Competition Among Universities and the Emergence of the Elite Institution

{Journals}Boer/54_3/f190/makeup/f190.3d

COMPETITION AMONG UNIVERSITIESAND THE EMERGENCE OF THE ELITE

INSTITUTION*

Gianni De Fraja and Elisabetta Iossa

University of York and C.E.P.R. and Brunel University

ABSTRACT

We consider an environment where two education institutionscompete by selecting the proportion of their funding devoted toteaching and research and the criteria for admission for theirstudents, and where students choose whether and where to attenduniversity. We study the relationship between the cost incurred bystudents for attending a university located away from their hometown and the equilibrium configuration that emerges in the gameplayed by the universities. Symmetric equilibria, where universitieschoose the same admission standard, only exist when the mobilitycost is high; when the mobility cost is very low, there is no purestrategy equilibrium. For intermediate values of the mobility cost,only asymmetric equilibria may exist; the final section of the paperprovides an example where asymmetric equilibria do indeed existfor a plausible and robust set of parameters.

I. INTRODUCTION

There are surprisingly few theoretical studies devoted to the universitysystem, despite its quantitative and qualitative importance, andresearchers’ direct interest in it. In this paper, we propose a theoreticalmodel of competition between universities. A sound understanding ofthis topic is important both as a possible way to explain the considerablydifferent manners in which the sector is organized in different countries,

275

# Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of EconomicResearch 2002. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF,UK and 350 Main Street, Malden, MA 02148, USA

Bulletin of Economic Research 54:3, 2002, 0307–3378

*We wish to thank two referees and Rich Romano, the editor for this paper, for theirextensive and extremely helpful comments on previous versions of the paper.


and as an aid to design policies aimed at improving the performance ofthe sector.There are several basic features that set the university sector apart

from other, better studied, industries. Firstly, the higher educationmarket does not typically clear in the usual sense: notwithstanding thepotential existence of a market price for university education, mostsystems allocate places to students by administrative rationing. Secondly,the performance of a university (measured along the dimension of thequality of the teaching provided) depends positively on the ability of itsown students: universities use a customer-input technology (Rothschildand White, 1995).1 Thirdly, the profit maximizing behaviour typicallyassumed for large commercial organisations,2 as well as for some not-for-profit private institutions,3 is not likely to be a good proxy for theobjective function of individual universities. The model of this papercaptures all three of these features: universities set an admissionstandard, and their performance depends positively on the ability oftheir students. With regard to their objective function, we choose a verygeneral formulation, and can therefore cover a potentially large range ofsituations: we assume that the universities aim at maximizing a measureof their prestige, which, in turn, we take to be positively affected by thequantity and quality of the students enrolled, and by the success of theuniversity’s research activities.The interaction between students and universities is schematically

modeled as follows: universities set admissions standards, and thosestudents who qualify for admission choose whether to attend universityor not; in addition, if a student achieves the admission standard at morethan one university, she chooses which university to attend. To make thischoice, students take into account the (current) travel and mobility costs(given by the monetary cost of travel and relocation, by the utility costsof being away from family and friends, and so on), and the (future)labour market income which can be obtained after attending a givenuniversity.We study the game between two ex ante identical universities. Our

main contribution is the characterization of the relationship between thelevel of the mobility costs, and the equilibrium configuration. Our results

1 See Clotfelter (1999) and Winston (1995) for a detailed discussion of the specificcharacteristics of the higher education sector.

2 Indeed much theoretical research shows how actions and decisions which would,superficially, appear to indicate that a commercial organization is not maximizing itsprofit, are in fact, once profit is properly defined and strategic interactions with anotheragents taken fully into account, part of a profit maximizing plan (Tirole, 1988).

3 For example, Dranove and White (1994) offer theoretical arguments, based on theconsequences of plausible assumptions on the objective functions of the relevant decisionmakers inside institutions, suggesting that not-for-profit private hospitals do in factultimately behave as if they maximized profit.

276 BULLETIN OF ECONOMIC RESEARCH

# Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2002.


can be summarized as follows.

* If the mobility cost is high, then the equilibrium is symmetric:universities are ex post, as well as ex-ante, identical. They set the sameadmission standard, admit the same number of students, and spendthe same amount on research.

* At lower levels of the mobility cost, if an equilibrium exists, it must beasymmetric: one university becomes an elite institution: it sets a higherstandard and enrols the best students, while the other sets loweradmission standards.

