Too Poor for School? Social Background E ects on School and University Opportunities

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1 Too Poor for Shool? Soial Bakground E ets on Shool and University Oortunities Alessandro Tamieri y University of Leiester June 7, 2010 Abstrat This aer studies how students soial bakground a ets shool teahing and university admission oliies. Students with disadvantaged bakground are enalised with reset to other students: they reeive less teahing and/or are less likely to be admitted at university. A surrising result is that a oliy aiming to subsidise eduation for disadvantaged students might derease their admission s oortunities. [Very Preliminary] JEL Numbers: C73, I21, J24 Keywords: Soial bakground, Ability. I would like to thank Gianni De Fraja, Ludovi Renou, Suresh Mutuswami, Javier Rivas, Miltos Markis, Carlos Carrillo-Tudela, Gaia Garino, Pierarlo Zanhettin, Tom Allen, Matteo Lii Bruni and Piero Pasotti for many suggestions that have led to substantial imrovement on revious drafts. Earlier versions of this aer has been resented to the Sottish Eonomi Soiety Annual Conferene, 2009, and to seminars in the Bank of Italy, 2009, Leiester, Lanaster, Konstanz and Goettingen, 2010, where I reeived helful omments. All errors are mine. y University of Leiester, Deartment of Eonomis, Leiester, LE1 7RH, UK, University of Rome La Saienza, Deartment of Publi Eonomis, Via del Castro Laurenziano 6, Rome, Italy; at198@le.a.uk.

2 1 Introdution There is substantial evidene that soial bakground in uenes eduational attainment (Haveman and Wolfe, 1995, Galindo-Rueda and Vignoles, 2005) and university admission oortunities (Marenaro-Gutierrez et al., 2007). This aer rooses a theoretial exlanation to it by examining how the shool and university behaviour hanges aording to the student s soial bakground. Our idea is that given the same distribution of innate ability within oulations with di erent soial bakground, an advantaged environment an heleveloing skills via arental and eer ressure. As a onsequene, students with advantaged soial bakground are more likely to have high ability. This is suorted by ast researh doumenting that family and environmental fators are major reditors of ognitive abilities (Cunha et al., 2006, Carneiro and Hekman, 2003, Joshi and MCullo, 2000). We onsider the interation between a shool and an university. The students they serve di er in ability and soial bakground. The shool reares students and wants the largest number of them to be admitted at the university. On the other side, the university would like to admit only high-ability students, but it annot observe ability. Our results suggest that the di erent ability distribution in di erent soial bakgrounds enalises the disadvantaged students, given the same ability: they will reeive less teahing and/or will be admitted to the university with less hane. The intuition is the following. The university refers advantaged students as they are more likely to have high ability. To inrease the exeted quality of disadvantaged students, the shool may devote less teahing e ort for disadvantaged and low-ability students. Desite that, the university may still refer advantaged students. As a result, disadvantaged students are enalised in any ossible ase. We then onsider a government that subsidises the ost of teahing disadvantaged students. Suh oliy may diminish the disadvantaged student s hane of being admitted, as it dereases the exeted quality of disadvantaged students. This aer ontributes to the literature on university admission (Fernandez 2

3 and Galí, 1999, Ele et al.,2006, Gary-Bobo and Trannoy, 2007) by introduing the role of the students soial bakground in the admission roess. The remainder of the aer is organised as follows. The model is resented in Setion 2. Setion 3 examines the equilibria. Setion 4 onsiders the government intervention. Setion 5 onludes. 2 The model We study the interation between a single shool and a single university 1. The interation ours as both serve a number of students, with measure normalised to one. Students an have high ( H ) or low ( L ) ability, whih is rivate information. In addition to ability, students an have either advantaged (a) or disadvantaged (d) soial bakground, whih is ubli information: this an be interreted as a one-dimensional measure of family environment, eer grous 2 and neighbourhood. We denote as 2 [0; 1] the roortion of advantaged students. Let a ; 2 [0; 1] be the robability that an advantaged or disadvantaged student has high-ability, resetively. Also, we assume that advantaged individuals are more likely to have high ability, i.e., a >. 2.1 University The university deides whether or not to admit a student. We all students demand the maximum amount ossible of admitted students and denote it by 2 [0; 1]. The university wants to admit only high-ability students. Indeed it would lose redibility if graduates had low-ability 3. Thus the university ayo is > 0 or 1 for admitting a high or low-ability student, resetively. The university observes the grade that a student obtains in a re-university 1 For simliity, we abstrat from fators suh as ometition between shools and between universities. 2 Peer grous means that students learn better if they are in a grou of abler students. 3 For simliity, we assume that the urose of university and shool is to signal the students ability and not to imrove their skills. 3

