GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 262

Inequality, Growth and the Politics of Education andRedistribution

Tetsuo Ono

May 2012

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Inequality, Growth and the Politics of Education andRedistribution
Tetsuo Ono
Osaka University
May 21, 2012

Abstract
This paper analyzes the political economy of public education and redistribution
in an overlapping-generation model of a two-class society in which growth is driven
by the accumulation of human capital. The levels of public education and lumpsum …nancial transfers are determined by voting, while private education which
supplements public education is purchased individually. The model, which includes
two-dimensional voting, demonstrates multiple steady-state political equilibria. One
is an equilibrium with a high share of public education in government expenditure;
the other is an equilibrium with a high share of lump-sum transfers. Numericalanalysis shows empirically plausible result of growth, inequality and the compositionof redistributive expenditures.
JEL Classi…cation Number: D72, D91, I24
Key words: Education, political economy, inequality, growth

Tetsuo Ono, Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

In most countries redistribution is carried out through in-kind transfer programs such as
public education for successive generations, as well as through in-cash transfer programs
such as social security and welfare budgets within the current generation. The size and thecomposition of redistributive policies in democratic countries are determined via voting,and redistributive expenditures a¤ect growth and income distribution, which in turn havean e¤ect on voting over in-cash and in-kind transfer programs. Therefore, it is natural
to expect some correlation among inequality, growth and size with the composition ofredistributive expenditures.
For the past decades, there has been an increasing amount of literature on the political
economy of growth, inequality and redistribution (see, for example, Persson and Tabellini(2000) for a survey). However, most of these focus on either redistribution in cash (SaintPaul and Verdier, 1996; Benabou, 1996; Bertola, 1993; Alesina and Rodrik, 1994; Persson
and Tabellini, 1994; Perotti, 1993) or redistribution in kind through public education(Glomm and Ravikumar, 1992, 1996, 2001; Glomm, 2004; Gradstein and Justman, 1996,1997). There are few studies that focus on both types of redistribution in the context ofgrowth and inequality.
Exceptions are Creedy, Li and Moslehi (2011) and Bernasconi and Profeta (2012).
Creedy, Li and Moslehi (2011) focus on …nancial transfer and pure public goods provision,and consider voting over the composition of government expenditure in an overlappinggeneration model. They successfully demonstrate the endogenous determination of thecomposition of redistributive spending. However, they show nothing about how the size ofgovernment spending is determined via voting because the tax rate is taken as exogenous
to ensure voting over one dimension. In addition, there is no link between growth andredistribution because income is assumed to grow exogenously in their framework.
Bernasconi and Profeta (2012) overcome the two limitations of Creedy, Li and Moslehi(2011). They develop a politico-economic model of a two-class society with human capital
accumulation and consider probabilistic voting over public education (i.e., redistribution
in kind) and lump-sum transfer (i.e., redistribution in-cash). However, private education
as an alternative to public education is abstracted away from their analysis because theirfocus is on the role of public education as a device to provide for poor-born children to
be recognized for their talent. Therefore, the following questions still remain unresolved:
how the politics of public education and lump-sum transfer a¤ect economic decisions overprivate education of agents; how the decisions in turn a¤ect the inequality among agentsand their preferences over the size and the composition of redistributive expenditures;
what is the long-run consequence of this interaction for growth and inequality.
In order to answer to these questions, this paper utilizes the framework in which
1Gradstein and Justman (1996) investigate the role of private education as an alternative
to public education. This model di¤ers from that of Gradstein and Justman (1996)
in that Gradstein and Justman (1996) consider public education as an only means ofredistribution, while this model allows for lump-sum transfer as an alternative to public
education. The presence of lump-sum transfer might give some agents an incentive toprefer lump-sum transfer to public education, and to make use of the transfer bene…ts forprivate education for their children. A model which includes this incentive enables us toanswer the abovementioned questions.
For the purpose of analysis, this paper develops a two-period overlapping-generation
model based on that by Gradstein and Justman (1996). There are two types of family
dynasties classi…ed according to their endowed levels of human capital: low and high.
An agent in each type of family enters adulthood with a stock of human capital invested
by his/her parents, earns after-tax labor income, and receives lump-sum transfer bene…ts
from the government. He/she decides the allocation of disposable income between current
consumption and private investment in his/her child’s further education. The private
educational investment combined with public education determines his/her child’s humancapital level.
Every adult agent votes over the tax rate as well as the allocation of tax revenue
between public education and lump-sum transfer. Given the bidimensional issue space,
the Nash equilibrium of a majority voting game may fail to exist. To deal with thisfeature, we use the concept of issue-by-issue voting; that is, notion of structure-induced
Nash equilibrium voting game formalized by Shepsle (1979) and applied by Conde-Ruizand Galasso (2003, 2005) for the framework of overlapping generations.
Voting may result in multiple political equilibria. One is the equilibrium where the tax
revenue is fully utilized for lump-sum transfer payments; no public education is provided.
Both high and low types of families privately invest in education to complement the lack ofpublic education. The other is the equilibrium where all the tax revenue is spent on public
education. Low-type families leave their children in the hands of public education andmake no private investments in their children. However, high-type families spend a part oftheir income on private educational investment to further improve their children’s humancapital level. The result suggests that the interaction between economic and politicaldecisions di¤er between the two equilibria.
The characterization of the political equilibria has the following two features. First, thetax revenues are exclusively used for either public education or lump-sum transfer. These
extreme usage patterns come from the simpli…ed assumption of the utility function and thehuman capital formation, both of which enable us to solve the model analytically. Second,
multiple political equilibria arise due to bidimensional voting. If we instead consider one
2dimensional voting as in Gradstein and Justman (1996), we …nd that multiple equilibria
do not arise.
Based on the characterization of the political equilibria, we undertake numerical analysis to demonstrate the long-term consequences of political equilibria in terms of growthand inequality. In particular, we focus on the case in which public education is lesse¢ cient than private education because the former provides standardized, rather than individualized, education to each child. In this situation, we …nd that an equilibrium which
exclusively uses tax revenue for public education attains a higher inequality level thanan equilibrium which exclusively uses tax revenue for lump-sum transfer. This result is
consistent with the empirical evidence in OECD countries: higher inequality is associatedwith a larger share of public education in redistributive spending (see Panel (a) of Figure1). The result also applies to the evidence demonstrated in Glomm (2004) where the setof countries is expanded to low and middle-income countries.
The evidence in OECD countries also suggests that higher inequality is associatedwith a lower growth rate (see Panel (b) of Figure 1). However, the result is not robust
as indicated by Persson and Tabellini (1994) and Alesina and Rodrik (1994). In fact, thepanel (b) of Figure 1 indicates a positive correlation between growth and inequality in somegroups of countries. The numerical analysis can demonstrate this mixed evidence in termsof growth and inequality by controlling the degree of e¢ ciency of public education. In
other words, the degree of e¢ ciency of public education is a key to explain the relationshipbetween growth and inequality.
The organization of this paper is as follows. In section 2, we set up the model andcharacterize economic equilibrium. Section 3 considers voting behavior of agents andcharacterizes political equilibria in each period. Section 4 demonstrates an equilibrium
path of the measure of inequality over time, and shows the existence and multiplicityof steady-state equilibria. Section 5 investigates how the e¢ ciency of public educationa¤ects the steady-state growth rates and inequality levels. Section 6 provides concluding
remarks.

