The existence of non-elite private schools

Transcription

1 Journal of Public Economic 90 (2006) The exitence of non-elite private chool Francico Martínez-Mora Department of Economic, Univerity of Leiceter, Univerity Road, LE1 7RH, Leiceter, United Kingdom Received 30 November 2003; received in revied form 29 June 2005; accepted 5 December 2005 Available online 27 April 2006 Abtract We provide an explanation to the puzzle of the exitence of paid-for private chool that offer lower quality education than ome tuition-free public alternative. We conider a model of a city compoed of two communitie: the urban area and the uburb. The uburban public chool provide higher quality education at an implicit price: the higher tax burden plu a houing rent premium. If that price i high enough and the urban public chool ha a ufficiently low quality, intermediate income houehold live in the urban area and ue a private chool. Intermediate quality private chool, then, exit to erve thee houehold' demand. Lower and higher income houehold ue different quality public chool. Therefore, perfect income tratification acro public and private education doe not characterize thi equilibrium Elevier B.V. All right reerved. JEL claification: H42; H73; I20; R13; R31 Keyword: Community choice; School choice; Stratification; Hierarchy of chool qualitie 1. Introduction Private chool do not alway provide higher quality education than public chool. Both caual obervation and empirical evidence (e.g. Figlio and Stone, 1999) confirm it. Thi tylized fact i, however, puzzling a it clahe with conventional widom, baic price theory and mot of the previou theoretical literature on competition between public and private chool (e.g. Epple and Romano, 1998). The puzzle i: why would any houehold pay for private chooling when higher quality tuition-free public education i available? The literature on the economic of education offer ome poible explanation. Figlio and Stone (1999) informally argue that private chool eem to offer ome non-academic advantage over public chool (e.g. more dicipline and ecurity, more opportunitie for extracurricular Tel.: addre: fmm14@le.ac.uk. URL: /$ - ee front matter 2006 Elevier B.V. All right reerved. doi: /j.jpubeco

2 1506 F. Martínez-Mora / Journal of Public Economic 90 (2006) activitie or, for ome parent, a religiou education). Along thee line, Brunello and Rocco (2005) demontrate theoretically that paid-for private chool of lower academic quality than the public alternative could uccefully enter the market by etting a lower educational tandard and thereby offering their tudent more leiure. In thi paper we provide an alternative explanation which doe not reort to non-academic difference between public and private chool. Intead, it i baed on the interaction between local public chooling, taxation and houing market. We build a general equilibrium model with two communitie which can, for the ake of fixing idea, be called the urban area and the uburb in which local public chool coexit with competitive private chool. We ue a computational verion of the model to contruct example where multiple equilibria may exit. Thee equilibria will be of one of two type, which we label urban-trap and urban-mixing equilibrium. In an urban-trap equilibrium, low income houehold live in the urban area and ue the local public chool, while high income houehold live in the uburb and end their children to the higher quality uburban public chool. There alo exit a et of intermediate income houehold who live in the urban area and opt out of public education. Intermediate quality (non-elite) private chool exit to erve their demand for private chooling. Finally, houehold from the top income clae may acquire elite private chooling or ue the uburban public chool intead. An economy may alo end up in an urban-mixing equilibrium, in which, by contrat, only houehold with income above a certain threhold acquire (elite) private chooling. Thoe with income below uch threhold perfectly tratify by income acro the urban area and the uburb and end their children to their repective local public chool. The intuition behind thee reult i the following. Local public chool follow reidencebaed admiion policie. Thi turn houing and public education into complement good, giving rie to a rent premium where public education i better the uburb ay. A a conequence, an implicit price for attending the tuition-free public chool in the uburb emerge a the um of the rent premium and the higher level of taxation neceary for funding higher quality public chooling. In an urban-trap equilibrium, the public chool in the urban area offer education of too low a quality. At the ame time, the very wealthy live in the uburb puhing the quality of the local public chool and it implicit price up. Intermediate income houehold, thu, prefer to live in the cheaper urban area, but becaue they value education, they acquire intermediate quality private chooling rather than ending their children in the low quality public chool. The cae where an urban-trap equilibrium arie contitute counterexample to a rather robut reult in the previou literature. The model in Stiglitz (1974), Epple and Romano (1996) and De Fraja (2001) predict that, if houehold differ along a ingle dimenion, either income or the ability of the tudent, they hould perfectly tratify acro the public and private educational ector along the relevant dimenion, with the better-off uing higher quality private chool. Thi property doe not hold in urban-trap equilibria: while intermediate income houehold opt for private chool, lower and higher income houehold prefer to ue a public alternative. The tratification pattern in thi type of equilibrium characterize ituation that are common to many US metropolitan area. While the central city provide low quality public chooling, the uburb are very expenive place in which to live. In that context, it i eay to find relatively low income houehold that live in the inner city and ue a non-elite private chool. That i, a we argue in the paper, the empirical evidence in Neal (1997), Figlio and Stone (1999) and Altonji et al. (2005) accord well with the prediction of urban-trap equilibria. The ret of the paper i organized a follow: Section 2 preent a two-community model which draw primarily on Epple and Romano (1996, 2002) and Nechyba (1999). Section 3 define urban-

