American Economic Association
Selective versus Universal Vouchers: Modeling Median Voter Preferences in Education
Author(s): Zhiqi Chen and Edwin G. West
Source: The American Economic Review, Vol. 90, No. 5 (Dec., 2000), pp. 1520-1534
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/2677864
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Selective Versus UniversalVouchers: Modeling Median Voter
Preferences in Education
By ZHIQI CHEN AND EDWIN G. WEST*
The literatureon educationvouchers appears
to be expanding.' Discussion hithertohas typically been based on the assumption that the
voucher system being analyzed is one in which
all families with children of school age receive
a voucherof a given monetaryvalue. Ourpaper
focuses instead on the availability of a choice
between the usual universal voucher as just
described and a selective voucher wherein recipients are restrictedto those who earnbelow a
given, and somewhatmodest, income level. Our
central purposeis not to present normativediscussion that ultimately recommends "correct"
policy. Instead we offer positive analysis that
attempts to predict majority preferences and
median voter choice between the two types of
vouchers in question.
The selective voucher system is beginning to
receive considerableempiricaltestimony in the
current literature. West (1997) finds recently
established selective pro-poor voucher plans
operating in Colombia, Chile, Guatemala, the
United States, Puerto Rico, and the United
Kingdom. Intellectual support,meanwhile, appears also to be growing. Vouchers to enable
low-income families exclusively to gain increased access to privateschools are advocated,
for instance, by William H. Oakland (1994),
West (1994), and Gary S. Becker (1995). Oakland maintainsthata case can be made for some
redistributiongenerally in the provision of social services, but suggests that it is better accomplished by extending the welfare system to
provide the poor with vouchers for selective
* Department of Economics, Carleton University, Ottawa, Ontario, Canada KIS 5B6. We would like to thank
Keith Acheson, Steven Ferris, Larry Schembri, and two
anonymous referees for their comments and suggestions.
Chen, who was a visitor at the National University of
Singapore (NUS) during the revision of this paper, would
like to thank the NUS for their hospitality and excellent
researchfacilities.
1
See West (1997) and Thomas J. Nechyba (1999).
1520
government services, such as education, than
using lump-sum and matching grants based on
tax-base characteristics.Becker's recommendation is based partly on fiscal considerations,
"butmainly because the bottom quarteror so of
the population are most in need of better education"(1995 p. 11). In addition,Becker quotes
studies demonstratingnot only the superiorperformanceof privateover public schools but also
the finding that "students from disadvantaged
backgroundstend to gain the most from attending private schools." This fact, he observes, is
not surprising "in light of the more extensive
choices available to middle class and rich students" (1995 p. 12).
Despite such argumentsfor selective vouchers, so far there has been no formal analysis on
the subject. Accordingly we attempt below a
theoreticalinvestigation,and one that offers potential explanationof the recent emergence and
growth of selective vouchers in the real world.
Indeed, our analysis shows that under the majority voting rule the selective vouchersystem is
politically more feasible than the universal
voucher system. Under fairly general conditions, families at the bottom half of the income
distribution, including the median voter, will
find the universal voucher system unattractive.
However, these families will generally accept a
switch to the selective voucher system, even
with the cost of education held constant. They
will endorse the switch if the increasedcompetition brought about by the voucher system reduces the cost of education.
We are, of course, not the first to analyze the
probable effects of introducingvouchers starting from a given equilibriumin traditionalpublic school provision. Examples of such studies
include Norman J. Ireland (1990), Charles F.
Manski (1992), Peter Rangazas (1995), and
Dennis Epple and RichardE. Romano (1998a).
All these studies however are confined to universal vouchers exclusively and they take the
value of vouchers as exogenously fixed. In con-
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CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
trast, we focus on the differential effects of
various types of vouchers, and we allow the
value of vouchers to be determined endogenously througha majorityvoting process.2
Section I of our paper analyzes government
provision of educationin terms of a democratic
political model. It assumes (a) that educationis
a single voting issue, and (b) that equilibrium
choice is determined by the majority voting
rule. The per-student cost of education is assumed to be fixed. Section II considers the
effects of switching first, to a system of universal vouchers and second, to a selective "propoor" system. Section III has two parts. In the
firstpartwe dropthe assumptionthatthe cost of
education is fixed and examine the cost implications of vouchers challenging the public
school via new and effective competition.In the
second part we present results from numerical
simulationsof the model. Section IV offers our
main conclusions.
I. The Model
Consider a community in which individuals
have differentincome levels. Income is distributed in the interval [.Y, Y1]with a probability
density function 4( Y). The average income
level in_this community can be expressed as
Y, = fJ YY(Y) dY. Distributionof income is
such that Ya is greaterthan the median income
level, denoted by YIm.People have identical,
strictlyconvex, homothetictastes. Each individual consumes two goods: education and a numeraire good (x). Education consumption can
be measuredby a single index, e, thattakes into
considerationboth the quantity and the quality
of education. The assumption of convex and
homothetic preferences implies that the utility
function of each individual takes the form:
(1)
U(e, x) = U(V(e, x))
2 Two additionalstudies that are complementaryto ours
are Epple and Romano (1998b) and Nechyba (2000).
Nechyba (2000) studies the effects of vouchers targetedon
low-income individuals, but the chief focus is on the consequential migration across districts or neighborhoods
rather than on public-choice issues. Epple and Romano
(1998b) departfrom a uniform-ratevoucher design (e.g., a
voucher value of $2,000, no more and no less, for all
recipients) such that stratificationby income and ability is
reduced.
1521
where V(e, x) is concave and homogenous
of degree one in (e, x). Furthermore, Uv >
0, Uvv < 0. Given these assumptions it can
be verified that U(V(e, x)) exhibits diminishing marginal rate of substitution between e
and x.
The price of the numerairegood x is normalized to 1. The price of a unit of e is denoted by
p. We assume thatp is the same for public and
private schooling.3 In this section and next, we
assume thatp remains constant as the community moves to a voucher system.
