Publi Shool Choie: An Eonomi Analysis Levon Barseghyan, Damon Clark and Stephen Coate May 25, 2018 Abstrat Publi shool hoie programs give households a hoie of publi shool and enourage shools to ompete for students. Proponents argue that, by the usual market logi, suh programs will improve shool quality. Critis retort that the usual market logi does not apply beause parents shool hoies are driven by preferenes over their hildren s peers. This paper advanes this debate by developing an eonomi model of publi shool hoie. The model reveals how peer preferenes ompliate the ase for hoie. In partiular, it shows that the equity and effiieny impliations of hoie depend ritially on the strength of peer preferenes. We thank John Friedman and four anonymous referees for very useful suggestions. For helpful omments and disussions, we thank Jean-Paul Carvalho, Jon Hamilton, Stergios Skaperdas, Steve Slutsky, David Sappington and various seminar partiipants. Barseghyan: Department of Eonomis, Cornell University, Ithaa NY 14853, lb247@ornell.edu. Clark: Department of Eonomis, UC Irvine, Irvine CA 92617, larkd1@ui.edu. Coate: Department of Eonomis, Cornell University, Ithaa NY 14853, s163@ornell.edu.
1 Introdution Publi shool hoie programs, also known as open enrollment or intra-distrit hoie programs, give households a free hoie of publi shool and provide publi shools with inentives to ompete for students. Supporters of these programs argue that by injeting market fores into the publi shool system, they will improve the quality of eduation that publi shools provide. Their opponents question this market logi. They argue that sine households pereptions of shool quality depend on the omposition of the student body as well as the efforts of shool personnel (i.e., parents have peer preferenes), then these programs might not play out as intended. First, they may not provide shools with strong inentives to exert effort. Seond, they may exaerbate eduational inequality, as more-advantaged households hoose shools that enroll moreadvantaged students. Empirial evidene on the effets of these programs ould shed light on their likely effets, but the evidene base is thin and findings are mixed. This lak of evidene suggests the potential value of theoretial work that aptures and illuminates the relevant fores. This paper provides a theoretial analysis of the effiieny and equity impliations of publi shool hoie programs in an environment in whih parents an have peer preferenes. We allow parents to have peer preferenes beause, as disussed in the next setion, these preferenes are supported by a large empirial literature (in addition to anedotal evidene and introspetion). We explore the impliations for effiieny and equity beause we wish to provide a omplete analysis and beause, with peer preferenes, we expet these to interat. For example, while a shool s inentives to inrease quality might depend on the soioeonomi omposition of the households that exerise hoie, the impats of hoie on eduational inequality will depend on the quality responses of shools. As disussed in the next setion, ours is the first paper to analyze how peer preferenes shape the inentives provided by publi shool hoie. The paper begins with a baseline model that makes strong assumptions but yields a tratable analysis and losed-form solutions. The model features one ommunity divided into two equal-sized neighborhoods, eah ontaining one shool. Households differ by their soio-eonomi status and one neighborhood ontains more advantaged students than the other. Without publi shool hoie, students attend their neighborhood shool. With hoie, households an enroll their hildren in either shool but fae a ost of attending the nonneighborhood shool. We assume that shools must admit any student that wishes to enroll and our baseline model abstrats from apaity onstraints. It follows that households will hoose the non-neighborhood shool if the ost is less than the gain in expeted shool quality. Quality depends on the efforts of shool personnel and on the omposition of the students enrolled, with a parameter that governs the strength of peer preferenes. Efforts of shool personnel are ommitted prior to enrollment deisions and are observed 1
by households. Sine households orretly antiipate the enrollment deisions of other households, they aurately predit shool qualities. Shools obtain utility from the revenue that omes with enrollment and they an inrease enrollment by exerting ostly effort (and thereby inreasing quality). This reates a game between the two shools and we study the equilibrium of this game. Absent peer preferenes, the baseline analysis is simple. Without hoie, shools exert zero effort. With hoie, shools exert positive effort. Sine the two shools are effetively symmetri, they exert the same effort. This means that no households exerise hoie and eduational inequality is unhanged. In short, hoie improves the quality of both shools and the welfare of all households. Peer preferenes hange this analysis. First, the stronger are peer preferenes, the less sensitive are household enrollment deisions to differenes in shool efforts. This weakens shools inentives to exert effort and redues the quality improvements that result from hoie. Seond, when peer preferenes are strong, hoie an derease welfare. Intuitively, while it is privately optimal for peer preferenes to drive enrollment deisions, it is not soially optimal: attending the non-neighborhood shool is ostly and one household s peer gain is another household s peer loss. Welfare dereases when the wasteful effets of peer-driven hoies overwhelm the benefits stemming from inreased shool efforts. Third, when peer preferenes are strong, hoie harms the more affluent neighborhood by dereasing the quality of its shool. This is beause the benefits of greater effort by shool personnel are offset by the osts of an inferior peer group. This baseline model also illustrates the onnetion between peer preferenes and neighborhood inequality. Peer preferenes are irrelevant when neighborhoods are equal but beome more onsequential as neighborhood inequality inreases and the more affluent shool beomes relatively more attrative. With the baseline analysis in hand, the paper onsiders three extensions to the baseline model that inorporate