funding schools for greater equity

www.elsevier.com/locate/econbase

Regional Science and Urban Economics 34 (2004) 203–224

Funding schools for greater equity

Elena Del Rey *

Departament d’Economia, Campus de Montilivi, 17004 Girona, Spain

Received 20 November 2000; received in revised form 4 December 2002; accepted 18 February 2003

Abstract

Countries that finance schools by means of uniform per-student allocations and allow free school

choice seem to recognize the need to regulate admissions at over-subscribed schools. In this paper,

we show that, without such regulations, (i) allowing free school choice leads to complete segregation

unless mobility costs are high, and (ii) higher allocations per disadvantaged student enrolled can help

achieve a unique and less segregated equilibrium, especially when mobility costs are low. The latter

instrument can make regulation unnecessary when the aim is to avoid cream-skimming by publicly

financed schools.

D 2004 Elsevier B.V. All rights reserved.

JEL classification: I22

Keywords: Educational finance; Cream-skimming; Vouchers; Positive discrimination

1. Introduction

Many schools all over the world receive public funding on a per student basis. Money

thus follows the pupil to the school where she enrolls as in a voucher system in which

vouchers were directly provided to schools. This finance scheme plays an important role in

guaranteeing the freedom to choose school provided that such freedom actually exists.

Still, policymakers tend to show a certain concern for the equity implications of free

school choice. In particular, the idea that publicly financed schools should be open to all

on equal basis is widely shared in democratic societies.

Unfortunately, as noted in the report School: a Matter of Choice (OECD, 1994), ‘‘the

freedom of parents to choose schools is often restricted by the necessity for schools to

0166-0462/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/S0166-0462(03)00027-9

* Tel.: +34-972-41-8749; fax: +34-972-41-8032.

E-mail address: [email protected] (E. Del Rey).

E. Del Rey / Regional Science and Urban Economics 34 (2004) 203–224204

choose pupils’’.1 In general, governments seem to recognize the need to regulate

admissions at over-subscribed schools. Indeed, different countries have set different rules

to determine who shall be admitted when there is excess demand.

For example, in Sweden, independent schools receiving government finance cannot

refuse to accept low ability students. Similarly, in Spain, public subsidies to schools are

conditioned on their accepting a minimum proportion of disadvantaged students. In New

Zealand, schools in danger of becoming overcrowded must apply to the government for

permission to limit admissions. In England, local authorities control admission criteria and

in the US admissions criteria and procedures tend to be determined by the district

authorities.2

In this paper we provide justification for such behavior and offer an instrument

alternative to regulation to those countries that want to avoid cream-skimming by publicly

financed schools. To this aim we construct a model with two publicly financed schools that

differ only in location and a continuum of parents who can freely choose school for their

children. Parents differ in skills (or education) and families are unevenly distributed in the

city. Schools care for the average performance of their pupils, which provides them with

recognition and prestige. Achievement of children at school is positively related not only

to the inputs provided by the school (which, unlike in Benabou (1993) and (1994) and

Fernandez and Rogerson (1996), is centrally financed) but also to their parents’ education

level.3

However, individual achievement is not observable. Instead, children leave the school

endowed with a signal: the public recognition of the school they attended, measured by the

average performance of all pupils. Due to this peer group effect, parents prefer the school

with more children of educated parents. In turn, given the aim to maximize the average

performance of pupils, the school confronted with an excess demand selects, among the

applicants, those children of educated parents. As a consequence, the initial difference

between school’s average performances grows deeper.

Although we do not address normative issues here, the aim to equalize school

performances clearly underlies this paper (see De Bartolome, 1990 for a normative

justification of the mixing of types at the social optimum in presence of peer group

effects).

In this framework we show that, without regulation of admission criteria, and unless

mobility costs are high, allowing free school choice when funding is based on a uniform

per student allocation leads to complete segregation. Moreover, higher grants aimed at

disadvantaged students and directly provided to the school where they enroll can

1 See pp. 39–40 and 51–52 of the cited report.2 To the best of our knowledge, only in The Netherlands is there a real full guarantee of entry into any public

school: if it becomes overcrowded, further accommodation must be found (classrooms may be borrowed from

neighbouring schools).3 Education of the family has of late been recognized as one of the main determinants of achievement of

children at school. A statistically significant advantage of 4–19% of the grade has been found in Science and

Mathematics exams of children whose fathers hold a university degree as compared to children whose fathers

have minimum education in 18 OECD countries or regions (TIMSS 1995, Third International Mathematics and

Science Study & IEA, Amsterdam, The Netherlands). See also Vandenberghe et al. (2000).

E. Del Rey / Regional Science and Urban Economics 34 (2004) 203–224 205

encourage schools to choose students who are more costly to educate. Hence, a unique and

more mixed equilibrium results, especially when mobility costs are low. This can make

regulation unnecessary, thus lowering the implementation costs (in terms of planning and

control) of making school choice a fair game.

For simplicity, family residence and school size are both assumed to be given. To partly

overcome this drawback of our model we exogenously change the distribution of family

types and analyze the effect of this change at equilibrium. We proceed similarly with

respect to school size. We thus obtain that, as expected, the greater the segregation of

families’ residential locations, the larger the segregation of children types within schools

may become. Also, for the reasons that will be made clear, the school placed in the

neighborhood populated by less favored families needs to be of a smaller size in order to

maintain a leading position with regards to the other school, i.e., in order to be preferred by

parents.4

To conclude, we point out that if popular schools passively accept all applications (as,

for instance, in The Netherlands) free school choice can be an equalizing measure

provided that all parent types react identically to the policy. It should be noted, however,

that mobility costs are likely to differ across parent types. Indeed, choice may only be

effectively available for better endowed families. This will undermine the equalizing effect

of free school choice when schools are passive.

