Education vouchers, the peer group problem, and the question of dropouts.

1. Introduction

Since 1990, there has appeared a burgeoning literature, both theoretical and empirical, on the effects of school vouchers (see Hoxhy 1996; Hoenack 1997; West 1997; Hoyt and Lee 1998; Rouse 1998b). Because the use of vouchers increases the size of the private school sector at the expense of the public and because of the public policy controversy that that result has created, considerable attention has been directed at measuring the efficiency of private schools relative to public (Evans and Schwab 1995; Sander and Krautman 1995). Here the early claim in the literature--that Catholic schools "do better" (Coleman, Hoffer, and Kilgore 1982)--has driven much of the subsequent analysis and has led critics to emphasize that early results that at first sight seem favorable to Catholic schools could result from student and/or parent selection bias (Murnane, Newstead, and Olsen 1985; Neal 1997). After controlling for various selection biases (Figlio and Stone 1999), researchers have accordingly tended to find that priv ate schools (as a group) do not perform any better than do their public counterparts (Neal 1998). (1)

Two aspects of this debate have attracted our attention. First, most participants use "student achievement" (i.e., the ability to score well on tests in math, English, science, and so on) as the benchmark for measuring educational performance and hence the basis for assessing the impact of vouchers. Our paper is both an addition to and a departure from this approach. We focus on the role of vouchers in relation to school dropouts, a somewhat different and, we believe, as important a measure of school performance. (2) Second, we have noticed that much of the evaluation of the performance of private and public schools compares only alternative benefits without invoking the caveat that efficiency issues cannot be resolved without a similar analysis of cost. Thus, while our analysis follows in this tradition by comparing only the effect of vouchers on alternative measures of education benefits, we recognize that greater voucher use may well bring about efficiency changes on the cost side that would dominate these considerations. (3)

These comments are not to detract from the rich variety of approaches to the question of how the mix of school types affects academic performance and how vouchers relate to that mix. The current literature analyzes many issues that complement those raised in this paper. They include the role of vouchers in relation to student mobility and geographic location (Neal 1997; Hoxby 2000; Nechyba 2000), the relative performance of religious versus nonreligious private schools on high school achievement and completion rates (Figlio and Stone 1999), and the relationship between school dropouts and both teenage pregnancy (Ribar 1994) and alcohol use/ abuse (Ribar 1999).

Our contribution to the voucher debate begins from a recent series of papers by Manski (1992), Epple and Romano (1998a, b), and Nechyba (2000), who have emphasized the presence of one particular educational spillover in the use of school vouchers--the "peer group" problem. This is an external effect that vouchers may have on public schools by helping private schools diminish the pool of higher-ability students that remain in the public system. By allowing lower-income students of high aptitude and ability to leave the public school system, it is argued, a voucher system encourages private schools to "skim the cream off" the public school system and so depreciate the average quality of students that remain behind. (4) Because the quality of education received by students is, in part, a function of the quality of their classmates, the loss of higher-ability students reduces the quality of education received by those remaining in the public system. On the other hand, private schools receiving the incoming benefi t are led by competition to internalize the externality through a fee structure that discriminates by ability (e.g., high fees to high-income students with scholarships given to higher-ability, low-income students). In this way, a voucher system, it is argued, will promote inequity across students even as it lowers education costs and improves the overall efficiency of the educational system.

Our analysis does not deny that voucher use may further this external effect on public schools. However, it is our concern that the current focus on the inequity of the peer group effect has distracted attention from what might be an even more pressing horizontal inequity. By this we mean the effects arising from the still lower level of education received by the roughly 12% of students who drop out of the education system by the end of high school. (5) Thus, even though a voucher program may produce a deterioration in the quality of the education received by those who do not leave the public school system, a voucher program may improve horizontal equity if it helps retain within the school system more of those students who would have dropped out entirely.

One measure of the importance of the peer group problem can be seen from the findings of the much-quoted work of Summers and Wolfe (1977).6 Using the difference in composite achievement scores between grade 3 and grade 6 as a measure of value added, Summers and Wolfe (p. 643) find that an increase in the percentage of high achievers in a student's school has two offsetting effects on student scores: one that significantly improves all students' scores and a second that reduces individual scores by an amount that depends on student ability. On net, average students were found to be largely unaffected by a peer group effect. However, students with the lowest levels of ability experienced as much as a half a year improvement in overall grade performance by the end of the three-year period, while students with the highest levels of ability were held somewhat back. (7)

Without minimizing the significance of Summers and Wolfe's findings for low achievers, one can appreciate that the size of the peer group loss experienced by such students could be small relative to the losses experienced by potential students who did not attend school at all. It is then possible that the loss in public school achievement due to the negative peer group effect could be dominated by an offsetting educational gain. This would require a voucher program to retain a sufficient number of those students who would otherwise

have left the school system. It is our objective, then, to reexamine vouchers from the perspective of the dropout problem and to highlight the resulting equity trade-off. We do this by extending Epple and Romano's (1998a, b) externality argument. Thus, while their analysis makes us more aware of the needs of those students who remain in the public system, the extended analysis allows us to highlight the potential benefits that vouchers can create for school dropouts in need. Becaus e the net effect will matter for an overall evaluation of a voucher program, our work also suggests that future voucher experiments should be designed to measure the effect of vouchers on the dropout rate as well as its effect on any associated peer group effect.

Our paper incorporates dropouts in two stages. In the first part of the paper, the model of Epple and Romano (1998a, b; henceforth ER) is adapted to provide an additional rationale for voucher use. In our model, vouchers allow low-income students to escape the frustration of having to conform to the educational uniformity of the public school system by attending a private school rather than dropping out of school entirely. The reasons why students choose to drop out of school will, of course, involve more factors than simply the level of their frustration, and these factors will be discussed at length later in the paper. However, the dimension of frustration allows us an immediate and straightforward way of modifying ER's model. This means that we need not rely on lower production costs to motivate the emergence of a private school system or a reason for voucher use. (8)

After abstractly modeling vouchers as an instrument for alleviating the dropout problem in sections 2 to 4, we turn to consider in section 5 whether greater voucher use can reduce the dropout rate. This involves a search of the education literature to see whether the greater use of the private school system could be expected to lower the dropout rate. At present, the dropout rate in the Catholic school system is lower than that of the public school system. This may arise either because private schools better provide the set of school characteristics that have been found to be most useful in reducing student dropouts or, as the more recent literature suggests, because the students and parents exercising choice have common characteristics that better match the private system. Ultimately, the size of the dropout effect relative to the peer group effect will be an empirical question. However, to the extent that voucher use can reduce the student dropout rate, it is clearly the case that the presence of a peer gro up externality is not in itself sufficient to prevent a reconsideration of the benefits of a voucher system on equity as well as efficiency grounds.