Consider first the public-school system currently adopted in most jurisdictions in North
America. Under this system, public schools
are financed through tax revenues and, therefore, are "free" to individuals. At the same
time, an individual has the option of going to
a private school, but he/she has to pay the full
price of the private education out of his/her
after-tax income. This is sometimes described
as "paying twice" for education. For ease of
presentation, we shall call this current publicschool system the PP regime. We shall then
analyze what would happen if the community
moved from PP to each of two modified regimes that involve vouchers. The first is a
universal voucher system (the UV regime)
under which all families and school-age children qualify for vouchers. The second is a
selective voucher system (the SV regime) under which only low-income individuals qualify for vouchers. All others in the school-age
population are entitled to the conventional
public schooling. They can also choose private schooling if they are prepared to "pay
twice." We focus on two issues: First, what
will happen if this education system is reformed through the introduction of vouchers?
Second, will a voucher system defeat the PP
regime under the majority voting rule?
We assume that public education is financed
by a flat tax on income, and that the level of
public expenditureon education is determined
by majority voting. It is now well known that
under the PP regime a majorityvoting equilibrium may not exist (see Epple and Romano,
3In reality there is evidence to the contrary(see Eugenia
Froedge Toma, 1996; West, 1997), but this assumption
serves to simplify our analysis.
1522
THEAMERICANECONOMICREVIEW
1996b). In this paper,however, the PP regime is
the referencepoint to which the UV regime and
SV regime are compared.Therefore,we assume
that, as the status quo, education is provided
under a PP regime that is the outcome of the
majority voting. To be consistent with reality,
we assume thata majorityof, but not all, people
use public education at the status quo, and that
families who use privateeducationhave aboveaverage incomes.
Epple and Romano (1996b) contains a comprehensiveanalysis of the equilibriumunderthe
PP regime.4 Here we reproduce some of their
results about the PP regime that are necessary
for our later analysis of vouchers. Let t be the
tax rate on income and g be the amount of
public education available to each person attending.The utility of a person who uses public
educationcan be writtenas U(V( g, ( 1 -t) Y)).
Then
YVe(g, ( 1 -t)
Y)
representsthe slope of this person's indifference
curve in (g, t) space. The assumptionson V(e,
x) imply that aMhag < 0 and aMhat < 0.
Following Epple and Romano (1996b), we focus on the following two cases:
LEMMA 1:5 Given any pair (g, t), let YP(g,
t) be the income of the individual who is
indifferent between public and private ed<
ucation. Then individuals whose Y YpKg,
t) use public education and individuals
whose Y > YP(g, t) use private education.
Furthermore, aYplag > 0 and aYplat > 0.
Therefore, given g and t, the fraction of
individuals in this community who use the
public-education system is fyP(g,t) 4(Y)dY. (To
simplify notation we will drop the Y under the
integral sign from now on.) Notice that g and
t are related through the government budget
constraint: pg fYP(g,t) 4f(y) dY = tYa. On the
left-hand side of the budget constraint is the
per capita expenditure on public education
and on the right-hand side the per capita tax
revenue. The budget constraint defines a relation between g and t, and we assume that
this relation yields a function denoted by t =
t( g). Then we can define a new function
(g, t(g)) and express the per
Yp(g)
YYp
capita expenditure on public education as a
function of g only: pg fYP(g) +(Y) dY. Obviously, as g increases, the per capita expenditure on public education will increase as well.
Define the increase in the per capita expenditure associated with a marginal increase in g
as "the marginal cost of public education,"
MC(g). Then
(a) aMhaY < 0 (Slope Declining in Income,
or SDI);
(b) aMhaY > 0 (Slope Rising in Income, or
SRI).
d[pg
(3)
It can be shown that among those who prefer
a positive g to g = 0, an individual's most
preferred g decreases with his income if SDI
holds but increases with his income if SRI
holds.
For a given level of public education g and
given tax rate t, some voters choose to go to the
public schools while others attend private
schools. As shown in Epple and Romano
(1996b), this choice depends on the income
level of each individual:
4Gerhard Glomm and B. Ravikumar(1998) study the
same issue, but their results can be considered as special
cases of Epple and Romano (1996b).
DECEMBER2000
MC(g)
=
j
yYp(g)
+(Y) dY]
dg
yYp(g)
=p |
(Y) dY
+ pg4(YP(g))Y, (g).
The marginalcost of public education has two
components. The first component, represented
by the first term on the right-handside of (3), is
the increasein expenditureon the existing users
S The proofs of all the lemmas and propositions in this
paper are presented in the Appendix.
of public education. The second component,
capturedby the second term on the right-hand
side of (3), is the additionalexpenditureon new
users who are attractedto public education because of a higher level of g. We assume that
MC(g) is nondecreasing in g in the relevant
range, and that income distribution is sufficiently uneven that (YmlYa)MC(g) < p in the
PP equilibrium.
Differentiation of the government budget
constraint pg fYP(g) +(Y) dY = tYa yields:
dtldg = MC(g)IYa. In other words, an increase in g by one unit has to be accompanied by an increase in the tax rate by
MC( g)IYa.
Let g* and t* be the level of public education
and tax rate prevailing under the PP regime. It
has been shown in Epple and Romano (1996b)
that a majorityvoting equilibriumalways exists
if SDI is satisfied and will frequently exist if
SRI is satisfied. Furthermore,g* and t* are
positive in the PP equilibrium.Let Y, denote the
income of the median voter who has chosen
these values. Then the identity of the median
voter depends on the sign of aMhaY. The following lemma is from Epple and Romano
(1996b).
LEMMA 2: Assume that a majority voting
equilibrium exists. Then Y, = Ym if SDI
is satisfied. In the case of SRI, Y, satisfies
fY, (Y) dY + fY(g*) eP(Y)dY = 1/2 where
Yp(g*) is the income of the family who is
indifferent between public and private education in the PP equilibrium.
In what follows, we develop a diagrammatic
illustrationof the equilibriumunder the PP regime. The diagrams will also be used in Section III to illustrate the effects of vouchers.
Since g* and t* are the most preferredchoice of
the median voter, they are the solution to the
problem:
(4)
1523
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
VOL.90 NO. 5
max U(V(g,x))
g,t
x
D
K
B
H
>
J
g*
e
F
OF THEMEDIANVOTER
FIGURE1. PREFERENCES
The first-ordercondition to this problem implies:
(6)
Ve(g*, X*)
=
V."(g*,X*) =,
Y
MC(g*).