* At lower still mobility cost, there is no pure strategy equilibrium.4

We believe that our model, though highly stylized, capturesappropriately the role of mobility cost in shaping the relationshipbetween institutions within the university sector. For example, theexistence of students grants (or cheap loans) and abundant andconvenient university accommodation seems likely to make themobility cost in the UK lower than in the rest of Europe, enablingan institution to view the entire nation as a potential source of students.Our analysis suggests that the equilibrium configuration which emergesin this case is ‘hierarchical’, with one institution requiring higherstandards than the other, and both institutions recruiting from theentire population: this tallies with the UK situation. In continentalEurope, by contrast, the institutions set similar standards and, by andlarge, students attend their local university.5 In the US, one may arguethat the benefit of attending a leading institution overcomes themobility cost only for brighter students (and=or that the latter havea lower mobility cost). Therefore, our model is consistent with theevidence of an increase in the stratification of the student population,with leading institutions recruiting an ever increasing number of thebrightest students from the whole country, while less bright studentsattend their local college (see Cook and Frank, 1993).6

4Del Rey (2000) develops a model related to ours. She obtains symmetric equilibria.Her analysis therefore corresponds to the case of high mobility cost in our model.

5 This is a point clearly deserving careful empirical investigation: some preliminarycomparison between Italy and England would however support it, suggesting that around8 per cent of English students live at home while studying for a university degree, when thecorresponding proportion is about 67 per cent for Italy; dividing each nation into nineregions, in England, 18 per cent of students attend a university in the same region, whereasin Italy, at least 80 per cent do (Naylor and Smith, 2001).

6Riesman (1998) describes the evolution of the American higher education market andraises the concern that competition among institutions for students will lead to loweracademic standards and marginal differentiation among universities. Hoxby (1997)studies the effects of the increased competition experienced by the American highereducation from 1940 to the present on college prices and quality. She finds empiricalevidence that greater competition led to an increase in the average quality and tuition feesas well as in product differentiation between universities. Recently Epple et al. (2001) findempirical evidence of a hierarchical stratification of US universities.

COMPETITION AMONG UNIVERSITIES 277



We present a general model in Section II: this details the behaviour ofthe universities (subsection II.1) and of the students (subsection II.2). InSection III we show that if a pure strategy equilibrium exists, then itmust be asymmetric, for the mobility cost below a certain threshold. InSection IV we provide an example with specific functional forms, whichshows that such equilibria do in fact exist. Section V concludes.

II. THE MODEL

II.1. The universities.

We consider an environment where a large population of potentialstudents is evenly distributed in two towns, A and B; in both towns thenumber of students is normalized to 1. In each town there is a universitywhich provides education services to the students it enrols.We assume that universities are run by a bureaucracy, or management,

who are interested in the prestige of their institution. Prestige, of course,is a rather vague term, and we let it be given here by a function with threearguments: (i) the number of students, n, (ii) the average ability of thestudent body, �, and (iii) the expenditure on research, R. Formally, wewrite the objective function of an institution as:

W(n; R;�) (1)

To justify (1), we begin by noting that, clearly, other things equal, alarge institution is more prestigious than a smaller one, because, forexample, it is more visible in the local community, it features more oftenin the national press, and so on. Therefore, other things equal, anincrease in the number of students improves the value of the objectivefunction of a university. If the quantity of students affects the prestige ofa university, so does their quality: to the extent that better studentsachieve more prestigious social positions, this also enhances the prestigeof that institutions.7 It is also the case that brighter students earn more inthe labour market, and this improves the university’s long-term potentialfor financial donations and bequests: if, on average, alumni donate orbequeath a proportion of their income to their alma mater, then anincrease in the total earnings of an institution’s graduates generates acorresponding increase in that institution’s potential for donations fromalumni. Finally, in addition to teaching, universities do research; the

7That the success of an institutions’ alumni matters is exemplified in the way universitymagazines and web pages boast of the success and achievement of their graduates.Another example is given by a typical career progression for the heads of UK institutions:several move from a smaller university to a larger one, and then to head an Oxbridgecollege, very small but extremely prestigious institutions. This suggests that size is not allthat matters.




prestige of an institution is clearly affected by the quality of its research.To the extent that expenditure on research increases the quality ofresearch, for example, because it allows institutions to recruit and toretain better researchers, to purchase more expensive equipment, and soon, then, ceteris paribus, an institution prefers to spend more on research.Note also that R could be re-interpreted as expenditure on improving thequality of education: after paying for basic tuition, the university can usewhat is left to improve the quality of research (which increases prestige)or the quality of education (which increases the earnings of the alumni).In view of this discussion, the first partial derivatives of W in (1) are allstrictly positive. With regard to the second derivative, we assume thatWnn(�);WRR(�);W��(�) < 0, where subscripts denote partial derivatives,and that the second cross derivatives are sufficiently small so that therelevant second-order conditions are satisfied. This implies that W(�) isapproximately separable.We assume that each university is assigned the same fixed budget,

b > 0, by the government agency in charge of the higher educationsystem: universities are unable to affect their revenues, and, inparticular, they are not free to choose what students are charged infees. This is a reasonable approximation of the current practice in mostEuropean countries; moreover, from a conceptual point of view, itallows us to analyse the decisions concerning expenditure on researchand academic standards separately from decisions about raisingrevenues. In this setting, each university chooses the required standardnecessary to be accepted as a student.8 We denote by xi, i¼ A; B, thisstandard and let it vary in the subset of the real line: xi 2 X Ð R,i¼ A; B. This implies that only students who reach at least standard xi

are accepted at institution i. On the financial side, we assume thatteaching n > 0 students carries a cost of c(n), naturally with c

