JOURNALOF
PUBLIC
ELSEVIER
ECONOMICS
Journal of Public Economics 62 (1996) 297-325
E n d s against the middle:
D e t e r m i n i n g public service provision w h e n there are
private alternatives
Dennis
E p p l e a, R i c h a r d
E. Romano
b'*
~Carnegie Mellon University, Pittsburgh. PA 15213. l I S A
bDepartment o f Economics. University o f Florida. Gainesville. F L 32611, U S A
Received April 1994; final version received July 1995
Abstract
Public provision of a service coexists with private market provision. The quality of
public provision is determined by majority vote. Preferences are not single peaked
owing to the presence of private alternatives. We identify two cases. In one, majority
voting equilibrium always exists and the median-income vote= is pivotal. In the
other, a necessary condition for equilibrium identifies the pivotal voter who must
have income below the median. When equilibrium exists, a coalition of middleincome househo!ds who consume the public alternative will be opposed by a
coalition of rich and poor households, with the rich choosing private consumptiun.
K e y w o r d s : Voting; Education; Public provision
JEL
classification: D72; H42; H52; 121
1. I n t r o d u c t i o n
M a n y publicly provided goods have privately available counterparts,
including education, health, crime prevention, postal service, sanitation, and
t r a n s p o r t a t i o n . Issues r e l a t e d to s u c h d u a l p r o v i s i o n s y s t e m s a r e i n c r e a s i n g l y
in t h e f o r e f r o n t o f p o l i c y d e b a t e . In e d u c a t i o n , m e c h a n i s m s such as v o u c h e r s
* Corresponding author.
romanor~dale.cba.ufl.edu.
Tel.:
(904)
392-4812: fax:
(904)
0~7-2727/96/$15.00 (~) 1996 Elsevier Science S.A. All rights reserved
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298
D. Epple. R . E . R o m a n o
/ Journal o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
a re p r o p o s e d to stimulate c o m p e t i t i o n b e t w e e n public and private providers.
In health care. m a n y countries provide a specified level of care at public
e x p e n s e , and individuals are r e q u i r e d to pay a p r e m i u m for a d d i t i o , a l
services.
E a c h dual-provision good or service has unique attributes. In health care,
individuals may c o n s u m e simultaneously both publicly and privately provided services. In e d u c a t i o n , by contrast, a given s t u d e n t in a given year
typically c o n s u m e s only the public or only the private alternative. In
t r a n s p o r t a t i o n , users of private a u t o m o b i l e s may benefit from an improved
public bus system that they n e v e r use if the bus system r e d u c e s highway
c ongestion. Despite these differences, dual systems of provision create a
d i l e m m a t h a t is c o m m o n to public service providers unless the public and
p ri vate alternatives are perfect substitutes. O n the one h a n d , the private
a l t e r n a t i v e reduces the d e m a n d on the public system, t h e r e b y reducing its
costs, to the benefit of users of the public system. O n the o t h e r h a n d , the
loss zff clientele to the private sector can be e x p e c t e d to reduce public
s u p p o r t for a high-quality public service, at least a m o n g those who do not
use t he public alternative. This is particularly true if those with the highest
d e m a n d for quality are the first to opt out of the public system.
This d i l e m m a has m a d e dual provision the t e x t b o o k example ( A t k i n s o n
a nd Stiglitz, 1980, p. 303) of n o n - s i n g l e - p e a k e d preferences. Intuitively,
n o n - s i n g l e - p e a k e d n e s s occurs b e c a u s e , at low levels of public service
quality, a h o u s e h o l d that prefers high-quality service may prefer the private
a l t e r n a t i v e . M o d e r a t e increases in quality from z. low base may m a k e the
h o u s e h o l d worse off b e c a u s e taxes rise while the increase in service quality
is not sufficient to induce the h o u s e h o l d to c o n s u m e the public alternative.
L a r g e increases in public service quality, by c o n t r a s t , may m a k e the
h o u s e h o l d b e t t e r off. T h e h o u s e h o l d may be induced to use the public
a l t e r n a t i v e , and the increased tax cost of that alternative may be offset by
t h e savings from forgoing the private service.
T h e implications of n o n - s i n g l e - p e a k e d n e s s are that a voting equilibrium
m a y not exist, a n d , if an equilibrium does exist, the s t a n d a r d a p p r o a c h
(in voc ation of the m e d i a n - v o t e r t h e o r e m ) does not generally apply to
c h a r a c t e r i z e that equilibrium. Since a majority-rule process is typical~ v the
simplest point of d e p a r t u r e for characterizing the political process that
d e t e r m i n e s the level of provision of public services, this has severely
h a m p e r e d m o d e l i n g a n d policy analysis relating to dual provision issues.
Barzel (1973) p i o n e e r e d the analysis of n o n - s i n g l e - p c a k e d p r e f e r e n c e s for
public e d u c a t i o n t h a t arise w h e n t h e r e is a private alternative, using a
n u m e r i c a l e x a m p l e calibrated to actual data. In his example the richest of
s e v e n income s e g m e n t s opts for private e d u c a t i o n and favors zero public
e x p e n d i t u r e , and the m a j o r i t y choice is d e t e r m i n e d by an income s e g m e n t
b e l o w that c o n t a i n i n g the m e d i a n - i n c o m e household. Stiglitz (1974) first
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
299
d e t a i l e d the t h e o r e t i c a l p r o b l e m of n o n - s i n g l e - p e a k e d ~9references in his
c o m p r e h e n s i v e investigation of the d e m a n d for e d u c a t i o n in public a n d
p r i v a t e school systems. I r e l a n d (1990) c h a r a c t e r i z e s m a n y p r o p e r t i e s of
d u a l - p r o v i s i o n s y s t e m s with v o u c h e r s , but he d o e s not a d d r e s s the p r o b l e m
o f e n d o g e n i z i n g the level of public service provi,,ion. G l o m m a n d
R a v i k u m a r (1996) are the first to e n d o g e n i z e public service provision in a
m o d e l with a d u a l - p r o v i s i o n s y s t e m u n d e r m a j o r i t y rule. Th~:y d o so by
a d o p t i n g specific functional f o r m s for p r e f e r e n c e s a n d the di:~tribution of
income.
In this p a p e r we carry on the w o r k of Barzel, a n d G i o m m a n d
R a v i k u m a r . We g e n e r a l i z e their .l l i.~.b .U .l t b " - ;r. t,~:9
,.-respeczs. ,=-~~ ,.=~, , : e sh. ......... .th~,
...
G l o m m a n d R a v i k u m a r ' s choice o f utility function satisfies a single-crossing
a s s u m p t i o n . ~ W e t h e n show t h a t a m a j o r i t y voting e q u i l i b r i u m exists for any
utility function t h a t satisfies this single-crossing c o n d i t i o n w i t h o u t restriction
o n the p a r a m e t r i c f o r m of the utility function a n d w i t h o u t any restriction on
t h e d i s t r i b u t i o n o f i n c o m e . W i t h this single-crossing a s s u m p t i o n , the m e d i a n i n c o m e v o t e r is pivotal. S e c o n d , the a p p r o p r i a t e single-crossing a s s u m p t i o n
d e p e n d s on the p r o p e r t i e s of d e m a n d for the service in q u e s t i o n . F o r s o m e
services (e.g. e d u c a t i o n ) , the o p p o s i t e single-crossing a s s u m p t i o n m a y be
m o r e appropriate.-" Such an a s s u m p t i o n is implicit in B a r z e l ' s e x a m p l e , a n d
o u r findings for this case are c o n s i s t e n t with his. We p r e s e n t a n e c e s s a r y
c o n d i t i o n for a m a j o r i t y - v o t i n g e q u i l i b r i u m for this a l t e r n a t i v e ahlgle-crossing condition_ T h e m e d i a n - i n c o m e v o t e r is pivotal only if, in e q u i l i b r i u m , no
h o u s e h o l d s c h o o s e the private a l t e r n a t i v e . It follows that the m e d i a n - i n c o m e
v o t e r is nt>t pivotal in significant d u a l - p r o v i s i o n cases, a n d we show t h a t the
level of public provision is g e n e r a l l y b e l o w that p r e f e r r e d by the m e d i a n i n c o m e v o t e r , iVioreover, in the result that m o t i v a t e s o u r title, we s h o w t h a t ,
if t h e r e is an e q u i l i b r i u m , a coalition of rich a n d p o o r p r e f e r r e d u c e d public
p r o v i s i o n , while the m i d d l e class p r e f e r an increase.
T h e p a p e r is o r g a n i z e d as follows. Section 2 p r e s e n t s the m o d e l a n d
t h e o r e t i c a l results on m a j o r i t y - v o t i n g are d e v e l o p e d . A c o m p u t a t i o n a l
c o u n t e r p a r t to the t h e o r e t i c a l m o d e l c o m p r i s e s Section 3. U s i n g a c o n s t a n t
elasticity o f s u b s t i t u t i o n ( C E S ) utility specification a n d U . S . d a t a on
e d u c a t i o n a l e x p e n d i t u r e a n d the i n c o m e d i s t r i b u t i o n , we c o m p u t e e q u i l i b r i a
for a r a n g e o f e d u c a t i o n a l d e m a n d elasticities e n c o m p a s s i n g b o t h cases o f
'Such conditions are used extensively in screening models (excellent references include
Cooper. 1984: Matthews and Moore. 1987; and Caillaud et ii . 1988) and in the analysis of
multi-community equilibrium (Ellickson, 1971; Westhoff, 1977: Epple et al.. 1984; Eppte and
Romer. 1991, Goodspeed. 1989; Fernandez and Rogerson. t996). Their power in voting
models was first established by Roberts (I977).
-"Fernandez and Rogerson (1t)96) analyze public education and assume that the direction of a
single crossing is opposite that of Glomm and Ravikumar (1996). The empirical evidence and
related theory is discussed below in Section 2.
3fJO
D. Epple. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
single crossing. O n e purpose is to examine w h e t h e r an equilibrium actually
exists w h e r e it must be of the 'ends against the middle" variety. We find that
the o u t c o m e that satisfies the necessary condition is, in fact, an equilibrium
in all cases. A n o t h e r purpose is to examine the consequences of much
d e b a t e d educational vouchers. L;ne interesting result is that, while private
school enrollmeqt is quite responsive to a voucher system, per student
public expenditure is not.
