Estimating the Effects of Private School Vouchers in Multi-District
Economies*
Maria Marta Ferreyra
Graduate School of Industrial Administration
Carnegie Mellon University
March 20, 2003
Abstract
This paper estimates a general equilibrium model of school quality and household residential
and school choice for economies with multiple public school districts and private schools. In the model,
households have heterogeneous religious preferences, and private schools can be either Catholic or
non-Catholic. The model is estimated with 1990 data from large metropolitan areas in the United
States, and the estimates are then used to simulate two large-scale private school voucher programs in
the Chicago metropolitan area: universal (non-targeted) vouchers, and vouchers restricted to nonCatholic schools. The simulation results indicate that both programs increase private school enrollment,
and affect household residential choice, with the largest school quality and welfare gains accruing to
low-income households. Universal vouchers increase enrollment at both Catholic and non-Catholic
private schools, yet when vouchers are restricted to non-Catholic schools, the overall private school
enrollment expands less at moderate voucher levels, and enrollment in Catholic schools tends to
decline, relative to the non-voucher equilibrium, as the dollar amount of the voucher increases. In
addition, fewer households benefit from non-Catholic vouchers.
*
This paper is based on my PhD dissertation at the University of Wisconsin-Madison. I wish to thank my advisors
Derek Neal, Steve Durlauf and Phil Haile for their guidance and encouragement. I am also grateful to Peter
Arcidiacono, Dennis Epple, Arthur Goldberger, John Kennan, Yuichi Kitamura, Bob Miller, Tom Nechyba,
Richard Romano, Ananth Seshadri and Holger Sieg for many helpful ideas and suggestions. I owe special thanks
to Grigory Kosenok for his help with the technical aspects of this project and for many insightful conversations. I
have also benefited from conversations with Oleg Balashev and Hiroyuki Kasahara, and wish to thank seminar
participants at Brown, Carnegie Mellon, Columbia Business School, Duke, Rutgers, U. of Illinois at UrbanaChampaign, Virginia, and Wisconsin, and participants at the 2002 CIRANO conference on education, at Camp
Resources X, and at the Stanford Conference on Empirical Boundaries for IO, Public Economics and
Environmental Economics, for their questions, comments and suggestions. I am grateful to the UW-Madison
Condor Team from the Computer Science Department, and in particular to Peter Keller, whose help has been
essential to carry out this project. Financial support from the UW Department of Economics to purchase data, and
from the UW Robock Award in Empirical Economics are gratefully acknowledged. All errors are mine.
1. Introduction
Vouchers play a central role in the debate about education reform in the United States.
Vouchers, it is argued, give households the opportunity to enroll their children in private
schools and access the type of education that they prefer for their children. Currently, many
households opt out of the public school system to send their children to private schools that
better meet their demands. These households pay private school tuition in addition to the
property taxes used to support local public schools. However, some households prefer private
schools over the public schools in their metropolitan areas, but face budget constraints that
restrict them to public schools.
Although public schools have no explicit tuition, there are costs associated with the
choice of public schools where residences and public schools are linked. In these places,
households must choose public schools and residences as bundles, whose costs are determined
by housing prices and property taxes. Therefore, in order to gain access to their preferred
public schools, households might choose to live in places they would not have selected in the
absence of bundling.
Vouchers, it is claimed, would break this bundling by allowing
households to choose private schools, which have no residence requirements.
Thus, vouchers may not only give households more school choices, but also alter
household residential decisions. As a result, public school districts may experience changes in
their property values, school funding, and the composition of their student populations. To
gain insight into the potential impact of large-scale private school voucher programs, these
general equilibrium effects should be examined.
In this paper I estimate a general equilibrium model that jointly determines school
quality and household residential and school choices within an economy with multiple public
school districts and private schools. I then use the parameter estimates to simulate two
different voucher programs.
Although models of household sorting across jurisdictions originated in 1956 with
Tiebout’s work, few attempts have been made to estimate them (Bayer (2000), Epple and Sieg,
(1999), Epple, Romer and Sieg (2001)). In particular, while researchers who have evaluated
the general equilibrium effects of vouchers have relied on calibration to simulate voucher
experiments (Epple and Romano (1998, 2000), Fernandez and Rogerson (1998, 2001), Glomm
and Ravikumar (1999), Nechyba (1999, 2000)), I use estimates to simulate such experiments.
2
The few voucher programs enacted to date in the U.S. have included a small number of
voucher recipients, and have often restricted the set of eligible private schools. As a result,
researchers have no data to evaluate the general equilibrium effects of potential large-scale
voucher programs, and therefore must turn to simulation to investigate them. In this context,
structural estimation using data from settings without vouchers can prove particularly helpful
by providing parameter estimates that capture the behavior of households and schools. The
parameter estimates can then be used to evaluate the general equilibrium effects of
hypothetical large-scale voucher programs.
One school attribute demanded by numerous households is religious education.
According to the 1989 Private School Survey, 85% of the private school enrollment in grades
9 through 12 was in religious schools. Moreover, the considerable variation in private school
markets among metropolitan areas is related to geographic differences in the distribution of
adherents to various religious denominations.1 In other words, to understand how private
school markets differ across metropolitan areas, and how voucher effects may differ
accordingly, the consideration of religious preferences appears to be relevant. Nevertheless,
previous research on the general equilibrium effects of vouchers has not incorporated religious
preferences as a determinant for the existence of some private schools.
Thus, I extend Nechyba’s (1999) theoretical work to include household religious
preferences and two types of private schools, Catholic and non-Catholic. Moreover,
households of a given religion in the model vary in their taste for each type of private school;
for instance, some Catholics like Catholic schools better than others. The inclusion of religion
in the model has important implications for voucher effects. For example, even among
households with the same endowment level, those with a high preference for Catholic schools
derive large rewards from vouchers relative to those with a weaker preference. In other words,
religious preferences affect the distributional effects of vouchers. In addition, a contentious
issue related to publicly-funded vouchers is their use to underwrite tuition at religious
schools.2 By including religious preferences and schools, the model I employ is able to
consider the effects of banning voucher use in religious schools.
Among private schools, I focus on Catholic schools for several reasons. First, Catholic
schools are the largest and most homogeneous group of schools within the private school
1
See Section 2 for more details.
The Supreme Court recently upheld voucher use at religious schools in the Cleveland voucher program. See the
Supreme Court’s ruling on Zelman v. Simmons-Harris (06.27.02).
2
3
market.3 Second, in states where it has been debated whether religious schools should be
eligible to participate in voucher programs or not, most of the students attending religious
schools are enrolled in Catholic schools. Third, the majority of the current voucher proposals
attempt to provide choices for low-income students in inner cities where Catholic schools have
historically had a strong presence, often times targeting economically disadvantaged students
through subsidized tuition and privately-funded vouchers.
To estimate the model I use a minimum distance estimator with 1990 data on the
metropolitan areas of New York, Chicago, Philadelphia, Detroit, Boston, St. Louis, and
Pittsburgh. In 1990 these areas were among the twenty largest metropolitan areas in the United
States, had a high level of dependence on local sources for public school funding and had
Catholic populations proportionately larger than the national average. To estimate the model, I
created a rich dataset that combines sources such as the 1990 School District Data Book, the
1989 Private School Survey, and the 1990 Churches and Church Membership in America
survey. The estimation strategy consists of simulating the equilibrium for each metropolitan
area and alternative parameter points, and then choosing the point that minimizes a welldefined distance between the data simulated from the model and the observed data.
Since there is no closed form solution for the equilibrium, the only available
mechanism for obtaining the predicted equilibrium for each parameter point is the simulation
of the equilibrium for each metropolitan area and parameter point. This paper is one of the first
attempts to use this procedure. It is a computationally intensive technique, but capable of
dealing with the simultaneity of the variables jointly determined in equilibrium. Although
many simplifications are needed in order to make the problem tractable, my estimates succeed
at capturing the pattern of income stratification, and the distribution of housing values across
districts within a metropolitan area. In addition, they replicate private school enrollment
patterns for large school districts quite well.
Currently, I have assessed the effects of two hypothetical policies, the introduction of
universal vouchers, and of vouchers for non-Catholic private schools. Although universal
(non-targeted) vouchers have not been implemented to date and most of the current voucher
proposals feature targeted vouchers, the analysis of universal vouchers gives insight into the
impact of an unrestricted voucher program, which is helpful given that any other program may
3
According to the 1989 Private School Survey, 58% of private school enrollment in grades 9 through 12 in
secondary and combined schools was in Catholic schools, 27% in other religious schools, and the remaining 15%
in non-sectarian schools in the United States in 1989. When restricted to secondary schools, the same survey
reports that Catholic schools captured 75% of the total private school enrollment.
4
be seen as a restricted version of universal vouchers. In addition, the simulation of vouchers
that may not be used for Catholic schools (“non-Catholic vouchers”) is particularly important
in light of the controversy over the legality of state-funded vouchers for private religious
schools, and of the Supreme Court’s ruling that the use of vouchers for religious schools does
not violate the Constitution. Therefore, when designing voucher programs, states can choose
whether or not to include religious schools.4 In view of this, the analysis of the general
equilibrium effects and welfare implications of not allowing vouchers for religious schools is
particularly pertinent. I simulate these policies for the Chicago metropolitan area, where 40%
of the population is Catholic and almost half of the population resides in the central district.
My simulation results indicate that universal and non-Catholic voucher programs
produce some qualitatively similar effects. Both increase private school enrollment, and both
affect where households choose to live. When the dollar amount of the voucher is low, both
vouchers provide incentives for some middle-income households from good public school
districts to migrate into districts where lower housing and public school quality is reflected in
lower property prices, and enroll their children in private schools. This result indicates that
vouchers may indeed break the bundling of residence and public schools. In addition, it
provides evidence that vouchers may attenuate the residential stratification generated by the
residence-based public school system.
While middle-income households use vouchers even when the dollar amount is low,
households in low-income districts only use them when the voucher level is above the aid per
student provided by the state to those districts. Yet, as the dollar amount of the voucher
increases, both high and low-income households adopt vouchers, and private schools arise in
poor and wealthy districts alike. Voucher availability reduces the housing premium in the best
public school districts but increases it in the others, thus causing capital losses or gains,
respectively, to the households in those places.
Both voucher programs induce comparable improvements in average school quality. In
addition, the largest proportional school quality gains accrue to low-income households.
Although public school quality falls in these simulations, these losses are spread throughout all
districts, and are more severe for the best districts as the voucher level grows. Interestingly, at
low and moderate voucher levels some households that remain in public schools also benefit
from the introduction of vouchers.
4
In certain states, allowing voucher money to be used for religious schools would require a change in the state
Constitution, as in the case of Michigan. See Mintrom and Plank (2001).
5
Both programs have redistributive effects. Since in these simulations vouchers are
funded with a state income tax, they tend to benefit the poor relative to the wealthy. Because
of the fiscal burden and capital losses, the average household that loses welfare under vouchers
is wealthier than the average winner. In addition, the average welfare change induced by each
program is small, yet the distributional effects are large. In both cases, low-income households
experience the largest gains relative to endowment (around 5%).
Although the programs are similar in the aforementioned effects, they are dissimilar in
others. The dissimilarities derive from the fact that a voucher program restricted to nonCatholic private schools changes the relative price of Catholic schools. This, in turn, may
induce some households, depending on their religious preferences and budget constraints, to
substitute non-Catholic private schools for Catholic schools. For example, consider a highincome Catholic household with a strong preference for Catholic schools that would attend
Catholic schools even without vouchers. Only a sufficiently high voucher level would induce
this household to enroll in a non-Catholic private school. On the other hand, consider a
Catholic household that only slightly prefers Catholic over non-Catholic schools, and whose
budget constraint does not permit it to choose private schools without vouchers. Even a
relatively low voucher level would induce this household to enroll in a private non-Catholic
school.
Vouchers that are targeted exclusively for non-Catholic schools reduce the extent of
private school enrollment and voucher use at moderate voucher levels, because many of the
households that would use vouchers if they were universal have a strong preference for
Catholic education. Moreover, whereas enrollment grows in all private schools under universal
vouchers, it grows at private non-Catholic schools but declines at Catholic schools under nonCatholic vouchers. At a sufficiently high level, universal and non-Catholic vouchers are used
at the same rate.
More households benefit from universal vouchers than from non-Catholic vouchers.
Furthermore, at least 20% of all households win under universal vouchers, yet lose under nonCatholic vouchers. These are households with a strong preference for Catholic schools, that
either attend Catholic schools in the absence of vouchers or choose them under a universal
voucher. As their average welfare gains drop between 2 and 4% of their wealth, these are the
households that experience the largest welfare change across voucher regimes. Furthermore,
whereas Catholic households gain the most relative to their wealth at each endowment level
under universal vouchers, they lose the most under non-Catholic vouchers.
6
The above results use structural estimates from a general equilibrium model and
provide insight into voucher policies. As such, my work contributes to several lines of
research. While this paper broadly complements the literature on school choice, it relates more
closely to research that evaluates the general equilibrium effects of vouchers in multi-district
settings (Nechyba (1999, 2000), Fernandez and Rogerson (2001)). In addition, my work
complements the literature that estimates general equilibrium models for multi-district
economies. For instance, Epple and Sieg (1999) estimate an equilibrium model of local
jurisdictions for the metropolitan area of Boston, and test the model’s predictions about the
equilibrium distribution of households across communities. However, their framework does
not include private schools, nor does it include multiple neighborhoods within jurisdictions.
Since the central districts of the largest metropolitan areas display a wide variety of housing
quality and neighborhood amenities that induce household sorting within districts, it seems
important to account for that variety in the model and the estimation. My work accomplishes
this task.5 Bayer (2000) estimates an equilibrium discrete choice model of residential and
school (public vs. private) decisions of households with elementary school-aged children in
California, in an attempt to clarify what drives the observed differences in school quality
consumption across households. Although private schools are included in his model, the
specific mechanisms by which they arise are not modeled. In contrast, such mechanisms are
part of the model presented in this paper. Additionally, the public school financial setting in
California permits the researcher to ignore the public choice process which funds public
schools,6 but this is not appropriate for metropolitan areas located in many other states where
property taxes are indeed endogenous. By allowing taxes to be determined by majority voting
rule, my work is suited to explore voucher issues in a wide variety of metropolitan areas and
public school financial regimes.7
This paper is organized as follows: section 2 provides descriptive statistics that
highlight the income-sorting of households among districts and the geographic variation in
private school markets among the metropolitan areas of interest. Section 3 presents the model.
5
While Nechyba (1999 and 2000) includes multiple neighborhoods within a given district, my research is the first
attempts to do so in an estimation setting.
6
In California, districts are fiscally dependent – namely, they do not rely on their own resources to fund their
schools. Counties levy property taxes, whose rate is legally limited to 1%. These revenues are then redistributed
by the state to the districts according to a legislative formula. In practice, California’s system resembles full state
funding.
7
In an extension of Epple and Romer (1999), Epple, Romer and Sieg (2001) investigate whether observed levels
of public expenditures satisfy the necessary conditions implied by majority rule in a general equilibrium model of
residential choice.
