The Supply of Education Quality in a Spatial Model with Asymmetric Moving Costs

Transcription

1 The Supply of Education Quality in a Spatial odel with Asymmetric oving Costs Patrizia Ordine a1, Giuseppe Rose b a University of Calabria, Department of Economics and Statistics, Italy. b Birkbeck College, University of London, UK. 1 Corresponding author. p.ordine@unical.it, Address: University of Calabria, Dept. of Economics and Statistics, 8706 Arcavacata di Rende (CS) Italy.

2 Abstract The paper analyzes the characteristics of the supply of high education in di erent geographical macroareas using a strategic interaction framework. The paper focuses on the quality of education showing that in equilibrium it is inversely related to students moving costs across areas. We show that in the presence of asymmetric information on workers ability and asymmetric costs of moving, the only PBE consistent with forward induction involves that only high ability workers acquire education and the quality of education is lower in macroareas where the moving costs are higher. In these areas high ability individuals need to acquire a higher education level in order to signal their ability. The theoretical model has many implications in terms of the existing debate about educational and growth policies. Our model predicts that policies must be regulated according to the speci c socioeconomic characteristics of the area. Direct subsidies to universities may be ine ective in improving the quality of education in the less developed areas. Jel classi cation: J24, D82, I21, I28. Key Words: Perfect Bayesian Equilibrium, Forward Induction, Spatial models.

3 1 Introduction The paper is motivated by the need of nding some possible explanations of quality di erentials among universities considering that the supply of education may be determined by the degree of competition among universities which in turns depends on students moving costs across areas. The paper analyzes the characteristics of the supply of high education in di erent geographical macroareas using a strategic interaction framework and highlights how in the presence of asymmetric information on workers ability and asymmetric costs of moving, the only perfect Bayesian equilibrium (PBE) consistent with forward induction involves that only high ability workers acquire education and the quality of education is lower in macroareas where the moving costs are higher. The model setup considers an economy that can be described as a circular city where a nite number of universities compete to attract students. The universities are located at the same distance from each other and give rise to di erent regional markets. In each market, there is a continuum of individuals choosing how much education to obtain conditional on their ability. In this economy two rms operate. They are exogenously located on the circle and set production on the basis of technology and compete to employ individuals. We model a strategic interaction process where there are strategic complementarities between the educational choice of individuals and the technological choices of rms. At the same time, we assume that the universities choices in terms of quality spill over on productivity and wages. The equilibrium is analyzed considering both symmetric and asymmetric costs of moving. In the presence of symmetric costs, we nd that the quality of education decreases as moving costs grow and all the individuals are paid with wages that re ect their productivity minus the moving costs. If we consider the case of a circular city where there are asymmetric costs of moving, we nd that the only PBE is characterized by universities that set di erent levels of quality. In particular, the universities located in the areas where the moving costs are relatively higher set a lower quality of education. Even if the quality of education is observable by all the agents, high ability individuals located in areas with relatively higher moving costs, decide to study in the low quality university although they know that this choice generates a low wage. oreover, in these areas high ability individuals need to acquire a higher education level in order to signal their ability. Another implication of the model is that in the presence of asymmetric moving costs productivity di erentials may arise. The theoretical model has many policy implications. The existing debate, especially in European countries, considers that the reform of the educational 1

4 system must confront the issues of the degree of governmental ownership, the degree of governmental subsidy, and the degree of competition (OECD, 2006). A crucial issue is then to evaluate if the government may a ect the quality by changing the level of spending for high-education. In particular, it is relevant to evaluate the right instruments to implement this policy: tax exemptions, educational vouchers or direct subsidies to universities. Our model predicts that policies must be regulated according to the speci c socioeconomic characteristics of the area. oreover, in general direct subsidies to universities may be ine ective in improving the quality of education in the less developed areas. This conclusion is consistent with the results of Aghion et al. (2005) who highlight that regions investing more in higher education do not necessarily grow faster. In our model it is crucial how governmental subsidies are distributed in order to improve the quality of education, the rms productivity levels and the regional macro performances. Further, although the degree of competition may be undoubtedly improved by raising the individuals information about universities characteristics and students performance in the labour market, we conclude that the monopoly power of local universities may be regulated more e ectively by modifying the structural characteristics and the institutions operating in the regional economies. Ine ciencies in the credit market and liquidity constraints, for example, may induce high mobility costs among areas and consequently in- uence the supply of quality. The labour market performance and the local rate of unemployment also might strongly in uence the individuals choice of moving and consequently the incentive of local universities to improve quality. 2 The model setup We consider an economy that can be described as a circular city (Salop, 1979) where a nite number J of universities, exogenously located along a circle of circumference J, compete to attract students supplying di erent levels of educational quality. The universities are located at the same distance from each other and give rise to J regional markets. In each market, there is a continuum of individuals whose total number is, which are assumed to be located uniformly along the circle arch of length 1. An individual location is indexed by z 2 [0; 1] in each regional market. The individuals choose to invest in human capital and may incur in moving costs to reach the universities. In this economy two rms operate (f = 1; 2). The rms are exogenously located on the circle and set production on the basis of technology T and compete to 2