The left-handside of (6) representsthe marginal
rate of substitutionbetween e and x when e is
equal to g* and is funded throughtax revenue.
The right-handside is the "price"of g to the
median voter. To see this, recall thatthe tax rate
has to be increased by MC(g)IYf in order to
finance a marginalincrease in g. To the median
voter, this increase in tax rate translatesinto an
increased tax burden by the amount of (Yl
Ya)MC(g). Notice that (YI/Ya)MC(g*) < p
because (Ym.Ya)MC(g*) < p by assumption
and Y, ' Y,mby Lemma 2.
The solution to this first-ordercondition is
illustrated in Figure 1. Point B represents the
median voter's consumption bundle under the
PP regime. Ic is the median voter's indifference
curve that goes throughthe point B. The slope
of this indifferencecurve at e = g* is given by
the left-hand side of (6).
If the median voter had chosen to use a private school, he would have to finance his consumption of both e and x out of his after-tax
income. Thus he would face a budget constraint:
Yp(g)
(5)
subjecttopg
f
,(Y) dY= tY,
and x = (1
-
t) Y,
(7)
x + pe-(1-t*)Yc.
This is representedby line HJ in Figure 1. Since
THEAMERICANECONOMICREVIEW
1524
x
t*)Yp(g*), implying that he is indifferent between either consuming g* units of public education or e* units of private education.
Diminishing marginal rate of substitution implies thatthe slope of Ippat g* is greaterthanp,
the slope of Ippat e*.
For an individual with income level Y' >
Yp(g*), a portionof his private-educationbudget line x + pe = (1 - t*) Y' will lie above the
indifference curve that goes through the point
(g*, (1 - t*)Y'). To him, privateeducationis
the better alternative.
(14-*)Y(g*)
(14*)Y+pg*
(14*)
1Y
'PP
g*
e*
OF THEINDIVIDUALS
WITHINCOME
FIGURE2. PREFERENCES
Y (g*) AND YUV
g* is positive in equilibrium,HJ must be ev-
erywherebelow the indifferencecurve Ic. (Otherwise, the median voter would be betteroff by
using privateeducation,in which case he should
have chosen g = 0.) The second line in Figure
1, line DKBF, is parallel to line HJ and goes
throughpoint B. It representsthe equation
(8)
x + pe = (1-t*)Y
DECEMBER2000
+ pg*.
Equation(8) would be the budget constraintof
the median voter if he were free to spend an
income of (1 - t*)Y, + pg*. The vertical
distance between line HJ and line DKBF is the
per capitapublic expenditureon educationpg *.
Since the slope of line DKBF is -p, the length
of the horizontalline HB is exactly equal to the
units of public educationpurchased:g*. Notice
thatat point B the slope of DKBF is steeperthan
that of IC.
Lemma 1 implies that Yp(g*) is the critical
income level that divides individuals who use
public education and individuals who use private education. It is clear that Yp(g*) > Yc.
The choice of the individual with this critical
income level is illustratedin Figure 2. If he uses
public education, he will consume g* units of
educationfor "free"and his utility level is given
by the indifferencecurve Ipp.On the otherhand,
if he uses privateeducation,he will purchasee*
units out of his after-tax income (1 t*)Yp(g*). As shown in Figure 2, the indifference curve that goes through the point (g*,
(1 - t*)Yp(g*)), Ipp,is tangentto the privateeducation budget line x + pe = (1 -
II. Vouchers
In this section we examine the effects of
modifying the PP regime to one that allows
education vouchers. We shall first consider a
universal voucher system (the UV regime) under which every family with school-age children is entitled to vouchers. We shall then
examine a selective voucher system (the SV
regime) under which only low-income families
receive vouchers. Under both regimes those entitled to a voucher have the option to supplement it with additional purchase of education,
bearing in mind that such supplementsmust be
purchasedin the private sector. The value of a
voucher is equal to the per-studentexpenditure
in public schools. Both public schools and
vouchers are financedby the tax on income. We
assume that a decision to switch regimes is
made through a two-stage voting process. At
stage 1, voters decide whether to switch from
the PP regime to a vouchersystem (eitherUV or
SV regime). If the outcome is "yes," they proceed to stage 2 and decide the values of g and t
under the voucher system. In the following
analysis of each voucher system, we shall begin
by studying the possible effects on voters of
different income levels should a switch to the
voucher system be made. Once we know how
different voters would fare under the voucher
system, it is easy to predictthe outcome of stage
1 voting.
A. UV Regime
The UV regime differs from the PP regime in
at least two aspects. First, it gives individuals
more choices in the kinds of schools they want
to attend.They can spend their vouchers in the
VOL.90 NO. 5
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
private schools or public schools. This will increase competitionamong schools and will tend
to improve the efficiency of schools. We shall
leave the implicationsof this aspect to the next
section (Section III). Second, the UV regime is
universal in the sense that every family with
school-age childrenis entitledto a voucher. Our
analysis in this section will focus on this second
aspect.
Before we presentthe results from our formal
model, it is instructive to conduct a heuristic
analysis of the UV regime using Figures 1 and
2. We first suppose that under the UV system
only those who were previouslyusing the public
education will be entitled to vouchers. In other
words, those who were previously using private
schools, or can be regardedas the "incumbent"
private-schoolpopulation,are for now administratively precluded from the voucher option.
(This assumptionwill later be dropped.)Under
this restricted version of the UV regime the
governmentbudget constraintremainsthe same
as under the PP regime, and at the tax rate t*,
the value of the voucherwill be exactly equal to
pg
As will be shown in Proposition 1, the median voter under the UV regime is not necessarily the median voter under the PP regime.
For simplicity, in this diagrammaticanalysis we
restrictour attentionto the case where the median voters under these two regimes have the
same income, as will be seen to result if SDI is
satisfied. In Figure 1 the value of the voucher,
pg*, can be represented by the length HD.
Under the UV regime, the median voter can use
the voucher to obtain the first g* units and pay
p dollars for each additional unit beyond g*.