2. The Model

In this model, we wish to motivate the emergence of a private school system from the demand rather than the supply side and so assume that private schools have no technical cost advantage relative to public schools. Similarly, we do not assume, as have others, that competition among private schools results in superior academic achievement, for example, via smaller school/classroom sizes, more student-centered learning, greater competition among schools, and so on. Rather, we assume that the private school system differs from the public only in that each private school can offer a bundle of education services to students and their parents that differs in its dimensions from the single-output mix produced by the public system.9 Competition among schools is assumed to lead to technical efficiency, while open entry into private schooling means that individual private schools may receive scarcity rents for a speciality but as a whole can earn only normal profits. (10)

Given this supply setting, we assume that the decision-making agent, the student and/or the student's family, maximizes a separable utility function of the following form:

[U.sub.i] = U([c.sub.i], [a.sub.i]) = U([c.sub.i]) + [a.sub.i], (1)

where [c.sub.i] represents the level of consumption received by the [i.sup.th] family and [a.sub.i] represents the level of educational attainment achieved by the student. The term U([c.sub.i]) is assumed to satisfy the Inada conditions (so that consumption exhibits diminishing returns in terms of utility). (11)

The level of consumption received by a household depends on individual income, [y.sub.i] (assumed exogenous in the analysis); the public school tax rate, t; and whether the student in the household attends a private school (that charges the tuition fee, p). That is,

[c.sub.i] = {[y.sub.i](1 - t) if in the public school system (or drops out), and

[y.sub.i](1 - t) - p if in the private school system. (2)

In general, the tax rate will depend on the number of public school students and hence on the scale of the public system. Here we assume that the individual's effect on the average tax rate is small so that each agent ignores the fiscal externality when choosing whether to attend a private school.

An individual's educational attainment, [a.sub.i], is assumed to have four (separable) parts: a minimum level of education arising from innate aptitude and life experience accumulated outside of school ([a.sub.0i]), a level of classroom education that is positively related to the individual's ability ([b.sub.i]), a peer group effect whereby the classroom education received is enhanced by the average quality of the students in the school attended ([[theta].sub.j] = [SIGMA][b.sub.i]/n (i = 1, ..., n; j = public or private), (12) and finally a term that reflects the dissipation of potential classroom achievement that arises through student frustration ([f.sub.i]).

Because the last term represents our departure from ER's model, further discussion is merited. Our analysis assumes that private schools are valued because they offer students and their families distinctive educational variety. Compared to the uniformity of the public system, individual private schools are assumed to provide a different education and lifestyle mix, with greater-than-average focus on, for example, the arts, sciences, mathematics, languages, sports, religion, and/or discipline. We model the demand for this variety through the private system's ability to reduce a consumption bad, called for convenience "frustration." With this assumption, we capture the idea that it is precisely the lack of educational uniformity in the private school system that allows for a better matching of individual students with the school most suitable for them, particularly those students whose tastes, temperaments, and/or abilities diverge widely from the mean. Setting frustration at zero for the private system is then purely a shorthand way of highlighting "better matching" as the important relative benefit that drives our analysis.

It is important to emphasize, however, that we are not assuming that the public school system is inherently frustrating, nor will it be frustrating to all students. Nevertheless, across a population of students with normally distributed tastes and abilities, any single program (even an optimally designed one) will necessarily produce some degree of frustration for students at the tails of the distribution. The private school system, by permitting individual specialization and achieving diversification in the aggregate, offers to individual students and their families a menu of possibilities that allows students to avoid having to conform to the uniformity offered by the public system.

To keep our model as close as possible to ER's analysis, we represent the student's frustration with the public system as a function of the degree to which the individual's ability diverges from the public system's mean. (13) More specifically,

[f.sub.i] = [lambda][([b.sub.i] - [[theta].sub.j]).sup.2] (3)

where

[lambda] = 1 for [b.sub.i] [less than or equal to] [[theta].sub.j] and 1 [greater than or equal to] [lambda] [greater than or equal to] 0 for [b.sub.i] [greater than or equal to] [[theta].sub.j].

The parameter [lambda] allows the degree of frustration experienced by public school students relative to their best private alternative to be asymmetric about the school mean. (14)

Overall, then, the level of educational attainment is assumed to take the following form:

[a.sub.i] = [a.sub.0i] + [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.j] - [b.sub.i]) - [a.sub.3i][f.sub.i], (4)

where [a.sub.0i], [a.sub.1], [a.sub.2], and [a.sub.3i] are all positive constants and, because frustration with the public system is measured relative to the most appropriate private school, [f.sub.i] = 0 if a private school is chosen.

Note that in Equation 4, students are allowed to differ in terms of their innate ability, [b.sub.i]; in terms of the degree to which they can learn outside the school environment, [a.sub.0i]; and in terms of the degree to which their frustration with the uniformity of the public school system dissipates their ability to learn, [a.sub.3i]. On the other hand, [a.sub.1] and [a.sub.2] are the same across individuals so that schools provide an equal benefit that varies directly with that student's ability and with the size of their peer group effect. Relative to ER's model, our peer group effect has education benefits that depend directly on the divergence between that individual's ability and school quality. This means that the size of the peer group benefit for students of lower-than-average ability is larger, the higher is the mean ability of the students in the school they attend.

Given the discreteness of the choices facing the family, the student/household would experience three distinctly different levels of utility, depending on whether the student is (i) in a public school, (ii) in a private school, or (iii) has dropped out of the school system. The levels of utility associated with each of these states are

a) if in the public system,

[U.sub.i](public) = U([y.sub.i](1 - t)) + [a.sub.0i], + [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.pub] - [b.sub.i]) - [a.sub.3i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2], (5)

b) if in the private system,

[U.sub.i](private) = U([y.sub.i](1 - t) - p) + [a.sub.0i] + [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.pri] - [b.sub.i]), (6)

c) if a dropout,

[U.sub.i](dropout) = U([y.sub.i](1 - t)) + [a.sub.0i]. (7)

where the average quality of public schools ([[theta].sub.pub]) and private schools ([[theta].sub.pri]).

Given these choices, the optimizing agent will choose the alternative that generates the highest level of (expected) utility. While there is no particular reason why one case could not dominate the others, we assume that the distribution of parameters across the community is such that some individuals fall into each category. This corresponds to the situation that arises in most jurisdictions when all three alternatives are present.

To utilize these categories analytically, we need to determine the characteristics of those individuals who are likely to change categories as the range of institutional possibilities is increased. To do this, we first isolate the net benefit that would accrue to an individual who, in the absence of a private school system, would be indifferent between dropping out rather than continuing to attend public school. This sets the characteristics of the decision agent at the margin of choice in our base-case institutional setting. We then assess how that individual would respond to institutional innovation--in this case, the arrival of private schools. Our interest is the Rawlsian concern with students who in the base case receive the least amount of education--the student dropouts--and thus how the characteristics of those on the school/dropout margin change once private schools enter. This allows us to predict the effect of introducing vouchers into this educational setting.