additional features that ould shape the effets of hoie. First, we relax the assumption that households neighborhood loations are exogenous. We do this by adding a housing market and allowing households to hoose neighborhood (i.e., Tiebout hoie). These neighborhood hoies take into aount the impliations for shool attendane. Again, we analyze this model with and without shool hoie. Sine Tiebout hoie gives shools an inentive to ompete for students without shool hoie, it is interesting to onsider whether this extension hanges our basi findings regarding the effets of hoie. Seond, we relax the assumption that shools have infinite apaity. Instead, we assume that enrollment in either shool annot exeed some threshold larger than the neighborhood student population but smaller than the ommunity student population. Third, we relax the assumption that the osts of exerising hoie are independent of soio-eonomi status. Instead, we assume that more affluent households are better able to take advantage of hoie. 2
Our findings largely survive these extensions, with two aveats. First, when peer preferenes are suffiiently strong that the apaity onstraint binds, hoie has no impat on shool quality. Nonetheless, it still inreases the welfare of households in the less affluent neighborhood - a weighted average of the welfare of the leavers and the stayers in this neighborhood. Seond, when more affluent households fae lower osts of exerising hoie, strong peer preferenes an inrease the differene between the soio-eonomi omposition of the two shools (for a given differene in shool efforts). This ontrasts with the baseline model in whih this differene is dereasing in peer preferenes (beause the households that exerise hoie are a representative subset of all households in the less-affluent neighborhood). In turn, this implies that stronger peer preferenes need not derease shool effort under hoie. It also implies that the quality of the shool in the less affluent neighborhood dereases as it loses its more affluent students. In the final part of the paper we disuss the impliations of our findings. From a poliy perspetive, our main finding is that the effets of hoie depend on peer preferenes and neighborhood inequality. If peer preferenes are weak, then our analysis provides support for the position that publi shool hoie is desirable. Even if peer preferenes are strong, our analysis supports this position provided neighborhood inequality is low. If peer preferenes are strong and neighborhoods are unequal, then our analysis onfirms that the ase for hoie is greatly ompliated. But this does not imply that hoie annot be justified in those settings. We provide a more nuaned disussion of that question and onlude that the answer will depend on the preferenes and priorities of the poliy-maker, and the nature of any auxiliary poliies. Sine the effets of hoie are shaped by peer preferenes and neighborhood inequality, we argue that our analysis an help to explain why empirial studies of the effets of hoie yield mixed findings. Furthermore, beause our findings suggest that affluent neighborhoods will hurt by hoie, we argue that our analysis an help to explain why these poliies are not more widespread. The rest of the paper is organized as follows. Setion 2 disusses related literature. Setion 3 analyzes the baseline model. Setions 4-6 analyze our three extensions of this model: in Setion 4 we allow households to hoose in whih neighborhood to live (i.e., Tiebout hoie); in Setion 5 we allow shools to be apaity onstrained; and in Setion 6 we allow for the osts of attending the non-neighborhood shool to be negatively orrelated with soio-eonomi status. Setion 7 identifies and disusses further limitations of the baseline model. Setion 8 onludes by summarizing and disussing our findings. 3
2 Related literature We model a publi shool hoie program in whih households an hoose any shool, shools must aept all appliants, and seats in over-subsribed shools are rationed by distane or lottery. This type of program operates in Israel, New Zealand, the UK, and many shool distrits in the US. 1 Given their poliy signifiane, these programs have attrated surprisingly little theoretial attention. Some researh has foused on their impliations for neighborhood omposition, but has abstrated from their inentive effets. 2 The losest paper to ours in approah is Lee (1997). He also employs a model featuring two neighborhoods with loal shools. As in our model, hoie allows households to enroll their hildren in non-neighborhood shools but it is ostly for them to do so. In his basi model, the quality of shools in eah neighborhood is determined solely by the level of spending that is hosen by neighborhood residents and the fous is on how this spending hanges when hoie is introdued. 3 Lee s paper is omplementary to ours in that he abstrats from peer preferenes by assuming that shool quality does not depend on student harateristis; we ignore the impliations of hoie for the level of shool spending hosen by voters. Empirial analysis of publi shool hoie programs has been foused on three questions. First, when given a hoie, whih shools do households hoose? The evidene reported by Hastings et al. (2009), Burgess et al. (2015) and others is onsistent with our framework: households fae a ost of attending non-neighborhood shools (as refleted in the revealed preferene for proximity), shools an attrat households by exerting effort that improves pereived quality (as refleted in the revealed preferene for shool average test sores) and pereived quality depends on peer omposition (as refleted in the revealed preferene for shools with more-advantaged students). 4 Further evidene of the relative importane of peer omposition is provided by Abdulkadiroglu et al. (2017) and Rothstein (2006). Seond, how does gaining aess to the first-hoie shool impat a hild s outomes? Exploiting the lotteries used to ration seats in the publi shool hoie systems, Cullen et al. (2006) find little impat on aademi ahievement, while Hastings et al. (2009) find positive effets for the households that plae more weight on aademi ahievement when hoosing shools. Our framework is silent on whether the first-hoie shool generates better aademi outomes than the 1 A growing literature onsiders the exat mehanism by whih households are alloated to shools (e.g., whether this indues households to express true preferenes or behave strategially). See Pathak (2011) for a review. 