4 exam 4 as a signal of ability. The ossible exam s outomes are a low (g D ) or a high (g U ) grade. 2.2 Shool The shool in uenes the admission by rearing students for the re-university exam 5. Unlike the university, it learns the students ability through their tests and assignments results. The shool obtains a bene t > 0 for every student who is admitted at the university. The reason is that eah student s admission inreases its reutation and redibility and it will be seen as an e etive institution for ahieving university admission. The rearation requires resoures: the quantity of teahing, the quality of buildings and lassroom equiment and the teahers e ort in teahing; we refer to all these asets as teahing. In addition to teahing, the shool an rovide some students with extra teahing, that is additional resoures, extra tuition, tris and more failities. Here we assume that, with teahing only, the student s robability of obtaining a high grade is 2 (0; 1) if she has high ability and zero if she has low ability. With extra teahing instead, the student s robability of obtaining a high grade is 1 if she has high ability and 2 (0; 1) if she has low ability 6. The shool ays a ost > 0 for eah student reeiving extra teahing. Hene the shool s ayo is given by the di erene between bene t and ost. 4 The exams whih we have in mind in the real world are the Sholasti Assessment Test in United States and the National Curriulum Assessment in United Kingdom. These exams are managed by the Eduational Testing Servie (Rourke and Ingram, 1991). 5 Note that the shool does not arrange the exam and hene it annot maniulate the students grades. 6 The two events high-ability with no extra teahing obtains a low grade and lowability with extra teahing obtains a high grade have the same robability to our in order to simlify the algebra. To give to these two events a di erent robability would just omliate the analysis without adding any insight. 4

5 2.3 The game between the shool and the university The interation an be desribed as a four-stage game of the following form. In the rst stage, nature draws a student tye. In seond stage, the student 7 attends shool and the shool hooses whether to give extra teahing to her. In the third stage, the student takes the re-university exam. Finally in the fourth stage, the student alies for the university, and the university deides whether to admit her. 3 Equilibria We study the erfet Bayesian equilibria of this game. In our set-u, a erfet Bayesian equilibrium is a ombination of shool and university strategies and beliefs. After observing a grade, the university must have a belief about the student tye, and this must be onsistent with the Bayes rule. For eah grade, the university must maximise its exeted ayo, given the belief and the shool strategy. The shool s strategy must maximize its exeted ayo, given the university strategy 8. Then, the students demand onstraint requires that the number of admitted students is at most. We start by making the following assumtions. Assumtion 1 < ( a (1 a )) (1 ) ( (1 ))). n Assumtion 2 > max 1 o. Assumtion 1 says that the students demand is lower than the highest ossible number of high-grade students. This assumtion fouses the attention on the equilibria where the role of soial bakground is straightforward. For instane, the university may want to admit high-grade students with advantaged and disadvantaged bakground. Sine there is no laement for all of them, the university needs to omare the di erent tyes and start admitting from the grou that gives the highest exeted ayo. The other tye will be admitted only for the remaining students demand. 7 For simliity, we assumed away the in uene of student s e ort. 8 Note that the shool has erfet information of the student tyes. 5

