Competition Between Private and Public Schools, Vouchers, and Peer-Group Effects

t American Economic Association Competition between Private and Public Schools, Vouchers, and Peer-Group Effects Author(s): Dennis Epple and Richard E. Romano Source: The American Economic Review, Vol. 88, No. 1 (Mar. , 1998), pp. 33-62 Published by: American Economic Association Stable URL: http://www. jstor. org/stable/116817 . Accessed: 01/02/2011 12:55 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www. jstor. org/page/info/about/policies/terms. jsp.

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By DENNIS EPPLE AND RICHARD E. ROMANO* A theoretical and computational model with tax-financed, tuition-free public schools and competitive, tuition-financedprivate schools is developed. Students differ by ability and income. Achievement depends on own ability and on peers' abilities. Equilibrium has a strict hierarchy of school qualities and twodimensional student sorting with stratification by ability and income. In private schools, high-ability, low-income students receive tuition discounts, while lowability, high-income students pay tuition premia.

Tuition vouchers increase the relative size of the private sector and the extent of student sorting, and benefit high-ability students relative to low-ability students. (JEL H42, 128) Discontent in the United States with the primary and secondary educational system has become the norm. The decline in SAT scores in the 1970's, embarrassinginternationalcomparisons of student achievement, slow growth in productivity measures, and increasing disparity in earnings all call into question the quality of the educational system. ' Education policy figured prominently in recenit presidential elections.

The debate has centered on issues of school choice, including voucher systems (Karen De Witt, 1992). Typical voucher proposals provide students attending private schools a tax-financed, school-redeemable voucher of fixed amount toward (or possibly covering) tuition. Although a 1993 California referendumfor vouchers was defeated, policy change at state and local levels abounds, as does change in the private educational sector. The state of Minnesota and school districts in 30 states allow residents to choose the public school their children attend. 2The city of Milwaukee introduced a voucher system in the 1989-1990 school year.

A -number f private o school and private-public school initiatives are developing (see e. g. , John F. Witte et al. , 1993; Steve Forbes, 1994; Steven Glazerman and RobertH. Meyer, 1994; Joe Nathan, 1994; Newsweek, 1994; Wall Street Journal, 1994; Steven Baker, 1995; Jay P. Green et al. , 1996). Educational reform emphasizing increased school competition with an increased * Epple: GraduateSchool of IndustrialAdministration, Carnegie Mellon University, Pittsburgh, PA 15213; Romano: Department of Economics, University of Florida, Gainesville, FL 32611.

We greatly appreciate the comments of Linda Argote, Richard Arnott, Lawrence Kenny, Tracy Lewis, David Sappington, Suzanne Scotchmer, and three anonymous referees, in addition to workshop participants at Carnegie Mellon University, Florida State University, Indiana University, Northwestern University, Princeton University, the University of Chicago, the University of Colorado, the University of Florida, the University of Illinois, the University of Kansas, the University of Virginia, Yale University, the 1993 Public Choice meetings, and the 1994 American Economic Association meetings.

We thank the National Science Foundation, and Romano thanks the Public Policy Research Center at the University of Florida for financial support. Epple acknowledges the supportof Northwestern University, where some of this research was conducted. Anv errors are ours. 2 Public funding of nonsecular schools and considerable freedom of school choice has been practicedfor years in England (Daphne Johnson, 1990) and much of Canada (Nick Kach and Kas Mazurek, 1986). These choice systems support horizontal differentiation in schooling and safeguards exist to limit vertical (quality) differentiation.

Our analysis is concerned primarily with the effects of a voucher system on vertical differentiation. 'The provocatively titled report of the National Commission on Excellence in Education (1983), A Nation at Risk, details the decline of performance of U. S. students in the 1970's. More recent data can be found in Daniel M. Koretz ( 1987). Modest gains in performanceon standardized achievement tests, followed by a leveling off, well below peak scores of the early 1960's, characterizes the late 1980's and 1990's. 33 34 THE AMERICANECONOMICREVIEW role of the private sector is at the forefront of he policy debate and recent policy initiatives. The modern economic case for vouchers and increased educational choice was made by Milton Friedman (1962). The academic educational and political-science professions have since considered the pros and cons of voucher systems and educational choice (John E. Coons and Stephen D. Sugarman, 1978; Myron Liberman, 1989; John Chubb and Terry Moe, 1990). Economic analysis of the interaction between public and private schools, and of related policy instrumentslike vouchers, is only beginning to emerge. This paper continues the study of the "market" for ducation by developing a model that focuses on the interactionbetween the public and private educational sectors and also examines the consequences of vouchers. We describe the equilibrium characteristics of the market for education with an open-enrollmentpublic sector and a competitive private sector. Our model embodies two key elements of the educational process. First, students differ in their abilities. Higher ability is assumed to increase a student's educational achievement and that of peers in the school attended. Second, households differ in their incomes, with higher income increasing the demand for educational achievement.

A studentin our model is then characterizedby an ability and a household income, a draw from a continuous bivariate distribution. A school's quality is determined by the mean ability of the student body, reflecting the model's peer-groupeffect. We characterizethe equilibriumdistributionof studenttypes across public and private schools and examine the tuition structure of private schools, assuming that student types are verifiable. We develop a theoretical and computational model in parallel, with the latter calibrated to existing estimates of parameter values. Equilibria are simulated for a range of voucher values.

Key characteristicsof an equilibriumare the following. A hierarchyof school qualities will be present, with the set of (homogeneous) public schools having the lowest-ability peer group and a strict ability-groupranking of private schools. The equilibrium student bodies of schools correspond to a partition of the ability-income-type space of students with MARCH 1998 stratification by income and, in many cases, stratificationby ability. As Figure 1 from our computational model illustrates, type space is then carved into diagonal slices with each higher slice making up a private school's student body and with the bottom lice comprising the public sector. The normality of demand for a good peer group leads relatively high-income studentsto cross subsidize the schooling of relatively high-ability students, producing the latter partition. Private schools attract high-ability, low-income students by offering them tuition discounts, sometimes fellowships. Even with free entry, schools price discriminate by income against students who are not on the margin between switching schools. The equilibrium differentiation of schools and economies of scale in education preclude perfect competition for every type of student.