3 F. Martínez-Mora / Journal of Public Economic 90 (2006) trap and urban-mixing equilibria. In Section 4 we provide the computational example of equilibrium and a dicuion of the reult. Some concluding remark are in Section The model We tudy a tatic, general equilibrium model of community and chool choice. The model repreent a city made up of two communitie C j, j=u,, which for expoition purpoe we label the urban area and the uburb, repectively. The city i inhabited by a continuum of houehold with unit meaure. All houehold conit of a head and a chool-aged child; they have identical preference defined over the quality of education the child receive (x) and conumption numeraire (b). With an eye toward our computational example of equilibrium, we adopt the trictly concave utility function, ued by Beare et al. (2001): uðb; xþ ¼ 1 1 r ½b1 r þ dx 1 r Š; r; dn0: ð1þ Each houehold belong to one of IN2 income clae indexed by i=1,,i, with y i 1 b y i, i=2,,i. The fraction of the population which belong to income cla i i denoted λ i. Communitie have a fixed tock of homogenou houe; each houehold rent out one unit of houing in the community in which they live at rent p j h. 1 We model education a a (publicly provided) private good. Public and private chool ue the ame technology to produce educational ervice from the numeraire. Quality unit are normalized uch that one unit cot one unit of conumption per tudent. 2 Each community impoe a proportional property tax on the value of houing and ue the proceed to provide public education of quality E. The pair (E, t), where t tand for the tax rate, i choen through a political proce, which we implify to majority voting. There are no tranfer from a central government: therefore, in community j, the local government' budget contraint i: E j ¼ t j p j N j h ; j ¼ u; : ð2þ n j where N j i the ma of houehold living in the community and n j the ma of thoe uing the local public chool; the remaining houehold ue private chool. Public chool follow a reidence-baed admiion policy: a houehold i required to live and pay taxe in a community in order to ue it local public chool. Houehold may opt out of the public ector and acquire private education of any level of quality in a competitive market. A in mot model of education, we conider public and private alternative a being mutually excluive: a child cannot receive public and private education imultaneouly. Deciion are made in two tage. In the firt tage, deciion-maker imultaneouly chooe among communitie and chool, taking into account their expectation over the equilibrium vector of public policie and houing price e =(E u, t u, p u h, E, t, p h ). In thi tage, local houing market clear. In the econd one, with reidence and chooling deciion committed, head vote on their community education policy. 3 1 A i tandard in the literature, we aume that all rent go to an abentee landlord to cloe the model. 2 For implicity ake, thi technology aume away the influence of peer group effect and other input uch a tudent effort. 3 Thi equence of deciion i alo found in Nechyba (1999). An alternative timing would allow houehold to chooe chool after voting (Epple and Romano, 1996, Beare et al., 2001). However, thi may reult in exitence problem, a preference are not ingle-peaked in that cae (Stiglitz, 1974).