2. T h e o r e t i c a l m o d e l a n d results
T h e r e are two goods, educational services and the n u m e r a i r e commodity.
We appeal to education in developing the theoretical results because it is an
i m p o r t a n t example, but the analysis applies m o r e generally. All households
are assumed to have the same strictly increasing, strictly quasi-concave, and
twice continuously differentiable utility function U(x, b) over educational
services, x, and the n u m e r a i r e bundle, b. The following additional assumptions are imposed on preferences:
Assumption A1. Educational services are a normal (or superior good).
Assumption A2. For x > O ,
b>O,
~ and / ~ > 0 ,
U(x,b)>U(O,b)
and
U (x, b) :> U (.17, 0).
A s s u m p t i o n A1 is non-controversial and accords with all existing empirical evidence. A s s u m p t i o n A2 is for technical convenience.
T h e following property of indifference curves, a consequence of A s s u m p tion A 1 , will be used frequently. Subscripts on functions denote partial
derivatives.
Diminishing marginal utility (DMU). A l o n g an indifference curve, the
marginal utility of the n u m e r a i r e declines as the n u m e r a i r e increases. T h a t
is, if U(x I , bl) = U(x 2, b2) and b 2 > b 1, then U2(x l, bl) > U2(x2, b,).
Proof. See Epple and R o m a n o (1994).
H o u s e h o l d s differ in e n d o w e d income (i.e. numeraire commodity), y. The
p.d.f, and c.d.f, of household income are d e n o t e d f ( y ) and F(y), respectively, with support [ y _ , ~ ] E [ 0 , ~ ) . We assume that f ( y ) is continuous and
positive over its support. We normalize the n u m b e r of households to one
and d e n o t e aggregate income by Y - ~j~.yf(y)dy, which is also then equal to
m e a n income.
D. Epple. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
30t
E d u c a t i o n a l services are p r o d u c e d from the n u m e r a i r e c o m m o d i t y . O n e
unit of publicly provided educational services is p r o d u c e d with one unit of
the n u m e r a i r e . All c o n s u m e r s of public school services obtain the same level
of e d u c a t i o n services. Public school inputs are financed by a p r o p o r t i o n a l
tax, t, on income. H e n c e , the public school budget constraint is
(1)
tY = NE ,
w h e r e N is the n u m b e r of households using public schools, and E is per
h o u s e h o l d public school services. T h e level of public school e x p e n d i t u r e is
d e t e r m i n e d by majority vote of all households, w h e t h e r or not they utilize
public schools.
Private school services are p r o v i d e d by price-taking suppliers. T h e cost
p e r ur, it of educational services provided by private schools is p units of the
n u m e r a i r e . A h o u s e h o l d c o n s u m i n g private school services can choose as
m a n y units as it desires at price p p e r unit. A h o u s e h o l d can c o n s u m e either
public or private school services, b u t not both. This follows the literature
a n d is a good app,-oximation for education. E p p l e and R o m a n o (1996) a n d
G o u v e i a (1996) analyze the alternative w h e r e public c o n s u m p t i o n can be
s u p p l e m e n t e d by private c o n s u m p t i o n , which m a y be a b e t t e r approxitriation for s o m e publicly p r o v i d e d services like health care.
A h o u s e h o l d that c o n s u m e s private school services chooses x to maximize
U ( x , b ) subject to the b u d g e t constraint y ( 1 - t ) = p x + b. Let
o ( p , y(1 - t)) = m a x U (x, y ( t - 1) - p x )
(2)
be the indirect utility function of a h o u s e h o l d with income y that chooses
private schooling, and let x * ( p , y(1 - t)) be the d e m a n d function for private
e d u c a t i o n a l services that solves the m a x i m i z a t i o n p r o b l e m in (2).
A h o u s e h o l d with income y choosing public schooling obtains utility:
U(E, y(t-
1)).
(3)
H e n c e , the induced utility function of a h o u s e h o l d with income y that can
choose b e t w e e n public and private alternatives is
V ( E , p , y(1 - t)) = m a x [ o ( p , y ( l - t)), U ( E , y(1 - t))].
(4)
T h e following observations are useful in sketching the indifference m a p in
the ( E , t ) plane c o r r e s p o n d i n g to the utility function V ( - ) . Note that
continuity of U ( - ) implies continuity of v ( - ) and V ( - ) . A s s u m p t i o n A 2
implies that for a given tax rate t ~ [0, 1), all households c o n s u m e positive
private school services w h e n the level of public school services, E, equals
zero. Likewise, for a given t and a h o u s e h o l d with a given i n c o m e , there is a
level of public schools services sufficiently large that the h o u s e h o l d will
p r e f e r public schools to private schools. In particular, this is clearly true if
t h e public school offers a level of services E as large or larger than the
302
D. Epple. R . E . R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
h o u s e h o l d w o u l d b u y if it utilized p r i v a t e schooling. Since the utility
f u n c t i o n (3) is c o n t i n u o u s l y i n c r e a s i n g in E, it follows t h a t t h e r e is a u n i q u e
positive v a l u e of E such t h a t , with a given t, t h e h o u s e h o l d is just indifferent
b e t w e e n p u b l i c a n d p r i v a t e schooling.
T h e locus of ( E , t) v a l u e s t h a t m a k e s a h o u s e h o l d y i n d i f f e r e n t b e t w e e n
p u b l i c a n d p r i v a t e school satisfies v(p, y ( 1 - t ) ) = U(E, y ( 1 - t)). D i f f e r e n t i a t i o n of this e x p r e s s i o n yields e q u a l i t y of the first t w o t e r m s below:
dt I
d E ,.(. )=U(-)
U,(E, y(1 - t))
y[Ue(E, y(1 - t)) -[Ue(x* ( . ) , y(1 - t ) - p x * ( . ) ) ] < 0 .
(5)
Since y(1 - t ) > y ( 1 - t) - p x * ( . ), the i n e q u a l i t y in (5) follows from D M U .
W e s u p p r e s s p as an a r g u m e n t a n d write the locus of ( E , t) pairs a l o n g w h i c h
h o u s e h o l d y is i n d i f f e r e n t b e t w e e n public a n d p r i v a t e school as E ( y ( 1 - t ) ) .
E x p r e s s i o n (5) implies t h a t / ~ ( - ) is d o w n w a r d sloping in the (E, t) p l a n e .
T h e s e o b s e r v a t i o n s imply t h a t for a n y h o u s e h o l d , a typical i n d i f f e r e n c e
m a p in t h e ( E , t) p l a n e will be as i l l u s t r a t e d in Fig. 1. L e t us c o n s i d e r first a
typical i n d i f f e r e n c e curve. F o r sufficiently low levels of public school
s e r v i c e s , i.e. for E < / ~ , t h e h o u s e h o l d will o p t to use p r i v a t e schools. F o r
v a l u e s of E such t h a t t h e h o u s e h o l d uses p r i v a t e s c h o o l i n g , the h o u s e h o l d ' s
utility is given by o(p, y ( 1 - t)), w h i c h d e p e n d s on t but n o t E. H e n c e , in
this r a n g e , a h o u s e h o l d ' s i n d i f f e r e n c e c u r v e in the ( E , t) p l a n e is fiat. F o r
sufficiently high E, i.e. for E > / ~ , the h o u s e h o l d will use public schools a n d
A
E
Fig. I
D. Epple. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
3f~
an indifference curve satisfies U(E, y ( l - t)) = constant. Here t increases
necessitate increases in E to maintain indifference. Using strict quasiconcavity of U(x, b), the increasing portion of the indifference curve is a
concave function in the (E, t) plane.Utility is increasing in the southeasterly direction in Fig. 1. S i n c e / ~ ( y ( l t)) is d o w n w a r d sloping, the indifference map has the property, illustrated in
Fig. 1, that the 'corner" shifts d o w n w a r d to the fight as we look across
different indifference curves in o r d e r of ascending utility.
H e n c e f o r t h , for expositional convenience, we set the price per unit of
private schooling, p, equal to I, and we suppress p as an argument in utility
and d e m a n d functions. All results below are valid for any p > 0.
A n analysis of voting requires comparing preference orderings across
individuals. The properties that facilitate such comparisons are developed in
several lemmas. L e m m a 1 shows that the level of public expenditures
necessary to induce a household to choose a public school increases with
income. Specifically, for a given tax rate, the "corners" of the indifference
curves of individuals with differing incomes shift to the right in the (E, t)
plane as income increases.
L e m m a 1. F.(y(1 - t ) ) is increasing in y.
Proof. Differentiate and use (5).
[]
Corollary 1. I f at any (E, ~), household y" weakly prefers private to public
schooling, then so do all households y > y: and if y" weakly prefers public to
private schooling, then so do all households y < y:
Proof. Let V" be the indifference curve through (/~. t ' ) o f a household y" that
prefers private schooling to the public alternative E. The "corner" of this
indifference curve is to the right of point (/~, t'), as illustrated by point A in
Fig. 2. By L e m m a 1, any household with a higher income has an indifference curve through (/~, t') with a corner to the fight of A. This is illustrated
by point B of the indifference curve V" in Fig. 2. where V" is the
indifference curve of some household with y" >y'. Thus. y" also prefers
private to public provision.
A similar argument establishes that if a household with y" prefers public
provision, then so do all households with y < y'. []
This concavity is proved in Westhoff (1977. p. 87). Our model is in the spirit of Westhoff's.
but differs in presuming that public educational services are congested and private alternatives
are available.
304
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
1
1
A
B
,,,n
E
Fig. 2
L e t E * ( t ) be e d u c a t i o n a l e x p e n d i t u r e p e r h o u s e h o l d for t h o s e a t t e n d i n g
p u b l i c s c h o o l w h e n all h o u s e h o l d s m a k e u t i l i t y - m a x i m i z i n g choices. W e will
call this t h e G o v e r n m e n t B u d g e t C o n s t r a i n t ( G B C ) . W e c a n n o w d e v e l o p its
key properties.