3. Educational Choices in a Community with a Public School System Only

Suppose, then, that we begin by considering an educational jurisdiction that permits only a single public school system. In this case, students and their families have only two choices: to attend a public school or to drop out entirely. (15) While many states have mandatory attendance laws (so that the dropout rate below a certain age might seem to be zero), such laws cannot be enforced costlessly so that large and persistent student dropout rates continue to plague many school districts. Should only public school education be available, our model states that the student will stay in school when the net benefit is positive, that is, [U.sub.i](public) [greater than or equal to] [U.sub.i](dropout). Inversely, the net benefit of dropping out of school is calculated by subtracting Equation 5 from Equation 7 and is positive for dropouts and zero for students indifferent between staying in school and leaving. That is,

NB(dropping out) [greater than or equal to] 0 [left and right arrow] [a.sub.i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2] [greater than or equal to] [a.sub.i][b.sub.i] + [a.sub.2]([[theta].sub.pub] - [b.sub.i]). (8)

Who, then, is the educational dropout in our model? First, students drop out only if the level of their frustration outweighs the value of their classroom experience as enhanced by the relevant peer group effect. Hence, the higher the degree of frustration with the uniformity of the public education program (i.e., the higher is [a.sub.i]), the more likely it is that the net benefit of dropping out will be positive and that the student will leave the public school system. Second, suppose that the distribution of student frustration with the public system is asymmetric about the mean ability level. Students with higher ability will then be better able to meet their education demands through the public school system (i.e., [lambda] < 1 for [b.sub.i] > [[theta].sub.pub]). Consequently, the proportion of the pool of school dropouts that have lower-than-average levels of ability will be larger.

If we expand the first term in Equation 8 and set the net dropout benefit equal to zero, we find a quadratic in [b.sub.i] that can be solved for the two ability levels at which such a student would be indifferent between dropping out and staying in school. (16) These values can be solved for as

b = [[theta].sub.pub] + [([a.sub.1] - [a.sub.2])/2[a.sub.3i][lambda]] [+ or -] [{[[([a.sub.1] - [a.sub.2])/2[a.sub.3i][lambda]].sup.2] + [a.sub.1][[theta].sub.pub]/[a.sub.3i][lambda]}.sup.1/2] (9)

Given that [a.sub.1] > [a.sub.2] and that [a.sub.3i][lambda][[theta].sub.pub] > [a.sub.2], (17) the net benefit of a student dropping out of school (from 8) begins positively at [b.sub.i] = 0 and falls as student ability rises. The net benefit becomes zero at the smaller of the two values of b < [[theta].sub.pub], (18) continuing to fall until it reaches its nadir at [b.sub.min] = [[theta].sub.pub] + [([a.sub.1] - [a.sub.2])/2[a.sub.3i][lambda]] > [[theta].sub.pub]. (19) Beyond that level, net benefits begin to rise until they become positive again at the higher value of b. If [lambda] were equal to one everywhere, the net benefit curve would be symmetrically U shaped about [b.sub.min]. As [lambda] [right arrow] 0, however, the net benefit-of-dropping-out curve flattens out above [[theta].sub.pub]. This implies that for a normal distribution of student abilities, any fall in [lambda] will increase the proportion of the potential student population that is below the lower b bound as compared to those above the upper bound. The more asymmetric the frustration effect (the closer [lambda] is to zero), the more the dropout problem is associated with lower rather than higher levels of ability. (20)

If the typical ability level of dropouts is below the population average, school quality or the average level of ability and performance of students who stay in the public system will be higher than the population average. (21) Somewhat ironically, then, the dropout problem has created a positive peer group effect for those students who remain in the school system. Such a finding also implies that if public policy does succeed in increasing educational productivity ([a.sub.1]) to recapture more dropouts, the reduction in dropout numbers would produce a negative peer group effect on the entire public system. (22) Confined to the institutional alternatives of either attending the public system or dropping out, higher completion rates could be achieved only by lowering the average level of student performance. (23)

4. The Entrance of Positive Tuition Private Schools

Suppose that fee-charging private schools are now allowed to enter and service niche markets. In our model, this means that private schools will enter at the ability extremes for those students for whom the utility cost of private schooling is low relative to the frustration received in the public school system. There are now three margins of choice: one in which students are indifferent between private and public schooling, a second where students are indifferent between a private school and dropping out, and a third where (as before) students are indifferent between staying in the public system and dropping out. Because the general form of the third remains unchanged from section 3 (Eqn. 8), we examine the net benefit of attending private school relative to the alternatives of attending a public school or dropping out. This allows us to isolate students who would be attracted to a private school out of the public system and the dropout pool.

Because private schools require the payment of an additional school fee, p, the level of income held by students and their families is now a key factor influencing private school choice. And, given diminishing returns to income, the utility cost of financing that private education will be lower for families with higher incomes. However, for private schools to become viable, they must compensate families for the higher cost of private education by providing some additional service (than can the public system). We assume that this advantage arises through the collective ability of the private school system to provide a nonstandardized education mix to students who have experienced frustration with the uniformity of the public school product. More formally, the net benefit of attending a private school relative to a public school is found by subtracting Equation 5 from Equation 6 to get

NB(Private/Public) = [U([y.sub.i](1 - t) - p) - U([y.sub.i](1 - t))] + [a.sub.2]([[theta].sub.pri] - [[theta].sub.pub]) + [a.sub.3i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2]. (10)

The first term in Equation 10, the utility difference within the square brackets, represents the utility loss associated with the private school fee, p. The second term, the net peer group effect, reflects the fact that the students attracted into the private system may have different levels of ability than those in the public system, so that [[theta].sub.pri] can differ from [[theta].sub.pub]. The last term, [a.sub.3i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2] reflects a net education gain from attending a private school--measured as the loss in frustration experienced by students who can now learn in an educational environment that more closely matches their own ability.

Note from Equation 10 that if we ignored the differential peer group term and assumed that the frustration cost was symmetric ([lambda] = 1), the net attraction of the private system to those disillusioned with the public system would be symmetrically distributed about the mean level of ability. Private schools would be just as likely to be designed to cater to low-ability students as to those at the high end of the ability spectrum. The presence of the differential school quality term, however, means that lower-ability students who group together in their own specialized private school lose the positive external effect of associating with higher-ability peers. For them, the peer group effect is perverse, a cost rather than a benefit of choosing the private system. For higher-ability students, the peer group term reinforces their desire to escape the frustration of the public system. By itself, this effect would imply that the distribution of private schools would be skewed toward higher-ability students. (24 ) On the other hand, if higher-ability students are better able to meet their educational objectives through the public system (i.e., if [lambda] is close to zero), the less will be their relative frustration. In the limit with [lambda] = 0, higher-ability students would gain only a peer group effect from attending private school, and the distribution of private schools by ability types would become increasingly skewed toward lower-ability students.

