Discussion Papers In Economics
And Business

Inequality and Education Choice
Tetsuo Ono and Yuki Uchida
Discussion Paper 16-17-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality and Education Choice
Tetsuo Ono and Yuki Uchida
Discussion Paper 16-17-Rev.

January 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality and Education Choice∗Tetsuo Ono† and Yuki Uchida‡
Osaka University

Abstract
This study presents a two-class, successive generations model with human capital accumulation and the choice to opt out of public education. The model demonstrates the mutual interaction between inequality and education choice and showsthat the interaction leads to two locally stable steady-state equilibria. The existence of multiple stable equilibria implies a negative correlation between inequality
and enrollment in public education, which is consistent with evidence from OECD
countries. This study also presents a welfare analysis using data from OECD countries and shows that introducing a compulsory public education system leaves thefirst generation worse oﬀ, though improves welfare for future generations of individuals in a lower class. The results also suggest that the two equilibria are not
Pareto-ranked.
•∗

JEL Classification Numbers: D70, H52, I24
Key words: Public education, opting out, inequality.

Tetsuo Ono acknowledges financial support from Grant-in-Aid for Scientific Research (C) from the
Ministry of Education, Culture, Sports, Science and Technology of Japan No. 15K03509. Yuki Uchidaacknowledges financial support from Grant-in-Aid for JSPS Fellows of the Japan Society for the Promotionof Science No. 15J01375.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: tono@econ.osaka-u.ac.jp‡
Corresponding author. Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: yuchida1988@gmail.com

1

Introduction

Compulsory school attendance laws, enforced in nearly all developed countries, requireparents to have their children attend public or private school for a designated period.
Public school is entirely funded by local and state taxes, whereas private schools obtainfunding by charging their students tuition. Parents can choose either option dependingon their income and preferences. Because public schooling is a kind of government intervention, higher income parents who benefit less from public schooling are more likely
to choose private schools. Therefore, we expect an association between higher inequality
and higher enrollments in private education institutions, as observed in data from OECD
countries illustrated in Figure 1.
[Figure 1 here.]de la Croix and Doepke (2009) develop a political economy theory that attempts to
explain the relationship between inequality and private education and show that their
theory is consistent with data from states in the U.S. and cross-country evidence. They
assume an exogenous income distribution and analyze the eﬀect of expanding inequality on
education choices; however, their analysis implies a reverse eﬀect, that is, that educationaﬀects inequality. Recent studies suggest that the reverse eﬀect is also important (SaintPaul and Verdier, 1993; Zhang, 1996) and exists across and within countries over time
(De Gregorio and Lee, 2002; Teulings and van Rens, 2008).
The presence of the reverse eﬀect implies that there is a mutual interaction of inequality
and education choice over time—inequality aﬀects adults’ choice of education and policy,
and this, in turn, determines inequality within the next generation. Cardak (2004a, 2004b)
attempts to demonstrate this mutual interaction of education choice and inequality in a
two-period overlapping-generations model. In particular, Cardak (2004a) calibrates themodel to the U.S. economy, and shows by simulation that the coexistence of public andprivate education polarizes the income distribution. He therefore focuses on education
choice and inequality within a country over time.
The present study instead focuses on the cross-country diﬀerences and aims to clarify
the causes of the diﬀerences observed in Figure 1 from the political economy point ofview. For this purpose, we follow a simple two-class, successive generations model withhuman capital accumulation as in Gradstein and Justman (1996) and de la Croix andDoepke (2004). We extend their frameworks by introducing the choice to opt out ofpublic education as Cardak (2004a, 2004b) does. In particular, the model in this study
has two types of family dynasties classified according to their level of human capital (i.e.,
low-type and high-type). Agents from either type of family enter adulthood with a stockof human capital invested by their parents, earn after-tax income, and obtain utility from1

consumption and their children’s human capital. Agents compare the maximized utility
under each type of education, and choose the one with the highest value.
Every adult agent votes on public education expenditures in each period. This study
assumes that most families are of the low-type. We compute the low-type’s preferredpublic education expenditure and analyze the corresponding education choice by adultagents. We show that the low-type adults always choose public education because they pay
less than they receive from public education. However, the high-type’s decision depends
on income inequality. As inequality increases, the income discrepancy between the two
types increases, and so does the high-type’s tax burden. Therefore, high-type adults opt
out of public education when inequality is high, while they choose public education wheninequality is low.
The current education choice and expenditures influence human capital formation,
which, in turn, determines inequality in the next generation. We demonstrate this mutual interaction between inequality and education across generations and show that theinteraction leads to two, locally stable steady-state equilibria. One steady state showsa polarized income distribution with high-type agents opting out of public education, inline with Cardak (2004a). The other steady state has perfect equality and full enrollmentin public education. This study is novel in that it shows the existence of multiple stableequilibria that imply that higher inequality is associated with lower enrollment in publicschools. This model prediction is consistent with the evidence presented in Figure 1.
To investigate the welfare implications of the model, we compare the two steady statesin terms of utility by considering an economic environment in which the equilibrium converges to the higher-inequality steady state. We then introduce an alternative education
system into this environment, that is, a compulsory public school system that prohibits
students from opting out of public school. This system forces the economy into thelower-inequality steady state. Therefore, we can evaluate the multiple, stable steadystate equilibria by comparing the higher-inequality steady state in the mixed education
system and the steady state in the compulsory public school system.
We show by simulation that every generation of the high-type is made worse oﬀ by the
introduction of compulsory public schooling, since expenditures on education depart fromtheir optimal levels. However, the new system has a mixed eﬀect on the low-type agents.
The first generation is made worse oﬀ since the per-capita public education expenditure
decreases. From generation 2 onward, there is a positive eﬀect from the compulsory publicschooling on human capital formation. This eﬀect may outweigh the negative eﬀect ofthe decrease in the per capita public education expenditure. The result suggests anintergenerational tradeoﬀ and that the two equilibria are not Pareto-ranked. The result
also suggests that the shift from a mixed education system to a compulsory public school2system that aims to improve equality, is not Pareto-improving.
This study is related to three strands of literature. The first is the static analysesof public and private education choices (e.g., Stiglitz, 1974; Epple and Romano, 1996;
Glomm and Ravikumar, 1998; Hoyt and Lee, 1998; Bearse, Glomm and Patterson, 2005;de la Croix and Doepke, 2009; Arcalean and Schiopu, 2015). This study advances thisearlier work by demonstrating the dynamic interaction of inequality and education choice,and in particular, complements Cardak’s (2004a, 2004b) work by showing the existenceof multiple stable equilibria that fit the cross-country evidence from OECD countries.
The second is the dynamic inequality analyses in given (public or private) education
regimes (e.g., Glomm and Ravikumar, 1992; Saint-Paul and Verdier, 1993; Gradsteinand Justman, 1997; Benabou, 2000; de la Croix and Doepke, 2004; Galor, Moav, andVollrath, 2009). However, this study departs from prior work by allowing for endogenous
education choice accompanied by voting on education policy. Gradstein and Justman(1996) and Ono (2016) conduct a similar analysis, though focus on private education asa supplement to public education. The present study instead focuses on the ability to
opt out of public education, which leads to novel implications for the multiplicity andeﬃciency of equilibria.
The third strand relates to political economy analyses of redistribution and private education (Hassler, Rodriguez Mora, Storesletten, and Zilibotti, 2003; Hassler, Storesletten,
and Zilibotti, 2007; Arawatari and Ono, 2009, 2013). In earlier frameworks, multiple, selffulfilling expectations of agents on future in-cash redistribution policies create two typesof equilibria: one characterized by low inequality and high redistribution, the other characterized by high inequality and low redistribution. This multiple-equilibria story implies
a negative correlation between inequality and redistribution. While this is relevant to ourpresent study, these earlier works consider private education and in-cash transfers, while
our study instead focuses on in-kind public education provision and allows for private
education as an alternative choice.
The remainder of this paper is organized as follows. Section 2 describes the model.
Section 3 considers agents’ voting behavior and describes the political equilibrium ineach period. Section 4 shows the existence and stability of a steady-state equilibrium
and clarifies the role of structural parameter values in the determination of inequality,
individual education choice, and policy. Section 5 presents a welfare analysis of thepolitical equilibria in addition to considering the welfare implications of a compulsory
public schooling system as an alternative. Section 6 provides concluding remarks. All
proofs are provided in Appendix.

3

2

Model

We consider a discrete-time, successive generations economy beginning at time 1. The
economy is populated by individuals who live in two periods (youth and adulthood) andbelong to one of two types of family dynasties indexed by i ∈ {L, H}. This assumption
simplifies the real economy, but it enables us to demonstrate the dynamic motion of
inequality in a tractable way.
A type-i adult in period 1 is endowed with hi1 units of human capital, where 0 < hL1 <hH
1 . Thus, type-L and type-H individuals in period-1 have low and high human capital,
respectively. As demonstrated below, members of the type-H endogenously choose moreeducation for their children than members of type-L, so there is always inequality within
the two types. However, the extent of this inequality is determined endogenously throughindividuals’ choices.
Each adult produces one child; thus, the population remains constant from generation
to generation. The fraction of type-L individuals within each generation is ϕ, leaving
1 − ϕ as the proportion of type-H individuals, where ϕ is constant across generations andsatisfies 0.5 < ϕ < 1. Therefore, type-L individuals are the majority in the economy inevery period, which reflects the real-world right-skewed income distribution.