F o r ( E , t) such t h a t all h o u s e h o l d s c h o o s e public e d u c a t i o n , E * = t Y f r o m
(1). W e c o n s i d e r ( E , t) such t h a t s o m e h o u s e h o l d s strictly p r e f e r p r i v a t e
s c h o o l i n g . L e t fi be t h e i n c o m e of t h e h o u s e h o l d i n d i f f e r e n t b e t w e e n public
a n d p r i v a t e schooling. T h e n f ( E , t) satisfies
U ( E , )~(1 -- t ) ) = v(fi(1 -- t ) ) .
(6)
C o r o l l a r y 1 implies:
N ( E , t) = F ( f i ( E , t)) .
(7)
U s i n g (1), E * ( t ) is t h e n defined implicitly in
tY
E * = N ( E * , t) "
(8)
N o w , f r o m (6) a n d (7) a n d using the e n v e l o p e a n d implicit f u n c t i o n
t h e o r e m s , we can find t h e tax elasticity of the n u m b e r of public school users:
ON t
¢N. , ---
Ot
N-
tPf(i)
(l -
t)N"
D. Epple. R . E . R o m a n o ! Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
305
Likewise, we can find the e x p e n d i t u r e elasticity of the n u m b e r of public
school users:
ON E
oe. N -
([~Ef(f~)
(1 - O N ( U * ,
-
&)"
w h e r e a tilde and an asterisk d e n o t e the c o n s u m p t i o n b u n d l e s for y = );
associated with public a n d private schooling, respectively. Both elasticities
a re positive, eN. ,, obviously, a n d ~N. e b e c a u s e D M U implies U z - U > O.
B o t h elasticities are of u n r e s t r i c t e d m a g n i t u d e since f 0 ~) is u n r e s t r i c t e d in
m a g n i t u d e . Finally, we can find the elasticity of the G B C from (8):
dE* t
m
dt E*
_
1 - ~N.,
1 +ex.E
.
(9)
L e m m a 2. E (t) is continuous f o r all t ~ (0. 1 ) and differentiable over this
range except at the p o i n t where the highest income household is indifferent
between the public and private alternatives.
Proof. If all h o u s e h o l d s prefer public to private provision at point (E*(t), t),
t h e n t he claims follow trivially from (1). F o r ( E * ( t ) , t ) , having some
h o u s e h o l d s choose t h e private alternative, E*(t) is defined by (6)-(8) with
t h e derivative described in (9). T h e positivity of the d e n o m i n a t o r on t h e
r i g h t - h a n d side of (9) a n d the implicit function t h e o r e m imply that E*(t) is
c o n t i n u o u s a n d differentiable o v e r this range. F o r t such t h a t .~ = 7 ,
c o n t i n u i t y follows from F 0 7 ) = 1. It is s t r a i g h t f o r w a r d to check t h a t E*(t) is
not, h o w e v e r , differentiable at this point. []
T h e o n e non-differentiability in E*(t) for t > 0 will be i n c o n s e q u e n t i a l t e
o u r analysis. If the m i n i m u m income in the p o p u l a t i o n is positive, t h e n E*(t)
is also d i s c o n t i n u o u s at t = 0, j u m p i n g from zero to a positive value.
L i k e w i s e , this is not of c o n s e q u e n c e , and o u r figures below h:::,e E*(t) go
t h r o u g h the origin w h e n y = 0.
P e r h a p s surprisingly, E*(t) n e e d not be e v e r y w h e r e increasing, by (9).
O n e effect of a higher tax rate is to increase a g g r e g a t e public e x p e n d i t u r e ,
a t t r a c t i n g m o r e s t u d e n t s into the public sector so long as p e r s t u d e n t
e x p e n d i t u r e s rises. This effect is c a p t u r e d by the d e n o m i n a t o r in (9) a n d
c a n n o t cause d E / d t < O . H o w e v e r , the relative p r e f e r e n c e for public over
private e d u c a t i o n at any E rises with the tax rate b e c a u s e the marginal utility
of disposable income rises. T h e n u m e r a t o r of (9) c a p t u r e s this effect, which
can be sufficiently s t r o n g in s o m e ranges of t for some income distributions
to c ause dE~dr <~O. We must t h e n c o n t e n d with a poorly b e h a v e d G B C , as
illustrated, for e x a m p l e , in Fig. 5 below.
T h e restriction on p r e f e r e n c e s that permits o u r d e v e l o p m e n t of the
306
D. Epple. R.E. Romano / Journal of Public Economics 62 (1996) 297-325
properties of voting equilibrium
indifference curve of U ( E , y ( 1 M ( E . y , t). H e n c e ,
U~(E, y ( 1
-
is n o w d e s c r i b e d . L e t t h e slope of an
t)) in t h e ( E , t ) p l a n e be d e n o t e d by
t))
M ( E , y . t) = y U z ( E " y(1 - t)) "
(10)
It will b e a s s u m e d f o r all y t h a t t h e slope o f t h e U ( E , y(1 - t)) f u n c t i o n in
t h e ( E . t) p l a n e is m o n o t o n e in y 4 In p a r t i c u l a r , we a s s u m e t h a t o n e of t h e
following alternatives holds:
A s s u m p t i o n A 3 ( S D I ) . O M ( E , y , t)/Oy ~ 0 for all y.
A s s u m p t i o n A 4 ( S R I ) . O M ( E , y , t)/Oy >1 0 for all y.
F o r e a s e o f r e f e r e n c e , we a d o p t t h e m n e m o n i c s S D I (slope d e c i,ining in
i n c o m e ) a n d S R I ( s l o p e rising in i n c o m e ) to r e f e r to t h e s e a s s u m p t i o n s .
I n c o m e c h a n g e s affect t h e m a r g i n a l willingness to b e a r a p r o p o r t i o n a l tax
rise f o r i n c r e a s e d p u b l i c e d u c a t i o n (i.e. M ( - ) ) t h r o u g h an i n c o m e a n d
s u b s t i t u t i o n effect. A s s u m p t i o n A1 e n s u r e s a positive i n c o m e effect. U n d e r
p r o p o r t i o n a l t a x a t i o n , c o u n t e r i n g this is t h e effective price i n c r e a s e o f p u b l i c
e d u c a t i o n a s s o c i a t e d with rising i n c o m e . F o l l o w i n g Kenny*s (1978) analysis,
S R I r e s u l t s if t h e i n c o m e elasticity o f t h e i m p l i e d d e m a n d f o r publ-~c
e d u c a t i o n e x c e e d s t h e ( a b s o l u t e v a l u e o f t h e ) price elasticity of t h e s a m e ,
a n d S D I r e s u l t s if t h e r e v e r s e h o l d s ~ M o s t o f t h e e m p i r i c a l analysis o f t h e
d e m a n d f o r e d u c a t i o n finds i n c o m e elasticities t h a t e x c e e d (tax) p r i c e
elasticities (e.g. see D e n z a u a n d G r i e r , 1984, a n d F i s c h e r , 1988), b u t
e s t i m a t i o n is c o n t r o v e r s i a l o w i n g to t h e possibilities o f T i e b o u t b i a s e s a n d
h o u s e h o l d p r o d u c t i o n o f e d u c a t i o n (see R u b i n f e l d a n d S h a p i r o , 1989). It is
p r u d e n t t o e x a m i n e b o t h possibilities t h e o r e t i c a l l y .
T h e f o l l o w i n g l e m m a e s t a b l i s h e s t h a t if t h e utility f u n c t i o n U ( - ) satisfies
t h e s i n g l e - c r o s s i n g - c o n d i t i o n S D I , t h e n t h e i n d i f f e r e n c e c u r v e s of t h e utility
f u n c t i o n d e f i n e d in E q . (4) also satisfy t h e s a m e s i n g l e - c r o s s i n g c o n d i t i o n .
L e m m a 3. I f S D I holds, then any indifference curve o f the utility f u n c t i o n
V ( E , y ' ( l - t)) crosses any indifference curve o f V ( E , y " ( 1 - t)) at m o s t
The exploitation of "single crossing" restrictions on preferences to analyze voting problems is
on the increase. Gans and Smart (1996) provide a very general analysis and interesting
applications, including a synthesis of earlier research.
-~An adaptation of Kenny's analysis to our model is in Epple and Romano (1994).
Alternatively. see Goodspeed (1986).
D. Epple. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
once,
307
I f a crossing occurs, and y " > y , then the indifference curve o f y"
crosses the indifference curve of y' from above.
Proof. T o establish the first claim, let us suppose the contrary. We consider
h o u s e h o l d s with incomes y" and y". There are two cases that must be ruled
out. O n e of these is illustrated by the solid curves in Fig. 3. Here an
indifference curve V" of household y" crosses an indifference curve V" of
h o u s e h o l d y" twice, once along the flat part and once along the upwardsloping part. The latter crossing and SDI imply y" > y ' . T o establish the
contradiction, w e draw the indifference curve of y' that has its fiat part
coinciding with the flat part of V" This is illustrated by the dasi~e.d curve V"
in Fig. 3. From Eq. (5), the c o m e r of 12" (point A ) is d o w n w a r d and to the
right of the c o m e r of V" (point B ) , and hence to the right of the corner of I/"
(point C). Since y " > y ' , this contradicts L e m m a 1.
V °
/
~
vx
..~,
#.
O°
C
A
Fig. 3
OT h e pc~oibility t h a t i n d i f f e r e n c e c u r v e s c o i n c i d e o v e r r a n g e s (e.g. a l o n g t h e h o r i z o n t a l
s e g m e n t s ) o r a r e t a n g e n t m a y c r e a t e c o n f u s i o n a b o u t w h a t w e m e a n by "a c r o s s i n g ' . A n y
*touching" o f i n d i f f e r e n c e c u r v e s t h a t v i o l a t e s e i t h e r S~¢,I o r S D I m u s t e n t a i l t w o c r o s s i n g .