7
Section 4 discusses the computable version of the model used for estimation purposes. Section
5 presents the data sources, section 6 describes the estimation procedure, and section 7
discusses the estimation results. Section 8 analyzes voucher effects in policy simulations, and
section 9 concludes.
2. Descriptive Statistics
My analysis focuses on the metropolitan areas of New York, Chicago, Philadelphia, Detroit,
Boston, St. Louis, and Pittsburgh. As of 1990, these areas were among the twenty largest
metropolitan areas in the United States, had a high level of dependence on local sources for
public school funding8 and had populations that were at least 25% Catholic. Ferreyra (2002)
describes in detail how these metropolitan areas were selected, and Table 1 lists some of their
relevant features. My analysis focuses exclusively on secondary and unified school districts.9
The school districts in these metropolitan areas, however, are far from homogeneous,
as shown by the summary statistics in Table 2.10 Large variation occurs in the fraction of
households with children in private schools, in average household income, and in average
housing rental value.11 Moreover, households with children in private schools tend to have
higher incomes and to live in houses with higher rental values than households with children in
public schools. From the point of view of public school finance, districts also differ widely in
their average spending per student and, to some extent, in their sources of public school
funding. While some of the variation among districts is driven by the fact that they belong to
8
Public school districts fund their operations through a mix of local (district level), state, and federal resources,
yet the share of each of these sources varies among metropolitan areas and districts.
9
As will be seen in Section 4, two of the simplifying assumptions made are: (1) public school funding mostly
relies on local property taxes, and is supplemented by a state flat grant per student, and (2) households in the
population are indifferent between public and non-Catholic private schools. While computational reasons explain
the imposition of both assumptions, (1) is also motivated by the extreme difficulty of replicating the actual public
school funding regimes with stylized formulas (see Section 2.4 for further discussion on this issue). Assumption
(2) is particularly restrictive with respect to religious non-Catholic households, many of whom may prefer schools
of their denomination over public schools. Thus, this assumption renders the model inappropriate to explain the
formation of such schools. However, given the need to impose assumptions (1) and (2), I select metropolitan
areas where public schools are indeed predominantly funded by local property taxes, and where most of the
population is Catholic. This selection criteria yields a set of metropolitan areas with high private (and Catholic)
school enrollment where proposals for publicly funded private school vouchers, and even privately funded
voucher programs, have flourished over the last ten years.
10
Summary data on households and enrollment pertains to grades 9 through 12.
11
The calculation of housing rental values is described in Ferreyra (2002).
8
different metropolitan areas, heterogeneity among districts also occurs within metropolitan
areas, providing evidence that households sort themselves non-randomly across jurisdictions.
In the metropolitan areas included in my sample, the inner city district is the largest
one, and is comprised of multiple neighborhoods with different housing qualities and
geographic amenities. The fraction of households with children in private schools is relatively
high in the inner city district and in some suburban districts. Households that live in inner city
districts and send their children to private schools include middle-income families who reside
in the district’s most expensive neighborhoods, and low-income families whose children
attend private schools with subsidized tuition.12
The evidence presented so far indicates that factors such as the distribution of housing
quality within and across districts and income distribution are related to private school
enrollment. In addition, Table 3 shows that religion is connected to private school enrollment.
Among the twenty largest metropolitan areas in the United States, those with a higher fraction
of students in private schools also have a higher fraction of private school students enrolled in
Catholic schools. Furthermore, these metropolitan areas have a higher fraction of Catholics,
and the correlation between the fraction of students enrolled in Catholic schools and the
fraction of Catholics is positive and high. Thus, the geographic variation in private school
markets also seems to be shaped by the geographic variation in the distribution of adherents to
different religions.
To summarize, the selected metropolitan areas display significant variation in the
dimensions of interest. The data also provides evidence of household sorting across districts
and into schools, and highlights the factors associated with this sorting. Thus, the model
presented in the next section captures all these elements.
3. The Model
In this section, I provide a theoretical model of school quality and household residential
choice.13 A computable version of the model aimed to simulate voucher experiments is
presented in the next section. In the model, an economy is a set of school districts that contain
neighborhoods of different qualities. There are three types of schools: public, private Catholic,
12
This is because numerous inner-city religious schools subsidize student tuitions. Often families do not belong
to the denomination of their children’s private school.
13
This is an extension of Nechyba’s (1999) model including household religious preferences and different types
of private schools.
9
and private non-Catholic. The economy is populated by households that differ in their
endowment level and religious preferences, yet maximize utility by choosing a location and a
school for their children. A school’s quality is endogenously determined by the average peer
quality of the school’s students and the spending per students. Public schools’ spending is
supported by a local property tax set by majority rule. Private schools, in turn, are modeled as
clubs formed by parents that set admission requirements based on peer quality and share the
school’s costs equally.14 Furthermore, the parental valuation of a school’s quality depends on
the match between the school’s religious orientation and the household’s religious preferences.
In equilibrium, no household wants to move, switch to a different school, or vote differently.
Households and Districts
The economy is populated by a continuum set of households N. Each household n ∈ N is
endowed with one house. Hence, N also represents the set of houses in the economy, which is
partitioned into school districts. Every district d is in turn partitioned into neighborhoods. The
set of houses in neighborhood h and district d is denoted by C dh , and C d denotes the set of
houses in district d. Houses may differ across neighborhoods, but within a given neighborhood
are homogenous and have the same housing quality and rental price.
Each household has one child, who must attend a school, public or private. One public
school exists in each district, and the child may attend only the public school of the district
where the household resides. If parents choose to send their child to a private school, Catholic
or non-Catholic, they are not bound to any rule linking place of residence and school.
In addition to a house, households are also endowed with a certain amount of income,
and there are I income levels. Households also differ in religious orientation, given by their
valuation of Catholic schools relative to non-Catholic schools. A household has one of K
possible religious types, where types 1,K, L are Catholic, and the others non-Catholic. Not all
the L Catholic types are necessarily identical, for they may differ in their relative valuation of
14
A club is a voluntary coalition of agents that derive mutual benefits from production costs, members’
characteristics, or a good characterized by excludable benefits – i.e., a good for which there is a mechanism that
prevents agents outside the club from consuming the good (Cornes and Sandler, 1996). Private schools fit this
description well, as parents freely join them and share the school’s cost through tuition payments. In addition,
they also share the characteristics of the student body at the school, and the educational services produced by the
school.
10
Catholic schools, and the same is true for the non-Catholic types.
Household preferences are described by the following Cobb-Douglas utility function:
U (κ , s, c) = s α c β κ 1− β −α , κ = k dh
(1)
where α , β ∈ (0, 1) , kdh is an exogenous parameter representing the inherent quality of
neighborhood h in district d, c is household consumption, and s is the parental valuation of the
quality of the school attended by the child, which, in turn, depends on the household’s
religious preferences. Household n seeks to maximize utility (1) subject to the budget
constraint:
c + (1 + t d ) p dh + T = (1 − t y ) y n + p n
(2)
where yn is the household’s income, ty is a state tax rate, and pn is the rental price of the
household’s endowment house. Given its per-period total income, represented by the righthand side of (2), the household chooses to live in a location (d , h) with housing price pdh and
local property tax rate td. It also chooses a school for its child with tuition T. Since public
schools are free, T = 0 in this case. Remaining income is used for consumption c.15
Production of school quality
All school types produce school quality ~
s according to the Cobb-Douglas production function:
~
s = q ρ x1− ρ
(3)
where ρ ∈ [0,1] , q stands for the school’s average peer quality, and x is spending per student at
the school. Denote by S the set of households whose children attend the school, and by y (S )
the average income of these households, y ( S ) = ∫ y n dn µ (S ) , where µ (S ) is the measure of
S
set S. Then the school’s average peer quality is defined as q = y (S ) .16 The spending per
student x equals tuition T if the school is private, and it equals the spending per student in
15
Note that if the household chooses to live in its endowment house, then (2) becomes
c + t d p n + T = (1 − t y ) y n , i.e. the after-tax income is spent on consumption, property taxes, and school
tuition.
16
The empirical findings on peer effects in education are quite varied, and there is no consensus on how peer
effects operate nor at which level (school, class, group of students within a class). Thus, theoretical models
usually pick a simple specification of peer effects that agrees with some of the empirical evidence. The current
model agrees with evidence that peer effects act primarily through parental involvement and monitoring of
schools, both of which increase with income (McMillan (1999)).
11
district d, xd, if the school is public and run by the local government.
The value that parents assign to school quality is represented in utility function (1) by
the variable s, and depends on their religious preferences. A household of religious type
k = 1,K, K whose child attends a school with religious orientation j = 1, 2 (Catholic or nonCatholic respectively)17 and quality ~
s j perceives the school quality as follows:
s kj = Rkj ~
sj
(5)
where Rkj > 0 is a preference parameter. Households of a given religious orientation may have
different tastes for Catholic and non-Catholic schools.
Public Schools
Let N pub be the subset of all households N in the economy that choose to send their children
to public schools, and define J d as the population residing in district d. The set N
pub
∩ J d is
then the set of all households residing in district d that choose public school for their children.
s = q ρ x1− ρ , where q is the average
The quality of the public school in district d is ~
d
d
d
d
income of households with children attending this school, q d = ∫
N
pub ∩ J d
y n dn µ (N
pub
∩ Jd ) .
Under a regime of local financing, the public spending in education is funded by local property
taxes, possibly aided by the state. Thus, the spending per student in district d is given by
xd = t d Pd µ (N
pub
∩ J d ) + AIDd , where AIDd is the amount of state aid per student for
district d, funded through a state income tax, and Pd = ∑ µ (C dh ) p dh is the local property tax
h
base.
Private Schools
Private schools are modeled as clubs formed by parents under an equal cost-sharing rule.
Because of the constant returns to scale of technology to produce education (see equation (3))
and the absence of fixed costs to open a private school, households of a given endowment type
have incentives to segregate themselves into a private school and reject lower endowment
17
Public schools are nonsectarian and, therefore, non-Catholic.
12
types. Therefore, a private school formed by households of income level yn has a peer quality
q = yn .
Parents of a given endowment who wish to set up a private school can choose to found
either a Catholic or a non-Catholic school. Households care about the makeup of the student
body because of peer quality and other households’ willingness to pay. However, households
do not care about the religious preferences of other students in the school; they only care about
the religious orientation of the school (see equation (5)).
Households of a given endowment type share costs equally at a private school. Thus,
the tuition equals the households’ optimal spending in education, holding their residential
locations fixed. That is, after choosing a location (d , h) with quality kdh, household n of
religious type k with income yn may send its child to a private school with tuition T and
religious orientation j = 1, 2 (Catholic or non-Catholic) that maximizes utility (1) subject to
the budget constraint (2) and the perception of school quality s = Rkj q ρ x1j− ρ , where q = y n , and
x j = T .18
The optimal tuition T determined by solving this optimal choice problem does not
depend upon Rkj, since changing Rkj only yields a monotonic transformation of the utility
function. Parents who decide to open a private school choose the school religious orientation
(Catholic or non-Catholic) for which they have a higher taste. Thus, if households of a given
endowment type with religious preferences Rk1 > Rk 2 ( Rk 2 > Rk1 ) decide to form a private
school, they will segregate themselves into a Catholic (non-Catholic) private school.19
Potentially, the model gives rise to as many private schools as households that exist in the
economy.
18
Modeling private schools as clubs is equivalent to modeling the private school market as a perfectly
competitive market where firms do not differentiate among students in their tuition policies, but do so in their
admission policies. Because of the constant returns to scale technology, all households in a given school are of the
same endowment type, and pay a tuition equal to the type’s most preferred spending in education. In equilibrium,
private schools make zero profits.
19
In particular, notice that some Catholic households may prefer non-Catholic over Catholic schools, other things
equal. The converse may also occur. In fact, almost 15% of the enrollment in Catholic schools in the United
States was accounted for by non-Catholics in 1990 (NCEA, 1990b). This might have been due not only to
preferences but also to the subsidized tuition offered by some Catholic schools, especially in inner cities. Such
subsidies can be incorporated into the model, although they have not been introduced here for computational
reasons.
13
Household Decision Problem
Households are utility-maximizing agents that choose locations (d , h) and schools
simultaneously, while taking tax rates td, district public school qualities sd, prices pdh,, and the
composition of the communities as given. Household n chooses among all locations (d , h) in
the budget set determined by the constraint (1 + t d ) p dh ≤ (1 − t y ) y n + p n . Migrating among
locations is costless in the model, and the household may choose to live in a house other than
its endowment house. For each location, the household compares its utility under public,
Catholic private, and non-Catholic private schools.
Absolute Majority Rule Voting
Households also vote on local property tax rates. At the polls, households vote for property
taxes taking their location, their choice of public or private school, property values, and the
choices of others as given when voting on local tax rates. Households that choose private
schools vote for a tax rate of zero, whereas households that choose public schools vote for a
nonnegative tax rate. Because voters choose the tax rate conditional on their school choice,
taking everything else as given, their preferences over property tax rates are single peaked, and
property tax rates are determined by majority voting. Since there is a one-to-one mapping
between property tax rates and spending per student for a given property tax base and number
of students in public schools, voting for property tax rates is equivalent to voting for school
spending.
State Funding
The state cooperates in funding public education in district d by providing an exogenous aid
amount per student, AIDd . This aid, which operates as a flat grant, is in turn funded by a state
income tax whose rate t y is set to balance the state’s budget constraint
∑ µ (N
pub
∩ J d ) AIDd = t y y (N )
d
14
(9)
where y (N ) = ∫ y n dn is the total income of all households in the state, which is equated to
N
the economy being modeled.
Equilibrium
An equilibrium in this model specifies a partition {J dh } of households into districts and
neighborhoods, local property tax rates td, a state income tax ty, house prices pdh, and a
partition of N into subsets of households whose children attend each type of school (public,
private Catholic, and private non-Catholic), such that:
(a) every house is occupied: µ ( J dh ) = µ (C dh ) ∀(d , h) ;
(b) property tax rates td are consistent with majority voting by residents who choose public
versus private school, taking their location, property values, and the choices of others as given
when voting on local tax rates;
(c) the budget balances for each district: xd = t d Pd µ (N
(d) the state budget balances:
∑ µ (N
pub
pub
∩ J d ) + AIDd ;
∩ J d ) AIDd = t y y (N ) ;
d
(e) at prices pdh, households cannot gain utility by moving and/or changing schools.
The existence of equilibrium with only public schools is proved in Nechyba (1997), and
extended to include private schools in Nechyba (1999). A similar logic can be applied to prove
the existence of equilibrium with several types of private schools. With sufficient variation in
mean neighborhood quality across districts, the equilibrium assignment of households to
locations and schools is unique (Nechyba, 1999).20
4. The Computable Version of the Model
In the computational version of the model the concept of “an economy” corresponds to a
metropolitan area. Moreover, the estimation strategy consists of simulating the equilibrium for
each metropolitan area at alternative parameter points, then choosing the point that minimizes
a well-defined distance between the equilibrium data simulated by the model and the observed
20
Due to the discreteness of housing types, there may be a range of housing prices, local property taxes, and
public school qualities associated with a given partition of households into districts, neighborhoods, and subsets
of households attending each type of school. The range decreases as the number of housing types increases.