0(n) > 0; it

is also convenient to assume c 00(n) � 0, at least in the relevant range, andthat limn 2 1 c

0(n)¼þ1.

II.2. The students

In addition to their location, students, the potential ‘customers’ of thetwo institutions, are differentiated according to their ability. This isdenoted by � 2 [0; 1] (an innocuous normalization). In each town, thestudents’ distribution by ability is F(�), with F(0)¼ 0, F(1)¼ 1, anddensity f (�)¼ F 0(�). We also assume that f 0(x) is bounded from above bya sufficiently large (positive) number, again to ensure that the second-order conditions are satisfied.

8We do not model explicitly the way in which the admission process operates. We can,however think of university requiring students to have a minimum grade at the schoolthey attend before entering university, or setting a specialist pre-entry exam.




We assume that there is a functional relationship linking the admissionstandard set by a university, x, and the lowest ability student that can beaccepted as a student, �. Let (x) be this function, so that �¼ (x). It isnatural to assume that there exists Zx such that (x)¼ 1 for x � Zx: auniversity can set an admission test so tough that nobody can pass it.With this assumption, it is also an innocuous normalization to measurethe admission standard as the lowest ability which is necessary to matchthat standard: (x)� x. It is of course possible to add a stochasticcomponent to this relationship allowing students whose ability is below(above) x to be admitted (rejected) with some probability; this would notalter the nature of the interaction between universities.If a student attends a university located in the town where she does not

live, she incurs a mobility cost. This is analogous to, but somewhat moregeneral than, the mobility cost in the location model in the horizontaldifferentiation literature, which measures the petrol cost of going to ashop far away and the cost of the time spent travelling. In our model, tothese costs there should be added the inconvenience of being away fromone’s home (and therefore missing parents, friends and so on), theadditional cost incurred in renting a room, and so on.If a student attends university, she receives a benefit summarized in a

pay-off function which depends on her own ability, �, the admissionthreshold of the university she attended as a student, x, and the mobilitycost T:

U(x; �)� T; (2)

where Ux(x; �);U�(x; �) > 0, and T 2 {0; t}, with T¼ 0 if the studentattends university in her home town, T¼ t > 0 otherwise.9The function (2) is the reduced form of some underlying process. A

way, by no means unique, to obtain (2) is the following. Let therealization of labour market earnings, w, be distributed in Rþ accordingto a function G(w j �), so that G(w j �) is the probability that the student’s(present discounted value of future) labour market income is below w.Suppose that the shape of G(w j �) depends on the two parameters, �and x, and naturally, that �1 > �2 implies that G(w j �1; x) first orderstochastically dominates G(w j �2; x) for every x, and, conversely, thatx1 > x2 implies that G(w j �; x1) first order stochastically dominatesG(w j �; x2) for every �. In words, given the quality of the university sheattends, a more able student is more likely to receive a higher wage thana less able student, and, a given student is more likely to receive a higherwage if she attends a university with a higher standard. Suppose also thatthe probability of a student obtaining a degree once admitted be an

9Del Rey’s model (2000) differs from ours in that a student’s demand for education ateach of the two universities is independent of her own ability: students only take intoaccount mobility costs and the quality of education.




increasing function of her ability � (�). Then, if a student maximizes herexpected utility, and has a von-Neumann-Morgerstern utility functionwith argument the present discounted value of future labour marketu(w), the first term in (2) would be given by

U(x; �)¼ � (�)ðw 2 Rþ

u(w)dG(w j �; x)þ (1� � (�))ðw 2 Rþ

u(w)dN(w j �; x):

In the above expression N(w j �; x) is the distributions of the futureearnings of a student whose ability is � 2 R who attends an institutionwhere the admission threshold is x, but is not awarded a degree. It isimmediate to show that, if obtaining a degree increases one’s earnings,that is, if G(w j �; x) first order stochastically dominates N(w j �; x), asseems natural, then the assumed signs for the partial derivatives,Ux(x; �);U�(x; �) > 0, obtain.10That ability should influence future wages positively is tautological. It

is also reasonable to assume Ux(�) > 0: it may be that a stricter admissiontest at an institution reduces the number of graduates from thatinstitution, and therefore increases their rent in the labour market, orthat it improves the reputation of the graduates of the institution. It mayalso be the case that a higher average ability of the students makesuniversity attendance more productive, for example, via a peer groupeffect or because it allows more advanced teaching.