2.1

Preferences and Budget Constraints

Upon entering adulthood at time t, a type-i individual has a stock of human capital hitthat defines his or her eﬀective labor capacity. He or she then inelastically supplies hisor her human capital to firms to receive wages. We assume that wages are normalized to
one in each period, implying that labor income is equal to the human capital level.
A type-i adult of generation t derives utility from his or her current consumption, cit ,and from his or her child’s anticipated future income, hit+1 . Consequently, we can express
the agent’s preferences with the following utility function
uit = ln cit + γ ln hit+1 ,where γ(> 0) is a common parameter that reflects the bequest motive. We employ thislogarithmic utility function to make our analysis more manageable.
Adults have a choice between public and private education for their children, which
they choose based on maximum utility. However, regardless of their choice, they must
pay income taxes to finance public education. Therefore, the budget constraint of a type-i
adult in period t iscit + eit ≤ (1 − τt )hit ,where eit (≥ 0) denotes type-i’s private education expenditure in period t, and τt is theperiod-t income tax rate.
4

Let qti ∈ {0, 1} denote a binary variable representing type-i’s education choice: qti = 0
when choosing private education, and qti = 1 when choosing public education. The child’slevel of education, hit+1 , is determined by his or her parents’ human capital, hit , and theparents’ choice of schooling, either xt or eit , where xt is per capita public education. In1−ηη
particular, we assume hit+1 = D (hit ) (qti xt + (1 − qti ) eit ) where D(> 0) is the totalfactor productivity of human capital and η ∈ (0, 1) is the elasticity of schooling. The
expression is reformulated as{1−ηη
D (hit ) (eit ) if qti = 0,iht+1 =1−ηD (hit ) (xt )η if qti = 1.
We assume the following with respect to γ and η.
Assumption 1. γη ∈ (0, 1).
Assumption 1 is satisfied as long as γ ∈ (0, 1). In Section 5, we estimate γ based ondata from OECD countries and find that γ = 0.151. This estimate fits well with Cardak’s(2004a) estimate of 0.13 and de la Croix and Deopke’s (2004) estimate of 0.169.

2.2

Education Choice

Given the tax rate, public education, and his or her human capital, each adult chooses
consumption and education to maximize his or her utility subject to the budget constraint.
In particular, he or she compares the maximum utility of each education choice andchooses the option with the highest value.
Suppose that a type-i adult chooses private education, qti = 0. He or she solves theutility maximization problem by allocating disposable income between private educationand consumption as in the following:γη
(1 − τt )hit ,1 + γη
1cit =(1 − τt )hit .
1 + γη

eit =The type-i adult’s utility from providing private education for his or her child, denotedi, isby Ve,t( )1−η
i+ γη ln µ,(1)= (1 + γη) ln(1 − τt )hit + γ ln D hit
Ve,t
where

γη

µ≡

.
(1 + γη)(1+γη)/γη
Alternatively, suppose that the type-i adult chooses public education, qti = 1. He or
she chooses eit = 0, and thus consumes all disposable income. In this case, the type-ii, isadult’s utility from choosing public education for his or her child, denoted by Vx,t( )1−η
i+ γη ln xt .
(2)
= ln(1 − τt )hit + γ ln D hit
Vx,t5Given a set of policies, (xt , τt ), each adult chooses between education alternatives forhis or her child to maximize utility. Therefore, type-i adults choose public education if
and only if the following condition holds:iiVx,t> Ve,t⇔ µ(1 − τt )hit < xt .

(3)

We assume that each adult chooses private education when the two alternatives are indiﬀerent. Therefore, the type-i’s education choice is{ii
1 if Vx,t> Ve,t⇔ µ(1 − τt )hit < xt ,
i
(4)
qt =
i
i
0 if Vx,t≤ Ve,t⇔ µ(1 − τt )hit ≥ xt .
The timing of events in period t is as follows: First, adult agents vote on public
education, xt . Given the voting outcome, the tax rate τt is set to satisfy the government’s
budget constraint. Second, given xt and τt , each agent chooses either public or private
education to maximize his or her utility. In choosing private education, agents decides how
to divide their disposable income between consumption and private education subject totheir budget constraints. We follow the backward induction approach to solve this multistage game. In particular, we first solve the second-stage problem in Section 3, and thensolve the first-stage problem in Section 4.

3

Economic Equilibrium

We define an economic equilibrium in the present model as follows.
Definition 1. Given a sequence of public education expenditure, {xt }∞, an economic{t=1i i i }t=1,...,∞
i t=1,...,∞
equilibrium is a sequence of choices and allocations, {qt }i=L,H and ct , et , ht+1 i=L,H ;iand a sequence of taxes, {τt }∞
t=1 , with the initial condition, h1 , i = L, H, such thatin each period,(i) given hit , xt , and τt , a type-i agent chooses qti and the corresponding cit and eit tomaximize his or her utility;}{(ii) given hit , xt , and qti , τt is set to satisfy the government budget constraint, qtL ϕ + qtH (1 − ϕ) xt =τt ht ;
(iii) given hit , xt , and (qti , eit ), which satisfy conditions (i) and (ii), hit+1 is determined byη1−η
hit+1 = D (hit ) (qti xt + (1 − qti ) eit ) .
To find the period-t economic equilibrium solution, we introduce an inequality index
ρt : Let ht denote the average human capital in period t, ht ≡ ϕhLt + (1 − ϕ)hHt , and let ρtdenote the ratio of hLt to ht ,
ρt ≡

hLt
∈ (0, 1] .
ht
6The index ρt suggests that a larger (smaller) ρt implies a lower (higher) income inequality
between the high-type and low-type groups, and thus a more equal (unequal) society.
Using this inequality index and the definition for average human capital, we can rewrite
the ratio hH
t /ht ashH
1 − ϕρt
t
=
.
ht
1−ϕ
Therefore, we replace the two state variables, hLt and hHt with ht and ρt in the following
analysis.
Using the definition of ht and ρt , we can reformulate the condition in (4) and obtainthe corresponding pair of education choices in the economic equilibrium. First, suppose()
that both types of adults choose private education, qtL , qtH = (0, 0). Condition (ii) in
Definition 1 implies that the government budget constraint is reduced to τt = 0 because
no agent will choose public education. Substituting τt = 0 in (4) and rearranging theterms, we obtain
qtL = 0 if xt ≤ µhLt and qtH = 0 if xt ≤ µhH
t .
()
Therefore, we have qtL , qtH = (0, 0) if xt ≤ µhLt , or
( L H)
qt , qt = (0, 0) if xt ≤ x00
t ≡ µρt ht ,(5)

( L H)
where the superscript “00” of x00= (0, 0).
t implies qt , qt()
Second, suppose that only type-H adults choose public education, qtL , qtH = (0, 1).
The government budget constraint is τt = (1 − ϕ)xt /ht . Substituting this into (4), we
obtain)(µρt(1 − ϕ)xt· hLt =ht ,1−
ht
1 + µ (1 − ϕ)()
(1 − ϕ)xtµ (1 − ϕρt ) /(1 − ϕ)H
qt = 1 if µ · 1 −· hH< xt .
t =ht
1 + µ (1 − ϕρt )qtL = 0 if xt ≤ µ ·There is no xt that satisfies both conditions. Therefore, there is no economic equilibrium()
with qtL , qtH = (0, 1).
)(
Third, suppose that only type-L adults choose public education, qtL , qtH = (1, 0). The
government budget constraint is ϕxt = τt ht . Substituting this into (4) and rearranging
the terms, we obtainµρt
ht < x t ,1 + µϕρtµ (1 − ϕρt ) /(1 − ϕ)qtH = 0 if xt ≤ x̄t ≡ht ,1 + µϕ (1 − ϕρt ) /(1 − ϕ)
( L H)= (1, 0), and the bar of x̄t indicates thewhere the superscript “10” of x10
t means qt , qt
qtL = 1 if x10
t ≡upper limit of xt , which induces the type-H adults to choose private education. Therefore,7we obtain( L H)
qt , qt = (1, 0) if x10
t < xt ≤ x̄t .