His budget constraintunder this system is represented by HBF in Figure 1. Recall that BF is
steeperthan Ic at point B. As a result, any point
on the segment BF (not including B) will generate a lower utility level than at point B. The
median voter will not buy any supplementeven
though he is allowed to do so. The assumption
of identical and homothetic preferences then
implies that individualswith lower income levels will not purchaseany supplement,either.
Intuitively,these individualswould not want
to purchase any supplements because their income levels are below the average income Ya
(see Lemma 2). The public education they receive is subsidized by the individuals with
1525
above-average income. As a result, their marginal rate of substitutionat g* is lower than the
true price of education p. Under a voucher
system, the median voter faces the true price of
education at the margin. Hence he has no incentive to increase his consumption of education. In fact, he would want to reduce it if he
could sell a portion of his voucher at price p
(see the BK line in Figure 1).
Let YUVbe the income level that divides
those who buy supplementsand those who do
not. The preceding discussion implies that
YUV> Y, The curve IUV in Figure 2 is an
indifference curve of the individual with income YUV.It is tangent to the budget line x +
pe = (1 - t*)Yuv + pg* at e = g*. This
individual has the highest income level among
those who do not purchase any supplements.
Individualswith a higher income want to consume more thanthe g* units purchasedwith the
voucher and hence purchase supplements. Recall that Ippin Figure 2 is an indifferencecurve
of the individual with income Yp(g*) and that
the slope of Ipp at g* is greater than p. The
assumptionof identical,homotheticpreferences
implies that IUVis below Ipp, which, in turn,
implies that Yuv < YP(g*).
Therefore,given that only the incumbentusers of public education receive vouchers, those
with income level below YUV,including the
median voter under the PP regime, will not
benefit from the switch to the UV regime. Now
we drop the assumption that vouchers are restricted to previous users of public education
and suppose instead that the community
switches to a truly universal voucher system in
which everyone is entitled to a voucher. This
means that a given amountof educationexpenditure has to be sharedequally among all individuals of the community.As a result, eitherthe
per capita expenditureon public education will
drop or the tax rate will increase, or both. These
changes will, in this particularsituation, make
individuals with income level below YUV,including the median voter, worse off. Hence, a
motion to replace the PP regime with the UV
regime will be defeated underthe majorityvoting rule.
These conjecturesare confirmedby our analysis of the formal model. Under the UV regime
where everyone is entitled to a voucher, the
governmentbudget constraintnow changes to:
1526
THEAMERICANECONOMICREVIEW
pg = tYa, which implies t = pglYa. We shall
use s to denote the supplementpurchasedby an
individual.Then an individual's most preferred
g is solved from
(9)
max U V g+sYVg+
Ug ? O'sO
y
ps
Yag
From this optimization problem we can determine the relationship between an individual's
most preferredg and his income level, and thus
the identity of the median voter under the UV
regime. Let YCVdenote the income of the median voter underthe UV regime. Recall that the
income of the medianvoter underthe PP regime
is Y'.
PROPOSITION 1:6 Under the UV regime,
assuming that SDI holds, the medianvoter has
the same income as the median voter under the
PP regime (i.e., Ycv = Yc). If SRI holds,
however, the median voter's income is lower
than that of the median voter under the PP
regime (i.e., Ycv < Yc).
Under the UV regime, therefore, the median voter has an income level equal to Yrn if
SDI is true and an income level less than Ym
if SRI is true. Since Ym is lower than the
average income Ya' the median voter's income is less than Ya as well. As a result, see
Proposition 2.
PROPOSITION2: Under the UV regime, individualswhoseincomelevels are less thanor equal
to that of the median voter (Y ? YCV)will not
supplementtheir vouchers(i.e., s = 0).
Under the PP regime, only a portion of the
populationuses public education.Underthe UV
regime, however, any given aggregate amount
of public expenditure on education has to be
spreadover all the families in the community.It
is then not surprisingthat a switch from the PP
regime to the UV regime will lead to a lower g
and/or a higher t.
6 Epple and Romano (1996a) obtain the same results as
our Propositions 1 and 2. In their model consumers can
purchase supplements to a government-provided good,
which is mathematicallyequivalentto the UV regime here.
DECEMBER2000
PROPOSITION 3:
(i) SupposeMC(g*) ' p. Underthe UV regime,
the level of public educationper family will
be lower than under the PP regime (i.e.,,
guv < g*). The tax rate will be higher (i.e.,
+ydY
tuv >t) if guv/g >IP
(ii) SupposeMC(g*) > p. Underthe UV regime,
the level of public educationper family will
be lower than under the PP regime (i.e.,
guv < g*) ifp > YaM(g*,pg*lYa,Yc).Thetax
rate will be higher(i.e., tuv > t*) if either(1)
SDI holds, or (2) SRI holds and guvlg* >
fYp(g*) +(Y) dY.
Notice that Proposition 3 covers both the
case where MC(g*) is below p and the
case where MC(g*) is above p. Intuitively,
under the UV regime the marginal cost of
public education ( g) becomes p. If the
marginal cost of public education under
the UV regime is higher than that under the
PP regime (i.e., p ' MC(g*)), the median
voter will respond to the switch from the PP
regime to the UV regime by reducing g but
not necessarily raising t. Hence, t will rise
only under certain conditions. On the other
hand, if the marginal cost of public education
under the UV regime is lower than that under
the PP regime (i.e., p < MC(g*)), the median voter will have stronger incentives to
preserve the level of g and will be more likely
to respond to the switch by raising t. In this
case, if the median voters under the two regimes have the same income (i.e., if SDI
holds), t will have to go up. Things are
slightly different in the case where SRI holds
because the median voter under the UV regime has lower income than under the PP
regime and an individual with a lower income
prefers a smaller g even in the absence of a
regime change. As a result, the median voter
may reduce g by a sufficient amount that no
increase in t will be needed. As in the case
MC(g*) ? p, t will rise only under certain
conditions.
4: SupposethateitherMC(g*) <
PROPOSMIION
p or MC(g*) > p > YaM(g*,pg*/Ya,Yc)is true.
Thereexists a critical incomelevel YUVsuch that
individuals whose Y > YUVsupplement their
VOL.90 NO. 5
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
vouchersand individualswhose Y < YUVdo not.
Furthermore,YCV< YUV< Yp(g*),and YUV>
(<) Yaif SDI (SRI) holds.