Nevertheless, this price discriminationdoes not disrupt the internalization of the peer-group externality by private schools. An equilibrium without a public sector is Paretoefficient given the equilibrium number of schools. Because free public schools do not price the peer-group externality, an equilibriumwith public schools is Pareto inefficient. In the computational model, we employ a Cobb-Douglas specification of utility and educational achievement which incorporatesthe peer-group effect. The parameters are calibrated to U. S. data from various sources. We compute approximate equilibria for voucher alues ranging from $0 to $4,200 per student ($4,222 equals the expenditure per student in public schools in 1988). ' With no vouchers, the predicted percentage of students in the public sector is 90 percent (the actual value for the United States is 88 percent). As the voucher is increased, the size of and mean ability in the public sector decrease. With a $2,000 voucher, for example, the percentage of students remaining in the public sector equals 70 percent, and the mean ability declines by 15. 8 percent. 3 The integer number of private schools in our model precludes existence of competitive equilibrium except in special cases.

This integer problem and our approximation approachare discussed later in the paper. VOL. 88 NO. 1 EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION The entry of private schools and consequent more efficient sorting of students across schools caused by vouchers increases average welfare (and achievement) only a little in our computational model, while having larger distributional effects. As we discuss in detail later, the magnitude of the aggregate effect depends on the extent of complementarity of peer ability and own ability in the educational production function. There is little empirical evidence to guide assessment of the extent of uch complementarity. The voucher increases the premium to ability in private schools. The largest proportionate gains from the voucher then accrue to low-income, high-ability students. For example, a household with income of $10,000 and student with ability at the 95th percentile has a welfare gain of about 7. 5 percent of income from a $2,000 voucher. Students of low income and low ability who remain in the public sector when a $2,000 dollar voucher is available experience small welfare losses but make up a majority. It bears emphasizing that our model takes public and private schools to be equally effective providers of education, however.

Some argue that private schools are more productive and that the competitive effect of a voucher program will increase public-school effectiveness. 4 For example, Hoxby (1996) concludes from her empirical investigation that competition-induced performance improvement would increase public-school achievement by more than enough to offset 4 Caroline M. Hoxby (1994, 1996) provides evidence thatprivate-school competition increasespublic-school effectiveness. William N. Evans and Robert M. Schwab (1995) find that Catholic private schools are more effective in inducing studentsto complete high school and also to attend college.

These studies take on the challenge of finding instrumentsthat predict well private-school attendance while being independent of unobserved determinants of educational achievement. Controversy exists concerning the quality of the instruments used. See Thomas J. Kane ( 1996) for a discussion of Hoxby's methodology. David N. Figlio and Joe A. Stone ( 1997) employ a different set of instrumentsthan Evans and Schwab and find that, at currentinput levels, religious private schools are less effective than public schools in producing achievement on standardizedexams in math and science but nonreligious private schools are more effective). See Witte (1996) and Figlio and Stone (1997) for references to other studies. 35 losses of the magnitude thiatemerge due to reduced peer quality in our computational model. Our analysis delineates the allocative effects of vouchers and demonstrates a potential for significant redistribution. A theoretical-economics literature on education is beginning to emerge. Charles A. M. de Bartolome ( 1990) develops a twoneighborhoodmodel of the provision of public educational inputs (quality) with two ability types and peer-group externalities. He shows hat the voting/locational equilibrium is inefficient because the median voter does not internalize the consequences of migration on peer groups in choosing the input level. No independent income variability characterizes students in his model. Raquel Fernandez and Richard Rogerson (1996) introduce income differences in a two-neighborhood model of the provision of inputs but abstractfrom peergroup effects. They examine the effects of redistributive policies and direct controls on inputs. Neither model has a private sector. Our analysis is differentiated by its consideration of a private sector and its two-dimensional, ontinuous type space. In a nolrnativeanalysis of student groupings in the presence of peergroup effects, RichardArnott and John Rowse ( 1987) show how a social plannerwould maximize the sum of achievements in allocating students of various abilities across classrooms. We analyze equilibrium outcomes, and most of our analysis is positive. Joseph E. Stiglitz ( 1974), Norman J. Ireland (1990), Ben Eden (1992), Charles F. Manski (1992), Michael Rothschild and Lawrence J. White (1995), Epple and Romano (1996), and GerhardGlomm and B. Ravikumar(1998) consider the consequences of a private sector for education. Stiglitz,

Glomm and Ravikumnar, and Epple and Romano are concerned with the existence and properties of voting equilibria over taxfinanced, public-school expenditure in the presence of a private alternative. Ireland analyzes the effects of vouchers on utilities and the quality of the public alternative,taking the tax rate as exogenous. Individuals differ only by income, and the private alternative can be purchased continuously in all these analyses. Hence, the private sector is relatively passive, and issues of financial aid and differences in 36 MARCH 1998 THE AMERICANECONOMICREVIEW studentability across schools do not arise.

Our model is distinguished by having differences in ability and related peer-group effects, and by providing an active role for private-sector schools. Eden ( 1992) analyzes vouchers in a purely private market system of provision of education having two ability types and peergroup effects. A voucher equal to the difference between the social and private benefit of education to each ability type is shown to induce socially optimal provision of education. Key differences in our analysis include our consideration of the interaction between the public and private sectors, our exploration of the implications of continuous ifferences in ability and income, and our attention to positive issues. Manski ( 1992) pursues a computational analysis of vouchers that also considers peer-group effects among other aspects of education (especially various objectives of public-school decision makers). Our models differ in a number of ways. Most importantly, we permit private schools to discriminate in their tuition policies, with many consequences. Rothschild and White ( 1995) analyze a competitive model with consumers also inputs to production (a peer-group effect), using higher education as their primary example.

We share a concern for market pricing in the presence of an externality. Differences in our model, among others, are the presence of a public sector, a more detailed specification of peer effects and demand for education, and student variation in both ability and household income. Our attention to the implications for pricing, profitability, and school qualities of a peer-group effect deriving from student abilities, the allocation of students according to ability and household income and the related distribution of educational benefits, and the effects of vouchers are not concerns in Rothschild and White.