The paper is organized as follows. Section 2 introduces the general model and analyzes

the behavior of parents and schools. Section 3 analyzes equilibria under uniform per-pupil

funding. Section 4 discusses the effect of positive discrimination of disadvantaged students

by means of higher per-student allocations. Concluding comments are given in Section 5.

The proofs of the propositions are contained in the Appendix.

2. The model

There is a continuum of P ¼ 1 parents in this economy. One half of them are high

skilled or educated ( j ¼ H) and the other half are low skilled ( j ¼ L ). Each parent has one

child that attends some school i. The achievement of this child, sji; is a function of the

inputs provided by both the school (investment per pupil) and the parent (home education).

Children leave school i to pursue higher education or work. In either case the exact value

of their achievement at school is not observed by future professors or employers. Instead,

the latter will know the value of the average performance of children at the school they

attended (Si; i ¼ 0; 1) and will use this information. In this sense, children can be assumed

to leave school endowed with Si: This assumption allows to account for the role of social

networks, which through peer effects, role models, job contacts or norms of behavior will

have an important effect on opportunities in adult life (Benabou, 1996).

4 Our conjecture is that, if residence location was endogenous, this would follow a socially stratified pattern

(see Hartwick et al., 1976). On the other hand, given the assumptions of our model, if school size was endogenous

both schools would choose to be smaller in order to exclusively enroll pupils of the good type. This would present

a problem since, generally, it is the right of each pupil to have a school to attend.


Families are located along the line 0� 1 and distributed in a way such that, when

moving towards the east, the proportion of highly educated parents increases at rate b.

Then, at location x; there are ð1� bÞ=2þ bxð Þ% families of type H and

(1� ð1� bÞ=2� bxÞ% families of type L: As b increases, the distribution of family

types in the district becomes more segregated. Each family is thus characterized by

j ¼ H ; L and xað0; 1Þ (see Fig. 1).

We consider two schools i ¼ 0; 1 placed at x ¼ 0 and x ¼ 1 , respectively. The

performance of children of j-parents attending school i is

sji ¼ c jki i ¼ 0; 1 j ¼ H ; Lð Þ

where ki is per-student investment at school i. In order to reflect the role of home education

or family background on achievement at school we assume

c H > c L > 0

Let Hi and Li be the number of children of H and L parents, respectively, enrolled at

school i of size Ri;with R0 þ R1 ¼ 1. Schools are not allowed to reject applications if they

still have some available capacity. Then Hi þ Li ¼ Ri at both schools and the average

productivity obtained by children at school i is

Si ¼HicH þ Lic L

Hi þ Liki ð1Þ

or, letting Hi ¼ hiRi and Li ¼ liRi; and since hi þ li ¼ 1

Si ¼ ð1� liÞc H þ licL

� �ki ð2Þ

Therefore, school performance is independent of school size: only relative admissions

matter. These relative admissions may be expressed in terms of L or H type children. We

arbitrarily choose to express the mix of children within schools in terms of L:

Fig. 1. Distribution of families in the city.


Schools care for the average performance of their pupils, S; that provides them with

recognition and prestige.

Parents choose school for their children. In order to do so, they anticipate and compare

the average performances of pupils at each school, information that will provide their

children with a signal in the market for higher education and/or labor.

We make the simplifying assumption that children’s innate abilities are identical. Thus

the only difference in performances is due to the different social origin of parents and per-

pupil investment by the school. Utility derived by a j-parent located at xwhen sending her

child to school 0 and to school 1 is, respectively:

v jðS0; xÞ ¼ S0 � cj x ð3Þ

and

v jðS1; xÞ ¼ S1 � cjð1� xÞ ð4Þ

where cj stands for mobility costs of j-parents:If xj is the location of the parent of type j who is indifferent among schools, then

xH ¼ 1

2þ S0 � S1

2cHð5Þ

xL ¼1

2þ S0 � S1

2cLð6Þ

Given the location of families, the number of applicants of type j at school i;D ji is given by

DH0 ¼ 1� b

2xH þ b

2x2H ð7Þ

DH1 ¼ 1

2� 1� b

2xH � b

2x2H ð8Þ

DL0 ¼ 1þ b

2xL �

b

2x2L ð9Þ

DL1 ¼ 1

2� 1þ b

2xL þ

b

2x2L ð10Þ

Total demand for school 0 is D0 ¼ DH0 þ DL

0 and total demand for school 1 is D1 ¼DH

1 þ DL1 :

Finally, schools are publicly financed. The government provides the funds in the form

of a per-student allocation f (i.e., ki ¼ f ):Each period the interaction between schools and parents takes place in two stages. First,

parents apply to schools. If demand equals capacity at both schools, all applications are

accepted and the game ends. Otherwise, the school facing the largest demand selects

(among applicants) the children it wants to keep. The other school is forced to take the


remaining children. Schools observe the type of parents, information that they use to

decide on admissions.

2.1. Choice of student mix by schools

In this model, since capacity is exogenously given, the school facing an excess demand

determines through its choice of H or L the mix of students in both schools. We arbitrarily

choose Li as the variable of reference. If D0 ¼ R0; then D1 ¼ 1� R0 ¼ R1 : demand

equals capacity in both schools and no choice is available to any of them. Otherwise, the

school with largest demand, henceforth the leading school, chooses who to keep and who

to reject.5 The other school has to accommodate, taking the rest of the children in the

market.