Assumption 1: Ux�(x; �) > 0.

The benefit of an increase in the admission threshold is higher forbrighter students. To the extent that x is positively correlated with thequality of teaching it implies that brighter students benefit more fromhigher quality teaching staff (it may also capture the idea that brighterstudents, ceteris paribus, need to exert less effort to be successful in theirstudies). Consequently, a student’s net benefits from more restrictiveadmission requirements and=or tougher exams increase with herability.We assume that a student who does not attend university has a

reservation utility normalized to U(0; 0). Therefore, a student living intown i would surely attend the university in town i if she is admitted; ifshe is not admitted, she would travel to town j if and only if U(xj; �) � t.While the following is an almost immediate consequence of Assump-

tion 1, it is worth stating formally, since it facilitates the analysis of therest of the paper considerably.

10 In this example, we could allow for the probability of a student obtaining a degree,� (�), to depend also on the admission threshold level x, which could be due to the fact thata tougher admission test makes a student of a given ability ‘closer’ to the lowest ability ofthe students admitted.




Proposition 1: Let university j set standard xj, j¼ A; B. Let xA > xB.(a) Let student of ability ~��k living in town k attend university A. Then all

students of ability � � ~��k living in town k also attend university A.(b) Let student of ability \�k living in town k attend university B. Then all

students of ability � � \�k living in town k attend either university A oruniversity B, and all students of ability � ˘ \�k living in town k either attenduniversity B or do not attend university.

(c) Let the student of ability ��k living in town k not attend university.Then all students of ability � ˘ ��k living in town k do not attend universityeither.

This follows from the fact that, given the mobility costs, a student ofhigher ability gains more from attendance to a university with a stricteradmission test. An analogue of this is established in Epple and Romano(1998). The proposition implies a straightforward stratification byability: within each town, the ablest students go to the best university.Of course it may happen that it is not the case that in a given town thereare students attending both universities.Figure 1 illustrates the proposition and shows the marginal values of

��k, \�k and ~��k for which statements (a), (b) and (c) in Proposition 1 hold.The horizontal axis measures students ability. Students whose abilityand location is along the dashed (dotted) line go to university A (B).Note that the lowest ~��B could be as high as 1, in which case no studentfrom town B goes to university A, and that the lowest ��A could be ashigh as the lowest \�A, in which case no student from town A goes touniversity B.

III. EQUILIBRIA

The assumptions of the previous two subsections allow us to simplifyconsiderably the interaction between universities and between univer-sities and students. We can set up a conceptually simple two-personnormal form game with universities A and B as the players and with their

θAθAˆ θB

~0 1

θ

A

B

θBˆ

Fig. 1. Students’ attendance at universities




strategy space given by the admission standards, xA 2 X, xB 2 X. Thepay-off functions of the two universities are given by:

�j(xA; xB)¼W(nj(xA; xB); b� c(nj(xA; xB));�j(xA; xB)); j¼ A; B;

where nj(xA; xB) and �j(xA; xB) are the number of students admittedto university j, and the average quality of the students at university j( j¼ A; B), respectively, given the admission standards xA and xB. We donot determine the exact shape of the functions nj and �j. It is, however,straightforward to establish that @�j=@xj > 0: an increase in a university’sadmission standards increases the average ability of that university’sstudents. This is obviously plausible. Finally, notice that we havesubsituted for R¼ b� c(nj(xA; xB)), where b > 0 is the university budget.A convenient benchmark case is obtained when universities are

monopolies. We can think of two cases: a university can be a monopolyin the whole sector, for example because it is the only university allowed,or it can be a monopolist in its own town, for example because thestudents do not travel. In the first case, if t¼ 0, the number of students itadmits if it sets a standard x is 2(1� F(x)), in the second it is (1� F(x)).The first order conditions for the two cases are:

2(Wn(�)�WR(�)c 0(�))f (x)¼W�(�)�x(�); (4)

(Wn(�)�WR(�)c 0(�))f (x)¼W�(�)�x(�): (5)