5 employ individuals. The economy lasts p = 0; 1; 2; :::1 periods, agents are risk neutral and they discount the future at a rate r x (with x = fj; ; fg for universities, individuals and rms respectively). We assume that r x is compatible with agents collusion only in the presence of strategies that contain credible punishment involving zero (pro ts/surplus) payo s in the presence of a relatively small number of players. We model a strategic interaction process where there are strategic complementarities between the educational choice of individuals and the technological choices of rms. 1 At the same time, we assume that the universities choices in terms of quality spill over on productivity and wages. Usually, the education quality is measured by some outcome or structure indicators such as the student-teachers ratios, the class size, the students marks or the proportion of students who terminate on time. However, these measures may contain some ambiguities. In our study we de ne the quality of education as the set of scienti c and technical skills provided by the university that rise the individuals productivity and we avoid to consider students facilities. We consider moving costs as determined by the characteristics of the local socioeconomic environment so that these costs may be in uenced by the degree of e ciency of the local credit market, the educational and cultural background of the local collectivity, the performance of the labour market. The individuals Individuals maximize an utility function, u (), expressed in terms of wages w() and the cost of education c(). Both functions w() and c() are determined by workers ability. We assume that there exist two types of workers with ability i, i = h; l with h > l, and = prob ( = h ) 2 (0; 1). The individual s ability is his own private information as in the Spence (197) model, but is common knowledge. Hence: u (w; e; q j j) = w (e; i ; q j ; T ) c (e; i ; q j ) u qj () > 0, u qj q j () < 0 (1) where q j 2 [0; q max ], q max > 0, indicates the quality of education supplied by the attended university j, j = 1; 2; :::J. Before entering in the job market the individual can obtain a level of education e 2 [0; e], e > 0, involving monetary and non-monetary costs represented by the twice di erentiable function c (e; i ; q j ). We assume that: 1 Strategic complementarity is de ned as in Acemoglu (1996).

6 c q () > 0; c e () > 0 (2) oreover, we assume that the so-called "single cross property" holds only if q > 0. Hence: c e (e; l ; q j ) > c e (e; h ; q j ) iff q j > 0 () This implies that if the individual acquires education of quality q = 0 the single cross property disappears and the marginal cost of education is identical for individuals with di erent ability. We assume also that: c qe (e; l ; q j ) > c qe (e; h ; q j ) (4) where c qe is the cross partial derivative of the cost function with respect to the level and the quality of education. The implications of (4) are illustrated in Figure 1 where we show that a reduction in the quality of education lowers the e ort of low ability more than the e ort of high ability individuals. The net e ect of a decrease in q results in the fact that the indi erence curves become less distant from each other. The rms Each rm employs workers in order to produce the nal output. Before hiring a worker the rm has to decide the technology to adopt. In particular, the rm can choose between high or low technologies. We indicate T = fht; LT g the rm investment in high or low technology respectively. The cost of technology HT is given by > 0 while the cost of technology LT is normalized to zero. Following Acemoglu (1997), the average productivity per worker is given by: y = y (e; i ; q j ; T ) = e 0 + e(q j + " 1 f=h ;T =HT g) (5) where e o is a constant and " > 0. From (5) it appears that the high technology is complementary only to the high ability workers. The worker s productivity is a linearly increasing function of education when q > 0. As illustrated in Figure 2, equation (5) assumes a scenario where the e ect of education on worker s productivity depends on the quality of education and on the match between high ability individuals and high technology. Ceteris paribus a high ability worker from a university of high quality produces a higher output. The rm s pro ts per worker are given by the following expressions: 4