At this stage demand Di is given. If school i is the leading school (Di > Ri) it chooses

the percentage li that maximizes, from (2) with ki ¼ f and li ¼ Li=Ri

Si ¼ cHf � cH � cL� �

fLi

Ri

ð11Þ

The choice of the optimal student mix may be constrained by the type of demand

addressed to the leading school or by the size of the school.

Consider first the situation where school 0 is the leading school. Admissions of L-

children are limited above by DL0 or by R0:

L0*Vmin DL0 ;R0

� �ð12Þ

where, as throughout the paper, * stands for optimal choice.

On the other hand, once all desired H have been accepted, schools are forced to fill in

capacity with L-children. Therefore,

L0*zmax R0 � DH0 ; 0

� �ð13Þ

Likewise, if school 1 is the leading school,

L1*Vmin DL1 ;R1

� �ð14Þ

and

L1*zmax R1 � DH1 ; 0

� �ð15Þ

5 As a result, some families may end up suffering mobility costs that are higher than what they would be

willing to pay. This families would in this case prefer not to send their children to school, but will be forced to by

law.


Differentiation of (11) with respect to Li yields:

ðcL � cH ÞfRi

< 0

Therefore, the lower limit on Li is binding. Then, school iwill decide to keep the fewest

possible L-children. That is to say, the admission of one more child of type L always

decreases the payoff of the leading school. As a result, the school facing an excess demand

will accept all the applications by students of type H : Only then, if the size of the school

allows it, capacity will be completed with L type children.

Note that since the objective function is the same, the optimal ratio of admissions is

identical in both schools. However, due to the uneven distribution of families in the city,

the constraints are not (see (12)–(15)) and for this reason the maximization programs are

not the same. In other words, although optimality would be achieved at the same point by

both schools, the constrained solution will differ.

2.2. Choice of school by parents

At stage 1 demand results from the anticipation of school performances by parents. If

D0 ¼ R0ðD1 ¼ R1Þ the mix of students will be determined by parental choice. Otherwise,

the leading school will reject some of the applicants and become the ultimate responsible

for the mix of children types within schools.

Recall that, from (7) to (10)

D1 ¼ 1� D0 ¼ 1� 1� b

2xH � b

2x2H � 1þ b

2xL þ

b

2x2L

where xH ; xL are defined by (5) and (6).

From (11)

S0 � S1 ¼ cH � cL� �

fL1

R1

� L0

R0

� �

Then, we can write

xj ¼1

2þ S0 � S1

2cj¼ 1

2þ ðcH � cLÞf

2cjl1 � l0ð Þ ð16Þ

The factor f ðcH � cLÞ in (16) thus transforms differences in relative children mix within

schools ðl1 � l0Þ into differences in school performances S0 � S1: Since 2ðcH � cLÞf > 0;the school with a higher proportion of H -type children always performs better. Then,

parents, in their search for school quality, will try to avoid schools with higher proportions

of L children (l).

On behalf of tractability, let us define

Cj ¼cj

ðcH � cLÞf z0 ð17Þ


for j ¼ L;H :A lower Cj implies a higher responsiveness of parents of type j to differences

in the mix of children types within schools (or anticipated difference in performances).

Hence, Cj represents some sort of weighted mobility cost taking account for both unit

transport costs c and school differences in average performance due to the socioeconomic

profile of the pupils.

On the other hand, since L0 þ L1 ¼ 1=2 and R0 þ R1 ¼ 1; we can write

l1 � l0 ¼l1R1 � ð1=2ÞR1

ð1� R1ÞR1

or l1 � l0 ¼l1 � 1=2

1� R1

ð18Þ

Hence (17) and (18) in (16) yield:

xj ¼1

2þ l1 � 1=2

1� R1ð Þ2Cj

ð19Þ

Note then that

xj ¼1

2þ l1 � l0

2Cj

z0Zl1z1

2� ð1� R1ÞCj ð20Þ

xj ¼1

2þ l1 � l0

2Cj

V1Zl1V1

2þ ð1� R1ÞCj ð21Þ

If the difference in L enrollments at each school is ‘too large’ ((20) and (21) not satisfied)

demand will react ‘shooting’ to a corner solution (zero or one). In other words, the large

difference in student mix within the two schools will make it worthwhile for all parents to

incur the mobility cost associated to taking their children to the school with the larger

average performance. Thus, all demand will turn to this one school.

We can then define,

xj ¼

1 if l1 z 12þ ð1� R1ÞCj

12þ l1�l0

Cjif 1

2� ð1� R1ÞCj < l1 <

12þ ð1� R1ÞCj

0 if l1 V 12� ð1� R1ÞCj

8>>>><>>>>:

ð22Þ

3. Equilibrium

At equilibrium, optimal admissions must be compatible with effective demands. In

order to determine the equilibrium mix of children within schools we proceed as follows:

first we look for a fixed point such that the choice of schools corresponds to available

demand and verify whether the solution is feasible; then, we check for stability.

For simplicity of exposition we first assume R0 ¼ R1 ¼ 1=2: In this symmetrical case

schools are not constrained by size, since a capacity of 1=2 allows to enrol all the students


of each single type as well as any desired ratio of types. In other words, Dji < Ri ¼ 1=2 for

all j ¼ H ; L and i ¼ 0; 1: In Section 3.1 we relax this assumption, in order to analyze the

effects of changes in the size of schools. Likewise we first assume b ¼ 1 and analyze the

effects of changes in b; the degree of segregation of families’ locations, in Section 3.2.

We have seen that the lower limit on L is binding. As a result, schools admit a minimum

of children of L-type parents. Since H -parents are able to register their children at the

school of their choice, their demand alone is responsible for the equilibrium mix of

children types.