2 For example, Epple and Romano (2003) analyze the equity impliations of publi shool hoie in a rih model in whih parents hoose neighborhoods, hoose shools and vote on taxes to support shools. Shool quality depends on student harateristis, but shools are passive. Avery and Pathak (2015) present a related analysis that examines how the introdution of hoie impats households neighborhood hoies. Their analysis also assumes student omposition determines shool quality but ignores shool effort. 3 An extension allows quality to also depend on shool effort under the assumption that shools are about the number of students they have enrolled. 4 These papers reah different onlusions as to whether preferenes are heterogeneous. Hastings et al. (2009) estimate that relative to lower soio-eonomi status parents, higher soio-eonomi status parents have stronger preferenes for shool average test sores than for proximity. Burgess et al. (2015) find no statistially signifiant differenes between the preferenes for shool average test sores among low and high soio-eonomi status households. 4
seond-hoie shool: we assume that households maximize utility but do not speify what households look for in shools (e.g., test sores, a safe environment, et). Third, how do publi shool hoie programs impat aademi and other student outomes? The evidene here is also mixed. 5 This is onsistent with our analysis, whih implies that the effets of publi shool hoie will be setting-speifi (i.e., shaped by peer preferenes and neighborhood inequality). 6 There are large theoretial literatures devoted to Tiebout hoie and private shool vouhers (see Epple and Nehyba (2004) and Epple and Romano (2012) for reviews). Although the institutional environments are very different, our fous on the impliations of peer preferenes for effiieny as well as equity is related to some of these analyses. 7 Epple and Romano (2008) analyze a onditional private shool vouher sheme that links vouhers to ability. They show that these an preserve the effiieny-enhaning effets of private shool vouhers without the ream-skimming that ours when vouhers are universal. 8 As we disuss in the Conlusion, our model has impliations for some auxiliary poliies suh as subsidizing the transportation osts of poorer students. This type of poliy targeting has also been onsidered in the vouher ontext (see Epple et al. (2017) for details of these poliies and Neilson (2013) for a theoretial analysis). Our model is related to models in the industrial organization literature that deal with swithing osts and network effets (Farrell and Klemperer, 2007). Whereas we onsider the impats of peer preferenes, papers in that literature onsider the impliations of size preferenes (i.e., how do two firms ompete when onsumers hoies depend on eah firm s market share). 9 The idea is that onsumers want to buy (e.g., teleommuniations) produts with larger market shares so that they an, for example, share the same network as friends and olleagues. An obvious and important differene is that there is no prie-setting in our model. Hene while this literature finds that these size-dependent preferenes an generate fat at effets that soften ompetition between firms and inrease prie-ost mark-ups, it does not neessarily follow that peer preferenes will blunt the inentive effets of publi shool hoie. 5 Studying a nationwide hoie program in New Zealand, Ladd and Fiske (2003) find no statistially signifiant orrelations between the prinipal reports of the impat of hoie on learning outomes and prinipal reports of the level of loal ompetition. Lavy (2010) evaluates an Israeli publi shool hoie reform implemented in a single shool distrit in Tel Aviv. He finds that relative to non-reforming ontrol distrits, a distrit that implemented a publi shool hoie program enjoyed signifiant shool produtivity gains as measured by dropout rates, test sores and behavioral outomes. Evidene from the UK is also mixed. Gibbons et al. (2008) exploit aross-region variation in hoie and find that hoie is assoiated with few produtivity gains. Bradley and Taylor (2002) find stronger produtivity gains using a differene-in-differene approah. 6 Interestingly, Ladd (2002) invokes peer preferenes to explain the apparently disappointing effets of publi shool hoie in New Zealand. 7 Models of private shool vouhers share the property that households trade higher quality against higher osts when deiding whih shool to attend, but must onsider how private shools admit students (e.g., whether or not they an turn vouher students away). Models of Tiebout hoie share the property that some households must pay a (housing) ost to attend higher-quality shools, but must onsider how distrits make tax and spending deisions. 8 MMillan (2005) also onsiders how publi shool efforts respond to private shool vouhers. In his model, more generous private shool vouhers an ause publi shool effort to fall. The intuition is that a private shool vouher an make the outside private shool option espeially attrative for higher-inome students, thereby inreasing the effort ost of keeping these students in the publi shool. This model does not onsider peer preferenes. 9 It would be interesting to develop industrial organization models of the impats of peer preferenes, sine these haraterize many industries (e.g., gym memberships, dating agenies). To our knowledge, no suh analyses exist. 5