2

The Model and Economic Equilibrium

We consider a discrete-time overlapping-generation economy that starts at time 0. The
economy is populated by two types of family dynasties, indexed by i 2 fL; Hg; of agents
whose lives consist of two periods, youth and adult age. A type-i adult agent in period
0 is endowed with hi0 units of human capital where 0 < hL0 < hH
0 . Thus, period-0 type-L
and type-H agents are endowed with low and high human capital, respectively.
Each adult agent produces one o¤spring, hence the population remains constant inevery generation. A fraction of type-i agents within each generation is given by i 2 (0; 1)
3

whereii

H

is constant across generations and satis…es 0 << 0:5 <

L

< 1 withPi= 1. The assumption implies that in every period, type-L agents are the majority inthe economy. This assumption re‡ects the right-skewed income distribution in the realworld.
A type-i adult agent at time t is endowed at the time he/she enters the adulthoodwith the stock of human capital hit which also de…nes his/her e¤ective labor capacity.
He/she receives lump-sum transfer from the government, bt . Given the income tax t andthe transfer bt , a type-i adult decides the allocation of disposable income between current
consumption, cit ; and private investment in his/her child’s further education, zti ; subject
to the budget constraint:
cit + zti

(1it )ht + bt :A type-i adult of generation t derives utility from current consumption, cit , and from
his/her child’s anticipated future income, hit+1 . A type i’s preferences are speci…ed by
the following utility function:uit = (1where) ln cit + ln hit+1 ;2 (0; 1) is a common parameter re‡ecting the bequest motive. A higher

impliesa greater incentive for educational investment. We employ logarithmic utility function for
the tractability of analysis.
The level of o¤spring’s education, hit+1 , is determined by public schooling, et ; as well
as by privately purchased supplementary education, zti . Following Gradstein and Justman(1996), we assume that the individual level of education is determined by the following
linear equation:
hit+1 = Ai

(1

) et + zti :The parameter Ai (> 0) represents a durable productive asset handed from generation
to generation such as genetic ability or cultural capital. The distribution of Ai is assumed
to be stationary over time and to be positively correlated to human capital, hit :AH = A > 0 and AL = A where2 (0; 1):This assumption implies that in average, children born in higher-income families areendowed with higher genetic ability or a higher level of cultural capital.1 The parameter2 [0; 1) implies that public education may be less e¢ cient than private education
1

A possible extension is to assume that children have the same genetic ability with a probability q.
For example, children born in higher-income families have high genetic ability, AH , with a probability q,
while they have low genetic ability, AL , with a probability 1 q. Bernasconi and Profeta (2012) assume
that this genetic probability of talent transmission, q, is not generally known in public, thereby resulting
in the talent mismatch. The current paper abstracts away the talent transmission and mismatch; instead,
it focuses on the interaction between public and private education choice.

