patently
unconstitutional.” Gutter v. Bollinger et. al. (2003, p. 3).
2
See Golden (2006) for a provocative account of the pervasive use of legacy preferences in college
admissions at many prestigious colleges.
3
Recently, President Bush called for an end to legacy preferences in college admissions. See Goldstein
(2004).
4
This corresponds to condition a) in Lemma 2.
5
These include Blacks, Hispanics and Native Americans.
6
This assumption essentially says that the difference in ability between any two adjacent admissions in any
well-ordered admissions subset is “small.”
7
As Loury (2002, p. 132) observes, “Selective institutions will naturally try to reject the least qualified of
the otherwise admissible nonblack applicants while admitting the most qualified of those black applicants
who would otherwise have been rejected.”
8
Note that if university admissions officers can be “bribed” to violate Assumption 4,  l will increase in
absolute value and the sufficient conditions for Lemma 2 are more likely to be satisfied, ceteris paribus.
9
Note that we adopt this numerical assumption to simplify the presentation. We experimented with a more
realistic assumption wherein both merit admissions and legacy-preference admissions share the eliminated
quota of the racial-preference admissions. The results are qualitatively similar, though not as pronounced,
and hence are not reported here.
10
Of course, one does not have to use the average student ability to measure the quality of student body.
Other measurements can be easily used here in the simulations. We chose to use the average/mean ability
to be consistent with the theoretical model presented in the previous section.
11
Admittedly there may be a correlation between family background and student ability.
12
Consider, for example, a university administrator with a utility function containing two arguments—the
average ability of the student body (A) and the university’s financial endowment (E). Suppose further that
the administrator’s utility function takes the simple form U ( A, E )  A  (1  a) E , where   (0, 1). How
much would the university administrator require in additional E to compensate for a one unit reduction in
A? This is given by the marginal rate of substitution of A for E, or MRSA-E =  / (1- ). Hence, the larger
the value of  , the greater the contribution to the endowment alumni would have to make in order to
compensate the administrator for the decline in the average ability of the student body resulting from the
admission of less-able students. This may explain why prestigious universities, those with relatively large
values of  , are able to extract large financial contributions from wealthy alumni who wish to gain
admission for their less-able offspring.
13
This, of course, is a variation on the General Theory of the Second Best—eliminating only one of
multiple constraints can [actually] lead to a lower level of social welfare. See Lipsey and Lancaster (1956).
14
See Loury (2002) inclusive of Tables 1-4 and Figures 1-2 in his Appendix.
17
18