7 = y (e; h ; q j ; HT ) w (6) = y (e; i ; q j ; LT ) w (7)! = y (e; l ; q j ; HT ) w! (8) where,, and! indicate the possible scenarios of rm s technology and worker s ability matches. The wage is set by rms competing à la Bertrand to hire workers. The universities The existing literature on education supply discusses when it is appropriate to introduce pro t maximizing behavior for the objective functions of universities. 2 The theoretical framework of this paper considers only economies that use a kind of high-education nancing method called cost-sharing where the mainly source of university funding is the government allowances and the student fees have only a marginal impact on the university resources (for a detailed discussion see Johnstone, 200). In this context, each university has the following objective function: max q j u j = (S C (q j )) X j (q j; q j ) + q j (9) s:t: (10) S C(q j ) 0 (11) X j (q j; q j ) > 0 (12) with X j () representing the enrollments demand for university j, C () the unit cost per student and S is the revenue per student given by the government subsides and by the student s fees. The cost function is increasing in quality: C q > 0 C qq 0 (1) 2 Rothschild and White (1995) consider the issue of competitive pricing and allocative e ciency for higher education. At the same time, Winston (1999) puts forward the limits of the use of standard pro t maximizing theory for the analysis of education supply. In any case, these approaches refer to universities that can set autonomously fees and admission rules. On the other hand, Dranove and White (1994) present theoretical arguments suggesting that no-pro t organizations behave as pro t maximizers and Kemnitz (200) considers the universities as pro t-maximizer agents because of the intent of having additional resources for teaching or research expenditures. The cost-sharing funding method became widespread in the European economies during the last ten years. Nevertheless, the impact of student fees on the total university funding is less than the 15% on average (see OECD, 2004). 5

8 and > 0 is a weight discussed below. The objective function in (9) derives from the strategic behavior of the internal political membership in period p who maximizes the probability (0 < 1) of being re-elected in period p+1. We assume that the probability of being re-elected in period p + 1 is given by the following expression: p+1 = h(q j;p ; j;p (q j;p )) (14) + + where h() is a generic "well behaved" function, q j;p is the quality xed by university j in period p and j;p (q j;p ) = (S C (q j;p )) X j;p (q j;p; q j;p ) is the surplus generated by the university governance given S. In equation (14) we are assuming that the probability of being re-elected in period p+1 is a function of the quality q xed in period p. This depends on two main reasons. First, an increase in q obtained by increasing the quality of the academic sta rises the utility of high quality professors since they prefer to be enrolled in a high quality institution and consequently they will sustain a membership investing in high quality. In this case, the higher is the quality set in period p the larger is the political consensus that the membership will have in period p + 1. At the same time, we assume that the probability of being re-elected is a function of the additional resources that the membership is able to generate given the subsidy S. We are assuming that the membership can spend this surplus in activities (larger administrative sta, standard teaching activities) that do not directly a ect the education quality but generate a general consensus and increase the probability of re-election. The objective function in (9) expresses the choice of q that maximizes the probability of being re-elected given q j;p and j;p (q j;p ) with weights > 0 and 1 respectively. In Figure we show how the university s quality is determined assuming that there exists a value qj 0 that maximizes j;p (q j;p ). Notice that given the quality choice qj that maximizes u j depends on the size of j;p (q j;p ). The demand is a continuous function of the quality q j and of the quality o ered by other universities, q j operating in the whole j > j < 0 (15) Since both the cost of education and the individual s productivity are positively related to the quality, then (15) implicitly assumes that for individuals: w q () > c q () (16) 6

9 The timing of the model The strategic interaction process is analyzed considering sequential moves that lead education to be a signal of individuals ability unless the quality is set to zero. Following the Harsanyi (1967/68) approach, we assume that the timing is as follows. Each period p is characterized by t = 0; 1; ::5 instants of time. At time t = 0 Nature chooses the ability of the individuals, at t = 1 elections take place in each university and then the elected membership sets the quality q that all agents can observe. At t = 2, given the quality choices of the universities, the individuals decide where they want to study and the level of education, considering the distance from the universities. At t =, the rms observe the level and quality of education of the individuals and make the technological choices. At the same time, the rms make their wage o ers. At t = 4 the individuals accept one of the wage o ers. The game is repeated for p = 1 periods. The equilibrium concept used to solve the model is the Perfect Bayesian Equilibrium (PBE) consistent with forward induction reasoning (Kohlberg and ertens, 1986) that allows us to exclude PBE characterized by unreasonable beliefs. This is also consistent with the Intuitive Criterion (Cho and Kreps, 1987). The game is illustrated in Figure 4. In what follows we study equilibria with and without moving costs and we highlight that the "one shot" Bertrand competition (both among universities and between rms) in a spatial model framework leads to a Nash equilibrium in which agents have positive surplus/pro ts. In Section we consider the implications of rms collusion in the absence of moving costs. In Section 4 and 5 we argue that with positive moving costs the only credible punishment that may characterize the "Nash reversion strategies" in order to generate agents collusion gives rise to positive surplus/pro t for each agent. Hence, given our assumptions on the discount rates, in the presence of moving costs we exclude the possibility of collusion. The equilibria without moving costs We rst consider the case where the individuals do not have costs of moving. We de ne the Bayesian equilibrium as the assessment = J ; ; f ; b(e) where J,, f, represent universities, individuals and rms strategies respectively and b (e) 2 [0; 1] is the rms beliefs function, i.e. the probability that a worker with education e is a high ability type derived using the Bayes rule. 7