6 Assumtion 2 says that the restige of having a former student at the university is onsiderably higher than the ost of roviding her with extra teahing. This rules out the ossibility that a student would not reeive extra teahing beause, aording to the shool tehnology, this is too ostly. Here we want to fous on the ase where the shool resonse deends on the university strategy omletely, and disregard the role of shool tehnology. After resenting Assumtion 1 and 2, we need to introdue a bit of notation: x La ; x Ha 2 [0; 1] are the robabilities that the shool gives extra teahing to an advantaged and low or high-ability student, resetively; x Ld ; x Hd 2 [0; 1] are the robabilities that the shool gives extra teahing to a disadvantaged and low or high-ability student, resetively; z Ua ; z Da 2 [0; 1] are the robabilities that the university admits an advantaged student with a high (U) or low (D) grade, resetively; z Ud ; z Dd 2 [0; 1] are the robabilities that the university admits a disadvantaged student with a high or low grade, resetively. If Assumtion 1 and 2 hold, the equilibrium will be one of three tyes, whih we label high-admission, middle-admission and low-admission equilibria. De nition 1 High-admission equilibrium: (i) shool: x Ha = x Hd = x La = (a(1 a)) x Ld = 1; (ii) university: z Ua = 1; z Ud = ; z (1 )( (1 )) Da = 0; z Dd = 0. De nition 2 Middle-admission equilibrium: (i) shool: n x Ha = x Hd = o x La = 1; x Ld = ; (ii) university: z (a(1 a)) (1 ) Ua = 1; z Ud = min ; (1 )( ; (1)) z Da = 0; z Dd = 0. De nition 3 Low-admission equilibrium: (i) shool: n x Ha = x Hd = 1; x La o= a ; x (1 a) Ld = ; (ii) university: z (1 ) Ua = z Ud = min ; ( a(1 ) ; )(1) z Da = 0; z Dd = 0. We then de ne the university beliefs about the students ability: they are based on the grade g j, where j 2 fu; Dg, the distribution of ability given the soial bakground i, where i 2 fa; dg and the shool strategy (x Ha,x La ; x Hd ; x Ld ). 6

7 η 1 Students demand higher than the number of high grade students Middle/Low admission η = ν η High admission Φ = ( 1 )η d Φ 1 Figure 1: Proosition 1. Equilibria De nition 4 Beliefs: ( H j g j ; i ; x Hi ) = x Li (1 i ) i x Hi x Li (1 i ). i x Hi i x Hi x Li (1 i ), ( L j g j ; i ; x Li ) = 3.1 Benhmark ase: one soial bakground In this setion we assume no di erenes in soial bakground. This allow us to highlight the role of the other harateristis, suh as the distribution of ability, the robability of obtaining a high grade and the tehnology. For onreteness, we onsider a oulation of disadvantaged students 9, i.e., = 0 (the analysis does not hange if we assume = 1). The following roosition shows the equilibria aording to. Proosition 1 Let Assumtions 1 and 2 hold and = 0. The high-admission equilibrium ours if ; the middle/low-admission equilibrium ours if <. Proof. See Aendix. 9 Note that, if = 0, the middle or the low-admission equilibrium are equivalent for a disadvantaged student. 7

8 Figure 1 illustrates Proosition 1. The horizontal axis is while the vertial axis is. Assumtion 1 holds below the downward-sloing area. The uwardsloing line reresents the threshold oints where =. The threshold determines whih equilibrium ours and deends on the arameters values and. As inreases, the university wants to admit more students as its exeted ayo inreases, thus making the threshold shifts u. As the robability of obtaining a high grade () inreases, the amount of lowability and high-grade students inreases and the university is willing to admit less students. The threshold shifts down and it is more likely to be in the middle/low-admission equilibrium. In the high-admission equilibrium, the university stritly refers to admit rather than not to admit a high-grade student, as she is very likely to have high ability. However, not all the high-grade students an be admitted (by Assumtion 1 ). Given the university strategy, the shool best resonse is to rovide every student with extra teahing, even though some of them are not to be admitted. This haens for the very high bene t of the shool from a student s admission (by Assumtion 2 ). The ost of teahing a student who will not be admitted is muh smaller than the loss of bene t given by a nonadmitted student who did not reeive extra teahing. So the shool does not want to take this risk. As falls below, the low-admission equilibrium ours. The university best strategy is not to admit all the high-grade students, beause now the roortion of low-ability and high-grade students is quite large. This number might be higher or lower than the students demand; in the former ase, the amount of admitted students is again. Then the shool rovides low-ability students with extra teahing only with a ertain robability smaller than one. This inreases the robability that a high-grade student has high ability. 3.2 General ase: di erenes in soial bakground Now we onsider 2 (0; 1). The following roosition shows the equilibria aording to a and. Proosition 2 Let Assumtions 1 and 2 hold. The high-admission equilibrium ours if a and d ; the middle-admission equilibrium ours if 8