(6)( L H)
Finally, suppose that both types of adults choose public education, qt , qt = (1, 1).
The government budget constraint is then xt = τt ht . Following the same procedure above,
we obtain

(
)
xt
1−
· hLt < xt ,
ht
(
)
xt
H
qt = 1 if µ · 1 −· hHt < xt .
ht
qtL = 1 if µ ·()
Therefore, we have qtL , qtH = (1, 1) if µ · (1 − xt /ht ) · hHt < xt , or
( L H)
qt , qt = (1, 1) if x11
t ≡µ (1 − ϕρt ) /(1 − ϕ)ht < x t ,1 + µ (1 − ϕρt ) /(1 − ϕ)
( L H)= (1, 1).
where the superscript “11” of x11
t indicates qt , qt

(7)

The analysis thus far suggests that there are three possible cases of education choice:( L H)
qt , qt = (0, 0), (1, 0), and (1, 1). The choice is aﬀected by the four threshold values of
10
11
xt , denoted by x00
t , xt , x̄t , and xt . The order of these values depends on the inequality
ρt . In particular, there are three critical values of ρt , denoted by ρl , ρm , and ρh , where0 < ρl < ρm < ρh < 1, such that 00l
 xt ≶ x11
t ⇔ ρt ≶ ρ ,
m
x00
(8)
t ≶ x̄t ⇔ ρt ≶ ρ , 1011
xt ≶ xt ⇔ ρt ≶ ρh .
We give the proof of (8) in Appendix A.1. In addition, direct calculation leads to{00
x10
t < xt ,11x10
t , xt < x̄t ,(9)Figure 2 illustrates the four cases of ρt that classifies the ordering of the four thresholdvalues of xt and the corresponding choice of education by each type of adult. We summarize the four cases in Figure 2 in Figure 3, which is precisely stated in the following
proposition:[Figure 2 here.][Figure 3 here.]
Proposition 1. There is a unique period-t economic equilibrium if any of the following
00
11three conditions hold: (i) xt ≤ min {x10t , xt } , (ii) max {x̄t , xt } < xt , or (iii) ρt ∈( l)
110, ρ and xt ∈ [x00t , xt ]. Otherwise, there are multiple period-t economic equilibria.

8

The result in Proposition 1 suggests that the economic equilibrium is unique if thelevel of public education is low or high. If not, there are multiple equilibria or a uniqueequilibrium. In order to understand this result, let us first consider the case of a low x such
11
that xt ≤ min {x10t , xt }. Due to the low level of public education expenditure, the tax
rate could be reduced to satisfy the government budget constraint, regardless of education
choice. This implies a low tax burden, making private education more aﬀordable even for
the type-L adults. Therefore, there is a unique economic equilibrium where both types of11
adults choose private education if xt ≤ min {x10t , xt }.
Second, consider the case of a high x such that max {x̄t , x00t } < xt . The government isrequired to set a high tax rate to satisfy the government budget constraint. This createsa negative income eﬀect, which in turn makes private education less aﬀordable, even for
the type-H adults. Thus, there is a unique economic equilibrium where both types ofadults choose public education if max {x̄t , x00t } < xt .
00
11
Finally, for the intermediate case such that min {x10t , xt } < xt ≤ max {x̄t , xt }, theuniqueness or multiplicity of equilibria depend on the tax rate that satisfies the govern()
ment budget constraint. For example, consider a low inequality case such that ρt ∈ 0, ρlholds, as illustrated in Panel (a) of Figure 2, and focus on the public education expenditure level with xt ∈ (x11t , x̄t ] . If the tax rate is low enough that τt = ϕxt /ht , then thetype-H adults can aﬀord to invest privately in education, and this choice is consistent
with the condition of τt = ϕxt /ht . However, if the tax rate is high enough that τt = xt /ht ,then they find it optimal to choose public rather than private education from the view ofutility maximization. This choice is actually consistent with τt = xt /ht .

4

Political Equilibrium

Based on the characterization of the economic equilibrium in Section 3, in this section, we
demonstrate voting on education policy. We assume that adults vote sincerely since every
agent has zero mass and thus no individual vote can change the outcome. In addition,in each period t, adult agents determine public education through a political process ofmajority voting. Assuming ϕ > 0.5, type-L adults constitute the majority. Therefore, thepolitical objective function in period t, denoted by Ωt , is the indirect utility function of
adult type-L agents.
Definition 2. Given ρt , a period-t political equilibrium is a level of public education
expenditure, xt such that xt maximizes the type-L adults’ utility subject to eachtype’s education choice, as well as the corresponding consumption functions andgovernment budget constraints.

9

We write the period-t political objective function according to the pair of education()
choices, qtL , qtH demonstrated in (5), (6), and (7). Recall that the government budget
constraint is()}{
ϕxt if (qtL , qtH ) = (1, 0),τ t ht =xt if qtL , qtH = (1, 1),
and substitute this into the indirect utility function for the type-L adults. Then, thepolitical objective function becomes:1−ηL+ γη ln µ
if (qtL , qtH ) = (0, 0), Ω00,t ≡ Ve,t τt =0 = (1 + γη) ln ht ρt + γ ln D (ht ρt )1−ηL
Ω10,t ≡ Vx,t= ln (ht − ϕxt ) ρt + γ ln D (ht ρt )+ γη ln xt if (qtL , qtH ) = (1, 0),Ωt =τt =ϕxt /ht Ω ≡ VL= ln (h − x ) ρ + γ ln D (h ρ )1−η + γη ln x
if (q L , q H ) = (1, 1).
11,tx,t τt =xt /ht

t

t

t

t t

t

t

t

(10)
The functions Ω00,t , Ω10,t , Ω11,t have the following properties: Ω00,t is independent of
xt because both types opt out of public education, whereas Ω10,t and Ω11,t depend on xt
because either or both types choose public education. In particular, the solutions thatmaximize Ω10,t and Ω11,t are, respectively:{γη
arg max Ω10,t = x∗t ≡ ϕ(1+γη)
ht ,
γη
∗∗
arg max Ω11,t = xt ≡ 1+γη ht (< x∗t ) .
In addition, the following conditions hold:Ω11,t < Ω10,t ∀xt > 0,
00xt < x∗∗t = arg max Ω11,t ,Ω
10,t |xt =x00 < Ω00,t ,tΩ00,t < Ω11,t |xt =x∗∗ < Ω10,t |xt =x∗ ,
t

(11)

(12)

t

where the proof is given in Appendix A.3.
Given ht and ρt , the period-t political equilibrium solution is xt = arg max Ωt and thecorresponding education choices in (5), (6), and (7). The tax rate is set to satisfy thegovernment budget constraint. In the following, we consider two cases: a low-inequalitymstate, ρt ∈ [ρm , 1), where x̄t ≤ x00
t holds, and a high-inequality state, ρt ∈ (0, ρ ), wherex00
t < x̄t holds.

4.1

A Low-inequality State: ρt ∈ [ρm , 1)

The education choice when [ρm , 1) is depicted in Panels (c) and (d) of Figure 2. Figure 4 shows the corresponding political objective function. For illustration, we use theproperties of Ωt in (11) and (12).
[Figure 4 here.]10When xt is below (above) the critical value x00
t , the government expects that bothtypes of adults will choose private (public) education, and the adults actually make thatchoice. We should note that when xt is below x00
t , there are at most three economic
equilibria. However, the government finds it optimal to expect that both types of adults
choose private education because the choice attains the highest utility for xt ≤ x00
t . Thus,
the political objective function is{Ω00,t if 0 < xt ≤ x00
t ,
Ωt =
<
x
<
ht .
Ω11,t if x00
tt> Ω00,t holds,
The solution that maximizes Ωt is x∗∗
t = arg max Ω11,t because Ω11,t |xt =x∗∗t
as shown in (12).
Lemma 1. For ρt ∈ [ρm , 1) , the period-t the voting solution is x∗∗
t = arg max Ω11,t .
Type-L adults pay less than they receive from public education and thus prefer public
education over private education. As decisive voters, they choose per capita public education expenditures given their expectations of type-H voters’ choices. Type-H adults
may prefer private education to public education because they pay more than they receive from public education. However, their costs to provide public education in termsof utility decrease as ρt increases; that is, as their income level relative to the average,mhH
t /ht = (1 − ϕρt ) / (1 − ϕ), decreases. In particular, if ρt is above ρ , the benefits interms of utility outweigh the costs of public education to type-H, and type-H adults findit optimal to choose public education. Therefore, when inequality is low enough thatρm ≤ ρt < 1, it is optimal for type-L adults to choose a per capita public education
expenditure of x∗∗
t = arg max Ω11,t , given the expectation that type-H adults also choose
public education.