Private schools are cases of clubs with nonanonymous crowding due to the abilitydependent externality and schools' power to price it. Suzanne Scotchmer (1994) provides an excellent synthesis of this literature. We follow this literaturein our competitive specification of private schools as further discussed below. The next section presents the model. Section II develops the theoretical results. The computational results comprise Section III. Concluding remarks follow. An Appendix contains some of the detail. I. The Model Household income is denoted y, and each household has a student of ability b. The joint arginal distributionof ability and income in the population is denoted f(b, y) and is assunmedto be continuous and positive on its . (0, bmax X (0, Ymax]All students ] support, S attend a school since we assume that free public schooling is preferredto no schooling. The household decision maker's utility function, U( ), is increasing in numeraireconsumption and the educational achievement of the household's student, and it is continuous and twice differentiable in both arguments. Achievement, a = a (0, b), is a continuous and increasing function of the student's ability and the mean ability of the student body in the school attended,O. Let Ytdenote after-taxincome and ' The influence of ability on own educational achievement is well documented and not controversial. Eric Hanushek( 1986) provides an excellent survey. In the economics literature, Anita A. Summers and Barbara L. Wolfe (1977) and Vernon Henderson et al. (1978) find significant peer-group effects. Evans et al. (1992) adjust for selection bias in the formationof peer groups and show that it eliminates the significance of the peer group in explaining teenage pregnancy and dropping out of school. They are careful to point out that their results should not be interpretedas suggesting that peer-groupeffects do not xist, but as demonstrating that scientific proof of those effects is inadequate. Note, too, that their work supports the notion that peer-group variables enter the utility function since a selection process does take place. The psychology literatureon peer-group effects in education also contains some controversy. In their survey paper, Richard L. Moreland and John M. Levine (1992) conclude: The fact that good students benefit from ability grouping, whereas poor students are harmedby it, suggests that the mean level of ability among classmates, as well as variability in their ability levels, could be an importantfactor.

The results from several recent studies . . supportthis notion. This squares with our reading of the literature(Summers e and Wolfe, 1977; Henderson. t al. , 1978; Chen-Lin Kulik and James A. Kulik, 1982, 1984; Aage B. Sorensen, 1984; VOL. 88 NO. I EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION p tuition expenditure, the latter equal to zero if a public school is attended. Thus, U = U(Y, - p, a(O, b)), with U1, U2, a,, and a2 all positive. The achievement function captures the peer-group effect in our model, discussed further below. To maintain simplicity and highlight the role of peer groups, a chool's quality is determined exclusively by the mean ability of its peer group. 6In ongoing work we are extending the model to include variation in educational inputs. U is also assumed to satisfy everywhere the "single-crossing" condition (SCI): (1) 0a OU/0 J yt > ?. Preferences for school quality might also depend on ability. We say preferences satisfy weak single crossing in ability if a Sorensen and MaureenT. Hallinan, 1986; Adam Gamoran and Mark Berends, 1987; Jennie Oakes, 1987; Gamoran, 1992). However, there are alternative interpretations (Robert E. Slavin, 1987, 1990). For simplicity, the possibility that dispersion in peer ability also affects achievement is not built into our model. Roland Benabou (1996b) explores the consequences for economic growth of dispersion in human capital. 7 We believe this to be uncontroversial. hile we know W of no empirical studies that use direct measures of educational quality, a substantial empirical literature on the demand for educationalexpenditureexists. Although considerable diversity in magnitudes of estimates of the income elasticity of demand for educational spending are present, estimates using a variety of approaches find the ncome elasticity to be positive (Daniel Rubinfeld and Perry Shapiro, 1989). 8 Households may consider education a consumption good, an investment good, or a combination of the two. Our formulation can be interpretedto accommodate any of these motives. However, for households not subject to borrowing constraints, a pure investment motive would imply a zero income elasticity of demand. For such households, this in turn would imply that the SCI condition in (1) would be only weakly satisfied. In light of the empir- OU/00 / Ob t OUIOyt which implies a weakly positive ability elasticity of demandfor quality.

However, because the pertinent empirical evidence is mixed and scarce, we postpone restricting preferences in this regard until necessary. 9 In our computational model and to illustrateour more general theoretical results, we adopt a Cobb-Douglas specification of the utility function: (2) Hence, for students of the same ability, any indifference curve in the (0, p)-plane of a higher-income household cuts any indifference curve of a lower-income household from below. This condition corresponds to an income elasticity of demand for educational quality that is positive at all qualities for all types. One set of sufficient conditions on U for SCI is U11 0 and U12 2 0, with at least I one having strict inequality. 8 37 U = (yt-p)a(O, a(O, b) b) = 0Yb'6 g ; O y ; O. While(2) satisfiesSCI,it embodies he "neut tral" assumption of zero ability elasticity of demand: at O? /0 9b 0. Our computational results are not driven by own-ability effects on the demand for education. Keep in mind, too, that the theoretical results do not assume specification (2). A school's costs depend only on the number of students it enrolls, since inputs vary only with size. All schools, public and private,have the simple cost function: (3)

C(k) = V(k) + F V' ;0 V">o ical evidence suggesting the income elasticity to be positive, we conserve space in the development that follows by assuming that SCI is strict for all households. 9 Henderson et al. (1978) find no interaction between own ability and the benefits to an improved peer group, corresponding to 2IU/00&b= 0 in our model. Summers and Wolfe ( 1977) find some supportfor higherpeer-group benefits to lower-ability students, that is, 02U/la6ab < 0. Thus the literatureprovides limited evidence from which to draw conclusions. 38 THE AMERICANECONOMICREVIEW where k denotes the number of attendingstudents.

Technical differencesamong schools are not an element of our model (for simplicity). Hence, vouchers cannot drive technically inefficient schools from the market,an effect predicted by some proponents of vouchers (see footnote4). Let k* denotethe "efficientscale," (4) k* ARGMIN[C(k)/k]. The presumptionof some economies of scale in education is realistic (Lawrence Kenny, 1982) and important. Otherwise, the private market would produce an infinite number of schools containing infinitely refined peer groups. Our model's equilibriumwill be consistent with the fact thatthe numberof types of studentsgreatlyexceeds the numberof schools.

