Journal of Public Economics 94 (2010) 410–420

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Journal of Public Economics

j ourna l homepage: www.e lsev ie r.com/ locate / jpube

Early decision and financial aid competition among need-blind collegesand universities

Matthew Kim ⁎

Department of Economics, University of St. Thomas, Mail 5029, 2115 Summit Avenue, St. Paul, MN 55105, USA

⁎ Tel.: +1 651 962 5677; fax: +1 651 962 5682.E-mail address: mkim@stthomas.edu.

1 I provide a list of need-blind admissions policies injournal's website.

2 The schools are 28 of the 31 member institutions oHigher Education (COFHE): Amherst, Barnard, Brown,Columbia, Cornell, Dartmouth, Duke, Georgetown, HMount Holyoke, Northwestern, Oberlin, PennsylvanRochester, Smith, Stanford, Swarthmore, Trinity, WWesleyan, Williams, and Yale.

0047-2727/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.jpubeco.2010.01.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 February 2008Received in revised form 15 December 2009Accepted 7 January 2010Available online 1 February 2010

JEL classification:I2L3

Keywords:Early decisionFinancial aidNeed-blind admissions

This paper presents a stylized theoretical model of competition among need-blind colleges and universitiesthat implement early decision admissions. Under need-blind admissions, an applicant's financial aid statuscannot affect their likelihood of admission. In the model, a need-blind school can use early decisionadmissions as a screening mechanism to indirectly identify a student's ability-to-pay, while superficiallymaintaining a need-blind policy. As a result, in equilibrium, non-financial aid students are more likely to beadmitted than financial aid students of comparable quality.

an appendix, available on this

f the Consortium on FinancingBryn Mawr, Carleton, Chicago,arvard, Johns Hopkins, MIT,ia, Pomona, Princeton, Rice,ashington (MO), Wellesley, 3 A student may a

4 Author's calculat

ll rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

With few exceptions, the most elite and prestigious colleges anduniversities in the U.S. share at least one feature: need-blindadmissions. A need-blind school makes two commitments: first,that an applicant's financial need does not directly affect theapplicant's probability of admission; and second, that an applicant'sdemonstrated need is fully met by financial aid.1 Need-blindadmissions signal a school's commitment to diversity and equalopportunity. However, maintaining need-blind admissions is a costlypolicy, which is why only the richest schools can afford to be need-blind. According to a recent paper by Hill et al. (2004), whichexamines financial aid records at 28 highly selective colleges anduniversities, only 55% of enrolled students pay full price—the restreceive some amount of financial aid.2 With rising educational costs,many need-blind schools (e.g., Brown University) have abandonedfully need-blind policies by explicitly including lack of need as a factorin admitting the last 5 to 10% of their classes (Ehrenberg, 2000).

Early decision (ED) is an admissions policy where an applicantcontracts to matriculate upon acceptance in exchange for earlynotification of admission.3 An early applicant is notified before theregular admission process begins; thus, an early applicant maypotentially avoid the regular application process. Early admissionsaccounts for a significant portion of first-year admissions at selectiveinstitutions. For example, among the Ivy League schools earlyapplicants accounted for nearly 20% of total applications; furthermore,the number of applicants admitted early was nearly one-half the totalnumber of students entering the fall class.4

ED is a controversial policy within higher education. Proponentslaud it as an innovation that benefits students and student choice: EDprovides a process in which talented students can express apreference for and match with their first-choice schools, andsimultaneously avoid the hassle, costs, and apprehension of applyingto and waiting for admission into other schools. Others, however,view ED as destructive of students' welfare: ED can be confusing forstudents, and forces students and parents to make premature anduninformed decisions. Particularly, opponents of ED argue that thecomplex decision-making environment of ED disproportionatelyhurts students from disadvantaged backgrounds because of, forexample, their limited (or lack of) access to trained college counselorsfor advice, compared to their relatively advantaged peers (Avery et al.,2003).

pply ED to only a single school.ions using College Board data for the Fall 2000 incoming class.

411M. Kim / Journal of Public Economics 94 (2010) 410–420

This paper presents a theoretical model of competition amongneed-blind schools that also implement ED admissions. Under need-blind admissions, schools do not observe students' incomes. However,themodel suggests that schools can use ED admissions to get studentsto self-sort by income. Thus, need-blind schools can use EDadmissions to indirectly identify students' incomes while superficiallymaintaining a need-blind admissions policy.5 When enrollment isbinding upon early admission–as it is in ED admissions–a financial aidstudent faces an implicit barrier to applying early from the cost oflosing the opportunity to compare and appeal multiple financial aidoffers.6 By admitting more students ED–where the applicant pool is,on average, wealthier and less likely to require financial aid–a schoolreduces its financial aid expenditures, alleviating the costliness ofneed-blind admissions, while completely maintaining the letter (ifnot the spirit) of a need-blind admissions policy. However, the modelimplies that ED results in a more efficient allocation of students thanan admissions process without ED admissions. I also find that ED isstrictly welfare-improving for lower-ability full-pay students andhigher-ability financial aid students, but strictly welfare-reducing forlower-ability financial aid students.

While it is possible to interpret the model as schools' attempts tocircumvent the trappings of need-blind admissions, the model doesnot require this interpretation. Consider this alternative interpreta-tion. Benevolent schools care about equality of opportunity withrespect to income, and implement need-blind admissions to pursuethis mission. Individually, schools compete for and admit students,based solely on merit. However, two results arise out of adecentralized equilibrium: 1) students self-sort across admissionperiods according to their incomes, and 2) competition forces schoolsto admit higher fractions of full-pay students (still strictly under need-blind admissions) just to maintain comparable student quality withtheir rival schools. Thus, even in this “naive” interpretation, schoolsthat are committed to need-blind admissions unwittingly discrimi-nate against financial aid students.7 Furthermore, any individualschool that recognized the equilibrium consequences for financial aidstudents could not unilaterally do anything about it. All this impliesthe paradoxical result: the only way to achieve “true” need-blindadmissions (when there is also early decision) is, in fact, to be non-need-blind, and give explicit preference to financial aid students.

The paper proceeds as follows. In the next section, I discussprevious research regarding ED admissions and modeling schoolbehavior. Section 3 presents the model. Section 4 characterizesequilibrium. In Section 5, I discuss the model's implications for theeffects of ED competition. Section 6 discusses the validity of the priceexogeneity assumption and the difficulties of modeling endogenousprice setting by schools. Section 7 concludes.

2. Background

While the economics literature that deals explicitly with EDadmissions is small, there is a substantial literature that attempts to

5 Ehrenberg (2000) and Avery et al. (2003) also suggest this idea without formaltreatment.

6 For example, in response to the frequently asked question, “What can admittedstudents do if they're not happy with their financial aid award?” the Dartmouth Officeof Admission writes: “A student should first contact the Financial Aid Office. In manycases, more information from the student or from his/her family, including copies ofbetter packages from other schools, will result in award adjustments consistent with afair and equitable treatment of all applicants” [italics added] (http://www.dartmouth.edu/apply/financialaid/faq.html).