15
data.21 Since the equilibrium cannot be solved analytically, I use an algorithm to compute it
iteratively for a tractable representation of each metropolitan area. This section describes the
setup of districts and neighborhoods in this representation, the construction of household
types, the state financial regime applied in computing the equilibrium, and the algorithm
employed.
Community Structure
In the computable version of the model there are an integer number of houses and households
in each metropolitan area, organized into districts and neighborhoods.22 I measure the size of
neighborhoods, districts, and metropolitan areas by the number of housing units.
Computational considerations prevent me from working with the actual district configuration
of each metropolitan area, splitting each district into neighborhoods of the size of
approximately a Census tract, having at least as many income levels as the ones provided by
the 1990 Census data, and including a large number of religious types.23 Thus, I aggregate the
real districts of each metropolitan area into quasi-districts in order to compute the equilibrium,
such that the largest district is a quasi-district in itself, while smaller, contiguous districts are
pooled into larger units.24 The actual 671 districts thereby yield 58 quasi-districts.25 For
illustrative purposes, Figure 1 depicts Census tracts, and school districts and quasi-districts for
the Chicago metropolitan area. Once the quasi-districts are constructed, I split them into
neighborhoods of approximately the same size, such that some districts26 have only one
neighborhood, while others have several. Larger metropolitan areas have larger
neighborhoods.
Neighborhood Quality Parameters
21
Section 6 explains the estimation strategy in detail.
Metropolitan areas are independent from one another. Each one is “an economy” in the sense of the model.
Households do not migrate across metropolitan areas.
23
The theoretical model includes K religious types, out of which L are Catholic. By a “large” number of religious
types I mean a “large” value for K, such that these K types are still split into Catholic and non-Catholic types. A
“large” number of religious types does not mean splitting K into more denominations – i.e., Catholic, Lutheran,
Baptist, Muslim, etc.
24
The size of each district is approximately a multiple of the size of the smallest district.
25
The aggregation of districts into quasi-districts produces an obvious loss of information, particularly for small,
suburban school districts.
26
From now on, “district” refers to “quasi-districts”.
22
16
In the theoretical model each neighborhood is composed of a set of homogeneous houses, such
that neighborhood h in district d has a neighborhood quality index equal to kdh. Since
neighborhood quality is not measured in standard datasets, I construct an index that captures
housing quality and neighborhood amenities, excluding public school quality.
The Census geographical concept that best approximates a neighborhood is the Census
tract. Therefore, I first compute the neighborhood quality index for each Census tract in a
metropolitan area. Then I pool tracts into neighborhoods subject to the size constraints
described above, and assign to each neighborhood the median neighborhood quality among all
the tracts that belong to the neighborhood.
To compute the neighborhood quality at the tract level, I regress the logarithm of the
tract's average rental price on a set of neighborhood characteristics and a school district fixed
effect, and then recover a metropolitan area constant.27 The neighborhood quality index for a
tract is the tract's fitted rental value, which is the sum of the "value" of the tract’s
neighborhood characteristics and a metropolitan area constant.28 The motivation for this
regression is that, roughly speaking, rental prices reflect housing characteristics, neighborhood
and metropolitan area-level amenities, and public school quality. The district fixed effect has
the purpose of netting out the school quality component from the measure of neighborhood
quality.
After obtaining the neighborhood quality index for each Census tract, I construct
neighborhoods of the desired size by pooling contiguous tracts with a similar value for the
neighborhood quality index. Finally, I assign each neighborhood the median neighborhood
quality of all the tracts belonging to that neighborhood.
For an example of the final representation of a metropolitan area through quasi-districts
and neighborhoods, see Figure 2 for the Chicago metropolitan area. The central city of
Chicago overlaps entirely with the central district. Unlike the suburban districts, which have
one neighborhood each, the central district has seven neighborhoods which differ in housing
quality. On average, the central district has the lowest housing quality in the metropolitan area.
27
The tracts located in the twenty largest metropolitan areas are used in this regression. See Ferreyra (2002) for
more details on how the neighborhood quality parameter is computed. Ideally I would use a set of variables that
reflect neighborhood physical amenities, such as quality of houses in the neighborhood, presence of parks and
open spaces, distance to major landmarks and business centers, and distance to public transportation, if
applicable. Housing quality variables are available from Census data, but the other variables have to be
constructed by the researcher. Thus, the current version employs only physical housing characteristics and
distance to the metropolitan area center.
28
Strictly speaking, the neighborhood quality index for a Census tract equals exp(fitted log of average rental
value) for that tract.
17
Household types
I consider five income levels, whose incomes are equal to the 10th, 30th, 50th, 70th and 90th
percentile of the income distribution of households with children in public or private schools
in grades 9 through 12 in a given metropolitan area.
The joint distribution of housing and income endowment is as follows. At the
beginning of the algorithm, the distribution of income in each neighborhood is initially the
same, and equal to the one that prevails for the metropolitan area as a whole. In other words,
income and housing endowments are independently distributed. In addition, religious
preferences and endowments are independently distributed.29
Recall that each household is characterized by two religious matches, one with respect
to Catholic schools and another with respect to non-Catholic schools. If household n is
Catholic, its religious preferences are described by its match with respect to Catholic schools,
RCn ,C , and its match with respect to Non-Catholic schools, RCn , NC . If household n is non-
Catholic, its religious preferences are given by its match with respect to Catholic schools,
n
n
R NC
,C , and its match with respect to non-Catholic schools, R NC , NC .
n
Since there are two types of schools, I make RCn , NC = R NC
, NC = 1 and focus on the
relative valuation of Catholic schools. Unlike income, whose distribution comes straight from
n
the data, the distribution of religious matches RCn ,C and RNC
,C needs to be estimated.
Therefore, I construct a discrete distribution of religious matches by assuming an underlying
continuous distribution. In particular, I assume that RCn ,C is distributed uniformly over the
n
interval [(1 + r )(1 − π ), (1 + r )(1 + π )] , and that R NC
,C is distributed uniformly over the interval
[(1 − r )(1 − π ), (1 − r )(1 + π )] , where
0 ≤ r < 1 and 0 < π ≤ 1 . The parameter r is both the premium
enjoyed by the average Catholic from attending a Catholic school, and the negative of the
discount suffered by the average non-Catholic from attending a Catholic school, while the
parameter π is proportional to the coefficient of variation of each distribution. Faced with the
choice between schools that are identical in everything except religious orientation, Catholics
29
For computational convenience I place ten households in each (house endowment, income) combination. For
instance, if the proportions of Catholics and Non-Catholics in the metropolitan area are 28% and 72%
respectively, then out of the ten households in each (house endowment, income) combination, I make three
Catholic and seven non-Catholic, respectively. The fraction of Catholics is the same for each endowment type,
based on the lack of empirical evidence against income and religion being independently distributed (see Ferreyra
(2002))
18
would choose non-Catholic schools if π > r /(1 + r ) , whereas non-Catholics would choose
Catholic schools if π > r /(1 − r ) .30
State Aid
Although the metropolitan areas included in my analysis fund their public schools primarily
through local sources, they still differ in the actual (and extremely complex) formulas used by
the states to allocate funds among districts.31 Therefore, I simplify the financial rules by setting
the same mechanism for all metropolitan areas – a local funding system with a state flat grant
per student, whose value equals the state aid reported by the data.
The Algorithm
In the model, the parameter vector is θ = (α , β , ρ , r, π ) . The algorithm that computes the
equilibrium for a given parameter point and metropolitan area is described in Ferreyra (2002).
Computing the equilibrium is an iterative process in which households make residential and
school choices and vote for the property taxes used to fund public schools. The process
continues until no household gains any utility by moving, switching to a different school, or
voting differently. In essence, the algorithm takes certain input, performs some operations over
it, and produces some output for a given parameter point. The input consists of the community
structure, initial distribution of household types, initial housing prices, and state aid for each
district.32 The output is the computed equilibrium, from which I extract the variables whose
predicted and observed values I wish to match.33
30
For an example on how I determine each household’s corresponding religious match, continue with the case
presented in footnote 29. Suppose that r = 0.2 and π = 0.1 . This implies that RC ,C is uniformly distributed
n
n
between 1.08 and 1.32, and RNC ,C is uniformly distributed between 0.71 and 0.88. Therefore, I assign a match
equal to 1.08, 1.2 and 1.32 to the first, second, and third Catholic household types, and a match equal to 0.72,
0.747, 0.773, 0.8, 0.827, 0.853, and 0.88 to the first through seventh non-Catholic household types. Since
religious preferences and endowment are independently distributed, the first, second, and third Catholic
household types initially placed in every (house endowment, income) combination have religious matches equal
to 1.08, 1.2, and 1.32 respectively, and a similar thing is true for the religious matches of non-Catholic
households.
31
See, for instance, Hoxby (2001).
32
Unless I state otherwise, state aid is always per student rather than the total over all students.
33
Throughout this work I assume a unique equilibrium. In future research I will explore the issue of multiple
equilibria.
19
5. The Data
I construct the distribution of income among households with children in grades 9 through 12
using data from the 1990 School District Data Book (SDDB). In addition, the state aid per
district comes from the SDDB, and the data sources used to compute the neighborhood quality
parameters are listed in Ferreyra (2002). The fraction of Catholics in a metropolitan area
comes from the 1990 Church and Church Membership in America survey. Furthermore, to
begin the algorithm I equate the initial price of all houses to the metropolitan area’s average
rental price, which I compute using the 1990 School District Data Book.
I match the observed and simulated values of the following district-level variables:
average household income, average housing rental value, spending per student in public
schools, and fraction of households with children in public schools. I calculate these variables
based on the 1990 SDDB. In addition, I would like to match the fraction of households from
each district that send their children to Catholic schools. However, the SDDB does not provide
this information; in fact, there is no data source that links households’ residence with different
types of private schools. Therefore, I am unable to match the district-level fraction of
enrollment in Catholic schools. However, I am able to match this fraction at the metropolitan
area level, and I compute the fraction of households with children enrolled in private schools
using data from the 1989 Private School Survey.
6. Estimation
I estimate the model by using a minimum distance estimator. I match the observed and
simulated values of the following district-level variables:
y1 = average household income,
y2 = average housing rental value,
y3 = spending per student in public schools,
y4 = fraction of households with children in public schools
In addition, I match y5, the fraction of households with children in Catholic schools at the
metropolitan area level. All these variables are scaled to have unit variance in the sample. Let
D denote the total number of districts in the sample (D=58), and M the number of metropolitan
20
areas (M=7). Let Nj denote the number of observations available for variable yj, j=1, …5, so
that N1= N2=N3=N4=D, and N5=M.
Assume that district i is located in metropolitan area m. Then, denote by Xi the set of
exogenous variables for district i, such that X i = xi ∪ x m ∪ x −i . Here, xi is district i’s own
exogenous data (state aid, number of neighborhoods, neighborhood quality in each
neighborhood), xm is exogenous data pertaining to metropolitan area m (10th, 30th, 50th, 70th
and 90th income percentiles, and the fraction of Catholic households in the metropolitan area),
and x −i is the “own” data from the other districts in metropolitan area m (their state aid,
number of neighborhoods, and neighborhood quality in each neighborhood). In addition, the
set of independent variables for y5m is Xm, which is the union of all the Xi sets corresponding to
the districts that belong to metropolitan area m. Finally, let ni denote the actual number of
housing units in district i, and let nm denote the actual number of housing units in metropolitan
area m.
I assume the following:
(10) E ( y ji ) = h j ( X i ,θ )
(11) E ( y 5 m ) = h5 ( X m ,θ )
j = 1,...4; i = 1,...N j
m = 1,...M
where the h's are implicit nonlinear functions. Since the yji’s are (district-level) sample means,
(12)
C ( y ji , y ki′
with V ( y ji ) =
σ jj
ni
σ jk

) =  ni
0

=
σ
2
j
ni
i = i′
i ≠ i′
= σ 2ji . Similarly, given that the y5m’s are also sample means (at the
metropolitan area level), C ( y ji , y 5 m ) =
σ j5
ni
C ( y ji , y 5 m ) = 0 otherwise. Also, V ( y 5 m ) =
if district i is located in metropolitan area m, and
σ 55
nm
=
21
σ 52
nm
= σ 52m .
Estimation Strategy
Because the number of observations is rather small, I estimate the model using Feasible
Weighted Least Squares to account for heteroskedasticity across observations, and then use the
cross-equation covariances to obtain correct standard errors.34
Feasible Weighted Least Squares proceeds in two stages. The first stage determines the
value for θ that minimizes the following sum of squared residuals:35
Nj
(13) L(θ ) = ∑∑ ( y ij − yˆ ij (θ )) + ∑ ( y 5 m − yˆ 5 m (θ ) )
4
2
j =1 i =1
M
2
m =1
and the residuals from this regression are used to compute σˆ 2j and σˆ 52 . The second stage runs
Nonlinear Least Squares on variables transformed to account for heteroskedasticity, and seeks
to minimize the following sum of squared residuals in the transformed variables:
N
j
M
4
2
2
~
(14) L (θ ) = ∑∑ ( y *ji − yˆ *ji (θ )) + ∑ ( y 5*m − yˆ 5*m (θ ) )
j =1 i =1
m =1
where * denotes division by σˆ ji or σˆ 5 m . The value of θ that minimizes this function, θˆ , is
my estimate for the parameter vector.
Computational Considerations
I use a refined grid search to find the parameter estimates. In a grid search, the objective
function can be evaluated at each point independently of its evaluation at other points. This
feature allows for parallel computing, which I exploit in order to estimate the model. However,
the disadvantage of a grid search is that the size of the grid grows exponentially with the
number of parameters, which limits the number of parameters that can be estimated within a
reasonable amount of time. This explains, for instance, why my parameterization of religious
preferences is extremely parsimonious.
34
35
More details on the estimation of parameters and standard errors are provided in Ferreyra (2002).
At times I refer to the sum of squared residuals as “loss function”.
22
I use Condor to estimate the model.36 On average, this lets me evaluate the objective
function at about 250 parameter points simultaneously. Since those function evaluations are
independent of each other, Condor uses one processor for each function evaluation. A function
evaluation takes about about ten minutes on an Intel processor at an average speed of 1 Gh,
and the full two-stage procedure takes between five and six days.
Identification
The model is identified if no two distinct parameter points generate the same equilibrium for
each metropolitan area. However, the discreteness of housing and household types entails a
positive probability that a sufficiently small change in the parameters leaves the equilibrium
unchanged in each metropolitan area. This probability falls as the number of metropolitan
areas, housing, and household types grows.
Thus, identification in this model refers to
identification up to an arbitrarily small neighborhood of the true parameter point.
A sufficient condition for local identification is that the matrix of first derivatives of
the predicted variables with respect to the parameter vector has full column rank when
evaluated at the true parameter points. This requires sufficient variation in the exogenous
variables across districts and metropolitan areas. In particular, identification requires sufficient
variation in neighborhood quality and state aid across districts, in the income categories
chosen as income types within a metropolitan area, and in the fraction of Catholic adherents
across metropolitan areas. Evaluated at my parameter estimates, the matrix of first derivatives
has full column rank in my sample.