Public-sector schools offer free admission to all students. This open-enrollment policy leads to homogeneous public schools in equilibrium because we assume no frictions in public-school choice are present. Without equalization of 0's in public-sector schools, students would migrate to higher-0 schools to reap the benefits of a better peer group. With equalized 0's, no incentives for switching schools within the public sector remain. We study the alternative of neighborhood school systems that impose residence requirementsin Epple and Romano (1995). Since all public schools will have the same , one can think of the public sector as consisting of one (possibly large) school. Publicsector schooling is financed by a proportional income tax, t, paid by all households, whether or not the household's child attends school in the public sector. Thus, Yt= (1 - t)y. The public sector chooses the (integer) number of schools and their sizes to minimize the total cost of providing schooling subjectto (3). The tax rate adjuststo balance the budget. Because households are atomistic, there is no tax consequence to a household's decision about school attendance. The public finance of chooling can then be largely suppressedin the analysis until the consideration of vouchers. The public sector is passive in this model for simplicity. Public-sector schools do not segment students by ability (track), increase educational inputs to compete more effectively with the private sector, or behave strategically MARCH 1998 in any way. More realistic alternativesare importanttopics for research, some of which are discussed in the final section. Private-sectorschools maximize profits, and there is free entry anidexit. 10Modeling private schools as choosing an admission policy and uitionpolicy is convenient andinvolves no loss of generality. Student types are observable, implying that tuition and admission can be conditioned on ability and income as competition permits. 1 Private schools are an example of clubs with "non-anonymous crowding" (Scotchmer and Myrna H. Wooders, 1987; Scotchmer, 1997) because of the peer-groupeffect, and we model private-schoolbehaviorfollowing the literature on competitive club economies. In particular,private schools maximize profits as utility tak-ers(see Scotchmer, 1994), a generalization of price-taking when consumers (types) and productsdiffer. Private chools believe they can attract any studenttype by offering admission at a tuition yielding at least the maximum utility the student could obtain elsewhere. Let an i subscript, i = 1, 2, .. n, indicate a value for the ith private school. A zero subscript does the same for "the" public school. Let pi (b, y) denote the tuition necessary to enter school i, with po(b, y) = 0 V (b, y). Let ai (b, y) C [0, 1] denote the proportionof type (b, y) in the population that school i admits, 10Consideration of alternative objective functions to profitmaximization is reasonable,especially given the significant proportion of nonprofit schools.

Some private pursuethe objective of quality schools might, for exa-mnple, maximization. Quality maximization, like profit maximization, is a member of a set of objective functions that are utility independent in the sense that they place no weight on offering any student types higher utility than the student's (equilibrium) reservation utility. Our preliminary analysis of this issue suggests that equilibtia where some private schools pursue objectives from this set other than profit maximization must also be competitive equilibria. Roughly, the failure of any school to maximize profits would permit ently by a profit-maximizingschool. The notion is that abilities can be determinedthrough testing, and required financial disclosures permit determination of household income. At least in the case of Cobb-Douglas utility, equation (2), students will have no incentive to underperformon exams, since tuition will be nonincreasing in ability in equilibrium (proved in Epple and Romano [1993]). Incentive compatibility in the reporting of income is more complex. EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION VOL. 88 NO. I with any ao(b, y) E [0, 1] "optimal" for the public school as determinedby the residualdemand for public education.

A private school's profit-maximizationproblem can be written as (5) MAX rri Oj,kj,pj(b,y),aj(b,y) [pi(b, y)ai (b, y) f - s X f(b, y) db dy]-V(ki) --F subject to ai (b, y) E [0, 1] V (b, y); (5a) (5b) (b, y)a(Oi, U(y,-pi MAX 2 j b) ) p1(b, y), a(0j, b)) j * i; aj(b,y) > O is in the optimal set of j U(yt I0,1, ... ,n that (5b) hold for all (b, y) as we have specified (i. e. , including for nonadmitted students). Tuition charged to students for whom ai (b, y) = 0 is school i's only optimal choice (i. e. , nonadmittedstudents) is irrelevant. Note, too, that tuition such that (5b) holds with strict quality will be optimal. Private schools enter so long as they expect to make positive profits as utility takers. Because incumbent private schools maximize profitsas utility takers,entryresults if and only if wri> 0 for some incumbent school. The public-sector/private-sectorequilibriumis described by the following five conditions in addition to the government balanced-budget condition presented below in Section II, subsection C, for the more general case with vouchers. Condition UM: U*(b, y) - MAX U(y,-pi(b, V (b, y); 39 ie E 0,1,... ,nI ai(b,y) y), a(0i, b)) > O is in the optimal set of i}

V (b, y). (5c) (b, y)f(b, y) db dy; ki =fai Condition VIM: s [Oi, ki, pi(b, y), ai (b, y) ] satisfy (5), (Sd) O kjfbai(b, y)f(b, y) db dy. s Constraints(5c) and (5d) define, respectively, the size of the school's student body and the mean ability. Constraint (5a) precludes a school from admitting a negative number of a type or more of a type than exits in the population. "2Constraint(5b) imposes the utilitytaking assumption. Students' alternatives are limited to schools where they are admitted. Students always have the option of attending the public school. It is innocuous to require i= 1,2,... ,n.

Condition ZfH: 7ri = 0 i = 1, 2, ... , n. Conditions PSP: po(b, y) = V (b, y) ao(b, y) E [0, 1] V (b, y) 12 One might object to the presumptionthat "competitive schools" recognize the limit to demand. The presumption is analogous to a monopolistically competitive firm's recognition of a limit on its demand curve. Dropping the presumptionwould lead to schools admitting infinite densities of some types. See Scotchmer (1994) for the analogue in the literatureon club goods. ao(b, y)f (b, y) db dy ko= s Go =-Af ko bao(b, y)f (b, y) db dy. s THE AMERICANECONOMICREVIEW 40 Condition MC: n xai (b,y)=1 V(b,y). i=O

Condition UM summarizes household utility maximization. Households choose a mostpreferred private or public school, taking admission/tuition policies, school qualities, and taxes as given. Profit maximization of private schools (VIM) and the public-sector policies (PSP) have been discussed. While the entry assumption above is formally part of the definition of equilibrium, it is convenient to substitute the implication that private schools must earn zero profits (ZH). The last condition is market clearance, which uses the simplifying assumption above that free public schooling is preferredto no schooling.

II. Theoretical esults R A. Solution to the Private School's Problem Using UM, the first-order conditions for problem (5) can be written as follows: U(yt- pi , a(Oi, b)) (6a) =U*(b, ai (b, y) (6b) piC as f (6c) We now turn to the properties of equilibrium, assuming one exists. Existence issues are discussed below. Heuristic argumentshave been substituted for formal proofs when reasonable. The first result concerns the qualities of schools. b0,db [ (b, si) L sho0 V'(ki) yd 0 +io (Oi b) = n7i- - X PROPOSITION 1: A strict hierarchy of school qualities results, with the public sector V (b y); i i J ith equality combined with the equilibrium condition UM; pe () is student-type (b, y)'s reservation price for attending school of quality 0i. Condition (6b) characterizes optimal admission policies. " The term 77i0i - b) may ( be thoughtof as the marginalcost of admission operating via the peer-group externality in school i. From (6c), 77i[the Lagrangianmultiplier on (5d) ] equals the per-studentrevenue change in school i deriving from a change in 0i. The appropriatelyscaled change in 0i due to admitting student of ability b equals (b t 0k); its negative is then multiplied by rqj o obtain the peer-externalitycost.