7 This implication is supported by recent independent announcements from threehigh-profile universities–Harvard, Princeton, and Virginia–to eliminate early admis-sions, all citing the inherent and unintended disadvantages to low-income students.For example, John Blackburn, Dean of undergraduate admission at the University ofVirginia, states, “Early decision has not been designed to keep out low-incomestudents, but what we've seen is that the population that were admitted under thisprogram are very homogenous” (Farrell, 2006).

model college behavior, in general. One of the difficulties of modelingcollege behavior is determining what a college's objective functionshould be. Winston (1997) outlines why the “economic analogy” ofmodeling colleges as for-profit firms may be not only inaccurate, butdangerously inappropriate, due to characteristics specific to the marketfor higher education. Ehrenberg (2000) discusses why it may benonsensical to attempt to model a college as having a single, unifiedobjective function since a college is comprised of faculty, administrators,and trustees, among many others, all with sometimes-conflictingobjectives. Despite these cautions, most researchers have proceededwith models of single-valued objective maximization: profit maximi-zation (e.g., Rothschild andWhite, 1995; Epple and Romano, 1998) andutility maximization (e.g., Ehrenberg and Sherman, 1984; Epple et al.,2002, 2003, 2006), whereby the college derives utility from differentinputs that the college is postulated to care about; e.g., student quality,diversity. The latter assumption is more flexible since it allows collegesto pursue policies that may not be necessarily profitable.

To my knowledge, Avery, Fairbanks, and Zeckhauser (2003) (hence-forthAFZ) are the only researchers to have empirically analyzedED in anysystematic manner; what is commonly “known” about ED and its effectsis largely anecdotal. AFZ's book,TheEarlyAdmissionsGame, representsfiveyears of research involving administrative data containing over 500,000applications—every single application during 1991–92 to 1996–97 from14 highly selective colleges and universities;8 survey data from the“College Admissions Project,”which oversamples college applicants fromprestigious high schools (i.e., the applicants most likely to apply to themost selective colleges and universities);9 and over 400 personalinterviews with students and college counselors. AFZ find what theydescribe as “a clear and consistentfinding” that the advantage of applyingearly is equivalent to an increase of 100points in SAT score (p. 9). AFZ alsosuggest a number of reforms to the early admissions system, and considerthe feasibility of implementation, as well as potential effects of theirproposed reforms.

The model presented in this paper is similar to the sorting modelpresented in Epple and Romano (1998). In their model, students differin income and ability, and schools (assumed to maximize profits)choose prices and admission policies. Theirmain result is the existenceof a sorting equilibrium, where the best schools admit both low-income high-ability students, and high-income low-ability students;the former students “earn” their admission, while the latter “buy” it.My model departs from Epple and Romano (1998) in three significantways. First, I explicitly incorporate anEDperiod into themodel in orderto investigate how ED affects the behavior of colleges and students.Second, Epple and Romano (1998) assume that schools are “utility-takers”: schools take students' choices as given when choosing theirown actions. This assumption removes any strategic behavior fromtheir model. In my model, I assume that schools recognize that theirchoices may affect competing schools' choices via students' choices.Third, I restrict tuition and admission policies to conform to need-blindadmissions policies (e.g., admission rates can differ by student quality,but not financial aid status), whereas Epple and Romano (1998)impose no such policy restrictions. This is because Epple and Romanodo not directly examine the issue of need-blind admissions.

In an independently-conceived work, Lee (2002) also presents atheoretical model of early admissions, and shows that ED results inlower admission standards in ED than in regular decision and that EDmay increase allocative efficiency. However, my model allowsstudents to differ in their abilities-to-pay, which allows me toevaluate the effects of ED for students of differing quality and income.

8 In return for access to administrative data, AFZ (2003) do not disclose theidentities of the 14 schools. However, each of the 14 schools rank in the top 20 of theU.S. News & World Report lists, “Best National Universities,” or “Best Liberal ArtsColleges.”

9 For more information about the College Admissions Project, see http://www.nber.org/~hoxby/collegeadmissions/.

412 M. Kim / Journal of Public Economics 94 (2010) 410–420

Furthermore, while Lee (2002) ignores price considerations, mymodel allows for prices (and specifically, financial aid awards) toaffect students' application and matriculation decisions. Withoutformally modeling prices, Lee (2002) discounts the idea presented inthis paper–that schools may admit students through ED admissions toreduce financial aid liabilities–by questioning why schools wouldemploy such a complex screening mechanism when they couldinstead allow “financial considerations [to] play a role in theadmission decision” (p. 31). The answer is that schools haveincentives to advertise and commit to need-blind admissions.Therefore, if it is possible for a school to commit to need-blindadmissions and indirectly observe income–to have its cake and eat ittoo–then it is clear why need-blind schools would use ED admissionsas a screening mechanism.

3. Model

I model strategic behavior in higher education as a game withstudents and schools as the players. Students choose where and whento apply, and schools choose prices and admission policies.

3.1. Admission periods

Students may either apply to a single school ED, or wait and applyto multiple schools regular decision (RD). Let t∈{e, r} denote theadmission period.

3.2. Students

Students vary along two dimensions: income and quality.10

Assume that the total mass of students is equal to one. Supposethere are two levels of income, y∈ {h, l}, where hN l .11 For simplicity,assume that l=0. Let ϕ∈(0,1) denote the proportion of h in theapplicant pool. Furthermore, suppose that students also differ inquality, q∈{g,b}, where gNb. To capture correlation between incomeand quality, let λy∈(0,1) denote the proportion of good-qualitystudents with income y; i.e., λh=Pr{q=g|y=h} and λl =Pr{q=g|y= l}. While schools perfectly observe quality, need-blind admissionsimply that they do not observe income.

3.3. Prices

Let pyqti denote the price charged to a student of income y and

quality qwho is admitted to school i in admission period t.12 I assumethat schools do not award merit-based aid but only need-based aid,which is a common practice for need-blind schools.13 Thus, whileschools cannot target financial aid at high-quality students, they canstill compete by increasing the overall generosity of their financial aidprograms.14 This assumption implies that prices can depend on

10 By “income” think “ability-to-pay,” and by “quality” think of some indicator ofability, e.g., SAT score.11 It is more common in the literature to make income and quality continuousvariables rather than discrete (e.g., Epple and Romano, 1998). However, in a two-period model, the optimal threshold rules are difficult to obtain. Using discretevariables greatly simplifies the solution, at little cost to the model's intuition.12 The highest price charged is called the “sticker price,” “full price,” or “list tuition.”“Financial aid” is the difference between the sticker price and any other price.13 While the majority of need-blind schools do not offer merit-based aid, a smallnumber do. In an appendix (available on the journal's website), I provide a list ofmerit-based aid policies.14 For example, in 2001, Princeton introduced its “no-loan” program, fulfilling itsfinancial aid commitments solely through grants and work-study. In 2004, Harvardeliminated the financial contribution of students' parents with annual incomes below$40,000. Yale did the same for parents with incomes below $45,000 starting in 2005,and Dartmouth replaced loans with scholarships/grants for all students with familyincomes below $30,000 starting in 2006. Recently, Stanford announced that it willeliminate required contributions from families with annual incomes of less than$45,000, while Amherst announced it will implement a no-loan financial aid program.

income but not on quality; i.e., pyqti =pyti ∀y. It also implies that high-

income students always pay the sticker price since, by definition, theynever qualify for financial aid; i.e., phti =ph

i ∀t. I assume phi =pN0,

where p is exogenously given. Extension to endogenous sticker priceis a challenging task. In Section 6, I discuss the validity of thisassumption and explain some difficulties of constructing a model thatincludes endogenous sticker price.

Low-income students face different prices than high-incomestudents. Need-blind admissions implies that schools guarantee tofully meet the demonstrated need, (p− l), of any admitted student.Therefore, the price charged to low-income students who apply toschool i in period t, can bewritten as piℓt = −dit , where dti isfinancial aidawarded by school i in admission period t in excess of demonstratedneed.