4

because the former provides standardized, rather than individualized, education to eachchild. The role ofwill be further investigated in Section 5.

In each period, the government raises tax revenue to …nance the provision of uniform
public schooling for all children, et ; as well as lump-sum transfer, bt : The fraction t 2 [0; 1]of the tax revenue is devoted to lump-sum transfer; and the rest is devoted to publicschooling. Thus, the government budget constraint is given by:
bt =

t (1

et = (1t ) t ht ;t )(1t ) t ht ;where the term ht is the average human capital in period t; which is equivalent to theaggregate income in that period. The term (1t ) denotes the distortionary factor that

represents e¢ ciency loss of taxation. This assumption is solely to ensure an interior
solution to preferred tax rates and otherwise plays no role.
The timing of events in period t is as follows. First, adult agents vote on the tax ratet as well as the fraction of tax revenue devoted to lump-sum transfer

t by majority vote.

Second, each agent decides on the allocation of disposable income between consumption
and private education subject to the budget constraint. We solve the model by backward
induction.

2.1

Economic Equilibrium

Given a sequence of tax rates and the sizes of redistribution and public education,t=0;:::;1

i i
i
with
f t ; bt ; et g1
t=0 , an economic equilibrium is a sequence of allocations, fzt ; ct ; ht gi=L;H
the initial condition hi0 (i = L; H), such that (i) in every period, a type-i agent maxi-

mizes his/her utility subject to the budget constraint and the non-negativity constraint
of investment in private education and (ii) the government budget is balanced in every
period.
Solving the utility maximization problem of a type-i agent leads to the following
private education decision:zti = max 0;(1it )ht + bt

1

(1

)et

:

(1)

Eq. (1) indicates that the investment decision depends on an adult’s human capital hit
as well as government policy variables,t , bt and et .

In particular, an agent chooses toinvest privately in education if his/her human capital is high, the tax rate is low, the sizeof redistribution is large and/or the level of public education is low; otherwise, he/shechooses no private investment in education and consumes all of his/her disposable income.
Therefore, the consumption function is given by:5

cit = min (1it )ht + bt ; (1

)

(1

i
t )ht + bt + (1)et:The utility obtained by agents in economic equilibrium is represented by their indirect
utility functions. We use the abovementioned investment and consumption functions to
obtain an indirect utility function of a type-i agent:8 iiVln [(1)et ] + (1) ln(1) + ln>t )ht + bt + (1>< t;z>0
1
i
if (1
(1
)et ;
t )ht + bt >
Vti =
i
i
V
(1
)
ln
[(1
)h
+
b
]
+
ln
e
+
ln(1
)
>
t t
t
t
t;z=0
>
:
1
i
if (1
(1
)et :
t )ht + bt

idenotes the indirect utility of a type-i agent when he/she invests some portion of
Vt;z>0i
his/her income in private education, and Vt;z=0
denotes the indirect utility when he/she

invests nothing in education.
With the use of the government budget constraints bt =(1t )(1follows:t (1t ) t ht

and et =t ) t ht , the condition that determines investment decisions is rewritten as
1(1)
zti > 0 , t >
1
1+
(1
)

1

1
t

hit
ht

1

1
(1

)

!

:

(2)

This condition states that a lower tax rate and a higher share of redistribution in government expenditure produce a larger income e¤ect, thereby giving an agent an incentive toinvest in education.
With the condition (2) in mind, we can write the indirect utility function in terms ofthe tax rate t and the fraction t as follows:
8 iiVt;z>0 ln(1) + tg tt ) + ln ht =ht + f(1
>
>
>
>
+
ln
h
+
(1
)
ln(1
)
+
ln
t
>
>
1
>
(1 )
hit
>
1
<
if t > 1+ 1 (1 ) 1
;
1
(1 )
t ht
i
Vt =
i
ln(1
) ln hit =ht + t t + ln (1Vt;z=0>t ) + (1t ) + ln t
>
>
>
+
ln
h
+
ln(1
)
>
t
>
>
1
>
(1 )
hit
1
:
if t 1+ 1 (1 ) 1
:
1
(1 )
t ht