The peer cost of admitting students with ability below the school's mean is positive because the resulting quality decline dictates reduced tuition to all students, while the peer "cost" of admitting above-mean-ability students is negative. Let ( MCi (b) V'(ki ) + r7i 0 - b), which we term effective marginal cost. Types with reservation prices below MCi (b) are not willing to pay enough to cover their effective marginal cost and are not admitted. The school admits all of a type that has a reservationprice above effective marginal cost, and any ai E [0, 1] is optimal if pi* = MCi. 1 B. Properties of Equilibrium

V (b, y); y) MARCH 1998 ai(b y) f(b y) db dy] Condition ( 6a) describes sclhooli' s optimnalut ition function, Pi* (b, y, Oi) and is just (5b) '3 Results (6b) and (6c) are found by substituting p* from (6a) into (5), and then forming a Lagrangianfunction to take account of (Sc) aind(Sd). Result (6b) is then derived by pointwise optimization over ai while taking account of the constraint (Sa). '4 In the upper and lower lines of (6b), the solution for ai is at a corner, and the first-orderconditions are also sufficient for a local maximum. In the middle line of (6b), where p * MCI and any ac (b, y) E [0, 11 satisfies the irst-orderconditions, V" sufficiently large implies local maximization. VOL. 88 NO. 1 EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION having the lowest-ability peer group: 0,, fJn-I > .. > > fJI > 00. Formal proof is in the Appendix. Here an economic interpretationis provided. All private schools must be of higher peer quality than schools in the public sector. Otherwise, no students would be willing to pay to attend any private school. Why must a strict hierarchy of private schools characterize equilibrium? If two private schools were of the same quality, then they would compete perfectly for students.

Consequently, they would have the same effective marginal costs of admitting all types, and their tuitions (to all admitted students) would equal effective marginal costs. An opportunity to increase profits would exist by varying admissions/tuitions in such a way to either: (a) increase quality and admit a student body that values quality by more, or (b) decrease quality and admit a student body that values quality by less. In either case, the school differentiates itself in quality, at the same time attractinga student body that permits profitable price discrimination over the quality change.

We sketch the example of a profitablequality improvement, beginning with schools having identical student bodies (the proof shows that this is without loss of generality). Let one school admit the same numberof (b2, Y2) types as it expels of (bl, yl) types, where b2 > b, and Y2> Yl, implying an increase in 0 but no change in production costs, V(k) + F. Further, choose the types (which is always feasible) such that Y2 - YI > b2 - bi by enough that, using SCI, the (b2, Y2)-types value increased quality by more than the (bl, Yi)types, even though their abilities differ.

This permits the school to charge the newly admitted students tuitions higher than their effective marginal costs because they are selected to value quality increases by more than the expelled students. The profit increase occurs because the new student body values the quality increase by more than would the original student body; 0 and rl rise in the school. It would not increase profits to substitute students in such a way that 0 rises without also changing the student body's average value of quality improvements, because tuitions equal 41 ffective marginal costs in the initially nondifferentiated schools. '5 This example assumes that a school substitutes students to increase quality, but alternativeprofitablesubstitutions exist that decrease quality, roughly, by also creating a lower-income studentbody. In either case, the argumentdepends on SCI. It also identifies the model's force for "diagonal stratification" (see the examples in Figure 1). As developed more fully below, this stratification results because students having relatively high income and low ability within a school cross subsidize relatively lowincome, high-ability students.

The strict hierarchy of Proposition 1 supports the equity-relatedconcerns of some that private schools operate to the detriment of public schools by siphoning off higher-ability students. Whethera strict hierarchyis efficient is analyzed below. First we develop furtherthe positive properties of equilibrium. Proposition 2 describes equilibriumpricing, and Proposition 3 describes the resulting partition of types. Some definitions are useful. Let { (b, y) E SIai (b, y) > 0 is optimal} denote the admission space of school i, i = 0, 1, ... , n (see Figure 1, for example). A locus of points (b,y) E A. n Aj, i ]j, assuming t exists, is referredto as a boundary locus between i and]. (Boundary loci have zero measure in S, as proved in Epple and Romano [1993]. ) Since any household prefers free public schooling to no schooling, the entire type space S is partitioned into admission spaces. Last, to avoid tedious qualification of statements for public-sector schools, we specify that MCo -0 for all (b, y). This notation is convenient since students see a zero cost of public education. PROPOSITION 2: (i) On a boundary locus between school i and j, pi MCi(b) and pj = MCj(b); pricing on boundary loci is strictly according to ability in private schools. ii) pi (b, y) > MCi (b) for off-boundarystudents who attend private school i; pricing off- '" Mathematically,beginning with equal O's,first-order effects on profits of varying admissions vanish, but the profit function is convex in some directions in [a(b, y), p(b, y)] -space, allowing a profit increase. 42 THE AMERICANECONOMICREVIEW boundary loci depends on income in private schools. (iii) Every student attends a school that would maximize utility if all schools instead set pi equal to equilibrium MCi Jor all students. The allocation is as though effective marginal cost pricing prevails in private schools. 16

See Epple and Romano (1993) for proof. Competition between private schools that share a boundary locus forces prices to effective marginalcosts for student-typeson the locus. These students are indifferentto attending the schools sharing the locus. Private schools then have no power to price discriminatewith respect to income on boundaryloci. Prices are, however, adjusted to differing abilities because private schools internalize the peergroup effect. Tuition to private school i decreases with ability at rate rRialong its boundary loci, reflecting the value of peergroup improvements of the school's student body.