Since a high-income student never receives financial aid, thestudent's utility upon ED admission is equal to the utility upon RDadmission. However, if dr

i Ndei , then a low-income student strictly

prefers admission via RD than ED. If, as some opponents of ED claim,the probability of ED admission is higher than RD admission(conditional on quality), then a low-income student must weigh thebenefit of receiving more financial aid in RD against the cost of areduced probability of admission.

3.4. Schools

There are two schools, indexed by i. In addition to choosing theprice discounts (dei ,dri), each school chooses admission ratesαqt

i , whereαqti is school i's admission rate for students of quality q in period t.15

Note that the admission rates only differ by quality and not by income;this is an implication of need-blind admissions.

Schools are limited in the total amount of financial aid they canaward to students: total expenditures cannot exceed total revenues.This is essentially a budget constraint for schools. For simplicity, Iassume that the marginal educational cost is zero.16 Since educationalcosts are zero, the only source of expenditures is financial aid. Thus,the total amount of financial aid awarded cannot exceed the totalamount of revenues collected in tuition.

3.5. Preferences

3.5.1. StudentsStudents maximize expected utility, taking schools' strategies as

given. Let VNp denote the value of schooling, and V0 denote the valueof no schooling, both of which are common to all students. Withoutloss of generality, we can set V0=0; thus V may be interpreted as thereturn of schooling over the alternative of no schooling.17

In addition to their quality-income types, students differ in theirpreferences over schools, which are not observed by schools.18 Thereare three types of students: those that prefer school 1 over school 2 bya magnitude zN0 (type 1), those that prefer school 2 over school 1(type 2), and those that are indifferent between schools 1 and 2(type 0). I assume that a proportion ω∈(0,1) of the students are type

15 A point of clarification: suppose that, for example, there are twenty good-qualityapplicants. If αge

i =0.5, then school i will admit exactly ten applicants. In contrast, itdoes not mean that school i admits each student with probability one-half. Rather, itmeans that school i chooses randomly from the set of all possible combinations of tenstudents.16 This assumption is akin to the zero constant marginal cost assumption in astandard Bertrand model with no fixed costs; having positive costs obscures theintuition of the model.17 Epple and Romano (1998) investigate competition among private schools, wherestudents have the outside option of attending a public school. Thus, an alternativeinterpretation of V could be the net return of attending a “brand-name” private school(as most need-blind schools are) over a public alternative, V0.18 Unobservable preferences for different schools have intuitive appeal, particularlyin the market for higher education where the “intangibles” weigh in heavily tostudents' decisions (Winston, 2000).

413M. Kim / Journal of Public Economics 94 (2010) 410–420

0, and (1−ω)/2 each are type 1 and 2. Furthermore, these preferencesare distributed independently across quality and income. In summary,while a type 0 student receives utility V at either school, a type 1student receives utility (V+z) at school 1, and V at school 2 (and viceversa for a type 2 student).

Assume that student utility is additively separable in price. LetVyjit denote the utility of a student of income y and preference type j

who attends school i admitted in period t. It is worthwhile to beexplicit in expressing the utility levels:

Vthji =

V + z−p if j = iV−p otherwise ∀t∈ e; rf g

�ð1Þ

Veℓji =

V + z + die if j = i

V + die otherwise

(ð2Þ

Vrℓji =

V + z + dir if j = i

V + dir otherwise

(ð3Þ

3.5.2. SchoolsSchools maximize average student quality, subject to a budget

constraint. Furthermore, I assume that each school has amatriculationtarget, M, that it must meet with equality.19 Assume that

ϕ + λℓ 1−ϕð Þ2

bM b12:

In any symmetric equilibrium, students will split evenly betweenthe two schools. The total number of high-income students is ϕ andthe total number of low-income good-quality students is λl(1−ϕ),thus, this assumption basically requires schools to admit some low-income bad-quality students in any symmetric equilibrium.

In summary, each school i maximizes average student quality bychoosing a vector of strategies (αge

i ,αbei ,αgr

i ,αbri , dei ,dri) subject to a

budget constraint and a matriculation constraint; schools take eachothers' strategies as given, but take students' best responses intoaccount. Therefore, the timing of choices in the model is as follows:1) colleges simultaneously commit to ED and RD admission andfinancial aid policies; and then 2) students observe schools' actionsand make their own admission and matriculation choices.

3.6. Decision rules for students

Students must decide to which schools to apply ED and RD; theyalsomust have a decision rule that determines which school to attend,conditional on admission to multiple schools. Students are notpermitted to apply ED to multiple schools; furthermore, studentsmust attend the schools into which they are admitted ED. I focus onlyon preference type 1 students (and suppress preference typesubscripts), as the decision rules for type 2 students may be solvedsymmetrically.20

First, consider a student's decision of which school to attend,conditional on admission into both schools. A student with income y

19 The matriculation constraint held at equality implies that a school is not onlycapacity constrained, but is required to reach a matriculation target. There are threemain reasons for this. First, requiring a school to reach a matriculation target prohibitsthe school (whose objective is to maximize average student quality) from maximizingits objective by shrinking class size. With no minimummatriculation level, each schoolcould absolutely maximize its objective by admitting a single good-quality student—and there would be no need for schools to compete for good-quality students. Second,requiring a matriculation target is an ostensibly more accurate depiction of the trueconstraints on admissions committees. Finally, the matriculation target providestechnical tractability—by fixing class size, maximizing average student quality isequivalent to maximizing the total number of good-quality students.20 The decision rules for type 0 students may be similarly derived by using z=0.

that is admitted into both schools prefers school 1 to school 2 if andonly if the utility of attending school 1 is larger than the utility ofattending school 2, i.e.,

V + z−p1yr ≥ V−p2yr ; ð4Þ

which simplifies to z≥pyr1 −pyr

2 . This decision rule is simple:conditional on being admitted to both schools, attend school 1 aslong as the price difference does not exceed z. Since I assume thatphr1 =phr

2 =p, this decision rule says that high-income students willattend the school that matches their preference type, conditional onadmission to both schools. For low-income students, for whompiℓr = −dir , this rule says that a student will attend school 1 as long asthe financial aid difference does not exceed z. In the event that Eq. (4)holds at equality (with type 0 students, for instance), I assume that themass of indifferent students will sort evenly across schools.

Next, consider a student's decision of which school to apply to ED,conditional on applying ED. A student will prefer to apply ED to school1 over school 2 if and only if

α1eV

ey1 + 1−α1

e

� �RDy ≥ α2

eVey2 + 1−α2

e

� �RDy; ð5Þ

where RDy=αr1αr

2max(Vy1r ,Vy2

r )+αr1(1−αr

2)Vy1r +αr

2(1−αr1)Vy2

r +(1−αr

1)(1−αr2)V0. I omit quality subscripts for notational conve-

nience. The left-hand side of Eq. (5) is the expected utility ofapplying ED to school 1 and the right-hand side is the expectedutility of applying ED to school 2. The first term (on either side) is theexpected utility of ED admission to school 1 (or 2). The second termis the expected utility of applying to both schools RD, conditional onED rejection.21 Similarly, if Eq. (5) holds at equality, I assume that themass of indifferent students will sort evenly across schools.

Finally, a student will apply ED if and only if her more preferred EDchoice dominates applying RD; i.e.,

max ED1y ; ED

2y

� �≥ RDy; ð6Þ

where EDyi denotes the expected utility of a student with income y

applying ED to school i. Similarly, if Eq. (6) holds at equality, I considerthe case where students sort evenly across admission periods; i.e., halfapply ED and half apply RD.