H e n c e . if i n d i f f e r e n c e c u r v e V m e e t s i n d i f f e r e n c e c u r v e V'. say f r o m b e l o w , c o i n c i d e s w i t h it
o v e r a r a n g e , a n d t h e n V rises a b o v e (falls b a c k b e l t ,~) V'. t h e n this c o u n t s as o n e ( t w o )
c r o s s i n g ( s ) . A t a n g e n c y o f i n d i f f e r e n c e c u r v e s w o u l d c t , r r e s p o n d to t w o crossings. T h e s i m p l e
h u e r i s t i c to d e t e r m i n e t h e n u m b e r o f c r o s s i n g s in any u a c l e a r c a s e is this: shift m a r g i n a l l y o n e
i n d i f f e r e n c e c u r v e in a w a y to c r e a t e t h e m a x i m u m n u m b e r o f crossings.
308
D. E p p l e . R . E . R o m a n |
/ J o u r n a l o f Public E c o n o m i c s 62 ( 1 9 9 6 ) 2 9 7 - 3 2 5
T h e o t h e r case to be r u l e d o u t is t h a t in w h i c h t h e flat p a r t s o f t h e
i n d i f f e r e n c e c u r v e s o f y" a n d y" a r e distinct o r o v e r l a p , while t h e u p w a r d s l o p i n g p a r t s cross m a n y t i m e s . T h i s is trivially a v i o l a t i o n o f S D I .
L e m m a 1 i m p l i e s t h a t t h e u p w a r d - s l o p i n g p a r t o f an i n d i f f e r e n c e c u r v e of
y " > y ' c a n n o t i n t e r s e c t t h e flat p a r t o f an i n d i f f e r e n c e c u r v e o f y'. It follows
f r o m S D I t h a t a n y c r o s s i n g m u s t be o n e in w h i c h an i n d i f f e r e n c e c u r v e o f y"
c r o s s e s a n i n d i f f e r e n c e c u r v e o f y" f r o m a b o v e .
[]
Definition. A n a l l o c a t i o n is a m a j o r i t y - v o t i n g e q u i l i b r i u m if it is o n t h e G B C
a n d it g a r n e r s 5 0 % o r m o r e o f t h e v o t e in a b i n a r y c o m p a r i s o n a g a i n s t a n y
alternative on the GBC.
Proposition I. W h e n S D I l~olds, a majority-voting equilibrium exists, a n d the
m e d i a n - i n c o m e voter is decisive. 7
Proof. T h e p r o o f is b a s e d o n R o b e r t s (1977). H e r e we a d a p t t h e g e o m e t r i c
p r o o f d e v e l o p e d in E p p l e a n d R o m e r (1991). L e t (/~, t-) be t h e p o i n t o n t h e
G B C m o s t p r e f e r r e d by t h e v o t e r with m e d i a n i n c o m e , a n d a s s u m e f o r n o w
t h a t (/~, t-)>:> (0, 0) a n d is u n i q u e . W e d r a w t h e m e d i a n - i n c o m e v o t e r ' s
i n d i f f e r e n c e c u r v e thr,~ugh this p o i n t (see t h e c u r v e l a b e l e d I~ in Fig. 4).
T h e r e c a n be n o p o i n t s o n t h e G B C b e l o w 17 since a n y such p o i n t w o u l d be
"~JfOfi
::
s
|
|
0
|
I
D
|
I
Fig. 4
T h i s g e n e r a l i z e s t h e result o f G l o m m a n d R a v i k u m a r (1996). w h o p r o v e a similar result by
i n v o k i n g restri~:fions o n t h e f u n c t i o n a l f o r m o f p r e f e r e n c e s a n d the d i s t r i b u t i o n o f i n c o m e .
D. Epple, R . E . R o m a n o
/ Journal o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
309
preferred to (/~, 7) by the median-income voter. Next, we consider any
points on the G B C that lie above I7 and the horizontal line through/" (points
i n region A). L e m m a 3 implies that all voters with income y t > f prefer
( E , / ' ) to any point in this region (e.g. the voter with dashed indifference
curve V" shown in Fig. 4). Since 37 is the median, then at least half the
electorate prefers (/~, t"). Hence, (/~, t") defeats all points in region A. A n
analogous argument establishes that (/~, t") defeats all points above V and
below the line through ?" because all voters with y ~< )7 prefer (E, t") to any
point i n this region.
If E = t = 0, a briefer version of the same argument establishes it as an
equilibrium. H e r e the horizontal line through (E, t") is the abscissa, and no
region below it exists to consider,
We have shown that the median-income household's most preferred
choice is always an equilibrium which establishes existence. Since the
a r g u m e n t s apply to any point in that household's most preferred set.
multiple equilibria are theoretically possible. H o w e v e r , a point not a
m e m b e r of this set cannot be an equilibrium. If the median voter preferred a
point on the G B C with both lower E and t than the candidate point, then,
by L e m m a 3, so too would all households with higher incomes. By the
continuity of V ( - ) in y, a positive measure of households with incomes
below but close to the median would also have such a preference. H e n c e ,
the candidate point would not garner a majority. A n analogous a r g u m e n t
rules out points where the median-income voter has a preference for a point
op the, G B C with both higher E and t. If there exists a point on the G B C
with (weakly) higher E and (weakly) lower t than a candidate point, then
there is unanimity of preferences for the former. Only a most preferred
point of the median-income household will be an equilibrium. []
Remarks. (1) Note that this result follows from the properties of the induced
utility function defined in Eq. (4) and does not rely on any properties of the
G B C . Intuitively, the result occurs for the following reason. If no private
alternative existed and all households consumed the publicly provided good,
the most-preferred level of provision would be inversely related to income
u n d e r SDI. The equilibrium would be the choice of the median-income
household. W h e n a private alternative is available, it is consumed by the
highest-income segment of the population. While this reduces the level of
the publicly provided good that this high-income segment prefers relative to
the case of no private alternative, it does not change the inverse relaticmst~ip
b e t w e e n income and the mosi-preferred level of public provision. H e n c e ,
the median-income household remains decisive.
(2) If the median-income voter's highest attainable indifference curve is
achieved at more than one point on the G B C , then all such points are
equilibria. Since such multiple equilibria are knife-edge cases, a unique
equilibrium is the generic outcome.
310
D. E p p l e , R . E . R o m a n o
/ J o u r n a l o f P u b l i c E c o n o m i c s 62 ( I 0 9 6 ) 2 9 7 - 3 2 5
GBC
f/',y
/
Em
[
~
Fig. 5
(3) A n i n t e r e s t i n g i m p l i c a t i o n o f P r o p o s i t i o n 1 is t h e f o l l o w i n g . S u p p o s e
t h a t t h e m e d i a n - i n c o m e v o t e r has a slight p r e f e r e n c e for pr i vat e schooling
o v e r p u b l i c s c h o o l i n g . F i g . 5 i l l v s t r a t e s s u c h a c a s e w h e r e V a n d V" a r e
i n d i f f e r e n c e c u r v e s o f t h e m e d i a n - i n c o m e h o u s e h o l d . F h e n t h e t a x r a t e will
b e s e t e q u a l t o z e r o a n d t h e r e will b e n o p u b l i c p r o v i s i o n . U n d e r t h e s e
c o n d i t i o n s , a s m a l l p e r t u r b a t i o n m a y l e a d t o l a r g e c h a n g e s in t h e l e v e l o f
public provision. For example, suppose that courts mandate that there be a
minimum level of public provision. Then the level of public provision
actually provided may be significantly greater than that mandated by the
c o u r t . I f t h e m a n d a t e d m i n i m u m l e v e l e x c e e d s E m in Fig. 5, t h e n t h e
e q u i l i b r i u m h a s p u b l i c p r o v i s i o n o f E.s
Next, we turn to a consideration of the voting equilibrium when the
m o n o t o n i c i t y c o n d i t i o n S R I h o l d s . F o r ( E , t) v a l u e s w h e r e t w o h o u s e h o l d s
prefer the public alternative and their indifference curves cross, the higher
income household's indifference curve crosses the lower income housebold's
indifference curve from below. Taking account of the private alternative and
u s i n g L e m m a 1 t o d r a w i n d i f f e r e n c e c u r v e m a p p i n g s , it is s i m p l e t o c o n f i r m
t h a t s i n g l e c r o s s i n g will fail t o h o l d , as w e i l l u s t r a t e s h o r t l y . U n d e r w e a k
asv:~,mptions, t h e m e d i a n v o t e r ' s p r e f e r e n c e will n o l o n g e r d e t e r m i n e p u b l i c
e x p e n d i t u r e in a v o t i n g e q u i l i b r i u m . A n e q u i l i b r i u m m a y a l s o fail t o exist. If
it e x i s t s , a l o w e r t a x - e x p e n d i t u r e c h o i c e t h a n t h e m e d i a n - i n c o m e v o t e r ' s
SAlternatively, suppose a policy change dictates seine minimum level of expenditure
financed outside of the jurisdiction (see, for example, "Can Big Money Fix Urban School
Systems? A Test Is Underway?', Wall Street J o u r n a l , 7 January 1992. p. i). This results in a
rightward shift in the GBC and, similarly, can lead to large changes in public expenJitures.
D. Epple. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
311
p r e f e r e n c e will prevail. A middle income group will prefer t a x - e x p e n d i t u r e
increases, but an equal-sized coalition of rich and p o o r households wilt
p r e f e r the opposite.
M a k i n g two realistic assumptions simplifies the p r e s e n t a t i o n a n d avoids
u n i n t e r e s t i n g cases.
Assumption A5. The m e d i a n - i n c o m e h o u s e h o l d ' s most preferred choice (or
choices) on the G B C has (have) E > 0.
T h e a p p e n d i x shows that a sufficient condition for A s s u m p t i o n 5 is that
the m e d i a n i n c o m e is below the m e a n income in the population.
Assumption A6. The highest-income individual strictly prefers a private
alternative if (any of) the m e d i a n - i n c o m e h o u s e h o l d ' s most p r e f e r r e d
choice(s) on the G B C prevails.
A s s u m p t i o n A 6 will preclude an equilibrium with no private c o n s u m p t i o n
of education. (The effect of relaxing this assumption is discussed later in
f o o t n o t e 9.) We have:
Proposition 2. The median-income voter's most preferred choice(s) cannot be
a majority voting equilibrium.