36
Condor is a project of the Computer Science Department at the University of Wisconsin-Madison
(http://www.cs.wisc.edu/condor). It is a software system that harnesses the power of a cluster of UNIX or NT
workstations on a network. Some of those workstations are specifically dedicated to highly intensive
computational jobs, and others are non-dedicated resources from offices and computer labs used by Condor
whenever they are idle. For a detailed discussion of how Condor was used to estimate the current model, see
National Center for Supercomputing Applications (2002) or http://access.ncsa/uiuc.edu/CoverStories/vouchers/
23
7. Estimation Results
In this section I analyze the parameter estimates and discuss what they imply in terms of
household and school behavior. In addition, I analyze the fit of the individual variables
matched in my estimation and discuss the limitations of these fits. Finally, I provide some
additional measures of goodness of fit.
Analyzing the parameter estimates
The parameter estimates are shown in Table 4. A simple test to determine whether the
estimates are in the right range is to compute the budget shares implied by the estimates, and
compare them with the ones implied in the data. Specifically, if housing quality were
continuous and if housing prices were proportional to housing quality, households’ optimal
choices would determine the budget shares for school spending, consumption, and housing as
follows:
(15) School spending share =
(16) Consumption share=
(17) Housing share =
α (1 − ρ )
1 − αρ
β
1 − αρ
1−α − β
1 − αρ
I use the data to compute an approximation to these shares. I calculate the school
spending share as the ratio between the average spending per student and the average
income,37 the housing share as the ratio between the average rental value and the average
income, and the consumption share as one minus the sum of the schooling share and the
housing share. This yields a schooling share of 0.13 and a housing share of 0.17, which are
reasonably close to the shares implied by the data, equal to 0.09 and 0.16 respectively.
In addition, the estimates imply that for Catholic households, RC,C is distributed
uniformly between 0.832 and 1.389, and for non-Catholic households, RNC,C is distributed
37
In this exercise I use weighted averages, with a district’s weight being equal to its size. I average over all
districts in the sample.
24
uniformly between 0.668 and 1.112. These estimates generate sufficient overlap that Catholics
may enroll in non-Catholic schools, and non-Catholics may enroll in Catholic schools.38
All the parameter estimates have very small standard errors, because my observations
are sample means computed from Census data based on thousands of observations. However,
the reported standard errors may understate the true variability to the extent that they have not
been adjusted for any of the numerical approximations employed throughout.
The model rejects the hypothesis that all moment conditions are satisfied.39 In other
words, the model fails to account for all the variation in the data,40 which is expected given the
large number of restrictions implied by the model. Furthermore, the fit of individual variables
face the limitations discussed below.
Analyzing the fit of the individual variables
Figures 3a through 3d, and figure 4 depict the predicted and observed values for each variable.
In analyzing the fit of the model, it is important to bear in mind that the aggregation and coarse
discretization imposed by computational limitations reduce the model’s ability to match the
data more closely.41
As can be seen in the figures, the model fits the district average rental value and
household income reasonably well, particularly for the large districts. The fact that the
parameter estimates capture the pattern of household sorting across jurisdictions is a promising
result, especially given the parsimonious parameterization of the model.
The model fits the district average public school spending per student less closely. This
may be due to the fact that although the metropolitan areas included in the estimation
predominantly rely on local sources for public school funding, they still differ considerably in
their financial regimes. Moreover, even districts within a state are often subject to different
38
The estimated parameters lead to the rejection of the hypothesis that the preferences of Catholics and nonCatholics for Catholic schools follow the same distribution. From the parameterization of religious preferences
presented in Section 4, it is clear that this hypothesis is equivalent to r=0, which is rejected at any standard
significance level. This provides evidence that allowing Catholics’ preferences for Catholic schools to differ from
those of non-Catholics significantly improves the fit of the data.
39
If the model is true, then the loss function evaluated at the parameter estimates follows an asymptotic chisquare distribution with degrees of freedom equal to the number of moment equations minus the number of
parameters. Since the degrees of freedom are large, the distribution of the statistic given by (loss function –
degrees of freedom)/(square root of twice the degrees of freedom), can be approximated by the standard normal
distribution.
40
Epple and Sieg (1999) report a similar result.
41
In Ferreyra (2002), I discuss these issues in detail.
25
school funding regimes.42 Yet, the funding rule used by the model ignores this heterogeneity
across districts and metropolitan areas, assumes that the state aid is exogenous with respect to
districts’ demographic conditions, and ignores the fact that when actual districts set their
property tax rates, they take into account the incentives contained in the school finance
formulas. Furthermore, the model assumes that a district’s property tax base equals the value
of residential property, and that each household that pays property taxes in a district also votes
for the property tax rate. These assumptions, however, do not hold in reality, for a district’s
property tax base usually includes non-residential property in addition to residential property,
and agents that pay property taxes on non-residential property in a district often do not
participate directly in the determination of the district’s property tax rate. While these
considerations are valid for all districts in a metropolitan area, they are particularly pertinent in
the central district, where non-residential property represents a large fraction of the total
property tax base. Since the property tax revenue is under estimated in the model and
estimation, so is the spending per student, particularly in the central districts. 43
Regarding the district-level fraction of households with children in public schools, the
fit of this variable is crucially affected by the representation of metropolitan areas and
household types. In particular, the low number of neighborhoods and households types
imposed by computational constraints implies that often all the households that choose a
particular district are relatively homogeneous, and make the same school choices. In these
cases, the district predicted fraction of households with children in public schools is either zero
or one. This feature, however, is less frequent for large districts. The limited number of
household types also affects the fit of the metropolitan area-level fraction of households with
children in Catholic schools. In particular, operating with only ten religious types seems to be
an important factor affecting the fit of this variable.44
42
For instance, all districts in the New York-Long Island metropolitan area set their own property taxes, with the
exception of the New York City School District, whose funding is provided by the government of New York City
from various sources and according to a mechanism not reflected in the model.
43
The inclusion of non-residential property in the central district does improve the fit of the district average
spending per student, but worsens the fit of other variables, most importantly the district fraction of public school
enrollment. This is because a higher spending in the central district improves the public school quality in that
district and discourages private school enrollment. In order to obtain the right level of private school enrollment,
one would need either a low productivity in public schools in the central district, or tuition subsidies for private
schools. In addition, such productivity differential or tuition subsidy should differ among districts and possibly
metropolitan areas, as suggested by some experiments I conducted. Since either possibility requires additional
parameters, they are computationally not feasible at this time.
44
For instance, 34% of the population in Philadelphia is Catholic, while 47% is Catholic in Pittsburgh. This
means that in Philadelphia, three out of ten religious types are Catholic, whereas five out of ten are Catholic in
Pittsburgh. Thus, a difference of 13% is translated into 20% of the types being different.
26
Finally, Table 5 shows the correlations between the matched variables, both for the
observed and fitted values for each version of the model. The correlations that do not involve
predicted spending per student resemble the actual correlations reasonably well.
8. Simulating Private School Vouchers
In this section I use the parameter estimates to simulate two types of voucher programs for the
Chicago metropolitan area. The first type is a universal voucher program in which every
household in the metropolitan area receives a voucher that may be used to attend any type of
private school. In the second program (“non-Catholic vouchers”), the voucher may only be
used to attend non-Catholic schools.45 In my simulations, a voucher is an amount of money
that a household receives from the state and may only be used for private school tuition.
Households may supplement the voucher with additional payments towards tuition, but cannot
retain the difference if the tuition is lower than the voucher level. Consequently, tuition is
never set below the voucher level.
Furthermore, in these simulations the voucher level is set exogenously by the state.
Vouchers are funded through a state income tax in the same manner as the state aid per
student. Thus, with vouchers, the state budget constraint (9) changes to:
(18) t y =
µ (N C ∪ N NC )v + ∑ µ (N
pub
∩ J d ) AIDd
d
y (N )
where v is the face value of the voucher. That is, the state has to fund both the flat grants given
to public school students and the vouchers given to private school students. Moreover, during
the policy simulations, household n’s budget constraint differs from the one given in (2):
(19) c + (1 +t d ) p dh + max(T − v,0 ) = (1 − t y ) y n + p n
Notice that when forming private schools, households of a given endowment level residing in
a certain neighborhood choose their optimal tuition taking into consideration the availability
45
My representation of religious preferences is not rich enough to evaluate voucher programs that prohibit the
choice of religious schools in general, so I limit the analysis to a hypothetical voucher program that prohibits only
the choice of Catholic schools.
27
and amount of vouchers. Other things equal, vouchers lead to a higher tuition (and school
quality) level while reducing the tuition share paid by parents. In addition, the definition of the
equilibrium provided in Section 3 changes under vouchers; condition (d) changes to (18), and
households maximize utility subject to the modified budget constraint (19).
The remainder of this section begins with a description of the Benchmark Equilibrium,
which is the equilibrium simulated using the parameter estimates for a non-voucher regime. It
then analyzes the outcome of a universal voucher program, and finishes with the study the
impact of a voucher program for non-Catholic private schools, and the comparison of the
effects of both voucher programs.46
Benchmark Equilibrium
The first column of Tables 6a and 6b illustrate the observed data, whereas the second column
of Tables 6a and 6b present the benchmark equilibrium. In addition, Figures 5a through 5d
contrast the observed and predicted values for some district-level variables. While the fit of the
data is obviously limited, such limitations are due to the same factors described in Section 7.
Nonetheless, the benchmark equilibrium replicates the overall pattern of income and
residential stratification observed in the data (see “Average Household Income Ratio”), with a
higher average spending per student in public schools in the districts with higher average
income and property values. Yet, the per-pupil spending in public schools is under estimated
in eight out of nine districts, likely because of the omission of non-residential property from
the property tax base (see section 7). The underestimation is proportionally larger for the
central district, whose predicted average household income and property values are so low that
the optimal voting strategy for these households in the simulated equilibrium is to vote for a
property tax rate of zero and have their public schools funded exclusively by the state (they
receive a per-student state aid of approximately $2,200).47 In addition, in the simulated
equilibrium these public schools have the lowest spending, peer quality, and overall school
quality in the entire metropolitan area.
Furthermore, in the simulated benchmark equilibrium, households that attend private
schools live in the better neighborhoods of the central district, whose tax-inclusive housing
46
The Benchmark Equilibrium is the set of initial conditions used to start the policy simulations.
According to the Financial Section of the 1990 School District Data Book. The under prediction of per-pupil
spending in the central district affects the outcomes of the voucher simulations, as will be seen later.
47
28
prices are low due to low housing and public school quality.48 The top panel of Figure 6
identifies the household types49 that choose private schools in the benchmark equilibrium.
Private school households are, on average, wealthier than public school households, but live in
houses of lower quality and property values (see Table 6b). Private school households also
have above-average taste for Catholic schools.50 On average, public schools have lower perpupil spending, peer quality and overall school quality than private schools, although with
higher variation around the mean because of their more heterogeneous student body.
In the benchmark equilibrium, Catholic schools are primarily attended by children
from Catholic households (84%) whereas public schools are mostly attended by children from
non-Catholic households. In fact, only 31% of the public school enrollment comes from
Catholic households.
Universal Vouchers
Below I analyze the effects of universal vouchers on school choices, residential decisions, and
school quality in both public and private schools. In addition, I discuss the fiscal and welfare
implications of universal voucher policies.
Household Sorting across Schools under Universal Vouchers
Table 6a presents some results from the simulation of universal vouchers for $1,000, $3,000,
$5,000 and $7,000.51 As the table shows, private school enrollment grows as the dollar amount
of the voucher increases and reaches a maximum of 0.93 of the entire population for a $7,000voucher. Although enrollment grows both in private Catholic and non-Catholic schools, the
market share for Catholic schools decreases with the voucher level.52
48
Although the central district has the lowest average housing quality, some of its neighborhoods are of even
higher quality than those in other districts. These are the neighborhoods chosen by private school families.
49
Recall that a household type is an (endowment, relative taste for Catholic schools) combination.
50
The metropolitan area-average taste for Catholic schools equals 0.978 according to my parameter estimates.
51
Just to put the choice of voucher amounts in context, in the simulated benchmark equilibrium, the highest perstudent public school spendings are $8,900 and $9,200 (the highest spending levels are $9,200 and $10,200 in the
data). Also, in the simulated non-voucher equilibrium the average private school tuition is $6,700. Regarding
actual voucher programs, voucher amounts are up to $5,300, $2,250 and $5,500 in the Milwaukee, Cleveland and
Florida programs, respectively. These values can be compared to the per-pupil spending in public schools in those
places, which in 2001 equaled $9,000, $6,413 and $6,049, respectively. For further information, see
www.heritage.org.
52
The decreasing market share for Catholic schools results partly from the parameterization of religious
preferences. The current parameterization implies that the non-Catholics who most dislike Catholic schools form
private non-Catholic schools only when the voucher level is high enough to guarantee a school quality higher
than what they could purchase in the public system. Furthermore, if the model allowed non-Catholics to have a
29
The second and third rows of Figure 6 depict household sorting across public, Catholic
and private non-Catholic schools for different voucher levels. Users of $1,000-vouchers are,
on average, wealthier than the rest of the population, have an above-average taste for Catholic
schools and supplement the voucher with their own tuition payments. Most of them attend
private (Catholic) schools prior to vouchers. Interestingly, vouchers for $1,000 do not reach
the lowest income population nor do they attract all of the middle and high-income population.
However, both the lower and higher income households choose private schools as the voucher
amount grows, as do households with weaker tastes for Catholic schools.
To understand the pattern of private school expansion under vouchers, we need to
recall that households in this model form private schools for at least one of following three
reasons: they either seek religious education,53 want to isolate themselves from peers of lower
quality, or seek a higher spending per student. On the low end of the income distribution there
are households that live in the central district in the non-voucher equilibrium and do not have
peer quality reasons to form private schools.54 Nor is spending an incentive for them until the
voucher amount gives them a per-pupil spending higher than the state aid for public schools.
Therefore, these households do not take vouchers for less than the state aid level of $2,200
unless their religious taste is sufficiently high.55 Yet when the voucher amount exceeds this
threshold, all low-income households take vouchers.56 Although they lose peer quality, the
gains in spending (and religious match with the private school in some cases) more than
compensate for the peer quality losses.
On the high end of the income distribution there are households that in the benchmark
equilibrium segregate themselves in high quality public or private schools. For those with a
low preference for Catholic education, vouchers are attractive only if the resulting private
school tuition is higher than their public school spending, which happens only at relatively
relative preference for non-Catholic private schools relative to public schools, voucher use would likely be higher
at each voucher amount.
53
Catholic education, given the parameterization of religious preferences in my model.
54
In the benchmark equilibrium, these households attend public schools in the central district, where they mix
with households of slightly higher income and therefore enjoy a school peer quality higher than their own.
55
In fact, none of these households takes $1000-vouchers.
