Draft
The Religious Factor in Private Education
Danny Cohen-Zada and Moshe Justman
Department of Economics
Ben-Gurion University
Beer-Sheva, Israel
June 2002
Corresponding author:
M. Justman, Department of Economics, Ben-Gurion University, Beer Sheva 84105, Israel. email:
[email protected], fax 972-8-6472941, tel 972-8-6472292.
The Religious Factor in Private Education
ABSTRACT
We develop a political economy model of education finance and school choice that
incorporates a religious motive: parents differ in the advantage they attribute to religious
education, as well as in their incomes. We then quantify this religious factor by calibrating
the model to enrolment shares by school type, the parameters of the income distribution, and
the share of income spent on public education. An implication of the analysis is that vouchers
are not likely to induce large numbers of low-income families to opt out of public education
if they cannot be used in religious schools.
Keywords: religious education, public education, school finance, school vouchers
JEL classification: H42, I22, I28
2
1. INTRODUCTION
In the United States, approximately 10 percent of schoolchildren (K-12) attend private
schools, and five of every six private school pupils are enrolled in religious schools.1 Opting
out of public education in the United States does not reduce one’s school-tax liabilities, hence
parents who choose to send their children to private schools must view these schools as
providing a better education. Yet average tuition in private religious schools is considerably
lower than spending per pupil in public schools. 2 This suggests that parents who send their
children to private religious schools implicitly value a dollar paid for tuition at a religious
school as considerably more effective than a tax dollar spent on public education. This
perceived advantage presumably reflects a combination of factors: religious parents’ specific
preference for the religious instruction and environment these schools offer; charitable
subsidization by teachers, religious organizations and external donations; and differences in
efficiency. 3
To investigate the extent of this perceived advantage, we define and calibrate a
political economy model of education finance and school choice that explicitly incorporates a
religious dimension. It posits that households differ in their preferences for religious
schooling as well as in their income levels, which determine the type of school they choose
for their children: public, private-religious or private-nonsectarian. Political decisions on
public school spending per pupil are determined by majority voting that anticipates these
household decisions on school choice, and a political-economy equilibrium is derived in
which these anticipations are correct.4
Calibrating the model to observed empirical values quantifies the strength of the
religious factor that underlies observed patterns of school choice, indicating that over 50% of
households would prefer religious to nonsectarian schooling if both were offered on equal
terms. This implies that when vouchers are restricted to nonsectarian schools they must be
3
very large to have a significant effect on private enrolment in general, and in any event are
not likely to have a significant effect on low-income households.5 However, vouchers similar
in size to those offered in current programs that are not restricted to nonsectarian schools but
may also be applied towards tuition in religious schools, can induce a substantial proportion
of low-income households to switch to private religious schools; and if means tests are
applied that offer these vouchers only to low-income families, spending per pupil in public
schools will also increase, holding the tax rate constant. Clearly, a voucher that increases
public spending per student while holding taxes constant is a Pareto improvement.
Of course, these are not decisive arguments for or against the use of vouchers to
support religious education. The fiscal effects we found are generally small, and there are
other large issues that must be considered. Proponents of voucher programs often emphasize
the competitive pressure they bring to bear on public schools as a key argument in their favor;
opponents fear that vouchers will serve as a substitute for dealing with the failings of public
schools. Peer group effects generated by the movement of pupils between public and private
schools are also a matter of concern, as are fears that vouchers may increase racial and ethnic
segregation. More generally, a wide-ranging voucher program may undermine the important
role of public education in reinforcing the fabric of society by building a core of shared
values, promoting effective communication, and so on. The point we make here is only that
vouchers restricted to be used only in nonsectarian schools are not relevant to meeting the
education needs of children from low-income households unless they are in very large
amounts.6
The structure of the paper is as follows: Section 2 defines the model, derives its basic
properties and characterizes its political-economic equilibrium; Section 3 calibrates it to
United States data; Section 4 applies the calibrated model to school voucher programs; and
Section 5 concludes.
4
2. FORMAL ANALYSIS
2.1 Basic definition of the model
Consider an economy with a continuum of households of measure one, indexed by i, each
comprising one parent and one child. Each household i is characterized by its income yi and
by a religious parameter ki > 0 that reflects the value it attaches to a religious education (ki
can be thought of as the strength of the household’s religious commitment). We let f (y, k)
denote their joint density function, ym denote median household income, and Y denote mean
income. Household utility then depends on consumption of a numeraire good c; on the
quality of education, measured as spending per pupil in the child’s school, x;7 and on the
religious orientation of the household’s school of choice (given the strength of its own
religious sentiment). To fix ideas, we set household utility equal to:
 ci α xi1− α
U (ci , xi , ki) = 
 ci α (ki xi)1− α
if household i chooses a secular school
if household i chooses a religious school
(1)
where 0 < α < 1 is a fixed, common parameter. Thus households with ki > 1 prefer private
religious schooling to secular schooling at similar spending levels, while households with ki <
1 view religious instruction as a drawback, and prefer nonsectarian schools to religious
schools.8 To simplify the exposition we assume that parents with ki = 1, who are indifferent
between secular and religious schooling, choose secular schooling for their children.
Public education is available free of charge to all households at a uniform quality x
funded by a proportional income tax rate t levied on all households and determined by
majority vote.9 Denoting by q the proportion of households that send their children to public
schools, and choosing quality units in such a way that the price of quality is equal to one, the
government’s balanced budget constraint implies that the quality of public schooling is
5
x=tY/q
(2)
Nonsectarian private schooling and religious private schooling are available as alternatives to
public schooling, and can be purchased from a competitively priced private sector in any
desired quality, though doing so does not reduce one’s tax liability,10 and we assume that the
cost of a unit of education quality is the same in all schools. 11
2.2 School choice
A household with income yi that sends its children to public school anticipates having indirect
utility:
Vp ( yi, t, qe) = [(1 – t) yi ]α [ t Y / qe ] 1 − α
(3)
where qe is the level of public enrolment that it anticipates when choosing a school. A
household with income yi that sends its children to a nonsectarian private school (such
households must have ki < 1) solves:
max c, x U(c, x, ki) = cα x 1− α
subject to
c + x = (1 – t) yi
(4)
and anticipates having indirect utility
Vn (yi, t) = α α (1 − α)1−α (1 – t) yi
(5)
And a household with income yi that sends its child to a religious private school (and
6
therefore must have ki > 1) solves:
max c, x U(c, x, ki) = cα (ki x) 1−α
subject to
c + x = (1 – t) yi
(6)
and anticipates having indirect utility12
Vr ( yi, ki, t) = ki1–α αα (1 − α)1−α (1 – t) yi
(7)
As opting out of public education does not reduce a household’s tax obligations it
must be aimed at obtaining a higher quality of education, and as education quality is a normal
good the households that opt out of public schooling will be those with higher incomes.
Comparing utility levels under public and private-secular schooling, given by equations (3)
and (5), we find that for a given tax level t and anticipated public enrolment qe, either all
households with ki < 1 prefer public education, or there exists a threshold income yn (t, qe)
implicitly defined by
(8)
Vp (yn, t, qe ) = Vn (yn, t)
such that those and only those households with ki < 1 and income above yn send their children
to nonsectarian private schools (see figures 1 and 2).13 The private nonsectarian enrolment
rate is given by
1
q n (t , q e ) =
∞
∫ ∫ f ( y , k )dydk
(9)
e
0 y n (t ,q )
7
Similarly, a household with ki > 1 chooses between public and religious schooling, and for a
given tax level t and anticipated public enrolment qe sends its child to religious schooling if
and only if it has income above a threshold income yr(ki; t, qe) implicitly defined by:14
(10)
Vp (yr, t, qe) = Vr ( yr, ki, t)
The private religious enrolment share then equals
∞
q r (t , q ) =
e
∫
∞
∫ f ( y , k )dydk
(11)
1 y r ( k ,t ,q e )
In equilibrium we require that, given the tax rate t, the actual public enrollment rate equals
the anticipated rate, i.e., we seek a value of q = q (t) that solves:
1 y n (t , q )
∞ yr ( k ,t , q )
0
1
q(t ) = 1 − qn (t , q ) − qr (t , q ) = ∫
∫
0
f ( y, k ) dy dk + ∫
∫ f ( y, k )dy dk
(12)
0
Differentiation of (8) and (10) with respect to q reveals that both yn and yr are decreasing in q,
and as the value of the right-hand side of (12) at q = 0 is non-negative and its value at q = 1
is no greater that 1, for each t there exists a unique equilibrium value of public enrolment q(t),
that equates anticipated and actual enrolment rates, which is implicitly defined by (12).