Proof. In Fig. 6, V m is an indifference curve of the m e d i a n - i n c o m e voter a n d
V an indifference curve of the highest-income (y-) voter. H e n c e , point A
r e p r e s e n t s the median-incon'le voter's m o s t - p r e f e r r e d alternative a n d V
GBC
Vm
8J
o~ ~J
s
~
_
Fig. 6
312
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f Public E c o n o m i c s 62 ( 1 9 9 6 ) 2 9 7 - 3 2 5
c o n f o r m s t o A s s u m p t i o n A 6 . ( T h a t V r e c r o s s e s t h e G B C is n o t r e l e v a n t to
t h e a r g u m e n t b u t will be u s e d to m a k e a n o t h e r p o i n t b e l o w . W e i g n o r e
p o i n t s B a n d C f o r n o w . ) A m a r g i n a l l y l o w e r t a x - e x p e n d i t u r e c o m b i n a t i o n is
p r e f e r r e d by a m a j o r i t y to p o i n t A. All h o u s e h o l d s with i n c o m e s b e l o w t h e
m e d i a n i n c o m e h a v e such a p r e f e r e n c e by C o r o l l a r y 1 a n d S R I . L i k e w i s e t h e
h i g h e s t i n c o m e h o u s e h o l d has such a p r e f e r e n c e as d o s o m e h o u s e h o l d s with
i n c o m e s n e a r y, t h e l a t t e r by c o n t i n u i t y of V ( - ) in y. T h e s a m e a r g u m e n t
a p p l i e s t o e a c h o f t h e m e d i a n - i n c o m e h o u s e h o l d ' s m o s t p r e f e r r e d c h o i c e s if
a m u l t i p l i c i t y o f such p o i n t s exists. []
T h e fact t h a t p o i n t A in Fig. 6 is n o t a m a j o r i t y v o t i n g e q u i l i b r i u m r e l a t e s
t o t h e f a i l u r e o f single crossing. U s i n g S R I , a p r e f e r e n c e f o r l o w e r taxes by
t h o s e h o u s e h o l d s with i n c o m e s n e a r , b u t b e l o w , t h e m e d i a n i n c o m e is
c o u n t e r e d by a p r e f e r e n c e f o r h i g h e r t a x e s by t h o s e with i n c o m e s n e a r , b u t
a b o v e , t h e m e d i a n i n c o m e . H o w e v e r , let us c o n s i d e r h o u s e h o l d s with
i n c o m e s sufficiently high t h a t t h e y m a k e a p r i v a t e c h o i c e at t h e m e d i a n i n c o m e h o u s e h o l d ' s m o s t p r e f e r r e d choice. I n d i f f e r e n c e c u r v e s o f such
h i g h - i n c o m e h o u s e h o l d s cross V m at p o i n t A f r o m a b o v e . T h e y . t o o , p r e f e r
t a x d e c r e a s e s , b r e a k i n g A as an e q u i l i b r i u m . S i m i l a r logic p r e c l u d e s a n y
p o i n t a b o v e A o n t h e G B C f r o m b e i n g an e q u i l i b r i u m :
Corollary 2. A majority voting equilibrium, i f it exists, entails less public
expenditure than the (minimum o f ) the median-income household's most
preferred choice(s).
Proof. ~We r e f e r to Fig. 6 w h e r e p o i n t A is t h e m e d i a n - i n c o m e h o u s e h o l d ' s
m o s t p r e f e r r e d c h o i c e . ( I / _~ multiplicity exists, let p o i n t A be t h e m e d i a n income household's minimum-expenditure most preferred choice.) Househ o l d s w i t h i n c o m e s b e l o w t h e m e d i a n i n c o m e strit.dy p r e f e r p o i n t A to a n y
p o i n t o n t h e G B C a b o v e A , by C o r o l l a r y 1 a n d S R I . By t h e c o n t i n u i t y o f
V ( - ) in y, p o i n t A is also p r e f e r r e d by a p o s i t i v e m e a s u r e o f i n c o m e s in t h e
vicinity o f ~ to all p o i n t s a b o v e A o n t h e G B C , e x c e p t p o i n t s o n o r in t h e
vicinity o f t h e arc BC. T h i s e s t a b l i s h e s t h a t p o i n t s a b o v e A on t h e G B C
o t h e r t h a n t h o s e o n o r in t h e vicinity o f t h e arc B C a r e d e f e a t e d by p o i n t A .
P o i n t A is also p r e f e r r e d by a p o s i t i v e m e a s u r e o f h o u s e h o l d s with i n c o m e s
a b o v e a n d in t h e vicinity of t h e m e d i a n i n c o m e to all p o i n t s o n o r in t h e
vicinity o f t h e arc BC, a g a i n by c o n t i n u i t y of V ( - ) in y. M a t c h e d a g a i n s t
p o i n t s o n o r in t h e vicinity o f t h e arc ~ C , p o i n t A w o u l d again g a r n e r a strict
m a j o r i t y . If t h e G B C wiggles in such a way t h a t o t h e r arcs p r e f e r r e d by y,
Like BC, a r e p r e s e n t , t h e n p o i n t s o n o r in t h e vicinity of t h e s e arcs c a n be
r e j e c t e d as e q u i l i b r i a a n a l o g o u s l y . C o m b i n i n g t h e s e results with P r o p o s i t i o n
2 i m p l i e s t h a t a n y m a j o r i t y - v o t i n g e q u i l i b r i u m m u s t h a v e less p u b l i c
e x p e n d i t u r e t h a n at p o i n t A . []
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
313
Proposition 3 contains the p a p e r ' s most novel result, n a m e l y necessary
conditions for an interior majority voting equilibrium (i.e. one with E > 0)
u n d e r SRI. T h e candidate point is majority preferred to local deviations on
the G B C u n d e r these conditions.
Proposition 3. I f (ft., i') is an interior majority voting equilibrium under S R I ,
then:
"'(i)'" there exists a household with income Yt that weakly prefers public
consumption at point (ft., ~) to public consumption at all other points on the
GBC;
'" (ii)'" there exists a household with incc, me Yh that is indifferent between
public and private consumption at point (ff.~);
""(iii)'" Yt < Yh ; and
Yh
""( i v ) " p ~- f y,
f ( y ) d y = 0.5.
Proof. T h e p r o o f co,~sists of two parts. First, we show that at points on the
G B C w h e r e househola2 y~ and Yh satisfying (i)--(iii) exist, p must equal 0.5
for the point to be an e q c i l i b r i u m . T h e seco, nd part shows that all candidate
points for an equilibrium must have such households.
Fig. 7 illustrates a potential equilibrium, w h e r e V~ and Vh are the
indifference curves of h o u s e h o l d s with incomes Yc and Yh, respectively.
H o u s e h o l d s with incomes y ~ (y~, Yh) strictly prefer the public alternative at
(/~, ~ using L e m m a 1 and Corollary 1. SRI implies that, relative to (/~, t"),
these h o u s e h o l d s prefer marginal e x p e n d i t u r e - t a x increases along the G B C .
If p > 0.5, then a marginal e x p e n d i t u r e - t a x increase defeats (/~, t").
Va
Fig. 7
314
D. E p p l e . R . E .
Romano
/ Journal of Public Econ.~mics 62 (1996) 297-325
R e l a t i v e to (/~, tO, h o u s e h o l d s with i n c o m e s b e l o w yz p r e f e r m a r g i n a l
t a x - e x p e n d i t u r e d e c r e a s e s a l o n g t h e G B C , by C o r o l l a r y 1 a n d S R I . H o u s e h o l d s w i t h i n c o m e s a b o v e Yn h a v e t h e s a m e p r e f e r e n c e , b X L e m m a 1. If
p < 0 . 5 , t h e n a m a r g i n a l t a x - e x p e n d i t u r e d e c r e a s e f r o m ( E , t") d e f e a t s it.
H e n c e , g i v e n t h e e x i s t e n c e of h o u s e h o l d s satisfying (i)-(iii) with i n c o m e s Yt
a n d yh at a p o i n t o n t h e G B C , p = 0 . 5 is n e c e s s a r y for it to be an
equilibrium.
W e n o w c o n s i d e r t h e c o n s e q u e n c e s for an e q u i l i b r i u m of t h e p o t e n t i a l
n o n - e x i s t e n c e o f s u c h h o u s e h o l d s . C o r o l l a r y 2 r u l e d o u t as a c a n d i d a t e
equilibrium those points abeve and including the (minimum) preferred
c h o i c e o f t h e m e d i a n v o t e r , i.e. p o i n t s a b o v e A in Fig. 6. ~ W e restrict o u r
a t t e n t i o n to p o i n t s b e l o w A o n t h e G B C . A t all such p o i n t s a h o u s e h o l d o f
t y p e Yn exists t h a t satisfies (ii). T h e a l t e r n a t i v e l e a d s t o a c o n t r a d i c t i o n .
A s s u m p t i o n A 6 a n d E x p r e s s i o n (5) i m p l y t h a t t h e h i g h e s t - i n c o m e h o u s e h o l d
c h o o s e s t h e p r i v a t e a l t e r n a t i v e at all p o i n t s b e l o w A o n t h e G B C . L e m m a 1
i m p l i e s t h a t t h e c o r n e r s o f t h e i n d i f f e r e n c e c u r v e s shift to t h e left as i n c o m e
d e c l i n e s . T h e n , if n o Yh t y p e exists at such p o i n t s , t h e n all h o u s e h o l d s m u s t
l i k e w i s e c h o o s e t h e p r i v a t e a l t e r n a t i v e . By (1), E = 0c, w h i c h c o n t r a d i c t s
L e m m a 2. A t y p e Yh will t h e n exist at all c a n d i d a t e p o i n t s . It also b e c o m e s
c l e a r m o m e n t a r i l y t h a t , f o r an e q u i l i b r i u m , t h e r i g h t - h a n d s l o p e o f Vh at
( E , tO m u s t e x c e e d t h a t o f t h e G B C (as Fig. 7 illustrates).
L e t us n o w c o n s i d e r t h e e x i s t e n c e o f Yt types. By C o r o l l a r y 1, all a n d o n l y
h o u s e h o l d s with i n c o m e s less t h a n Yh strictly p r e f e r t h e p u b l i c a l t e r n a t i v e at
t h e p o i n t in q u e s t i o n , a n d t h e y m a k e u p t h e set of c a n d i d a t e h o u s e h o l d s .