At equilibrium, admissions equal demand. From (13), (15) and (19) for j ¼ H ; we canwrite the following equilibrium conditions for the case in which the lower limit on Li is

binding

l0 ¼ 1� 1

2þ 1� 2l0

2CH

� �2

ð23Þ

l1 ¼1

2þ 2l1 � 1

2CH

� �2

ð24Þ

where CH ¼ cH=ðcH � cLÞf > 0:Once we account for the fact that l0 þ l1 ¼ 1; it is clear that (23) and (24) are the same

condition: independently of which school has the leading position, the equilibrium

condition is the same. We perform the analysis in terms of l1 (equilibrium levels of l0directly follow). Since demand for school 1 will be larger when it enrolls less L-children,

the choice of a proportion of disadvantaged students l1 < 1=2 will guarantee that school 1is the leading school:

We find two values of l1a½0; 1� that satisfy (24) with equality when R0 ¼ R1 ¼ 1=2 andb ¼ 1:

lA1 ¼ 1

2þ CH

2CH � 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð1� CH Þ2

q� �ð25Þ

lB1 ¼ 1

2þ CH

2CH � 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð1� CH Þ2

q� �ð26Þ

Fig. 2 depicts the roots of (24) together with the limiting values of l1 that, for a given

CH ; support an interior xH according to (22). We use this figure to explain what will be

shown formally in Proposition 1.

It can be shown (see Appendix) that candidate equilibria represented by lB1 are either

unfeasible (if CH > 1) or unstable (if CH < 1). We then focus on candidate equilibria

represented by lA1 :Note first that lA1 does not satisfy the conditions for interior demand ((20) and (21))

when CH < 1. In this case, lA1 is too small relative to weighted mobility costs to be an

equilibrium. In other words, if lA1 is chosen by school 1 when CH < 1 it will attract all

demand. Then, since the lower limit on l1 is binding, l1* ¼ 0 p lA.
1

Fig. 2. Candidate equilibrium with lump-sum finance.


We then investigate whether corner solutions for demand x ¼ 0 (l1 ¼ 0Þ and x ¼ 1

(l1 ¼ 1Þ may take place at equilibrium when CH < 1. This is indeed the case, since

once all H type children are in the same school no parent has an incentive to change.

Finally, when CHz1; the equilibrium share of L type children at school 1 is unique

and described by lA1 .

Proposition 1. If CH < 1, there exist two stable equilibria which involve total segregation

of children types. If CHz 1, there exists a single stable equilibrium. This equilibrium is

such that school 1 is the leader and involves more segregation than that resulting when

parents cannot choose school for their children.

When schools are financed by lump-sum transfers and CH < 1; stable equilibria involvetotal segregation: either all H attend school 1 and all L school 0 or vice versa. The

difference in school’s average performances is extreme.

As weighted mobility costs CH start to increase some H families living furthest from

school 1 start finding it more convenient to send their children to school 0. Thus, school 1

is forced to accept some L type children in order to fill capacity. At the unique and stable

equilibrium corresponding to any CHz1 (or cHzðcH � cLÞf Þ the leading school 1 will

admit only lA1 < 1=4. When mobility costs of H type parents tends to infinity, lA1 ! 1=4:Note that the outcome l1 ¼ 1=4 is attained when parents are forced to send their children tothe closest school (xH ¼ 1=2Þ:

We can now depict (stable) equilibria as a function of weighted mobility costs of H -

families when schools are equally financed and parents are free to choose school for their

children (see Fig. 3).

Note that, in our model, different parent types agree on what is good for their children

and suffer identical (unit) mobility costs. Total segregation and the maximum difference in

school average performances result from the fact that schools can choose their pupils

among all applicants.

Fig. 3. Equilibria with lump-sum finance.


3.1. Changes in b

The degree of concentration of families’ residential locations has an effect on the

equilibrium mix of students within schools. In particular, and as it may be expected,

segregation of children within schools is larger the larger the segregation of family types in

the city. To see this we maintain the assumption R0 ¼ R1 ¼ 1=2 and go back to (15) and

(19), where DH1 is now given by (8), to obtain:

l1 ¼ ð1� bÞ 1

2þ 2l1 � 1

2CH

� �þ b

1

2þ 2l1 � 1

2CH

� �2

This equilibrium condition has, as before, two roots which now also depend on the

concentration of families (bÞ :

l AV1 ¼ 1

2þ CH

2bCH � 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ð1� CHÞ2

q� �

l BV1 ¼ 1

2þ CH

2bCH � 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ð1� CHÞ2

q� �

Following the same steps that allowed us to identify equilibria in the previous case we

may conclude that candidate equilibria represented by lBV1 are either unstable (if CHV1) or

unfeasible (if CH > 1).

The other candidate equilibrium, lAV1 ; crosses now the horizontal axis twice: at CH ¼ 1

and at CH ¼ b=ð2� bÞ: Between these two points, it remains negative, i.e., unfeasible.

Therefore, if CH < maxf1; b=ð2� bÞg there exist two stable equilibria which involve total

segregation of children types. The range of CH supporting total segregation of children

types thus increases as the population segregates in residential location (b increases). When


CH > b=ð2� bÞ; the ratio of admissions of L type children l1 is given by l AV1 < 2� bð Þ=4:Finally, as CH tends to infinity

limCH!l

lAV1 ¼ 2� b

4

Therefore, the larger b; the lower the percentage of L students enrolled at school 1 when

CH > maxf1; b=ð2� bÞg.