It follows that the private system attracts out of the public system nonconformist students from both tails of the ability distribution, with a bias that depends on the strength of the peer group effect relative to the frustration asymmetry. Should the peer group effect dominate, high-ability-recruitment bias introduces into the public school system the negative peer group effect highlighted by ER. On the other hand, if the public system better responds to the wishes of higher-ability students and/or their parents, the public system would experience a positive peer group effect as private schools cater more to lower-ability students. Whether or not the private school system will focus on one ability type, the students who do withdraw from the public system will typically have experienced higher-than-normal levels of frustration in school and come from families with higher-than-average levels of income and wealth (for whom the utility cost of private education is low).

The effect produced by the entrance of private schools on the dropout problem is found by subtracting Equation 7 from Equation 6. That is, the net benefit of now attending a private school rather than dropping out is

NB(Private school/dropout) = [U([y.sub.i](1 - t) - p) - U([y.sub.i](1 - t))] + [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.pri] - [b.sub.i]). (11)

As might be expected, a dropout will choose to attend a private school only if the utility loss associated with paying the additional private school fee is lower than the education gain that can be received by attending a private school. Because private schools also attract out of the dropout pool those students who have already chosen not to attend the public school system, Equation 11 can be combined with Equation 8 to produce

[U([y.sub.i](1 - t)) - U([y.sub.1](l - t) - p)] < [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.pri] - [b.sub.i]) < [a.sub.3i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2]. (12)

In essence, private schools attract public school dropouts for whom education is economic only if it can be paid for with income rather than with frustration. As discussed previously, the dropout pool that private schools recruit from consists of both high- and-low ability students but one that becomes more heavily weighted toward the lower-ability potential students as 1 [right arrow] 0.

The introduction of a competitive private school system then produces three types of allocative effects. First, private schools recruit out of both the public system and the dropout pool those students who can afford the fee and have experienced a relatively high degree of frustration with public education. These students will come from families with relatively high income levels. Second, the existence of the peer group effect, ceteris paribus, does make it relatively more attractive for private schools to recruit students with higher rather than lower levels of ability. However, this tendency is offset by the degree to which the public school system itself modifies its structure in the directions demanded by higher-ability students. To the extent that it is successful, lower- rather than higher-ability students form a larger proportion of the dropout pool and primary market for private schools. In this case, average ability levels within the public school system would rise as lower-ability students (with hig her-income parents) leave for private schools. This would imply a higher level of frustration for lower-ability students who remain and a reversal of the ER peer group result. Third, for the education system as a whole, the aggregate level of educational achievement rises by reducing the dissipation caused by student frustration and by recovering students who would otherwise fall into the dropout pool. Note that depending on the relative size of the peer group and frustration effects, the average level of student ability and hence the average quality of all schools could rise or fall as private schools absorb more students from one end of the ability spectrum or the other. (25)

In both ER and our analysis, the peer group problem and its interaction with the dropout problem can be seen to have begun well before the arrival of a voucher system. This is because in both analyses, there is a tendency for private schools to attract particular types of students from the public system. For ER, this arises because higher-income, higher-ability students (and their parents) place a higher value on education. This encourages private schools to attract higher-ability students and internalize through higher average school fees the positive peer group effect their presence will generate. Lower-ability students are less willing to pay for the higher quality of education and are left behind. In our analysis, income gives families the ability to rescue their nonconformist children from dealing with the uniformity of a public system. This is equally true for parents at both ends of the ability spectrum. (26)

Stated alternatively, the allocative role of the private schools is to allow the education system as a whole to match more closely the tastes and aptitudes of students and their families. In this sense, education is provided more efficiently. In terms of the distribution of benefits, however, the market basis for allocating education variety (through fees) means that students from higher-income families will be favored. Because the benefits of variety are not valued equally by students of equivalent ability, lower-income families are less likely to exercise private school choice and hence will benefit less from its availability. (27) Dropouts from low-income families, in particular, find that additional choice has not been to their advantage. Their relative position in the welfare hierarchy falls, falling behind those who now leave the dropout pool for private school and remaining well below those who stay within the public system.

5. The Role of Vouchers

What, then, does the introduction of a voucher system do in our model? In large part, of course, the effects will depend on the particular characteristics of the voucher system (see West 1997 for a survey of types). However, the fact that a voucher is a grant conditional on attending private school means that the voucher has no direct effect on the margin between attending public school and dropping out. Vouchers affect that choice indirectly and only if average school quality is altered. Along the other two margins, however, the giving of a grant conditional on private school attendance tilts education choices toward the private system. A voucher system will encourage the private school system to recruit students out of both the dropout pool and the public school system.

Suppose, then, that the voucher system consists of a uniform grant to all families choosing to attend a private school and that the grant is funded out of general tax revenues raised through income taxes. In this case, the voucher system will be progressive ex ante in its effect on the distribution of income (in that families with higher incomes receive the same benefit but pay more than their share of the cost). (28) More often, a voucher program will be even more restrictive in its coverage--for example, available only to inner-city and/or lower-income families--so that the program in practice is more progressive. Finally, because of diminishing marginal utility, a voucher program is more progressive in utility than in relation to income. In short, whether or not the voucher program redistributes education according to ability, the program will work to diminish the effect of income on the choice of education alternatives.

While the income redistribution effects have often been recognized, it is the unforeseen allocative consequences of a voucher program that impact on the "least well off" that lies at the root of the ER peer group concern and hence merits closer attention. We begin with the effect of the voucher program on the private/public school margin. Using [nu] to represent the size of the conditional grant, the net benefit condition from Equation 10 now becomes

NB(Private/voucher) = [U([y.sub.i](1 - t) - p + v) - U([y.sub.i](1 - t))] + [a.sub.2]([[theta].sub.pri] - [[theta].sub.pub]) + [a.sub.3i][lambda][([b.sub.i] - [[theta].sub.pub]).sup.2]. (13)

From Equation 13, it is apparent that receiving a voucher is equivalent to having the private school fee reduced and thus lowering the utility cost to all potential entrants. (29) The concavity of the utility function, however, means that a uniform voucher has a larger proportional effect on lower-income families. In relative terms, more low-income students and their families (whose level of frustration with the public school system was previously not quite high enough to merit the utility cost of the private school fee) now choose to attend a private school.