4 1508 F. Martínez-Mora / Journal of Public Economic 90 (2006) In thi economy, an equilibrium i a partition of houehold acro communitie and chool, an allocation (b, x) acro houehold and a vector of community policie and houing price e atifying: (i) Rational choice: every houehold maximize utility; (ii) houing market equilibrium: houing demand equal houing upply in every community; (iii) majority voting equilibrium: forj=u,,the pair (E j, t j ) atifie the local government budget contraint and it i majority-preferred by voter in community j, given the partition of houehold acro chool and the price of houing in the community. A pair (E j, t j ) i majority-preferred in community j if it i preferred by at leat half the electorate of that community in a pair-wie contet with any other bundle atifying the local government budget contraint. The notion of tratification we adopt are the following: an equilibrium exhibit perfect income tratification acro public and private education if every houehold uing a private chool ha weakly higher income than all houehold ending their children to a public chool; an equilibrium how perfect income tratification acro communitie if every houehold living in one community ha weakly higher income than all thoe living in the other community. We will alo ay that a community i mixed when ome of it inhabitant ue the local public chool while other opt for a private alternative. Thi kind of model ha been extenively analyzed in previou literature. Several reult are relevant for the analyi below and extend readily to our etting 4 :(i)within a mixed community any equilibrium exhibit perfect income tratification acro public and private education (Epple and Romano, 1996); (ii) if communitie differ in the quality of public chooling in equilibrium, the community with a better public chool ha a higher gro-of-tax price of houing 5 ; (iii) houehold that ue public chooling perfectly tratify by income acro communitie, with higher income houehold living in the community that provide higher quality public education 6 ; (iv) houehold that acquire private education live in the community with lower gro-of-tax price of houing 7 ; (v) a majority voting equilibrium exit in every community Urban-trap v. urban-mixing equilibrium In the computational example of equilibrium below both communitie provide public education of different quality and private chool emerge. We aume the urban area offer lower quality public chooling, which implie that the gro-of-tax price of houing i lower there. In uch cae, an equilibrium will be of one of two type, which we label urban-trap and urban-mixing equilibria. Definition 1. An urban-trap equilibrium i an equilibrium in which: (i) low income houehold live in the urban area and ue the low quality local public chool; (ii) a et of intermediate income houehold live in the urban area and acquire intermediate quality private education; (iii) (a et of) 4 The Working Paper verion of thi paper (Martínez-Mora, 2003) formally prove that all thee reult hold in our model. There, we ue a general trictly quai-concave and non-eparable utility function. 5 Thi i due to the reidential requirement of public chool, which make houing and public chooling complement good (e.g. Epple and Romano, 2002). 6 Thi reult hold becaue the increae in the willingne to pay for a houe in repone to an increae in public education quality rie monotonically with income (e.g. Bénabou, 1996). 7 Houing and chooling are not complement for thee houehold, a there i no link between community and private chool choice. Thu, they chooe to live where their net income (income after paying for the taxe and the houe) i maximum (Beare et al., 2001; Nechyba, 1999). 8 Under the equence of deciion in our model, preference over the tax rate are ingle-peaked at the voting tage. The median voter theorem (Black, 1948), thu, applie. Note that the voting equilibrium in a mixed community will exhibit the end againt the middle property invetigated in Epple and Romano (1996). Conequently, the median voter will have lower income than the median income houehold in the community.

5 School quality F. Martínez-Mora / Journal of Public Economic 90 (2006) Low and middle income Interm. inc. High income Very high inc. Urban area Urban area Suburb Urban area Public chooling Priv. chool. Public chooling Priv. chool. Income Fig. 1. Urban-trap equilibrium. houehold with higher income live in the uburb and end their children to the high quality local public chool; and (iv) houehold from the top income clae may either live in the urban area and acquire elite private chooling or live in the uburb and ue the local public chool. 9 Definition 2. An urban-mixing equilibrium i an equilibrium in which: (i) low income houehold live in the urban area and end their children to the local public chool; (ii) intermediate income houehold live in the uburb and ue the higher quality public chool; (iii) rich houehold live in the urban area and acquire elite private chooling. Fig.1,2 and3 illutrate the allocation of houehold acro communitie and chool in both type of equilibrium. Taking community and chool choice into account, they plot the quality of chooling received by the tudent againt their houehold' income level. Urban-trap equilibria contain the main contribution of the paper. Our example of thi type of equilibrium erve to prove two novel reult on competition between public and private chool: in a two-community equilibrium, (i) the allocation of houehold acro public and private education doe not necearily exhibit perfect income tratification; and (ii) non-elite private chool (i.e. private 9 Equilibria that have ome imilarity have been found by Bénabou (1993), Rangaza (1995) and Epple and Romano (2002). Epple and Romano analyze a ingle-juridiction model with multiple neighbourhood, each with one public chool. Private chool are not allowed. Quality differential among chool emerge due to difference in the quality of the peer group. Under an open-enrolment policy, if a poitive tranport cot exit, perfect income tratification arie acro neighbourhood. However, a et of intermediate income houehold that live in the poor neighbourhood are willing to pay the tranport cot in order to receive higher quality education. Rangaza provide an example of a complete tratification equilibrium in which exogenou difference among communitie lead to perfect income tratification acro them. In thi equilibrium, the better-off houehold in each community opt for private chooling. In Bénabou (1993), the poor community may not be able to upply education of enough quality. In that cae, the whole population may chooe not to exert any educational effort and join the informal economy.