4. Equilibrium

For notational convenience, let αti=(αgt

i ,αbti ), αi=(αe

i ,αri), α=

(α1,α2), di=(dei ,dri), and d=(d1,d2). Furthermore, let Nyqti denote the

mass of students of income y and quality q that apply at time t toschool i.22 Let ψy

i denote a coefficient that adjusts what proportion ofstudents of income y will attend school i conditional on admission toboth schools, as described in the previous section. Finally, let Xi denoteutility (institutional “excellence”) for school i. School i's problem is

Xi = maxαi ;di

∑y

αigeN

iyge + ψi

yαigr 1−αi

ge

� �Niyge + Ni

ygr

h in oð7Þ

subject to

M = ∑q

∑y

αiqeN

iyqe + ψi

yαiqr 1−αi

qe

� �Niyqe + Ni

yqr

h in o; ð8Þ

21 The implicit assumption here is that the admissions process is independent acrosstime periods; a student that is rejected ED (i.e., “deferred”) is not more or less likely tobe admitted RD to a school than if she applied directly RD to that same school.22 For example, N1

ℓgr denotes the mass of students of low-income good-qualitystudents that apply RD to school 1. However, this mass of students excludes any low-income good-quality students that applied ED to school 1 and were rejected (andtherefore automatically deferred into the RD application pool).

23 For the remainder of the paper, I will implicitly assume that Eq. (10) holds.

414 M. Kim / Journal of Public Economics 94 (2010) 410–420

∑q

αiqeN

ihqe + ψi

hαiqr 1−αi

qe

� �Nihqe + Ni

hqr

h in op

≥∑q

αiqeN

iℓqed

ie + ψi

ℓαiqr 1−αi

qe

� �Niℓqr + Ni

ℓqr

h idir

n o:

ð9Þ

The notation hides the fact that ψyi and Nyqt

i are functions of (α,d);thus, a school's objective function is a very complicated (and noteverywhere differentiable) function of schools' strategies and stu-dents' best responses.

Eq. (8) is the matriculation target, which requires that each schooladmit an entering class size of exactly M. Eq. (9) is the budgetconstraint; the left-hand side is total revenues generated by tuitionpayments made by high-income students, and the right-hand side istotal financial aid expenditures (in excess of demonstrated need)awarded to low-income students.

A Nash equilibrium is defined as a vector of policies (α,d) thatsolves the schools' problems Eq. (7), satisfying constraints Eqs. (8)and (9) and solves students' application and matriculation decisions,as defined by Eqs. (4), (5), and (6).

4.1. A benchmark: RD equilibrium

In order to talk about the effects of ED, it is necessary to definesome standard against which to measure the ED outcome. Thus,before solving for the equilibrium of the ED game, it is useful to solve,as a benchmark, the simpler case of when there is no ED (i.e., imposeαei =0∀i); henceforth, this game is called the RD game.For simplicity, I search for only symmetric pure-strategy equilibria.

Denote the equilibrium strategies with a * superscript; e.g., αgr⁎. Thesymmetry of the equilibrium (specifically d1=d2) implies that low-income students admitted to both schools will attend their morepreferred schools. Any strategy without αgr

i =1 is a dominatedstrategy. Therefore, both schools will choose αgr⁎=1, and theremaining strategies are uniquely characterized by the constraints.Thus, I obtain the following proposition:

Proposition 1. There exists a unique symmetric equilibrium of the RDgame with the admission rates αgr

⁎=1, and

α⁎br = 1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−2M

1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ

s

d⁎r =1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ− 1−λhð Þ 1−2Mð Þ1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ− 1−λℓð Þ 1−2Mð Þ

ϕp1−ϕ

:

Proof. All proofs are available in an appendix (available on thejournal's website), unless noted otherwise. □

In equilibrium, all students, regardless of type, apply to bothschools and are admitted according to the schools' equilibriumstrategies. Both schools matriculate exactly half of the good-qualitystudents, i.e., school utility is X ⁎ = 1

2λhϕ + λℓ 1−ϕð Þ½ �. Note that

each half consists of all of the students that have a preference forthat school (type 1 or 2), plus half of the indifferent (type 0)students. The remaining seats are filled with bad-quality students,admitted in the same relative high-to-low-income ratio as thepopulation income ratio. The matriculant income ratio is equal tothe population income ratio since need-blind schools do notobserve income type.

4.2. ED equilibrium

Next, I consider any symmetric pure-strategy equilibrium whenED admissions are allowed. Denote the ED equilibrium strategies with

a ** superscript; e.g., αge⁎⁎. By using similar logic as in the RD game, Iobtain the following proposition and corollary:

Proposition 2. For every (ϕ,λl ,M,p,V,z) such that

ϕp2M−ϕ

≥ 1−2Mð ÞV2M−ϕ−λℓ 1−ϕð Þ +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−2Mð Þ 1−λℓð Þ 1−ϕð Þp2M−ϕ−λℓ 1−ϕð Þ z; ð10Þ

there exists a symmetric equilibrium of the ED game with the admissionrates αge

⁎⁎=αbe⁎⁎=αgr

⁎⁎=1, and

α**br = 1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−2M1−λℓð Þ 1−ϕð Þ

s

d**e = 0

d**r =ϕp

2M−ϕ:

Furthermore, the set of parameters that satisfies Eq. (10) is not empty.

Corollary 3. Suppose Eq. (10) holds. Even though ED competition resultsin the same number of bad-quality students admitted to some school (ascompared to the RD game outcome), the group of admitted students is lesslikely to be low income (as compared to the RD game outcome).

In equilibrium, students choose the following strategies:

• Of the high-income good-quality students, half apply ED, and therest apply to both schools RD. Of those that apply ED, half apply toeach of their (weakly) preferred schools;

• All of the high-income bad-quality students apply ED, half to eachschool;

• None of the low-income students apply ED, but instead apply toboth schools RD.

As in the RD equilibrium, both schools matriculate exactly halfof the good-quality students; i.e., X** = X* = 1

2λhϕ + λℓ 1−ϕð Þ½ �.

The difference is that, while in the RD game high- and low-incomestudents of bad-quality were admitted with equal probability (αbr⁎), inthe ED game high-income bad-quality students are admitted withprobability αbe⁎⁎=1 and low-income bad-quality students are admit-ted with probability αbr

⁎⁎b1. Since αbe⁎⁎=1, high-income bad-quality

students will apply ED (where they can guarantee themselvesadmission to some school). On the other hand, even though low-income bad-quality students could also apply ED and be admittedwith certainty, they choose to wait and apply RD. Condition (10)guarantees that low-income bad-quality students prefer applying RDthan ED.23 The left-hand side of Eq. (10) is the amount of financial aidawarded to each low-income student, conditional on low-incomestudents applying RD instead of ED. The right-hand side is thereservation financial aid value that a low-income bad-quality studentmust receive to induce her to apply RD.

Recall that the matriculation constraint implies that at least oneschool (and both, in a symmetric equilibrium) must admit some bad-quality students. However, since schools compete on financial aidawards for low-income good-quality students, schools would like toadmit as many high-income (i.e., tuition-paying) students as possible.Specifically, each school wants to maximize the number of high-income students within the group of admitted bad-quality students.Since schools can induce low-income bad-quality students to applyRD, schools can infer that any students that apply ED must be high-income types; therefore, schools increase their ED admission rates.The result is that ED admissions allows need-blind schools toindirectly infer income type, by screening out low-income studentsfrom the ED applicant pool.