Corollary 2 . The larger the segregation of families types’ residential locations, b, the

larger the segregation of children within schools.

3.2. Changes in R

Due to the assumptions of the model, schools’ payoffs are unaffected by size.

Moreover, schools are not constrained by capacity when they have identical size. This

is due to the fact that 1/2 is precisely the mass of H as well as L type students. Therefore,

Dji VRi for all i; j . The only constraint faced by schools regards the type of demand

addressed to it.

In order to explore the role of school size, we now consider exogenous changes in R0

and R1.

We could expect, when relative school size changes, to observe the larger school

performing better. Indeed, while the smaller school faces an additional capacity constraint,

the larger school, with capacity larger than 1/2 has one less constraint. However, this is not

the case.

To see why, recall that, in the present context, the payoff of the schools is maximum

when Li ¼ 0: With size Ri ¼ Li þ Hi larger than 1/2 the fact that schools must accept

applications until full capacity is met makes Li ¼ 0 an unfeasible outcome. It is therefore

the smaller school that enjoys an advantage. Indeed, the smaller the school, the easier it is

for it to accept only H type students.

Yet, due to the uneven distribution of families in the city, small is relative. Indeed, the

following proposition establishes that, while school 0 will lead if R0 < ð2� bÞ=4; school1 only needs that R1 < ð2þ bÞ=4:

Proposition 3 . If R0 < (2�b)/4 there exists a stable equilibrium in which more than half of

the students enrolled at school 0 are of type H. If R1 < ð2þ bÞ=4 there exists a stable

equilibrium in which more than half of the students enrolled at school 1 are of type H.

The intuition underlying this result is simple. When mobility costs are large,

parents send their children to the closest school. Once mobility costs start falling,

parents start considering the possibility of changing their children’s schools. Since

schools are lump-sum financed, schools prefer children of type H : In turn, parents

prefer schools where a majority of H type children enrol. As a result, the first parents

who move their children when mobility costs fall are H type parents (L type parent’s

applications will be rejected). They choose the school that enrols more than a half of

H type children. This school is school 1 [resp. 0] provided that its size is not larger


than ð2þ bÞ=4 [ð2� bÞ=4]. The moving of these first few children can only make the

school of their choice more desirable to all, hence reinforcing the leading position of

this school.

In principle, given b and CH ; we can find the relative school size yielding full equa-

lization of schools when mobility costs are large: R0 ¼ ð2� bÞ=4 and R1 ¼ ð2þ bÞ=4:We know, however, that mobility costs may fall and that families tend to choose

strategically their residential location, not to mention the incentives for schools to reduce

capacity and accept only H type students. For all these reasons we may be more interested

in changing the incentives for schools to accept L type children. In order to do this we

propose a system of grants to disadvantaged children that, paid directly to schools, modify

the costs and benefits of their enrolment. The next section shows how far we can get in our

quest for equality by using this strategy.

4. Positive discrimination of disadvantaged students

In order to compensate disadvantaged students for their lower productivity in the

educational process the government may provide each school with an additional grant /per each L type child enrolled. Per student investment at school i; ki becomes then

ki ¼F þ /LiHi þ Li

¼ f þ /li

From (2), the first-order condition for the maximization of the school’s output is

hicH þ lic

L� �

/ � f þ /lið ÞðcH � cLÞ ¼ 0 ð27Þ

and hence, the percentage of L type students enrolled at school i at an interior solution is

li* ¼ cH/ � f ðcH � cLÞ2ðcH � cLÞ/

In order for the planner’s budget for schools B ¼ f þ ðL0 þ L1Þ/ to remain balanced

(i.e., dB ¼ 0) we need marginal increases in /; d/; to be accompanied by

df ¼ � 1

2d/

A balanced budget rise in / then has the following effect on admissions of L-children by

schools:

dl i*

d/� 1

2

dl i*

df

� �d/ ¼ f

2/2þ 1

4/¼ 2f þ /

4/2> 0

We can then conclude that l i* is monotonically increasing in budget-balanced increases of

/:When / is small, the lower limit on Li is binding and school imaximizes its payoff at the

lower possible admissions of L. When / is large, the upper limit on Li is binding and

school i maximizes its payoff at the highest possible admissions of L: Section 3 analyzed


equilibrium when the lower limit on Li is binding. In this section, we consider the case

where the upper limit in Li is binding.6

Consider then the case in which, not only we allow for a grant /; but also it is

sufficiently large relatively to the lump-sum transfer to make the upper limit on Li binding

for the school enjoying the leading role. From (27) this is the case when7

/zf ðcH � cLÞ

cH � 2l i*ðcH � cLÞ

In this case, the benefits in terms of larger funds associated to the admission of L

children outweigh the effects of their lower marginal achievement. The leading school

would like to admit at least as many L students as have applied. As we did before, we

consider first the particular case where R0 ¼ R1 ¼ 1=2 and b ¼ 1: Deviations from this

particular case will be studied in Sections 4.1 and 4.2.

From (2) with ki ¼ f þ /li :

S0 � S1 ¼ ðcH � cLÞf � cL/� �

ðl1 � l0Þ

Then, we can write (6) as

xL ¼1

2þ l1 � 1=2

1� R1ð Þ2CL

ð28Þ

where CL ¼ cL= ðcH � cLÞf � cL/ð Þ:Note that CL can now be positive or negative. In other

words, parents may now prefer the school with more H -children as before, or they may

prefer the school where there are more L (from the effect of a positive or negative CL on

(28)): It will certainly depend upon whether the better school has more students of one or

the other type. In turn, this will depend upon the size of / and the ability of this variable to

completely overcome the negative effect of disadvantaged cultural backgrounds on

average performances of the schools ðcH � cLÞf � cL/ð Þ.The case in which schools may end up preferring L children as a result of a large /

(CL < 0) shall be considered with caution.8 On the other hand, it can be shown that CL < 0

trivially leads to the mirror image of equilibria resulting in Section 3. Indeed, with schools

accepting only L type children and parents searching schools with a majority of L type

children, it suffices to rewrite Section 3 exchanging H by L (and CH by CL) to obtain the

corresponding equilibria. For these reasons, we assume henceforth that ðcH � cLÞf > cL/(i.e., Cj > 0; j ¼ H ; L).