2.3 Voting on the tax level
Under our Cobb-Douglas utility function, all households that anticipate sending their children
8
to public school prefer the same tax rate, characterized by the first-order condition
dVp /dt = (1 – t)α –1 (y / x )α [ –α x + (1 – α) (1 − t ) d x / d t ] = 0
(13)
where x (t) = t Y / q (t) , and q (t) is defined by the solution of (12). We restrict our attention
to cases in which a majority of households choose public education,15 so that (13) determines
the tax rate t, subject to equation (12) identifying the margin between public and private
education; and we assume that this determines t uniquely.
3. CALIBRATION
3.1 Calibration to National United States Data
We calibrate the model to average United States data, to gain an indication of the effects
generated by differently designed voucher programs, which we consider in the following
section.16 We approximate the actual distribution of income in the population by a lognormal
distribution of income, ln y ~ N (µy, σy2), implying a median income of ym = exp(µy) and a
mean income of Y = exp(µy + σy2/2). To solve for µy and σy we set Y equal to $52,513, mean
household income in the United States in 1998; and the median equal to median household
income in that year, $38,885;17 this gives µy = 10.57 and σy 2 = 0.601. We posit that the
religious parameter k also follows a lognormal distribution, ln k ∼ N (µk, σk2), and assume that
it is uncorrelated with the distribution of ln y, so that the joint distribution of ln y and ln k is
bivariate normal with zero correlation, with the joint density of y and k given by
f ( y,k ) =
1
2 πσ
k
σ
e

 ln k − µ
− 0 . 5 ⋅  

σ k

y
9
k
2
 ln y − µ


 + 
σ y


y




2




(14)
The parameters of the distribution of k, µk and σk2, along with α, the parameter of the
utility function are calibrated from data on tax rates and enrolment shares in private-secular
and private-religious education. Incorporating the lognormal specification in equations (9)
and (11) we obtain analytical expressions for the share of households that opted for privatesecular and private-religious education, and set them equal to actual enrolment shares in
1997/1998,
1
qn =
∞
∫ ∫ f ( y , k ) dy
dk = 0.0156
(9a)
0 y n (t , q )
∞
qr =
∞
∫ ∫ f ( y , k ) dy dk
= 0.0836
(11a)
1 y r ( k ,t , q )
Then q = 0.901 percent is the public enrolment share in 1997/8.18 The tax rate on household
income is derived by multiplying public expenditure per pupil in 1997/8, $6,189, by the ratio
of pupils to households, m = 0.507, multiplying it by q and dividing it by average household
income, giving a tax rate of t = 5.38 percent.19 Substituting equation (8) in (9a) and (10) in
(11a), and requiring that equation (12) is satisfied gives three equations in the three
unknowns α, µk and σk . The calibrated values of the household preference parameters that
we obtain from solving these equations are: α = 0.933, µk = 0.148, and σk = 0.232. This
implies that the mean value of k is 1.190, its median value is 1.158, and its standard deviation
is 0.279.
10
3.2 Sensitivity Analysis
We checked the sensitivity of our calibration to the choice of parameter values in two ways:
by individually varying the parameters around United States values, and by calibrating the
model to state-level data. The sensitivity of the calibration results to moderate individual
variation of parameter values can be gauged from Table A1 in Appendix A. The mean value
of k ranged from 1.165 to 1.235 and its median value varied between 1.131 and 1.191:
approximately ±3% around the calibrated values of 1.190 and 1.158. Its standard deviation
varied much more, between 0.194 and 0.361, about ±30% around the calibrated value 0.279.
Calibrating the model to state level data was successful for 37 states. Basic descriptive
data for all states is presented in Table A2, and the calibration results are reported in Table
A3. The calibrated mean value of k has an average value of 1.196 over the 37 states, very
close to the value calibrated from national data, 1.190, ranging between 0.988 and 1.467, and
with a standard deviation of 0.134. It is closely correlated with the share of religious schools
in enrolment, with a correlation coefficient of 0.91.20 The median value of k behaves
similarly, with a mean value of 1.126, compared to the calibrated value derived from national
data of 1.158, ranging between 0.807 and 1.328, and with a standard deviation of 0.131. The
standard deviation of k varies more widely. Its mean value is 0.384, compared to 0.279 from
national data, and it ranges between 0.052 and 1.396 with a standard deviation of 0.272.
Among the thirteen states for which we were not able to calibrate the model there is a
large group of states with very low private enrolment rates (Arkansas, Nevada, North and
South Dakota, Texas, Utah, West Virginia, Wyoming). This may stem from low population
densities, which increase the relative cost of private education, especially for secular schools,
or from a very low level of income, neither of which factors enters in the model. The
remaining uncalibrated states have much higher than average private enrolment rates
(Alabama, Maine, Maryland, Mississippi, Vermont) which may be attributed to historical or
11
geographical factors that have led to the location of a high concentration of private schools in
these states attended by children whose permanent home is in another state (state data on
private enrolment are tallied by the location of the school, not by the home town of the pupil).
Again, this possibility does not enter in the model.
4. SCHOOL VOUCHERS
In this section we apply our results to gauge the effect of differently structured school
voucher programs on enrolment shares and public spending per pupil. We assume throughout
that the tax rate is fixed at 5.38 percent of household income, and that the amount of the
voucher is exogenously determined but financed from the same tax base as school spending,
so that funds allocated to funding the voucher program are deducted from the public
education budget. The voucher program is allowed to vary in three dimensions: the size of the
voucher; whether its use is restricted to non-sectarian schools or “unrestricted”; and whether
it is means-tested, i.e., offered only to households below some income threshold, or offered to
all households.
4.1 Unrestricted vouchers offered to all households for use in all schools
Consider a voucher of exogenously specified magnitude s funded from the same tax base as
expenditure on public schools and available to all households that opt out of public education
for use towards tuition in any private school they choose. We require a balanced budget, so
that tax revenues fully fund both expenditures on public schooling and the cost of the voucher
program, and limit our attention to vouchers that are smaller in size than spending per pupil
in public schools. Denoting by m the number of children per household (all households have
the same number of children and each treats all its children equally) spending per pupil in
public schools is:
12
x (t, q, s) = [t Y – (1 – q) s m ] / (q m)
(15)
and the indirect utility of a household that chooses public education equals
Vps (yi) = [(1 – t) yi ]α x 1−α
(16)
Households that send their children to nonsectarian private schools have indirect utility:21
 1−α
α
s [(1 − t ) y i ]
Vns (yi, ki) = 
α α (1 − α )1−α [(1 − t ) yi + sm] / m1−α

yi < α s m / [(1 − α )(1 − t )]
yi ≥ α s m / [(1 − α )(1 − t )]
(17)
And households that send their children to religious private schools have indirect utility:22

α
1−α
(k i s) [(1 − t ) yi ]
Vrs (yi, ki) = 
α α (1 − α )1−α k i 1−α [(1 − t ) yi + sm ]/ m1−α

yi < α s m / [(1 − α )(1 − t )]
yi ≥ α s m / [(1 − α )(1 − t )]
(18)
Equating (16) and (17) determines a threshold income level yns (t, qe, s) defined by23
Vps (yns) = Vns (yns)
(19)
such that all households with ki < 1 and income below yns send their children to public schools
while those with ki < 1 and income above yns send their children to nonsectarian private
13
schools.24 The private-nonsectarian enrolment share is then
∞
1
∫ f ( y , k )dydk
∫
qn (q ) =
e
(20)
0 y ns ( t , q e , s )
Next, comparing (16) and (18) for a given value of ki > 1, either there exists a threshold
income level yrs (ki, t, qe, s) defined implicitly by
Vps(yrs) = Vrs(yrs , ki)
(21)
(see figure 3a), or if ki > x / s then all households with that value of k choose religious
schooling (figure 3b), in which case we set yrs (ki, t, qe, s) = 0. Then all households with the
given value of ki and income below yrs send their children to public schools while those with
income above yrs send their children to religious schools, and the enrolment share of religious
schools satisfies
∞
qr (q e ) =
∫
∞
∫ f ( y , k ) dy dk
(22)
1 y rs ( k , t , q e , s )
Realized public enrollment must then equal its anticipated value, i.e., we seek q that solves:
q = 1 – qn (q) – qr (q)
(23)
for a given voucher amount s, the given tax rate t = 5.38 percent, and the parameter values
calibrated in the preceding section.