415M. Kim / Journal of Public Economics 94 (2010) 410–420

4.3. Non-uniqueness

There exists a second symmetric pure-strategy equilibrium to theED game (denoted ***), which always exists, regardless of whethercondition (10) holds.24 One key difference between the two equilibriaprovides a testable implication to identify the correct equilibrium: theoriginal (**) equilibrium predicts that low-income bad-qualitystudents will apply RD, while the alternative (***) equilibriumpredicts that they will apply ED.25 AFZ (2003) present evidence thatearly applicants are, on average, wealthier than regular applicants.Therefore, for the purposes of subsequent analysis, I focus on the **equilibrium instead of the *** equilibrium.While the following resultsare based on the ** equilibrium, it should be noted here that the ***equilibrium generates identical qualitative implications. Thus, thefollowing analysis is not sensitive to the choice of equilibrium.

5. Effects of early decision

5.1. Admission standards

It is well-attested that schools, as a general rule, admit a higherproportion of their early applicants than their regular applicants (see,e.g., AFZ, 2003; Ehrenberg, 2000). Two competing theories explainwhy schools admit a higher proportion of students ED than RD. On onehand, the higher ED admission rates could be the result of loweradmission standards—admitting a student who applies ED, butrejecting an identical (quality) student who applies RD. On theother hand, the higher ED admission rates could be the result of the EDapplicant pool being of higher quality than the RD applicant pool. Ingeneral, colleges and universities argue that the latter story is thecorrect one.26

The model presented in this paper implies that ED admission ratesare higher than RD admission rates because ED admission standardsare lower. There are two ways to observe this result. First, note that(αbe⁎⁎/αge⁎⁎)N(αbr⁎⁎/αgr⁎⁎). In other words, the relative admission rate ofbad-quality students to good-quality students is higher in ED than inRD, which implies that it is relatively easier for a bad-quality studentto be admitted in ED than in RD. Second, note that αqe

⁎⁎≥αqr⁎⁎∀q. In

other words, if a student is more likely to be admitted ED than RD,conditional on student quality, this implies that ED admissionstandards must be lower. This latter result agrees with the empiricalresults found by AFZ (2003). Since schools' admission probabilities, α,are conditional on quality, a higher admission probability isequivalent to a lower admission standard. Since αbe

⁎⁎Nαbr⁎⁎, a bad-

quality student that applies ED is more likely to be admitted than anequally-qualified student that applies RD. The model also predictsthat, in equilibrium, αge

⁎⁎=αgr⁎⁎=1; i.e., that admission standards are

no different between ED and RD for good-quality students. This resultsuggests that the observation that overall ED admission rates arehigher than RD admission rates masks differences in standards acrossstudent quality. This matches an empirical observation by AFZ (2003),who find that applying early is most advantageous for “a student whois just at or below the admission standard as a regular applicant”(p. 157).

24 The equilibria are not Pareto-rankable, since different types of students preferdifferent equilibria.25 A complete description of the alternative equilibrium, as well as a brief discussionof some its properties, is available upon request.26 For example, on its Frequently Asked Questions webpage, the Princeton AdmissionOffice writes, “A candidate to whom we otherwise would not offer admission is notgoing to be offered admission simply because he or she applied Early Decision.However, it is the case that the rate of admission of early applicants is invariablyhigher than our overall admission rate. In part, that's simply a matter of there being arather large number of compelling candidates in the early applicant pool” (http://www.princeton.edu/pr/admissions/u/QandA.html).

5.2. Allocative efficiency

In the previous section, I showed how ED competition does notaffect the average quality of matriculating students at either school.However, while the average quality remains the same, the allocationof students changes in two significant ways. First, there is an “equity”issue: as stated in Corollary 3, ED competition results in fewer low-income students admitted to any school.27

Second, there is an efficiency issue: the allocation of studentschanges from the sorting of different preference type students acrossschools. This is because ED offers students a method of signaling theirpreference types to schools. Consider the equilibrium allocation in theRD game. Since αgr

⁎=1, all good-quality students are admitted intoboth schools, and therefore self-select into their more preferredschools (i.e., type 1 students attend school 1, and type 2 studentsattend school 2). On the other hand, since αbr

⁎b1, there is a positivemass of αbr

⁎(1−αbr⁎)(1−ω) bad-quality students who are admitted

only to their less-preferred schools. We can interpret this allocation asinefficient, since the students matriculating at their less-preferredschools could all switch places and be made better off, while thismakes the schools and the other students no worse off. The failure ofan efficient allocation of students in the RD game stems from schools'inabilities to observe students' preference types. In contrast, the massof students matriculating at their less-preferred schools is smaller inthe ED game than the RD game. This is because high-income bad-quality students may now signal their preferences for their preferredschools by applying ED to those schools. Low-income bad-qualitystudents do not apply ED, and therefore some proportion of themwillstill be admitted to their less-preferred schools. Thus, while the EDallocation is still inefficient, it is more efficient than the RD allocation.

5.3. Selectivity ratings

Opponents of ED argue that ED provides a mechanism for a schoolto inflate objective measures of selectivity, such as its overallacceptance rate and yield rate. A school cares about these measures,in large part, because they are explicitly incorporated into itsU.S. Newsranking (Ehrenberg, 2000).28 Opponents of ED feel that thesemeasures of selectivity are vulnerable to gaming. For instance, AFZ(2003) write that “an obvious way for a college to improve itsselectivity [acceptance rate] and yield is to accept more earlyapplicants” (p. 181).29 The purpose of this subsection is to formallyexamine these claims.

5.3.1. Yield rateLet υ denote a school's yield rate, which is defined as the total

number of students matriculated divided by the total number ofstudents admitted. Since both schools have a matriculation target, M,the total number of matriculating students is constant across games,which greatly simplifies the analysis; a school can increase its yieldrate only by admitting fewer students. The equilibrium yield rates inthe RD and ED games are, respectively,

υ* = M λhϕ + λℓ 1−ϕð Þ + α⁎br 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ �

n o−1;

υ** = M34λhϕ +

12

1−λhð Þϕ + λℓ 1−ϕð Þ + α**br 1−λℓð Þ 1−ϕð Þ� �−1

:

27 I discuss the welfare effects of ED competition below in Section 5.4.28 Ehrenberg (2000) comments, “Even though colleges and universities constantlycriticize the rankings and urge potential students and their parents to ignore them,every institution pays very close attention to the ratings and tries to take actions toimprove its own ranking” (p. 52).29 Starting in 2003, U.S. News no longer considers yield rate in its calculations ofranking in response to, in large part, complaints from college and high-school officials(Young, 2003).

Table 1Model predictions: price effect.

Price paid conditional on admission

Student type RD game ED game ED–RD Effect on welfare

High-income p p 0 0Low-income −dr⁎ −dr⁎⁎ – +

416 M. Kim / Journal of Public Economics 94 (2010) 410–420

Since ED essentially replaces students who may matriculate withstudents who will definitely matriculate, ED necessarily increasesyield rates. My model also predicts this: ED increases schools' yieldrates over their RD rates; i.e., υ⁎⁎Nυ⁎.