Propositions 3 and 4 imply that individuals
with Y ' YUVwill most likely lose from the
switch from the PP regime to the UV regime.
Under the UV regime, they will obtain no benefit from the option of buying supplements,
receive less public education, and possibly pay
more tax. Therefore,if asked to choose between
the PP regime and the UV regime, these individuals will likely vote against the switch from
the PP regime to the UV regime. If these individuals constitutea majority,a move to the UV
regime will be defeated.
PROPOSMIION
5: SupposethateitherMC(g*) '
p or MC(g*) > p> YaM(g*,pg*lYa,Yc)is true.
A motion to replace the PP regime by the UV
regimewill be rejectedby a majorityof voters if
either(1) SDI holds, or (2) SRI holds, Yuv> YIm,
and gvlg* > f Yp(g )Y) dY.
Proposition5 confirms our earlier discussion
about the UV regime based on Figures 1 and 2.
If SDI holds, the median voters under the two
regimes have the same income level, which is
an assumptionmaintainedin the diagrammatic
analysis. In this case everyone with income
below Ym will indeed vote against the UV
regime.
It is interestingto note the incentives of individuals with different income levels. The
high-income individuals who use private education under the PP regime (whose Y >
Yp(g*)) no longer have to "pay twice" for
education and they receive a windfall in the
form of vouchers.They are generally supporters
of the UV regime.8 Individuals with income
between YUVand Yp(g*) benefit from the freedom of buying supplements but lose from the
7 In a model of educationwith peer-groupeffects, Epple
and Romano (1998a p. 50) present a numerical example
where a majority of voters reject a universal voucher
system.
8 However, if the tax rate is expected to rise after a
switch to the UV regime, individuals with extremely high
incomes may vote against the switch because their gain in
vouchers is offset by the increase in the amountof tax they
have to pay.
1527
sharing of public expenditure with the highincome individuals. Finally, individuals with
income below YUVare not interestedin taking
advantage of the freedom of buying supplements. In the case where SDI is true, they are
unquestionablyopponentsof the UV regime. If
SRI is true, however, their solidarityagainstthe
UV regime may be weakened somewhat.In this
case, the median voter underthe UV regime has
a lower income than that under the PP regime.
Since lower-income individuals prefer smaller
values of g in this situation,the smaller guv is
closer to their most preferredlevels of public
education than g *. As a result, we may see a
situationwhere individualsat the bottom of the
income spectrum join the individuals at the
other end of the income spectrumin supporting
the UV regime, a situation similar to that of
"endsagainstthe middle"in Epple and Romano
(1996b). Such a situation, however, will not
occur if the conditions in Proposition 5 are
satisfied.
Epple and Romano (1996a) examine voters'
choices between a regime where a good is provided by governments only (the GO regime)
and an alternativeregime where the good supplied by governmentsmay be supplementedby
private-market purchases (the GM regime).
They demonstratethat a majorityof voters prefer the GM regime to the GO regime. Since the
GM regime in Epple and Romano (1996a) is
equivalentto the UV regime in our model, their
conclusion is in sharp contrast to our Proposition 5, which says that a majority of voters
would reject the UV regime that allows private
supplements.The key difference between their
analysis and ours is the different alternativesto
the GM/UV regime. In their model the alternative to the GM regime is the GO regime where
only governmentprovision is allowed, while in
ours it is the PP regime underwhich both public
provision and private purchase coexist. As a
result, in their analysis a move from the GO
regime to the GM regime tends to be Pareto
improving, but in ours a move from the PP
regime to the UV regime makes a majority of
voters worse off.
B. SV Regime
Consider now a selective voucher system
where only low-income individuals are entitled
1528
THEAMERICANECONOMICREVIEW
to receive vouchers.Here low income is defined
as the level below Y, the income level of the
median voter under the PP regime. From the
above analysis, the effects of this SV regime are
easy to ascertain.Because Y, < Yp(g*), users
of privateeducationdo not receive vouchers.As
a result,the governmentbudget constraintis not
affected by a move from the PP regime to the
SV regime. The median voter under the SV
regime has the same income level Y, and
chooses the same g* and t*. Those who receive
vouchers under the SV regime do not want to
buy any supplements(see Figure 1). A switch
from the PP regime to the SV regime, therefore,
does not lead to any change in public and private spending on education. Everyone is indifferent between the two regimes.
PROPOSITION6: Given the price p, the SV
regime is tied with the PP regime under the
majorityvoting rule.
Notice that given the assumptionof constant
price p, everyone is as well off under the SV
regime as underthe PP regime. In other words,
there is no gain from moving from the PP regime to the SV regime. This, of course, will not
be true if the increased competition brought
about by the voucher system brings down the
price of education (or improves quality). We
discuss this issue in next section.
III. The ProductionSide and SimulationResults
So far we have assumed that the supply side
of the educationsystem is fixed. Hence the price
of education,p, has been assumed to be constant before and after the introductionof the
voucher system. In this section we discuss the
effects of a voucher system on the supply side.
We view schools as firms that produce output
(education)using a productiontechnology subject to decreasing returnsto scale. We assume
that the public schools and the private schools
have the same cost structure.
Under the PP regime, the public schools as a
whole have monopoly power over the segment
of the marketin which individuals' demandfor
education is less than or equal to g* units (see
Figure 1). This is because public education is
"free" to everyone in the community. Private
schooling, on the other hand, has to be paid
DECEMBER2000
through after-tax income. As a result, private
schools can only supply the residual marketin
which buyers want more than g* units of education. Since the public schools have a market
shareof more than 50 percent,it is reasonableto
assume that they are the price setters in the
market.The private schools, on the other hand,
are price followers in the sense that they take
the price as given when deciding their output.
We view this situation as similar to the Stackelberg leadership model where the public
schools form a cartel and act as a price leader
while the private schools form a competitive
fringe. It is clear that in this situationthe equilibrium price will be above marginalcost.
One consequence of the voucher system is
the introductionof competition into the market
segment previously monopolized by public
schools. Private schools can now compete with
public schools to supply g* units paid for by the
vouchers. The increased competition will bring
down the price of educationp.9 To the incumbent users of public education,a lower p means
that a given amountof tax revenue can be spent
on a higher level of g, or alternatively,a given
level of g can be supported by a lower t. A
lower price of education,therefore,should benefit the incumbent users of public education.