As we have seen, private schools will recruit both high- and low-ability students. Because the peer group effect in the second term of Equation 13 raises the educational benefit of operating a private school targeted at higher-ability students, recruitment on this basis alone would drain from the public system relatively more higher-ability students. A voucher system would then increase the negative peer group effect on the public system even as it spreads the benefits of differentiated education more evenly across income. On this margin, then, our model is consistent with ER's results. On the other hand, the asymmetry in frustration cost could offset the peer group effect and result in relatively more lower-ability students receiving subsidized private school education. In this case, average ability levels in the public school system would rise and generate a positive peer group effect.

In addition to its effects on public school enrollment, the voucher system will also alter the marginal condition facing students who had previously dropped out of the education system altogether. These were students who found that the frustration of having to conform to the public system dominated any positive education experience and could not afford the additional fee to attend private school. For them, the voucher now means that the net benefit of attending a private school rises to become

NB(Private/Dropout\voucher) = [a.sub.1][b.sub.i] + [a.sub.2]([[theta].sub.pri] - [b.sub.i]) - [U([y.sub.i](1 - t)) - U([y.sub.1](1 - t) - p + v)]. (14)

As in Equation 13, the introduction of a voucher system lowers the income cost of attending private school. Because the voucher system leaves the public school/dropout margin unaffected, the reduction in private school fees lowers a binding constraint on educational choice and allows the education system to recapture some students who would otherwise remain lost. The voucher system increases the scale of the private school system unambiguously but not primarily by raiding the public system of its best students. Rather, the private system expands by attracting high- and low-quality students out of both the dropout pool and the public system.

In the absence of the dropout problem and with the assumed ability to price discriminate among students, ER find that the benefits of the voucher program will be appropriated initially by private schools through their recruitment of high-ability students. Moreover, under open entry, the subsidization of private schools results ultimately only in an increase in the degree to which the private student population is segmented among schools by ability. For students who remain behind in the public system, there are only peer group losses. Hence, to forestall the dissipation of rent through excessive segmentation, ER propose a redesign of current income-based voucher programs to redirect funds toward the public schools that currently cater to low-ability students. In this way, benefits can be retained for those students most in need.

In our analysis, however, further subsidization of the public system would do little to assist those who we would qualify for the Rawlsian designation of "least advantaged," that is, those who are most at risk of dropping out of the education system entirely. For them, the traditional voucher strategy of conditionally subsidizing students by income is entirely appropriate. Such a voucher program diminishes the underlying inequality directly and results in greater education choice, more aggregate education, and less individual frustration with the education system.

The new benefit highlighted by our analysis is that vouchers will lead many students who would otherwise have chosen to drop out to remain in school. It does so via the creation of new private schools that compete to better sort students by ability, and this results in lower overall levels of frustration for almost all students within the education system. In our analysis, then, a voucher program promotes greater educational diversity and directs those gains to those students who were most disadvantaged under the previous system. Moreover, because student frustration is likely to arise at both ends of the ability spectrum, the primary beneficiaries need not be those with the highest levels of ability. To the extent that the beneficiaries lie closer to the bottom of the ability distribution, the ER peer group problem is reversed, and net external benefits arise within the public system. On the other hand, the use of vouchers may well increase the negative peer group effect experienced by those students who rem ain in the public system. Thus, even though we question whether ER's peer group concern truly identifies the Rawlsian most deserving, our analysis does suggest a legitimate concern for the potential loss that could be experienced by students within the public system. (30)

6. Vouchers and the Scale of the Dropout Problem: A Plausible Connection?

In the previous sections, we used differences in ability across students to investigate how a voucher program that could create a negative peer group migration effect on the public school system might also alleviate the student dropout problem. It is immediately apparent, however, that if a voucher system does lead students to exit the public system on the basis of some other characteristic, the size of the negative peer group effect within the public system would be reduced, perhaps substantially. To use one specific example, if a voucher program allowed disruptive students who need greater-than-normal levels of attention and discipline to receive better attention in a private school, the peer group migration effect on students remaining in the public system would be positive rather than negative. (31)

Similarly, when discussing the effect of inputs on education output, the measure of school achievement most often used is average test scores received by students. This is, of course, not necessarily the same thing as measuring the effectiveness of those inputs in preventing school dropouts. Moreover, to the extent that test scores do measure the achievement of those actually in school, a change in test results may mean the opposite of what they appear. As Wenger (2000) has emphasized, the view that schools are producers of multiple competing outputs means that "if schools increase expenditure in ways that increase education levels rather than test scores, increasing expenditures per pupil could even lead to falling test scores" (p. 34). Applying this directly to the issue of peer group effects, she writes that "it seems likely that keeping below-average students in school longer may decrease average test scores because the retained students have relatively low test scores and/or because these students may sl ow down the rate of learning in the classroom" (p. 34). (32)

In this section, then, we look to the education literature for a more complete description of the characteristics held by dropouts and the type of school characteristics that have been found to be helpful in stemming the incidence of dropping out. (33) Our purpose is to ask whether a subsidized expansion of the current private school system through vouchers would better retain or reabsorb school dropouts and, second, whether the private system has in fact been any more successful than the public system in preventing dropouts. (34) Note that this need not mean that private schools are inherently more productive than public schools. Rather, as our analysis has emphasized, the greater choice offered by an expanded private system may in itself allow both students and their parents to self-select into schools that better match individual needs.

We begin our discussion with the set of individual and family characteristics that the education literature has found to be associated with student dropouts. Disproportionately, they consist of students who come from single-parent families and families with low income and socioeconomic status and of students who have performed poorly in school, who have low self-esteem, who lack positive relationships with adults and peers, who have experienced pregnancy and/or delinquency, and who have a history of substance abuse (Natriello 1994). Two things are apparent. First, academic ability is only one of the factors that affects the dropout decision. Second, it is likely to be prohibitively costly or even impossible for any schools to substitute for all social deficiencies. Nevertheless, the education literature does suggest that the school environment can substitute for at least some, and this philosophy has encouraged education research to focus on school characteristics that could alleviate the problem.

For education researchers, many of these dropout characteristics (low self-esteem, lack of care and support, and the absence of positive adult and peer relationships) suggest that student dropouts come from dysfunctional family units with an absence of supportive social networks, a lack of what is now called "social capital." Fostering social capital, that is, the development of stronger supportive ties between students and adults, can take place in a positive school environment, and this can help reestablish motivation and interest in students. Not surprising, then, supportive relationships between teachers and their students and an established climate of purpose and concern have been identified as important elements for keeping students in school (Rumberger 1987).