10 Proposition 1 The following pro le of strategies and beliefs = J ; ; f ; b(e) is a PBE of the game described in Figure 4 consistent with reasonable beliefs: J = fset q j = q max 8 jg (17) 8 9 choose d = d >< min, e = e s iff i = h >= choose e = 0 iff = i = l accept the highest wage o er >: >; choose randomly in case of tie 0 iff e < e s b(e) = 1 iff e e s (18) (19) 8 >< f = >: choose HT iff e e s 8 f choose LT iff e < e s 8 f w = e 0 iff e < e s 8 f w = y (e s ; h ; q max ; HT ) iff e e s 8 f 9 >= >; (20) where e s indicates the generic level of education consistent with a separating equilibrium as shown in Figure 5. The proof of proposition 1 is given in the Appendix. Intuitively, Bertrand competition among universities drives education quality at its maximum level and a separating signaling equilibrium, where only high ability individuals invest in human capital, arises. The single cross property is preserved. In this case, the maximum level of education quality is determined by the amount of subsidy S, because the Bertrand competition among universities implies a zero pro t equilibrium with S = C (q j ) in (9). Since the quality of education is similar in all universities, the high ability individuals are indi erent about location. We assume that they choose the closest university so that d = d min. All rms generate the highest level of productivity given, the share of high ability individuals on the total labour force, and S. We should consider that this may be just one of the possible equilibria since collusion among rms and universities may occur. However, this is not the case in the presence of a large number of universities given the assumptions on the discount rate r J. Firms cooperation may lead to wages lower than worker s productivity. As shown in the Appendix the main features of the equilibrium described in proposition 1 are unchanged. 8

11 4 The equilibrium with symmetric moving costs We consider the equilibrium when individuals incur in moving costs. Since in this economy J > 2 universities operate, we consider a circular city model as in Salop (1979). In Figure 6, we describe a location of rms and universities when J =. The total cost of acquiring education in university j, for an individual located at a distance d; is c (e; i ; q j ) + d, (with ; d > 0) and =2 can be thought of as the cost per unit of distance travelled by the individual in going to and from university j s location. We assume that the cost per unit of distance is identical for each individual and it is independent on the area where he is located. At the same time, the net wage obtained in rm f is w (e; i ; q j ; T ) v, where v is the distance of an individual from the rm 1 s location. In order to nd the PBE of the game we need to nd the Nash equilibrium in the simultaneous move game among universities. To keep things treatable, we assume that it is always worthwhile for a high ability individual to acquire education so that: w (e; h ; q j ; HT ) c (e; h ; q j ) (v + d) > 0 (21) Proposition 2 The following pro le of strategies and beliefs = J ; ; f ; b(e) is the only PBE of the game described in Figure 4 consistent with reasonable beliefs in the presence of symmetric moving costs: J = fset q j = q j = q g (22) 8 9 choose d = d >< min, e = e s iff i = h >= choose e = 0 iff = i = l accept the highest net wage o er >: >; choose randomly in case of tie (2) 0 iff e < e s b(e) = 1 iff e e s (24) 9

12 8 >< f = choose HT iff e e s 8 f choose LT iff e < e s 8 f w = e 0 ( 1 2 J v) iff e < es for f = 1 w = e 0 v iff e < e s for f = 2 w = e 0 iff e < e s and v = J 4 8 f 9 >= >: w = y (e s ; h ; q ; HT ) ( 1 2 J v) iff e es for f = 1 w = y (e s ; h ; q ; HT ) v iff e e s for f = 2 w = y (e s ; h ; q ; HT ) iff e e s and v = J 4 8 f where q is the level of quality satisfying the following condition: >; (25) (S C (q))u q 2C q + 2 = 0 (26) The expression in (26) represents the best response function in the simultaneous move game among universities in a symmetric equilibrium. We derive analytically (26) in the proof given in the Appendix. Using the implicit function theorem, it is possible to show =@ < 0 =@S > 0. This means the the quality level set by universities decreases as the cost of moving rises. oreover, with symmetric moving costs, the quality of education increases with subsidies. It is interesting to notice that it may be more e ective a reduction in moving costs than an increase of subsidies as long as ju q ()j < C q + 2 since in this case we have that j@q =@j > j@q =@Sj. As shown in the Appendix, in the PBE equilibrium 0 < q < q max. From (26) it may be easily seen that q = q max only if = 0. Each university has a demand equal to and we observe high ability individuals uniformly distributed among universities. The Nash equilibrium for the simultaneous move game between rms is determined by considering that rms o er a di erent wage to each worker. 4 The presence of travel costs introduces di erentiation between the two rms since individuals may now strictly prefer to work in one of the two rms even when wages are the same. Only individuals located at the same distance from the two rms (v = J ) earn a wage equal to their productivity since they are indi erent 4 about the direction of moving. The level of productivity in turns depends on q. In this type of economy, the level of subsidies and the dimension of the travelling costs are the crucial determinants of rms productivity and individuals utility. However, the equilibrium levels of these variables are always inferior to those reached in the case of absence of moving costs. 4 The main results do not change if rms o er the same wage conditional on education to all workers. The proof of this result is available on request. 10