56
However, if the benchmark equilibrium spending per student were higher (closer to the observed values), then
the minimum voucher level necessary to shift low-income households in the central district households into
private schools would also be higher. Therefore, when the remainder of this section refers to the voucher level
threshold that makes low-income households indifferent between public and private schools, it should be kept in
mind that this threshold would be higher if the property tax base were appropriately computed. In general, if the
benchmark equilibrium spending per student were higher in each district, then there would be less private school
enrollment at each voucher level, and the privatization of the system would occur at voucher levels higher than
$7,000.
30
high voucher levels of approximately $5,000 or more. However, households with a higher
preference for Catholic education use vouchers even at lower voucher levels. Other things
equal, the higher the household wealth, the higher the voucher amount necessary for a
household to choose private schools.57
Between the low and high end of the income distribution there are middle-income
households for whom vouchers may provide better peers, higher spending per student, or a
better religious match with a school. Once again, the stronger a household’s preference for
Catholic schools, the lower the voucher amount needed by the household to shift to private
schools.
As for the religious composition of private schools, throughout these simulations,
Catholic schools are predominantly attended by Catholic households, and non-Catholic
schools are mostly chosen by non-Catholic households.
Household Sorting across Neighborhoods and Districts under Universal Vouchers
Voucher-using households choose the best neighborhoods of the central district when the
dollar amount of the voucher is low (approximately below $2,200), as vouchers provide
incentives for some middle-income households from good public school districts to migrate
into the central district and enroll their children in private schools. Since this migration raises
the average household income in the central district, it severs the residential stratification
created by the residence-based public school system, as evidenced by the drop in the “Average
Household Income Ratio” in Table 6a.
Because housing quality is a normal good, private school households choose
neighborhoods of higher quality as the dollar amount of the voucher grows,58 so that private
schools progressively appear in higher quality neighborhoods (see Figure 7).59 Meanwhile,
low-income households take up the voucher as well when the voucher amount is sufficiently
high yet stay in the central district. Thus, a growing voucher level brings forth less income
mixing within each district, and more inter-district stratification. In particular, for vouchers of
57
As can be seen in Figure 7 Non-Catholic private schools arise for higher voucher levels than Catholic schools.
This is due to my parameterization of religious preferences, and it can explained by the reasoning presented in
footnote 52.
58
Holding everything else constant, with higher vouchers households can choose higher tuitions while paying a
lower portion of them, and re-allocate money from educational expenditures into consumption and housing.
59
Statements about the location of private schools should be interpreted as statements about the location of
households that form private schools, since private schools do not have a geographic address in this model.
31
$7,000, the geographic distribution of income closely mirrors that of housing quality, and
residential stratification is even stronger than in the absence of vouchers.60
The counterpart of the increased demand for housing in neighborhoods with low taxinclusive property values is the reduced demand for housing in the more expensive
neighborhoods and districts as some households move away. Since housing appreciates in the
neighborhoods chosen by private school households and depreciates in the other ones,61 the
variation in housing prices falls with vouchers (see the “Average Property Value Ratio” in
Table 6a), and housing prices no longer reflect public school quality but merely housing
quality. Interestingly, through all these price changes vouchers create migration opportunities
for households that choose private schools, and for those that stay in the public system, some
of whom get the opportunity to enjoy higher housing quality in a less expensive neighborhood,
and sometimes a higher public school quality as well.
Universal Voucher Effects on School Quality
By affecting household residential and school choices, vouchers affect the quality of public
and private schools. Table 6b shows the evolution of school quality indicators under vouchers.
The average quality grows with the voucher level; under a $7000-voucher the average school
quality is 28% higher than in the benchmark equilibrium. This school quality gain amounts to
4.4% of the average household endowment.
As the voucher level grows and most households choose private schools, the average
spending grows, but since vouchers defray a growing fraction of tuition, the dispersion of
spending around its mean falls. Therefore, although the average school quality and spending
per student increase with the voucher amount, their variation around the mean falls.
A look at public versus private schools reveals some differences between both groups
and the households that choose them (see Table 6b). As the voucher amount grows, public
school spending, school quality and peer quality increase on average but display less variation,
given the fact that households that remain in public schools are progressively wealthier and
60
Because vouchers affect residential and school choices, they impact the geographic distribution of average
religious preferences across neighborhoods. See Ferreyra (2002) for a discussion on this issue.
61
California’s Proposition 174 was a statewide voucher initiative voted in 1993, which unlike other proposals,
featured universal vouchers. Using precinct returns on the ballot, Brunner, Sonstelie and Thayer (2001) find
evidence that homeowners in good public school districts were less likely to vote for the voucher, as they
perceived it as a threat to their property values. In other words, their findings agree with my simulation outcomes.
The issue of capital gains/losses surfaces again in this section when talking about winners and losers from
vouchers. As will be seen there, my simulations also provide evidence that households in good public school
districts suffer capital losses and often lose with voucher policies.
32
more homogeneous. The opposite happens in private schools as low-income households join
them.62 However, once vouchers exceed $3,000 and more middle and high-income households
attend private schools, the trend reverses.
Even though public schools are not responsive to competition from private schools in
this model, vouchers still have the potential to improve public school quality.63 For instance, if
the increased demand for housing on the part of private school households in the central
district resulted in higher property tax revenues per student in public schools, the loss of the
best peers in public schools might be offset by a higher spending, which might lead to higher
public school quality.64 However, this hypothetical improvement in public school quality does
not occur in the current simulations. Since the pre-voucher property tax rate in the central
district is zero, public schools in the district cannot benefit from the voucher-induced higher
residential values. Moreover, at sufficiently high voucher levels the district’s median voter
chooses private schools, so that the equilibrium property tax rate remains zero.
Although public school quality falls in these simulations, these losses are spread
throughout all districts. Public schools in the suburbs suffer either because of lower property
values, loss of public school constituency, or loss of good peers to private schools.65 Lower
property prices not only hurt a district’s property tax revenue per student, but also attract
households of lower income than the original residents, which hurts the peer quality of the
district’s public schools and may lower the median voter’s income.66
To evaluate how gains and losses in school quality are distributed in the population,
Figure 8 depicts the average school quality for the 10th, 30th, 50th, 70th and 90th percentile
endowments in the benchmark and voucher equilibria as well as the proportional gain in
school quality for these wealth levels. Not surprisingly, school quality is increasing in
endowment. The hierarchy of schools by quality stays the same for all voucher levels, so that
the households with the worst schools in the benchmark equilibrium also attend the worst
(albeit better) schools in the voucher equilibria.
62
Based on a 1995 survey designed to explore the voucher issue, in which households were asked about their
desire to go to private schools, Moe (2001) characterizes the public and private sectors that would result if
vouchers were available. His conclusions mirror those presented here.
63
Nechyba (2001) considers the effects of competition on public schools, among others. For empirical evidence
on the positive effects of vouchers on public schools, see, for instance, Hess (2001).
64
This reasoning is explained in detail in Nechyba (1999).
65
The “cream skimming” of the best peers by private schools is an argument often used against vouchers. It is
interesting that in a multi-district setting the phenomenon is spread across districts.
66
The public school quality of a district that loses residential value may actually rise with vouchers. Lower
property values lead to higher property tax rates, holding everything else constant, and under certain conditions
this may lead to higher property tax revenue per child and a higher school quality in public schools.
33
The households that gain the most in school quality at all voucher levels are those
whose endowment is around the 30th percentile (about $35,000). In the absence of vouchers,
these households attend public schools in the central district. With vouchers, some of them
initially take advantage of lower housing prices in other districts and move there yet remain in
the public system. However, as vouchers increase all these households switch into private
schools. Unlike other household groups, this one experiences gains in peer quality and
spending at all voucher levels. This occurs because in the benchmark equilibrium they attend
public schools with an average peer quality that is lower than their own, thus leaving room for
improvement in peer quality. Additionally, the per-pupil spending in the central district is the
lowest in the metropolitan area, which leaves room for improvement in spending as well.
Finally, the wealthiest households almost invariably lose school quality, although
relatively little. These losses are primarily due to a lower per-pupil spending in public schools
induced by lower residential prices and higher income tax rates.
Universal Voucher Fiscal Effects
Since vouchers are funded by a state income tax in these simulations, fiscal burdens
progressively shift from district property taxes onto the state income tax. Whereas the average
property tax rate falls, largely as a result of fewer people in the public system,67 the income tax
rate rises (see Table 6c). For each child that switches from public to private schools the state
saves the state aid for public school students, but it also incurs the voucher cost. Given the
state aid levels for the districts in the Chicago metropolitan area, the state incurs a higher net
expense in these simulations. Moreover, even though households pay less in property taxes in
a voucher regime, on average, their fiscal burden is higher because of their higher income tax
liabilities.
Welfare Implications of Universal Vouchers
Some of the most important questions regarding vouchers concern their effects on social
welfare, and who wins and who loses when vouchers are introduced.68 As Table 6d shows, the
67
68
Recall that for households that choose private schools, the optimal property tax rate is zero.
Gains and losses are computed relative to the benchmark equilibrium.
34
majority of the population benefits from vouchers of (at least) $3,000.69 The fraction of
households that are better off with vouchers reaches a peak of 0.85 for vouchers of $5,000,
when 80% of the population attends private schools, but drops to 0.64 for a voucher of $7,000
mainly as a consequence of the high fiscal burden imposed by this voucher level.
Furthermore, the average welfare gain70 is maximized at $250 for vouchers of $3,000,
and minimized at -$650 for $7,000-vouchers. These outcomes amount to 0.5 and 1.2% of the
average household endowment respectively. Although the average welfare gain is small
relative to the average wealth, the distributional effects are large, as explained below. In
addition, notice that although the average school quality is monotonically increasing in the
voucher amount, the average welfare gain is not. The reason is that the fiscal burden, public
school and capital losses of the losing minority break the upward trend for the average welfare
gain and drive it down for vouchers above $3,000.
Figure 9 identifies the winners and losers from each voucher amount. While the
average gain among winners is no larger than $1,200, the average loss among losers can be as
high as $5,800 (see Table 6d). With a higher taste for Catholic education than the average
loser, the average winner enjoys school quality gains and a better religious match with his
school relative to the non-voucher equilibrium.71 A low-level voucher benefits middle-income
households with a strong taste for Catholic education, many of whom are in private schools
before vouchers. This makes the average winner (slightly) wealthier than the average loser (see
Table 6d). However, as soon as low-income households use the voucher the opposite occurs,
and the average loser continues to be wealthier than the average winner as the voucher level
grows, primarily due to the rising income tax burden.
In order to analyze the welfare implications of vouchers it is useful to focus on four
groups: households in public schools before and after vouchers, households that switch from
public to private schools, households in private schools before and after vouchers, and
69
While $3,000 is the closest voucher amount to $2,200 included in my simulations, a majority of winners is
likely to arise for vouchers approximately larger than $2,200. In what follows, references to the voucher threshold
of $3,000 can also be interpreted as references to the voucher amount that induces all the low-income population
to use vouchers. Bear in mind, however, that if the spending per student were better predicted in the Benchmark
Equilibrium, this threshold would be higher (around $5,500, which is the observed per-pupil spending in the
central district).
70
Welfare gains and losses are measured using compensating variation.
71
If the model included some households that preferred other religious schools, or private non-sectarian schools
over public schools, then among the winners there would be not only households with a high taste for Catholic
education, but also households with a high taste for other religious or private non-sectarian schools. This
hypothetical model may predict more winners than the current one, hence the current universal vouchers
simulation may underestimate the number of winners and their welfare gains.
35
households that switch from private to public schools (see Tables 7d and 7e). Interestingly
enough, each of these groups presents its own winners and losers at certain voucher levels.
Among households in public schools before and after vouchers, no more than half of
them win with vouchers. The average winner takes advantage of housing price changes and
moves into better neighborhoods, which also gives him access to better peers in public schools
and eventually to a higher public school quality. At the same time, because his initial location
is now sought after by wealthy private school households, he reaps capital gains on his nonvoucher equilibrium house.72 As for the average loser, his profile depends on the dollar
amount of the voucher. For a voucher of $1,000, losers belong to two groups. One is lowincome households that stay in the central district and experience lower public school quality.
The other group is middle and high-income households that tend to stay in their initial
neighborhoods and experience capital losses, the higher fiscal burden imposed by vouchers,
the loss of good peers to private schools, and the loss of property tax revenue that funds public
schools. As the voucher level rises, the first group of losers shifts to private schools and
actually benefits from vouchers, while the second group shrinks to only include progressively
wealthier households.73
Most households that move from public to private schools win with the introduction of
vouchers. Furthermore, winners as well as losers in this group enjoy a higher school quality
and a better religious match with their school. A sizeable fraction of these households move to
neighborhoods of lower housing quality where they save in housing costs and property taxes.
The higher the voucher amount, the lower the loss in neighborhood quality.74 At high voucher
levels losers are generally wealthier than winners. Since the public schools that losers attend
before vouchers are among the best in the metropolitan area, only a very high voucher level
produces a higher school quality for them. However, such a high voucher level also demands a
relatively high fiscal burden and low consumption, which offsets their gains in school
quality.75
Households that remain in private schools win for vouchers below $7,000. However,
when the voucher reaches $7,000, the income tax burden is high enough to make some of
72
The gains experienced by some households that remain in the public system is an example of how a multidistrict general equilibrium model produces results that would not necessarily arise in a partial equilibrium model
or in a single-district framework.
73
If the effects of competition on public schools were considered, there would likely be more winners among
households that remain in public schools.
74
Recall that private schools arise in higher quality neighborhoods as the dollar amount of the voucher rises.
75
A similar result for households that switch to private schools yet lose with vouchers is reported by Epple and
Romano (1998).
36
these households lose. Among all the households in the population that benefit from vouchers,
households in private schools before and after vouchers are those that reap the largest average
gains at each voucher level. Their highest average gain is for $3,000-vouchers, when they gain
$1,800, which represents 3% of their average wealth.
Only four out of seven hundred and fifty household types leave private schools after
vouchers. Furthermore, this occurs only for the $1000-voucher. These four households lose
because of vouchers. Induced by rental value changes, they move into better neighborhoods
outside their original central district, and in so doing they pay higher tax-inclusive housing
prices. At voucher levels greater than $1,000 all these households remain in private schools.
As previously mentioned, the average welfare effects of vouchers are modest relative
to the average household’s wealth at all voucher levels (see Table 6d, third row). There are,
however, important distributional effects. Figure 10 depicts the compensating variation
relative to endowment (“relative gain”) at each taste for Catholic schools, for the 10th, 30th,
50th, 70th and 90th percentile endowment levels. The welfare gain is non-decreasing in the taste
for Catholic schools for each endowment level, reflecting the gains from a better religious
match with a private school and the fact that, non surprisingly, households with a strong
preference for Catholic schools are the most favored by this voucher regime.
Besides experiencing the largest proportional school quality gains, households with the
30th percentile endowment level reap the largest relative welfare gains (around 5%).76 Most of
these households stay in the public system with a $1,000 voucher, but move to districts with
better schools. With higher voucher levels, they move to neighborhoods slightly better while
reaping the benefits of private schools. Since their income level is low, the higher income tax
rate has little effect on them.