9 η 1 Students demand higher than the number of high grade students a η = ν η Low admission η = ν η Φ = λ ( η ( 1 ) ( 1 λ ) ( η ( 1 ) a a d d Middle admission High admission Φ Figure 2: Proosition 2. Equilibria with differenes in soial bakground 1, d a a and d < <. ; the low-admission equilibrium ours if a < and Proof. See Aendix. Figure 2 illustrates Proosition 2. The dashed and the ontinuous uwardsloing lines reresent the threshold oints of a and, resetively. To interret Proosition 2, begin by looking at the key assumtion, a >. This imlies that the university obtains, given the same shool strategy, a higher exeted ayo by admitting advantaged students. In the high-admission equilibrium the university admits students while the shool rovides every student with extra teahing. Given the same teahing strategy for advantaged and disadvantaged students and a >, the university exeted ayo will be higher by admitting the former rather than the latter. Thus the disadvantaged and high-grade ones are admitted only for the remainder of the students demand. In the middle-admission equilibrium, the university wants not to admit all the disadvantaged and high-grade students, sine is quite low. This grou an be higher or lower than the remainder of the students demand. In turn, the shool rovides with extra teahing a lower number of disadvantaged and low-ability students. This inreases the robability that a disadvantaged and 9

10 high-grade student has high ability. However, the university still refers to admit advantaged and high-grade students, beause the e et of a > is stronger than the shool strategy. In the low-admission equilibrium, the university does not admit all the highgrade and advantaged students. The shool gives extra teahing to less disadvantaged than advantaged and low-ability students. In this ase the e et of a > is erfetly o set by the shool strategy in the university exeted ayo. In other words, the university is indi erent between admitting an advantaged or a disadvantaged and high-grade student. Aording to Proosition 2, disadvantaged students are enalised omared to advantaged students, given the same level of ability: they may reeive less teahing, or be admitted with lower robability to the university, or both. This result an be linked to the analysis of e ient rovision of eduation. In the resene of di erenes in soial bakground, the otimal rovision of eduation requires that disadvantaged and high-ability students reeive more eduation than high-ability and advantaged students 10. Following Proosition 2, a oliy intervention would be neessary to reah an e ient level of eduation. 3.3 Analysis of equilibria In this setion we study the roerties of equilibria. The following orollary shows some omarative statis results. Corollary 1 An inrease in the robability of obtaining a high grade diminishes the admission and the rovision of extra teahing; an inrease in the university ayo inreases the rovision of extra teahing; an inrease in the ost of extra teahing or a derease in the shool bene t inreases the admissions. Finally, an inrease in the roortion of advantaged students inreases the admissions. 10 De Fraja (2005) studies the rovision of eduation when students di er in ability and soial bakground. In the resene of asymmetri information (the government does not know the student s ability) and externality (the ubli rovision of eduation makes the students aquire more eduation than they would aquire rivately) the otimal rovision of eduation is a seond best result where high-ability and disadvantaged students reeive more eduation than high-ability and advantaged students. Hene the introdution of reverse disrimination oliies, like a rmative ation, are justi ed on an e ieny grounds, and the trade-o between equity and e ieny disaears. 10

11 Proof. See Aendix. An inrease of makes the number of high-grade students inrease. Thus their robability of being admitted diminishes 11. In turn this makes the robability of reeiving extra teahing diminish. With an inrease in, the shool gives extra teahing to more low-ability students, as there are more hanes of admission. Intuitively, the higher and/or the lower, the higher the robability that the university admits a high-grade student. The robability that she has high ability inreases, as it is more ostly for the shool to give extra teahing 12. An inrease in has two ontrasting e ets in high-admission equilibrium: the amount of disadvantaged and high-grade students diminishes and the laements in the students demand net of advantaged students are lowered. With the rst e et, the robability of a disadvantaged and high-grade student being admitted inreases. With the seond e et, it diminishes, beause there are less laements. Nevertheless, the rst e et more than o sets the seond e et. The reason is the following: the derease in the amount of disadvantaged and high-grade students inreases the relative students demand ( (1 )( (1 )) ( laements ( a(1 a)) ). (1 )( (1 )) ) with more intensity than diminishes the relative admission 4 Subsidising disadvantaged students In many ountries, governments send substantial resoures to ght unequal eduational outomes 13. We an deit suh an intervention by onsidering a government that subsidises eduation for disadvantaged students. 11 This does not our for advantaged and high-grade students in the ase of high and middle-admission equilibrium, as when a is higher than the threshold, the university admits any of them irresetive of. 12 This does not haen in high-admission equilibrium to disadvantaged students and in high and middle-admission equilibrium to advantaged students. Indeed, when a or are over the threshold, the otimal shool strategy is to rovide every student with extra teahing, regardless of and. 13 To ite some examle, in the United States, reent measures of funding eduation for disadvantaged have been onsidered the No Child Left Behind At of 2001, and in the Amerian Reovery and Reinvestment At of In the United Kingdom, the Eduation Manteinane Allowane (EMA) funds low-inome students who deide to kee studying after