3

(3)

Period t Political Equilibrium

In each period t, the tax ratet and the proportiont are determined by period-t adultagents through a political process of majority voting. Type-L and type-H adult agents
cast a ballot over t ; the income tax rate, and t , the share of lump-sum transfer ingovernment expenditure. Individual preferences over the two issues are represented by
the indirect utility function in (3) for i = L; H. Every agent has zero mass and thus noindividual vote can change the outcome of the election. Thus, we assume agents votesincerely.
6The current majority voting game is characterized by a bidimensional issue space,and . Thus, a Nash equilibrium may not exist within the majority voting game. To
deal with this characteristic, we use the concept of issue-by-issue voting, or the structureinduced Nash equilibrium, as formalized by Shepsle (1979) and applied by Conde-Ruizand Galasso (2003, 2005) to the framework of overlapping generations. Under the conceptof the structure-induced Nash equilibrium, a su¢ cient condition for ( t ;t ) to be a period-t political equilibrium of the voting game is that t represents the outcome of majority
voting over t when the other dimension is …xed at its level t ; and vice versa, provided
that preferences are single peaked along every dimension of the issue space.
Under the current framework, type-L agents are the majority for each issue; and thepreferences of type-L agents, speci…ed in Eq. (3), are singled-peaked for each issue. We
can apply the concept of the structure-induced Nash equilibrium to the current framework.
Let

L
t (

L
t (

) denote type-L’s most preferred tax rate as a function of) denote type-L’s most preferred share as a function of the tax ratet.

t , and let

The point wherethese two reaction functions cross corresponds to the structure-induced Nash equilibriumoutcome of the voting game.
In the following analysis, we focus on the low-to-mean ratio of income as the measureof inequality:thLt =ht 2 (0; 1):

This captures the extent of income inequality in the economy; a higher implies lessinequality. Given the assumption that type-L agents are the majority for each issue, weinvestigate two sorts of equilibria, an equilibrium with ztL > 0 and an equilibrium with
ztL = 0, respectively.

3.1

ztL > 0 equilibrium

Suppose that the type-L agent invests a part of his/her income in education. The conditionof ztL > 0 in (2) is rewritten in terms of t :
!
1
(1
)
1
t
ztL > 0 , t >
1
:
(4)
1
1
1+
(1
)
(1
)
t
Under the condition (4), the type-L agent, as a decisive voter, choosesLhis/her indirect utility Vt;z>0.
L
The …rst derivative of Vt;z>0
with respect toL@Vt;z>0
1
=
+
@ t
1
t

t to maximize

t is:

(1
t + f(1

)+
)+

t
tg

:
t

The …rst term on the right-hand side shows the marginal cost of taxation; the second termshows the marginal bene…t of taxation. The above equation indicates that the marginal7

cost is independent of the share of redistribution in government expenditure, t , while
the marginal bene…t is increasing in t . Therefore, there is a critical value of t , andtype L agents will …nd it optimal to owe no tax burden whenvalue, that is,
when

( t

t

(1

t is below this critical)) = , while they …nd it optimal to owe some tax burden)) = . The optimal choice of t by type-L agents is summarized as:(
if t 1 ( t (1
)) ;
t = 0
i
h
L
(5)
t = z>0 ( t ) ,11
t(1) if t >( t (1
)) :
t =
1 2 t

t > ( t(1

Next, consider the choice oft by typeL agents with ztL > 0. A marginal increase int results in an increase of redistribution by one unit whereas it results in a decrease inpublic education by 1
they always choose

units. The net bene…ts are positive: 1(1) > 0. Therefore,t = 1 :t =L
z>0 ( t ) = 18 t 2 [0; 1]:

The period t political equilibrium when ztL > 0 is the point where the two reaction
Lfunctions, t = z>0( t ) and t = Lz>0 ( t ), cross (see Figure 2). The period-t equilibrium
policy when ztL > 0 is calculated as:

( t;

t) =

1
(1
2

t ); 1:

We substitute this solution into the condition of ztL > 0 in (4) and …nd that the condition
folds for any
ztL holds.

HHt 2 [0; 1]. We also …nd that zt > 0 in the current equilibrium because zt >[Figure 2 about here.]
Proposition 1. There exists a period-t political equilibrium of the voting game with
ztL > 0 and ztH > 0 such that ( t ;

t) =

1
(1
2

t ); 1

8 t 2 [0; 1]:

Two remarks are in order. First, type L agents prefer a higher tax rate ast becomes

lower, that is, as their income becomes lower compared to the average income. Second, fort 2 [0; 1], there always exists a voting equilibrium where type L agents invest privately
in education and prefer no public education provision. In other words, they prefer redis-

tribution to public education regardless of whether they are endowed with high income ornot. However, there is also the equilibrium where type L agents do not invest privately
in education and prefer public education to redistribution when their income is below thecritical value, which will be demonstrated in the next subsection.