The definition of the function � is analogous to that given above for thecase where two universities compete. We may denote by xM and xm thesolution to the above expressions; and note that the assumption that thecross derivatives are sufficiently small ensures that the relevant second-order conditions identify xM and xm as the unique maxima for theproblems.In general the relationship between xM and xm is ambiguous. Consider

the optimum choice of x when the distribution of students is given byF(�) (xM and xm are the optima for ¼ 2 and ¼ 1, respectively).Total differentiation of the first-order conditions (leaving out the crossderivatives) gives:

dx

d¼

f (�)Wn(�)�WR(�)c

0(�)þ [Wnn(�)�WRR(�)c 0(�)�WR(�)c 00(�)](1� F(�))

d2W=dx2

this would be negative if W(�) and c(�) were linear (Wnn(�)¼WRR(�)¼c 00(�)¼ 0). Figure 2 depicts the indifference map in the (x; n) plane (usingthe constraint given by research): the budget set is given by the curves1� F(x) and 2(1� F(x)) for the two cases. In the picture xM > xm,




although it should be obvious that the opposite relationship could wellhold. Intuitively, when the population increases the same reduction inthe admission threshold is compensated for by a larger increase in thenumber of students. IfWn(�),WR(�) and c 0(�) are approximately constant,then the cost reduction of this increase in the number of students isapproximately constant, but the benefit increases as the populationincreases. Vice versa, when W is concave, the benefit of an increase instudents number is lower, and the cost in reduced research expenditure ishigher, for a large university.We can now begin the analysis of competition between universities.

Our first result may appear surprising.

Proposition 2: A symmetric equilibrium in pure strategies exists if and onlyif t � U(xm; xm).

Proof: At a symmetric equilibrium in pure strategies, it must be the casethat xA ¼ xB ¼ xm: any value of common threshold xA ¼ xB differentfrom xm necessarily violates the first-order condition (5). Noticemoreover, that, when t is high enough, xA ¼ xB ¼ xm does indeedidentify a Nash equilibrium, in view of the definition of xm and of theassumptions we made on the second derivatives of the function W.Now suppose, by contradiction, that there is in fact a symmetric

equilibrium with t < U(xm; xm). We now show that one university, say B,has an incentive to deviate from {xm; xm}. Let L be such that

U(xm � L; 1)¼U(xm; 1)� t

That is, if a university, say university A, reduces its standard by L, itmakes the brightest student living in town A indifferent betweenattending university A and travelling to B. Clearly L > 0, and byAssumption 1, U(xm � L; �) > U(xm; �)� t for every � 2 [xm; 1). Now let

1

2

0 1

xxm xM

n

Fig. 2. The standard chosen by a monopolist




university B choose xm � ", for some " 2 (0; L). Its change in pay-off isapproximated by:

�" (Wn(�)�WR(�)c 0(�))@nB(xm; xm)

@xB

��xB < xm

þW�(�)�x(�)

0@

1A (7)

where

@nB(xm; xm)

@xB

��xB < xm

¼ limh 2 0�

nB(xm; xm þ h)� nB(xm; xm)

h

is the left partial derivative of nB(xm; xm). Note that, for xB < xm,nB(xm; xB)¼ 1� F(xm)þ 2(F(xm)� F(xB)) and, therefore

@nB(xm; xm)

@xB

��xB < xm

¼ limh 2 0�

2(F(xm)� F(xm þ h))

h¼�2f (xm)

and (7) becomes:

�" (�2(Wn(�)�WR(�)c 0(�))f (xm)þW�(�)�x(�))

using (5) this is:

"W�(�)�x(�) > 0

Therefore a small reduction from xm increases university B’s pay-off; thisestablishes the Proposition.On the other hand, if t � U(xm; xm), then this deviation is not possible,

because, if a student of ability below xm travelled to the other town toattend university, she would get a negative utility. &

The argument underlying the proof of the Proposition may bedescribed informally as follows. If xA ¼ xB and xA, say, is increasedmarginally, then the objective function for university A is the same aswhen there is local monopoly since no one would travel to town A: Thisimplies that if xA ¼ xB < xm we are not at an equilibrium. Moreover, ifxA ¼ xB and again xA is decreased marginally, then either there is notravel to town A (t � U(xA; xB)), or there is some travel to town A bythose who now have the opportunity to go to a university (t < U(xA; xB)).In the former case, if initially xA ¼ xB < xm, then one can check thatuniversity A’s payoff rises more rapidly as xA is decreased marginallythan under local monopoly. Hence, xA ¼ xB ¼ xm is necessary for asymmetric equilibrium.11 However, a symmetric equilibrium does notexist for low mobility costs. This is because of the asymmetric effect of a

11We thank an anonymous referee for suggesting this argument.