3 The baseline model The model features a single ommunity with a population of households of size 1. The ommunity is divided into two neighborhoods, A and B, eah ontaining 1/2 of the population. There are two shools serving the ommunity, one in eah neighborhood. The shool in neighborhood J {A, B} is referred to as shool J. Households differ in their soio-eonomi status. There are a ontinuum of types indexed by s. Types are uniformly distributed on [ µ, µ], where µ > 0, so that the average soio-eonomi status is 0. 10 The neighborhoods are stratified and A is the more affluent neighborhood. Thus, households of type [0, µ] live in neighborhood A, while neighborhood B is omprised of types [ µ, 0]. The parameter µ measures the degree of neighborhood inequality. Eah household has a hild whih it must send to one of the two shools. Households are about shool quality (as they pereive it) but inur a utility ost if using the shool not in their neighborhood. This ost aptures the additional transation osts arising from using the non-neighborhood shool and varies aross households. 11 In our baseline speifiation, for all household types, osts are uniformly distributed on the interval [0, ], so that the fration of households with ost less than or equal to [0, ] is /. Later in the paper, we allow osts to be orrelated (negatively) with soio-eonomi status. Letting q J denote the quality of shool J, a household living in neighborhood A with ost obtains a payoff q A from using shool A and a payoff q B from using shool B. Similarly, a household living in neighborhood B with ost obtains a payoff q B from using shool B and a payoff q A from using shool A. The quality of shool J depends on the effort it exerts and on the average soio-eonomi status of its hildren. Thus, q J = e J + αs J, (1) where e J is shool J s effort, s J is the average soio-eonomi status of its students, and α is a parameter measuring the importane of peer omposition. 12 Turning to shools, we assume that the ommunity provides shools with a per-student payment that exeeds the osts that an additional student reates. We normalize the per student surplus to one, so that shool J s payoff is given by E J γ e2 J 2, (2) 10 The average soio-eonomi status being 0 is just a normalization and involves no loss of generality. 11 These inlude any additional time taken to travel to shool, additional expenses arising from higher transport osts, psyhi osts resulting from loss of ommunity, et. 12 Peer omposition will be important if parents pereive there to be peer effets in the outomes that they are about. Some of these outomes (e.g., aspets of hildren s behavior) may be unrelated to objetively measurable shool quality. This is why it is important to distinguish between shool quality as households experiene it (whih is q J ) and objetive measures of shool quality (suh as test sores) that are used in the empirial literature. 6
where E J denotes enrollment in shool J and γ is a parameter measuring the marginal ost of effort. The timing of the interation between shools and households is as follows. First, the two shools simultaneously ommit to their effort levels e A and e B. Seond, knowing shool effort levels, households simultaneously hoose in whih shool to enroll their hildren. In making this deision, they are assumed to orretly antiipate the enrollment deisions of other households. 13 For now, we ignore apaity onstraints, assuming that both shools an aommodate all students who hoose to enroll. Capaity onstraints will be introdued in a later setion. 3.1 Shool hoie We are interested in how shool hoie impats shool quality and household welfare. Our benhmark for omparison is a no-hoie poliy under whih households must enroll their hildren in their neighborhood shool. Under this assumption, eah shool s enrollment onsists of the students in its neighborhood and thus is fixed at 1/2. Sine enrollment is fixed and effort is ostly, shools exert zero effort. Eah shool s quality is therefore determined by the average soio-eonomi status of its students (see (1)). Thus, shool A s quality is αµ/2 and shool B s is αµ/2. It follows that a household living in neighborhood A obtains a payoff αµ/2 from enrolling their hild in shool A and a household living in neighborhood B obtains a payoff αµ/2 from enrolling their hild in shool B. To understand what happens under hoie, we work bakwards, first analyzing the seond stage when households simultaneously hoose where to enroll their hildren, knowing shool effort levels e A and e B. Assume that households antiipate that the quality of shool A will be higher than that of shool B and let the antiipated quality differential be denoted q (i.e., q = q A q B ). Then, all households in neighborhood A will use shool A and households in neighborhood B will use shool A if their osts are less than q and shool B otherwise. Assuming that q is less than or equal to, it follows that the average soio-eonomi status of those enrolling in shool A is while that of shool B s students is s A = µ 2 [ ] 1 q 1 + q, (3) s B = µ 2. (4) 13 This formalization of the relationship between shool efforts and household enrollment deisions is admittedly highly stylized. In reality, this is a dynami proess in whih shools understand that working harder with urrent students may bring enrollment gains in the future. Capturing this requires a dynami model in whih prospetive parents observe today s shool efforts and shool ompositions and these observations guide tomorrow s enrollment deisions. In an earlier version of this paper (Barseghyan et al., 2014), we present a dynami model that aptures this proess. The drawbak with this model is that it is signifiantly more ompliated and less amenable to extensions. Both dynami and stati models generate similar onlusions regarding the impat of publi shool hoie. 7
Using (1), this means that, if households orretly antiipate other households deisions, q must satisfy the equation q = e + αµ 1 + q, (5) where e is the effort differential (i.e., e = e A e B ). This is a quadrati equation with solution q( e) = ( + e) 2 + 4αµ + e. 2 (6) Equation (6) gives us a losed form solution for the equilibrium quality differential. The solution will lie in the interval [0, ] if e + αµ is non-negative and if e + αµ/2 is less than or equal to. Given this, with effort levels e A and e B, the two shools will antiipate enrollments of E A ( e) = 1 2 [ 1 + q( e) ], (7) and E B ( e) = 1 2 [ 1 q( e) ]. (8) Aordingly, if we define an equilibrium to be a pair of effort levels (e A, e B ) suh that eah shool J is maximizing its payoff (2) given its rival s effort level, the equilibrium effort levels will be idential and given by e A = e B = 1 γ [ 1 2 q ] (0). (9) This ondition just reflets the requirement that, for eah shool, the marginal benefit of an inrease in effort must equal the marginal ost. The marginal benefit is the resulting inrease in enrollment. Computing the derivative q (0) from (6), we find that the equilibrium effort level is e S (effort under shool hoie) whih is defined to equal [ ] e S 1 1 γ 4 2 + 4αµ + 1. (10) 4 Notie immediately that the parameters measuring strength of peer preferenes (α) and the extent of neighborhood inequality (µ) enter into this expression multipliatively (i.e., as αµ). Moreover, an inrease in αµ redues equilibrium effort beause it redues the responsiveness of enrollment to effort. Intuitively, when either peer preferenes are strong or neighborhood inequality is high, households enrollment deisions are more driven by peer group onerns than shool efforts. Given (10), the equilibrium qualities of the two 8