6 School quality 1510 F. Martínez-Mora / Journal of Public Economic 90 (2006) Low and middle income Interm. inc. High income Urban area Urban area Suburb Public chooling Priv. chool. Public chooling Income chool that offer lower quality education than the high quality public alternative) may exit (Fig. 1 and 2). Note alo that urban-trap equilibria will exhibit perfect income tratification acro communitie if houehold from the top income clae live in the uburb and ue the local public chool. In that cae, private chooling exit but doe not induce any income mixing within communitie (Fig. 2). Urban-mixing equilibria, in turn, exhibit perfect income tratification acro public and private education. That i to ay, only the rich opt for (elite) private chooling in thi type of equilibrium. They all live in the urban area, where the gro-of-tax price of houing i lower. There they mix with low income houehold who end their children to the local public chool intead (Fig. 3). The propertie of what we call urban-mixing equilibria have already been analyzed in previou literature (Beare et al., 2001). 4. Computational example and dicuion Fig. 2. Urban-trap equilibrium. We now preent and dicu the computational example of equilibrium. 10 To better highlight the intuition and implication of the analyi, we chooe two economie for which both type of equilibrium exit. 11 Let u decribe what thee economie look like: the firt three row in Table 1 contain the value of the parameter. Our income ditribution are dicretized verion of truncated lognormal ditribution. They have twenty income clae ranging from y min =5 to y max = 100 in contant interval. The proportion of houehold in the i-th income cla i given by 12 : λ i =Φ 10 Calculation that prove the example are indeed equilibria of our model are available at fmm14.html. 11 Note of coure that multiple equilibria do not exit for every economy. 12 F i the cumulative denity function of the lognormal ditribution, Φ i a normalizing contant that aure P i k i ¼ 1.

7 School quality F. Martínez-Mora / Journal of Public Economic 90 (2006) Low and middle income Interm. inc. High income Urban area Suburb Urban area Public chooling Public chooling Private chool. Income Fig. 3. Urban-mixing equilibrium. (F (y i ) F (y i 1 )), i 1 withλ 1 =Φ F (y 1 ). The utility function i given by Eq. (1), with σ and δ et at the value hown in the table. In our example, the urban area i large relative to the uburb: H u i equal to 0.83 in example 1 and to 0.91 in example 2. Finally, the cot of contruction of a houe i et at c h =2.5. Thi aure that every houehold in the economy can at leat afford to pay the gro-of-tax price of a houe in C u. The remaining of Table 1 preent the value of key endogenou variable. For each equilibrium, it report the quality of public and private chool, the median voter income, the equilibrium tax rate, the implicit price of the uburban public chool (τ) and the percentage of tudent attending private chool. Moreover, y 1, y 2 and y 3 characterize the allocation of houehold acro communitie and chool: houehold with income y [y min, y 1 ] live in the urban area and ue the urban public chool; houehold with income y (y 1, y 2 ] live in the urban area but end their children to a private chool; houehold with income y [y 2, y 3 ] live in the uburb and ue the uburban public chool; finally, houehold with income y (y 3, y max ] live in the urban area and ue an elite private chool. 13 The poibility that intermediate income houehold chooe non-elite private chooling i due to the interaction among local public chooling, taxation and houing market and to the convexity of preference. Local proviion of public education give rie to difference acro communitie with repect to the level of taxation and expenditure in education, and thu in public chool quality. Moreover, the reidence-baed admiion policy of local public chool make 13 Note that in urban-mixing equilibrium, y 1 =y 2 and no intermediate income houehold opt for private chooling. Note alo that if y 3 =y max in urban-trap equilibrium, houehold from the top income cla do not opt for private chooling and the equilibrium i characterized by perfect income tratification acro communitie.