14
Table 1 presents the effect on enrolment and spending of unrestricted vouchers in
increments of $1,000, from $1,000 to $5,000. The impact on private enrolment is substantial,
especially in religious schools, and each has a positive, though small, impact on public
spending per pupil, indicating Pareto improvement over the no-voucher case.25 The fall in
public enrolment below 50% when a voucher of $5,000 is offered points to the possibility of
large unrestricted vouchers ultimately undermining the viability of public education. It
suggests that on purely fiscal grounds, and ignoring the beneficial social externalities of
public education, a majority of households may prefer to replace public education altogether
with a pure voucher program. We consider this issue further in section 5.
To test the sensitivity of these results to our calibration of the parameters values to
national data we also computed the impact of a $3,000 unrestricted voucher taking the
parameter values from the state-level calibrations in Appendix A, for a selection of twelve
states. The results are presented in table A4. Public enrolment falls by 7.2 to 14 percentage
points (compared to a fall of 9.5 based on the national calibration), religious enrolment
increases by 5.4 to 13.2 percentage points (compared to 8.4), and nonsectarian private
enrolment increases by 0.5 to 1.8 percentage points (compared to 1.1).
4.2 Vouchers restricted to non-sectarian schools (without a means test)
The constitutional separation of church and state may preclude the use of tax dollars to
finance vouchers for religious schools. We next consider the effect of explicitly incorporating
such a restriction in the model, assuming for the moment that such vouchers are available to
all, regardless of income. Spending per pupil in public schools, in this case, is given by
x = (t y – qn s m) / (q m)
(24)
15
and indirect utility from public education is
Vp1 (y) = [(1 – t) y]α x 1−α
(25)
Indirect utility from private nonsectarian schooling is
 1−α
α
s [(1 − t ) y i ]
Vn1 (yi) = 
α α (1 − α )1−α [(1 − t ) yi + sm] / m1−α

yi < α s m / [(1 − α )(1 − t )]
yi ≥ α s m / [(1 − α )(1 − t )]
(26)
and indirect utility from religious schooling is
Vr1 (y, k) = αα (1 − α)1−α k 1–α(1 – t) y / m1–α
(27)
As in the preceding section, all households with ki < 1 prefer private non-sectarian schooling
to religious schooling, and hence send their children to nonsectarian private schools if and
only if their income exceeds a threshold level yn1(t, qe, qne, s) defined by Vp1 (yn1) = Vn1 (yn1).26
However, households with k > 1 may now prefer private secular to private religious schooling
because only the former allows them to take advantage of the voucher program. A household
with income y and k > 1 sends its children to private religious school if it prefers it to both
public and private nonsectarian schooling. It prefers religious to public schooling if its
income exceeds the threshold level yrp1(k, t, qe, qne , s ) defined by (the positive root of) Vp1
(yrp1) = Vr1 (yrp1,k); and it prefers religious to private nonsectarian schooling if its income
exceeds the threshold level yrn (k, t, s ) defined by Vn1 (yrn1) = Vr1 (yrn, k).27 Denoting yr1(ki ;
16
t, qe, qne, s) = max { yrp1, yrn}, private religious enrolment is then
∞
q r (q , q ) =
e
e
n
∫
∞
∫ f ( y , k ) dy
dk
(28)
1 y r 1 ( k , t , q e , q ne , s )
A household with income y and k > 1 chooses private-secular education if it prefers it to both
public and religious education (figure 4a), which holds if
yn1(t, qe , qne , s ) < y < yrn(k, t, s )
(29)
Inspection of (27) reveals that yrn is decreasing in k so that for sufficiently high values of k,
we may have yn1 > yrn1 in which case households with such values of k never choose privatesecular schooling (figure 4b); let k1 (t, qe, qne , s ) be the smallest value of k for which yn1 >
yrn. The share of private nonsectarian enrolment is then:
1
q n (q , q ) =
e
e
n
∫
∞
∫ f ( y , k ) dy dk
+
0 y n 1 ( k ,t , q e , q ne , s )
k 1 ( t , q e , q ne , s )
y rn ( k , t , s )
1
y n 1 ( t , q e , q ne , s )
∫
∫ f ( y , k ) dy dk
(30)
The model is then solved by requiring that anticipated enrolment shares in public and privatesecular education accord with household decisions, i.e., we seek q* and qn* such that q* = 1
– qr (q*, qn*) – qn (q*, qn*) and qn* = qn (q*, qn*) where the functions qr (qe, qne) and qn (qe,
qne) are defined by equations (29) and (30).
The results of these calculations for vouchers restricted to non-sectarian schools, in
increments of $1,000 between $1,000 and $5,000, holding the tax rate fixed at t = 5.38
percent, are presented in Table 2. The relative effect on private non-sectarian enrolment is
17
substantial—a $2,000 voucher more than doubles the private non-sectarian share—but
because of its small absolute size, and because some of the increase is drawn from private
religious enrolment, the impact on public enrolment is small. Moreover, the beneficiaries of
such a program are exclusively higher income households, many of which would have opted
for private education without the voucher; consequently spending per pupil in public
education falls slightly. This highlights the importance of restricting vouchers to lowerincome households, as has generally been the practice in experimental programs.
4.3 Means-tested vouchers
Means-tested vouchers are arguably the type of voucher policy most likely to be implemented
in practice. They are targeted at low-income families, which these policies are principally
designed to help, and so are not likely to support households that would have chosen private
education anyhow, thus increasing the likelihood that they will not be a drain on the public
purse (Chen and West, 2000; Bearse et al., 2000). Means-tested vouchers available for use in
any school, religious or secular, can induce extensive enrolment in private religious schools
among low-income households.