5.3.2. Acceptance rateLet σ denote a school's acceptance rate, which is defined as the

total number of students admitted divided by the total number ofapplications. Then the equilibrium acceptance rates in the RD and EDgames are, respectively,

σ* = λhϕ + λℓ 1−ϕð Þ + α*br 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ �;

σ** =34λhϕ + 1

21−λhð Þϕ + λℓ 1−ϕð Þ + α**br 1−λℓð Þ 1−ϕð Þ34λhϕ + 1

21−λhð Þϕ + 1−ϕð Þ

:

While comparing these acceptance rates requires only an algebraiccalculation, it is ambiguous which acceptance rate will be larger.While it is true that admitting more students ED means that a schoolcan admit fewer students RD, the net effect on acceptance rate isunclear because the school will also receive fewer applications in RD:other schools' ED admission policies reduce the total number of RDapplicants. Numerical analysis confirms that the effect is ambiguous.30

To calibrate themodel, I obtain estimates from a variety of sources.Hill, et al. (2004) find that 55% of students enrolled at COFHEmemberschools pay full price, with the rest receiving some amount of financialaid; therefore, I set ϕ=0.55.31 Calibrating λh and λl requires a moreindirect process. Let ρ denote the correlation coefficient betweenincome and quality. Epple and Romano (1998) borrow the pointestimate for ρ, obtained by Solon (1992) and Zimmerman (1992) whoboth estimate ρ=0.4. For the purposes of calibration, I assume thatλh=1−λl =λ, which means that ρ=0.4 implies λ=0.7.32

Under these parameter conditions, my model predicts thatacceptance rates increase as a result of ED competition, across allpossible values ofM. However, this increase is nomore than one-half apercentage point; i.e., the increase is small.

5.4. Student welfare

The effects of ED competition on student welfare can bedecomposed into three separate effects: ED could affect the pricesthat students pay to attend school (the “price effect”); ED could affectthe probability of being admitted to a school (the “admission effect”);and ED could affect the ability of students to sort efficiently, therebysecuring higher utilities for themselves (the “preference effect”).Thus, in order to analyze the welfare effects of ED competition, it isuseful to separately determine the magnitudes of these three effects.

Table 1 summarizes the price effect of ED competition. Since priceenters negatively into student utility, a positive change in priceimplies a negative effect on welfare. In both the RD and ED games,high-income students always pay a price of p; the price effect is zero.In contrast, low-income students always pay a negative price: dr⁎ inthe RD game, and dr⁎⁎ in the ED game. Simple algebra shows thatdr⁎⁎Ndr⁎; i.e., the price effect is positive. This makes intuitive sense: in

30 Results available upon request.31 Technically speaking, ϕ denotes the population proportion of high-income types. Ifincome and ability are positively correlated, then the proportion of matriculating high-income types will be higher than the population proportion. Thus, this calibrationchoice is made for simplicity.32 I provide a derivation of the relationship between ρ and (λh,λl ) in Appendix B.There are multiple (infinite) pairs of (λh,λl ) that map into any single value of ρ. Thus,given an estimate for ρ, I cannot separately “identify” (λh,λl ). Therefore, I assume thatPr{q=g|y=h}=Pr{q=b|y= l}; i.e., λh=1−λl ≡λ. This assumption is the mostnatural, in the sense that it assumes that the correlation structure is symmetric overall qualities and incomes. For more details regarding the calibration and numericalanalysis, see Appendix A.

the ED game, schools admit more high-income students (relative tothe RD game) and fewer low-income students—which means morefinancial aid is distributed to a fewer number of students.

Table 2 summarizes the admission effect of ED competition. Let πyqdenote the equilibrium probability of admission to any school for astudent with income y and quality q. Then, for example, πhb⁎=αbr

1 +αbr2 −(αbr

1 αbr2 ); by symmetry, this simplifies to πhb⁎=αbr

⁎(2−αbr⁎). In

both the RD and ED games, all good-quality students are alwaysadmitted with probability one; the admission effect is zero. Incontrast, in the RD game, all bad-quality students are admitted withprobability αbr

⁎(2−αbr⁎)b1; however, in the ED game, high-income

students (applying ED) are admitted with probability one, while low-income students (applying RD) are admitted with probabilityαbr

⁎⁎(2 −αbr⁎⁎)bαbr

⁎(2−αbr⁎). Thus, the net result of ED competition

admits more high-income students, but fewer low-income students(i.e., Corollary 3).

Table 3 summarizes the preference effect of ED competition. Recallthat all students share a common return to school, V, and studentswho attend their preferred schools gain additional utility of z. In boththe RD and ED games, all good-quality students attend their preferredschools; thus, they have the same utility (net of price) as a result of EDcompetition, V+(1−ω)z, on average. The effect for bad-qualitystudents, however, depends on income. In the RD game, bad-qualitystudents realize utility, on average, of V+[(1−ω)/(2−αbr

⁎)]z. Thisaverage reflects the fact that some students attend their preferredschools, while others attend their less-preferred schools. In the EDgame, high-income students signal their school preferences byapplying ED, and thus attain higher utility of V+(1−ω)z. In contrast,low-income students–who do not apply ED–cannot signal theirpreferences to schools. As a result, they realize an average utility ofV+[(1−ω)/(2−αbr

⁎⁎)]z. This utility level is smaller than in the RDgame: the admission probability to a given school is smaller, thus theadmission probability to one's preferred school is also smaller.

The total welfare effect is the sum of the price, admission, andpreference effects, all of which are summarized in Table 4. As a resultof ED competition, low-income good-quality students and high-income bad-quality students are unambiguously better off, whilehigh-income good-quality students are no worse off. However, thewelfare change for low-income bad-quality students is ambiguous:the direction of the change depends on the relative magnitudes of thecomposition effects, which, in turn, depend on the magnitudes of theparameters. Furthermore, the effect is ambiguous even whencondition (10) holds.

Let Uyq denote the ex ante average expected utility of a student ofincome y and quality q; i.e.,Uyq=πyqVy+(1−πyq)V0, where Vy is fromEqs. (1)–(3). Low-income bad-quality students experience negativeadmission and preference effects; however, those who are admittedto a school receive strictly more in financial aid—the price effect is

Table 2Model predictions: admission effect.

Probability of admission to some school

Student type RD game ED game ED–RD

High-income/good-quality 1 1 0Low-income/good-quality 1 1 0High-income/bad-quality αbr

⁎(2−αbr⁎) 1 +

Low-income/bad-quality αbr⁎(2−αbr

⁎) αbr⁎⁎(2−αbr

⁎⁎) −

34 As mentioned earlier, I provided an appendix containing the supportingdocumentation.35 See, for example, Hill and Winston (2006).36 This framework accommodates various unmodeled economic, political, or socialinstitutions that may create rigidity in sticker price determination. For example, giventhe large number of colleges and universities that compete for students, sticker pricecould be the result of some degree of price-taking.37 This interpretation would also justify the assumption of a rigid matriculationtarget, M, from which the admissions committee cannot deviate.38 In an appendix (available upon request), I describe more fully why the currentmodel cannot accommodate price competition, and propose some ideas for how onemight be able to. I provide an example that illustrates the difficulties of includingendogenous price setting.39 This result slightly differs from similar games of Bertrand competition with

Table 3Model predictions: preference effect.

Utility (net of price) conditional on admission

Student type RD game ED game ED–RD

High-income/good-quality V+(1−ω)z V+(1−ω)z 0Low-income/good-quality V+(1−ω)z V+(1−ω)z 0High-income/bad-quality V + 1−ωð Þz

2−α⁎br

V+(1−ω)z +

Low-income/bad-quality V + 1−ωð Þz2−α⁎

br

V + 1−ωð Þz2−α⁎ ⁎

br

−

Table 4Model predictions: summary of effects.