7 Some caution is required when, like in this case, limiting levels of / are expressed as a function of Li;which

depends itself on /: However, this way of proceeding greatly simplifies the exposition. We then stick to it while

carefully verifying ex-post the validity of the inequality (see Appendix).8 No relation between attainment and per-pupil expenditure seems to have been systematically found. See

Vandenberghe (1996) for an insightful discussion of this literature.

6 Intermediate levels of / yield interior solutions which provide little additional insight. A full description of

such interior equilibria is provided in Del Rey (2001) together with the detailed range of / yielding corner and

interior equilibria when b ¼ 1, and R0 ¼ R1:


At equilibrium in this corner solution, admissions of L type students equal their

demand. Hence, from (12), (14) and (28) (and since l0 � l1 ¼ l1 � 1=2ð Þ= 1� R1ð Þ):

l0 ¼ 1� 1

2� 1� 2l0

2CL

� �2

ð29Þ

l1 ¼1

2� 2l1 � 1

2CL

� �2

ð30Þ

As before (29) and (30) are equivalent equilibrium conditions. When the upper limit on L1is binding, we find two roots of l1a½0; 1� that satisfy (30) with equality:

lC1 ¼ 1

21þ CL þ C2

L � CL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2CL þ C2

L

q� �ð31Þ

lD1 ¼ 1

41þ CL þ C2

L þ CL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ CL þ C2

L

q� �ð32Þ

Fig. 4 depicts these roots of (30). Clearly, from the figure, the only interior equilibrium

is given by lC1 ðlD1 does not satisfy (20) nor (21)). Proposition 4 shows that lC1 is in fact the

only equilibrium percentage of L type students at school 1 when the upper limit of Li is

binding.

Fig. 4. Candidate equilibria with positive discrimination.


Proposition 4. If /z f (cH�cL) /(cH�2l i*(cH�cL)), there exists a single stable equilibrium

for all CL>0. This equilibrium is such that school 1 is the leader and involves less

segregation than that resulting when parents cannot choose school for their children.

Since CL > 0; performance of schools continues to be larger the larger the proportion of

H type children it enrols. However, / is large enough to make admissions of L type

children more desirable. Therefore, school 1 accepts the larger possible number of L

children compatible with its keeping the leading position. However, as weighted mobility

costs CL fall, the difference in enrolments of L children required to ensure school 1 its

leading role is smaller.

Note that the proportion of L type children in school 1 is always larger than 1/4 when

/ is large: We have seen in Section 3 that, without this element of positive

discrimination, the proportion of L children in school 1 is always smaller than 1/4

and, for sufficiently low mobility costs, it is zero. Hence, a level of / sufficiently large

to make the upper limit on Li binding, combined with the subsidization of transport costs

of L type families (jcL ) may prove efficient in reducing differences in schools’

performances at a stable equilibrium.

4.1. Effect of the grant for different levels of b

Let us now consider the effect of changes in the degree of segregation of family types in

the city (b) on the equilibrium mix of children within schools. From (14) and since R1

¼ 1=2 we know that l1R1 ¼ DL1, or, using also (28):

l1 ¼ 1� ð1þ bÞ 1

2þ l1 � 1=2

1� R1ð Þ2CL

� �þ b

1

2þ l1 � 1=2

1� R1ð Þ2CL

� �2

This equilibrium condition is satisfied by

l C V1 ¼ 1

2þ CL

2bCL þ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ð1þ CLÞ2

q� �

l DV1 ¼ 1

2þ CL

2bCL þ 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ð1þ CLÞ2

q� �

The only difference between lC1 and lCV1 is that, when we let b vary, the percentage of L

students in school 1, l1; tends to ð2� bÞ=4 instead of 1=4 when CL tends to infinity:Therefore, the relative share of L type children at school 1 (l1) is lower the larger b:

On the other hand, lDV1 remains larger than 1=2þ CL=2b; i.e., unfeasible, for all b (see

(21) for R1 ¼ 1=2).

Corollary 5 . The ability for the grant / to achieve a larger mix of children in schools is

smaller the larger the segregation of family types’ residential locations, b.


4.2. Effect of the grant for different school sizes

When / is large, schools accept all applications by L type parents. However, parents

still prefer the school where more H students enrol. Therefore, in order to keep a leading

position schools need to be larger than twice the demand by L type parents (which

guarantees that a larger proportion of H students enrol). It can be shown that this will

always be the case for school 1 provided that its size, R1; is no smaller than ð2� bÞ=4:Assoon as R1 ¼ ð2� bÞ=4; it will accept as many L as H students and lose the leading

position in favor of school 0.