The role of social capital in preventing school dropouts has for many years been the research interest of Coleman and Hoffer (hereafter CH). In their widely known study of high school dropout rates in private and public schools, CH (1987) argue that Catholic schools have a stronger interest in countering what they call "the depressive effects of functional or structural deficiencies among families" (p. 97). By providing a more supportive learning environment and broader sense of social concern, the dropout rate in Catholic schools was expected to be lower, especially relative to public schools. Their results, presented as Table 1, are consistent with this prediction. Catholic school students in 1987 experienced roughly one-quarter the dropout rate of public school students and one-third the dropout rate experienced by students in other private schools. (35) Only high-performance private schools that devote considerable resources to students have succeeded in producing a lower dropout rate.

Other analysts have gone beyond the dysfunctional family to focus on the dysfunctional school. Ekstrom et al. (1986) and Legters, McDill, and Partland (1992), for example, have drawn attention to particular organizational features of the school system that have been particularly effective in producing dropouts. School dropouts have typically attended schools that are relatively large with specialized departments of instruction. In these schools, teaching roles become defined in terms of their subject matter rather than in terms of their ability to create interest for students. Additionally, individual teachers tend to have little direct involvement with either their students or their students' families.

Because the typical private school is less than half the size of the typical public school and students are taught by teachers who have smaller average class sizes and who stay longer each day with the same set of students, private schools would seem to have an advantage in providing the kind of positive support relationships lacking in a dropout's environment. (36) To the extent that smaller school sizes and more focused personal attention can enhance performance, realized academic success would reinforce the higher probability of staying in school (Natriello 1994). On the other side, the typical public school has continued to grow in size with students being taught increasingly by a number of different teachers in different subject specialties. The consolidated district public school is, in other words, increasingly different from the older neighborhood school. (37) This trend may well improve the content and quality of education received by most students who encounter few adjustment problems and who receiv e greater stimulation in an academic environment. However, the greater concern with subject area and professionalism has made the public system, in relative terms at least, more impersonal and less individually flexible. In such an environment, teachers are given less motivation to anticipate and react to non-subject-related learning difficulties and find that less is expected of them in substituting for other family and social deficiencies. In short, it is argued, the educational focus is increasingly on the school's scholastic achievement and not on the retention of its potential dropouts. (38)

More recent research, however, has tended to place quite a different interpretation on the superior performance of the Catholic and, to a lesser extent, non-Catholic private school system. Rather than attribute superior achievement and higher completion rates to the characteristics of the school or school system, many recent writers now suggest that a proper accounting of student and parental characteristics would explain much of the differential (Murnane, Newstead, and Olsen 1985). Tests for whether Catholic schools still produce superior performance after accounting for socioeconomic selection criteria have had mixed results (Neal 1998). For example, even after controlling for selection issues, both Evans and Schwab (1995) and Neal (1997) continue to find evidence of superior student achievement in Catholic schools. On the other hand, both Sander and Krautman (1995) and Figlio and Stone (1999, p. 134) find that "public schools compare at least as well as private in test performance and high school completio n rates." Perhaps most important, however, all these studies illustrate the general point that student and family characteristics may account for much of the superior achievement of the private system.

If differential school productivity cannot fully explain the difference, what explains why these students and their parents choose to attend private schools? (39) Figlio and Stone suggest that "parents may seek a more disciplined environment for their children, might desire for their children to have a religious education, or might desire a higher probability that their children will be able to participate substantially in extracurricular activities. In addition, parents might wish for their children to interact with certain peer groups" (1999, p. 135). All these reasons, we would argue, are consistent with the avoidance of frustration hypothesis outlined in this paper. From our perspective, private schools need not be more academically productive than their public counterpart for a voucher program to reduce the overall dropout rate. The expansion of the private system need only permit the sorting of students in ways that result in greater student compatibility with their learning environment. In our terms, s uch sorting reduces individual frustration with having to conform to the public program and so enhances the probability of the student staying in school. (40)

These comments seem to suggest that the current private school system and the selection preferences of parents currently choosing private schools will mean that voucher use will favor performance improvements only for higher-ability, more socially privileged students. And if greater voucher use expanded the education opportunities of only such advantaged students, vouchers could not be expected to address the difficulties that the public school system has had in dealing with less privileged subgroups of society. As Toma (1999) has observed, "It is the lower socioeconomic groups in the United States whom the system has failed most noticeably over the past three decades. This is the category where dropout rates are greater than graduation rates. In certain inner city schools such as that of Washington, D.C. and Detroit, Michigan, [only] 36 percent of the students complete four years of high school" (p. 7). (41)

However, it is precisely in relation to less advantaged students, that is, in relation to the students of large-city urban minorities, that private schools have been found to have achieved their greatest success (Neal 1997; Figlio and Stone 1999). (42) Such empirical findings have then given impetus to some education advocates who now see vouchers as a way of overcoming income barriers and so alleviating chronic educational deficiencies by targeting explicitly these groups for vouchers. For these students and their parents, one clear constraint on current school choice has been income. In addition, greater voucher use will induce entry into the private school market, and the resulting competition for students can be expected to expand the variety and character of both private and public schools (Hoxby 1998, 2000). Florida Governor Bush's recent plan, the Bush/Brogan A+ Plan, to target vouchers at students who currently attend "failing" schools is one example of the creative experimentation taking place with vouchers.

7. Conclusion

Current writers often suggest that the greater use of school vouchers will bring about the unfavorable trade-off of economic efficiency for social inequity (ER 1998a, b; Hoxby 2000). The potential efficiency gains arising from greater school competition and students who are better streamed, they argue, may be dominated by the negative peer group losses experienced by those who are most vulnerable, namely, those lower-ability students who remain in the public school system. In contrast, our analysis adopts an alternative definition of the Rawlsian "least advantaged" and instead focuses on the effect vouchers have on low-income school dropouts. Our model suggests that vouchers work by lowering the cost of education to low-income nonconforming students and so increasing their probability of remaining in school. Should voucher use result in education gains to previous dropouts that exceed the peer group losses, there would arise a net equity gain that would reinforce the potential efficiency case usually made for voucher use.

The desirable effects of vouchers on student dropouts highlighted here do not require private schools to be any more productive than public, nor does the case deny that the current private schools segment the student pool by ability and produce adverse effects on those students who remain in the public system. (43) Rather, we argue that a voucher system could help restore a distributive balance. In relative terms, a voucher system will benefit those students from lower-income families who are least able to conform to the uniformity of a public school system and will improve their position in the education system by allowing them to choose a school whose curriculum and student body more closely suit their ability and temperament. With vouchers subsidizing private schooling, new entrants can be expected to compete across more dimensions of student alienation and so assist in rescuing from the dropout pool and the current public system more of those who were previously least likely to realize success. A selectiv e voucher that is targeted at lower-income students will only increase the size of these gains.