12 The government annot observe the student s ability and subsidises for all disadvantaged students. We onsider the roblem from a artial equilibrium ersetive, in the sense that the government s taxation is not integrated into the eduation oliy. The following roosition shows the oliy results. Proosition 3 Let us assume that the government funds for every disadvantaged student reeiving extra-teahing: (i) if a and d, the high-admission equilibrium ours (as before); (ii) if a and d <, the strategies are: shool, x Ha = x La = 1; x Hd ; x Ld 2 (0; 1), university, z Ua = 1; z Da = 0; z Ud = 0; z Dd = 0; (iii) if a < and d <, the strategies are: shool, x Ha = 1; x La = n o a ; x (1 a) Hd; x Ld 2 (0; 1), university, z Ua = min ; a(1) ; z Da = 0; z Ud = 0; z Dd = 0. Proof. See Aendix. The oliy might worsen their oortunities of being admitted at the university. The reason is intuitive. The shool has no osts in roviding disadvantaged students with extra-teahing, so the university does not believe in their grade as a signal of ability if is low. Providing extra teahing only to high ability student is not a redible strategy: if the university believed it, the shool ex ost would give extra teahing even to low-ability students. This analysis suggests are should be taken in the oliy hoies, sine the attemt to imrove the shooling attainment of disadvantaged students might diminish their university admission oortunities. 5 Conluding remarks This aer examines how soial bakground a ets shool s teahing and university admission oliy. We analysed the interation between one shool and one university when students attend shool and then aly to the university. Our results suggest that disadvantaged students are enalised omared to advantaged students, as they reeive less teahing and/or are less likely to be admitted at the university. A oliy reform aiming to inrease the amount of 12

13 teahing rovided to disadvantaged students might derease their admission oortunities. Our analysis foused on the signalling role of eduation, by leaving aside the eduation e ets on students rodutivity. However, this aset may be onsidered in our setting. This an be seen by observing our equilibria. In every equilibrium, any admitted student has reeived extra teahing: all highability students always reeived extra teahing, while low-ability students need to reeive extra teahing to have the hane of being admitted. If we assume that extra-teahing inreases the rodutivity of a high and low-ability student by the same magnitude, the university ayo an be interreted as the ayo obtained by a student who reeived extra teahing. In other words, a greater rodutivity given by extra teahing an already be onsidered in the university ayo. The oliy onsiderations an be extended in many diretions. For instane, the government might imose some restrition on the university strategy in order to favour disadvantaged students, like in the ase of a rmative ation 14. Some oliy aliation an be investigated in the analysis of grade in ation and eduational standards: grade in ation haens when good grades are awarded too easily 15, while the literature on eduational standards 16 examines the riteria adoted by shools in evaluating students. From a welfare analysis ersetive, a oliy an be onsidered where the standard grade () is set so as to maximize welfare. 14 The term a rmative ation refers to oliies that attemt to inrease the resene of individuals who belong to minorities in areas of emloyment and eduation. These oliies generate ontroversy when they involve referential seletion on the basis of rae, gender or ethniity. 15 To ite some examles in the literature, Chan et al. (2005) rooses a signalling model where emloyers know only the students grade but not the students ability and the state of the world (that is, the roortion of talented students). Shwager (2008) examines the imat of grade in ation in the job market in resene of workers with di erent ability and soial bakground. Regardless of the soial bakground, it is ossible that mediore students reeive a high grade aused by grade in ation. Also, grade in ation may bene t from advantaged bakground as it devalues the good grades earned by disadvantaged students. 16 The theoretial frameworks on eduational standards are rovided by Costrell (1994, 1997) and Betts (1998). In the ontext of eduational standards, the issue of soial bakground has been introdued by Himmler and Shwager (2007), who show that a shool with a large roortion of disadvantaged students alies less demanding standards sine its students have less inentives to graduate. 13