Non-Catholic Vouchers
Non-Catholic vouchers raise the price of Catholic schools relative to non-Catholic private
schools. To see this, notice that if the voucher is universal, households of a given endowment
forming a private school set the same optimal tuition regardless of the religious orientation
chosen for the school. In contrast, when vouchers may only be used at non-Catholic schools,
the optimal tuition (and school quality) is lower at Catholic than private non-Catholic schools,
other things equal, and Catholic school households must bear the full tuition and reduce their
76
In fact, they usually attain the largest compensating variation in absolute values.
37
consumption of all other goods. As a result, only households with a high taste for Catholic
schools and the ability to pay for them remain in Catholic schools once the voucher is
introduced. As the dollar amount of the voucher increases, Catholic schools lose not only
market share, but also enrollment in absolute terms.
Restricting the eligibility of private schools also alters the religious composition of
private Catholic and non-Catholic schools. On the one hand, as the dollar amount of the
voucher grows, enrollment in Catholic schools (while they still exist) becomes progressively
more Catholic, since households with a lower taste for Catholic schools (including some nonCatholic households) choose other options. On the other hand, the religious composition of
enrollment in and public and private non-Catholic schools diversifies as the dollar amount of
the voucher increases, given that many Catholic households now attend private non-Catholic
schools.
Household Sorting Across Schools under Non-Catholic Vouchers
As Figure 11 shows, among the households that choose Catholic schools in the benchmark
equilibrium, those with the highest preference and ability to pay for Catholic schools remain
there when the $1,000-voucher is implemented, while the others either switch to public or
private non-Catholic schools. In addition, some public school households switch to private
non-Catholic schools as well. Although the fraction of households that use vouchers for $1,000
is lower under non-Catholic than universal vouchers, the fraction of households in private
schools is approximately the same precisely because many households continue to attend
Catholic schools in spite of the fact that they cannot use the voucher there.
A $3,000-voucher shifts additional Catholic school households to public and private
non-Catholic schools. As in the universal voucher case, all low-income households use the
voucher at this level, because it guarantees them a higher school quality than the public
schools in the central district. However, some middle and high-income households that would
use the voucher if it were universal for the sake of attending Catholic schools do not find it
large enough now. Hence, at the $3,000-voucher private school enrollment and voucher use
are lower under non-Catholic than universal vouchers.
At higher voucher levels of the non-Catholic voucher, Catholic schools are not formed
because the gains in school quality attained by former Catholic school households in either
private non-Catholic schools through vouchers, or in public schools, compensate the loss in
38
religious match. Yet, a $5,000-voucher reaches a vast sector of the population regardless of the
voucher regime because two-thirds of all households have a per-pupil spending below $5,000
in the benchmark equilibrium. In other words, this voucher level is high enough not only to
guarantee a higher school quality to a large fraction of the population, but also to compensate
some Catholic school households for a worse religious match at a private non-Catholic school.
This explains why the private school enrollment and voucher use are the same for $5,000
universal and non-Catholic vouchers.
Whereas universal vouchers for $7,000 enroll 93% of the population in private schools,
the whole population chooses private schools under non-Catholic vouchers for the same
amount. While the logic exposed in the previous paragraph certainly applies to a $7,000voucher, an additional factor must be mentioned. Under universal vouchers, housing prices in
the only district that remains public drop by about 50%, whereas they fall by 10% under nonCatholic vouchers. Such disparity is due to the fact that under universal vouchers, Catholic
households have stronger incentives to leave expensive districts even at a $7,000-voucher
level. Thus, houses in these districts depreciate more, to the point that households that remain
find it optimal to stay in public schools rather than forming private schools, and raise property
taxes to prevent spending from falling in their public schools.
To summarize, vouchers for $1,000 are too small to generate a difference in private
school market share between universal and non-Catholic vouchers, whereas vouchers for
$5,000 or $7,000 are too large. A difference, however, arises for $3,000-vouchers, which
generate less private school enrollment and voucher use in the non-Catholic regime. In other
words, at moderate voucher levels, non-Catholic vouchers yield less private school enrollment
(and voucher use) than universal vouchers.
Non-Catholic Voucher Effects on Household Residential Sorting and School Quality
Both universal and non-Catholic vouchers induce residential changes, reduce residential
stratification, and produce a similar impact on property values. However, these effects are not
quantitatively identical. At low non-Catholic voucher levels, the fact that Catholic households
do not have the voucher-induced incentive to migrate to the central district reduces their
demand for housing in the central district relative to universal vouchers. Thus, voucher-using
non-Catholic households find it easier to migrate there, while Catholics move to other
locations yet remain in Catholic schools. Moreover, since the rate of voucher use is lower as is
39
the fraction of households that move from the suburbs into the central district, the depreciation
rate of property values in suburban districts is also lower.
Even at higher levels for the non-Catholic voucher, the fact that Catholics lose the
incentive to live in certain locations in order to attend private schools changes the dynamics of
the housing market. For vouchers above $3,000 Catholics move out of the central district,
collect capital gains on their benchmark equilibrium location and move to suburban districts
where they spend more in housing and attend private non-Catholic or public schools.
Furthermore, appreciation rates differ from those of universal vouchers, mainly depending on
the location of the new private schools.
The overall effects on school quality are very similar for both voucher policies.
Households with the 30th percentile endowment experience the largest proportional gains in
school quality, for the same reasons they do so under universal vouchers. Although Catholic
school households suffer some loss of school quality if they switch to public schools, on
average their loss is lower than 5%. While they exist, Catholic schools have higher quality,
spending and peer quality than private non-Catholic schools because the households that stay
there are wealthier, on average, than those that take up vouchers.
As in the case of universal vouchers, non-Catholic vouchers also affect public school
spending, peer quality and overall school quality in the different districts. Especially at low
voucher values, the losses experienced by public schools in the suburbs are slightly smaller
than those induced by universal vouchers, given that fewer households leave those districts for
the sake of private schools elsewhere. 77
Welfare Implications of Non-Catholic Vouchers
Another difference between both voucher programs is related to welfare implications. For
vouchers above $1,000, fewer households benefit from non-Catholic than universal
vouchers.78 Moreover, the average welfare gain from non-Catholic vouchers is also smaller. In
77
Fiscal effects of non-Catholic vouchers are very similar to those induced by universal vouchers.
If some households in the model had a preference for other religious schools and chose them under universal
vouchers, then the effects described here for households with a strong taste for Catholic schools would apply to
them as well. As a result, even more households would lose from the introduction of non-religious vouchers.
Therefore, the fraction of loser households from this simulation may be seen as a lower bound of those that would
arise from a more complete model.
78
40
fact, while the average welfare gain is negative for non-Catholic vouchers above $1,000, it is
only negative for universal vouchers above $5,000.79
When comparing welfare under universal and non-Catholic vouchers, one seeks to
answer questions such as: do the same households win/lose in both experiments? If not, who
wins in one case but loses in the other? How different are the welfare effects experienced by
households under both policies? Figure 12 identifies winners and losers from non-Catholic
vouchers. Both voucher programs have similar redistributive effects - at sufficiently high
levels vouchers benefit the poor but create capital losses and heavy fiscal burdens for the
wealthy.
Most households either win or lose in both voucher programs. Furthermore, the highest
persistence in the winner state occurs for $5,000-vouchers, when two thirds of the population
gains in both experiments.80 However, at least 20% of all households win under universal
vouchers above $1,000, yet lose under non-Catholic vouchers. These households would either
attend Catholic schools in the benchmark equilibrium or would choose them under universal
vouchers. The fall in their relative gains, which averages between 2 and 4% of their wealth,
makes them the household group with the largest change in relative welfare effects.
As in the case of universal vouchers, non-Catholic vouchers induce, on average, a
small gain relative to household endowment (see Table 7c). However, the distributional effects
are significant here as well. Figure 13 shows that the highest relative gains accrue to the 30th
percentile endowment households for the same reasons as under universal vouchers. Among
households of a given endowment level, Catholic households lose the most – particularly those
that attend Catholic schools in the non-voucher equilibrium. While universal vouchers do not
achieve better outcomes for every household in the population than those brought about by
non-Catholic vouchers, households with a strong preference for Catholic schools are better off
under universal than non-Catholic vouchers. At the same time, the few households that are
worse off suffer only relatively small losses.
Once again, it is convenient to partition the household population into school choice
groups for the analysis of welfare effects (see Table 7d). Beginning with households in public
schools in the benchmark equilibrium, the qualitative effects for them are the same as under
universal vouchers, except that losses among the wealthy for low voucher levels are smaller
79
In addition, whereas the maximum and minimum average welfare gain brought about by universal vouchers are
$250 and -$650 respectively, the corresponding non-Catholic voucher figures are $10 and -$1,040, respectively.
80
At this voucher level, most of the population gains school quality, and their gains are not offset by capital
losses and the higher income tax burden.
41
now, given that the smaller number of people attracted into private schools reduces their
capital and peer quality losses.
Among households in public schools before vouchers and private (non-Catholic)
schools after vouchers, at least half of these households win with non-Catholic vouchers. Their
average winner usually gains school quality, despite sometimes losing peer quality from good
public schools. However, the fraction of households in this group that benefits from nonCatholic vouchers is lower than for universal vouchers due to the absence of religious
motivations to form private schools.
As for Catholic school households, most of them lose for non-Catholic vouchers above
$1,000 regardless of the school they choose. They are outbid in the central district by those
that use the voucher for non-Catholic schools, and need to move into more expensive
neighborhoods. Since they are relatively wealthy, the growing pressure derived from the
income tax affects them negatively as well. Finally, although they sometimes gain school
quality by switching to public or private non-catholic schools, the higher school quality fails to
compensate them for the loss of the best possible school religious match.
9. Concluding Remarks
Few policies are as controversial in the United States as private school vouchers. Although no
data is available to analyze the impact of a large-scale voucher program in the United States,
one can learn about their hypothetical effects by simulating programs on the basis of structural
estimates from a general equilibrium model. In this paper I estimated a general equilibrium
model with multiple public school districts, and private schools. This is one of the first
attempts to estimate a multi-jurisdictional model, and the first one to include private schools
and their formation. A novelty in this paper is the inclusion of religious schools and household
religious preferences, which has enabled me to compare the effects of a universal voucher
program with those of a program excluding religious schools.
The model used here can be used to explore a variety of issues that are relevant for the
voucher policy debate. These issues include the effects of voucher targeting, child-centered
funding policies, and changes in the public school financial regime. For example, every
publicly funded voucher program that has been enacted in the United States targets a specific
42
population.81 Income and/or district targeting can be easily incorporated into the current
model, so that actual programs could be simulated and their predictions compared to the
observed outcomes. In addition, some expansions of actual programs could be simulated as
well, which would let us answer questions about the effect of broadening the program’s target.
Furthermore, proposals for targeted vouchers have sprouted in several states in the last ten
years although most of them have not been implemented to date.82 The current estimates
permit the simulation of these specific proposals, and the identification of their welfare effects.
Proposals of child-centered funding have also been presented to several state
legislatures.83 In this case, parents are entitled to a given sum of money that can be used in any
school (public or private) they choose. This is another type of program that can be simulated
using the current model, by having all households vote on a state-wide per-pupil spending and
letting them select how to use it.
In some states, school reform has called for a change in public school financial regime.
Such changes can be examined using the current model as well. For instance, Proposal A,
approved in March 1994 in Michigan, shifts the main burden of school funding from local
property taxes to the state sales tax (Mintrom and Plank, 2001) and makes public school
funding fully portable, so that students moving from one public school to another may take
their funding with them. Using data from the 2000 Census would permit the comparison
between the effects predicted by the model and those that have actually taken place since the
legislative change. Moreover, an issue of current policy debate in Michigan is whether the
portable funding could also be used as a private school voucher. Yet, the Michigan
Constitution explicitly prohibits public funding for non-public schools and places a cap on the
tax revenue collected by the state relative to the state total personal income. The current model
can also be used to analyze the welfare implications of both these constitutional provisions.
81
The A+ program in Florida focuses on children in failing public schools, the Milwaukee Parental Choice
Program targets children from low-income families in Milwaukee, and the Cleveland Scholarship Program targets
children from low-income families residing in the Cleveland city district.
82
See, for instance, New York City’s former Mayor Giuliani’s 2001 proposal of a $12 million program, modeled
after the Milwaukee choice program, to reach students in some school districts for a three-year period. For more
details, see www.heritage.org/schools/new_york.html
83
See, for instance, California’s Proposition 38, introduced in 2000 to provide universal vouchers worth $4,000
per child for use at any public or private school. For further information on Proposition 38, see
www.heritage.org/schools.
43
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TABLE 1
Selected Metropolitan Areas
Metropolitan Area
1990 Census Population
(in thousands)
No. of
School
Districts
Largest
District’s
Relative
Size
Fraction
of
Catholic
Population
Share
of
Local
Sources
87
.158
.49
.72
Boston, MA
2,871
50
.472
.41
.68
Chicago, IL
6,070
110
.261
.35
.78
Detroit, MI
4,382
167
.644
.43
.76
New York-Long Island, NY
11,156
106
.315
.34
.57
Philadelphia, PA—NJ
4,857
80
.141
.47
.63
Pittsburgh, PA
2,057
71
.146
.26
.65
St. Louis, MO-IL
2,444
Notes:
Secondary and Unified School Districts included.
District Relative Size = number of housing units in district / number of housing units in metropolitan
area .
Share of local sources for public school funding is the average share over districts in each metropolitan
area.
Source: 1990 Census and School District Data Book (SDDB), and 1990 Churches and Church
Membership in America.
TABLE 2
School Districts in Selected Metropolitan Areas: Summary Statistics
Mean
Std. Dev.
1st
Percentile
115
37
.009
.096
28,870
24,792
39,764
3,870
3,748
4,189
3,541
.001
.143
.011
.005
99th
Percentile
10,698
10,619
.348
.268
145,046
131,120
220,896
35,226
32,803
40,781
13,315
.644
.979
.754
.120
Fall Enrollment
1,988
10,741
No. Households
1970
11,829
Fraction Private
.150
.074
In Central District
.194
.042
Avg. Household Income ($)–All Households
53,469
22,872
Hhs. w/ Children in Public Schools
49,743
22,036
Hhs. w/ Children in Private Schools
74,560
34,354
Avg. Housing Rental Value ($)–All Households
10,667
6,390
Hhs. w/ Children in Public Schools
10,295
6,255
Hhs. w/ Children in Private Schools
12,772
7,622
Avg. Spending per Student in Public Schools ($)
6,866
2,066
District Size Relative to Metro Area
.210
.262
Share of Local (District) Revenues
.590
.196
Share of State Revenues
.351
.165
Share of Federal Revenues
.058
.042
No. observations: 671 school districts
Notes:
Household data and Fall Enrollment is for grades 9 through 12.
“Fraction private” is short for fraction of households with children in private schools.
District size = number of housing units in the district / number of housing units in the metropolitan area
Shares of Local, State and Federal Revenues add up to 1. Shares with respect to district total revenues.
Weighted statistics. Weight = number of households for income and rental value and fraction private,
Fall enrollment for everything else.
Source: 1990 SDDB and author’s own calculations.