Since the incumbent users of public education
account for a majorityand a move from the PP
regime to the SV regime does not change the
formula of the government budget constraint
(i.e., it only lowers p), we expect that the SV
regime will defeat the PP regime.
A move from the PP regime to the UV regime
will bring down p. But to the incumbentusers
of public education this effect is partially or
even completely offset by the sharingof public
expenditure with the rich. A move to the SV
regime, on the other hand, does not have this
second negative effect. Therefore,as long as the
9 Recall that e is an index that takes into consideration
both quality and quantityof education.A fall in the price p
can be interpretedboth as a fall in the cost of educationand
as an increase in the quality of education. In reality, it is
more likely that a fall in p will manifest itself in the form of
improvementin quality. Empiricalevidence was published
in the 1990's showing that the introductionof competition
via increased use of private schooling leads to improved
public-school performance.(See Jim F. Couch et al., 1993;
Caroline M. Hoxby, 1994; M. V. Borland and R. M.
Howsen, 1996.)
VOL.90 NO. 5
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
TABLE I-p
SRI
SDI
1 UNDER ALL REGIMES
UV regime
PP regime
t*
Parameters
Cases
=
f
p
g*
(percent)
Y,
Yp
guv
tuv
(percent)
0.02
0.1
0.5
-0.2
3,085
3,529
4.7
6.0
32,774
42,093
101,192
184,188
2,735
3,724
4.8
6.5
TABLE
2-p
Voters who
prefer SV to
UV (percent)
Voters who
prefer SV to
PP (percent)
(p
=
SRI
0.5, ,B = 0.02)
YCV
YUV
17,239
42,093
38,455
62,075
(percent)
69.9
81.1
p = 0.94 under the UV regime
69.8
98.7
SDI
(p = -0.2, ,B = 0.1)
SRI
(p = 0.5, ,B = 0.02)
SDI
(p = -0.2, B - 0.1)
81.4
68.8
72.2
98.7
100
SV regime and the UV regime generate price
reductions of similar magnitudes, we expect
that the SV regime will defeat the UV regime.
Indeed, we can show that the above conjectures are true under a set of technical assumptions. However, instead of going into the
technical details (which are available upon request), we use numericalsimulationsto demonstrate the validity and robustness of these
results.
In the remainderof this section we present
the results of numerical simulations of our
model. These simulations serve two purposes:
(1) to illustratethe conclusions from our general
model, and (2) to illustrate that the conditions
that we have imposed in order to obtain these
conclusions are sufficient but are by no means
necessary.
In the numericalsimulations,we assume that
the income in our hypotheticalcommunity follows the Dagum distribution.10To be more specific, the cumulative distribution function of
income takes the form:
(10)
Voters who prefer
PP (SV) to UV
= 1 UNDER THE PP REGIME, p = 0.95 UNDER THE SV REGIME
p = 0.95 under the UV regime
Voter preferences
1529
F'(Y) = (1 + Ay-a)-b.
10
It has been shown that the Dagum distributionfits the
observed income distributionbetter than the Gamma, the
lognormal,and the Singh-Maddaladistribution(S. Kotz and
N. L. Johnson, 1982).
100
We set A= 25,000, a = 2.5, b = 0.6. These
parametersimply that the median and the average household incomes in this community are
$42,093 and $57,435, respectively.
Preferences are representedby the constant
elasticity of substitution(CES) utility function:
(11)
U(e, x) = [e
-P + (1
-O
l-
/0.
This function satisfies the SDI when p < 0 and
the SRI when p > 0.
Tables 1 and 2 report two sets of simulation
results, one for a case of SDI and the other for
a case of SRI.11 The preference parameters
for the SDI case are (, = 0.1, p =-0.2), and
those for the SRI case are (,3 = 0.02, p = 0.5).
Table 1 presents the equilibria under different
regimes given the assumption that the price of
education p remains constant at 1. In Table
2, we study a situation where the introduction
" The simulations are done using Maple V. The equilibrium under the PP regime is calculated using the same
method as the simulation exercises in Epple and Romano
(1996b). The equilibriumunder the SV regime is obtained
in the same way. The equilibriumunder the UV regime is
computed from the following three equations:(1) the firstordercondition to the median voter's optimizationproblem
(9), (2) the government budget constraint, and (3) in the
case of SRI, a condition that determinesthe income of the
median voter.
1530
DECEMBER2000
THEAMERICANECONOMICREVIEW
of a voucher system reduces the price of
education. Two cases of price reductions are
considered. In the first case, we assume that
the price is reduced by 5 percent under both
the UV regime and the SV regime. In the
second case, we suppose that price reduction
is larger under the UV regime than under the
SV regime: 6 percent under the UV regime
but 5 percent under the SV regime.
The results in Tables 1 and 2 confirm the
main conclusions from our model. Table
1 shows that holding the price of education
constant, a majorityof voters would reject the
UV regime in favor of the PP regime (or equivalently, the SV regime). In the case of SRI,
close to 70 percent of voters would reject the
UV regime;and the numberis 81 percentfor the
SDI case. The first two columns of Table
2 show that for choice between the UV regime
and the SV regime the voting pattern would
remainthe same if both regimes are expected to
produce a 5-percent price reduction. The last
two columns of Table 2 demonstratethat the
conclusion would continue to hold even if the
SV regime and the UV regime generate somewhat different magnitudes of price reductions.
Here we consider a case where the UV regime
generates a slightly larger price reductionthan
the SV regime (6 percent as opposed to 5 percent). We see that about 69 percentof voters in
the case of SRI and 72 percent of voters in the
case of SDI prefer the SV regime to the UV
regime despite the largercost savings associated
with the UV regime. We also see in Table 2 that
the price reduction from the SV regime leads
virtually everyone to prefer the SV regime to
the status quo.
Table 1 also demonstratesthatthe restrictions
we have imposed on the model are sufficientbut
not necessary for our main conclusions. Because the theoretical model we have used is a
fairly general one, a number of restrictionson
the parametershad to be imposed in order to
derive our results. For example, in Propositions
3, 4, and 5 we requirethat either MC(g*) < p
or MC(g*) > p > YaM(g*, pg*/Ya, Y,) hold.