shools under shool hoie (qa, q B ) will be given by q A = e S + αµ 2 [ ] 1 q(0), (11) 1 + q(0) and q B = e S αµ 2. (12) This analysis suggests what equilibrium under shool hoie must look like. However, it stops short of proving that both shools hoosing effort level e S is an equilibrium. For this, we have to hek that, for eah shool, e S is a genuine best response to the other shool hoosing e S. There are two tehnial issues to worry about. First, for eah shool, is e S a global maximum in the set of effort levels that give rise to a shool quality differential desribed by (6)? All the analysis so far establishes, is that for both shools e S satisfies the first order neessary ondition for maximizing eah shool s payoff given the other shool is hoosing e S and the effort differential e is suh as to generate a shool quality differential given by (6). Seond, for eah shool, does e S dominate effort levels that would generate an effort differential giving rise to a negative shool quality differential whih would not be desribed by (6)? When the two shools hoose effort level e S, the quality differential will be positive beause of shool A s natural advantage stemming from its loation in the more affluent neighborhood. However, if, for example, shool B dramatially inreased its effort level above e S it ould make the quality differential negative. With suh an effort hoie, the flow of students between shools would be reversed: households from the more affluent neighborhood A would be enrolling their hildren in shool B. Different expressions for the quality differential and the enrollments for the two shools would apply. In the on-line Appendix, we provide a omprehensive disussion of both issues. We learly identify the types of deviations that might threaten the existene of equilibrium. We show that for any given values of the parameters α, µ, and γ, the effort levels desribed in equation (10) are indeed equilibrium effort levels if the upper bound of the ost distribution exeeds a ritial level. This ritial level is given by { } αµ max αµ, 3 γ αµ, 1 1 2γαµ, γ + (αµ)2. (13) We find the assumption that is relatively large natural, sine there will likely exist households that would not exerise hoie under almost any irumstanes. Nonetheless, readers who would prefer not to make suh restritions, should be reassured that in parameter ranges not satisfying this suffiient ondition, we did not find any examples in whih the effort levels desribed in equation (10) were not equilibrium effort levels. We 9
will therefore heneforth assume that the shool qualities under hoie will be (11) and (12). 3.2 The impat of shool hoie We are now ready to study the impat of introduing shool hoie on shool quality and household welfare. The poliy debate and the empirial literature have foused on the quality impats of hoie, both in the aggregate and aross shools and ommunities. The fous on household welfare is more in the spirit of traditional publi eonomis. 14 Measures of household welfare inlude the additional osts that households inur when they hoose their non-neighborhood shool, whih we view as a legitimate part of the soial alulus. We begin by defining the preise quantities of interest. Reall that without hoie, shool A s quality is αµ/2 and shool B s is αµ/2. Using (11) and (12), the hanges in the two shools qualities are and dq A = e S αµ ( q(0) ), (14) 1 + q(0) dq B = e S. (15) The enrollment-weighted average hange in shool quality, whih we denote by dq, is dq = E A (0)q A + E B (0)q B = e S. (16) The expression in (16) is so simple beause hanges in shool qualities resulting from student omposition are zero sum and hene wash out of the analysis. Turning to welfare, three variables are of interest: the average hange in welfare of households in the two neighborhoods, whih we denote dw A and dw B, and the average hange in welfare, whih we denote by dw. The welfare hange from hoie for households in neighborhood A is just dw A = dq A. (17) This reflets the fat that, sine q is positive, all households in neighborhood A ontinue to send their hildren to shool A and hene the only impat on their welfare is how the quality of their shool hanges. 14 This said, we do not onsider shool payoffs in our welfare measure. This is beause we see the poliy problem to whih hoie is one possible answer as improving shool performane for given levels of eduational spending. Eliiting more effort from shool personnel is not onsidered a soial loss. In addition, our model of shools payoffs is too redued form to permit a satisfatory aounting of the surplus aruing to shool personnel and stakeholders. 10
Households in neighborhood B are more ompliated beause some swith to shool A and some do not. The non-swithers obtain a welfare hange of dq B. Those who do swith obtain a welfare hange of q A +αµ/2 = dq A + αµ. Averaging over swithers and non-swithers, we obtain dw B = ( 1 q(0) ) dq B + q(0) 0 (dq A + αµ ) d. (18) Using (17) and (18), it is straightforward to show that the average welfare gain is ( dw = dq E A (0) 1 ) q(0). (19) 2 2 This shows that the average welfare gain depends on the differene between two terms. The first is the hange in shool average quality, whih we know from (16) is just the hange in shool effort. The seond represents the additional osts inurred by households in neighborhood B who use shool A. For hoie to generate positive average welfare gains, the inrease in average quality must outweigh the additional osts inurred by swithing households. We are interested in the sign and magnitude of the six variables defined in (14)-(19) and in how they hange with the importane of peer preferenes and the extent of neighborhood inequality. We will assume that > 6/γ and onsider the impat of varying αµ from 