Change in welfare

Student type Price Admission Preference Total

High-income/good-quality 0 0 0 0Low-income/good-quality + 0 0 +High-income/bad-quality 0 + + +Low-income/bad-quality + − − −/+

417M. Kim / Journal of Public Economics 94 (2010) 410–420

positive. In general, ED competition is strictly welfare-reducing if andonly if

πℓb** dr**−dr*ð Þb πℓb* −πℓb**ð Þ V + dr*ð Þ + αbr*−αbr**ð Þ 1−ωð Þz: ð11Þ

The left-hand side measures the price effect—the increase inexpected utility from receiving a larger financial aid award conditionalon being admitted to a school. The first term on the right-hand sidemeasures the admission effect—the decrease in expected utility fromthe lower probability of admission in the ED game than in the RDgame. The second term on the right-hand side measures thepreference effect—the decrease in expected utility from the decreasedlikelihood of forming efficient student–school matches. If we assumethe same parameter values from the previous sections–that ϕ=0.55and λ=0.7–then numerical analysis suggests that inequality Eq. (11)holds for all values of M, V, p, z, and ω; i.e., ED competition isstrictly welfare-reducing for low-income bad-quality students. (SeeAppendix A for details.)

5.5. Institutional quality

Winston (2000) and Ehrenberg (2000) both describe the marketfor elite higher education as an arms race: schools, concerned less bythe absolute level of student quality (or library volumes, percent offaculty with Ph.D.s, etc.) and more by their relative position to otherschools, spend money to try to gain rank, or sometimes, just to keepup with the competition. As Winston (2000) puts it: “In an arms race,there's a lot of action, a lot of spending, a lot of worry but, if it's asuccessful arms race, nothingmuch changes” (p. 2). In much the sameway, ED competition yields the same result: for all of the potentialgaming and maneuvering, schools end up with the same level ofstudent quality after ED competition as they had without it; i.e.,X⁎=X⁎⁎: ED competition is something schools do in order to keep up,and not get ahead. This result has significant implications for potentialreform of the ED system: even if schools all preferred to end EDpolicies, no single school could unilaterally abstain from ED policies(Ehrenberg, 2000; AFZ, 2003).33

33 In a recent public address, Richard Levin, president of Yale University, explains:“The problems [associated with ED competition] would more or less disappear if allearly admissions programs were abandoned... Unfortunately, it is not so easy to getfrom here [the ED game] to there [the RD game], unless all colleges and universitiesmoved simultaneously in this direction...Any single school opting to abstain from earlyadmissions while others persisted would run the risk of losing qualified applicantswho simply couldn't or wouldn't wait” (Levin, 2003).

6. Discussion of price assumptions

For analytical tractability, I make several strong behavioralassumptions regarding how schools compete for students on price:1) schools do not awardmerit-based financial aid (i.e., all financial aidis need-based); 2) schools are permitted to compete for low-incomestudents on financial aid; and 3) sticker price is exogenous.

The first two assumptions imply that financial aid can only dependon income and not quality and that financial aid to low-incomestudents may exceed demonstrated need. The first assumptionreflects the fact that there is only a small number of need-blindschools that offer merit aid—and within these schools, the number ofmerit aid awards is also small.34 To the extent that schools do notaward merit aid, they cannot compete for high-income students onfinancial aid (since, by definition, high-income students do not needfinancial aid). The second assumption reflects the possibility thatschools may compete on “need-based” financial aid. For example, aneed-blind school can meet demonstrated need using a mixture ofgrants, loans, and work-study opportunities. However, schoolscompete on need-based financial aid when they choose to increasethe proportion of grants in a student's financial aid package.35

Regarding the exogenous sticker price assumption, as long as theprice paid by high-income students is the same across time periods andEq. (10) is satisfied, the model's results do not change at all. The keyelement of the screeningmechanism is thatwhile low-income studentspay different prices across time periods, high-income students pay thesameprice. If theprice is the sameacross timeperiods, it doesnotmatterhow that price is determined (i.e., exogenously or endogenously).

The primary purpose of this model is to explain the admissionspolicy behavior of these schools, and not necessarily the optimalpricing behavior. For example, an exogenous sticker price could bethought of as determined by some independent equilibriumprocess.36

Instead of modeling the actions of schools, the model presenteddescribes the actions of schools' admissions committees. At privateschools, tuition is often chosen by a board of trustees. In contrast,admissions decisions–but not tuition decisions–are made solely by anadmissions committee that is essentially told how many students itmust admit and the size of its financial aid budget. Thus, from theadmissions committee's point of view, sticker price is exogenous.37

To include endogenous price in the current environment, it is notsimply enough to allow schools to choose price.38 Each student type(across income, quality, and school preference) represents a singlehomogeneous unit whose demand is perfectly elastic with respect toprice. Put differently, since every student type has a positive mass ofstudents, thereexists an incentive todeviate fromanyequilibriumprice inorder to steal away (good-quality) students from the competing school.39

Thus, if an equilibrium exists, it can only exist in mixed strategies.40

capacity constraints (e.g., Kreps and Scheinkman, 1983; Davidson and Deneckere,1986) because the objective of the firms (or schools) is not profit maximization, bututility maximization. In a profit-maximizing context, a firm will not undercut price bytoo much, as too low of prices results in (by definition) lower profits. However, in autility-maximizing context, price is merely instrumental to obtain some otherobjective.40 Though given the non-continuity of schools' objective functions, it is not clear thateven a mixed strategy equilibrium exists.

418 M. Kim / Journal of Public Economics 94 (2010) 410–420

7. Conclusion

This paper presents a stylized theoretical model of ED competitionin the market for higher education. The model, though relativelysimple, generates many theoretical implications that are consistentwith beliefs held by researchers and high-school and college officials.First, applying ED improves a student's chances of admission. Second,financial aid students are less likely to be early applicants; thus, theyare overall less likely to be admitted to need-blind schools. Third, EDcompetition results in a more efficient allocation of students acrossschools through students signaling of school preferences. Fourth,while ED competition unambiguously increases yield rates, it has anambiguous effect on acceptance rates. My calibrated results suggestthat ED competition actually increases acceptance rates, though themagnitude of the increase is quite small. Fifth, strictly in terms ofobtaining student quality, ED competition does not make schoolsbetter off than they would be without ED; ED allows schools to keepup with the competition, and not get ahead of it, in equilibrium.41

Sixth, with respect to full-pay students, ED competition has weaklypositive welfare effects: higher-ability students are no worse off,while lower-ability students are strictly better off. Finally, withrespect to financial aid students, while ED has positive welfareconsequences for higher-ability students, ED most likely has negativewelfare consequences for students of lower ability. Put differently, EDconstructs a barrier to access for these students: full-pay students aremore likely to be admitted under ED competition than financial aidstudents of comparable quality. This is remarkable since under aneed-blind admissions system, financial aid status is not supposed toaffect the probability of admission.

The model presented takes need-blind admissions as given, thenexplains why need-blind schools have incentives to implement EDpolicies. However, it is also important to recognize that schools mayhave incentives to advertise need-blind admissions to potentialapplicants. Therefore, it would be interesting to develop a model thatrecognizes the simultaneous choices of need-blind admissions and earlyadmissions policies. Other directions for future research includeexpanding the model to include a richer structure of idiosyncraticstudent–school match values, schools' choices of early action (with EDor no early admissions as alternatives), dynamics, and the explicittreatment of peer effects in the education production technology.

Acknowledgments

I thank Jim Andreoni for helpful guidance, and Luke Davis, DennisEpple, Sarah Hamersma, Craig Marcott, Rich Romano, AnanthSeshadri, an anonymous referee, and seminar participants at CarletonCollege, University of Florida, Swarthmore College, and University ofWisconsin for comments on earlier versions of this paper. Allremaining errors are my own.

Appendix A. Numerical analysis

A.1. Calibration

In this section, I provide preliminary numerical evidence for theresults that depend on the relative magnitudes of the parameters in themodel. In order to reduce thedimension of the parameter search space, Icalibrate the model with estimates obtained from previous research.