small reduction and a small increase in a university’s choice of x. Thereasoning is as follows. When xA ¼ xB, all students located in townk¼ A; B with ability � � xA go to the university in their home town; whileall students with ability � < xA do not attend university. Local optimalityimplies that the beneficial effects of an increase in x by " > 0 (given by theimprovement in the quality threshold) are exactly compensated for by itsnegative effect of the reduction in size (dampened by the resources freedfor research). However, when the admission threshold is lowered by ", itis possible to obtain an increase in the number of students whose positiveeffects have the same absolute value as an increase by 2".Mixed strategy equilibria, in the present situation, lack appeal; we

therefore prefer to investigate whether asymmetric pure strategyequilibria exist. For the sake of definiteness, we consider the casexA > xB (so that A and B are mnemonics for ‘Alto’ (high) and ‘Basso’(low)). Clearly, if {xA; xB} is an equilibrium, then the mirror strategy pair{xB; xA} is too.An important concept in the rest of the paper is the function

~��(xA; xB; t), which is the ability level such that the student with thatability who lives in town B (the low admission university) is indifferentbetween going to her local university and travelling to town A, where sheincurs travel cost t, but can expect higher future labour market income.~��(xA; xB; t) is therefore the solution in � of

U(xA; �)� t¼U(xB; �); (8)

if there is a solution to (8) in (xA; 1). If there is no such solution, then~��(xA; xB; t) is the appropriate extreme of the interval (xA; 1):~��(xA; xB; t)¼ 1 if t > U(xA; 1)�U(xB; 1), and ~��(xA; xB; t)¼ xA ift < U(xA; xA)�U(xB; xA). Note that, by Assumption 1, there is at mostone value of � which satisfies (8). Moreover, denote by ��(xB; t) the abilitylevel such that the student with that ability who lives in town A isindifferent between travelling to town B and not attending the university.Then, ��(xB; t) is the solution of

U(xB; �)� t¼U(0; 0) (9)

if there exists � 2 [xB; xA] satisfying the above. Otherwise, if the solutionto (9) is such that � < xB then ��(xB; t)¼ xB, and if the solution to (9) issuch that � > xA, then the �(xB; t)¼ xA, and no student from university Aattends the university in town B.An immediate consequence of Proposition 1 and of the definitions of

~��(xA; xB; t) and ��(xB; t) is the following.

Corollary 1. At any equilibrium where xA > xB: students from town Aattend university A if � 2 [xA; 1] and university B if � 2 [��(xB; t); xA).Students from town B attend university A if � 2 [~��(xA; xB; t); 1] anduniversity B if and only if � 2 [xB; ~��(xA; xB; t)).




The next Proposition is the main result of this section. It characterizeshow the equilibrium configuration changes as the mobility cost changes.This is important: casual empiricism suggests that the parameter t variesconsiderably from country to country; moreover, education policiestargeted to the university sector are likely to affect this parameter,perhaps even unintentionally.

Proposition 3: There exist t1 < U(xm; xm), such that:For t 2 [0; t1) there is no pure strategy equilibrium.For t 2 [t1;U(xm; xm)] there are, at most, asymmetric pure strategy

equilibria.For t > U(xm; xm) the pure strategy equilibrium is symmetric, with

xA ¼ xB ¼ xm.

Proof: The third statement has been proved in Proposition 2, and thesecond follows trivially from it. Consider the first statement. We showthat there is no pure strategy equilibrium for t¼ 0, using an argumentwhich can be extended by continuity to a (sufficiently) small t > 0. Bycontradiction, let {xA; xB} with xA > xB be a Nash equilibrium whent¼ 0. In this case all students with � 2 [xB; xA) attend university in townB, and all students with � � xA attend university in town A. Note that thepay-off of university B must be at least as big as university A’s:

�B(xA; xB) � �A(xA; xB): (10)

The reason is that, otherwise, university B could deviate to xA þ ", take(almost) all the students from university A, and therefore improve itspayoff. Notice also that ��(xB; t)¼ xB < 1, because of t¼ 0. Now let ~xx begiven by: 1� F(~xx)¼ F(xA)� F(xB): ~xx is such that there are, overall,2(F(xA)� F(xB)) students of ability ~xx or higher. Clearly if university Achose ~xx, it would have the same number of students as university B has atthe candidate equilibrium, and therefore the same research expenditureas university B at this candidate equilibrium. However, since ~xx > xB itwould have a bigger pay-off:

�A(~xx; xB) > �B(xA; xB) (11)

(10) and (11) together imply �A(~xx; xB) > �A(xA; xB) but this in turnimplies that xA cannot be university A’s maximizing choice. Thisestablishes the first statement. &

IV. AN EXAMPLE

In this section we specify the functional relationships of the model inorder to show that asymmetric equilibria do in fact exist for plausible




and robust functional forms and parameter sets. We modify the modelas may be necessary to achieve explicit and not too cumbersomesolutions (and therefore some of the results obtained in the generalset-up considered above may not hold). Specifically, we assume thefollowing

U(x; �)¼ wxþ � (12)