0 to. 15 Our findings onerning the impats of hoie on shool quality are summarized in: 16 Proposition 1 i) Choie inreases average shool quality, but stronger peer preferenes and greater neighborhood inequality redue the extent of the inrease. ii) In the shool in the more affluent neighborhood, hoie inreases quality when peer preferenes are weak and neighborhood inequality is low, but redues it when the produt of these variables exeeds a ritial level. Moreover, stronger peer preferenes and greater neighborhood inequality inrease the redution. iii) In the shool in the less affluent neighborhood, hoie inreases quality but stronger peer preferenes and greater neighborhood inequality redue the extent of the inrease. Part i) of the Proposition desribes how hoie impats average shool quality. The first omponent is unsurprising: average shool quality depends solely on shool effort and shools exert zero effort without hoie. The seond omponent is more interesting. The intuition is that stronger peer preferenes and greater neighborhood inequality redue the responsiveness of enrollment to shool effort and therefore lead to lower efforts. Parts ii) and iii) of the Proposition desribe how hoie impats the distribution of quality aross shools. 15 The assumption that > 6/γ plays a role in the findings that hoie an redue quality in shool A and redue average welfare. 16 The proofs of Propositions 1 and 2 an be found in the Appendix. 11
The findings reflet the fat that when peer preferenes are weak and neighborhood inequality is low, few households in the disadvantaged neighborhood exerise hoie in equilibrium. It follows that in both shools, hoie inreases shool effort without hanging shool peer groups. As suh, hoie inreases quality in both shools. Stronger peer preferenes or greater neighborhood inequality mean that more households in the disadvantaged neighborhood exerise hoie. This household swithing dereases peer quality in shool A and leaves it unhanged in shool B. Hene for shool B, stronger peer preferenes and greater neighborhood inequality redue the quality gain (beause they redue the effort inrease) but the quality gain remains positive. For shool A, stronger peer preferenes and greater neighborhood inequality generate a larger redution in the quality gain (beause they redue effort and peer quality) and the quality gain is negative when the produt of these variables exeeds a ritial level. Turning to welfare, our findings onerning the impats of hoie on household welfare are summarized in: Proposition 2 i) Choie inreases average welfare when peer preferenes are weak and neighborhood inequality is low, but redues it when the produt of these variables exeeds a ritial level. ii) In the more affluent neighborhood, hoie inreases welfare when peer preferenes are weak and neighborhood inequality is low, but redues it when the produt of these variables exeeds a ritial level. iii) In the less affluent neighborhood, hoie inreases welfare and stronger peer preferenes and higher neighborhood inequality first derease this welfare gain but then, at some point, inrease it. Again, part i) of the Proposition desribes how hoie impats average welfare, and parts ii) and iii) desribe the impat on the distribution of welfare aross neighborhoods. To gain intuition for part i), reall that the average welfare gain from hoie is just the differene between the average quality gain and the osts inurred by the swithing households (equation (19)). In the ontext of Proposition 1, we explained how peer preferenes and neighborhood inequality impat the average quality gain. Thus, we just need to understand the swithing osts. When peer preferenes are weak and neighborhood inequality is low, shool A s peer advantage has little attration. As a result, few households will exerise hoie and hene these osts will be low. When peer preferenes are stronger and neighborhood inequality greater, shool A s peer advantage will drive more households to exerise hoie. These swithing households obviously benefit from their deision, but their benefit omes at the expense of households in the more affluent neighborhood shool and thus there is no aggregate gain. Welfare is redued by the osts that these households inur and, for high enough peer preferenes and neighborhood inequality, these osts overwhelm the benefits of higher average quality. For part ii), sine no households in neighborhood A exerise hoie, the hange in welfare in neighborhood A equals the hange in quality in shool A. Thus, the intuition underlying the welfare hange in the more 12
affluent neighborhood is just that used to explain the quality hange. For part iii) and neighborhood B, the main differene between the welfare and the shool quality results, is that stronger peer preferenes and higher neighborhood inequality an inrease the welfare gain enjoyed by neighborhood B despite dereasing the quality of shool B. This is beause the benefits enjoyed by those households who swith to shool A outweigh the osts (of lower shool quality) experiened by those do not. Figure 1 illustrates the theoretial results established in Propositions 1 and 2. The Figure assumes that µ = 1, γ = 2, and = 2.5, and illustrates how the impat of publi shool hoie varies as α ranges from 0 to 2. 17 In eah panel, α is measured on the horizontal axis and the outome variable of interest on the vertial. The first panel illustrates how enrollment in the two shools hanges as α inreases and shows that enrollment is higher in the advantaged shool with stronger peer effets. The seond panel illustrates the hange in shool effort reated by hoie and shows that it is positive but dereasing in α. The third panel illustrates the hange in shool quality reated by hoie, showing that the average hange is positive, but the advantaged shool experienes a derease in quality when α is high. The fourth panel illustrates the hange in welfare, showing that itizens in both neighborhoods experiene an inrease in welfare when α is small, but only itizens in the disadvantaged neighborhood benefit when α is high. Moreover, the hange in aggregate welfare turns negative for α high enough. 