14 Furthermore, the framework an be develoed in several ways, two of whih we disuss brie y. First, it seems natural to onsider di erent shools for eah soial grou by taking into aount di erenes in quality of teahing. Seond, it would be interesting to examine this framework alongside di erent generations for exlaining segregation mehanisms. The analysis of an extended model in these diretions is left for future work. Referenes Betts, J The imat of eduational standards on the level and distribution of earnings, Amerian Eonomi Review 88, Carneiro, P. and Hekman, J.J Human aital oliy. In Hekman, J. J., and Krueger, A.B Inequality in Ameria: What role for human aital oliies? The MIT Press. Chan, W., L., Hao, and Suen, W A signalling theory of grade in ation, International Eonomi Review 48, Costrell, R A simle model of eduational standards, Amerian Eonomi Review 84, Costrell, R Can entralized eduational standards raise welfare? Journal of Publi Eonomis 65, Cunha, F., Hekman, J.J., Lohner, L., and Masterov, D. V Interreting the Evidene on Life Cyle Skill Formation. NBER De Fraja, G Reverse Disrimination and E ieny in Eduation. International Eonomi Review. 46 (3): Ele, D. N., Romano, R., Sara, S. and Sieg, H Pro ling in Bargaining over College Tuition. The Eonomi Journal 116: F459-F479. Fernández, R., and Galí, J To Eah Aording To...? Markets, Tournaments, and the Mathing Problem with Borrowing Constraints. Review of Eonomi Studies 66:

15 Galindo-Rueda, F., and Vignoles, A The delining relative imortane of ability in rediting eduational attainment. J. Human Resoures XL(2): Gary-Bobo, R. J., and Trannoy, A E ient tuition fees, examinations, and subsidies. Journal of Euroean Eonomi Assoiation 6(6): Gibbons, R A rimer in game theory. Harvester-Wheatsheaf. Hemel Hemstead. Haveman, R., and Wolfe, B The determinants of hildren s attainments: A review of methods and ndings. Journal of Eonomi Literature 33 (4): Himmler, O. and R. Shwager Double standards in eduational standards: Are disadvantaged students being graded more leniently? ZEW Disussion Paer Joshi, H. E., and MCulloh, A Neighbourhood and family in uenes on the ognitive ability of hildren in the British national hild develoment study. Soial Siene and Mediine 53 (5): Marenaro-Gutierrez, O., Galindo-Rueda, F., and Vignoles, A Who atually goes to university? Emirial Eonomis 32, (2): Rourke, E. and Ingram, F Eduational Testing Servie. International Diretory of Comany Histories. Shwager Grade in ation, soial bakground and labour market mathing. ZEW Disussion Paer Stanford Enyloedia of Philosohy:htt://lato.stanford.edu/entries/a rmative-ation/. Aendix Proof of Proosition 1 and 2 The roof follows Proosition 2. Proosition 1. By setting = 0 we obtain the roof of 15

16 Case 1. a ; University. The university strategies are z Ua = 1; z Da = 0; z Ud = z Dd = 0. The university s beliefs are ( H j g U ; a) = (a(1 a)) (1 )( (1 )) ; a and ( a(1 a) L j g U ; a) = (1 a), if the student has a high grade and ( a(1 a) H j g D ; a) = 0 and ( L j g D ; a) = 1 if the student has a low grade. Thus the exeted ayo s 17 for admitting a (1 a) a(1 a) a(1 an advantaged and high-grade student is E Ua =. This a) must be a (1 a) 0 and, after few assages, a(1 a) a(1 a) a. The exeted ayo s for admitting and not admitting an advantaged and low-grade student are E Da = 1 and N Da = 0, resetively, thus E Da < N Da. The university s beliefs are ( H j g U ; d) = d and ( (1 ) L j g U ; d) = (1 ) if (1 ) the student has a high grade and ( H j g D ; d) = 0 and ( L j g D ; d) = 1 if the student has a low grade. The exeted ayo for admitting one disadvantaged and high-grade student is E Ud =. This must be (1 ) (1 ) (1 ) a (1 a) 0 and, after few assages, a(1 a) a(1 a) d. The exeted ayo s for admitting and not admitting a disadvantaged and low-grade student are E Dd = 1 and N Dd = 0, resetively, thus E Dd < N Dd. Now, the university needs to omare the exeted ayo obtained by high grade students with di erent soial bakground 18 : this is E Ua > E Ud, as a >. As a onsequene,the university admits all the advantaged and high-grade students and the disadvantaged ones only for the remainder of the students demand. Given the restritions on the students demand, the number of disadvantaged (a(1 a)) and high grade who are to be admitted is. (1 )( (1 )) Shool. The shool strategies are x La = 1; x Ha = 1; x Ld = 1; x Hd = 1. The exeted ayo s for giving or not giving extra teahing to an advantaged and high-ability student are T Ha = and NT Ha =, resetively. This must be T Ha > NT Ha, that is, and therefore. The exeted ayo s for 1 giving or not giving extra teahing to an advantaged and low-ability student are T La = and NT Ha = 0, resetively. This must be T La NT La, that is 0, and therefore. The exeted ayo s for giving 17 The suersrit of the university s exeted ro t indiates the strategy erformed by the emloyer, where E indiates to admit and N not. The subsrit sei es the student s grade, where U indiates a high grade and D a low grade, while a and d indiates the student s soial bakground. 18 This is not neessary for low-grade students as none of them are admitted. 16