4.2

A High-inequality State: ρt ∈ (0, ρm )Panels (a) and (b) of Figure 2 show the education choice when ρt ∈ (0, ρm ). Figure 5
illustrates the corresponding political objective function. Because the government finds
it optimal to expect the education choice that attains the highest utility, the politicalobjective function when ρt ∈ (0, ρm ) is: Ω00,t if 0 < xt ≤ x00
t ,
<
x
≤
x̄t ,
Ω10,t if x00
Ωt =
t
t

Ω11,t if x̄t < xt < ht .
A main diﬀerence from the previous case is that the political objective might be maximized
at xt ∈ (x00
t , x̄t ], where type-H adults opt out of public education while type-L adults
do not. Either type-L adults could opt out, or neither could opt out depending on theinequality.
11

[Figure 5 here.]To find a political equilibrium solution, consider the following two cases: x∗t ≤ x̄t as
illustrated in Panel (a) of Figure 5, and x∗t > x̄t as illustrated in Panel (b) of Figure
5. Consider first the case when x∗t ≤ x̄t ⇔ ρt ≤ ρ∗ ≡ (1 − γη (1 − ϕ) /µϕ) /ϕ. As thefigure shows, Ωt is maximized at x∗t = arg max Ω10,t . At this public education level, thetype-H adults opt out of public education while the type-L do not. This case arises asa political equilibrium outcome when inequality is high enough that ρt ∈ (0, ρ∗ ). Thisset is non-empty if and only if ϕ > γη/ (µ + γη). Therefore, there is a period-t politicalequilibrium with (qtL , qtH ) = (1, 0) and xt = x∗t if ρt ∈ (0, ρm ) and ϕ ∈ (γη/ (µ + γη) , 1).
Next, consider the case when x∗t > x̄t ⇔ ρt > ρ∗ . As the figure shows, there are twocandidates for the period-t voting solution: one is xt = x̄t , where the type-H opt outof public education, and the other is xt = arg max Ω11,t = x∗∗t , where both types choose
public education. The type-L adults, as decisive voters, choose either to attain the highest
utility. Appendix A.4 shows that there is a critical value of ρt , denoted by ρ∗∗ ∈ (ρ∗ , ρm ),such that Ω10,t |xt =x̄t ≶ Ω11,t |xt =x∗∗ ⇔ ρt ≷ ρ∗∗ . The following lemma summarizes thetresults thus far.
Lemma 2. Assume ρt ∈ (0, ρm ) . Given ρt and ht , the period-t voting solution is ∗∗ρt ∈ (ρ∗∗ , ρm() , xt = arg max Ω11,t if)γη
∗∗arg max Ωt =xt = arg max Ω10,t if ρt ∈ (0, ρ ] and ϕ ∈ µ+γη , 1 ,


x̄t
if
ρt ∈ (max (0, ρ∗ ) , ρ∗∗ ] .
()γη
Consider first a situation where ρt ∈ (0, ρ∗ ] and ϕ ∈ µ+γη, 1 hold: type-H adults areendowed with a suﬃciently high income level but they have a low share of the populationin their generation. They prefer private to public education due to its positive income
eﬀect. Given this type-H choice, Ω10,t is the type-L’s indirect utility function. As decisive
voters, they choose per capita public education expenditure xt that maximizes Ω10,t ,x∗t = arg max Ω10,t . As Panel (a) of Figure 5 shows, this choice is feasible if x∗t ≤ x̄t . In
the current situation, the condition x∗t ≤ x̄t actually holds because a low share of type-H
individuals in the generation is equivalent to a high share of those of type-L and thus
implies a low per-capita level of public education.
Next, consider a situation where ρt ∈ (max (0, ρ∗ ) , ρ∗∗ ] holds: inequality is high but lesssevere than that observed in the first case. The type-H adults still prefer private to public
education, but the type-L adults cannot choose an “interior” solution, x∗t = arg max Ω10,t .
Their choice is constrained by the upper limit, x̄t . We hereafter refer to x̄t as a “corner”
solution. Finally, if inequality is low enough that ρt ∈ (ρ∗∗ , ρm ) , the type-H adults choose
public education and the political objective is maximized at x∗∗
t = arg max Ω11,t .
12

We should note that the corner solution arises when the proportion of type-L adults
is low enough that ϕ ∈ (1/2, γη/ (µ + γη)]. A low ϕ implies a small tax burden foreach agent for a given level of public education expenditure x. This lowers the marginal
cost of public education, thereby inducing the type-L adults to prefer a higher public
education expenditure. However, type-H adults will opt out when the public education
expenditure is below x̄t . If the expenditure is above x̄t , type-H adults prefer public
education to private education. Therefore, the upper limit, x̄t , constraints the type-L’s
choice of public education as long as type-H adults opt out of public education.

4.3

Voting Outcome and Education Choice

Summarizing the results in Lemmas 1 and 2, we obtain the voting solution in period t
and the corresponding education choice.
Proposition 2. Given the inequality index ρt , the period-t voting solution is ∗∗ρt ∈ (ρ∗∗ , 1)(, xt if)γη
,1 ,arg max Ωt =x∗t if ρt ∈ (0, ρ∗ ] and ϕ ∈ µ+γηx̄t
otherwise.
The corresponding education choice is{( L H)(1, 1) if
q ,q =(1, 0) if

ρt ∈ (ρ∗∗ , 1) ,ρt ∈ (0, ρ∗∗ ] .

Proposition 2 states that type-H adults choose public education if inequality is low
enough that ρt ∈ (ρ∗∗ , 1); otherwise, they choose private education. A small ρt implies
a high income disparity between the two types of adults, so type-H adults owe a large
tax burden. In particular, if ρt ≤ ρ∗∗ , the negative income eﬀect dominates the positiveeﬀect of public education provision. This incentivizes type-H adults to opt out of public
education. However, if inequality is low enough that ρt > ρ∗∗ , the positive eﬀect of public
education provision dominates the negative income eﬀect, and type-H adults choose public
education.

5

Steady-state Equilibrium

The analysis in the previous section demonstrates public education expenditure as apolitical outcome for a given inequality index (ρt ). The current public education influences
human capital formation, which in turn determines inequality in the next generation(ρt+1 ). To consider the dynamic interaction between inequality and public education, we
demonstrate the inequality index movement across periods and the existence and stability13

of a steady-state equilibrium in which ρt+1 = ρt holds along the equilibrium path. Basedon the description of the equilibrium, we examine how the structural parameters ϕ, γ,and η aﬀect steady-state inequality and the choice of education.
Given an initial condition, ρ1 (> 0), the political equilibrium sequence {ρt } is characterized by the first-order diﬀerence equation, ρt+1 = P (ρt ), where[)1−η ]−1
(


1
η

if ρt ∈ (ρ∗∗ , 1) ,P11 (ρt ) ≡ ϕ + (1 − ϕ) ρt − ϕ




[

( )1−η ]−1
(
)
)η
(
γη
ϕ
P (ρt ) =if ρt ∈ (0, ρ∗ ] and ϕ ∈ µ+γη
,1 ,
P10 (ρt ) ≡ ϕ + 1+γη (1 − ϕρt ) ρ1t


[

(
)1−η ]−1



1/γ
1−ϕρt

otherwise.
 P̄10 (ρt ) ≡ ϕ + (1 − ϕ) (1 + γη)(1−ϕ)ρt()
where the subscripts “11” and “10” in P (·) imply q L , q H = (1, 1) and (1, 0), respectively. The three cases correspond to those in Proposition 1. Appendix A.5 provides thederivation of P11 (ρt ) , P10 (ρt ) and P̄10 (ρt ).
A closer analysis of P (·) reveals that the function has the following properties (seeAppendix A.6 for the formal proof of the following statement). First, P11 (·) , P10 (·) andP̄10 (·) are strictly increasing in ρt . Second, P10 (·) ⋛ P̄10 (·) if and only if ρt ⋛ ρ∗ . Third,′P̄10 (·) < P11 (·) ∀ρt ∈ (0, 1). Fourth, P11 (·) satisfies P11 (1) = 1 and P11(1) = 1 − η ∈()(0, 1). Fifth, P̄10 (·) satisfies P̄10 (0) = 0 and limρ→0 ∂ P̄10 (·) /∂ρt = ∞; P10 (·) satisfies
P10 (0) = 0, and limρ→0 (∂P10 (·) /∂ρt ) = ∞. These properties imply that (i) there is alocally stable steady-state equilibrium with ρ = 1, and (ii) P (·) is strictly increasing inρt but discontinuous at ρt = ρ∗∗ . Figure 6 illustrates the possible patterns of P (·) whenϕ ∈ (γη/ (µ + γη) , 1). The ϕ ∈ (1/2, γη/ (µ + γη)] case is qualitatively similar, but thethreshold value ρ∗ is negative and thus irrelevant. From the figure, we obtain the following
proposition:[Figure 6 here.]
Proposition 3. If P̄10 (ρ∗∗ ) > ρ∗∗ , there is a unique stable steady-state equilibrium with
ρ = 1 ; if P̄10 (ρ∗∗ ) ≤ ρ∗∗ , there are two locally stable steady-state equilibria, onewith ρ ∈ (0, ρ∗∗ ] and the other with ρ = 1.
The unique stable steady-state equilibrium is distinguished by perfect equality between
the two types of agents and 100% enrollment in public school. However, another typeof equilibrium exists when the multiple stable steady states are realized, distinguishedby the presence of income inequality and type-H agents’ opting out of public education.
Thus, the multiple steady states imply that higher inequality is associated with lower
enrollment in public school. This model prediction is consistent with empirical evidence
from OECD countries.
14