13 5 The equilibrium with asymmetric moving costs We now discuss the case when the costs of moving depend on the origin area of the individuals. We assume that there are J = universities as described in Figure 6. We indicate the area delimited by universities j = 1, and j = 2 as area A1; the area delimited by universities j = 2, and j = as area A2; the area delimited by universities j = 1, and j = as area A. In particular we assume that individuals located in the area A1 incur in higher moving costs per unit of distance travelled, denoted by, than people located in the area A2 and A, which have costs indicated by, with <. We rst assume that rms are located symmetrically with respect to A1. Later we relax this assumption and discuss the implications of an asymmetric rms positioning. Proposition The following pro le of strategies and beliefs = J ; ; f ; b(e) is the only PBE of the game described in Figure 4 consistent with reasonable beliefs: set qj = ^q if j = 1; 2 J = set q j = q (27) > ^q if j = 8 >< = >: choose d = d min, e = e sj iff i = h in A1 choose d = d min ; e = e sj iff i = h, z 2 [0; ^z] [ 1 ; 1 in A2 or A 2 choose d = d max, e = e sj iff i = h ; z 2 ^z; 2 1 in A2 or A choose e = 0 iff i = l accept the highest net wage o er choose randomly in case of tie (28) 9 >= >; 0 iff e < e sj b(e) = 1 iff e e sj 8 j (29) 11

14 8 >< f = choose HT iff e e sj 8 f choose LT iff e < e sj 8 f w = e 0 ( 1 2 J v) iff e < esj for f = 1 w = e 0 v iff e < e sj for f = 2 w = e 0 iff e < e sj and v = J 4 8 f 9 >= w = y (e; h ; q j ; HT ) ( 1J v) iff e 2 esj, for f = 1 w >: = y (e; h ; q j ; HT ) v iff e e sj for f = 2 >; w = y (e; h ; q j ; HT ) iff e e sj and v = J 8 f 4 (0) where e sj indicates the level of education that generates a separating equilibrium with: e s < e s2 and e s2 = e s1 and d max represents the maximum distance between the individual s location and the contiguous universities. The interesting result derives from the analysis of the equilibrium strategies of the universities. As shown in the Appendix, the quality level set by the universities located in the area where individuals face high travelling costs is lower than the quality level set elsewhere. The low quality of education implies a separating equilibrium in which the high ability individuals need to acquire a higher education level in order to signal their ability. Higher moving costs induce a sort of stronger monopoly power of universities which prefer to loose part of the potential demand of education and lower the education quality. An interesting exercise is to evaluate the impact of di erent policy measures in order to rise quality in the di erent areas. First of all, we may show that an increase of governmental subsidies S, is more e ective in increasing the education quality in universities facing the demand of individuals from lower moving costs area. As proved in the Appendix we ^q At the same time we show that rising subsidies or changing moving costs have an impact on quality which depends on the relative size of moving costs and on the margin of pro ts per student of the universities. We show that it is worthwhile to rise subsidies instead of reducing moving costs only if: 2 + (S C(^q)) > 0 (2) 12

15 The values of for which condition (2) is satis ed are graphically illustrated in Figure 8 where A = scenarios may arise: + p 2 +4(S C(q)) 2. Since > must hold, two A ) It is always worthwhile reducing the marginal moving cost instead of increasing the direct subside to the universities. A > ) In case of extremely wide regional disparities ( > A) an increase in S generate a greater improvement in education quality than a reduction in : In case of small regional disparities ( < < A) reducing the marginal moving cost yields to a greater quality improvement than the one obtained by increasing the direct subside. On top of that, the presence of travel costs makes some individuals strictly prefer working for one of the two rms even when wages are the same. Only individuals located at the same distance from the two rms (v = J 4 ) earn a wage equal to their productivity since they are indi erent about the direction of moving. Of course, the symmetry in rms location with respect to the area with the higher travel costs generates no di erences in rms productivity. If we remove this symmetry assumption, the market power of the rm closer to the highest travel costs area can be used only on individuals that have a lower productivity, because of the low education quality received. In this case a rm always performs a low productivity. 6 Conclusions In this paper we analyze the supply of high education considering the implications of individuals moving costs and rms technological choices in determining the quality of education. We de ne the quality of education as the set of scienti c and technical skills provided by the university that rise the individuals productivity. We consider moving costs as determined by the characteristics of the local socioeconomic environment so that these costs may be in uenced by the degree of e ciency of the local credit market, the educational and cultural background of the local collectivity, the performance of the regional labour market. We show that the quality of education and the average productivity of workers is lower in the presence of moving costs. oreover, we nd that if moving costs are asymmetric, in the areas where they are relatively higher, the quality of education is lower than the quality supplied elsewhere. In these areas high ability individuals need to acquire a higher education level in order to signal their ability. oving costs in these 1