8 1512 F. Martínez-Mora / Journal of Public Economic 90 (2006) Table 1 Computational example of equilibrium Parameter Example 1 Example 2 Income ditribution y mean =40.91; y median =35 y mean =41.72; y median =35 y min =5; y max =100 y min =5; y max =100 Utility function σ=1.83; δ= σ=1.71; δ= Houing market c h =2.50; H u =0.83; H =0.17 c h =2.50; H u =0.91; H =0.09 Urban-trap (1a) Urban-mixing (1b) Urban-trap (2a) Urban-mixing (2b) Public policie E u E t u t Median voter income, urban area Median voter income, uburb Private education Minimum private chool quality Maximum private chool quality Student, non-elite priv. chool Student, private chool Houing price u p h p h p u =p u h (1+t u ) p =p h (1+t ) τ=p p u Border income y y y Definition of border income: y 1, highet income of houehold living in the urban area and uing the local public chool; y 2, lowet income of houehold living in the uburb and uing the local public chool; y 3, highet of houehold living in the uburb and uing the local public chool. houing and public education to be complement good, which in equilibrium reult in a rent premium where public chool quality i higher. 14 Therefore, the higher quality (and tuition-free) uburban public chool ha an implicit price equal to the um of the rent premium and the higher tax levied on that community' reident: τ =p h (1+t ) p h u (1+t u ). Becaue preference are convex, that implicit price may make ome intermediate income houehold find the chool quality-conumption numeraire bundle aociated with the public alternative unatifactory. Thi i hown in Fig. 4, which illutrate the optimal choice of houehold with income y = 45, 55, 65 in the urban-trap equilibrium of example 1. The alternative 14 For thi to happen houing upply cannot be perfectly elatic. We aume a fixed houing upply for implicity. However, uch rent premium alo arie in equilibrium in multi-community model in which the upply of houing i elatic (e.g. Epple et al., 1984). Note alo that the exitence of houing market i not a conditio ine qua non for urbantrap equilibria to exit. It i eay to find thi kind of equilibrium in a verion of the model with head taxation and no houing market.

9 F. Martínez-Mora / Journal of Public Economic 90 (2006) Example 1 - urban-trap equilibrium Optimal choice y = Conumption numeraire y = 55 y = School quality Indifference curve Suburb, public chool Urban area, public chool Budget contraint Urban area, private chool Fig. 4. Example 1 urban-trap equilibrium. Optimal choice. Indifference curve; Budget contraint; Suburb, public chool; Urban area, public chool; Urban area, private chool. thee houehold face are: (i) to live in the urban area and ue the local public chool, the triangular point, (ii) to live in the urban area and acquire their mot deired level of private education, the circle, and (iii) to live in the uburb and end the child to that local public chool, the quare. 15 A the figure how, houehold with income y = 55 maximize utility by chooing the econd alternative. Thi choice provide them with a preferable combination of chool quality and conumption numeraire. Fig. 4 alo how the optimal choice of houehold with income y = 45, who prefer to live in the urban area and ue the local public chool, and of houehold with income y=65, who in turn chooe to live in the uburb and end their children in the uburban public chool. It i remarkable that, a our example reveal, both type of equilibrium may exit for the ame economy. The emergence of one or the other hinge, on the one hand, on the level of political upport for public chooling in the urban area, and on the other hand, on the reult of the competition for living in the uburb which in turn depend on the reult of the local political proce and on how the willingne to pay for a houe there varie with income. Let u further expand on thi: let p h (y, E u, t u, p h u, E, t ) be the maximum rent a houehold with income y i willing to pay for a houe in the uburb given E u, t u, p h u, E and t. Thi can be written a: p h ðdþ ¼y ½ðy p uþ 1 r þ deu 1 r de 1 r Š 1 1 r 1 þ t if y a½y min ; y r u Š; p h ðdþ ¼y ½ðy p u x u ðy; p u ÞÞ 1 r þ dx u ðy; p u Þ 1 r de 1 r Š 1 1 r 1 þ t if y a½y r u ; y maxš: ð3þ 15 Keep in mind that all houehold opting-out of the public ector trictly prefer to live in the urban area.