Let y denote the maximal income for participating in the voucher program. Then, as
vouchers are unrestricted with regard to type of school, households with ki > 1 that choose
private schooling choose religious schooling, while households with ki < 1 that choose private
schooling choose nonsectarian schooling. Spending per pupil in public education is then a
function of public enrolment q and of the share of households meeting the means test that
choose private education, π:
x ( q, π ) = [ t Y – π s m ] / (q m)
(31)
18
Comparing indirect utility levels across school types within each of the four types of
households (ki < 1, y > y;), (ki < 1, y < y;), (ki >1, y > y) and (ki >1, y < y;) yields four
threshold income levels: ynh (t, s, qe, πe), ynl (t, s, qe, πe), yrh (t, s, qe, πe) and yrl (k, t, s, qe, πe)
such that each household chooses private education (of the type it prefers) if and only if its
income exceeds the relevant threshold.28 Private secular enrollment is then:
qn (q , π ) =
e
e
1
∫
y
∫
f ( y , k ) dy dk
∞
1
+
0 y nl ( t , s , q e ,π e )
∫ f ( y , k ) dy dk
∫
(32)
0 y nh ( t , s , q e ,π e )
private religious enrollment is
qr (q , π ) =
e
e
∞
y
∫
∫
∞
f ( y , k ) dy dk
∞
∫ f ( y , k ) dy dk
∫
+
1 y rl ( k ,t , s , q e ,π e )
(33)
1 y rh ( k ,t , s , q e ,π e )
and the share of households that use a voucher is
π (q , π ) =
e
e
1
∫
∞
y
∫ f ( y , k ) dy dk
+
0 y nl ( t , s , q e ,π e )
∫
y
∫ f ( y , k ) dy dk
(34)
1 y rl ( k ,t , s , q e ,π e )
The model is then solved by requiring that anticipated public enrolment and voucher use
accord with household decisions, i.e., we seek q* and π* such that q* = 1 – qr (q*, π *) – qn
(q*, π *) and π * = π (q*, π *) where the functions qr (qe, π e), qn (qe, π e) and π (qe, π e) are
defined by equations (32) – (34).
The results are presented in Table 3 for vouchers of $3,000 and $4,000, and means
tests between $20,000 and $80,000. Religious enrolment increases substantially, with the size
of the increase depending strongly on the size of the voucher and the stringency of the means
19
test. For the larger voucher, religious enrolment increases by more than half when the means
test is set at $40,000 and more than doubles when it is set at $80,000. Public spending per
pupil increases throughout, indeed increasing more the larger the voucher and the higher the
threshold (within the stated range), though the largest increase is no more than 4 percent. This
increase incidentally causes a slight decline in nonsectarian private enrolment. Clearly, in
terms of the model, an unrestricted means-tested voucher of $4,000 is a Pareto improvement
over no voucher: households that choose to remain in the public school system benefit from
higher spending per pupil without an increase in taxes; households that take advantage of the
voucher clearly gain; and those that do not meet the means test are no worse off than
before.29
Restricting vouchers to nonsectarian schools limits their use by lower-income
households that are not likely to use them even if they are offered in very generous amounts.
Our calculations indicate that even a $6,000 voucher restricted to use only in non-sectarian
schools would not be used by households earning less than $40,000 annually.30
These results bear directly on arguments presented to the Supreme Court of the United
States regarding the use of public funds by the State of Ohio to fund unrestricted school
vouchers for low-income households in Cleveland—a similar program is operating in
Milwaukee—which are said to violate the constitutional principle of separation of church and
state. Defenders of the program have argued that making the funds available to parents, and
not directly to the schools, severs the link between church and state.31 However, the vast
majority of parents who participated in the program chose to send their children to religious
schools, and those arguing against the Cleveland program maintain that these results imply
that the parents had no choice, and hence that the tax funds are in effect directly supporting
organized religion. The numerical analysis presented above shows that the very high share of
religious enrolment by voucher recipients in Cleveland (and Milwaukee) stems from the
structure of demand for private schooling rather than from a lack of local nonsectarian private
20
schools.
5. DISCUSSION
In this section we discuss in more general terms various considerations that bear on the
preceding analysis and conclusions, but do not enter in the formal model.
5.1 Endogenous determination of the sum of the voucher
In calibrating the effect of different voucher programs we assumed that the tax rate and
voucher amounts are exogenously determined and focus on their affect on enrolment.
Relaxing this assumption first to allow endogenous determination of the voucher amount, we
restrict our attention to vouchers that leave a majority of households attending public schools,
and allow the amount of the voucher to be determined endogenously while holding the tax
rate fixed. Then the majority of voters who anticipate choosing public education for their
children all prefer the voucher amount that maximizes spending per pupil in public schools.
Letting t0 denote the fixed tax, this is the amount s that satisfies ∂ x (t0 , s ) / ∂ s = 0 , where
∂ x (t0 , s) / ∂ s is obtained by total differentiation of the relevant equilibrium conditions.32
Applying this observation to the voucher programs considered in the preceding section, it
indicates that of the different configurations presented in Tables 1-3, a voucher of $4,000
available to households with an income below $80,000 for use in any type of private school
would command a popular majority over any other voucher program described in these
tables. Alternatively, if voters are constrained to spend a given amount x0 per pupil in public
schooling in voting on the amount of the voucher, the tax rate and voucher amount are linked
by the equation x (t0 , s ) = x0, which implicitly defines t as a function of s, and households
21
anticipating sending their children to public schools seek to minimize the tax rate subject to
this constraint, which similarly implies ∂ x (t , s ) / ∂ s = 0.
5.2 Endogenous determination of the tax rate
Assume now that the tax rate is determined endogenously before the voucher amount is
chosen by popular vote, and let s*(t) denote the voucher amount so chosen. We retain, for the
moment, our assumption that a majority of households will continue attending public schools
after the voucher program is implemented. Our choice of utility function then implies that all
these households prefer the same tax rate, which satisfies (after applying the envelope
theorem)
t / ( 1 − t ) = [(1−α) / α] [∂ x (t , s * (t )) /∂ t ] / [ x (t , s * (t )) / t]
(35)
Numerical simulations indicate that the partial elasticity [∂ x (t , s ) /∂ t ] / [ x (t , s * (t )) / t] is
small for our calibration, implying little variation in the tax rate—as long as there is a
majority in favor of public schooling. 33
Whether such a majority exists will depend on the distribution of preferences for
religious education in the population, and on possible exogenous limitations, say a lower
bound on public spending or an upper bound on the size of the voucher. Absent such bounds
and where there is substantial demand for religious schools, a coalition of households
favoring religiously education and higher income households may constitute a majority in
favor of discontinuing public schooling altogether. Consider a given tax rate t funding public
education without vouchers, and let q be the fraction of households attending public
education. Households with income above some threshold ys would prefer that public
22
education be discontinued and all tax revenues used to fund an unrestricted voucher of sum
tY, which they could apply towards tuition in schools that offer higher quality than current
public schools;34 and so would households with values of k > 1 / q, who would apply the
voucher towards tuition in religious schools.35 Our calibrated finding of µk = 0.148 implies
that over half the population has k values in excess of 1/q = 1.1, indicating that such a
majority may well exist. This suggests that continuing public support for public education
must rest on other considerations: an appreciation of its external benefits in reducing crime,
raising property values, and promoting communal values; constitutional objections to the use
of public funds to subsidize religious education; wide subscription to the principle of equal
opportunity embodied in public education; and so on.
5.3 The cost of education
We assume in our analysis that the cost of “education quality” absent the religious dimension
is uniform across school types, but there is considerable evidence to the contrary. Empirical
evidence suggests that tuition at parochial schools may be subsidized by as much as 50
percent, through religious donations, institutional support from the church and reduced
salaries paid to teachers in religious orders; and they appear to achieve better results. Such
variation may also be observed in public education: in poorly managed school districts the
imputed cost of quality is much higher than in well-managed districts. While a theoretical
extension along these lines is easily done, relating the parameters of the model to observed
variables is less straightforward. Parents in parochial schools are often expected to
supplement tuition with contributions of money or time that raise the cost of schooling;
dimensions of quality other than academic achievement are always difficult to measure, and
is separating their religious and nonsectarian components; and self-selection introduces
systematic variety in student motivation and parental support that may be difficult to identify.
23
Supply side factors may also affect the cost of schooling as voucher programs change
enrolment patterns. In small school districts, especially, scale effects will lower average costs
where enrolment expands, and raise them where it contracts. Where tuition in parochial
schools is subsidized from church funds or other private sources, it may not be possible to
maintain current subsidy levels if enrolment expands substantially. Moreover, even where
average costs are stable in the long run, it may well be that cost savings from reducing public
enrolment materialize slowly, generating a negative fiscal impact in the short term, though
the long-term effect is positive.