16 areas generate a sort of stronger monopoly power of universities which prefer to loose part of the potential demand of education and lower the education quality. In our model this implies that the level of rms productivity in less developed areas with high moving costs is also low. These results are relevant when discussing educational and growth policies of economies with regional disparities and where there exists a high degree of governmental ownership and a centralized governmental funding based on cost-sharing formulas. This is the case of many European economies. We argue that when the system of funding is centralized and based on a xed amount of subsidy per student, increasing this subsidy may not be e ective in improving the education quality in less developed areas. The right instruments to be used in order to rise the quality of high education and then to promote local growth and to reduce regional disparities depend on a set of variables which include relative moving costs across areas and the margin of pro t per student. An increase in governmental subsidies might just rise the monopoly power of local institutions and might not be useful in reducing disparities. This conclusion is consistent with the results of Aghion et al. (2005) who highlight that regions investing more in higher education do not necessarily grow faster. In our model it is crucial how governmental subsidies are distributed in order to improve the quality of education, the rms productivity levels and the regional macro performances. Further, although the degree of competition may be undoubtedly improved by raising the individuals information about universities characteristics and students performance in the labour market, we conclude that the monopoly power of local universities may be regulated more e ectively by modifying structural characteristics and institutions operating in the regional economies. Ine ciencies in the credit market and liquidity constraints, for example, may induce high mobility costs among areas and consequently in uence the supply of quality. The labour market performance and the local rate of unemployment also might strongly in uence the individuals choice of moving and consequently the incentive of local universities to improve quality. References [1] Acemoglu, D., Credit Constraint, Investment and Growth. In: Booth, A.L., Snower, D.J. (Eds.). Acquiring Skills, Cambridge: Cambridge University Press, [2] Acemoglu, D., Training and Innovation in an Imperfect Labour arket. Review of Economics Studies 64,

17 [] Aghion, P., Boustan, L., Hoxby, C., Vandenbussche, J., Exploiting States istakes in Identify the Causal Impact of Higher Education on Growth. European Economic Association Annual Conference, Amsterdam. [4] Cho, I-K., Kreps, D Signaling Games and Stable Equilibria. Quarterly Journal of Economics 102, [5] Dranove, D., White, W. D., Recent Theories and Evidence on Competition in Hospital arkets. Journal of Economics and anagement Strategy, [6] Harsanyi, J.C., 1967/68. Games with incomplete information played by Bayesian players. anagement Science, , 20-4, [7] Johnstone, D.B., 200. The Economics and Politics of Cost Sharing in Higher Education: a Comparative Perspective. Economics of Education Review 2, [8] Kemnitz, A., 200. Funding, Competition and Quality in Higher Education, mimeo. [9] Kohlberg, E., ertens, J.F., On the Strategic Stability of Equilibria. Econometrica 50, [10] OECD, Education at a Glance, Paris. [11] OECD, Education at a Glance, Paris. [12] Rothschild,., White, L.J., The Analytics of the Pricing of Higher Education and Other Services in Which the Customers are Inputs. Journal of Political Economy 10, [1] Salop, S., onopolistic Competition with Outside Goods. Bell Journal of Economics 10, [14] Spence, A.., 197. Job arket Signaling. Quarterly Journal of Economics 87, [15] Winston, G., Subsidies, Hierarchy and Peers: The Awkward Economics of Higher Education, Journal of Economic Perspectives, 1,

18 Appendix Proof of proposition 1. First we show that q j = q max for all j is a Nash equilibrium in the simultaneous move game among universities. At level of quality q max all the universities earn zero pro ts. Neither university can gain by lowering quality 0 < q j < q max because it will loose all students due to the assumption in equation (16). We need to show that setting q j = 0 also implies that university j looses all students. Suppose that q j = 0. Since q j is common knowledge, education acquired in university j is not a signal on individuals ability. The rms choose T = HT and pay a wage w p = (e 0 e") + (1 ) e 0 only if the expected pro t from this choice is higher than the pro t when T = LT and w p = e 0. Hence, the following condition must hold: (e 0 e" w p ) + (1 ) (e 0 w p ) 0 () where the LHS represents expected pro ts in case of T = HT while the RHS are pro ts in case of T = LT. Condition () requires 0. Since > 0 rms never choose to invest in HT and always o er a wage e 0 when individuals are from university j. Consequently, as shown in Figure 7 no one chooses to attend university j. The single cross property holds and the other strategies and beliefs of proposition 1 represent a separating equilibrium similar to that discussed in Spence (197). Q:E:D: In the presence of cooperation among rms which leads to a positive pro t per worker = =! =, > 0, equation () becomes: (e 0 e" w p + ) + (1 ) (e 0 w p + ) (4) since cancels out, the main results from () hold. Proof of proposition 2 Consider the simultaneous move game among universities. The universities are located at the same distance from each other along the circle of length J. In this spatial model the universities compete only locally. We rst consider competition between contiguous universities as in a linear model. Given the assumption of symmetry, we can extend the essential features of the results to the case in which J > 2. Indicating with ~z the location of an individual who is indi erent between studying in university j or j: we have: u (; q j ) ~z = u (; q j ) (1 ~z) (5) 16