Moving inside a boundarylocus in a private school's admission space, students' preferences change in such a way that they would strictly prefer the school attended if it practiced effective marginal-cost pricing. Part (ii) of Proposition 2 establishes that private schools exploit this by increasing price. These students are also indifferent between the private school attended or their best alternative by (6a), but this is a result of discriminatory pricing. Generally, then, price depends both on ability and income within admission spaces. "7 Part (iii) of Proposition 2 follows because it is profitable for a private school to be sure o attractany student whose reservation price 16 The statementsregard the equilibriumeffective marginal cost. Income effects would cause these costs to change if tuition equaled effective marginal cost for all students. This has distributional (but not efficiency) implications. '7 While there are no published studies of the allocation of financial aid by income and ability among private elementary and secondary schools, there is evidence on the allocation of financial aid by colleges and universities. There the evidence is that both ability and family income are significant determinantsof whether and how much financial aid is received (J.

Brad Schwartz, 1986; Sandra R. Baum and Saul Schwartz, 1988; Charles T. Clotfelter, 1991). MARCH 1998 exceeds the school's effective marginal cost. The student allocation's link to effective marginal costs, and hence abilities, will be shown to be efficient (except for the public sector). The income-related price discrimination that occurs does not disrupt the allocation consistent with effective marginal-cost pricing; rather,it is purely redistributive. While this income-related price discrimination is of the first degree (a la Pigou), its magnitude is limited by competition for students among the differentiatedschools.

Near a boundaryin a school's admission space, a student's preference for the school attended would be slight under effective marginal-cost pricing, so that the admitting school can capture little rent. The numberand sizes of private schools then determine their powier to price discriminate over income. All private schools have student bodies less than k* by a similar argumentto that in more standardmonopolistically competitive equilibria. "8Here school i's marginal-revenuecurve can be constructed by ordering from highest to lowest students' reservation prices minus peer costs [i. e. , p* + rbi(b - Of)], and thus the associated ownward-slopingaverage revenue curve may be derived. Zero profits then implies a scale below k*. If we let k* decline, then private schools become more numerous and less differentiated (have closer 0's), and incomerelated price discriminationdeclines. Now consider the partition of types into schools. We say stratificationby income (SBI) holds if, for any two households having students of the same ability, one household's choice of a higher-O school implies it has a weakly higher income than the other household. Analogously, stratification by ability (SBA) is present if, holding income fixed, the household that chooses a higher-Oschool must ave a student of weakly higher ability. The combination of SBI and SBA implies a diagonalized partitionas, for example, in Figure 1. PROPOSITION 3: (i) SBI characterizes equilibrium. (ii) If preferences satisfy weak c single crossing in ability (W-SCB) and m7, 18 The points made here are proved in Epple and Romano (1993). EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION VOL. 88 NO. I ?72 ? '- ? 71, then SBA also characterizes equilibrium. '9 To confirm part (i), consider two households with students of the same ability but dif- feringincomes y2 ; yI. In the (0, p) -plane, indifference curves of a household are upward loping. For the same ability, SCI implies that any indifference curve of y2 cuts any indifference curve of yl from below. Allocations are as if tuitions equal effective marginal costs [part (iii) of Proposition 2]. Thus, the choice between schools i andj may be representedin the ( 0, p ) -plane as a choice between ( Oi, MCi (b)) and (0j, MCj(b)). If Oj; Oi, it must be that MCj(b) ; MCi (b) if either type chooses i. A standard single-crossing argument then applies to complete the proof. Part (ii) is proved in the Appendix; here we provide some intuition. Assume first that the demand for quality is independent of ability (e. . , as in the Cobb-Douglas specification) and that all private schools give the same discount to ability along their boundary loci (i. e. , schools' 7's are the same). Holding nominal household income fixed, real income would rise with student ability due to tuition discounts at all private schools. SBA would then result by the same logic explaining SBI. Hence, the combination of a positive income elasticity (SCI) and discounts to ability alone would cause both SBI and SBA, the diagonalized partition as in Figure 1. Relatively high-income and lowability students cross subsidize relatively low-income and high-ability students in rivate schools. The argument holds more strongly if the 7's strictly ascend or if WSCB holds strictly. However, neither condition is necessary for SBA, nor do any of our other results require these conditions or SBA. It may be possible absent these conditions to get cases having nonmonotonic boundary loci in the (b, y) -plane. 20 9 We thank an anonymous referee for encouraging us to investigate bid-rentfunctions (see e. g. , MasahisaFujita, 1989), which ultimately led to part (ii) of Proposition 3. 20 The alternativeto W-SCB implies that lower-ability types are willing to pay more for a better peer group, and he alternativeto weakly ascending q's implies that lower- 43 We now turnto normativeresults which are quite intuitive. Again, see Epple and Romano (1993) for the formal analysis. Pareto efficiency requires: (i) a student allocation that internalizes the peer-group externality given the nmtmberf schools, and (ii) entry as long o as aggregate household net willingness to pay for an allocation with one more school exceeds the change in all schools' costs. An equilibrium without a public sector would satisfy condition (i) but not condition (ii). Effective marginal cost includes the marginal value of he peer group externality, implying that MCi (b) equals the social marginal cost of attendance at school i by a student of ability b. A purely private-school equilibrium then satisfies efficiency condition (i) by part (iii) of Proposition 2. However, entry to the point of zero profits entails externalities so that efficient entry [condition (ii)] fails to hold in a fully private equilibrium. An entrantcapturesthe full value of its product to the studentbody it admits but ignores utility changes of nonadmitted students and profit changes of other schools resulting from the reallocation. ' Fixed costs, quality schools give bigger discounts to ability. Either would tend to work against pure ability stratification, though Proposition I implies that some degree of ability stratificationwould be present. It is desirable to demonstrate SBA without assuming ascending q's, since these values are endogenous. However, providing general primitive conditions for SBA independentof assumptionsconcerning the equilibrium q's is difficult, because their equilibrium values depend on the entire distribution of types in the population. For the Cobb-Douglas case and ssuming independence of income and ability in the population, we (Epple and Romano, 1993) have shown SBA without assuming weakly ascending 7's. 21 The comparison of the equilibrium number of schools in a fully private equilibrium to the Paretoefficient number entails a trade-off. The entrant ignores the lost revenues and cost savings to other schools from the students that; t admits. Since almost every student ati tracted away from incumbent schools is inframarginal (i. e. , tuition exceeds effective marginal cost), the net effect here of entry is negative, tending to cause too much entry.