10 1514 F. Martínez-Mora / Journal of Public Economic 90 (2006) A Bid p h p h p > p h h B Income p h 3.2 Bid p h p > p h h Income Fig. 5. a: Bid function. Example 2 urban-trap equilibrium. b: Bid function Example 2 urban-mixing equilibrium. where p j =p h j (1+t j ) i the gro-of-tax price of houing in C j, x j (y, p j ) i the private education demand function and y u r i the level of income that make a houehold living in the urban area indifferent between uing the local public chool and ending the child to their mot preferred private alternative. 16 Houehold with income y uch that p h (y, ) p h live in the uburb in equilibrium. Next, let u define ŷ a the level of income at which a houehold demand for private education, were they living in C u, would jut equal E, i.e. x u (y, p u )=E. Lemma 1 analyze the behavior of p h a a function of income (Fig. 5). Lemma 1. For E u be,p h ( ) i quaiconcave in y and the global maximum i at y=y if y u r by, and at y=y u r if y u r y. Proof. For given E u, t u, p h u, E and t, let u denote the firt and econd piece of p h ( ) a ρ a (y) and ρ b (y) repectively. Note firt that for E u b E, the maximum gro-of-tax rent a houehold with 16 From Epple and Romano (1996), we know that all houehold with income above thi threhold ( y u r ) trictly prefer to opt out over uing the public chool in C u and vice vera. Taking thi into account, the bid function i eaily deduced from the equality between u( y p u, E u )=u(y p h ( ) (1+t ), E ) in the income interval in which yby u r and from the equality among u(y p u x u ( y, p u ), x u ( y, p u ))=u ( y p h ( ) (1+t ), E ) for y y u r. It i continuou, a by definition u (y p u x u (y, p u ), x u (y, p u ))=u ( y p u, E u )aty=y u r, and differentiable except at y=y u r.

11 F. Martínez-Mora / Journal of Public Economic 90 (2006) income below y u r i willing to pay for a houe in C, ρ a (y) (1+t ), i alway larger than p u. Thi make ρ a to be increaing in income in all it upport: qv a ðyþ ¼ ½y q aðyþð1 þ t ÞŠ r ðy p u Þ r ½y q a ðyþð1 þ t ÞŠ r ð1 þ t Þ N0 ð4þ The econd piece of p h ( ), ρ b (y), ha a tationary point at ẙ uch that ẙ ρ b (ẙ)(1+t )= ẙ p u x u (ẙ, p u ), a in that cae: qv b ðy Þ¼ ½y q b ðy Þð1 þ t ÞŠ r ½y p u x u ðy ; p u ÞŠ r ½y q b ðy Þð1 þ t ÞŠ r ð1 þ t Þ After ome manipulation, the econd derivative of ρ b at ẙ can be written a: h i r Ax uðy ;p u Þ Ay qwðy Þ¼ ½y q b ðy Þð1 þ t ÞŠð1 þ t Þ b0 ¼ 0 ð5þ ð6þ which i alway negative becaue the demand for education i normal. Thi implie the exitence of a unique tationary point and that it i a global maximum. Now conider y. By definition x u (y, p u )= E, hence ρ b (y ) mut be uch that the level of numeraire conumption and thu utility of houehold with income equal to ŷ i equalized acro the relevant alternative, i.e. uch that: y ρ b (y )(1+t )=y p u x u (y, p u ). Thi i the condition for ρ b (y) to have a tationary point at y. Therefore, a by definition ρ a (y r u )=ρ b (y r u ), p h ( ) i quaiconcave in income and it ha a ingle maximum at y=y if y r u by, and at y=y r u if y r u y. By Lemma 1, the willingne to pay for a houe in the uburb increae with income in the range within which houehold prefer the urban public chool over the private ector. Once income i above the threhold y r u, however, a houehold' bid i higher the cloer their demand for chool quality i to the quality of the uburban public chool. Therefore, it increae with income for all y uch that x u (y, p u )be, decreae with income for all y uch that x u (y, p u )NE, and i maximum at y. A we oberve in Table 1, inanurban-trap equilibrium, the level of quality of the uburban public chool i very high. Thi induce wealthy houehold, whoe demand for chool quality i cloe to that level, to outbid lower income houehold with a lower demand for chool quality from the uburb (Fig. 5a). At the ame time, the urban public chool offer education of too low a quality. Conequently, ome middle income houehold who are trapped in the urban area opt out of the public ector and acquire private education of intermediate quality. The reulting allocation of houehold acro communitie and educational ector utain the level of quality in both public chool a a voting equilibrium. In the urban area a large hare of the population opt for a private chool. Therefore, the political proce reult in a low level of expenditure per tudent and in low quality public chooling. The rich population of the uburb, in turn, upport a high level of expenditure per tudent and thereby high quality public chooling. In an urban-mixing equilibrium, in turn, the wealthy are not willing to pay a much a ome lower income houehold for a houe in the uburb (Fig. 5b). Thi i becaue they demand higher quality education than that offered by the uburban public chool. Thu, they live in the urban area and acquire higher quality private chooling. Intermediate income houehold, by contrat, do not opt out of public education. On the one hand, ome of them are willing to pay the price for living in the uburb in order to ue the local public chool there. On the other hand, the quality of the urban public chool i not low enough to induce thoe of them living in the urban area to opt out.