5. Passive schools

In order to underline the importance of school choice, consider, to conclude, the

case in which schools passively accept all applications. Assume, for convenience, that

cL ¼ cH ¼ c:Schools of unlimited capacity enroll now every applicant. Thus, school 1 enrols

L1 ¼ DL1 and H1 ¼ DH

1 ð33Þ

where DL1 and DH

1 are given by (10) and (8), respectively. The size of school 1 is

R1 ¼ L1 þ H1:Under these circumstances families located at x* are indifferent among schools

(see (19)):

x* ¼ 1

2þ L1=ðL1 þ H1Þ � 1=2

ð1� H1 � L1Þ2Cð34Þ

where C ¼ c=ðcH � cLÞf : Substituting L1and H1 from (33) into (34) we obtain

x* ¼ 1

2� b

4C

Then, x*z0ZCzb=2 and is always smaller than 1. Therefore, if C < b=2; x*

remains constrained at zero and, from (33), we get L1 ¼ 1=2 and H1 ¼ 1=2: This

implies that school 0 disappears from the market. When Czb=2 from (33) and (10)

and (8):

L1* ¼ 1

2� 1þ b

2

1

2� b

4C

� �þ b

2

1

2� b

4C

� �2

ð35Þ

H 1* ¼ 1

2� 1� b

2

1

2� b

4C

� �� b

2

1

2� b

4C

� �2

ð36Þ


Note that, as C tends to infinity, H 1* ! ð2þ bÞ=8 and L1* ! ð2� bÞ=8: Then, school

0 tends to disappear as mobility costs fall, since the global amount of pupils in

school 1 tends to 1 (all the market) when mobility costs tend to zero. If mobility

costs tend to infinity, school sizes are equal. However, as the size of the better

school increases from 1/2 when mobility costs start to fall, the number of L

children that get to attend this school is larger than the number of new H children:

L1*�2� b

8> H 1

*� 2þ b

8ð37Þ

Hence, if schools passively accept all applications (and all families are equally

mobile) free school choice may provide disadvantaged children with better oppor-

tunities to attend the better school. This is what has traditionally been claimed by

free school choice advocates.

However, as we said before, there is some evidence that mobility could be more costly

for disadvantaged families. If this is the case, then choice may only be effectively available

for better endowed families. As a result we can no longer be sure that (37) will be true.

And if (37) is not true, disadvantaged children are not enjoying better opportunities to

attend the better school.

6. Concluding remarks

It has been our goal in this paper to underline the role of school’s choices as opposed to

differences on the demand side (like differences in mobility costs, preferences or ability to

pay) in explaining unequal results in public school performances. In order to do so, we

have constructed a very simple model of public schooling provision. In this model, schools

aim at maximizing the average performance of their pupils and parents choose the better

school for their children provided that it is not too far away from their homes. Since

children of high skilled parents perform better, they are preferred by schools. In turn,

schools whose pupils perform better in average are preferred by parents. In this

framework, school’s reactions to excess demand have been shown to intensify segregation

of children types within schools, preventing all possible desirable effects of free school

choice from taking place.

We have shown how an adequate level of grants to disadvantaged pupils directly

provided to schools and eventually combined with transport subsidies to disadvantaged

families may take us arbitrarily close to the equalization of average school’s performances

by modifying the school’s incentives to accept applications of disadvantaged children.

Also, if schools passively accept all applications, free school choice may improve the

opportunities of disadvantaged children, provided that they do not face higher mobility

costs.

The model remains a simplified abstraction. Some of its features, like the technology of

education production or the partial and static nature of the analysis deserve to be

challenged by future research.


Acknowledgements

This research was mainly conducted during my stay at CORE, Universite Catholique de

Louvain (Belgium) as a Ph.D. student. Funding from Fundacion Ramon Areces (Spain)

and the European Commission through the TMR Program is gratefully acknowledged. I

appreciate the helpful comments of Nicolas Boccard, Isabel Grilo, Maurice Marchand,

Pierre Pestieau, Konrad Stahl, Jacques Thisse, Vincent Vandenberghe, Xavier Wauthy and

one anonymous referee.

Appendix

Proof of Proposition 1

We first show that lB1 is not an equilibrium when CH < 1: Then we show that both

l1 ¼ 0 and l1 ¼ 1 are equilibrium distributions of children among schools when CH < 1:Finally, we show that, when CHz1; the equilibrium is unique and described by lA1 :

Step 1. l1B < 1=2� CH=2 implying, from (22), that xH ¼ 0: Then, l 1* ¼ 0 p lA1 :

Step 2. If l1 ¼ 0; the utility to the H-families living furthest from school 1 (x ¼ 0) is

v1 ¼ S1ðl1 ¼ 0Þ � cH ¼ cH f � cH

If one of these families decided to move her child to the closest school 0, she would then

benefit from utility

v0 ¼ S0 l0 ¼ 1ð Þ ¼ cLf

Since cH < ðcH � cLÞf this family does not have an interest to change school.

A similar argument shows that if all H-children attend school 0 no single H-family will

have an incentive to change school when CH < 1: Then, l 1* ¼ 1 is also an equilibrium:Both l 1* ¼ 0 and l 1* ¼ 1 are robust to e-deviations (see Appendix B in Del Rey, 2001).

Step 3. lB1 > 1 when CHz1; hence it cannot take place. Further, neither l1 ¼ 0 nor

l1 ¼ 1 are equilibrium outcomes when CHz1: To see this just note that when CHz1

xH ¼ 1

2þ l1 � l0

2CH

< 1 if l1 ¼ 1

> 0 if l1 ¼ 0

8<:

When CHz1 the equilibrium, determined by

lA1a1

2� CH

2;1

2þ CH

2

� �;


is unique and stable: Also, lA1 < 1=4 with

limC!l

lA1 ¼ 1

4

Proof of Proposition 3

When mobility costs are large (xL ¼ xH ¼ 1=2), each family chooses the school closest

to her residence. Schools, on their side, accept applications by H type parents first and fill

their capacity with L type children. This implies that

H0 ¼ min2� b

8;R0

�

H1 ¼ min2þ b

8;R1

�

L0 ¼ max 0;R0 �2� b

8

�

L1 ¼ max 0;R1 �2þ b

8

�

As mobility costs decrease, they will reach a point in which H parents near the center of

the district may consider the possibility of changing school. If R0Vð2� bÞ=8we know that

L0 ¼ 0 since, when CH ! l;DH0 ¼ ð2� bÞ=8 : school 0 is already full of H type children

and cannot accept any more due to a capacity constraint.