Opposing this is the entrant's failure to capture the full returnsfrom increased varie-tyof school qualities that results. Altlhoughthe entrant fully price discriminates to the students it admits, it cannot tax other students for the adjustments in the incumbent schools' qualities. A net benefit to other students is likely to result because the incumbent schools will better accommodate preferences. 44 THE AMERICANECONOMICREVIEW hence the finite size of an entrant,underlie the entry externalities as in many models of monopolistic competition. Introductionof the free public sector implies eviations from both efficiency conditions. In general, the public sector displaces multiple differentiatedprivate schools, substitutingthe equivalent of one "large" homogeneous school. This effective reduction in the number of schools is without attention to costs and benefits, generally implying a deviation from efficiency condition (ii). Holding fixed the number of schools in the public-private equilibrium (and counting the public sector as one school), zero pricing of public schooling violates condition (i). By just reallocating students between the public sector and private school 1 near their shared oundary locus, Paretian gains are feasible. Reference to the upper panel of Figure 1 from our computational equilibrium may help clarify the argument. Gains would result from shifting into private school I relatively lowerability students below but near the boundary locus, students for whom the marginal social cost in the public sector is positive. These students are nearly indifferent between the two schools when facing the social cost of attending the private school but a tuition (zero) below the social cost of attending the public school. Students near the boundary locus and ttending the private school may also be of sufficiently high ability that the social "cost" of attending the public school is negative. Gains from shifting such studentsinto the public sector are then also feasible. Such students exist in our computational model, the rough prescriptionbeing to rotatethe boundarylocus counterclockwise at the point of ability having zero social marginal cost in the public school. Collecting these results, we have the following proposition. efficient. (ii) The public-private-sector equilibrium has neither an efficient number of schools, nor an efficient student allocation iven the number of schools. When fixed costs of schooling are small, the departure from efficiency in a fully private equilibrium will be correspondingly small. Part (i) of Proposition 4 can then be interpreted as making a case for private schooling and the vouchers we study. However, we have some reservations concerning this efficiency result. First, we are sympathetic to the view of many that access to a quality education is a right and serves as a means to limit historical inequities. Second, longer-run externalities from education not considered by private schools, like reduced crime, may be present.

For these reasons, we explore the consequences of vouchers on all types instead of just providing aggregate measures. A somewhat distinct concern arises because exact equilibrium exists only in special cases. The interpretationof the efficiency results in the approximateequilibriumwe study is discussed in subsection D, below. C. Vouchers We examine tax-financedcash awardsto all those attending private school. 22 No role for vouchers is present in the tuition-free public t sector. Refo-rmulate he model by everywhere adding the amount of the voucher, v, to yt for households that choose a private school.

The government's budget constraintis: tyf (b, y) db dy (7) s This positive externality will tend to cause too little entry. We believe that too many or too few private schools are possible, but we have not proved this. vf(b,y)dbdy fJ - PROPOSITION 4: (i) The allocation in a fuilly private equilibrium is (Pareto) efficient given the number of schools; the equilibrium number of schools is not, however, generally MARCH 1998 . U AlU UA2U . +sr[s N U . +r , / B 7(k 22 Our model permits households to retain as income any excess of the voucher amount over the tuition paid to he private school of choice, thereby avoiding considerable complication. VOL. 88 NO. 1 45 EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION where N and k denote, respectively, the costminimizing number and size of schools in the public sector that satisfy demandfor public education. Vouchers lower the real price of private education and increase the demand for it. We examine the effects of vouchers in our computationalmodel. the results will provide at least suggestive evidence about the impact of policy interventions. However, scant empirical evidence exists on some important parameters of the odel. D. Existence of Equilibriumand an ApproximateEquilibrium We require specifications for the density of income and ability, the utility and achievement functions, and the cost function for education. As we discuss in the Appendix, exact equilibrium generally fails to exist due to the integer number of private schools. We examine an approximate equilibrium in our computational analysis. Our "epsilon-competitive equilibrium' requires that no (utility-taking) private school, incumbentor entrant,could increase profitsby more than s. Let 7rax and .. min denote the maximum and minimum profits arned by incumbent schools [which maximize profits overp(b, y) and a (b, y) locally], and replace Zfl in the definitionof equilibrium with MAX [irmax, Xrmax -lrmin - rminl c Here lrmax equals the maximum potential profits to an entrant,and the maximum of the second two terms in the brackets equals the largest feasible profit increase by an incumbent school. The revised definition of equilibrium continues to require UM, PSP, MC, and local profit maximization by incumbent private schools [i. e. , (6a) - (6c)]. Last, the number of private schools is the minimum number satisfying these requirements.

The epsilon equilibriumretains all the positive propertiesof an exact equilibriumexcept that private schools could gain s in profits via global adjustments. The allocation of students in a fully private equilibriumwould then continue to satisfy efficiency condition (i). M III. Computational quilibrium odeland E R Illustrative esults We develop a computationalmodel to illustrate our results, to examine vouchers, and to explore issues for which comparative-static analysis may yield ambiguous results. We calibrate it to existing empirical evidence so that A. Specification and Calibration