8

3.2

ztL = 0 equilibrium

Suppose that the type-L agent invests nothing in education: ztL = 0. The condition of
ztL = 0 in terms oft is given by:
1

ztL = 0 ,

(1
)
1
1+
(1
)

1
(1

t

!

:

(6)

Under the condition (6), the type-L agent, as a decisive voter, chooses

t to maximize

t

1

1

t

)

L
.
his/her indirect utility Vt;z=0
Lwith respect to
The …rst derivative of Vt;z=0

t is:

L
@Vt;z=0
1
=
+ (1
@ t
1
t

t

)

+

t+

:

t

t

The …rst term on the right-hand side shows the marginal cost of taxation; the second
term shows the marginal bene…t of taxation via redistribution; and the third terms shows
the marginal bene…t of taxation via public education.
L
Corner solutions, t = 0 and 1, are not optimal for type L agents because @Vt;z=0=@ tL@Vt;z=0=@ t+1 > 0 and=t =1L@Vt;z=0 =@ t = 0, which results in:

t =

t =0

1 < 0 hold. Thus, the optimal solution satis…esL
z=0 ( t ) ,1(1 + ) t 1
t =

1

1+

t

:

2 t

Next, consider the choice oft by the type L agents when they invest nothing inLLwith respect to t is:
education, zt = 0. The …rst derivative of Vt;z=0
L@Vt;z=0
= (1
@ t

t

)

t+

(7)

:

1

t t

t

The …rst term on the right-hand side shows the marginal bene…t from an increase inredistribution; and the second term shows the marginal cost from a decrease in spending
on public education.
When t > (1)(1)=(1 + ), the latter e¤ect overcomes the former one; type L

agents prefer no redistribution,

t = 0.

However, when

two opposing e¤ects are o¤set at some level ofLsatis…es @Vt;z=0=@ t = 0, i.e.,t = (1(1t)(1t 2 (0; 1). In this case, the optimal sharet = t . Therefore, the preferred share)type L agents is summarized ast =L
z=0 ( t )max 0; (1Figure 3 illustrates the reaction functions)=(1 + ), the

t

)

:

t
t =

9

L
z=0 ( t ) andt =Lz=0 ( t ).

t for

=

[Figure 3 about here.]
The crossing points of the two reaction functions may correspond to the period-t
structure-induced Nash equilibrium of the voting game. Figure 3 demonstrates that there
are two possible solutions: one characterized by no redistribution, and the other characterized by some redistribution. They are given by:
( t;

t) =

1+

;0

and

^t ; ^ twhere ^ t > 0:

The former solution ( =(1 + ); 0) satis…es the ztL = 0 condition if t (1 )(1 )=(1+ );the latter solution ^t ; ^ t does not satisfy the z L = 0 condition.2 Therefore, the solution
is limited to ( t ;t) = (The solution ( t ;low such that

t

(1

=(1 +

t) = (tL); 0) when zt = 0.

=(1 + ); 0) is feasible if the low-to-mean income ratio,)(1

t , is)=(1 + ). Because of a low ratio, the marginal bene…t ofredistribution is always greater than the marginal cost of redistribution as demonstrated
in the …rst term on the right-hand side in Eq. (7). Type L agents …nd it optimal toset= 0, that is, no redistribution, from the viewpoint of their utility maximization if
(1

t

)(1

)=(1 + ) holds.

The absence of income redistribution has a negative impact on private education. Dueto this e¤ect, type L agents cannot a¤ord to undertake private investment in education.
However, type H agents can still undertake private educational investment because they
are endowed with high income.3
The results established so far are summarized as in the following proposition.
Proposition 2. There exists a period-t political equilibrium of the voting game with
ztL = 0 and ztH > 0 such that ( t ;

t) =

1+

; 0 8 t 2 [0; (1)(1)=(1 + )] :The results established in Propositions 1 and 2 imply that the tax revenues are exclusively used for either public education or lump-sum transfer in the political equilibria.
The extreme usage patterns come from the simpli…ed assumption of the utility function
and the human capital formation. The results also imply that there are multiple period t
Suppose that the solution ^t ; ^ t satis…es the z L = 0 condition in (6). We substitute ^ t = (1)(1 + t = t ) 0, which is a contradiction.
t = t into (6) and obtain
3
The proof of ztH > 0 is as follows. We substitute ( t ; t ) = ( =(1 + ); 0) into the condition ztH > 0in (3) to obtain:
!
1
(1
)
1 ytH
1
0>
1
;
1 yt 1 (1
1 + 1 (1))
2that is,1
1+(1) yt < ytH :This condition holds for any yt and ytH (> yt ).