The advocacy of school vouchers, however plausible, cannot substitute for the more detailed empirical work needed to quantify the scale of these theorized effects. Only with a better grasp of the types of student sorting induced by voucher use and with better knowledge of the comparative magnitudes of the peer group and dropout effects can a more conclusive case be made for or against voucher use. Our reading of the literature is that the hypothesis that voucher use will reduce the scale of the dropout problem is not inconsistent with empirical evidence. Whether the scale of this effect leads to benefits that are sufficient to offset the magnitude of the ER peer group losses remains an important unanswered question. We conclude, then, with a call for more empirical work on both the efficiency and equity effects of existing voucher programs and the better design of voucher experiments to allow the better isolation and measurement of all these separate effects.

Table 1.

Dropout Rates Across Sectors

                           Percentage
Sector                    Dropping Out

Public                        14.4
Catholic                       3.4
Other private                 11.9
High-performance private       0.0

Source: Coleman and Hoffer (1987, p. 99).

Received April 2000; accepted March 2001.

(1.) Figlio and Stone (1999, p. 136) conclude that religious (private) schools benefit some subsectors of students, highlighting particularly African-Americans in big cities. This is consistent with Neal's (1997) earlier finding that Catholic schools benefited urban minorities.

(2.) There is considerable evidence that the dropout rate is large (particularly in inner-city areas) and that the consequences of dropping out are substantive. In relation to the first, the status dropout rate for individuals between 20 and 21 in 1997 was 12.7% (U.S. Department of Education, 1998, table 106, p. 125). It is also clear that dropping out has a substantive effect on lifetime earnings. Current evidence suggests that not finishing high (elementary) school results in both an annual income that is only 70% (56%) of median (full time) income and a much higher probability of unemployment (U.S. Department of Education, 1998, tables 380 and 383).

(3.) The stylized fact in the literature is that public schooling may produce similar input quality adjusted output but at a somewhat higher expenditure per student, at least on the margin. See Figlio and Stone (1999, p. 136). Caroline Hoxby finds the same net difference in favor of private schools but believes that subsidization of private schooling would raise achievement within public schools while not significantly impacting spending per student (1998, p. 54). On the other hand, Ruggiero and Vitaliano (1999) find private school cost savings of up to 15%.

(4.) Different types of vouchers would include general school vouchers (available to all students) to selective vouchers targeted at specific types of students, such as those in low-income families (the Milwaukee Program) and/or those who attend schools that consistently underperform (Florida). The distributive benefits highlighted by our analysis are larger the more the voucher program is restricted to lower-income students/families.

(5.) U.S. Department of Education (1998, table 106). The text uses the 12.7 percentage for the 20- and 21-year-old (all races, both sexes) age-group in 1997.

(6.) The generality of this analysis has been criticized for the limitations in sample coverage.

(7.) The work of Henderson, Mieszkowski, and Sauvageau (1978) on Canadian data supports Summers and Wolfe's (1977) finding that lower-ability students are helped by being with a higher-average-ability student group but dispute whether the peer group effect experienced by higher-than-average-ability students becomes negative. In our model that follows, we allow for the possibility of a similar asymmetry in effect about the mean,

(8.) Most writers on vouchers tend to agree that the competition introduced by vouchers will result in "technical efficiencies" (see Epple and Romano 1998b). White recognizing the importance of this consideration, we wish to focus on the equity thrust given by Manski (1992). Accordingly, we abstract from the efficiency considerations to highlight Rawis's "difference principle," a concern with the equity of the least able in society.

(9.) The analysis assumes that there are no integer problems arising from insufficient numbers of students capable of supporting private schools targeted at each particular student type.

(10.) We note here that the existence and type of voucher could have an important effect on the public's willingness to fund schooling. See, for example, Bearse, Glomm, and Ravikumar (2000).

(11.) The values of the first derivative at zero and infinity guarantees that individuals will choose some positive level of consumption whether or not they choose to acquire education, that is, U'([c.sub.1] = 0) > [a.sub.i] and U,([c.sub.i] = [infinity]) = 0.

(12.) This also implies that the peer group effect becomes negative for students who have a level of ability that is higher than the school average.

(13.) While we measure the degree of frustration along the dimension of ability, ([b.sub.i]) could be thought of as a vector of student attributes. The squaring of ([f.sub.i]) recognizes that frustration will increase at an increasing rate for students on either side of the school average.

(14.) If the degree of frustration experienced by high- and low-ability students in the public school system relative to the private school system were symmetric about the mean, [lambda] = 1. However, it is often argued that students with above-average abilities are better able to articulate their degree of frustration to teachers and/or the school board and so receive more teachers, classes, and/or programs that more closely match their abilities even within the public system. To the extent that the public school system does track higher-ability students into special classes with more appropriate curriculum to a degree equal to that of the private system, [lambda] = 0 for [b.sub.i] > [[theta].sub.j], and frustration (in our sense) will arise only for students with lower-than-average abilities.

(15.) In this case, the frustration parameter will likely be lower than in the cases where students have the alternative of attending a private school, hence the use of the primed coefficient in this particular case.

(16.) At the margin, NB(dropping out) = 0 = [a.sub.3][lambda][b.sup.2] - [2[a.sub.3][lambda][[theta].sub.pub] + ([a.sub.1] - [a.sub.2])]b - ([a.sub.2][[theta].sub.pub] - [a.sub.3][[theta].sub.ub]).

(17.) The former implies that the direct effect of teaching dominates the peer group effect, and the taller implies that frustration dominates the peer group effect at the ability extremes.

(18.) Note that the square root term in Equation 9 is larger in absolute size than [([a.sub.1] - [a.sub.2])/2[a.sub.3i][lambda]] so that the two values of b straddle [[theta].sub.pub]. With [lambda] < 1 for [b.sub.i] > [[theta].sub.pub], the larger value of b will be more than proportionately above the public school mean, [[theta].sub.pub], than the lower value of b is below.

(19.) From footnote 16, [partial](dropping out)/[partial][b.sub.i] = -([a.sub.1] - [a.sub.2]) + 2[a.sub.3i][lambda]([b.sub.i] - [[theta].sub.pub]. The slope of the net benefit curve is negative at [b.sub.i] = 0 as long as the direct benefit of learning ([a.sub.1]) exceeds the indirect peer group effect ([a.sub.2]) and then rises in value to become zero at [b.sub.i] = [[theta].sub.pub] + ([a.sub.1] - [a.sub.2])/2[a.sub.3i][lambda] > [[theta].sub.pub]. For larger values of [b.sub.i], [partial]/[[partial].sub.i] becomes increasingly positive.

(20.) In terms of the other parameters driving the dropout choice, an increase in educational productivity ([a.sub.1]) and a diminishing of the size of the peer group effect ([a.sub.2]) will spread further apart the two values of b. These effects will reduce the scale of the dropout problem.