2
TABLE 3
Private School Enrollment and Catholic Population in the Twenty Largest Metropolitan Areas
Metropolitan Area
Atlanta
Baltimore
Boston
Chicago
Dallas
Detroit
Houston
Los Angeles - Anaheim
Minneapolis-St. Paul
New York - Long Island
Philadelphia
Phoenix
Pittsburgh
Riverside-San Bernardino
San Diego
San Francisco-Oakland
Seattle
St. Louis
Tampa-St. Petersburg
Washington
Private
Schools
Enrollment
Share
Fraction of
Catholic
Population
0.06
0.15
0.16
0.17
0.05
0.09
0.04
0.10
0.11
0.18
0.24
0.05
0.09
0.03
0.06
0.15
0.09
0.17
0.06
0.12
0.05
0.17
0.49
0.41
0.10
0.35
0.18
0.32
0.26
0.43
0.34
0.15
0.47
0.16
0.16
0.24
0.12
0.26
0.15
0.17
Fraction in each type of private
school
Catholic
0.25
0.68
0.73
0.84
0.43
0.75
0.38
0.62
0.57
0.67
0.80
0.61
0.77
0.26
0.46
0.60
0.46
0.79
0.38
0.50
Other
Non
Religious Sectarian
0.49
0.26
0.16
0.16
0.06
0.21
0.12
0.03
0.46
0.11
0.23
0.02
0.45
0.16
0.27
0.11
0.34
0.09
0.24
0.09
0.11
0.08
0.39
0.00
0.09
0.14
0.65
0.08
0.41
0.13
0.18
0.22
0.42
0.12
0.14
0.07
0.45
0.17
0.29
0.21
Correlation between the fraction of Catholics and the fraction of students enrolled in Catholic schools
is 0.79
Notes:
Sources: 1989 Common Core of Data, 1989 Private School Survey, and 1990 Churches and Church
Membership in America.
Enrollment in grades 9 though 12.
3
TABLE 4
Parameter Estimates
Parameter
Estimate
α
0.12
(0.001)
0.72
(0.001)
0.24
(0.001)
0.11
(0.009)
0.25
(0.001)
376.56
β
ρ
r
π
Sum of Squared Residuals
Notes:
Standard Errors in parentheses.
Number of observations: see Section 6.
TABLE 5
Goodness of Fit : Some Correlations
a. Observed Data
Average Hh. Income
Average Rental Value
Spending per Student
Fraction Public
Average Hh.
Income
1
0.98
0.52
0.25
Average Rental
Value
Spending per
Student
1
0.61
0.21
1
-0.13
Average Hh.
Income
1
0.82
0.95
0.41
Avg. Rental
Value
Spending per
Student
1
0.78
0.37
1
0.63
Fraction Public
1
b. Fitted Data
Average Hh. Income
Average Rental Value
Spending per Student
Fraction Public
Notes:
-58 districts
-weighted correlations - weight: number of households in the district
4
Fraction Public
1
TABLE 6a
Universal Vouchers in Chicago: Effects on School Choice and Basic Demographics
Data
$0
Private School Enrollment
Fraction Households in Private Schools
Fraction Hhs. in Catholic Schools (w.r.t. Hhs. in
Private Schools) (1)
0.16
0.17
0.27
0.73
0.87
0.93
0.84
1.00
1.00
0.62
0.58
0.53
0.73
0.11
0.01
0.16
0.27
0.56
0.00
0.17
0.13
0.70
0.00
0.17
0.07
0.76
0.00
0.17
0.84
0.61
0.62
0.60
0.60
n/a
0.31
n/a
0.32
0.80
0.23
0.80
0.20
0.80
0.20
2.83
2.20
2.60
2.18
0.56
3.38
1.69
0.47
3.63
1.34
0.60
3.38
0.99
0.64
School Choice before and after Vouchers
Fraction Hhs. choosing Public-Public (2)
Fraction Hhs. choosing Public-Private
Fraction Hhs. choosing Private-Public
Fraction Hhs. choosing Private-Private
Religious Composition of Public and Private
Schools
Catholic Schools: fraction of Catholic students
Private Non-Catholic Schools: fraction of nonCatholic students
Public Schools: fraction of Catholic students
Demographics
Average Household Income Ratio (3)
Average Housing Rental Value Ratio (4)
Fraction of Hhs. that move
Voucher Amount
$1,000 $3,000 $5,000 $7,000
2.42
3.37
Notes:
(1) “Hhs.” stands for “households”.
(2) “Public-Public” is short for “public schools before vouchers, and public schools after vouchers”.
(3) Average Household Income Ratio = avg. hh. income in highest housing quality district / avg. hh.
income in lowest housing quality district.
(4) Average Housing Rental Value Ratio = id. Avg. Hh. Income Ratio, but for housing rental value.
5
All Schools
TABLE 6b
Universal Vouchers in Chicago: Effects on School Quality
Data
Voucher Amount
$0
$1,000 $3,000 $5,000
Avg. Quality (1)
Avg. Spending
Avg. Peer Quality
Fraction of Hhs. with higher School
Quality (2)
Public Schools
Avg. Quality
Avg. Spending
Central District
Suburbs (average)
Avg. Peer Quality
Public School Households
Avg. Wealth
Avg. Housing Rental Value
Avg. Neighborhood Quality
$8,500 $8,600 $8,800 $9,500 $10,800
$5,000 $5,200 $5,300 $6,000 $7,300
$45,000 $45,000 $45,000 $45,000 $45,000 $45,000
0.47
0.61
$7,900 $8,100 $12,300
$4,700 $4,800 $7,100
$5,500 $2,200 $2,200
n/a
$7,800 $6,100 $5,900 $7,100
$49,200 $41,900 $42,000 $69,100
$9,900
Private Schools
Avg. Quality
Avg. Spending
0.65
0.81
$12,600
$7,300
n/a
$7,300
$71,800
$14,200
$8,300
n/a
$8,300
$80,900
$50,800 $51,000 $78,800 $82,200 $90,900
$9,000 $9,000 $9,900 $8,300 $6,800
0.72
0.75
0.96
1.01
1.07
$11,400 $10,200 $7,500
$6,700 $6,000 $4,700
Avg. Peer Quality
Avg. Fraction of Tuition covered by
Voucher
Private School Households
Avg. Wealth
Avg. Housing Value
Avg. Neighborhood Quality
$7,000
$8,900 $10,600
$5,800 $7,200
$67,700 $60,700 $53,200 $36,300 $40,900 $42,400
0.19
0.73
0.90
0.94
$69,300 $61,700 $44,800 $49,500 $51,100
$11,300 $7,800 $7,100 $7,500 $8,100 $8,400
0.66
0.62
0.62
0.67
0.69
Notes:
(1) this and all dollar values rounded to the closest hundred.
(2) “higher” relative to Benchmark Equilibrium. All comparisons are relative to the Benchmark Equilibrium
TABLE 6c
Universal Vouchers in Chicago: Fiscal Effects
$0
0.03
0.24
$4,000
Income Tax Rate
Avg. Property Tax Rate
Avg. Tax Burden (property tax + income tax)
6
Voucher Amount
$1,000 $3,000 $5,000
0.03
0.05
0.10
0.24
0.17
0.11
$3,900 $4,100 $5,300
$7,000
0.15
0.08
$7,100
TABLE 6d
Universal Vouchers in Chicago: Welfare Implications
Voucher Amount
$1,000 $3,000 $5,000 $7,000
Fraction of Households that Win with Vouchers 0.35
0.63
0.85
0.64
Avg. Welfare Change
$80
$250
-$70
-$650
Avg. Welfare Change / Avg. Hh. Wealth
0.001
0.005 -0.001 -0.012
Winners
Avg. Wealth
$57,000 $49,300 $46,600 $39,200
Avg. Taste for Catholic Schools
1.11
1.03
0.99
1.01
Fraction Hhs. w/ higher School Quality
0.82
0.73
0.75
1.00
Fraction Hhs. w/ higher School Utility (1)
0.85
0.76
0.78
1.00
Avg. Welfare Change
$600 $1,200 $900
$800
Avg. Welfare Change / Avg. Hh. Wealth
0.01
0.02
0.02
0.02
Losers
Avg. Wealth
$52,200 $61,400 $96,500 $79,700
Avg. Taste for Catholic Schools
0.91
0.88
0.88
0.93
Fraction Hhs. w/ higher School Quality
0.28
0.41
0.22
0.49
Fraction Hhs. w/ higher School Utility
0.28
0.44
0.14
0.49
Avg. Welfare Change
-$200 -$1,200 -$5,800 -$3,200
Avg. Welfare Change / Avg. Hh. Wealth
-0.00
-0.02
-0.06
-0.04
Notes:
(1) “School Utility” is the parental valuation of school quality ( = religious match b/w
household and school * school quality).
TABLE 6e
Universal Vouchers in Chicago: Welfare Implications by Household Group
Voucher Amount
$1,000 $3,000 $5,000 $7,000
Hhs. choosing Public-Public
Fraction w.r.t. Hhs. in Public Schools in Benchmark Eqbm.
0.87
0.30
0.16
0.08
Avg. Wealth
$51,000 $78,800 $82,200 $90,900
Avg. Taste for Catholic Schools
0.89
0.84
0.81
0.81
Fraction Winners
0.15
0.26
0.50
0.00
Hhs. choosing Public-Private
Fraction w.r.t. Hhs. in Public Schools in Benchmark Eqbm.
0.13
0.70
0.84
0.92
Avg. Wealth
$49,900 $37,600 $44,800 $47,300
Avg. Taste for Catholic Schools
1.10
0.95
0.94
0.93
Fraction Winners
0.72
0.69
0.89
0.69
Hhs. choosing Private-Private
Fraction w.r.t. Hhs. in Private Schools in Benchmark Eqbm. 0.97
1.00
1.00
1.00
Avg. Wealth
$69,900 $69,300 $69,300 $69,300
Avg. Taste for Catholic Schools
1.28
1.28
1.28
1.28
Fraction Winners
0.99
1.00
1.00
0.67
Hhs. choosing Private-Public
Fraction w.r.t. Hhs. in Private Schools in Benchmark Eqbm. 0.03
0.00
0.00
0.00
Avg. Wealth
$53,200
n/a
n/a
n/a
Avg. Taste for Catholic Schools
1.30
n/a
n/a
n/a
Fraction Winners
1.00
n/a
n/a
n/a
7
TABLE 7a
Non-Catholic Vouchers in Chicago: Effects on School Choice and Basic Demographics
Data
$0
Private School Enrollment
Fraction Households in Private Schools
Fraction Hhs. in Catholic Schools (w.r.t. Hhs. in Private
Schools) (1)
Fraction Hhs. Using Vouchers
0.16
0.17
0.27
0.60
0.87
1.00
0.84
1.00
0.51
0.13
0.03
0.58
0.00
0.87
0.00
1.00
0.70
0.13
0.03
0.14
0.14
0.00
0.33
0.51
0.07
0.09
0.02
0.07
0.11
0.72
0.02
0.15
0.00
0.15
0.00
0.83
0.00
0.17
0.00
0.17
0.84
0.93
1.00
n/a
n/a
n/a
0.31
0.72
0.32
0.61
0.39
0.60
0.38
0.60
n/a
2.83
2.20
2.62
2.17
0.52
3.32
1.84
0.67
3.60
1.36
0.61
3.66
1.93
0.56
School Choice before and after Vouchers
Fraction Hhs. choosing Public-Public (2)
Fraction Hhs. choosing Public-Private
Fraction Hhs. choosing Private-Public
Fraction Hhs. choosing Private-Private
Fraction Hhs. Choosing Catholic-Catholic
Fraction Hhs. Choosing Catholic-Non-Catholic
Religious Composition of Public and Private Schools
Catholic Schools: fraction of Catholic students
Private Non-Catholic Schools: fraction of non-Catholic
students
Public Schools: fraction of Catholic students
Demographics
Average Household Income Ratio (3)
Average Housing Rental Value Ratio (4)
Fraction of Hhs. that move
Voucher Amount
$1,000 $3,000 $5,000 $7,000
2.42
3.37
Notes:
(1) “Hhs.” stands for “households”.
(2) “Public-Public” is short for “public schools before vouchers, and public schools after vouchers”.
(3) Average Household Income Ratio = avg. hh. income in highest housing quality district / avg. hh.
income in lowest housing quality district.
(4) Average Housing Rental Value Ratio = id. Avg. Hh. Income Ratio, but for housing rental value.
8
TABLE 7b
Non-Catholic Vouchers in Chicago: Effects on School Quality
Data
Voucher Amount
$0
All Schools
Avg. Quality
Avg. Spending
Avg. Peer Quality
Fraction of Hhs. with higher School Quality
Public Schools
Avg. Quality
Avg. Spending
Central District
Suburbs (average)
Avg. Peer Quality
Public School Households
Avg. Wealth
Avg. Housing Rental Value
Avg. Neighborhood Quality
Private Schools
Avg. Quality
Avg. Spending
Avg. Peer Quality
Private School Households
Avg. Wealth
Avg. Housing Value
Avg. Neighborhood Quality
Catholic Schools
Avg. Quality
Avg. Spending
Avg. Peer Quality
Catholic School Households
Avg. Wealth
Avg. Housing Rental Value
Avg. Neighborhood Quality
Non-Catholic Private Schools
Avg. Quality
Avg. Spending
Avg. Peer Quality
Avg. Fraction of Tuition covered by Voucher
Non-Catholic Private School Households
Avg. Wealth
Avg. Housing Rental Value
Avg. Neighborhood Quality
$1,000
$3,000
$5,000
$7,000
$45,000
$8,500
$5,000
$45,000
$8,700 $8,900 $9,500 $10,900
$5,200 $5,400 $6,000 $7,300
$45,000 $45,000 $45,000 $45,000
0.54
0.70
0.62
0.81
$5,500
$7,800
$49,200
$7,900
$4,700
$2,200
$6,100
$41,900
$8,200
$5,000
$2,200
$6,000
$41,900
$12,700
$7,400
n/a
$7,400
$71,100
n/a
n/a
n/a
n/a
n/a
$50,800
$9,000
0.72
$51,200 $70,400 $82,200
$9,100 $9,200 $8,400
0.75
0.90
1.02
n/a
n/a
n/a
$11,400
$6,700
$60,700
$10,100 $7,400 $9,000
$6,000 $4,600 $5,800
$53,400 $35,100 $41,000
$10,900
$7,300
$45,000
$69,300
$7,800
0.66
$60,900 $42,800 $49,500
$7,200 $7,000 $8,100
0.62
0.59
0.67
$53,800
$8,700
0.71
$11,400
$6,700
$60,700
$11,600 $16,400
$6,900 $9,700
$61,100 $87,500
n/a
n/a
n/a
n/a
n/a
n/a
$69,300
$7,800
0.66
$69,900 $96,300
$7,100 $8,600
.65
.71
n/a
n/a
n/a
n/a
n/a
n/a
$9,900
$67,700
$11,300
9
$11,100
$6,600
n/a
$6,500
$59,900
n/a
n/a
n/a
n/a
$8,500 $7,100 $9,000
$5,000 $4,400 $5,800
$45,300 $33,300 $41,000
0.23
0.76
0.90
$10,900
$7,300
$45,000
0.96
n/a
n/a
n/a
$51,600 $41,000 $49,500
$6,800 $6,900 $8,100
.59
.59
0.67
$53,800
$8,700
0.71
TABLE 7c - Non-Catholic Vouchers in Chicago: Welfare Implications
Fraction of Households that Win with Vouchers
Avg. Welfare Change
Avg. Welfare Change / Avg. Hh. Wealth
Winners
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Hhs. w/ higher School Quality
Fraction Hhs. w/ higher School Utility
Avg. Welfare Change / Avg. Hh. Wealth
Losers
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Hhs. w/ higher School Quality
Fraction Hhs. w/ higher School Utility
Avg. Welfare Change / Avg. Hh. Wealth
$1,000
0.59
-$10
0.000
Voucher Amount
$3,000
$5,000
0.45
0.67
-$320
-$550
-0.006
-0.010
$7,000
0.39
-$1,040
-0.019
$66,500
1.00
0.73
0.71
0.00
$45,500
0.98
0.87
0.75
0.02
$46,600
0.96
0.75
0.78
0.02
$33,000
0.97
1.00
1.00
0.02
$35,500
0.95
0.27
0.27
-0.01
$60,800
0.97
0.55
0.49
-0.02
$96,500
1.04
0.22
0.14
-0.06
$67,400
0.98
0.69
0.58
-0.03
TABLE 7d - Non-Catholic Vouchers in Chicago: Welfare Implications by Household Group
Voucher Amount
Hhs. choosing Public-Public
Fraction w.r.t. Hhs. in Public Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
Hhs. choosing Public-Private
Fraction w.r.t. Hhs. in Public Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
Hhs. choosing Private-Private
Fraction w.r.t. Hhs. in Private Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
Hhs. choosing Catholic-Catholic
Fraction w.r.t. Hhs. in Catholic Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
Hhs. choosing Catholic-Non-Catholic
Fraction w.r.t. Hhs. in Catholic Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
Hhs. choosing Private-Public
Fraction w.r.t. Hhs. in Private Schools in Benchmark Eqbm.