19 ~z = u (; q j) u (; q j ) + 2 and we can write the demand for universities j as follows: 8 9 < 0 if ~z < 0 = X j (q j; q j ) = ~z if ~z 2 [0; 1] : ; if ~z > 1 which implies (6) (7) 8 < X j (q j; q j ) = : 0 if u (; q j ) < u (; q j ) ~z if u (; q j ) 2 [u (; q j ) ; u (; q j ) + ] if u (; q j ) > u (; q j ) + 9 = ; utatis mutandis the demand for university j is given by: 8 < X j (q j; q j ) = : 0 if u (; q j ) > u (; q j ) ~z if u (; q j ) 2 [u (; q j ) ; u (; q j ) + ] if u (; q j ) > u (; q j ) + Note that in searching for the best response to quality choice of its rival q j, each university j can restrict itself to the interval [u (; q j ) ; u (; q j )+] since in the other scenarios it can never rise pro ts. Thus the best response to q j solves the following maximization problem: 9 = ; max(s C(q j )) u (; q j) u (; q j ) + q j 2 s:t: u (; q j ) 2 [u (; q j ) ; u (; q j ) + ] + q j (8) The necessary and su cient Khun-Tucker FOC for (8) are: 2 C q j [u (; q j ) u (; q j ) + ] + (9) 8 + < 0 if u (; q j ) = u (; q j ) 2 (S C(q j))u qj + = 0 if u (; q j ) 2 [u (; q j ) ; u (; q j ) + ] : 0 if u (; q j ) = u (; q j ) + 17

20 Since we look for a symmetric equilibrium where qj = q j = q we can see from (9) that it may be reached only in the middle case. So q is the solution of: with: 1 2 [(S C (q))u q C q + 2] = @S = C q + 2 u q q (S C(q )) C q u q C q q < 0 (41) u q u q q (S C(q )) C q u q C q q > 0 (42) because of the assumptions on rst and second order derivatives and: C q + 2 = (S C (q ))u q (4) As in the case without moving costs, if quality is set equal to zero no one invests in human capital and the demand for education collapses to zero. Consequently we must have q > 0. If we consider the circular city we must have that each university has a demand equal to. Q:E:D Proof of proposition Consider rst the simultaneous move game among universities. We show that the quality level of education of university j = 1 and j = 2 is lower than the level of quality supplied by university j =. Consider the demand for education of university j = 1. This university competes to attract students from areas A1 and A. As explained in the proof of proposition 2 the location of an individual who is indi erent between studying in university j or j is: ~z A1 = u (; q 1) u (; q 2 ) + 2 ~z A = u (; q 1) u (; q ) + 2 We may have the following seven scenarios: (44) (45) 18

21 X 1 (q 1 ; q 2 ; q ) = (46) u (; q1 ) < u (; q = 0 if 2 ) u (; q 1 ) < u (; q ) = if u (; q1 ) > u (; q 2 ) + u (; q 1 ) < u (; q ) = if u (; q1 ) < u (; q 2 ) u (; q 1 ) > u (; q ) + = 2 if u (; q1 ) > u (; q 2 ) + u (; q 1 ) > u (; q ) + = ~z A1 if u (; q1 ) 2 [u (; q 2 ) ; u (; q 2 ) + ] u (; q 1 ) < u (; q ) = ~z A if u (; q 1 ) < u (; q 2 ) u (; q 1 ) 2 [u (; q ) ; u (; q ) + ] = ~z A + ~z A1 if u (; q1 ) 2 [u (; q 2 ) ; u (; q 2 ) + ] u (; q 1 ) 2 [u (; q ) ; u (; q ) + ] Note that in searching for the best response to the quality choice of its rivals, university j = 1 can restrict itself to the interval [u (; q ) ; u (; q )+ ] and [u (; q 2 ) ; u (; q 2 )+]. The maximization problem of university j = 1 considers only the last scenario described in (46). Since universities j = 1, and j = 2 are symmetric mutatis mutandis the same scenarios are faced by university j = 2. For university j = the only relevant scenario to consider is the following: X (q 1 ; q 2 ; q ) = ~z A + ~z A2 where, in equilibrium, ~z A = ~z A2. The maximization problems are the followings: max(s C(q 1 )) u (; q 1) q 1 if u (; q ) 2 [u (; q 1 ) ; u (; q 1 ) + ] u (; q ) 2 [u (; q 2 ) ; u (; q 2 ) + ] (47) u (; q 2 ) + 2 s:t: u (; q 1 ) 2 [u (; q 2 ) ; u (; q 2 ) + ] u (; q 1 ) 2 [u (; q ) ; u (; q ) + ] 19 + u (; q 1) u (; q ) q 1 (48)