This condition, however, is not met in the SDI
case in Table 1. In fact, p (= 1) is less than
YaM(g*, pg*/Ya, Y,) (= 1.047). As a consequence, we have guv > g* (as opposed to guv
< g* predicted by Proposition 3). Despite the
failure of this condition, it is still true that an
overwhelming majority of voters (81 percent)
prefer the PP regime to the UV regime.
IV. Conclusion
Ourmodel predictsthatthe medianvoter will
reject a proposedmove from the currentsystem
of public plus private schools (PP) to a regime
of vouchers that are utiiversally available to
families with school-age children (UV). The
median voter will be indifferentto any proposal
for a selective voucher (SV) available only to
low-income families, but only if we assume that
the cost of educationcan be expected to remain
fixed. Because, however, the selective voucher
will allow private schools to compete for lowincome studentstraditionallyin public schools,
the public-school monopoly will be weakened
and costs will fall all around(and/orqualitywill
improve all around). The median voter in this
situationwill be in favor of the adoptionof SV.
Finally, since the introductionof the SV regime will be an efficient move that reduces the
price and improves the quality of education,the
question arises why the median voter approved
of the inefficient alternativein the first place.
One answer may be offered in terms of initially
imperfectvoter information.The public monopoly in educationtook time to develop, as did the
growth of the education bureaucracy and its
constant pressurefor consolidation.
APPENDIX
PROOF OF LEMMA 1 AND LEMMA 2:
See Epple and Romano (1996b) Corollary 1,
Proposition 1, and Proposition3.
PROOF OF PROPOSITION 1:
Considereach individual's most preferredg.
The first-orderconditions to the maximization
problem (9) are
(Al)
g=
g9'?0;
(g
(A2)
UVVe- UvVxP(Y/Ya)? 0;
a
g= 0
= UVVe- UvVIvP'< ?;
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
VOL.90 NO. 5
s? 0;
au
as
PROOF OF PROPOSITION 3:
s = 0.
For an individual with Y > Ya P(YIYa) > P
The above two conditions imply that g = 0
and s > 0. Conversely, for an individual with
Y < Ya' (Al) and (A2) imply that g > 0 and
s = 0.
Consider those whose Y < Ya and hence
whose optimal g > 0. Condition (Al) implies
that YaM(g, pglY,
Y)
-
p = 0. Comparative
statics on this condition reveals that
(A3)
ag=_
aY
Ya(aMlaY)
+ p(aMlat)Yla(aMlag)
The denominatoris negative. Then the sign of
ag/aY is negative if aMiaY < 0 (SDI) and
positive if aMiaY > 0 (SRI).
Therefore, if SDI holds, an individual's
most preferred g decreases with his income,
with g = 0 for the individuals with aboveaverage income. The median voter has an
income YCV = Y,, which is the same as
under the PP regime.
On the other hand, if SRI holds, an individual's most preferredg increaseswith his income
for those who prefer g > 0 to g = 0. In this
case, individuals at the two ends of the income
distributionwant a small or zero g but individuals in the middle prefer a large g. The
median voter has an income level defined
by fYcv 4(Y)dY
+ fY
4(Y)dY
1531
=
/2. Re-
call from Lemma 2 that Yc is defined by
f '(Y) dY + f Yp(g*) k(Y) dY = 1/2. Since
all users of private education have aboveaverage income, we have Yp(g*) 2 Ya. It
follows that YCV< Y, < YmPROOF OF PROPOSITION2:
Proposition 1 implies that YCV< Ya. Conditions (Al) and (A2) imply that the median voter will choose s = 0. Thus, a Ulas ?
0 when Y = YCV.It can be shown, by differentiating (A2), that a2UiaYas > 0. Then
for individuals whose Y < YCV, aU/as < 0
when evaluated at s = 0 and g chosen by
the median voter. They will also choose
s = 0.
(i) MC(g*) ' p. -The assumptionof nondecreasing marginal cost implies that
MC(g) ? p for any g < g*. Since aM/
at < 0 and fYp(g) 4(Y) dY < 1, M(g,
(pglYa) f YP(g) 4Y) dY, Yc) > M(g, pg/
Ya Yc). If SRI is true, YC > YCV,in which
case M(g, (pgl ra) f Yp(g) 4(Y) dY, YC)>
M(g, pglYa, YCV).This last inequalityalso
holds in the case of SDI (where Ycv = Yc).
On the otherhand, equation(6) implies that
YaM(g, (pglYa) f Y;/g) tf(Y) dY, YC) =
MC(g) at g = g*, and from the Proof of
Proposition 1 that YaM(g, pg/Ya, Ycv) =
p at g = guv. Since MC(g) ? p, aM!
ag < 0, and aM/at < 0, to satisfy these
two equationsit is necessarythatg* > guv
The government budget constraints under
the two regimes imply that t* = (pg*lYa)
fYp(g*) 'k(Y) dY and tuv
Pguvl/Y.
Hence t* < tuv if fY,(g ) O(Y) dY <
gUV/g*.
(ii) MC(g*) > p.-Given the condition p >
YaM(g*, pg*lYa, Yc), the median voter's
optimizationconditionYaM(g,pglYe,,Ycv) =
p implies that M(guv, pguv/Ya, Ycv) >
M(g*, pg*lYa, YC),which in turn implies
gUV< g* for both the case of SDI (YCV=
Yc) and the case of SRI (Ycv < Yc).
To prove that t* < tuv in the case where
SDI holds, rewrite the median voters' optimization conditions as YaM(g*, t*, YC) =
MC(g*) and YaM(guv, tuv Ycv) = p. Recall that YCV= Yc = Y,M in this case. The
condition MC(g*) > p implies that YaM(g*,
t*, Ym) > YaM(gUV, tUV, Y,,m). Rewriting
this inequality using the definition of M(g, t,
Y), we obtain:
(A4)
Ve(g*, (1
t-)y.