Avg. Wealth
Avg. Taste for Catholic Schools
Fraction Winners
10
$1,000
$3,000
$5,000
$7,000
0.84
$50,600
0.92
0.55
0.39
$71,600
0.90
0.40
0.13
$82,000
0.89
0.28
0.00
n/a
n/a
n/a
0.16
$51,600
0.91
0.40
0.61
$37,400
0.93
0.46
0.87
$46,000
0.92
0.80
1.00
$50,800
0.92
0.44
0.84
$69,800
1.30
0.92
0.56
$72,800
1.29
0.51
0.87
$67,100
1.28
0.37
1.00
$69,300
1.28
0.15
1.00
6.98
1.30
0.92
0.12
9.63
1.39
0.00
na
n/a
n/a
n/a
na
n/a
n/a
n/a
n/a
n/a
n/a
n/a
0.44
6.62
1.26
0.65
0.86
6.71
1.28
0.37
1.00
6.93
1.28
0.15
0.16
$66,800
1.15
0.85
0.44
$65,000
1.27
0.51
0.13
$83,300
1.27
0.18
0.00
n/a
n/a
n/a
FIGURE 1
Chicago: Census Tracts, School Districts, and School Quasi-Districts
Lake M ichigan
N
W
4
E
0
4
8 Miles
S
Note: The small partitions in the figure are Census tracts, and different colors are used to identify Census tracts located in different
school districts. The thick black lines are the boundaries of the quasi-districts.
FIGURE 2
Chicago: Housing Quality by Neighborhood
Lake Michigan
Housing Quality
0.45 - 0.5
0.5 - 0.55
0.55 - 0.6
0.6 - 0.65
0.65 - 0.7
0.7 - 0.75
0.75 - 0.8
0.8 - 0.85
0.85 - 0.9
0.9 - 0.95
0.95 - 1
1 - 1.05
1.05 - 1.1
1.1 - 1.15
N
W
E
3
0
3
6 Miles
S
11
FIGURE 3
Fitted vs. Observed Values
(observed values on the horizontal axis, fitted values on the vertical axis – Circle size is proportional
to the observation’s total number of households).
Figure 3a - Fraction of Households w/ children in
Public Schools
Figure 3b - Average Household Income
(in $10,000)
11.5
Fitted Fraction Public
Fitted Avg. Hh. Income
1
0
.5
0.00
Observed Fraction Public
0.50
1.00
Figure 3c - Average Rental Value
(in $10,000)
Observed Avg. Hh. Income
11.50
Figure 3d – Spending/Student in Public Schools
(in $10,000)
3.3
Fitted Spending
Fitted Avg. Rental Value
1.4
.1
0
0.10
Observed Avg. Rental Value
0.00
3.30
No. of observations (figures 4a through 4d): 58 quasi-districts
12
Observed Spending
1.40
FIGURE 4
Fraction of Households w/ Children in Catholic Schools
.2
Fitted Fraction Catholic
Chicago
Detroit
New York
St. Louis
Pittsburgh
Boston
Philadelphia
0
0
Observed Fraction Catholic
.2
14
FIGURE 5
Chicago: District-level Observed and Predicted Data
a2. District Predicted Fraction of Hhs. w/ Children
in Private Schools
a1. District Observed Fraction of Hhs. w/ Children
in Private Schools
Lake Michigan
Lake Michigan
Fraction Private
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
Fraction Private
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
N
W
N
E
4
0
4
8 Miles
W
S
E
4
0
4
8 Miles
S
b2. District Predicted Average Household Income (in dollars)
b2. District Predicted Average Household Income (in dollars)
Lake Michigan
Lake Michigan
Avg. Hh. Income
7,500-10,000
10,000-20,000
20,000-30,000
30,000-40,000
40,000-50,000
50,000-60,000
60,000-70,000
70,000-80,000
80,000-90,000
90,000- 100,000
Avg. Hh. Income
7,500-10,000
10,000-20,000
20,000-30,000
30,000-40,000
40,000-50,000
50,000-60,000
60,000-70,000
70,000-80,000
80,000-90,000
90,000- 100,000
N
N
W
E
4
0
4
8 Miles
W
E
S
S
14
4
0
4
8 Miles
15
FIGURE 5
Chicago: District-level Observed and Predicted Data (cont.)
c1. District Observed Average Rental Value (in dollars)
c2. District Predicted Average Rental Value (in dollars)
Lake Michigan
Lake Michigan
Avg. Rental Value
5,000-6,000
6,000-7,000
7,000-8,000
8,000-9,000
9,000-10,000
10,000-11,000
11,000-12,000
12,000-13,000
13,000-14,000
14,000-15,000
15,000-16,000
16,000-17,000
17,000-18,000
18.000-19,000
19,000-20,000
20,000-22,000
Avg. Rental Value
5,000-6,000
6,000-7,000
7,000-8,000
8,000-9,000
9,000-10,000
10,000-11,000
11,000-12,000
12,000-13,000
13,000-14,000
14,000-15,000
15,000-16,000
16,000-17,000
17,000-18,000
18.000-19,000
19,000-20,000
20,000-22,000
N
W
N
E
4
0
4
8 Miles
W
S
E
4
0
4
8 Miles
S
d2. District Predicted Avg. Spending Per Student - Public Schools
(in dollars)
d1. District Observed Avg. Spending per Student - Public Schools
(in dollars)
Lake Michigan
Lake Michigan
Avg. Spending Public Schools
2,000-3,000
3,000-4,000
4,000-5,000
5,000-6,000
6,000-7,000
7,000-8,000
8,000-9,000
9,000-10,000
10,000-11,000
Avg. Spending Public Schools
2,000-3,000
3,000-4,000
4,000-5,000
5,000-6,000
6,000-7,000
7,000-8,000
8,000-9,000
9,000-10,000
10,000-11,000
N
W
N
E
4
0
4
8 Miles
W
S
E
S
15
4
0
4
8 Miles
16
FIGURE 6 – Chicago: Household Sorting Across Schools
(Benchmark Equilibrium and Universal Vouchers - Endowment in $10,000)
Benchmark Equilibrium
12
Endowment
10
8
Public
6
Catholic
4
Private Non-Catholic
2
0
0.5
0.7
0.9
1.1
1.3
1.5
Taste for Catholic Schools
Voucher Amount = $3000
12
12
10
10
8
8
Endowment
Endowment
Voucher Amount = $1000
6
6
4
4
2
2
0
0
0.5
0.7
0.9
1.1
1.3
1.5
0.5
0.7
Taste for Catholic schools
1.1
1.3
1.5
1.3
1.5
Voucher Amount = $7000
Voucher Amount = $5000
12
12
10
10
8
8
Endowment
Endowment
0.9
Taste for Catholic Schools
6
4
6
4
2
2
0
0
0.5
0.7
0.9
1.1
1.3
0.5
1.5
0.7
0.9
1.1
Taste for Catholic Schools
Taste for Catholic Schools
16
FIGURE 7
Universal Vouchers in Chicago: Neighborhood Fraction of Hhs w/Children in Private Schools
voucher = $ 1,000
voucher = $ 3,000
Lake Michigan
Lake Michigan
fraction Private
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
fraction Private
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
N
W
N
E
4
0
4
8 Miles
W
S
E
4
0
4
8 Miles
S
voucher = $ 7,000
voucher = $ 5,000
Lake Michigan
Lake Michigan
fraction Private
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
fractionPrivate
0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
0.7 - 0.8
0.8 - 0.9
0.9 - 1
N
N
W
E
S
4
0
4
8 Miles
W
E
S
4
0
4
8 Miles
18
FIGURE 8
Universal Vouchers in Chicago: Effects on School Quality
Average school quality for the 10th, 30th, 50th, 70th and 90th percentile endowment
(a)
School Quality (in $10,000)
1.8
1.6
1.4
Benchmark Equilibrium
Voucher = $1000
Voucher = $3000
Voucher = $5000
Voucher = $7000
1.2
1
0.8
0.6
0.4
0.2
0
1.56
3.69
5.19
6.56
9.56
Endowment (in $10,000)
Average gains in school quality for the 10th, 30th, 50th, 70th and 90th percentile
endowment, relative to the Benchmark Equilibrium
(b)
Relative Gain in School Quality
2
1.5
Voucher = $1000
Voucher = $3000
Voucher = $5000
Voucher = $7000
1
0.5
0
1.56
3.69
5.19
6.56
-0.5
Endowment (in $10,000)
18
9.56
Voucher Amount = $1000
Voucher Amount = $3000
Winner
Winner
Loser
12
12
10
10
Endowment
Endowment
FIGURE 9
Universal Vouchers in Chicago: Household Winners and Losers
(Endowment in $10,000)
8
6
4
8
6
4
2
2
0
0
0.5
0.7
0.9
1.1
1.3
0.5
1.5
0.7
Voucher Amount = $7000
Winner
Winner
Loser
10
10
Endowment
12
8
6
4
0
1.1
1.3
1.5
1.3
1.5
Loser
4
0
0.9
1.5
6
2
Taste for Catholic Schools
1.3
8
2
0.7
1.1
Voucher Amount = $5000
12
0.5
0.9
Taste for Catholic Schools
Taste for Catholic schools
Endowment
Loser
0.5
0.7
0.9
1.1
Taste for Catholic Schools
FIGURE 10 – Universal Vouchers in Chicago: Welfare Gains by Household Type
Depicted for households in the 10th, 30th, 50th, 70th, and 90th percentile of wealth (wealth in $10,000)
Voucher Am ount = $ 3,000
0.08
0.06
0.04
1.56
0.02
3.69
5.19
0
6.56
-0.02
9.56
-0.04
-0.06
-0.08
Comp. Variation / Endowment
Comp. Variation / Endowment
Voucher Amount = $ 1,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
-0.04
9.56
-0.06
-0.08
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
Taste for Catholic Schools
Taste for Catholic Schools
Voucher Am ount = $ 7,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
-0.04
9.56
-0.06
-0.08
Comp. Variation / Endowment
Comp. Variation / Endowment
Voucher Am ount = $ 5,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
-0.04
9.56
-0.06
-0.08
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
Taste for Catholic Schools
Taste for Catholic Schools
FIGURE 11 – Chicago: Household Sorting Across Schools
(Benchmark Equilibrium and Non-Catholic Vouchers - Endowment in $10,000)
Benchmark Equilibrium
Endowment
12
10
8
Public
6
Catholic
4
Private Non-Catholic
2
0
0.5
0.7
0.9
1.1
1.3
1.5
Taste for Catholic Schools
Voucher Amount = $3000
12
12
10
10
8
8
Endowment
Endowment
Voucher Amount = $1000
6
6
4
4
2
2
0
0
0.5
0.7
0.9
1.1
1.3
1.5
0.5
0.7
Taste for Catholic schools
Voucher Amount = $5000
1.1
1.3
1.5
1.3
1.5
Voucher Amount = $7000
12
12
10
10
8
8
Endowment
Endowment
0.9
Taste for Catholic Schools
6
6
4
4
2
2
0
0
0.5
0.7
0.9
1.1
Taste for Catholic Schools
1.3
1.5
0.5
0.7
0.9
1.1
Taste for Catholic Schools
Voucher Amount = $1000
Voucher Amount = $3000
Winner
Winner
Loser
12
12
10
10
Endowment
Endowment
FIGURE 12
Chicago: Household Winners and Losers (Non-Catholic Vouchers - Endowment in $10,000)
8
6
4
2
8
6
4
2
0
0
0.5
0.7
0.9
1.1
1.3
1.5
0.5
0.7
Taste for Catholic schools
Voucher Amount = $7000
Winner
Winner
Loser
12
10
10
8
6
4
0
1.1
1.3
1.5
1.3
1.5
Loser
4
0
0.9
1.5
6
2
Taste for Catholic Schools
1.3
8
2
0.7
1.1
Voucher Amount = $5000
12
0.5
0.9
Taste for Catholic Schools
Endowment
Endowment
Loser
0.5
0.7
0.9
1.1
Taste for Catholic Schools
FIGURE 13 – Non-Catholic Vouchers in Chicago: Welfare Gains by Household Type
Depicted for households in the 10th, 30th, 50th, 70th, and 90th percentile of wealth (wealth in $10,000)
Voucher Am ount = $ 3,000
0.08
0.06
0.04
1.56
0.02
3.69
5.19
0
6.56
-0.02
9.56
-0.04
-0.06
-0.08
Comp. Variation / Endowment
Comp. Variation / Endowment
Voucher Amount = $ 1,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
-0.04
9.56
-0.06
-0.08
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
Taste for Catholic Schools
Taste for Catholic Schools
Voucher Amount = $ 7,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
9.56
-0.04
-0.06
-0.08
Comp. Variation / Endowment
Comp. Variation / Endowment
Voucher Amount = $ 5,000
0.08
0.06
0.04
1.56
0.02
3.69
0
5.19
-0.02
6.56
9.56
-0.04
-0.06
-0.08
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
0.67 0.76 0.83 0.85 0.93 1.02 1.02 1.11 1.20 1.39
Taste for Catholic Schools
Taste for Catholic Schools
2