For an intuitive interpretation of the condition P̄10 (ρ∗∗ ) ≷ ρ∗∗ , we reformulate it asP̄10 (ρ∗∗ ) ≷ ρ∗∗ ⇔1 − ϕρ∗∗ 1· ∗∗ ≷ (1 + γη)1/γη ,
1−ϕ
ρ

(13)

where ρ∗∗ , defined in Subsection 4.2, satisfies Ω10,t |xt =x̄t = Ω11,t |xt =x∗∗ , ort(

1 − ϕρ∗∗1−ϕ)γη
= 1 + µϕ ·γη1 − ϕρ∗∗, µ≡.
1−ϕ(1 + γη)(1+γη)/γη(14)Eq. (14) indicates that ρ∗∗ is a function of ϕ and γη, ρ∗∗ = ρ∗∗ (ϕ, γη). Thus, we canillustrate the condition in (13) in a ϕ − γη space, as Figure 7 shows. The figure suggeststhat the model is more likely to produce multiple steady-state equilibria if ϕ and γη arelower.
[Figure 7 here.]
To explain this argument, we first consider the eﬀect of γη. A low γ means a low weightattached to the utility of children’s human capital, and a low η means low elasticity inhuman capital with respect to public education expenditure. These factors imply that
type-H agents determine a low benefit from public education in terms of utility. This inturn induces type-H agents to opt out of public education. Therefore, a low γη encourages
an equilibrium with ρ < 1.
Next, we consider the role of ϕ in the steady-state equilibria outcome. Recall thedefinition of ρt+1 ≡ hLt+1 /ht+1 , or,ρt+1 ≡

hLt+1
hLt+1
=
.
ht+1
ϕhLt+1 + (1 − ϕ)hH
t+1This expression indicates that the parameter ϕ has two eﬀects in the determination ofρt+1 . First, given hH
t+1 , a lower ϕ implies a larger proportion of type-H agents. This leads
to a higher average human capital, ht+1 , and thus a lower ρt+1 ≡ hLt+1 /ht+1 for a given
hLt+1 . Second, a lower ϕ implies a lower aggregate public education expenditure and thus
a lower tax burden on type-H agents. This produces a positive income eﬀect on private
education expenditure by type-H agents. This, in turn, increases type-H’s human capital,hH
t+1 , and the average human capital, ht+1 , and thus decreases ρt+1 . Because of these two
negative eﬀects, the model produces an equilibrium with ρt+1 < 1 if ϕ is low.
However, an economy with low γη and ϕ also has an equilibrium with ρ = 1 provided
that the initial condition of ρ is high. A higher ρ implies a lower income gap and thus
lower income for type-H agents. Because of this negative income eﬀect, type-H agentsfind it optimal to choose public education over private education. Therefore, there is also
an equilibrium with ρ = 1 for low values of γη and ϕ.
15

Thus far, we assume that human capital productivity, represented by D, is common
between the two types of agents. However, D may represent a durable productive asset
such as generic ability, technology transfer, or business succession that children inheritfrom parents. Based on this view, we can alternatively assume that the distribution of
D is positively correlated with human capital: DH > DL , where Di (i = H, L) is typei’s human capital productivity (Gradstein and Justman, 1996). This assumption implies
that, on average, children born to higher-income families are endowed with greater humancapital productivity (Behrman and Taubman, 1989).
Under this alternative assumption, the law of motion of human capital when (q L , q H ) =(x, x) is reformulated as:[DHρt+1 = ϕ + L (1 − ϕ)η
D

(

1
−ϕ
ρt

)1−η ]−1.

This equation implies a stable steady-state equilibrium with ρ < 1, which seems more
realistic than the equilibrium with ρ = 1, which assumes DH = DL = D. However, thequalitative results remain unchanged. Therefore, for the tractability of analysis, we keep
the assumption of DH = DL = D in the following analysis.

6

Welfare Analysis

We use simulations to investigate the model’s welfare implications. In the analysis, we
set parameters ϕ, η, γ, and D as in the following. First, recall that 1 − ϕ is the fractionof type-H agents and only type-H agents opt out of public schooling. The fraction 1 − ϕtherefore represents enrollment in private institutions as a percentage. We set 1 − ϕ
at 0.147 because the average rate in OECD countries was 14.7% in 2011. Second, theestimate in Card and Krueger (1992) implies an elasticity of school quality of 0.12. In
addition, recent simulation studies suggest that η is in the range of 0.1 − 0.3 (Cardak,2004) and 0.05 − 0.15 (Glomm and Ravikumar, 1998). Following these earlier results, wehere set η = 0.15.
For γ, we focus on the public education expenditure-GDP ratio in the steady-state
equilibrium distinguished by ρ = 1. The ratio in this equilibrium is x/h = γη/(1 + γη).
Given η = 0.15, we can estimate γ by using the average ratio x/h observed in selectedOECD countries. Table 1 lists 25 OECD countries and their Gini coeﬃcients (Panel (a))
and the percentage of enrollment in private secondary education (Panel (b)) in 2011. We
chose the countries in the top 50% in Panel (a) and those in the top 50% in Panel (b)and define this set as those that attain the steady state with ρ = 1. These countries are
Slovenia, Norway, the Czech Republic, Finland, and the Netherlands. The average ratio16

in these countries was 2.22% in 20111 , so we can now determine γ by solving 0.022 =0.15γ/(1 + 0.15γ) for γ: γ ≃ 0.151. This estimate fits well with that of 0.13 by Cardak(2004) and that of 0.169 by de la Croix and Doepke (2004). Finally, we normalize D as
D = 1.
[Table 1 here.]

6.1

Utility Gap

We set the initial condition hL1 and hH
1 to attain an equilibrium path that converges tothe unequal steady state with ρ < 1. Figure 8 plots the utility gap between type-L
and type-H agents along this equilibrium path. The ratio of type-L’s utility to type-H’s
utility represents the gap. Given the logarithmic form of the utility function, both types’
utility functions take negative values in the following numerical analysis. Hence, a ratio
greater than one implies that type-L’s utility is lower than that of type-H. For example,
if V L = −1.2 and V H = −1.0, then the ratio is 1.2. A higher ratio implies a wider utility
gap.
[Figure 8 here.]
We look at the utility gap along the transition path and investigate how the utility gap
changes in response to the spread of the initial inequality by considering a mean-preserving
reduction of type-L’s initial human capital, where a reduction in hL1 is associated with anincrease in hH
1 , keeping the average h1 unchanged. Figure 8 plots the ratio from generation 1 to generation 80 for three cases of initial conditions. The solid curve illustrates
the baseline case and the dashed (dot-and-dash) curve illustrates a higher (lower) initialinequality case. The figure shows that the utility gap widens as the initial inequality
increases, but the diﬀerence between the three cases disappears in the long run.

6.2

Compulsory Public School System

To further investigate the welfare implications, we introduce a compulsory public school
system as an alternative education regime and compare it to the mixed education systemanalyzed thus far in terms of utility. In the present framework, the compulsory publicschool system prohibits students from opting out of public school. Due to the limited
choice of education, the system forces the economy into a steady state with ρ = 1. Thisis identical to the steady state with ρ = 1 in the mixed education system. Therefore,
we can evaluate the multiple stable steady-state equilibria in Proposition 2 in terms of1

Source: UNESCO statistics (http://data.uis.unesco.org/, February 14, 2016).

17

utility by comparing the equilibrium with ρ < 1 in the mixed education system and theequilibrium with ρ = 1 in the compulsory public school system.
For the analysis, we take the ratio of utility in the mixed education system to thatin the compulsory public school system and plot it from generation 1 to generation 80 in
Figure 9. Panels (a), (b), and (c) illustrate the ratios for type-L agents, type-H agents,and social welfare (the population-weighted average utility of the two types of agents),
respectively. As in Figure 8, the ratio of more (less) than 1 implies that the utility inthe compulsory public school system is higher (lower) than that in the mixed education
system.
[Figure 9 here.]
To interpret the result in Figure 9, we first note the indirect utility of type-L agents:() hLt( L )1−η
L
+
γ
ln
D
ht
+ γη ln xHVmix,t= ln ht − ϕxHtt ,ht( L )1−ηhLt
L
Vcomp,t= ln (ht − x∗∗
)
+
γ
ln
D
ht
+ γη ln x∗∗
t
t ,
ht
L
L
where Vmix,tand Vcomp,t
are the indirect utility in the mixed education system and thatin the compulsory public education system, respectively.
The expressions show that introducing compulsory public education has two opposing∗∗
eﬀects on type-L’s utility. First, the tax burden decreases from ϕxHt to xt . Second,∗∗
per capita public education expenditure decreases from xHt to xt . The numerical result
in Panel (a) shows that the latter negative eﬀect outweighs the former positive one, so

introducing the compulsory public education system makes type-L agents in generation
1 worse oﬀ.
From generation 2 onward, there is an additional positive eﬀect via the human capital formation generated by the compulsory public school system. The terms hLt /ht and( )1−η
D hLt
in the above expressions illustrate this eﬀect. This positive eﬀect increases as
the initial inequality decreases. In addition, this eﬀect amplifies the tax reduction eﬀect.
Therefore, for the baseline case and the low initial inequality case, introducing compulsory
public education makes generations from 2 onward better oﬀ. However, for the case ofhigh initial inequality, it takes a long time to realize this welfare improvement because
the negative eﬀect remains stronger as the initial inequality increases. Type-L agents’
welfare improves only from generation 70 onward.
Panel (b) plots the ratio of type-H from generation 1 to 80. In the current setting,they choose private education in the mixed education regime. Thus, their indirect utility