10political equilibria if

t 2 [0; (1)=(1 + )]. The multiplicity arises because of)(1

the bidimensional voting.
If instead we consider one dimensional voting, we …nd that the multiple period t
political equilibria never arise. To con…rm this, let us consider voting over the tax rate
provided that the share of redistribution

is exogenously given at a certain level. Eq. (5)indicates that the tax rate is uniquely determined via voting at t = 0 if t
it is uniquely determined at t = (1=2) f1))g if t <t= (t + (1
The two ranges of

of

t

(1

)(1

t do not overlap when)=(1 + ) whent +(1t + (1);).

t is …xed. However, they overlap for the range

t is chosen endogenously via voting.

Therefore,

bidimensional voting is a key to the multiple equilibria.

4

Dynamic Equilibrium

Given the characterization of the period t political equilibrium in the previous section,
we are now ready to consider the dynamic equilibrium which presents the motion of t
over time. Following the results in Propositions 1 and 2, we …rst derive separately the
dynamic paths when z L > 0 and z L = 0. After that, we combine two cases and derive
the condition for the existence of the steady-state equilibrium wheretime.

t is stationary overFirst, suppose that the type L agents privately invest in education: z L > 0. The
period t equilibrium policy is ( t ;Lt ) = ((1Ht )=2; 1); and both types of agents invest

privately in education, z > 0 and z > 0 (Proposition 1). Under this situation, thenext-period income levels of the both types of agents are:hLt+1 = A(1hH
t+1 = A(1t)t)

hLt + f t t + (1)(1hH
t + f t t + (1

)(1

tg

t)

tg

t)

ht ;
ht ;

respectively; and the mean income level is:
ht+1 =

H H
ht+1

L L
ht+1 +

=A

(1

t)

(1

)

L

ht + f t t + (1

A

(1

t)

)(1

hLt + f t t + (1The low-to-mean ratio of income in period t + 1,
t+1 =

1

( t)

3 t
1+ t

t)

(1

11

)

tg

ht

)(1

t)

t+1 , becomes:Lfort 2 [0; 1]:

tg

ht :

1

where the superscript “1” infunction( t ) implies the presence of private investment. The

1( ) is strictly increasing and strictly convex in

1

(0) =

3

(1

L

)

1

2 (0; 1) and(1) =t with:

1

(1

L

)

2 (0; 1):

Therefore, there always exists a steady-state equilibrium with z L > 0 and z H > 0; and it
is unique.
Alternatively, suppose that the type L agents make no private investment in education: z L = 0. The period t equilibrium policy is ( t ; t ) = ( =(1 + ); 0); and thetype H agents still invest privately in education (Proposition 2). Under this situation,
the next-period income levels of both types of agents are:hLt+1 = A (1

)

t

hH
t+1 = At)hH
t + (1

(1

(1

t)

ht ;
)

ht ;

t

respectively; and the mean income level is:
ht+1 =

L L
ht+1 += A (1
L

A

H H
ht+1
t)

(1

ht

L

t)

hLt :

(1

)

t+

+ (1

L

for

t 2

) (1

)

t

The low-to-mean income ratio when ztL = 0 is:
0

t+1 =

( t)

( L + (1L)

) + 11+

1+
1

L

t

0;

(1

)(1
1+

)

;

where the superscript 0 means the absence of private investment by type L agents. The
function0( ) is strictly increasing and strictly convex in
0

0

(1

(0) =

)(1
1+

)

=

( L + (1+ 11+

L)t with:2 (0; 1);

) + 11+L (1)

2 (0; 1):

Therefore, there exists a steady-state equilibrium with z L = 0 and z H > 0 if:
0

(1

)(1
1+

)

)(1
< (1 1+);that is, if:L<1(1 + )(1
(1
)(1

)
)(1

)

:

The results established so far are summarized in the following proposition.
12(8)

Proposition 3. The dynamic equilibrium of f t g satis…es:8hi< f 1 ( t ); 0 ( t )g for t 2 0; (1 )(1 ) ;
1+
i
t+1 =
(1
)(1
)
1
:
;1 :
( t)
for t 2
1+(i) If (8) holds, there exist multiple steady-state equilibria: one is featured by z L > 0;
and the other is featured by z L = 0: (ii) If (8) fails to hold, there exists a unique,locally stable steady-state equilibrium with z L > 0.