17 or not giving extra teahing to a disadvantaged and high-ability student are T Hd = z Ud and NT Hd = z Ud, resetively. This must be T Hd NT Hd, that is z Ud z Ud, and therefore z Ud. The exeted ayo s for (1 ) giving or not giving extra teahing to a disadvantaged and high-ability student are T Ld = z Ud and NT Hd = 0, resetively. This must be T Ld NT Ld, that is z Ud 0, and therefore. z Ud Demand onstraint. The total number of students admitted are: ( a (1 a )) (1 ) ( (1 )) ( a (1 a )) (1 ) ( (1 )) : Case 2. a ; < As a, the university and shool strategy for advantaged students does not hange omared to the revious ase. University. n The ouniversity strategies are z Ua = 1; z Da = 0; z Ud = min ; z Dd = 0. The university s beliefs are ( H j g U ; d) = ; (a(1 a)) (1 )( (1)) and ( x Ld (1 ) L j g U ; d) = x Ld(1 ) x Ld (1, if the student has a high grade ) and ( H j g D ; d) = 0 and ( L j g D ; d) = 1 if the student has a low grade. Thus the exeted ayo for admitting an advantaged and high-grade student is E Ud =. This must be x Ld (1 ) x Ld (1 x Ld (1 ) = 0 and, after few assages, x x Ld (1 ) Ld = (1. To be a robability, ) then d < 1, by whih (1 ) d <. The exeted ayo s for admitting and not admitting a disadvantaged and low-grade student are E Dd = 1 and N Dd = 0, resetively, thus E Dd < N Dd. Now, the university needs to omare the exeted ayo obtained by high grade students with di erent soial bakground 19 : this is E Ua > E Ud, as E Ua > 0, while E Ud = 0. Shool. The shool strategies are x La = 1; x Ha =; x Ld = ; x (1 ) Hd = 1. The exeted ayo s for giving or not giving extra teahing to a disadvantaged and high-ability student are T Hd = z Ud and NT Hd = z Ud, resetively. This must be T Hd NT Hd, that is z Ud z Ud, and therefore. z Ud (1 ) The exeted ayo s for giving or not giving extra teahing to a disadvantaged and low-ability student are T Ld = z Ud and NT Hd = 0, resetively. This must be T Ld = NT Ld, that is z Ud = 0, and therefore z Ud =. x Ld (1 ) x Ld (1 ) ) 19 This is not neessary for low-grade students as none of them are admitted. 17

18 Demand onstraint. The total number of students admitted 20 are: ( a (1 a )) (1 ) ( (1 )) ; n thus the students demand onstraint imlies z Ud = min Case 3. a < ; < o (a(1 a)) ; (1 )(. (1)) As <, the university and shool strategy for disadvantaged students does not hange omared to the revious ase. University. n We assume o that the university strategies are z Ua ; z Ud = min ; z Da = 0; z Dd = 0. The university s beliefs are ; ( a(1 ) )(1) a ( H j g U ; a) = and ( ax La (1 a) L j g U ; a) = x La(1 a) ax La (1 a), if the student has a high grade and ( H j g D ; a) = 0 and ( L j g D ; a) = 1 if the student has a low grade. Thus the exeted ayo for admitting an advantaged and high-grade student is E Ua = a. This must be x La (1 a) ax La (1 a) ax La (1 a) a x La (1 a) = 0 and, after few assages, x ax La (1 a) ax La (1 a) La = a be a robability, it is neessary that a < 1, by whih (1 a) a < (1 a). To. The exeted ayo s for admitting and not admitting an advantaged and low-grade student are E Da = 1 and N Da = 0, resetively, thus E Da < N Da. Now, the university needs to omare the exeted ayo obtained by high grade students with di erent soial bakground: this is E Ua = E Ud, as both E Ua = 0 and E Ud = 0. Shool. The shool strategies are x La = a ; x (1 a) Ha = 1; x Ld = ; x (1 ) Hd = 1. The exeted ayo s for giving or not giving extra teahing to an advantaged and high-ability student are T Ha = z Ua and NT Ha = z Ua, resetively. This must be T Ha NT Ha, that is z Ua z Ua, and therefore. z Ua (1 ) The exeted ayo s for giving or not giving extra teahing to a disadvantaged and low-ability student are T La = z Ua must be T La = NT La, that is z Ua and NT Hd = 0, resetively. This = 0, and therefore z Ua =. 20 Note that the number of disadvantaged and high-grade students in this equilibrium is (1 ) ( (1 ) x Ld ), in this equilibrium x Ld = (1 ), by substituting we obtain (1 ) (1 ) d, whih an be simli ed in (1 ) ( (1 )). (1 ) 18