Suppose then that R0 > ð2� bÞ=8 so that some L type students attend school 0

(L0 ¼ R0 � ð2� bÞ=8). Consider H type families located at x ¼ 1=2þ e: If mobility costs

are large, they choose school 1. However, if they move from school 1 to school 0, L0 will

be reduced in the amount of new H admissions. Then

L0V ¼ R0 �2� b

8� 1� b

2e � b

2ðe2 � eÞ

Let us compare the utility they obtain when staying at school 1 to that obtained when

moving their children to school 0.

uðS0V; xÞ � uðS1; xÞ ¼ ðcH � cLÞf ðl1 � l0VÞ � 2CH e

since L1 ¼ R1 � ð2þ bÞ=8; then l1 ¼ L1=R1 ¼ 1� ð2þ bÞ=8R1 and, similarly

l0V¼ 1� 2� b

8R0

� 1� b

2R0

e � b

2R0

ðe2 þ eÞ


Then,

l1 � l0V ¼ �2þ b

8R1

þ 2� b

8R0

þ 1� b

2R0

e þ b

2R0

ðe2 þ eÞ

¼ �ð2þ bÞR0 þ ð2� bÞð1� R0Þ þ 4ð1� R0Þ ð1� bÞe þ bðe þ 1Þeð Þ8R0ð1� R0Þ

uðS0V; xÞ � uðS1; xÞz0 if and only if

�ð2þ bÞR0 þ ð2� bÞð1� R0Þ þ 4ð1� R0Þ ð1� bÞe þ bðe þ 1Þeð Þ16R0ð1� R0Þ

� 2 CH ez0

where CH ¼ cH=ðcH � cLÞf :We are interested in the case where e tends to zero (families living very close to the

center of the district). Then, this condition becomes

�ð2þ bÞR0 þ ð2� bÞð1� R0Þz0ZR0 <2� b

4:

Thus, if R0 < ð2� bÞ=4;H parents near the center and sending their children to school

1 prefer to move them to school 0. Once they do so, school 0 will become the leader and

receive most applications. If R0 > ð2� bÞ=4; school 1 is preferred.

Proof of proposition 4

lD1 does not satisfy (20) nor (21), hence it cannot be an interior equilibrium. In contrast

with the case when the lower limit for L1 was binding, the corner solution l1 ¼ 1 is no

longer an equilibrium. The reason is that, if school 1 admits all L; its average performance

will be lower than that of school 0 (/ is not large enough to make the best school choose

l i* > 1=2Þ:Then, demand will turn to school 0, that will become the leader. If, on the other

hand, school 0 is the leader, it will not choose l0 ¼ 0 (which would imply that l1 ¼ 1) since

/ is large. A similar argument shows that l1 ¼ 0 ðl0 ¼ 1Þ is not an equilibrium when the

upper limit on L1 is binding.

Equilibrium is then represented by 1=2 > lC1 ðCLÞ > 1=4; which satisfies (20) and (21)

and is robust to deviations. Finally

limC!0

l C1 ¼ 1=2

limC!0

LC1 ¼ 1=4

On the other hand, it is important to verify that there exists a range of / that supports

this equilibrium. First, for the upper limit on L1 to be binding, we need, from (27)

/ >ðcH � cLÞf

cH � 2lCi ðcH � cLÞ


But also, it must be the case that

CL > 0Z/ <ðcH � cLÞf

cL

Therefore, we need that

2lCi ðcH � cLÞ < cH � cL

which is always true at the candidate equilibrium, since lC1 < 1=2:

References

Benabou, R., 1996. Equity and efficiency in human capital investment: the local connection. Review of Eco-

nomic Studies 63, 237–264.

Benabou, R., 1994. Human capital, inequality and growth: a local perspective. European Economic Review 38,

817–826.

Benabou, R., 1993. Workings of a city: location education and production. Quarterly Journal of Economics 108,

619–652.

OECD, 1994. School: a Matter of Choice. Centre for Education Research and Innovation (CERI).

De Bartolome, C., 1990. Equilibrium and inefficiency in a community model with peer group effects. Journal of

Political Economy 98–1, 110–133.

Del Rey, E., 2001. Persistent Inequality through Schooling: the role of Limited School Capacity. CORE DP

2001/10.

Fernandez, R., Rogerson, R., 1996. Income distribution, communities and the quality of public education.

Quarterly Journal of Economics 111, 135–164.

Hartwick, J., Schweizer, U., Varaiya, P., 1976. Comparative statics of a residential economy with several classes.

Journal of Economic Theory 13, 396–413.

Vandenberghe, V., Zachary, M.-D., Dupriez, V., 2000. Is There an Effectiveness-Equity Trade-off? A cross-

country comparison using TIMSS test scores. Report on Education Equity, Harvard Education Review Press

& AIR/ESSI.

Vandenberghe, V., 1996. Functioning and regulation of educational quasi-markets, Ph.D. Dissertation, CIACO

Louvain-la-Neuve (Belgium).

funding schools for greater equity

Documents