W(n; R;�)¼ wxnþ R (13)

c(n)¼ cn2 (14)

F(�)¼ � (15)

Note that (12) implies Ux� ¼ 0 against Assumption 1. This simplifiesthe analysis, without altering the qualitative features of the solution. (14)and (13) are specific functions which allow the determination of anexplicit solution. According to (13), a university’s pay-off is an increasingfunction of the number of graduates, n, and of its own admissionstandard, as well as the amount spent on research. This simplifies theanalysis, and is justified on the grounds that an increase in the admissionstandard increases the average ability of its graduates. Finally, (15) saysthat the distribution of abilities is uniform.Since our aim is to show that asymmetric equilibria exist, we further

simplify the model by assuming that the utility from obtaining auniversity degree is always greater than any mobility costs the studentmight have to incur in order to attend the university. That is, for everyx; � 2 (0; 1]

U(x; �)� t � U(0; 0)

Hence U has a discontinuity at (0; 0), with a jump greater than t.Given the normalization (x)¼ x, this implies that ��B ¼ ��A ¼ xB:students from both towns with ability at least xB will be admitted touniversity and will be willing to attend.12

Towards the analysis of the equilibrium, note that students in town Aattend university A, if their ability is at least xA, and university B, if theirability is less than xA, but at least xB. Students in town B, on the otherhand, attend university B, if their ability is less than xA, but at least xB;and if their ability is xA or above, attend university A if xA � xB ˘ t=w;university B in the opposite case.Note that the parameter space (t; w; c) can be simplified by defining

g¼ w=c so that the equilibrium can be fully characterized in a Cartesiandiagram (t=w ; w=c). This we do in Figure 3. In the diagram, the lines are

12 This assumption is made for simplicity and implies the non-existence of symmetricequilibria. It is therefore a special case of the result in Proposition 2, which holds becauseof U(xm; xm)� t > U(0; 0) for all t:




given by:

f1 ¼g(gþ 4þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g(gþ 1)

p)

8(gþ 2)(gþ 1); f2 ¼

g(gþ 4�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig(5gþ 16)

p)

8(gþ 2)2;

f3 ¼gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig(3gþ 8)

p4(gþ 2)2

:

These lines identify a number of regions and the following propositiondescribes the nature of the equilibrium in each of them.

Proposition 4: In the white areas in the figure, there is no pure strategyequilibrium. In the remaining areas there exists an asymmetric equilibrium.In the areas labelled I, university A and B choose respectively

xA ¼gþ 22(gþ 1)

¼ xm (16)

xB ¼(4þ 3g)(gþ 4)8(gþ 1)(gþ 2)

(17)

In the areas labelled II

xA ¼gþ 42(gþ 2)

¼ xM (18)

xB ¼(gþ 4)2

4(gþ 2)2(19)

( )( )28

2241 +

++=

( )1+gg

gggf

( )1+g

( )( )( )22

28

1654

++−+=

g

ggggf

( )( )23

24

83

++=

g

gggf

0.01

0.02

0.03

0.04

0.05

0.06

0.07

00.1 0.2 0.3 0.4 0.5g

w/c=g

t /w

I

II

Fig. 3. Equilibria in the parameter space




The proof of this result is analytically very cumbersome, and isrelegated to an appendix available on request from the authors or fromwww-users.york.ac.uk/�gd4/elite.htm. Note also that area Iextends beyond the boundary of the drawing.However, a verbose description of the two type of equilibria can be

given relatively straightforwardly: when the parameter combination isrepresented by a point in area labelled I, students’ attendance touniversities is described in Figure 4: students who can be admitted touniversity A go there only if they live in town A; all other students, ofability xB or above go to university B. The admission standard set byuniversity B is sufficiently close (relative to the mobility cost) to make itimpossible for university A to capture all the high ability students: thoseliving in town B go to town B. So university A chooses the optimallocation given distribution F(x), xm, determined in (5). Consider nowuniversity B: could it profitably deviate from standard xB? Suppose thatuniversity B lowers its standard so as to lose its high quality students andgain more low quality students from both towns. Since the higher themobility costs the lower the level of xB that induces the high qualitystudents in town B to move to town A; it is clear that (given g) for highmobility costs, this deviation is too expensive in terms of lost prestige. Adifferent type of deviation is a sufficiently large increase in admissionstandard, so as to leapfrog university A. This, again, will not beprofitable if, given g; the mobility cost is high, since the higher themobility cost the higher the level of xB that is necessary to attract thehigh quality students and the lower the number of students. If aparameter combination is an equilibrium of type I, then a sufficientreduction in the mobility cost will destroy it by making it profitable foruniversity B to deviate in one of the two ways described above.Consider now parameter combinations represented by points in the

area labelled II. Here, all the students who can attend the high qualityuniversity go to university A: This situation is described in Figure 5,where, as before, the dashed (dotted) line denotes students whoselocation and ability is such that they attend university A (B). Tounderstand why this constitutes an equilibrium, note first of all that,given that university B has lower quality than university A, the latter will