(21.) High-income families often congregate in geographic areas and use their political influence to redistribute resources within the public system. As Toma (1999) has written, upper-income students "live in nice suburban homes and consume the best the public system has to offer. They get the best buildings, the best teachers, the most rigorous curriculum, and the widest variety of extracurricular activities" (p. 7). This implies that in practice the dropout problem will fall that much more inequitably on lower-income students and their families.

(22.) Not coincidentally, the same factors imply that it is in the teacher's and the school's interest to discourage low achievers (costly to teach students) from continuing to attend school.

(23.) Greater success with the dropout problem implies lower measured success on average. But a lowering of average achievement scores will correspond to lower overall achievement only if the same number and type of students are involved. If the incremental achievement of previous dropouts were recognized or the reduction in dropout rates (or rise in the graduation rates) were valued independently, the aggregate outcome would appear quite different, So this, see Wenger (2000).

(24.) ER (1998b) argue that income and education are complements so that the demand for specialized education increases with both income and ability. If this were so, our distribution of private school types would be more skewed in favor of high-ability students.

(25.) Should the peer group effect dominate the frustration asymmetry in favor of high-ability students, the average level of ability in the school system as a whole will rise somewhat at the expense of the public system.

(26.) Because higher-ability students provide private schools with an output spillover, some private schools will find it profitable to specialize in higher-ability types. On the other hand, low-ability students who are less able to influence the public school board increase the demand for specialized private schooling to overcome the peer group loss of leaving the public system. These are the students whom our analysis highlights relative to ER. In both cases, private schools allow income to register intensity. Hence, rather than necessarily favoring one ability type, it will be primarily students of higher-income families who get skimmed from the public system. Lower-income students who experience the same degree of frustration are more likely to choose to stay behind in the public system.

(27.) Depending on the relative size of the frustration asymmetry relative to the peer group effect, some types of low-income students in the public system will gain from the larger proportion of either high- or low-ability students who depart for private schools.

(28.) Ex post vouchers could be used primarily by higher-income families that would have attended private schools anyway. In this case, higher-income families would benefit at the expense of the general public, and the voucher program becomes regressive. This arises because the program can have intramarginal as well as marginal effects. Such effects will be minimized if the voucher is restricted to low-income families (as in the Milwaukee example).

(29.) Again we assume that individuals ignore the consequences of their schooling decision on their tax bill so that t is viewed as independent of school choice.

(30.) Ironically, one means of promoting productivity improvement within the public system may be the voucher system itself. That is, the additional competition that vouchers generate both among schools in the same system and across school systems encourages education reform and has sometimes been found to reduce the real cost of providing public education. Ruggiero and Vitaliano (1999, p. 329) suggest the availability of gains of up to 15%.

(31.) More generally, if the voucher program attracts students by features that are independent of ability, there would be no peer group effect in the sense of Manski (1992).

(32.) Similarly, when commenting on the usual finding that teacher experience is unimportant, Summers and Wolfe (1977) report, "We found that students whose third-grade score was above average benefitted from more experience, but those who were very much below grade level were negatively affected. In fact this group did best with newer teachers who perhaps have undampened enthusiasm for teaching those who find it hard to learn" (p. 644).

(33.) This discussion implicitly assumes that the characteristics of the students in the two programs are identical so that the difference in dropout experience can he attributed to the school system. More recent work, however, has investigated whether the ability to self-select into the private school system gives that system higher-quality students and hence attribute "too much" of the achievement result to the system (rather than to the special characteristics of the student). See Evans and Schwab (1995), Neal (1997, 1998) and Figlio and Stone (1999).

(34.) Given the characteristics of the dropout pool and the public/private school margin, one might expect that the additional competition provided by the entry of new schools would lead private schools to better target programs that would retain students. This reinforces the positive aspect of Hanushek's (1998, p. 23) reassessment that "existing work does not suggest that resources never matter. Nor does it suggest that resources could not matter. It only indicates that the current organization and incentives of schools do little to ensure that resources are used effectively" (p. 23). See also Hoxby 2000.

(35.) In more recent empirical work, Sander and Krautman (1995) also conclude that "Catholic schooling has a significant negative effect on the odds of dropping out" (p. 232).

(36.) The average public school size was 527 students with a pupil-to-teacher ratio of 17.3 in 1995. The corresponding numbers for private schools were 182 and 14 (U.S. Department of Education, 1998, tables 60 and 89).

(37.) Between 1988 and 1995, the total number of public school districts declined in each year (15,376 to 14,766) as the proportion of students in the largest-sized districts continued to grow. Over 30% of students are now taught in school districts of over 25,000 in size (U.S. Department of Education, 1998, table 91).

(38.) The increased focus on subject disciplines is seen in she rise of the average educational achievement required of public school teachers relative to private. Similarly, the increased disadvantage that public school teachers face in dealing with low-achieving (difficult) students can be seen in the ever higher student-to-teacher ratio and other such indicators as the larger average distance they live from their students and the rising average age and job tenure of public school teachers compared to their private counterparts (U.S. Department of Education, 1998, tables 65 and 70).

(39.) Note that by the fall of 1995, Catholic school enrollment (2,519,205) was roughly 50% of total private school enrollment (5,032,200), with enrollment in other religious private schools (1,743,791) accounting for roughly two-thirds of the other 50%. Recent court decisions have allowed some states (e.g., 1998 for Wisconsin) to extend the benefits of voucher programs to Catholic and other religious schools.

(40.) The perceived superior performance of private schools is reinforced by the perceptions of teachers. While one must make full allowance for the subjective nature of such evidence, a 1993-1994 U.S. Department of Education survey of teachers' perceptions of the seriousness of different issues in their respective schools does report that 14.1% of public school teachers identify dropping out as a serious problem in their secondary school, while only 1.3% of private school teachers come to the same conclusion. This concern is also reflected in answers given to broadly similar questions, such as the seriousness of student absenteeism (27.1 to 5.2, respectively) (U.S. Department of Education, 1998, table 91).

(41.) Rumberger (1987) notes that the 63 high schools in Chicago have dropout rates that vary between 10% and 62%.

(42.) Rouse (1998a) finds a similar but more modest achievement effect for private schools in the Milwaukee school choice experiment.

(43.) This is particularly likely if, as ER assume, income and ability are complementary in the demand for education.

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J. Stephen Ferris (*)

Edwin G. West (+)

(*.) Department of Economics, Carleton University, Ottawa, Ontario KIS 5B6, Canada; E-mail [e-mail]; corresponding author.

(+.) On October 6, 2001, Eddie West passed away in his eightieth year after a long struggle with cancer.

We would like to thank Keith Acheson, Zhiqi Chen, Ambrose Leung, and two referees from this Journal for confirming the presence of positive peer effects through the qualitative improvements they have helped make in this paper. They are, of course, not responsible for the errors that might remain.