4 Tiebout hoie A limitation of the baseline model, is that it by assuming that the alloation of households aross neighborhoods is exogenous and independent of shool quality, it ignores the inentives reated by Tiebout hoie. If households hoose the neighborhood in whih they live taking into aount the quality of its shool and there is some elastiity in housing supply, then enrollment will be sensitive to shool efforts even when households must use their neighborhood shools. When shool payoffs are inreasing in enrollments, this provides shools with an inentive to provide effort even without shool hoie. Moreover, intuitively, it seems possible that introduing shool hoie ould undermine the inentives reated by Tiebout hoie and in this way might possibly bakfire. Given this limitation, it is desirable to extend the analysis to inorporate Tiebout hoie. We do this in the simplest possible way by adding an initial stage to the model in whih households hoose in whih neighborhood to live. Thus, we ontinue to assume there are a ontinuum of household types uniformly distributed on [ µ, µ], but now assume that eah household must hoose in whih neighborhood to buy 17 Obviously, equivalent results are obtained by fixing α equal to 1 and letting µ vary from 0 to 2. 13
0.8 Enrollment with hoie 0.6 0.4 Enrollment in A Enrollment in B 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 Change in shool effort 0.05 de 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 Change in shool quality 0-0.5 dq A dq B dq -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 Change in welfare 0.5 0-0.5 dw A dw B dw -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 1: The impat of shool hoie in the baseline model 14
a house. Houses are idential within neighborhoods. Neighborhood A is more attrative than B, perhaps beause it has better houses or superior amenities. 18 This differene in neighborhoods is aptured by assuming that a household of soio-eonomi status s who buys a house in neighborhood A obtains a housing-related payoff of b (ξ s)p A if he buys a house in neighborhood A and a payoff of (ξ s)p B if he buys in neighborhood B. Here P J is the prie of houses in neighborhood J and b is a positive parameter apturing how muh more attrative neighborhood A is than neighborhood B. The parameter ξ is greater than µ, so that ξ s is positive for all types. This formulation implies that higher soio-eonomi status types inur a lower disutility from paying for a house whih aptures the idea that they have more resoures and hene a lower value of the dollars left-over after purhasing a house. Higher values of ξ make households more sensitive to house prie differenes and thus redue the relative demand for houses in the more expensive neighborhood. The supply of housing in neighborhood J is assumed to be S J (P J ) = β + δ J P J, (20) where β (0, 1/2) and 0 < δ A < δ B. A higher value of β makes housing supply in both neighborhoods more inelasti at any prie, while a higher value of δ J makes housing supply in neighborhood J more elasti. Assuming supply is more responsive in neighborhood B is neessary to ensure that the two neighborhoods end up equally sized in equilibrium despite the fat that neighborhood A is more desirable. We ontinue to assume that, with shool hoie, if households use the non-neighborhood shool, they inur a utility ost and that this ost is uniformly distributed on the interval [0, ]. Moreover, we further assume that at the time households hoose whih neighborhood to live, they are unertain exatly what the ost of using the non-neighborhood shool would be. All they know is the distribution. This assumption means that at the time they hoose neighborhoods, households of the same soio-eonomi status will all behave in the same way, whih substantially simplifies the analysis. Introduing Tiebout hoie in this way, means that there is an additional fator to be determined in equilibrium. This is the alloation of households aross the two neighborhoods. Under the assumption that the neighborhoods are stratified and that, beause it is more attrative, neighborhood A is more affluent, this alloation an be desribed by the fration of households living in neighborhood A, whih we denote x. The fration living in neighborhood B is then 1 x. Given stratifiation, neighborhood A will onsist of those 18 This differene in non-shool amenities, ombined with our assumption that all households share the same preferenes, ensures that our model does not feature the multiple equilibria that are ommonly found in other models of loational sorting (Bayer and Timmins, 2005). 15
households with soio-eonomi status [µ(1 2x), µ] and neighborhood B will onsist of those households with soio-eonomi status [ µ, µ(1 2x)]. Given x, the workings of the rest of the model are exatly as desribed in the previous setion exept that the neighborhood populations are desribed by the intervals [µ(1 2x), µ] and [ µ, µ(1 2x)] as opposed to [0, µ] and [ µ, 0]. Aordingly, the key task in solving the extended model is to haraterize the equilibrium size of neighborhood A. 4.1 Solving the model with Tiebout hoie 4.1.1 Without shool hoie We first solve the model without shool hoie. This is neessary to provide the benhmark with whih to ompare shool hoie. It is also relatively simple and provides insight into the solution of the more general ase. A household of type s buying a house in neighborhood A obtains a payoff of q A + b (ξ s)p A, (21) while if it buys in neighborhood B, it obtains a payoff q B (ξ s)p B. (22) Comparing these payoffs and letting P denote the housing prie differential P A P B, we see that the soio-eonomi status of the indifferent type (i.e., the type who gets the same payoff from buying in either neighborhood) is given by ξ ( q + b)/ P. Assuming that the shool quality differential is non-negative, neighborhood A will onsist of all those households with soio-eonomi status higher than the indifferent type. Aordingly, the size of neighborhood A is given by ( ) µ ξ q+b P x =. (23) 2µ We an make further progress by tying down the housing prie differential P and the shool quality differential q. Given (20), if the size of neighborhood A is x, the house pries in the two neighborhoods must be (x β)/δ A and (1 x β) /δ B. The prie differential is therefore given by P = (δ A + δ B ) x (β (δ B δ A ) + δ A ) δ A δ B. (24) 16