θ

A

B

0 1xB xA

Fig. 4. Type I Equilibrium




choose the admission standard that an ‘undisturbed’ monopolist wouldchoose, xM. Consider university B next. It is at a local optimum: giventhat it cannot attract students of ability xA or above, xB is the preferredstandard. However, it could choose an admission standard sufficientlyclose to xA so that all the students living in town B prefer to go touniversity B. This has the benefit of attracting (1� F(xA)) students(the high ability students who attend university A in the candidateequilibrium), but at the cost of losing low ability students from bothtowns, who are no longer capable of passing the admission test. Whenthe mobility cost is low, in order to attract the high ability students,university B needs to choose an admission standard quite close touniversity A, implying a high cost of the deviation: thus an increase in themobility cost would destroy this equilibrium by allowing the university Bto position its standard below xA but sufficiently close to attract the highability students living in town A. Instead of attracting only the highability students in its own town, university B could deviate from thecandidate equilibrium trying to attract all the high ability students: it cando so by leapfrogging university A, choosing an admission standardsufficiently higher than xA. For this deviation to be profitable, it must bethe case that the mobility cost is sufficiently low that university B’sdeviation standard is close enough to xA (otherwise there are not enoughstudents in the two towns).13 To sum up, when an equilibrium in area IIexists, it is destroyed both by an increase in the mobility cost – whichmakes a deviation to a standard below xA profitable – and by a decreasein the mobility cost – which makes a deviation to an admission standardabove xA profitable.Some further features of the above equilibria may be worth

mentioning. First, in the equilibrium in area I, the number of studentsat each of the two universities is nA ¼ (1� xA) and nB ¼ (1� 2xB þ xA)and, therefore, nB > nA: that is, university A is an elite institution: it hasfewer students and higher admission standards. Instead, in the equilibria

θ

A

B

0 1xB xA

Fig. 5. Type II Equilibrium

13This is the same type of deviation we discussed for the previous equilibrium. Noticethough that the mobility cost does not need to be as high as in the previous case since thelevel of xA is lower.




in areas II the number of students at each of the two universities is:nA ¼ 2(1� xA) and nB ¼ 2(xA � xB). A simple calculation shows that, inthis case, nA > nB; this is robust to change in the functional forms. It isalso the case that, in our example, the degree of differentiation betweenthe two universities (as captured by the difference between xA and xB)tends to increase with the level of competition (as captured by areduction in the mobility costs t (relative to w)).

V. CONCLUSION

In this paper we use a simple industrial organization framework to analysethe strategic interaction between two non-profit education institutions.They compete by unilaterally choosing their own admissions standard,and each aims to maximize prestige, which we let depend on the number ofstudents enrolled, their ability, and the success of its research.We concentrate on the relationship between the equilibrium config-

uration and the mobility cost for students. When this cost is low, there isno pure strategy Nash equilibrium. When the cost is high there are purestrategy equilibria; for intermediate values of the mobility cost, ifequilibria exist, then they are asymmetric. The example in Section IVshows that asymmetric equilibria do in fact exist for reasonableparameter combinations.Throughout the paper we have restricted attention to the case where

the universities are allocated a fixed and identical budget from thegovernment and where tuition fees play no role in the determination ofthe allocation of students between the institutions. Further researchshould extend the model to the more general and realistic case whereuniversities are also able to charge a tuition fee to their students and thegovernment optimally designs the subsidy to be given to the universities,taking into account their interaction.

REFERENCES

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Del Rey, E. (2000). ‘Teaching versus research: a model of state universitycompetition’, Journal of Urban Economics, 49, pp. 356–73.

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Epple, D. and Romano, R. E. (1998). ‘Competition between private and publicschools, vouchers, and peer-group effects’, American Economic Review, 88,pp. 33–62.

Epple, D., Romano, R. E. and Sieg, H. (2001). ‘Peer effects, financial aid, andselection of students to colleges and universities: an empirical analysis’,Journal of Applied Econometrics, forthcoming.

Hoxby, C. (1997). ‘How the changing market structure of U.S. higher educationexplains college tuition’, N.B.E.R., New York: Working Paper 6323.

Naylor, R. and Smith, J. (2001). Personal communication, University ofWarwick.

Riesman, D. (1998). ‘On higher education: the academic enterprise in an era ofrising student consumerism’, New Brunswick, N J. and London: Transaction.

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Competition Among Universities and the Emergence of the Elite Institution

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