In addition, as the calibration to state data indicated, there may be other cost factors,
such as density of population, that affect the relative costs of private and public education,
and geographic or historical factors may affect the location of private schools that draw their
student populations from near and far.
Finally, the process of school choice may itself require additional resources to be
expended, both by schools and by parents. Experience with open enrolment and charter
schools suggests that if schools become dependent on voucher income they will find a need
to devote substantial resources to marketing efforts (Wilgoren, 2001). At the same time, as
parents face wider choices they need to collect more information to inform their choices,
monitor school performance more closely, and generally deal with a school administartion
that has at least one eye on the bottom line. The heavy responsibility that this involves may
deter all but the most committed and enterprising parents from opting out of the public
system they know.36
5.4 Other extensions of the model
The joint distribution of income, religious preferences and family size. Other variables that
could be incorporated in the model are the correlations between religion, income and the
24
number of children in the family, replacing our assumptions of a zero correlation between
religion and income and a fixed number of children per household. Variation in these
parameters among communities and religious denominations should affect the impact of
different voucher programs, and could be incorporated in the model with little change. More
generally, the model could be extended to allow for more complex income affects, which
may affect demand in circumstances of extreme poverty or affluence.
Institutional factors. Our simple analysis ignores important institutional detail, such as state
and federal sources of external funding, the tax base, or the electoral process through which
education budgets are approved, which vary substantially from one school district to another.
Moreover, our assumption that vouchers are funded by the school district itself, though
experience with past programs suggests that funding is more likely to originate at the state
level, as a means of aiding low-income, ailing school districts. This raises a set of issues
regarding the relation between local and state jurisdictions that have been addressed
elsewhere,37 which could be incorporated in applying the present analysis to a specific
context.
Other important elements. Other elements that weigh heavily in the public debate on
education vouchers in general and on their application to religious schooling in particular are
less readily incorporated in the analysis. Supporters of voucher schemes since Adam Smith
and Thomas Payne have emphasized the advantages of increased competition through school
choice in promoting education efficiency (Friedman, 1962), an effect recently measured by
Hoxby (2002). Peer-group effects are often cited by opponents of voucher programs as
promoting inequality, because they create a more hierarchical separation of students, thus
benefiting the strong but hobbling the weak; such effects have been incorporated in school
choice models by Epple and Romano (1998), among others. Constitutional barriers to the use
of vouchers to support religious schools are grounded in fears of the polarization that may
25
result from increased enrolment in schools that promote different value systems (Gradstein
and Justman, 2001), though Glazer (2001) has recently argued that some parochial schools
are more faithful guardians of traditional American values than some multi-cultural public
schools. Finally, the localized structure of school finance in the United States implies that
school funding and school choice are closely linked to property values, housing choice and
competition between local jurisdictions. These issues that have been recently integrated in
formal models and detailed quantitative analyses by Epple and Sieg (1999) and Nechyba
(2000), could be extended to accommodate the religious dimension of education demand that
we address in the present paper.
6. CONCLUDING REMARKS
There is growing recognition of the need to examine alternative modes of education finance
as a means of improving the quality of education, especially in low-income communities. The
difficulties of experimentation imply that evaluating the impact of proposed reforms on
enrolment and spending must rely on estimates of the underlying parameters of household
utility derived from observations on school choice and public spending, absent reform. By
incorporating a religious dimension in a political economy model of education finance and
school choice, the present paper offers an improved methodology for calibrating these
parameters, as well as a framework for assessing the financial implications of allowing only
non-sectarian private schools to benefit from public support. Our results indicate that voucher
programs restricted to non-sectarian schools are not likely to have a substantial effect on
enrolment, especially of children from lower-income households, while means-tested
vouchers, available for use in either religious or nonsectarian private schools, can have a
substantial impact on private enrolment among low and middle income households while also
modestly raising spending per pupil in public schools without raising taxes.
26
Our focus on the public finance aspect of school vouchers ignores other important
dimensions of such programs such as the competitive pressures they exert on public schools;
the growing role of the states in funding local public school systems and other important
institutional details of the political decision-making process; peer-group effects that result
from the changing composition of school populations; property values, migration and
competition between local jurisdictions; and the impact of increased private schooling on
communal values. These various avenues leave wide scope for further research.
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29
Table 1. Universal unrestricted vouchers
90.1%
Nonsectarian
private
enrolment
1.56%
$6,195
88.1%
1.84%
10.09%
$2,000
$6,195
85.2%
2.20%
12.57%
$3,000
$6,197
80.6%
2.70%
16.74%
$4,000
$6,249
70.0%
3.37%
26.60%
$5,000
$6,221
47.1%
4.63%
48.26%
no voucher
Public
spending per
pupil
$6,189
$1,000
Voucher amount
Public
enrolment
Religious
enrolment
8.36%
Table 2. Universal vouchers restricted to non-sectarian schools
Voucher amount
no voucher
Public
spending per
pupil
$6,189
90.1%
Nonsectarian
private
enrolment
1.56 %
Public
enrolment
Religious
enrolment
8.36 %
$1,000
$6,186
89.74%
2.34%
7.91%
$2,000
$6,176
89.10%
3.59%
7.30%
$3,000
$6,152
87.88%
5.63%
6.49%
$4,000
$6,099
85.41%
9.14%
5.45%
$5,000
$5,986
79.49%
16.33%
4.18%
30
Table 3. Means-tested, unrestricted vouchers
Public
spending per
pupil
Public
enrolment
Nonsectarian
private
enrolment
Religious
enrolment
$20,000
$6,192
90.0%
1.56 %
8.47 %
$40,000
$6,203
89.6%
1.55 %
8.80 %
$60,000
$6,241
88.5%
1.53%
10.00%
$80,000
$6,306
86.3%
1.48 %
12.20 %
$20,000
$6,228
88.3%
1.53 %
10.22 %
$40,000
$6,290
85.5%
1.50 %
13.02 %
$60,000
$6,360
82.5%
1.45 %
16.00 %
$80,000
$6,435
79.3%
1.41 %
19.27 %
Maximum
qualifying income
$3,000 voucher
$4,000 voucher
31
Figure 1. School Choice
without Vouchers
Vr(k,y)
V
Vn(y)
Vp(y)
yn
yr(k)
32
y
Figure 2.
The Distribution of Households Among
School Types (Schematic Depiction)
y
private
nonsectarian
yn
religious
public
1
k
1
33
Figure 3.
Unrestricted Vouchers
V
Vns(y)
Vrs(k,y)
Vps(y)
(a)
αsm
(1–α)(1–t)
yrs(k)
yns
y
Vrs(k,y)
V
Vns(y)
Vps(y)
(b)
yrs(k) = 0
αsm
(1–α)(1–t)
yns
34
y
Figure 4: Vouchers Restricted
to Nonsectarian Schools
V
Vp1(y)
(a)
Vn1(y)
Vr1(k,y)
αsm
(1–α)(1–t)
V
yn1
yrn(k) = yr1(k)
y
yrp1(k)
Vn1(y)
Vr1(k,y)
Vp1(y
(b)
αsm
yrn(k)
(1–α)(1–t)
yn1
yrp1(k) = yr1(k)
35
y
Appendix A. Sensitivity Analyses
Table A1.