18

is:
(
) hH
( H )1−η
(
) H
t
H
H ht
Vmix,t= ln ht − ϕxH
+
γ
ln
D
h
+
γη
ln
h
−
ϕx
+ γη ln µ,
t
t
t
t
ht
ht
( )1−η
hHt
H
Vcomp,t= ln (ht − x∗∗)+ γ ln D hH
+ γη ln x∗∗
t
t
t ,
ht
H
H
where Vmix,tand Vcomp,t
are the indirect utility in the mixed education system and thatin the compulsory public education system, respectively.

As in the case with type-L, introducing compulsory public education has two opposing
eﬀects on type-H’s utility in the initial period: the tax burden decreases from ϕxH
t to
(
) H
∗∗
H
xt and the expenditure for human capital formation decreases from ht − ϕxt ht /ht
to x∗∗
t . The numerical result in Panel (b) shows that the latter negative eﬀect outweighs
the former positive one, so the change makes type-H agents in generation 1 worse oﬀ.
From generation 2 onward, these agents are also worse oﬀ because there is an additional
negative eﬀect through the delay of human capital formation generated by the compulsory( H )1−ηpublic school system. The terms hH
in the above expressions illustrate
t /ht and D ht
this eﬀect. The numerical result suggests that the shift from the mixed education to thecompulsory public school system is not Pareto-improving.
Finally, we investigate the eﬀect of the compulsory public school system on social
welfare. The introduction decreases social welfare in period 1 because both types of
agents are worse oﬀ. However, the eﬀect on welfare from period 2 onward depends on theinitial inequality: welfare improves earlier as the initial inequality decreases, as illustrated
in Panel (c). The result suggests that the social welfare ranking of the multiple equilibria
depends on the initial inequality condition.

7

Conclusion

This study presents a political economy theory to explain why countries with higher inequality are associated with lower enrollment in public education. We base the theory on
a two-class (high and low), successive-generations model with human capital accumulation and the choice to opt out of public education accompanied by voting on education
policy. This condition creates multiple, locally stable steady-state equilibria: one with
low inequality and high enrolment in public education and the other with high inequality
and low enrolment in public education. This study is novel in that it shows the negative correlation observed in OECD countries in the mutual interaction of inequality andeducation.
From the equity viewpoint, it is desirable to attain the low-inequality steady state.
One path to this steady state involves introducing compulsory public schooling. We useda simulation to investigate the welfare implications of introducing this reform and find19

that it makes high-income families worse oﬀ, while improving the lot of future generationsof low-income families at the expense of the current generation. The results suggest thatthe multiple equilibria are not Pareto-ranked, and that the shift from the existing mixed
education system to a compulsory public school system is not Pareto-improving.
We demonstrated these results by making several assumptions that make the analysis
tractable. In particular, we assume two classes and that the low-type agents constitute themajority in every period. This assumption enables an analytical solution to the model andan illustration of the multiple locally stable steady-state equilibria observed in empiricalstudies. A future extension could include probabilistic voting, a la Lindbeck and Weibull(1987), to reflect the preferences of both types of agents. While this type of analysis isideal, the analysis is rather complicated and best left for future research.

20

A

Appendix

A.1

Proof of (8)1011Recall the definition of x00
t , xt , x̄t , and xt in the text. We compare these as follows:211x00
t ≷ xt ⇔ 0 ≷ f (ρt ) ≡ µϕ (ρt ) − (1 + µ) ρt + 1,
2
2x00
t ≷ x̄t ⇔ 0 ≷ g (ρt ) ≡ µϕ (ρt ) − (1 + µϕ) ρt + 1,
2
11x10
t ≷ xt ⇔ 0 ≷ h (ρt ) ≡ µϕ (1 − ϕ) (ρt ) − (1 + µ (1 − ϕ)) ρt + 1,

where (i) f (0) = g(0) = h(0) > 0, (ii) f (·) < g (·) < h (·) for any ρt ∈ (0, 1] , and (iii)f ′ (·) < 0, g ′ (·) < 0, and h′ (·) < 0 for any ρt ∈ (0, 1). As illustrated in Figure A.1, there
are three critical values of ρt , denoted by ρl , ρm , and ρh , where 0 < ρl < ρm < ρh < 1,( )( )such that f ρl = 0, g (ρm ) = 0, and h ρh = 0. From Figure A.1, we obtain (8).
[Figure A.1 here.]■

A.2

Proof of Proposition 1

(
)
11
00
Suppose that ρt ∈ 0, ρl . Figure A.1 shows that in this case, x00
t < xt , xt < x̄t , and
11
10
00
11
x10
t < xt holds. With the condition in (9), we obtain xt < xt < xt < x̄t as illustrated
in Panel (a) of Figure 2. The figure shows that there is a unique economic equilibrium if
00
11xt ∈ (0, x10t ), (xt , xt ] , or (x̄t , ht ); otherwise, there are multiple economic equilibria.
Following the same procedure, we can show the uniqueness or multiplicity of the
[
) [
)
[
)
economic equilibria for the remaining three cases, ρt ∈ ρl , ρm , ρm , ρh , and ρh , 1 .
There is a unique economic equilibrium if any of the following three conditions hold:[)(i) ρt ∈ ρl , ρm and xt ∈ (0, x10
t ] or (x̄t , ht ) as illustrated in Panel (b) of Figure 2; (ii)[ m h)
10ρt ∈ ρ , ρ and xt ∈ (0, xt ] or (x00
t , ht ) as illustrated in Panel (c) of Figure 2; and (iii)[ h )00
ρt ∈ ρ , 1 and xt ∈ (0, x11t ] or (xt , ht ) as illustrated in Panel (d) of Figure 2. Proposition

1 summarizes the results established thus far.
■

A.3

Proof of (12)The first condition, Ω11,t < Ω10,t ∀xt > 0, is immediate from the definition of Ω11,t and∗∗Ω10,t . We show the second condition, x00
t < xt , with a direct comparison:∗∗x00
t < xt ⇔ µρt ht <γη
ht ⇔ ρt < (1 + γη)1/γη ,1 + γη
21

which holds for any ρt < 1 and γη ∈ (0, 1) .
To show the third condition, Ω10,t |xt =x00 < Ω00,t , we compare Ω10,t |xt =x00 and Ω00,t ,ttand obtainΩ10,t |xt =x00 < Ω00,t ⇔ ln (1 − ϕµρt ) ρt ht + γη ln µρt ht < (1 + γη) ln ρt ht + γη ln µt

⇔ ln (1 − ϕµρt ) < 0.
The last inequality holds since ln (1 − ϕµρt ) < ln 1 = 0.
To show the fourth condition, we first compare Ω00,t and Ω11,t |xt =x∗∗ and obtaint(Ω00,t < Ω11,t |xt =x∗∗ ⇔ (1 + γη) ln ρt ht + γη ln µ < ln 1 −tγη
1 + γη

)ρt ht + γη lnγη
ht
1 + γη

⇔ γη ln ρt < 0,where the last inequality holds since ln ρt < ln 1 = 0. The inequality Ω11,t |xt =x∗∗ <Ω10,t |xt =x∗ is immediate since Ω11,t |xt =x∗∗ < Ω10,t |xt =x∗∗ < Ω10,t |xt =x∗ .
t

t

t

t

t

■

A.4

Proof of Lemma 2

The text provides the following statement:(arg max Ωt = x∗t if ρt ∈ (0, ρ∗ ]and ϕ ∈)γη,1 .
µ + γη

The remaining task is to show that there is ρ∗∗ ∈ (ρ∗ , ρm ) such that
Ω10,t |xt =x̄t ≶ Ω11,t |xt =x∗∗ ⇔ ρt ≷ ρ∗∗ ,t

whereΩ10,t |xt =x̄t = ln (ht − ϕx̄t ) ρt + γ ln D (ht ρt )1−η + γη ln x̄t ,1−η+ γη ln x∗∗Ω11,t |xt =x∗∗ = ln (ht − x∗∗t .
t ) ρt + γ ln D (ht ρt )
t

22

(15)