19 Demand onstraint. The total number of students admitted 21 are: (1 ) ( a (1 ) ) ; n thus the students demand onstraint imlies z Ud = min ; ( a(1 ) )(1) o. Proof of orollary 1 High-admission equilibrium. Di erentiation of to, and 0, (a(1 a)) (1 )( (1 with reset )) ( a(1 a)) = 1 a (1 )( ( a(1 (1 )( (1 )) 1 (1 < ) ( 1)( ( 1)) 2 ( a(1 a)) (a(1 (1 )( (1 = > 0, resetively. )) ( 1) 2 ( (1 )) with reset to, < 0, re- 2 Middle-admission equilibrium. Di erentiation of and < 0, = > 0, setively. Di erentiation of d 2 ( (1 ) with reset to @ < 0, and d = 1 (1 ) 1 > 0, resetively. Low-admission equilibrium. Di erentiation of and a (1 a) 2 a < 0, and Di erentiation with reset to,, and yields 1 > 0 2 Proof of Proosition 3 < 0, resetively. (1 ) a = 1 a (1 a) = (1 ) with reset to 1 a > 0, resetively. < = The government subsidises. Thus to rovide disadvantaged students with extra teahing is weakly dominant. This does not hanges anything in the ase 1: in fat, the shool strategy was x Ld = 1; x Hd = 1. In ases 2 and 3, the equilibria are a eted by this reform. Case 2. a ; < For advantaged students we refer to the roof (ase 2) of Proosition 2. University. The university strategies are z Ua = 1; z Da = 0; z Ud = 0; z Dd = x 0. The university s beliefs are ( H j g U ; d) = Hd and ( x Hd x Ld (1 ) L j g U ; d) 21 Note that the number of advantaged and high-grade students in this equilibrium is ( a (1 a ) x La ), in this equilibrium x La = a (1 a), by substituting we obtain a (1 a ) a, whih an be simli ed in ( a (1 )). (1 a) 19

20 x Ld (1 ) x Hd x Ld (1 =, if the student has a high grade and ( ) H j g D ; d) = 0 and ( L j g D ; d) = 1 if the student has a low grade. Thus the exeted ayo for admitting an advantaged and high-grade student is E Ud = x Ld (1 ). This must be x Hd x Hd x Ld (1 ) after few assages, < to have < x Hd x Ld (1 ) x Ld (1 ) x Hd x Hd x Ld (1 ) x Hd x Ld (1 < 0 and, x ) Ld x x Hd x Ld. The ondition Ld x Hd x Ld is su ient. After few assages, this is veri ed if x Hd x Ld, whih always holds 22. The exeted ayo s for admitting and not admitting a disadvantaged and low-grade student are E Dd = 1 and N Dd = 0, resetively, thus E Dd < N DdṠhool. The shool strategies are x La = 1; x Ha = 1; x Ld 2 (0; 1); x Hd 2 (0; 1). The exeted ayo s for giving or not giving extra teahing to a disadvantaged and high-ability student are T Hd = 0 and NT Hd = 0, resetively. This must be T Hd = NT Hd, and it is veri ed for every value of x Hd. The exeted ayo s for giving or not giving extra teahing to a disadvantaged and low-ability student are T Ld = 0 and NT Hd = 0, resetively. and it is veri ed for every value of x Hd. Demand onstraint. It is veri ed as ( a (1 a )) < : Case 3. a < ; < For advantaged students we refer to the roof (ase 3) of Proosition 2. As <, the university and shool strategy for disadvantaged students does not hange omared to the revious ase. 22 As the ayo of giving extra teahing is higher for high-ability rather than low-ability students, >. 20