H e n c e , (iii) m u s t be satisfied. T h e r e a r e t w o possibilities f o r t h e n o n e x i s t e n c e o f a t y p e Yt, as d e f i n e d by (i). O n e h a s n o h o u s e h o l d w i t h a
t a n g e n c y at t h e p o i n t in q u e s t i o n , with t w o s u b c a s e s . T h e first s u b c a s e is
i l l u s t r a t e d in Fig. 8, w h e r e t h e l o w e s t i n c o m e (y_) h o u s e h o l d ' s i n d i f f e r e n c e
c u r v e ( V ) is flatter t h a n t h e G B C at t h e p o i n t in q u e s t i o n . T h e r i g h t - s i d e
s l o p e o f t h e Yh t y p e ' s i n d i f f e r e n c e c u r v e m u s t t h e n also be flatter t h a n t h e
G B C , o t h e r w i s e s o m e y E (y_, Yn) w i t h a t a n g e n c y exists, b3, S R I a n d t h e
c o n t i n u i t y of V ( - ) in y. S u c h a p o i n t c a n n o t be an e q u i l i b r i u m b e c a u s e a
u n a n i m o u s p r e f e r e n c e f o r m a r g i n a l l y r e d u c e d t a x e s is i m p l i e d . N o t e also
t h a t a slightly m o d i f i e d v e r s i o n o f t h e l a t t e r a r g u m e n t c a n be u s e d to r e j e c t
a n y p o i n t s o n a n y n o n - i n c r e a s i n g r a n g e s of a G B C (i.e. p o i n t s w h e r e
d E * ~dr <~ 0).
T h e s e c o n d s u b c a s e o f n o t a n g e n c y , i l l u s t r a t e d in Fig. 9, p r e s u m e s t h e
l o w e s t i n c o m e h o u s e h o l d ' s i n d i f f e r e n c e c u r v e (_V_~) is s t e e p e r t h a n t h e G B C
at t h e p o i n t in q u e s t i o n . T h e p r o p e r t i e s of t h e i n d i f f e r e n c e m a p p i n g s i m p l y
A n e q u i l i b r i u m with t h e m e d i a n - i n c o m e h o u s e h o l d p i v o t a l m a y result if t h e h i g h e s t - i n c o m e
h o u s e h o l d w e a k l y p r e f e r s p u b l i c c o n s u m p t i o n at p o i n t A (i.e. if A s s u m p t i o n 6 is d r o p p e d ) .
S u c h a n e q u i l i b r i u m satisfies a c o r n e r v e r s i o n o f t h e n e c e s s a r y c o n d i t i o n s in P r o p o s i t i o n 3.
D. Epple. R . E . R o m a n o / Journal o f Public E c o n o m i c s 62 (1996) 2 9 7 - 3 2 5
3|5
GBC
I
!
!
/
/
/
/
vh
_v
Fig. 8
t h a t all y E ~ , Yh) prefer a marginally higher tax, a n d all y >Yh prefer a
marginally lower tax. Point B cannot be an equilibrium if ~,,~ is o t h e r than
the m e d i a n income. If yh equals the m e d i a n i n c o m e , then points like C will
d e f e a t B. All h o u s e h o l d s with i n c o m e s below Yh prefer C to B, as does a
positive m e a s u r e of households with incomes g r e a t e r t h a n , and in the
vicinity of, Yh"
T h e r e m a i n i n g possibility of the non-existence of a y~ type, illustrated in
Fig. 10, p r e s u m e s a t a n g e n c y at the c a n d i d a t e point (B), but fails to satisfy
the r e q u i r e m e n t that the point is a most p r e f e r r e d point of public c o n s u m p tion of the y~ h o u s e h o l d . T h e a r g u m e n t s we m a k e apply w h e t h e r or not the
G B C is concave at B; Fig. 10 illustrates the "more difficult" case. If p ~: 0.5,
GBC
Vh
y1
B
Fig. 9
316
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f Public E c o n o m i c s 62 ( 1 9 9 6 ) 2 9 7 - 3 2 5
Vh
GBC
Fig. I0
t h e n B is d e f e a t e d by a local d e v i a t i o n o n t h e G B C by t h e first a r g u m e n t o f
t h e p r o o f . If p = 0.5, t h e n t h e ye h o u s e h o i d ' s m o s t p r e f e r r e d p u b l i c
a l t e r n a t i v e c a n be a b o v e p o i n t B (e.g. p o i n t C) o r b e l o w p o i n t B (e.g. p o i n t
D ) . In t h e f o r m e r case, C d e f e a t s B, since all h o u s e h o l d s with i n c o m e s
y E [ Y t , Y h ) p r e f e r C t o B, as d o a p o s i t i v e m e a s u r e o f h o u s e h o l d s w i t h
i n c o m e s a b o v e a n d in t h e vicinity o f Yh" T h i s is c o n f i r m e d by d r a w i n g t h e
i n d i f f e r e n c e c u r v e s o f such t y p e s t h r o u g h B, u s i n g t h e i r r e q u i r e d p r o p e r t i e s .
I n t h e l a t t e r c a s e , D d e f e a t s B, since all h o u s e h o l d s w i t h y ~< y~ a n d y > Yh
p r e f e r D t o B, as d o a p o s i t i v e m e a s u r e of h o u s e h o l d s with i n c o m e s b e l o w
a n d in t h e vicinity o f Yh- []
R e m a r k s . (1) T h i s r e s u l t h a s m u c h intuitive a p p e a l f o r services such as
e d u c a t i o n . W h e n a p r i v a t e a l t e r n a t i v e is a v a i l a b l e , h i g h - i n c o m e h o u s e h o l d s
p r e f e r l o w p u b l i c s c h o o l e x p e n d i t u r e b e c a u s e t h e y d o n o t use p u b l i c schools.
L o w - i n c o m e h o u s e h o l d s p r e f e r low p u b l i c s c h o o l e x p e n d i t u r e s b e c a u s e t h e y
a r e less willing t h a n h i g h e r - i n c o m e h o u s e h o l d s to s u b s t i t u t e p u b l i c s c h o o l
e x p e n d i t u r e s f o r o t h e r g o o d s . M i d d l e - i n c o m e h o u s e h o l d s , by c o n t r a s t , use
p u b l i c s c h o o l s a n d p r e f e r t h a t t h e y be o f r e l a t i v e l y high quality. H e n c e , a
coalition of m i d d l e - i n c o m e households prefers higher public school expendit u r e at t h e m a r g i n , while a c o a l i t i o n o f high- a n d l o w - i n c o m e h o u s e h o l d s
p r e f e r s a r e d u c t i o n . In e q u i l i b r i u m , t h e s e t w o c o a l i t i o n s a r e e q u a l in size
a n d b a l a n c e e a c h o t h e r in v o t i n g ) 9
~°The e n d s - a g a i n s t - t h e - m i d d l e p r o p e r t y of an e q u i l i b r i u m is r e m i n i s c e n t o f D i r e c t o r ' s I ,nw o f
r e d i s t r i b u t i o n . Public r e d i s t r i b u t i o n is f r o m t h e rich a n d the p o o r to the m i d d l e class a c c o r d i n g
t o t h e L a w ( s e e Stig|er, 1970). O u r m o d e l p r o v i d e s s o m e t h e o r e t i c a l s u p p o r t for this e m p i r i c a l
phenomenon.
D. Epple. R . E . R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
317
(2) The proposition provides a necessary condition for an interior
equilibrium. The condition ensures that local deviations will not defeat the
candidate point on the G E C . If preferences obeyed single crossing, then
satisfaction of the local condition for a voting equilibrium at the pivotal
v o t e r ' s most preferred choice would imply that the point defeats all
alternatives. The presence of double crossings eliminates the guarantee of
no majority-preferred alternatives. Let us consider, for example, a "large"
tax increase from the candidate point. Some higher income households that
choose a private alternative at the candidate point and would vote against
marginal tax increases, would favor a tax increase that causes them to switch
to the public alternative. U n d e r single crossing, no alternative points like
the latter could exist. These "switchers" are, however, countered by some
middle-income households who would prefer a marginally better public
alternative to the candidate point and would vote for marginal tax increases,
but would vote against a large tax increase. Equilibrium requires that the
size of the latter g r o u p exceeds the former, and analogously for all "large"
tax deviations. W h e t h e r a candidate point passes the global test will then
d e p e n d on the specifics of preferences and the distribution of income. It may
be surprising that, in our computational analysis reported below, we found
the point that satisfies the necessary condition to be an equilibrium in all
cases.
(3) E p p l e and R o m a n o (1996) and G o u v e i a (1996) have independently
analyzed the alternative dual provision e n v i r o n m e n t where a proportional
income tax finances the provision of a good c o n s u m e d by all households, but
w h o can frictionlessly supplement consumption with private m a r k e t purchases. These papers show that a voting equilibrium exists generally, and,
assuming SRI and a median income below the m e a n , an equilibrium is
characterized by the ends-against-the-middle property. The middle-income
g r o u p that favors a tax increase consists of the half of the population that
has an income below but closest to the mean. In contrast, the present p a p e r
shows that when the public and private alternatives cannot be jointly
c o n s u m e d , the middle-income group that favors a tax increase has no fixed
u p p e r b o u n d ; rather, it depends on preferences, technology, and income
distribution.
(4) A corner equilibrium at the origin (i.e. t = E = 0) or the non-existence
of an equilibrium are, of course, other possibilities. We can probably
contrive cases with a multiplicity of equilibria. Since the alternative equilibria would n e e d to tie, empirical relevaace is unlikely.
We have assumed that households can purchase as many units of private
school services as they desire at a constant unit price, p. In reality, of
course, there are a finite n u m b e r of private schools, each offering a given
level of educational services per pupil. We might then consider voting over
public school inputs when there are a discrete n u m b e r of private
318
D. Epple. R.E. R o m a n o I Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
alternatives. We have shown (Epple and Romano,
a b o v e e x t e n d e a s i l y t o s u c h a case.