VX
1(g*, 0
>
t*) Y11
)
Ve(guv, (1 -tUV)Ym)
t (g9UV9 ( 1
tuv) Y11t)
which, along with the assumptionof homothetic
preferences, implies that (1 - t*)Yn,/g* >
( - tuv) Ym/guv.Rewritingthis inequalityby
substitutingthe governmentbudget constraints
1532
THEAMERICANECONOMICREVIEW
for g* and guv, we obtain: [(1 - t*)/t*]
fYp(g*) ?(Y) dY > (1 tuv)ltuv. Since
fYp(g*) ?(Y) dY < 1, we know that (1 t*)/t* > (1 - tuv)ltuv, which implies that
t* < tuv.
In the case where SRI holds, the result about
t* < tuv is proved using the governmentbudget constraintsin same way as in the proof of
part (i).
PROOF OF PROPOSITION4:
Given guv and tuv an individual's utility
maximizationproblem is:
(A5) max U(V(guv + s, (1
-
tuv)Y-
ps)).
s2 0
The first-orderconditions are that a U/as ' 0,
s ? 0, and s(aUlas) = 0. Define YUVas the
income level at which a U/as evaluated at s =
0 is equal to 0. (We will prove the existence of
YUVlater.)
First, we establish that under the UV regime
the individual with Y = Yp(g*) will choose
s > 0. To simplify notations in this proof we
will use Yp as a shorthandfor Yp(g*). Recall
that Yp is the income level of the individual
who, in the PP equilibrium, is indifferent between public education and private education.
Hence Ypsatisfies
(A6)
U(V(g*, (1
=
-
t*)Yp))
maxe U(V(e, (1 - t
Yp-pe)).
Let e* be the solution to the maximization
problem on the right-hand side of (A6). It is
obvious from equation (A6) that e* > g*.
Under the UV regime, the individual has an
effective after-tax income of (1 - tuv)Yp +
pguv, which he can spend on educationguv +
s (s ? 0) and the numerairegood x. Substituting tuv = pguvIYa for tuv, we can rewriteit as
Yp-pguv(Yp - Ya)lYa. On the other hand,
the individual's after-tax income under the PP
regime is (1 - t*)Yp, which after substituting
the government budget constraint for t*, can
be written as Y - pg*(Yp/Ya) f ; ?(Y) dY.
Since Yp fV +(Y) dY < fVY YY(Y) dY
< Ya,we haveYp -Ya < fYP
dY.Recall
+f(Y)
fromProposition3 thatguv < g*. Thus,guv Yp-
DECEMBER2000
Ya) < g*Y, fY ?(Y) dY. We can then conclude
> (1 -t)yp(g*);
that (1 - tuv)Yp(g) + pg
the individualhas a higher effective after-taxincome under the UV regime than under the PP
regime. The assumptionof homothetic preferences implies that his consumptionof education
underthe UV regime,guv + s exceeds e* which,
in turn,is greaterthan g*. Since g* > guv, we
have s > 0.
Second, we prove that YUVexists and that all
individualswith Y < YUVchoose s = 0 and all
individualswith Y > YUVchoose s > 0. Since
s > 0 at income level Yp(g*), a U/as > 0 at
s = 0. From the proof of Proposition 2 we
know that at income level YCV,a U/as < 0 at
s = 0. By continuity there exists an income
level YUVsuch that a Ulas = 0 at s = 0. It can
> 0. Thus, for
be verified that (a2U/aYas)
individualswhose Y < YUV,a Ulas < 0 at s =
0, in which case theirutility-maximizingchoice
is s = 0. Conversely, for individuals whose
Y > YUV,aUlas > 0 at s = 0, in which case
their utility-maximizingchoices satisfy s > 0.
Third,we show thatan individualwith Y = Y
will choose s = 0 (and hence YUV> Y?)if SDI
holds and s > 0 (and hence YUV< Y?)if SRI
holds. To this individualthe marginalcost of an
extra unit of g under the UV regime is (Ya Ya)p =
p, which is the same as the marginalcost of s.
Hence,he is indifferentbetweenpubliceducation
(g) and privatesupplements(s). Note that Ya>
YCV.In the case where SDI is true, individuals'
most preferredg's decreasewith income, which
impliesthatthis individual'smost preferredg + s
is less thangul, Thus,he will set s = 0. In the case
where SRI holds, individuals'most preferredg's
increasewith income,which impliesthathis most
preferredg + s is greaterthan guv. He will set
s > 0.
PROOF OF PROPOSITION5:
Consideran individualwho chooses s = 0 in
the UV equilibrium.Rewriting his budget constraint, x = (1 -
t) Y, using the government
budget constraints under the two regimes, we
obtain: x + g(pYlYa) f YP(g) ?(Y) dY = Y
under the PP regime and x + g(pYI/Y) = Y
under the UV regime. We can see that the unit
cost of g under the UV regime, (pYIYa), is
higher than that under the PP regime, (pYIY,a)
fYp(g) O(Y) dY.
Consider the case where SDI is true. The
VOL.90 NO. 5
CHENAND WEST:SELECTIVEVERSUSUNIVERSALVOUCHERS
median voters under the PP regime and the
UV regime have the same income level Ym.
For any given g, the median voter's utility is
given by U(V(g, Y,r - (pgYm/Ya) f
?(Y) dY)) under the PP regime, and is given
by U(V(g, Ym - pgYm/Ya)) under the UV
regime. Since the former is always larger than
the latter for the same g, the median voter is
better off under the PP regime than under the
UV regime. By continuity, those with income
levels slightly above Ym also prefer the PP
regime to the UV regime. As to the voters
whose income is below Ym' their most preferred g's under the PP regime are larger than
g*. In other words, within their PP budget
sets they can be made better off only if g is
larger than g*. Under the UV regime their
utility levels are lower because their budget
sets are smaller (due to higher unit cost of g)
and the equilibrium value of g is further reduced. Therefore, a majority of voters prefer
the PP regime to the UV regime.
Next, consider the case where SRI is true.
Proposition3 states that guv < g* and tuv >
t*. Individuals whose Y ? Yuv are worse off
underthe UV regime than underthe PP regime.
These individuals constitute a majority given
the condition that YUV> Ym.They will defeat
the motion to replace the PP regime with the
UV regime.
PROOF OF PROPOSITION6:
Obvious from the discussion in the text.
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