12 1516 F. Martínez-Mora / Journal of Public Economic 90 (2006) Some further remark are worth making. Firtly, the empirical evidence on the poitive effect of (Catholic) private chool on urban minority tudent (e.g. Neal, 1997; Figlio and Stone, 1999; Altonji et al., 2005) i conitent with the tratification pattern that characterize urban-trap equilibria. Thi i not urpriing. Thi kind of equilibrium ketche a picture that i common to many metropolitan area in the US: ome urban tudent attend and, epecially thoe from minoritie, benefit from Catholic private chool becaue (in Neal' word) their local public chool alternative are poor (and) becaue their familie can afford the modet tuition that are common in Catholic chool, but ( )can eldom afford houing in the excluive neighborhood with the bet public chool. Thi wa already oberved by Friedman (1962). 17 Secondly, urban-trap equilibria have clearly undeirable propertie from an equity point of view. The quality of chooling tudent from low income houehold receive i lower. Thi i aociated to a higher level of private chool attendance in the economy, which in turn increae income orting acro individual chool. Furthermore, the argument that private chooling tend to reduce reidential income tratification (e.g. Nechyba, 1999; Beare et al., 2001) i weaker when referred to thi kind of equilibrium: a example 2 illutrate, perfect income tratification acro communitie may characterize an urban-trap equilibrium. Thirdly, the poibility that the economy i in an urban-trap equilibrium hould be conidered in the analyi of voucher policy propoal. Nechyba (1999, 2000) and Luengo-Prado and Volij (2003) have invetigated the conequence of different voucher policie in a multi-community etting. It eem important to extend thi reearch to an economy which i in thi type of equilibrium before the voucher policy i implemented. In thi repect, Rangaza (1995) took a firt tep by providing an example of hi complete tratification equilibrium in which private chool voucher lead to an increae in public chool quality in both the rich and the poor communitie. Finally, our reult hed ome light on how private chool ditribute along the quality pace. In particular, urban-trap equilibria explain the tylized fact that ome private chool are very good while other are not (Mark et al., 2001), and upport the notion that private chool emerge to fill the gap in the menu of public alternative. 5. Concluding remark The market for education in a multi-community economy i complex. According to conventional widom one would expect an equilibrium with public and private chool to be uch that the rich ue private chool and the ret of the population ue public chool. Moreover, all private chool hould be of higher quality, a no one would pay tuition for private education when higher quality free education i available. We how that thi i not necearily the cae in a multicommunity context. An implicit price for high quality public education exit in the form of a higher tax bill and a premium in the price of houing. Conequently, ome middle cla houehold may not be able to afford to live in wealthy communitie. If they are unatified with the quality of public chooling in their community, they will acquire private education of intermediate quality intead. 17 Another branch of empirical reearch focue on the determinant of houehold' choice between public and private chool. One reult of thi literature alo eem to upport the empirical relevance of urban-trap equilibria. Buddin et al. (1998) obtain that Black and Hipanic white are, ceteri paribu, more likely to attend a private chool. A plauible explanation i that, ceteri paribu, a Black or Hipanic white tudent i more likely to live in egregated part of citie where public chool are poor.