I994) t h a t all t h e r e s u l t s
3. Computational model
W e d e v e l o p a c o m p u t a t i o n a l m o d e l w i t h t w o o b j e c t i v e s . O n e is to
investigate, for a range of p a r a m e t e r values, w h e t h e r the point satisfying the
n e c e s s a r y c o n d i t i o n in P r o p o s i t i o n 3 is a m a j o r i t y - v o t i n g e q u i l i b r i u m . T h e
o t h e r is t o e x p l o r e i m p l i c a t i o n s o f t h e m o d e l f o r a p o l i c y t h a t p r o v i d e s
vouchers for private education.
T h e c o m p u t a t i o n a l m o d e l r e q u i r e s a s p e c i f i c a t i o n f o r t h e i n c o m e dist r i b u t i o n and the utility function. We a s s u m e that h o u s e h o l d incomes are
l o g - n o r m a l l y d i s t r i b u t e d , ! n y - - - N ( / z , tr2). G i v e n o u r o n e - j u r i s d i c t i o n a l
m o d e l a n d t h e l i k e l i h o o d o f T i e b o u t s o r t i n g o v e r m u l t i p l e j u r i s d i c t i o n s in
t h e r e a l w o r l d , o u r p r e s u m e d d i s t r i b u t i o n is a d m i t t e d l y a c r u d e a p p r o x i m a t i o n . In 1989, m e a n a n d m e d i a n U . S . h o u s e h o l d i n c o m e s w e r e $ 3 6 , 2 5 0
a n d $ 2 8 , 9 0 6 , r e s p e c t i v e l y . M e a s u r i n g i n c o m e in t h o u s a n d s , t h e s e i m p l y
/~ = 3 . 3 6 a n d tr = 0 . 6 8 ) ~
W e a s s u m e that p r e f e r e n c e s are given by the following C E S function:
U ( x , b ) = [ f i x - ° + (1 - / 3 ) b - ° ]
t-'/p)
W h e n p < 0, this f u n c t i o n satisfies a s s u m p t i o n S D I . W h e n p > 0, a s s u m p t i o n
S R I is s a t i s f i e d . 12
W e c a l i b r a t e t h e u t i l i t y f u n c t i o n as f o l l o w s . E x p e n d i t u r e p e r s t u d e n t in
U . S . p u b l i c s c h o o l s in 1988 w a s $4,222 a n d t h e r e w e r e 0.5 s t u d e n t s p e r
h o u s e h o l d (i,e. e x p e n d i t u r e p e r h o u s e h o l d w a s $2,111). In o u r c a l i b r a t i o n ,
we r e q u i r e that the p a r a m e t e r s be such that the necessary conditions for
v o t i n g e q u i l i b r i u m a r e s a t i s f i e d at a p u b l i c e x p e n d i t u r e o f $ 4 , 2 2 2 p e r
s t u d e n t . A n a d d i t i o n a l c o n d i t i o n is o b t a i n e d b y fixing t h e v a l u e o f t h e p r i c e
e l a s t i c i t y o f d e m a n d f o r e d u c a t i o n ( e v a l u a t e d at t h e p o i n t s a t i s f y i n g t h e
necessary conditions for equilibrium). Values for the two utility function
p a r a m e t e r s , p a n d /3, a r c d e t e r m i n e d b y t h e s e t w o c o n d i t i o n s .
R e s u l t s a r e r e p o r t e d in T a b l e 1 f o r f o u r d i f f e r e n t p r i c e elasticities. W e
c h o s e a b r o a d r a n g e o f p r i c e e l a s t i c i t i e s in o r d e r t o i l l u s t r a t e h o w o u t c o m e s
~' If In(x) ~ N(/z. o.2 ), the mean ofx is E ( x ) = exp(/~ + (o.z/2)), and the median ofx is e ~' .
Given the mean and median of x, these can be solved for ~ and o.:.
~zThis function is homothetic, implying an income elasticity of demand equal to one. The use
of such a homothetic function greatly simplifies the computations. While this income elasticity is
consistent with macro estimates of the demand for education, it is considerably larger than
results obtained from micro studies (Rubinfeld and Shapiro, 1989). Consistent with the
discussion in Section 2, p > ( < ) 0 implies a price inelastic (elastic) demand.
D. Epple, R.E, R o m a n o / Journal o f Public Economics (i2 (1996) 2 9 7 - 3 2 5
3~9
Table I
7/D
y,
Yh
t dE
E de
$18.190
$45.519
0.59
88.0
23,497
62,241
97.8
98.9
28,854
28,854
/3
E
t
0.006
$2.110
Voucher = 0
0.043
74.9
...... 0 . ....
54
0.02
2,110
0.051
-0.21
-0.35
0.10
0.134
2,110
2,110
0,057
0.057
-0.5
g .2
0.006
$2,098
I! - 0 . 6 7
0.54
0.02
2,t31
0.053
0, I0
0.134
2,100
2,090
0.056
0.057
$2,056
-0.5
iI - 0 . 6 7
-- 1.25
-- 1.5
p
1.2
% Public
[
0.74 - l
NA
NA
0.93
0.95
$16,815
$42,366
0.68
85.9
22,607
59,982
0.79
96.8
98.2
28,854
28,854
Voucher = $1,000
0.045
71.4
!
- 1.25
- 1.5
-I).21
-0.35
Voucher = $2,000
0.047
65.7
NA
NA
0.91
0.95
$14,555
$37,995
0.82
55.,498
-0.5
! .2
0.006
I-0-67
0.54
0.02
2,166
0:054
83.2
21.473
-1.25
- 1.5
-0.21
--0.35
0.I0
0.134
2.111
2,080
0.056
0.056
95.1
96.9
28,854
28,854
NA
NA
[
J
0.83[
0.91
0.94
change as the elasticity is changed. Price elasticities less than o n e in absolute
value correspond to assumption SRI, while those greater than o n e in
absolute value correspond to SDI (see footnote 12). H e n c e , the first two
rows o f results correspond to SRI and the second two rows correspond to
SDI.
W h e n SDI holds, Proposition 1 establishes that a voting equilibrium
exists. W h e n SRI holds, Proposition 3 provides the necessary conditions. ~3
W e checked whether the allocation in Proposition 3 was an equilibrium for
each case in which the parameters correspond to SRI. We did this by
computing the proportion of voters favoring (/~, t") against a dense grid o f
alternatives along the G B C . For an a~.!c.cation to be an equilibrium, it must
garner at least half the votes against every alternative. Fig. i 1, plotted for
Z~All computations were done using the algorithm for solving non-linear simultaneous
equations in G a u s s . These equations s i m u l t a n e o u s l y specify t h e G B C . t h e s l o p e o f t h e G B C .
".he s l o p e o f v o t e r y,'s indifference curve, e q u a l i t y o f the latter two slopes, the indifference o f
voter Yb over public and private consumption, and one-half the population between y, and yh.
W h e n a p o i n t satisfying the necessary conditions is f o u n d , a second program is then used to
calculate the vote favoring that point against each o f a dense grid o f p o i n t s along the budget
constraint. More details are available from the authors upon request.
320
D. E p p l e . R . E . R o m a n o
/ J o u r n a l o f P u b l i c E c o n o m i c s 62 ( 1 9 9 6 ) 2 9 7 - 3 2 5
Vote
1
0.8
0.7
-
0.6
0.5
0.4
0
1,000
2,0C~
3,000
4.000
E~pend~ure
F~O = 0.54: Be,Ca ~ 0.02; Vou~"mr = 0
Fig. 11. Percentage of voters that favor an equilibrium o u t c o m e against alternative expenditure
levels.
p a r a m e t e r s in t h e h i g h l i g h t e d r o w o f t h e case with a v o u c h e r e q u a l to z e r o in
T a b l e 1, is typical o f t h e r e s u l t s w e o b t a i n e d . In e v e r y case r e p o r t e d in t h e
t a b l e , t h e a l l o c a t i o n satisfying P r o p o s i t i o n 3 is an e q u i l i b r i u m . Thi~ is o n e
k e y f i n d i n g o f o u r c o m p u t a t i o n a l analysis. W h i l e we b e l i e v e t h a t t h e r e m a y
b e c i r c u m s t a n c e s u n d e r w h i c h an a l l o c a t i o n satisfying P r o p o s i t i o n 3 fails to
b e a n e q u i l i b r i u m , t h e r e s u l t s in T a b l e 1 s u g g e s t t h a t failure of e x i s t e n c e
m a y n o t b e a significant p r o b l e m for e m p i r i c a l l y i n t e r e s t i n g cases.
W e n o w t u r n to a d i s c u s s i o n of t h e s u b s t a n t i v e r e s u l t s o t o u r c o m p u t a t i o n s . C o m p a r i n g t h e first f o u r r o w s o f t h e c o m p u t a t i o n a l results, we s e e
t h a t t h e m a i n e f f e c t o f c h a n g i n g t h e price elasticity o f d e m a n d is in p u b l i c
s c h o o l e n r o l l m e n t . T h e g r e a t e r t h e price elasticity (in a b s o l u t e v a l u e ) , t h e
l a r g e r is p u b l i c s c h o o l e n r o l l m e n t . T h i s is intuitively p l a u s i b l e . T h e h i g h e r
t h e p r i c e elasticity, t h e g r e a t e r t h e sensitivity t h a t h o u s e h o l d s h a v e to t h e
p r i c e d i f f e r e n t i a l r e q u i r e d f o r p r i v a t e s c h o o l i n g c o m p a r e d with f r e e p u b l i c
education.
I n T a b l e 1, we h a v e h i g h l i g h t e d t h e results c o r r e s p o n d i n g to a price
elasticity o f - 0.67. W e d o this for t w o r e a s o n s . First, t h e e q u i l i b r i u m p u b l i c
s c h o o l a t t e n d a n c e with this elasticity is 8 8 % , a n d this is t h e o b s e r v e d U . S .
p e r c e n t a g e . S e c o n d , this v a l u e is within t h e r a n g e f o u n d in e c o n o m e t r i c
s t u d i e s , t4
A m u c h - d e b a t e d policy issue with r e g a r d to e d u c a t i o n is t h e effect o f
i n t r o d u c i n g v o u c h e r s . H o w will a subsidy for p r i v a t e e d u c a t i o n affect p u b l i c
~aFor example. Rubinfeld and Shapiro (1989) report price elasticity estimates ranging
f r o m - 0.43 t o - 0.719 using various specifications applied to data from Massachusetts and
Michigan. O u r b e n c h m a r k price elasticity o f - 0 . 6 7 lies near the upper end of this range.