The dynamic equilibrium is una¤ected by the parameter representing the ine¢ ciencyof public education as long as z L > 0, while it is a¤ected by if z L = 0. The di¤erencein voting. When z L > 0, the equilibrium

comes from the di¤erence in the choice ofvalue ofbecomes one (Proposition 1): all the tax revenue is spent on redistribution,and thus there is no provision of public education. The e¤ect of via public education
disappears in the equilibrium with z L > 0. However, when z L = 0, the e¤ect of remains
valid because all the tax revenue is spent on public education.
In order to observe the e¤ect ofon the dynamic equilibrium, we illustrate three

numerical examples of the law of motion of t in Figure 4 : the cases of = 0 (panel(a)); = 0:2 (panel (b)); and = 0:4 (panel(c)). In producing the …gure, the values of, , A andLare set to be 0:7, 0:3, 4:5 and 0:7, respectively. From the …gure, we can…nd that a higher , which implies a lower e¢ ciency of public education, leads to a lower
low-to-mean ratio and thus a higher inequality level in the economy with z L = 0. This
point will be further investigated in the next section.
[Figure 4 about here.]
In closing this section, we brie‡y consider the dynamic motion oft by utilizing Figure

4. All the three panels demonstrate two steady-state equilibria. However, the propertiesof their stability in panel (a) di¤er from those in panels (b) and (c). In panel (a), theeconomy has no locally stable steady-state equilibrium: the economy may move back andforth between the two states, z L > 0 and z L = 0. In panels (b) and (c), such movement
still arises when the initial level of is below the critical value. However, when the initiallevel is above the critical value, the economy displays a monotone convergence toward thesteady-state equilibrium with z L > 0. Therefore, the (in)e¢ ciency of public education
critically a¤ects the dynamic equilibrium in the economy.

5

Growth and Inequality in Steady States

This section analyzes the e¤ects of the ine¢ ciency of public education, represented by
the parameter on growth and inequality. In particular, we focus on the growth rate13

of average income, ht+1 =ht and the low-to-mean income ratio, t . Because the model
may attain multiple values of t+1 for a given t , it is generally unable to demonstrate theevolution of growth and inequality over time without imposing any additional assumptions
that select an equilibrium. Given this limitation, we here focus our attention on thetwo steady-state equilibria and discuss the empirical plausibility of the multiple politicalequilibria.
Let

0

1

and

The growth rates of the economy at
ht+1
ht
ht+1
ht

=A
= 0

=
= 1

when z L = 0 and z L > 0, respectively.

denote the steady-state level of

L

1+

A
(1 +
4

1

)

(1

)

3

(1

t =

1
1+

0

1

and

are computed as:L+ 1 + (1
) L

1

L

)(1

)

1 + (1) L

respectively. Similar to the dynamic equilibrium of

1+

A

0

1+

;

;

t , the steady-state growth rate isLa¤ected by if and only if there is no private education, z = 0.
In order to keep the result shown below comparable to the result in the previous
section, we set the values of , , and A to be 0:7, 0:3 and 4:5, respectively; and we consider
the three cases:= 0; 0:2 and 0:4. Because of the assumption of the low-majority, wetake the value of L from 0:5 to 0:99. Figure 5 demonstrates the low-to-mean ratios ofincome (panel (a)) and the growth rates (panel (b)) for each L . The numerical result
indicates that the two similar economies may attain di¤erent education and redistribution
policies, and experience di¤erent growth and inequality.
[Figure 5 about here.]First, consider the case of= 0: there is no e¢ ciency di¤erence between private

and public education. The equilibrium with z L = 0 attains a higher growth rate and ahigher equality level than the equilibrium with z L > 0. The equilibrium with the absenceof private education, z L = 0, is superior to the equilibrium with the presence of private
education, z L > 0, in terms of growth and equality.
If the e¢ ciency of public education is 20 percent points lower than the private education, that is, if= 0:2, then the z L = 0 equilibrium attains a higher growth rate but

a lower equality than the equilibrium with z L > 0: there is a trade-o¤ between growthand equality. In addition, the z L = 0 equilibrium is characterized by a higher inequality
and a higher share of public education in redistributive spending. The numerical result
predicts a positive correlation between inequality and the share of public education inredistributive expenditure. This prediction is consistent with the empirical evidence inOECD countries as demonstrated in Panel (a) of Figure 1.
14

Finally, when the e¢ ciency of public education is su¢ ciently low such that

= 0:4,

the z L = 0 equilibrium realizes a lower growth rate and a higher inequality level thanthe equilibrium with z L > 0. The result still predicts the positive correlation between
inequality and public education expenditure. However, the z L = 0 equilibrium attains alower growth rate and a higher inequality level than the z L > 0 equilibrium: this resultis opposite to that in the case of= 0:2. The growth-inequality relationship is criticallya¤ected by the e¢ ciency of public education; and the di¤erent results as regards growthand inequality might give one possible explanation for the mixed evidence of growth andinequality in OECD countries as illustrated in Introduction.