Solon (1992) and Zimmerman (1992) estimate that the correla-tion between fathers' and sons' incomes is ρ=0.4. Epple and Romano(1998) argue that intergenerational income correlation arises from

41 AFZ (2003) provide many other incentives schools may have to have ED policies,other than obtaining high-quality students. For example, ED may identify “enthusias-tic” students who have higher match values with the schools to which they apply, withthe hypothesis that having more “enthusiastic” students creates a better student body.

two sources: correlation between income and student quality; and,for a given ability, the correlation between income and school quality.Thus, they conclude that 0.4 should be considered as an upper boundon ρ. As a matter of simplicity, I assume that ρ=0.4. As I explain inAppendix B, in order to move from an estimate of ρ to values of(λh,λl), I must make an assumption about their relative magnitudes;i.e., λ=λh=1−λl . A correlation of ρ=0.4 implies λ=0.7.

Hill et al. (2004) estimate that fifty-five percent of students atCOFHE schools pay the full sticker price, hence I assume ϕ=0.55. Hillet al. (2004) recognize that financial aid students differ in ability-to-pay, and do not all have zero ability-to-pay. Since I assume thatstudents are either full-pay or no-pay students, 0.55 should beconsidered as a lower bound on ϕ.42

In addition, Hill et al. (2004) also calculate that the average stickerprice at the COFHE schools (in 2001–02) is $33,831; the total stickerprice over four years equals $135,324 (assuming no increase in price).Thus, I assume p=135. I use the cumulative four-year price sinceanother parameter, V, represents the total lifetime value of schoolingover no schooling; therefore, the price paid must also represent thetotal price. Similar to Epple and Romano (1998), an alternativeinterpretation for V is the total lifetime value of elite (private)schooling over a non-elite (public) alternative. As a sensitivityanalysis, I also try alternative values of p={55,95}.

After setting these parameters, only three parameters remain: M,the matriculation target, V, the value of schooling, and z, themagnitude of school preference. Quite arbitrarily, I chose a “small”value for z, e.g., z=2. This says that other things equal, a studentwould require a compensating differential of $2000 to attend, forexample, Yale instead of Princeton. The results are not at all sensitiveto the choice of z, so long as it is not too large relative to p.

In the model, M is a school's matriculation target—but since in thereal world there exist more than two schools, M roughly representsthe “tightness” of admissions into elite need-blind schools; smallervalues of M correspond to tighter, or more selective admissions; i.e.,lower overall acceptance rates. Both M and V are difficult to interpret,and therefore estimate (or find estimates). Therefore, I allow thesetwo parameters to vary, taking the other parameter values as given,and report results as functions of a range of values for M and V. Theparameter choices for p, ϕ, and λ imply that M∈(0.3425,0.5) andVN135. Therefore, the search grid contains these boundaries (with anadditional upper bound of Vb5000); the resolution of the search gridis set at increments of 0.0005 forM and 1.0 for V. GAUSS programs areavailable upon request for any of the analysis.

A.2. Equilibrium exists

Fig. 1 shows in which regions of values for V andM there exists the** ED equilibrium (EQ). In this figure, p=135, however, the graph isqualitatively identical for alternative values of p. The curve representsthe set of points at which low-income students are indifferentbetween applying ED and applying RD. In the region below and to theright of the curve is the region where an equilibrium exists; i.e., d⁎⁎,the equilibrium financial aid award, exceeds d̂, the equilibriumreservation value for low-income students to prefer applying RD. Asthe value of schooling increases, only sufficiently “loose” admissionswill support an equilibrium; if admissions is tight, then acceptanceprobability for low-income students is lower, which implies that thereservation financial aid award must be higher to compensate for thereduced acceptance probability.

42 The parameter ϕ denotes the population proportion of high-income types. Ifincome and ability are positively correlated, then the proportion of matriculating high-income types will be higher than the population proportion. Thus, this calibrationchoice is made for simplicity.

Fig. 1. Equilibrium space over M and V.

419M. Kim / Journal of Public Economics 94 (2010) 410–420

A.3. Student welfare

ED competition is strictly welfare-reducing for low-income bad-quality students if and only if

πℓb** dr**−dr*ð Þb πℓb*−πℓb**ð Þ V + dr*ð Þ + αbr*−αbr**ð Þ 1−ωð Þz:

I introduce one simplification to the analysis. Since ω and z neverenter separately in the inequality, I define a new variable, z ̃=(1−ω)z.In addition to searching over values ofM,V, and z ̃, I also searched overp∈(0,200) in increments of 10. This inequality held at every point inthe search grid. This suggests that ED competition is strictly welfare-reducing for low-income bad-quality students.

Appendix B. Derivation of ρ

The important thing to realize is that quality and income areBernoulli random variables. Then, calculating the covariance, and inturn, the correlation coefficient, ρ, immediately becomes amechanicalexercise.

First, calculate the first and second moments of y and q:

E yð Þ = hϕ + ℓ 1−ϕð ÞE qð Þ = g λhϕ + λℓ 1−ϕð Þ½ � + b 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ �E y2� �

= h2ϕ + ℓ2 1−ϕð ÞE q2� �

= g2 λhϕ + λℓ 1−ϕð Þ½ � + b2 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ �E yqð Þ = hgλhϕ + hb 1−λhð Þϕ + ℓgλℓ 1−ϕð Þ + ℓb 1−λℓð Þ 1−ϕð Þ:

Then, the variances of y and q are, respectively:

var yð Þ = E y2� �

− E yð Þ½ �2 = ϕ 1−ϕð Þ h−ℓð Þ2;

var qð Þ = E q2� �

− E qð Þ½ �2

= λhϕ + λℓ 1−ϕð Þ½ � 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ � g−bð Þ2;

and the covariance between y and q is:

cov y; qð Þ = E yqð Þ−E yð ÞE qð Þ = λh−λℓð Þϕ 1−ϕð Þ h−ℓð Þ g−bð Þ:

Putting all of these parts together results in the correlation, ρ:

ρ = cov y; qð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffivar yð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

var qð Þp

=λh−λℓð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ϕ 1−ϕð Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλhϕ + λℓ 1−ϕð Þ½ � 1−λhð Þϕ + 1−λℓð Þ 1−ϕð Þ½ �p :

There are two things to point out here. First, since both square rootterms are always strictly positive, the correlation is positive if and onlyif λh−λl N0. The second point, which may be less obvious, is thatthere are multiple (infinite) pairs of (λh,λl) that map into a single ρ,given any value forϕ. This is problematic since in order to calibratemymodel, I take some established value for ρ and back out what values λh

and λl must take in the model. Since multiple (λh,λl) pairs map intoany single ρ, I have essentially an identification problem. Therefore, Imust make some “identification assumption” to pin down themagnitudes of λh and λl .

The most natural assumption (to me) is to assume symmetryacross the 45-degree line of quality and income. Formally, thisassumption is

Pr q = g jy = hf g = Pr q = b jy = ℓf g= 1−Pr q = g jy = ℓf g;

which in the model's notation is

λh = 1−λℓ:

Let λ=λh=1−λl . Applying this identifying assumption to theformula for ρ yields

ρ =2λ−1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ϕ 1−ϕð Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2λ−1ð Þ2ϕ 1−ϕð Þ + λ 1−λð Þ

q= h λð Þ;

where the function h:[0,1]→ [0,1] is a bijection (and specifically, one-to-one); therefore, any value of ρ uniquely determines λ.

Appendix C. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.jpubeco.2010.01.003.

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