A direct comparison of Ω10,t |xt =x̄t and Ω11,t |xt =x∗∗ leads totΩ10,t |xt =x̄t ≶ Ω11,t |xt =x∗∗t(
⇔ ln ht − ϕ

tµ 1−ϕρ
1−ϕ)tµ 1−ϕρ
1−ϕht + γη lnh
1−ϕρt tt
1 + µϕ 1−ϕρ
1
+
µϕ
1−ϕ
1−ϕ
(
)
γη
γη
≷ ln ht −ht + γη ln
ht
1 + γη
1 + γη
(
)γη
1−ϕρt
µ 1−ϕ
(γη)γη
⇔ ln ()1+γη ≷ ln
(1 + γη)γηt
1 + µϕ 1−ϕρ
1−ϕ[]γη/(1+γη)
1 − ϕρt
1 − ϕρt⇔≷ 1 + µϕ
,
1−ϕ
1−ϕ
{z
}
|
{z
} |

(16)

RHS

LHS

where LHS and RHS in (16) are increasing in ρt .
At ρt = ρ∗ ≡ (1 − γη (1 − ϕ) /µϕ) /ϕ,( )γη/(1+γη)
1
LHS|ρt =ρ∗ > RHS|ρt =ρ∗ ⇔> 1,
ϕwhich holds for any ϕ ∈ (0, 1) and γη ∈ (0, 1). It also holds thatlim LHS < limm RHS ⇔ Ω10,t |xt =x̄t ,ρ=ρm < Ω11,t |xt =x∗∗ ,ρ=ρm ,

ρ→ρm

ρ→ρ

t

where the second inequality condition holds as shown in Lemma 1. Therefore, there is aunique ρt , denoted by ρ∗∗ ∈ (ρ∗ , ρm ) , that satisfies (16) with an equality.
To summarize, the results thus are: ∗∗mρ∗∗ < ρt < ρ(, xt = arg max Ω11,t if
)
γη
∗
∗
,1 ,
arg max Ωt =xt = arg max Ω10,t if ρt ∈ (0, ρ ] and ϕ ∈ µ+γη


x̄t
if
max (0, ρ∗ ) < ρt ≤ ρ∗∗ ,wheremax (0, ρ∗ ) =


 0
 ρ∗

if ϕ ∈(1, γη
2 µ+γη

],)γη
if ϕ ∈ µ+γη
,1 ,
(

γη
because ρ∗ ≷ 0 ⇔ ϕ ≷ µ+γη
.

■

A.5

Derivation of P11 (·) , P̄10 (·) , and P10 (·)

First, assume ρt ∈ (ρ∗∗ , 1): both types of agents choose public education, (q L , q H ) = (1, 1).
The average human capital in period t + 1 is
ht+1 = ϕhLt+1 + (1 − ϕ)hH
t+1
( L )1−η( )1−η
η= ϕD ht(xt ) + (1 − ϕ)D hH(xt )η .
t
23

Using this expression, we can reformulate ρt+1 = hLt+1 /ht+1 asρt+1 =( )1−η
D hLt(xt )η1−η

1−η

(xt )η + (1 − ϕ)D (hHt )( H )1−η ]−1ht= ϕ + (1 − ϕ)hLt[
)1−η ]−1(1−ϕ= ϕ + (1 − ϕ)η
,ρtϕD (hLt )[(xt )η

L
where the equality in the third line comes from hH
t /ht = (1/ρt − ϕ) /(1 − ϕ).
Next, assume ρt ∈ (0, ρ∗∗ ]: type-L agents choose public education and type-H agentschoose private education. Type-H’s human capital equation is()η( H )1−η
γηHH(1 − τt ) ht
ht+1 = D ht
1 + γη
)η (
(
)η
γη
ϕxt
H
1−
,
= Dht
1 + γη
ht
Hwhere the first equality comes from the private education function, eH
t = γη (1 − τt ) ht / (1 + γη),and the second equality comes from the government budget constraint, ϕxt = τt ht . With( )1−η
hLt+1 = D hLt(xt )η , the period t + 1 inequality index, ρt+1 , becomes( )1−η
D hLt(xt )η()η ()η
ρt+1 =
η
γη
ϕxt
L 1−η
H
ϕD (ht ) (xt ) + (1 − ϕ)Dht 1+γη
1 − ht
)
(
)
(

η
η −1
ϕxt
γη
1
−
hH
t
1+γη
ht
 .
= ϕ + (1 − ϕ)ηL 1−η(ht ) (xt )(17)Assume the corner solution,
xt = x̄t ≡tµ 1−ϕρ
1−ϕt
1 + µϕ 1−ϕρ
1−ϕ

ht .

Substituting this into (17) and rearranging the terms, we obtain P̄10 (·), as in the text.
Alternatively, assume the interior solution, xt = x∗t = γηht /ϕ(1 + γη). Substituting this
into (17) and rearranging the terms, we obtain P10 (·), as in the text.
■

A.6

Properties of P11 (·) , P̄10 (·) , and P10 (·)(i) Claim 1: P11 (·) , P̄10 (·) , and P10 (·) are strictly increasing inρt .
24This claim is immediate from the expressions of P11 (·) , P̄10 (·) , and P10 (·) in the text.
(ii) Claim 2: P10 (·) ⋛ P̄10 (·) if and only if ρt ⋛ ρ∗ .
We directly compare P10 (·) and P̄10 (·) and obtain()1−η ()η ()
1 − ϕρt
1 − ϕρtϕρt1/γ
P10 (·) ⋛ P̄10 (·) ⇔ (1 − ϕ) (1 + γη)⋛(1 − ϕ) ρt
1 + γη
ρt[]
1
1 − ϕ γη⇔ ρt ⋛ ρ∗ ≡ · 1 −
·
.
ϕ
ϕ
µ
(iii) Claim 3: P̄10 (·) < P11 (·) ∀ρt ∈ (0, 1).
We directly compare P̄10 (·) and P11 (·) and obtainP̄10 (·) < P11 (·)⇔ (1 − ϕ)

(
η

1
−ϕ
ρt

)1−η

(
< (1 − ϕ) (1 + γη)1/γ

1 − ϕρt(1 − ϕ) ρt)1−η⇔ 1 < (1 + γη)1/γ ,which holds for any γη ∈ (0, 1).
′(iv) Claim 4: P11 (·) satisfies P11 (1) = 1 and P11(1) = 1 − η ∈ (0, 1).
P11 (1) = 1 is immediate from the definition of P11 (·) in the text. The first diﬀerentiation of P11 (·) with respect to ρ is[)1−η ]−2
(
)−η
(
1
1
1
η
′
η
−ϕ
(1 − ϕ) (1 − η)−ϕ.
P11 (ρt ) = ϕ + (1 − ϕ)ρtρt(ρt )2′
We evaluate this at ρt = 1 to obtain P11(1) = 1 − η ∈ (0, 1).
()(v) Claim 5: P̄10 (·) satisfies P̄10 (0) = 0 and limρ→0 ∂ P̄10 (·) /∂ρt = ∞; P10 (·) satisfies

P10 (0) = 0 and limρ→0 (∂P10 (·) /∂ρt ) = ∞.
We obtain P̄10 (0) = 0 and P10 (0) = 0 by directly substituting ρt = 0 into P̄10 (·) and()
P10 (·). To show limρ→0 ∂ P̄10 (·) /∂ρt = ∞, we diﬀerentiate P̄10 (·) with respect to ρt .
After rearranging the terms, we obtain∂ P̄10 (·)=[∂ρt(1 − ϕ)η (1 + γη)1/γ (1 − η)((
)1−η ]2
)η ,
2
1/γ
1−ϕρt
1−ϕρt
· (ρt ) ·ϕ + (1 − ϕ) (1 + γη)(1−ϕ)ρtρtor,[∂ P̄10 (·)= (1 − ϕ)η (1 + γη)1/γ (1 − η) × (ϕ)2 (ρt )2−η (1 − ϕρt )η
∂ρt
]−1
{
}2 ( 1 )2(1−η)
ρ
(1
−
ϕρ
)
t
t
+2ϕ(1 − ϕ) (1 + γη)1/γ+ (1 − ϕ) (1 + γη)1/γ(ρt )η
.
1−ϕ
1−ϕ′(·) = +∞.
Evaluating this at ρt = 0, we obtain limρt →0 P̄1025

To show limρ→0 (∂P10 (·) /∂ρt ) = ∞, we follow the same procedure described above.
Diﬀerentiating P10 (·) with respect to ρt yields)η(∂P10 (·)
ϕ=[ϕρt + (1 − η) (1 − ϕρt )]∂ρt
1 + γη
[
]−1
(
)η
(
)2η
ϕ
ϕ
2
2−η
2
η
× (ϕ) (ρt )+ 2ϕ(1 − ϕρt ) ρt +(1 − ϕρt ) (ρt ).
1 + γη
1 + γη
We evaluate this at ρt = 0 and obtain()η∂P10 (·)
ϕ
lim=[0 + (1 − η) (1 − 0)] × (0)−1 = +∞.
ρt →0
∂ρt
1 + γη
■26