6

Conclusion

How the politics of public education and lump-sum transfer a¤ect economic decisions overprivate education of agents; how the decisions in turn a¤ect the inequality among agentsand their preferences over the size and the composition of redistributive policies; whatis the long-run consequence of this interaction for growth and inequality. In order toanswer these questions, this paper developed an overlapping-generation model with private educational investments which are supplemented by public education and …nanced
by lump-sum transfer. The tax on income as well as the allocation of tax revenue between public education and lump-sum transfer are determined by majority voting in every
period.
Under this framework, we obtain the following results. First, there might be multiple
political equilibria: one is the equilibrium featured by the exclusive use of tax revenue forpublic education; the other is the equilibrium featured by the exclusive use of tax revenue
for lump-sum transfer. The multiplicity of equilibria comes from multidimensional voting.
Second, the inequality level in the former equilibrium is higher than the latter equilibrium when the e¢ ciency of public education is lower than the private education to someextent. The result predicts the positive correlation between inequality and the share of
public education in government transfer expenditure, which is consistent with the empirical evidence in OECD countries.
Third, the relationship between growth and inequality depends on the e¢ ciency of
public education. When the degree of e¢ ciency is above a certain level, the numericalanalysis predicts a positive correlation between growth and inequality. However, it predicts a negative correlation when the degree of e¢ ciency is below a certain level. The two
opposing results might provide one possible explanation for the mixed evidence of growthand inequality in OECD countries.

15

References[1] Alesina, A., and Rodrik, D., 1994. Distributive politics and economic growth. Quarterly
Journal of Economics 465–490.

[2] Benabou, R., 1996. Inequality and growth, NBER Chapters, in: NBER Macroeconomics
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[3] Bernasconi, M., and Profeta, P., 2012. Public education and redistribution when talents
are mismatched. European Economic Review 56, 84–96.

[4] Bertola, G.,1993. Market structure and income distribution in endogenous growth models.
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[5] Conde-Ruiz, J.I., and Galasso, V., 2003. Early Retirement. Review of Economic Dynamics
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[10] Glomm, G., and Ravikumar, B., 1996. Endogenous public policy and multiple equilibria.
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[11] Glomm, G., and Ravikumar, B., 2001. Human capital accumulation and endogenous public
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16

Panel (a) of Figure 1

ratio of public education expenditure

0.90

KOR

MEX

0.85
0.80
0.75

USA

0.70

ISL

JPN
CAN

0.65

ITA

0.60
0.55

SWE

0.50

DNK

NOR
CZE
FRA
GBR
NZL
DEU IRL
AUT
SVK
FIN
AUS
HUN ESP
CHE
NLD
BEL

POL

0.270

0.370

PRT

0.45
0.40
0.220

0.320

0.420

0.470

Gini

Panel (b) of Figure 1
12

SVK

growht rates

10
8
6
4
2
0
0.200

CHE
ISL
FIN

POL
IRL
KOR

NLD AUSESPCZESWE AUT
BEL
NZL
GBR
NOR
JPN
FRADEUCAN
DNK
ITA
HUN
0.250

0.300

0.350
Gini

MEX
PRT
USA

0.400

0.450

0.500

Figure 1. Panel (a) illustrates the cross-country data on the mid-2000 Gini coefficient and the ratio of public
education expenditure per GDP in 2008 to the sum of the public education expenditure per GDP in 2008 andthe income support to the working age population per GDP in 2007: Panel (b) illustrates the cross-country
data on the mid-2000 Gini coefficient and growth rates in 2007. We compare data from different years
because of the limited availability.
Source: OECD (2008) for the Gini coefficient, OECD (2011a) for public education expenditure per GDP andgrowth rates, and OECD (2011b) for income support to the working age population per GDP.

17

Panel (a) of Figure 2

1

1

(

− (1 − ))

=

( )

=

( )

( )

=

1
2

0

Panel (b) of Figure 2

1

( )

=

1

0
(

1
2

− (1 − ))Figure 2. The reaction functions,=( ) and))/γ > 0; Panel (b) illustrates the case of (=( ). Panel (a) illustrates the case of (− (1 − ))/γ ≤ 0.

18

− (1 −

( )

=

( )

=

0
δ/(1 + δ)

Figure 3. The reaction functions,=( ) andδ /(1 − δ)

=

19

( ).

1
2

Panel (a) of Figure 4

Panel (b) of Figure 420

Panel (c) of Figure 4

Figure 4. The figure illustrates the law of motions of the low-to-mean ratio of income for the case of γ = 0.0(Panel (a)), = 0.2 (Panel (b)), and γ = 0.4 (Panel (c)), respectively. The solid curve indicates the law of
motion in the equilibrium with the presence of private educational investment by low-income type agents;
the dotted curve indicates the law of motion in the equilibrium with the absence of private educational
investment by low-income type agents.

21

Panel (a) of Figure 5

Panel (b) of Figure 5

Figure 5. Equality levels (Panel (a)) and growth rates (Panel (b)) in steady states.
22