Sensitivity of the Calibration to Variation in Individual Parameter Values
Parameter
value
Mean k
Std dev k
Median k
Share of public spending
in income
4.38%
1.187
0.271
1.157
6.38%
1.194
0.287
1.161
0.64
1.165
0.208
1.147
0.76
1.195
0.300
1.159
Private religious
enrolment share
7.4%
1.148
0.206
1.130
9.4%
1.235
0.341
1.191
Private nonsectarian
enrolment share
1.0%
1.184
0.194
1.169
2.0%
1.187
0.362
1.135
Median to mean income
ratio
36
Table A2. Descriptive State Level Data
State
Alabama
Alaska
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
Florida
Georgia
Hawaii
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Maryland
Massachusetts
Michigan
Minnesota
Mississippi
Missouri
Montana
Nebraska
Nevada
New Hampshire
New Jersey
New Mexico
New York
North Carolina
North Dakota
Ohio
Oklahoma
Oregon
Pennsylvania
Rhode Island
South Carolina
South Dakota
Tennessee
Texas
Utah
Vermont
Virginia
Washington
West Virginia
Wisconsin
Wyoming
Mean income
(household)
41,815
54,373
45,673
39,678
58,454
54,357
73,608
57,687
48,840
50,127
57,143
43,199
58,286
47,415
45,564
48,526
41,940
43,254
43,361
59,990
62,960
51,213
54,068
38,704
47,078
37,955
47,897
52,630
56,597
69,543
41,004
63,095
46,507
40,948
48,823
40,451
47,165
51,840
51,646
41,880
42,957
45,226
51,521
48,996
45,303
53,196
53,676
36,579
49,290
44,826
Median to
mean ratio
0.867
0.932
0.812
0.697
0.700
0.857
0.632
0.719
0.715
0.771
0.714
0.849
0.741
0.838
0.812
0.757
0.864
0.734
0.822
0.834
0.673
0.817
0.886
0.752
0.854
0.832
0.760
0.755
0.794
0.716
0.769
0.593
0.771
0.740
0.797
0.834
0.828
0.753
0.788
0.794
0.763
0.754
0.695
0.904
0.869
0.815
0.883
0.730
0.838
0.786
Public education
share in income
5.22%
9.35%
4.65%
5.58%
4.90%
4.58%
5.23%
5.07%
4.43%
5.45%
4.85%
5.96%
4.82%
5.89%
5.98%
5.54%
5.56%
5.83%
6.75%
5.11%
4.99%
6.35%
5.63%
5.61%
5.15%
7.08%
5.72%
4.41%
4.87%
5.86%
6.41%
5.93%
4.85%
5.93%
5.47%
6.02%
5.73%
5.50%
6.26%
5.81%
5.59%
4.64%
5.78%
5.78%
7.17%
4.91%
5.04%
7.28%
6.46%
7.28%
37
Religious
enrolment
6.05%
4.29%
4.19%
4.78%
7.54%
5.41%
8.36%
14.66%
8.78%
4.94%
13.17%
3.45%
12.07%
9.09%
8.93%
7.35%
8.63%
13.26%
3.84%
10.99%
8.51%
9.25%
8.89%
5.56%
10.50%
4.56%
12.05%
3.43%
6.53%
12.18%
4.04%
12.15%
4.82%
5.65%
11.11%
3.88%
6.35%
14.37%
11.80%
5.45%
6.09%
7.03%
4.65%
1.62%
4.04%
5.95%
6.05%
4.23%
13.11%
2.14%
Private secular
enrolment
2.77%
0.23%
1.04%
0.74%
1.97%
1.70%
3.10%
3.11%
1.88%
2.28%
1.75%
0.35%
0.93%
0.55%
0.17%
0.62%
0.93%
2.17%
3.64%
2.53%
3.31%
0.68%
0.69%
4.19%
1.10%
0.33%
0.23%
0.72%
2.96%
1.92%
1.44%
1.89%
1.83%
0.18%
0.88%
0.37%
1.21%
1.53%
2.51%
2.40%
0.34%
1.63%
0.77%
0.93%
5.23%
2.18%
1.16%
0.40%
0.89%
0.46%
Table A3. Calibration of the Model to State-Level Data (selected states; see text)
State
Alaska
Arizona
California
Colorado
Connecticut
Florida
Georgia
Hawaii
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Massachusetts
Michigan
Minnesota
Missouri
Montana
Nebraska
New Hampshire
New Jersey
New Mexico
New York
North Carolina
Ohio
Oklahoma
Oregon
Pennsylvania
Rhode Island
South Carolina
South Dakota
Tennessee
Virginia
Washington
Wisconsin
US
Religious
enrolment
4.29%
4.19%
7.54%
5.41%
8.36%
8.78%
4.94%
13.17%
3.45%
12.07%
9.09%
8.93%
7.35%
8.63%
13.26%
8.51%
9.25%
8.89%
10.50%
4.56%
12.05%
6.53%
12.18%
4.04%
12.15%
4.82%
11.11%
3.88%
6.35%
14.37%
11.80%
5.45%
6.09%
7.03%
5.95%
6.05%
13.11%
8.36%
Mean k
1.095
1.062
1.144
1.035
1.173
1.201
1.056
1.418
1.070
1.347
1.262
1.239
1.140
1.240
1.424
1.159
1.258
1.264
1.287
1.098
1.337
0.988
1.370
1.029
1.413
1.061
1.314
1.065
1.152
1.467
1.300
1.051
1.076
1.137
1.069
1.074
1.388
1.190
Standard
Median k
dev k
0.521
0.988
0.090
1.058
0.251
1.118
0.476
0.941
0.524
1.071
0.332
1.157
0.230
1.032
0.641
1.293
0.061
1.069
0.352
1.303
0.247
1.239
0.143
1.231
0.118
1.134
0.378
1.186
0.875
1.213
0.655
1.010
0.255
1.232
0.383
1.210
0.472
1.208
0.077
1.095
0.212
1.320
0.700
0.807
0.624
1.247
0.052
1.028
0.596
1.302
0.143
1.051
0.337
1.273
0.054
1.064
0.262
1.123
0.689
1.328
1.396
0.886
0.371
0.991
0.052
1.075
0.224
1.116
0.419
0.996
0.534
0.961
0.477
1.313
0.279
1.159
38
α
0.890
0.945
0.940
0.944
0.936
0.945
0.935
0.937
0.932
0.937
0.923
0.924
0.933
0.927
0.924
0.939
0.918
0.925
0.931
0.917
0.925
0.940
0.924
0.926
0.926
0.942
0.928
0.930
0.929
0.926
0.920
0.929
0.934
0.943
0.939
0.938
0.910
0.933
Mean ln k Std dev
ln k (σk2)
(µk)
-0.012
0.452
0.056
0.085
0.111
0.216
-0.061
0.438
0.069
0.427
0.146
0.271
0.032
0.215
0.257
0.431
0.067
0.057
0.265
0.257
0.214
0.194
0.208
0.115
0.126
0.103
0.171
0.298
0.193
0.566
0.010
0.526
0.209
0.201
0.190
0.296
0.189
0.355
0.091
0.070
0.278
0.157
-0.215
0.637
0.220
0.435
0.027
0.050
0.264
0.405
0.050
0.134
0.242
0.253
0.062
0.051
0.116
0.225
0.284
0.446
-0.121
0.876
-0.009
0.343
0.072
0.048
0.110
0.195
-0.005
0.378
-0.040
0.470
0.272
0.334
0.148
0.232
Table A4. Effect of a $3,000 unrestricted voucher on public spending per
pupil and enrolment rates, selected states
Proportionate
change in public
spending
per pupil
Public
enrolment
Nonsectarian
private enrolment
0.1%
-9.5%
1.14%
8.4%
-1.0%
-9.7%
1.52%
8.2%
Florida
0.3%
-13.0%
1.48%
11.6%
Illinois
1.1%
-13.5%
0.63%
12.9%
Indiana
2.0%
-11.7%
0.49%
11.2%
Massachusetts
0.4%
-7.9%
1.33%
6.6%
Michigan
1.4%
-9.3%
0.51%
8.8%
Minnesota
3.6%
-13.7%
0.78%
12.9%
-0.4%
-7.2%
1.82%
5.4%
New York
0.2%
-7.4%
0.57%
6.8%
Ohio
1.8%
-13.8%
0.68%
13.1%
Pennsylvania
2.2%
-14.0%
0.75%
13.2%
Tennessee
0.0%
-13.4%
1.83%
11.7%
United States
California
New Mexico
Percentage points change in:
39
Enrolment in
religious
schools
Appendix B. Threshold levels for unrestricted means-tested vouchers
The threshold levels are determined by setting the indirect utility values offered by the
different school types equal to each other. In the first three cases we solve:
[(1 – t) ŷnh ]α [ (t Y –πe s m ) / (qe m)] 1−α = αα (1 − α)1−α (1 – t) ŷnh / m1−α
[(1 – t) ŷnl ]α [ (t Y – πe s m) / (qe m)] 1−α = αα (1 − α)1−α [(1 – t) ŷnl + s m] / m1−α
[(1 – t) ŷrh ]α [ (t Y – π e s m ) / (qe m)] 1−α = ki1–α αα (1 − α)1−α (1 – t) ŷrh / m1−α
and set ynh = max {ŷnh , y};
ynl = min {ŷnl , y} ; and yrh = max {ŷrh , y}. In the fourth case
we take into account the possibility that a low-income household with k > 1 may take
advantage of the voucher without adding to it. This happens if and only if ki > x / s , in which
case all households with this k and income below the means test use a voucher, and we set ŷrl
= 0. If 1 < ki < x / s then ŷrl is the larger root that solves
[(1 – t) ŷrl]α [(tY – πesm) / (qem) ]1−α = ki1–α αα(1−α)1−α [(1 – t) ŷrl + sm]
and we set yrl = min {ŷrl , y}.