References[1] Arawatari, R., and Ono, T., 2009. A second chance at success: A political economy perspective. Journal of Economic Theory 144(3), 1249–1277.
[2] Arawatari, R., and Ono, T., 2013. Inequality, mobility and redistributive politics. Journalof Economic Theory 148(1), 353–375.
[3] Arcalean, C., and Schiopu, I., 2015. Inequality, opting-out and public education funding.
Social Choice and Welfare 46(4), 811-837.
[4] Bearse, P., Glomm, G., and Patterson, D.M., 2005. Endogenous public expenditures on
education. Journal of Public Economic Theory 7(4), 561–577.
[5] Behrman, J.R., and Taubman, P., 1989. Is schooling ‘mostly in the genes’ ? Nature-nurturedecomposition using data on relatives. Journal of Political Economy 97, 125–146.
[6] Benabou, R., 2000. Unequal societies: income distribution and the social contract. American Economic Review 90, 96–129.
[7] Card, D., Krueger, A.B., 1992. Does school quality matter? Returns to education and thecharacteristics of public schools in the United States. Journal of Political Economy 100,
1–40.
[8] Cardak, B.A., 2004a. Education choice, endogenous growth and income distribution. Economica 71(281), 57–81.
[9] Cardak, B.A., 2004b. Education choice, neoclassical growth, and class structure. Oxford
Economic Papers 56(4), 643–666.
[10] de la Croix, D., and Doepke, M., 2004. Public versus private education when diﬀerential
fertility matters. Journal of Development Economics 73(2), 607–629.
[11] de la Croix, D., and Doepke, M., 2009. To segregate or to integrate: education politics anddemocracy. Review of Economic Studies 76(2), 597–628.
[12] De Gregorio, J., Lee, J-W., 2002. Education and income inequality: new evidence fromcross-country data. Review of Income and Wealth 48(3), 395–416.
[13] Epple, D., and Romano, E.R., 1996. Public provision of private goods. Journal of Political
Economy 104, 57–84.
[14] Galor, O., Moav, O., and Vollrath, D., 2009. Inequality in land ownership, the emergenceof human-capital promoting institutions, and the great divergence. Review of Economic
Studies 76(1), 143–179.
[15] Glomm, G., and Ravikumar, B., 1992. Public versus private investment in human capital:
Endogenous growth and income inequality. Journal of Political Economy 100, 818–834.
[16] Glomm, G., and Ravikumar, B., 1998. Opting out of publicly provided services: A majority
voting result. Social Choice and Welfare 15(2), 187–199.
[17] Gradstein, M., Justman, M., 1996. The political economy of mixed public and private
schooling: A dynamic analysis. International Tax and Public Finance 3(3), 297–310.
[18] Gradstein, M., and Justman, M., 1997. Democratic choice of an education system: implications for growth and income distribution. Journal of Economic Growth 2(2), 169–183.
[19] Hassler, J., Rodriguez Mora, J.V., Storesletten, K., and Zilibotti, A., 2003. The Survival of
the welfare state. American Economic Review 93(1), 87–112.

27

[20] Hassler, J., Storesletten, K., and Zilibotti, F., 2007. Democratic public good provision.
Journal of Economic Theory 133(1), 127–151.
[21] Hoyt, W.H., and Lee, K., 1998. Educational vouchers, welfare eﬀects, and voting. Journalof Public Economics 69(2), 211–228.
[22] Lindbeck, A., and Weibull, J.W., 1987. Balanced-budget redistribution as the outcome ofpolitical competition. Public Choice 52(3), 273–297.
[23] Ono, T., 2016. Inequality and the politics of redistribution. International Tax and Public
Finance 23(2), 191–217.
[24] Saint-Paul, G., Verdier, T., 1993. Education, democracy and growth. Journal of Development Economics 42(2), 399–407.
[25] Stiglitz, J.E., 1974. The demand for education in public and private school systems. Journalof Public Economics 3(4), 349–385.
[26] Teulings, C., van Rens, T., 2008. Education, growth, and income inequality. Review of
Economics and Statistics 90(1), 89–104.
[27] Zhang, J., 1996. Optimal investments in education and endogenous growth. Scandinavian
Journal of Economics 98(3), 387–404.

28

percentage of enrollment in private institutions (%, 2011)
0
10
20
30

GBR
ESP
FRA

SWE
LUX
PRT
DNK
ISL

ISR
NZL

NOR

SVK
FIN
CZE

AUTCHE
DEU

ITA
CAN
POL

NLD
SVN

.25

GRC
EST

IRL

.3
.35
Gini coefficient (2011).4

Figure 1: Gini co-eﬃcient and enrollment (percentage) in private secondary education institutions in 2011. Source: World Bank Indicator (http://data.worldbank.org/indicator);
OECD Statics (http://stats.oecd.org/).

29

(a)

(b)

(c)

(d)

(
)
[
)
[
)
Figure 2: ρt ∈ 0,[ ρl case(Panel (a)); ρt ∈ ρl , ρm case (Panel (b)); ρt ∈ ρm , ρh case](Panel (c)); ρt ∈ ρh , 1 case (Panel (d)).

30

(
)
Figure 3: Education choice qtL , qtH classified according to xt .

31

(a)

(b)

[ m h)
Figure 4: Illustrationofthepolitical
objectivefunction
for
the
ρ
∈
ρ , ρ case (Panelt[](a)) and the ρt ∈ ρh , 1 case (Panel (b)).

32

(a)

(b)

( l)
Figure 5: Illustrationofthepolitical
objectivefunction
for
the
ρ
∈
0, ρ case (Panelt[ l m)(a)) and the ρt ∈ ρ , ρ case (Panel (b)).

33

(a)

(b)

(c)

Figure 6: P̄10 (ρ∗∗ ) > ρ∗∗ case (Panel (a)) and P̄10 (ρ∗∗ ) ≤ ρ∗∗ case (Panels (b) and (c)).

3410.8018020.601601

0.401401

0.2012

0.001
0.5

0.599

0.699

0.799

0.899

0.999

nz = 129420

Figure 7: Multiple steady-state equilibria for the shaded area; a unique steady-state
equilibrium for the non-shaded area.

35

(a)
Slovenia
NorwayDenmark
IcelandCzech Republic
Slovak Republic
Finland
Belgium
SwedenLuxembourg
Austria
Netherland
Switzerland
Germany
Ireland
PolandFranceCanadaNew Zealand
Italy
Estonia
GreeceSpain
Portugal
United Kingdom
Israel

(b)
0.248
0.25
0.251
0.257
0.262
0.263
0.264
0.273
0.273
0.279
0.282
0.283
0.289
0.291
0.302
0.305
0.309
0.315
0.323
0.324
0.334
0.337
0.342
0.343
0.344
0.371

Ireland
Slovenia
Netherland
Estonia
PolandGreeceCanadaNorwayCzech Republic
Italy
Germany
Finland
Austria
Switzerland
Slovak Republic
New ZealandIsrael
IcelandDenmark
Portugal
Luxembourg
SwedenFranceSpain
United Kingdom
Belgium

0.69839
1.49191
3.32049
3.417224.344244.66068
7.35733
8.165498.446048.520178.75589.361469.488439.727119.81426
10.92371
11.93098
12.39742
13.8019516.32851
17.92853
19.38773
26.1503827.30868
29.42805
68.60273

Table 1: Gini coeﬃcients (Panel (a)) and enrollment (percentage) in private secondary education institutions (Panel (b)) in selected OECD countries in 2011. Source: World Bank
Indicator (http://data.worldbank.org/indicator); OECD Statics (http://stats.oecd.org/).

36

Type-L's utility / Type-H's utility

1.5
1.45

hL1 =0.0165, h H1 =0.05
hL1 =0.0145, h H1 =0.061605

1.4

hL1 =0.0185, h H1 =0.038395

1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0

10

20

30

40

50

60

70

80

generation

Figure 8: Ratio of type-L’s utility to type-H’s utility from generation 1 to 80.

37

1.04

Type-L's utility ratio: mixed / public

1

1.03

0.98

1.02

0.96

1.01

0.94

1

0.92

0.99

0.9
0

20

40

60

80

0

generation
1.03

Type-H's utility ratio: mixed / public

20

40

60

80

generationsocial welfare ratio: mixed / public

1.02

hL1 =0.0165, h H1 =0.05

1.01

hL1 =0.0145, h H1 =0.061605

1

hL1 =0.0185, h H1 =0.038395

0.99
0.98
0.97
0

20

40

60

80

generation

Figure 9: Ratio of mixed education systems to compulsory public school systems in termsof type-L’s utility (Panel (a)), type-H’s utility (Panel (b)), and social welfare (Panel (c)).

38

○ ●

○ ●

○ ●

Figure A.1: Illustration for Condition (8).

39