Turning to the quality differential, given that the indifferent type is ξ ( q + b)/ P, the average soioeonomi status of those living in neighborhood A is [µ + ξ ( q + b)/ P ] /2 and the average soio-eonomi status of those living in neighborhood B is [ µ + ξ ( q + b)/ P ] /2. Using (1), it follows that the shool quality differential is q = e + αµ. (25) This is partiularly simple, sine it is independent of the alloation of households aross neighborhoods. Substituting (24) and (25) into (23), yields a quadrati equation in the equilibrium size of neighborhood A. This has solution x( e) = [(δ A + δ B ) (ξ µ) + 2µ (β (δ B δ A ) + δ A )] 2 + 8µ (δ B + δ A ) ( e + αµ + b)δ A δ B +2µ (β (δ B δ A ) + δ A ) (δ A + δ B ) (ξ µ) 4µ (δ B + δ A ). (26) Equation (26) provides a losed form solution for the equilibrium size of neighborhood A for any given shool effort levels. This equation aptures the expeted relationships between the demand and supply of housing and equilibrium neighborhood size. 19 Given this, with effort levels e A and e B, the two shools will antiipate enrollments of x( e) and 1 x( e) respetively. Defining equilibrium as in the previous setion, we find that the equilibrium effort levels are idential and satisfy e A = e B = 1 γ x (0). (27) Notie that equilibrium effort levels are positive, onfirming the intuitive idea that Tiebout hoie provides shools with inentives to exert effort. Compared with (9), the impat on enrollment is oming from the expansion of the size of the neighborhood rather than the inrease in enrollment from the other neighborhood. Computing the derivative in question, we find that the equilibrium effort level with Tiebout hoie but no shool hoie is e T (effort under Tiebout hoie) whih is defined by e T 1 δ A δ B. (28) γ [(δ A + δ B ) (ξ µ) + 2µ (β (δ B δ A ) + δ A )] 2 + 8µ (δ B + δ A ) (αµ + b)δ A δ B Effort is dereasing in the importane of peer effets (α) and the natural advantage of neighborhood A (b). 19 To see this more learly, suppose that δ A = δ A δ (i.e., the elastiity of housing supply is the same in both neighborhoods). In that ase equation (26) an be written x( e) = 1 2 + ξ 2 +4µδv ξ, where v = e + αµ + b aptures the relative value of 4µ neighborhood A (i.e., the differene in effort, omposition and the non-shool amenity). If this value is zero then x = 1 (i.e., the 2 population is split equally into neighborhoods A and B). If this value is positive then the size of neighborhood A is inreasing in the elastiity of housing supply δ and dereasing in households sensitivity to prie differenes ξ. 17
Effort is also dereasing in β, refleting the idea that more inelasti housing supplies dampens inentives. Finally, effort is dereasing in ξ. Reall that higher values of ξ influene the demand side of the model, by making households more prie sensitive. This inreases the natural advantage of the heaper neighborhood, and thus makes neighborhood demand less responsive to shool quality differenes. A natural question to ask is how the inentives provided by Tiebout hoie ompare with those provided by shool hoie. This is a diffiult question to answer, beause they are driven by ompletely driven fores. Tiebout hoie inentives are limited by the elastiity of housing supply and the substitutability of the neighborhoods (whih is measured by b). Shool hoie inentives are limited by the willingness of households to hoose a non-neighborhood shool (whih is measured by ). It is possible to hoose these parameters in suh a way as to make the differene between e S and e T either positive or negative. It is notable that both efforts are redued by stronger peer preferenes. In the Tiebout ase, this is beause stronger peer preferenes redue the substitutability of the two neighborhoods. To provide a benhmark with whih to ompare shool hoie, we will assume that the parameters are suh that the equilibrium alloation without shool hoie is suh that the two neighborhoods are equally sized (i.e., x(0) = 1/2). In the on-line Appendix, it is shown that this requires that ξ = 2(αµ + b)δ Aδ B (1 2β) (δ B δ A ). (29) Under this ondition, the qualities of the two shools are given by q A = e T + αµ 2, (30) and q B = e T αµ 2. (31) 4.1.2 With shool hoie If shool hoie is possible, the payoff from buying a house in neighborhood A remains as desribed in (21). The payoff from buying a house in neighborhood B is hanged beause a household will hoose to exerise hoie if its ost is less than q. Under the assumption that the household does not know its ost at the time of the loation deision, its expeted payoff from buying a house in neighborhood B is given by q 0 (q A ) d + q q B d (ξ s)p B. (32) 18
Comparing (21) and (32), we see that the soio-eonomi status of the indifferent type is given by ξ ((1 q 2 ) q +b)/ P. Sine neighborhood A will onsist of all those households with soio-eonomi status higher than the indifferent type, its size is given by x = µ (ξ q (1 2 ) q+b P 2µ ). (33) Using (24) to substitute in for the prie differential, we find that the equilibrium size of neighborhood A, given a shool quality differential q, is x( q) = [(δ A + δ B ) (ξ µ) + 2µ (β (δ B δ A ) + δ A )] 2 + 8µ (δ B + δ A ) ((1 q 2 ) q + b)δ Aδ B +2µ (β (δ B δ A ) + δ A ) (δ A + δ B ) (ξ µ) 4µ (δ B + δ A ). (34) Turning to the determinants of the shool quality differential, following the same steps that led to (5), while reognizing that the neighborhood populations are desribed by the intervals [µ(1 2x), µ] and [ µ, µ(1 2x)] as opposed to [0, µ] and [ µ, 0], we find that, given x and e, the quality differential is q(x, e) = (x + e(1 x)) 2 + 4αµx(1 x) (x e(1 x)) 2(1 x) (35) Given e, the equilibrium size of neighborhood A and shool quality differential x ( e) and q ( e) are impliitly defined by the system of equations x = x( q ) q = q(x, e) (36) It is straightforward to show that there exists a solution to this system of equations for all e in the relevant range (see the on-line Appendix). A suffiient ondition for uniqueness is that at any solution (x, q ) it is the ase that the produt x ( q ) q(x, e)/ x is less than 1. While it is diffiult to find simple onditions on the primitives to guarantee this ondition holds, it is easily satisfied in our simulations. Thus, we will assume it is true in what follows. Given all this, with effort levels e A and e B, the two shools will antiipate enrollments of E A ( e) = x ( e) + q ( e) (1 x ( e)), (37) 19