40
Appendix C. Means-tested vouchers restricted to nonsectarian schools
In this case, spending per pupil in public education is a function of public enrolment q and of
the share of households meeting the means test that choose private nonsectarian education π:
x ( q, π) = ( t Y – π s m ) / (q m)
Utility from public education is then
Vp2 = [(1 – t) yi ]α [ (t Y – πe s m ) / (qe m)] 1−α
As households that opt for religious schooling are now not eligible for the voucher, utility
from religious education is
Vr2 = αα (1 − α)1−α k1-α (1 – t) y / m1−α
All households with ki < 1 prefer private nonsectarian schooling to religious schooling, and
again, for these households we have two thresholds between public and private nonsectarian
schooling: one for lower-income households who meet the means test, y < y, and are eligible
for the voucher; and another for higher income households who choose private education
though not eligible for a voucher. The threshold income level between public and private
nonsectarian schooling for households who are eligible for the voucher, ŷnl, is implicitly
defined by
[(1 – t) ŷnl ]α [ (t Y – πe s m) / (qe m)] 1−α = αα (1 − α)1−α [(1 – t) ŷnl + s m] / m1−α
and, as before, let ynl = min {ŷnl , y}. The threshold income, ŷnh, between public and private
nonsectarian schooling for households not eligible for the voucher, is implicitly defined by
[(1 – t) ŷnh ]α [ (t Y – πe s m ) / (qe m)] 1−α = αα (1 − α)1−α (1 – t) ŷnh / m1−α
and denote ynh = max {ŷnh , y}.
The difference between this case and the case of unrestricted means-tested vouchers is
that in this case households with k > 1 and y < y may prefer private secular to private
religious schooling because only the former allows them to take advantage of the voucher
41
program. A household with income y < y and k > 1 sends its children to a private
nonsectarian school if it prefers it to both public and private religious schooling. It prefers
private nonsectarian schooling to public schooling if its income exceeds the threshold level
ynl (t, qe, πe, s) defined above; and it prefers private nonsectarian schooling to religious
schooling if its income is lower than the threshold level yrn (k, t, s ) defined by
αα (1 − α)1−α [(1 – t) yrn + s m] / m1−α = αα (1 − α)1−α k1-α (1 – t) yrn / m1−α .
Private nonsectarian enrollment is then
qn (q , π ) =
e
e
∞ min{ y rn , y }
∫
∫
y nl
1
1
f ( y , k )dydk + ∫
y
∫
0 y nl
1 ∞
f ( y , k )dydk + ∫
∫ f ( y , k )dydk
(C1)
0 y nh
A household prefers religious schooling to public schooling if its income exceeds the
threshold yrp2 defined implicitly by
αα (1 − α)1−α k1–α (1 – t) yrp2 / m1−α =[(1 – t) yrp2 ]α [ (t Y – πe s m ) / (qe m)] 1−α
Private religious enrollment is then
∞
∞
y
∞
∫ f ( y , k )dydk + ∫1 max [y∫ f, (y ]y , k )dydk
1 max [ y rp 2 , y rn ]
rp 2
qr ( q , π ) = ∫
e
e
(C2)
and the share of households that use a voucher is
π (q , π ) =
e
e
1
∞ min{ y rn , y }
y
∫ ∫ f ( y , k )dydk
0 y nl
+
∫ f ( y , k )dydk
∫
1
(C3)
y nl
The model is then solved by requiring that anticipated public enrolment and voucher use
accord with household decisions, i.e., we seek q* and π1* such that q* = 1 – qr (q*, π*) – qn
(q*, π *) and π* = π (q*, π*) where the functions qr (qe, πe), qn (qe, πe) and π (qe, π e) are
defined by equations (C1) – (C3).
42
1
Of the 5,076,119 students enrolled in private schools in the United States 1997/8, 2,514,699
students were enrolled in Catholic parochial schools and 1,764,447 in other religious private
schools, representing 84.2% of total private enrolment (Digest of Educational Statistics,
2000, Table 60). The religious dimension of private education is also supported by
econometric estimates of the demand for private schooling that attribute a prominent role to
religious factors, in the United States and in other countries (Clotfelter, 1976; James, 1987;
Long and Toma, 1988; Hamilton and Macauley, 1991; Buddin et al., 1998).
2
In 1993/4 spending per pupil in public elementary and secondary school was $5,767;
average tuition in Catholic schools was $2,178; and average tuition in other religious schools
was $2,915 (Digest of Education Statistics, 2000, Tables 170, 62).
3
Hoxby (1998) estimated that charitable subsidies from all sources reduce tuition costs by as
much as 50%. (See also Evans and Robert (1995) on the relative efficiency of instruction in
Catholic schools.) This is roughly equal to the ratio of average tuition in religious schools to
public school spending per pupil (note 2, above). However, tuition levels may not fully
reflect private costs if parents are expected to supplement tuition with donations of their own
time or money, as they often are in religious schools; and any meaningful measurement of
tuition costs must control for quality.
4
This theoretical model extends earlier work on the political economy of education finance
and school choice by Rangazas (1995), Epple and Romano (1996) and Glomm and
Ravikumar (1998), to which we have added a religious dimension. Sonstelie (1982)
recognized that the data imply a difference in effectiveness between private and public
schooling, interpreting his empirical findings as indicating that tax dollars spent on public
education are only 37% as effective as private tuition dollars. However, he did not distinguish
43
between religious and non-sectarian schools, and assumed a uniform difference for all
households. These assumptions are not consistent with the large share of religious education
in private schooling or with the wide range of incomes represented in private religious
education.
5
Low-income households will use a voucher to pay for private schooling only if it covers a
large majority of costs. Assuming the “cost of quality” in private non-sectarian schools is no
lower than in public schools, this implies offering a subsidy at least twice the size of vouchers
offered in recent voucher programs in Cleveland or Milwaukee. This is especially significant
because most voucher proposals are aimed at offering a lifeline to low-income families that
are poorly served by public education (Chen and West, 2000) .
6
Direct evidence on the effect of school vouchers is limited. Private voucher programs are
generally small in scale while public experimentation is politically controversial, especially
with regard to religious education: the separation of church and state in the United States
generally precludes the use of public funds to support religious institutions. International
experience is also limited and inconclusive (West, 1997). Hence the need to gauge the effect
of voucher programs by calibrating theoretical models. Calibrations of this nature, without
the religious factor, have been carried out by Rangazas (1995), Epple and Romano (1996),
Martinello and West (1988), Bearse et al. (2000), Nechyba (2000) and others.