22 max(s C(q 2 )) u (; q 2) q 2 u (; q 1 ) + 2 s:t: u (; q 2 ) 2 [u (; q 1 ) ; u (; q 1 ) + ] u (; q 2 ) 2 [u (; q ) ; u (; q ) + ] + u (; q 2) u (; q ) q 2 (49) max(s q C(q )) 2 [u (; q ) u (; q 1 ) + u (; q ) u (; q 2 ) + 2] + q s:t: (50) u (; q ) 2 [u (; q 1 ) ; u (; q 1 ) + ] u (; q ) 2 [u (; q 2 ) ; u (; q 2 ) + ] where q j indicates the rivals choice in each q j best response search process. We indicate with q j the best response to q j. Since universities one and two are symmetric the Nash equilibrium must yield q 1 = q 2 = ^q satisfying the following FOC: C^q + 2 (u (; ^q) u (; q )) + u^q 2 + [S C (^q)] + = 0 (51) 2 The maximization problem of university j = gives the solution for q : C q [(u (; q ) u (; ^q) + )] + u q [S C (q )] + = 0 (52) We have that (51) is equal to (52) since they are both equal to zero, hence: C^q + 2 (u (; ^q) u (; q )) + u^q 2 + [S C (^q)] = (5) 2 C q [(u (; q ) which leads to: (S C (q )) u q (S C (^q)) u (; ^q) + )] + u q [S C (q )] u^q = (54) 2

23 (u (; q) u (; ^q)) C q + 2 C^q + C q C^q By contradiction assume that ^q > q. In this case the RHS is negative since: u (; q ) u (; ^q) < 0 C q + 2 C^q > 0 C q C^q 0 given the assumptions on rst and second order derivatives of u() and C(). The LHS can be negative only if: (S C (q)) u q < (S C (^q)) 2 + u^q (55) 2 Since: and (S C (q )) > (S C (^q)) (56) > (57) we should have that u q < u^q. Since q < ^q we have that u q > u^q and we have a contradiction. Consider now : As in the previous cases if quality is set equal to zero no one invests in human capital and the demand for education collapse to zero. Consequently we must have q > ^q > 0. Single cross property always holds and we have only separating equilibria. Since q > ^q the location of an individual who is indi erent between studying in university one and three is ^z A < 1=2 and the same holds for ^z A2 : As a consequence, individuals located in the interval (^z A ; 1=2) and (^z A2 ; 1=2) choose to study in the university that is located at the maximum distance. In simultaneous move game between rms, the presence of travel costs makes some individuals strictly prefer working for one of the two rms even when wages are the same. Only individuals located at the same distance from the two rms (v = J ) earn a wage equal to their productivity since 4 they are indi erent about the direction of moving. Q:E:D: 21

24 Proof of (1) In order to prove (1) we derive the = C q q uq u^q + u q u q C q + u q q (S C (q )) C q u q = C^q ^q + 2 u^q u q u^q h u^q C^q + u^q ^q Ignoring high order derivatives we may show that: i h i h i (S C (^q)) C^q u^q (59) This implies that: u q u^q h i < 2 u q C q u^q h 2 + i C^q (61) Proof of (2) In order to prove (2) we We derive the following expressions: ^q = C^q ^q + 2 u^q u q u^q h u^q C^q + u^q ^q i h i h i (S C (^q)) C^q u^q = u^q (S C (^q)) h 22 C^q ^q + When: 2 u^q u q u^q C^q + u^q ^q i h i (S C (^q)) C^q u^q i.e. 2 + (S C (^q)) > 0

25 we have: Q:E:D: (S C (^q)) < 2 (62) (6) 2

26 Figure 1: The E ect af a Decrease of Education Quality 24

27 Figure 2: High and Low Ability Individuals Productivity Conditional on the Firm Technological Choice 25

28 Figure : The university s maximization problem 26

29 Figure 4: The Individuals -Universities-Firms Bayesian Game 27

30 Figure 5: A Separating Equilibrium in the Signaling odel 28

31 Figure 6: The Circular City with Firms and Universities 29

32 Figure 7: The Individuals Utility aximzzation when q=0 0

33 Figure 8: Increase in subsidy versus reduction in moving costs 1