13 F. Martínez-Mora / Journal of Public Economic 90 (2006) Thi equilibrium, which we label urban-trap equilibrium, reflect a ituation that i common to many metropolitan area in the US: the central city provide low quality education and the uburb are very excluive. Thi make ome relatively low income houehold, who cannot afford to live in the uburb, acquire non-elite private chooling. A picture like thi eem to be behind the evidence of Neal (1997), Figlio and Stone (1999) and Altonji et al. (2005). Thi branch of empirical reearch could therefore benefit from our reult. The poibility that an urban-trap equilibrium emerge give rie to ome equity concern: central citie public chool and not elite public chool could be thoe mot damaged by the cream-kimming effect of private chooling. Therefore, tudent left-behind in inner-city public chool after a voucher policy could face inurmountable obtacle to ucceed at compulory education. Thi hould be taken into account when deigning and analyzing voucher propoal. Acknowledgement I am deeply indebted to Gianni De Fraja for helpful comment, dicuion and guidance. Thank for their comment are due to Sam Bucovetky, Charle de Bartolome, María Cubel, Elena del Rey, Manuel Martín Rodríguez, Diego Martínez, Fabien Moizeau, Thoma Nechyba, Socorro Puy, Javier Rodero, Daniel Santín, Albert Solé, my PhD advior Rogelio Velaco and epecially Richard Romano and an anonymou referee of thi journal. All remaining error are my own. Financial upport by Junta de Andalucía, Centro de Etudio Andaluce and FUNCAS i gratefully acknowledged. Reference Altonji, J.G., Elder, T.E., Taber, C.R., Selection on oberved and unoberved variable: aeing the effectivene of Catholic chool. Journal of Political Economy 113 (1), Beare, P., Glomm, G., Ravikumar, B., Education Finance in a Dynamic Tiebout Economy. Bénabou, R., Working of a city: location, education and production. Quarterly Journal of Economic 108 (3), Bénabou, R., Equity and efficiency in human capital invetment: the local connection. Review of Economic Studie 63 (2), Black, D., On the rationale of group deciion making. Journal of Political Economy 56, Brunello, G., Rocco, L., Educational tandard in private and public chool. CESIFO Working Paper, vol Buddin, R.J., Corde, J.J., Kirby, S.N., School choice in California: who chooe private chool? Journal of Urban Economic 44, DeFraja, G., Private and public chool: theoretical conideration. Lavoro e Relazioni Indutriali 2/2001, Epple, D., Romano, R., End againt the middle: determining public ervice proviion when there are private alternative. Journal of Public Economic 62, Epple, D., Romano, R., Competition between private and public chool, voucher and peer group effect. American Economic Review 88, Epple, D., Romano, R., Neighbourhood chool, choice and the ditribution of educational benefit. In: Hoxby, C.M. (Ed.), The Economic of School Choice. National Bureau of Economic Reearch, Univerity of Chicago Pre, pp Epple, D., Filimon, R., Romer, T., Equilibrium among local juridiction: toward an integrated treatment of voting and reidential choice. Journal of Public Economic 24, Figlio, D., Stone, J., Are private chool really better? In: Polachek, S.W. (Ed.), Reearch in Labor Economic, vol. 17. Amterdam North-Holland, pp Friedman, M., Capitalim and Freedom. The Univerity of Chicago Pre. Luengo-Prado, M.J., Volij, O., Public education, communitie and voucher. Quarterly Review of Economic and Finance 43, Mark, J., Burn, J., Pilkington, P., Thompon, P., Faith in Education. The Intitute for the Study of Civil Society, London.

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