7
There are conflicting opinions regarding the extent in which material resources—such as
reduced class size—affect scholastic achievement and classroom behavior (Krueger, 1998;
Card and Krueger, 1996; Hanushek, 1986, 1996; among others). However, for the purpose of
our positive analysis it is parents’ perceptions that matter, i.e., it is sufficient that parents
believe that their children will benefit the more is spent on their education (within school
types).
44
8
That religious households value a dollar spent on religious schooling more than a dollar
spent on public schooling is evident from the substantial levels of enrolment in private
religious schools, despite lower spending per student in these schools (notes 1 and 2, above).
That nonsectarian parents do not generally share this valuation is evident, e.g., from the high
rate of Catholic children in Catholic schools, 87.9% in 1989/90 (National Catholic
Educational Association, 1990). However, some religious households may choose
nonsectarian private schooling if offered vouchers that can only be used in nonsectarian
schools, as we show below.
9
Public schooling in the United States is largely financed by a combination of property taxes
and state grants, with local taxes determined by referenda on proposals set by a school board
(Romer et al., 1992). We ignore these important institutional factors in the analysis, and
implicitly assume that incomes are perfectly correlated with property values.
10
Thus we abstract from the possibility of purchasing private education as a supplement to
public schooling. We also ignore the fixed costs of education, which limit the variety of
private schooling options in smaller communities.
11
Formally, the model attributes the revealed advantage of religious schooling entirely to
demand-side factors. This is readily extended, in the model, to allow the cost of quality to
vary across school types. However, empirically separating the respective effects of supplyside and demand-side factors would require us to measure the cost of school quality in the
different types of schools, which is beyond the scope of the present paper.
12
Vr ( yi, ki)= ki1–α Vn ( yi) implies that households choose private secular schooling over
private religious if and only if ki < 1.
13
From (3) and (5), yn ( t, qe) = t Y / [qe (1 – t) (1 − α) αα/(1−α) ] ; it is possible of course that
there are no households beyond this threshold, i.e., that f(y,k) = 0 for y > yn and k < 1.
45
14
From (3) and (7), yr (ki , t, qe) = t Y / [ki qe (1 – t) (1 − α) αα/(1−α) ] , and again there may be
no households beyond this threshold for some or all values of k > 1..
15
We discuss popular support for public education in Section 5.
16
Operational calibrations for predicting actual policy outcomes can only be undertaken in
the concrete context of specific school districts in which idiosyncratic institutional detail can
be taken into account as well as important peer-group, housing and migration effects
(Nechyba, 2000, is a prominent example). We discuss some such extensions to the model in
Section 5.
17
Per capita money income in that year was $20,120 and there were 2.61 persons per
household (Statistical Abstract of the United States, 2000, Tables 737, 753, 63).
18
The number of children enrolled in public schools in 1997/8 was 46,126,897. The
corresponding number for private schools was 5,076,119, of whom 2,514,699 were enrolled
in Catholic parochial schools and 1,764,447 in other religious private schools, comprising
together 8.357 percent of total enrolment; and 796,973 were enrolled in non-sectarian private
schools, accounting for 1.556 percent of total enrolment (Digest of Educational Statistics,
2000, Tables 41 and 60).
19
The tax rate is given by t = x m q / Y , where spending per student, x , is taken from the
Digest of Educational Statistics (2000, Table 169), and m-q is the ratio of public school
students to households; there were 101,041,000 households in 1998 (Statistical Abstract of
the United States, 2000, Table 63).
20
This is the R-square value of the regression equation
k = 0.86 + 4.08*(religious share)
estimated across the 37 states.
21
If vouchers are unrestricted, only households with ki < 1 choose private nonsectarian
schooling. They maximize U(c, x, ki) = cα x
46
1− α
subject to c + x m = (1 – t) yi + s m
and x > s. As we have assumed that the subsidy is smaller than spending per pupil in public
school and can only be used for private education, the second constraint is never binding:
parents prefer nonsectarian private school to public school only if they intent to spend more
than public spending per pupil. Hence such parents have yi > [α s m + (t Y – s m) / q] /
[(1 − α) (1 – t)] > α s m / [(1 − α) (1 – t)].
22
They maximize U(c, x, ki) = cα (ki x)1−α subject to c + x m = (1 – t) yi + s m and x > s,
and must have ki > 1. A household with ki > x / s > 1 may choose to opt out of public
education without adding to the sum of the voucher.
23
Note that yns (t, qe, s) is the larger root of (19); there is also a smaller root, but it is
irrelevant as it is less than the threshold indicated in note 18.
24
We rule out the possibility that all households with k < 1 choose private schooling, but not
the reverse: there may be no households with ki < 1 and income greater than the threshold yns.
25
In the context of the model, an increase in public spending per pupil holding the tax rate
fixed indicates that the voucher represents a Pareto improvement over the no-voucher case.
This obtains if savings to the public system as a result of the reduced pupil load are greater
than the cost of vouchers paid to pupils who would have attended private schools without the
vouchers. For a voucher of size s, public enrolment after the voucher is implemented can be
no greater than a threshold value q* given by t Y / q0 = [t Y – (1–q*) s m] / q* , where q0 is
public enrolment before the voucher program is implemented. Hoyt and Lee (1998, p. 224)
calculate a related threshold, concluding that vouchers were likely to reduce taxes holding
public spending per pupil fixed. Of course, there may be other benefits or losses, through
peer-group effects, competitive pressures, greater curriculum choice, etc., which our present
framework does not address; and the process of school choice may itself require additional
resources to be expended (Wilgoren, 2001).
47
26
As above, yn1 is the larger root of this equation.
27
This threshold value is yrn = s m / [( k 1–α –1) (1 – t) ] .
28
29
See Appendix B for details of the derivation.
The large majority of households prefer means-tested vouchers to universal vouchers, as
they generate a greater improvement in public school quality.
30
While this result should not be taken literally—there are, of course, other factors that
determine school choice beyond income and religious belief; and nonsectarian schools may
offer reduced tuition to voucher holders, either from a sense of communal service or possibly
because their marginal costs are below their average costs—they are consistent with the very
low level of enrolment in private secular schools under the Cleveland voucher program. See
Appendix C for detailed derivations of the case of restricted means-tested vouchers.
31
The G.I. Bill that provided World War Two veterans with public funds to attend the
college of their choice—including religiously affiliated institutions—is a precedent that is
cited in support of this argument.
32
Letting θ (t0, s) denote the share of households receiving a voucher—the definition of θ (t0,
s) will vary with the type of voucher program—spending per pupil in public schools is
x (t0 , s) = [ t0 Y – θ (t0, s) s m ] / [q (t0, s) m ], which is maximized when – (∂q / ∂s ) x (t0 , s)
= θ (t0, s) + s (∂ θ / ∂ s ), where the derivatives of q and θ with respect to s are obtained by
total differentiation of the relevant equilibrium conditions.
33
Further details are available from the authors on request. Increases in the tax rate are offset
by increases in public enrolment, which dampen the effect of the tax rate on spending per
pupil. The small size of the effect is also indirectly indicated by the small variations in
spending, in Tables 1-3, when the voucher amount is changed: as the voucher has little effect
on spending per student when the tax rate is held fixed, allowing the tax rate to vary should
48
not result in much change in the chosen rate.
34
This value is implicitly defined by [tY + (1 – t) y] αα(1 − α)1−α = [(1 – t) y]α (tY / q)1−α
35
If k > 1 / q then k t Y > t Y / q which implies that household utility from an education
voucher funded by all tax revenues is greater than utility from public schooling without
vouchers, for a given tax rate.
36
Parents of weaker pupils are especially wary of privately managed schools, as was evident
in the recent electoral defeat of an initiative to transfer five failing schools in New York City
to private-sector management (Wyatt and Goodnough, 2001).
37
See, for example, Fernandez and Rogerson (1999) on state and local funding in California.
49