D. Epple, R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
321
school attendance and public school expenditures? We 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 choose
private schooling. We investigate how the equilibrium public expenditure
per capita is affected by this voucher. O n e motivation for this approach is to
suppose that a voucher is m a n d a t e d by a higher level government. W h e n
voting over local expenditures, voters take this federal policy with respect to
vouchers as given. In equilibrium, the tax rate must be high enough to fund
local expenditures on public schools and the cost of the vouchers. The utility
of houseliolds that choose private school is now v ( y ( l - t ) + s ) and the
g o v e r n m e n t budget constraint satisfies t Y = N E + (1 - N ) s . The model is the
same otherwise, and the equilibrium results carry over. 15
T h e middle panel of Table 1 reports results with a voucher of $1,000 per
student, and the b o t t o m panel a voucher of $2,000 per student. The most
striking feature of these computations is that expenditure per student in
public schooling is r e m a r k a b l y insensitive to the introduction of vouchers.
We might have thought that a v o u c h e r as large as $2,000 would draw
students from public schooling into p,~'ate schooling, and that this in turn
w o u l d lead to a substantial reduction ,.q the a m o u n t voted for public
education. The first part of this intuition is correct, but the second part
proves to be incorrect. For example, for our b e n c h m a r k price elasticity
o f - - 0 . 6 7 , per student expenditure on public schooling rises slightly from
$4,222 to $4,332 as the voucher is increased from 0 to $2,000. This increase
occurs despite the fall in public school attendance and the associated fall in
income of the voter whose indifference curve is tangent to the G B C at the
equilibrium allocation.
Increasing the voucher affects a houschold's indifference mapping only by
shifting out the E ( - ) locus. While the voucher increases the likelihood of
private school choice, preferences over (E, t) are unchanged as long as
public schooling remains the household's optimal choice. Changes in the
v o u c h e r then impact the equilibrium only through the effect on tbe G B C
and on the changing identity of Yc in the case of SRI, and only through the
f o r m e r effect in the case of SDI. Raising the voucher from 0 to $2,000 in the
b e n c h m a r k case of Table 1 does lower the income of the pivotal voter Yt,
implying a decreased preference for public expenditure. Offsetting this,
h o w e v e r , is an increased tax elasticity of the G B C . This elasticity rises
because the voucher attracts some students to private schooling, reducing
the incremental revenue required for a given per student increase in public
schooling expenditure. Loosely, a voucher tends to flatten the G B C in (E, t)
space. This is illustrated in the last column of Table 1. In our b e n c h m a r k
1_,T h e G B C h a s a t - a x i s i n t e r c e p t , s / Y , w h i c h i n c r e a s e s t h e c h a n c e o f a c o r n e r e q u i l i b r i u m
w i t h E = O, b u t t h i s n e v e r o c c u r s in o u r s i m u | a t i o n r e s u l t s .
322
D. Epp!e. R.E. R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
c a s e , t h e elasticity of p e r s t u d e n t public schooling e x p e n d i t u r e with r e s p e c t
t o t h e tax i n c r e a s e s f r o m 0.74 to 0.83 as the v o u c h e r i n c r e a s e s f r o m 0 to
$2,000.
W h i l e e x p e n d i t u r e p e r p u b l i c school s t u d e n t is not very sensitive t o t h e
v o u c h e r r e g a r d l e s s of t h e price elasticity, t h e s a m e is not t r u e of public
s c h o o l e n r o l l m e n t . W h e n the price elasticity is low in a b s o l u t e value ( - 0.5),
e n r o l l m e n t d r o p s f r o m 7 5 % to 6 6 % w h e n t h e v o u c h e r is i n c r e a s e d f r o m 0 to
$2,000. H o w e v e r , if t h e price elasticity is as high a s - 1.5, t h e effect of
v o u c h e r s o n e n r o l l m e n t is q u i t e small. Enrollment. d r o p s f r o m 9 9 % t o 9 7 %
w h e n t h e v o u c h e r is i n c r e a s e d f o r m 0 to $2,000.
4. C o n c l u s i o n
O u r goal in this p a p e r is to u n d e r s t a n d t h e d e t e r m i n a t i o n of public c h o i c e
a n d p a t r o n a g e for a service w h e n a p r i v a t e a l t e r n a t i v e is p r e s e n t . E d u c a t i o n
is a n i m p o r t a n t e x a m p l e . W e find it plausible t h a t , for e d u c a t i o n , highi n c o m e h o u s e h o l d s are m o r e willing to s u b s t i t u t e public e d u c a t i o n for o t h e r
g o o d s t h a n l o w - i n c o m e h o u s e h o l d s (i.e. S R I applies). W e s h o w for this case
t h a t t h e pivotal v o t e r h a s a b e l o w - m e d i a n i n c o m e ( P r o p o s i t i o n 2) a n d t h a t
a n e q u i l i b r h l m is c h a r a c t e r i z e d by a b a l a n c i n g of a m i d d l e - i n c o m e c o a l i t i o n
p r e f e r r i n g h i g h e r public e x p e n d i t u r e a g a i n s t a coalition of high- a n d lowi n c o m e h o u s e h o l d s p r e f e r r i n g l o w e r e x p e n d i t u r e ( P r o p o s i t i o n 3). T o t h e
b e s t o f o u r k n o w l e d g e , n o p r e v i o u s m a j o r i t y - v o t i n g m o d e l h a s this e n d s a g a i n s t - t h e - m i d d l e p r o p e r t y as t h e g e n e r i c f e a t u r e of an e q u i l i b r i u m .
W e also c h a r a c t e r i z e t h e v o t i n g e q u i l i b r i u m ( P r o p o s i t i o n 1) w h e n lowi n c o m e h o u s e h o l d s are m o r e willing t h a n h i g h - i n c o m e h o u s e h o l d s to
i n c r e a s e t a x e s to p a y for t h e publicly p r o v i d e d g o o d . T h i s is a p l a u s i b l e
c h a r a c t e r i z a t i o n of p r e f e r e n c e s for services such as public bus t e a n s p o r t a t i o n
w h e n p r i v a t e t r a n s p o r t a t i o n is an a l t e r n a t i v e . In this case w e s h o w t h a t t h e
m e d i a n - i n c o m e v o t e r is decisive.
W e d e v e l o p a c o m p u t a t i o n a l m o d e l to investigate a v o u c h e r policy. O u r
r e s u l t s s u g g e s t t h a t public school e x p e n d i t u r e p e r s t u d e n t is relatively
i n s e n s i t i v e to t h e i n t r o d u c t i o n of v o u c h e r s , b u t t h a t public school a t t e n d a n c e
is r e l a t i v e l y sensitive to t h e i r i n t r o d u c t i o n .
T h e m o d e l we study in this p a p e r is relatively s p a r t a n . It is a useful
s t r u c t u r e t o i l l u m i n a t e t h e issues involved in c h a r a c t e r i z i n g a v o t i n g
equilibrium. At the same time, however, the model abstracts from potentially i m p o r t a n t f e a t u r e s of c o m p e t i t i o n b e t w e e n public a n d p r i v a t e prov i d e r s (e.g. p e e r - g r o u p effects in e d u c a t i o n ) . W e b e l i e v e , h o w e v e r , t h a t it
will p r o v e feasible to g e n e r a l i z e the s t r u c t u r e to i n c o r p o r a t e s u c h f e a t u r e s .
W e see this as a n a v e n u e for r e s e a r c h o n e d u c a t i o n a l c o m p e t i t i o n a n d
e d u c a t i o n p~licy.
D. Epple. R . E . R o m a n o I Journal o f Public Economics 62 (199,',)297-325
323
Acknowledgements
This research was s u p p o r t e d by the National Science F o u n d a t i o n . W e
t h a n k Larry K e n n y and two a n o n y m o u s referees for helpful c o m m e n t s . T h e
first a u t h o r t h a n k s the Kellogg G r a d u a t e School of M a n a g e m e n t at N o r t h w e s t e r n University w h e r e parts of this research were c o n d u c t e d .
Appendix
H e r e we show that a m e d i a n income (Ym) below the m e a n income (E[y])
is sufficient for A s s u m p t i o n 5. Recall that we have n o r m a l i z e d the n u m b e r
of c o n s u m e r s in the p o p u l a t i o n to 1, implying E[y] = Y, the a g g r e g a t e
i n c o m e . We find the hypothetical g o v e r n m e n t p r o d u c t i o n frontier, G B C ,
w h e r e e v e r y o n e always c o n s u m e s the public alternative. A l o n g it, E = t Y , as
d e p i c t e d in ~ig. 12. We show that the m e d i a n - i n c o m e h_._ousehold's indifference curve t h r o u g h the origin (Vm) cuts above the G B C . T h e n , since the
actual G B C is n o w h e r e a b o v e G B C (and is the same only w h e r e n o
h o u s e h o l d s c o n s u m e the private alternative or t ~-0), A s s u m p t i o n 5 is
implied.
T h e m e d i a n - i n c o m e h o u s e h o l d ' s utility at the origin is given by U ( y m x*(ym),X*(Ym) ). Let t~ d e n o t e the tax along the G B C that yields p e r
h o u s e h o l d public e x p e n d i t u r e equal to x*(Ym). It satisfies troy = t~E[y] =
x * ( y ~ ) . T h e n , Vm cuts above G B C , if U ( Y m ( 1 - - t m ) , X * ( y m ) ) > U ( y m - x * ( y ~ ) , x * ( Y m ) ), or if tmYm<X*(ym). T h e latter is implied by y ~ <
E[y].
[]
t •
GBC
GBC
tm
~* CYm3
Fig.
12
324
D. Epple, R . E . R o m a n o / Journal o f Public Economics 62 (1996) 2 9 7 - 3 2 5
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