We assume that [n (b) is distributedbivar- iate normal with mean LbJand covariance matrix 2 01b P UbUy P bUy aY J To calibratethe distributionof income, we use mean ($36,250) and median ($28,906) income for households from U. S. census datafor 1989. With units of income in thousands of dollars, these imply that ,uy = 3. 36 and ay = 0. 68. We adopt specification (2) for the combined utility-achievement function. To calibrate the ability distribution we presume that educational achievement determines futureearnings and that the benchmarkeconomy is in a steady state. First, define normed achievement, aN, s our achievement function raised to the power 1/3 and multiplied by a constant, aN Y Ka "' = KO l'b. 23 Then, a studentwith ability b attending a school with a peer quality of 0 is presumedto have futureannualearnings (E) given by ln E = ln aN= In K + (y/o/)ln 0 + ln b. This normalization is such that a percentage change in ability leads to the same percentage change in dollars earned. Henderson et al. ( 1978) reportthe change in achievement percentile that results from moving students from classes stratified by ability to mixed 23 The constant of proportionality, K, is arbitrary. A onvenient scaling is to set K = E[ b -'. This scaling has the propertythat, if all students in the populationwere to attend the same school (i. e. , 0 = E[b]), then normed achievement would equal ability (i. e. , aN = b). 46 THE AMERICANECONOMICREVIEW classes. An elasticity of achievement with respect to peer ability that is 30 percent as large as the elasticity with respect to own ability is representative of the results they report. We adopt the somewhat conservativevalue of y/l 0. 2. To complete the calibration of the distribution of ability, we then assume that the observed household-income distributionis the ncome distribution that emerges in a steadystate equilibrium in our benchmark model. 24 This yields Ilb = 2. 42 and b = 0. 61. Thus, mean and median ability are 13. 6 and 11. 3, respectively, and the standard deviation of ability is 9. 1. 25 GarySolon (1992) and David J. Zimmerman (1992) provide evidence on the correlationbea tween father's income and son's incomrre,nd they both find that the best point estimate of this correlationis approximrately. 4. Intergen0 erationalcorrelationin income arises from two sources:correlationbetween householdincome nd student ability and, for given ability, correlation between income and quality of school attended. Hence, SBI suggests that the intergenerationalcorrelationin incomes is an upper bound on the correlationbetween parent's income and child's ability. For purposes of sensitivity analysis, we then assume that p E [0, 0. 4]. For our benchmarkcase, we set p = 0, More precisely, we let the distribution of ability be lognormal, and we approximateby assuming that this generates a lognormal distributionof earnings. We set the first two moments of the distribution of earnings equal to the irst two moments of the distribution of income. That is, a we choose 11h nd cb such that our benchmarkequilibrium has E[aN] = E[y]lm and Var[aN] = Var[y]/m2. The constantm is the ratio of employed workersper household to the number of students per household (m = 2. 6 in 1990). The distribution of earnings will not be exactly lognormal because of the discrete difference in schools attended, even though the distribution of ability is presumed to be lognormal. If every student attended public school in the benchmarkmodel, and hence faced the same 0, earnings would be exactly lognormal.

The approximation is a good one because 90 percent of the students do attend public schools as we will see. 25 Ability can be related to IQ. Using IQ - X(100, 256), one obtains In b = -1. 38 + 0. 038(IQ). In our novoucher steady state, this implies that a workerwith an IQ of 100 has expected income of $22,074, and a 10-point increase in his IQ increases expected income to $32,510. See the discussion in what follows relatingto Figure 6 and the calculation of expected steady-state income conditional on ability. 24 MARCH 1998 which is particularly onvenient for our steadyc state calibrationof the model.

This completes the calibrationof f(b, y). We now complete the calibration of preferences. The Cobb-Douglas specification implies unitary price and income elasticities for school quality, 0. Given the absence of empirical evidence on the demand for quality, these are plausible focal values and are consistent with estimates of demand for school expenditure (see e. g. , Theodore Bergstrom et al. , 1982). This function also implies thatthe marginal rate of substitutionbetween school quality and the numeraire is invariant to own ability. Empirical evidence is mixed about whether an improvementin peer group is more eneficial to high- or low-ability students. Hence, our model's assumption that the effect of peer group is not biased toward either highor low-ability types seems an appropriate choice for a baseline model. If school quality could be purchasedat a constant price per unit of quality, each household's expenditure on education relative to total expenditureon other goods would be y/( 1 + -y). The existing share of aggregate disposable personal income in the United States that is spent on education is approximately 0. 056. Hence, we set y = 0. 06. Using y/P = 0. 2 from above, the calibrated tility-achievement function is then U = (Yt - P)0006b0. 30. We chose a cost function that is quadratic in the percentage of students (or households) a school serves: F + V(k) = 12 + 1,300k + 13,333k2, with parameters set as follows. Expenditure per student in public schools in 1988 was $4,222 (Statistical Abstract, 1991 p. 434) and there was 1/2studentper household (Statistical Abstract, 1992 pp. 46, 139). We specified our benchmark case to have four private schools and chose parametervalues such that average cost in equilibriumwas approximately$4,200 per pupil. 26 Experimentation indicated that 6 We have presented the cost function in terms of the percentage of students served or, equivalently, the per- VOL. 88 NO. I EPPLE AND ROMANO:PRIVATE-PUBLIC SCHOOLS COMPETITION equilibriumpropertiesare not very sensitive to the benchmark number of schools, but rather are sensitive to the minimum of the average cost of schooling. We set e = 4. 2. This is the minimum value sufficient to assure existence of epsilon equilibrium for voucher values varying from zero to $4,200 per student. 27 B. Results For our benchmark equilibrium with no voucher, the public sector has 90 percent of the student population, and the four private chools combined serve the remainder. The actual U. S. percentage of students enrolled in public schools during this period equaled 88 percent. Increasing p from 0 to 0. 4 reduces public-sector attendance to 88 percent. Effects on other variables of so changing p are also small, and the results that follow are for p = 0. centage of households served, k. In terms of k, average cost reaches a minimum at $2,100, with k* = 0. 03; $2,100 can then be interpretedas the average cost per household. There are twice as many households as students in the United States. Letting s denote the numberof studentsand ubstituting s = k/2, one sees that the minimum of the average cost per student is $4,200. In our presentation,we focus on per-student measures of tuition and costs; the related per-household measures are simply half those of the per-studentvalues. 27 This value is about 7 percent of the cost of a school operating at a scale that minimizes cost per student. Relative to fixed cost, ? is approximately 35 percent. Of course, a minimal E, however measured, is desirable. We have studied how the minimum e varies as we vary efficient school scale, k*, while holding average cost constant.

We find that the requisite e to support equilibrium varies approximatelyproportionatelywith k* if fixed cost is varied proportionately with k*. This suggests, as we would expect, that e can be made as small as desired if k * is made sufficiently small. We have also investigated increasing fixed cost while holding k* and minimum average cost constant. This tends to reduce the ratio of e to fixed cost but increases the absolute magnitude of e required to sustain equilibrium. Our investigation reveals that substantive findings from the computational model re not sensitive to the choice of k* or the relative magnitude of fixed to variable cost. Rather, the key aspect of costs is the value of average cost at the minimum, and as discussed in the text, this value is based on observed school costs. The problem with pursuinga calibrationthat further lowers e is that it leads to a computationally unmanageable number of schools for large vouchers. 47 Othercomputationalresults are presentedin Figures 1-6. The upperpanel of Figure 1 presents the boundary loci and admission sets in type space, in addition to the equilibrium O's nd k's. Here and in some other figures, both absolute and percentile ability scales are provided for perspective. The lower panel displays the allocation for a voucher of $1,800. The linear boundary loci derive from the Cobb-Douglas specification. For results we present, intersections of boundaryloci, if any, occur very near the bounds of the support of type space. S

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