Discussion Papers In Economics
And Business

Public Education and Social Security:
A Political Economy Approach
Tetsuo Ono
Discussion Paper 13-06-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Education and Social Security:
A Political Economy Approach
Tetsuo Ono
Discussion Paper 13-06-Rev.2

September 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Education and Social Security: A Political
Economy Approach
Tetsuo Onoy
Osaka University

Abstract
This study presents an overlapping-generations model with altruism towards
children. We characterize a Markov-perfect political equilibrium of voting over two
policy issues, public education for the young and social security for the old. The
model potentially generates two types of political equilibria, one favoring public
education and the other favoring social security. One equilibrium is selected by
the government to maximize its objective. It is shown that (i) longevity a¤ects
equilibrium selection and relevant policy choices; and (ii) private education as analternative to public education and a Markov-perfect political equilibrium can generate the two types of equilibria.
Keywords: Public education; Social security; Intergenerational con‡ict
JEL Classi…cation Number: H52, H55, I22.

The author would like to thank Yuki Uchida for his research assistance. This work was supported inpart by a Grant-in-Aid for Scienti…c Research (C) from the Ministry of Education, Science, and Cultureof Japan No. 24530346.
yGraduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

With increasing life expectancy, many developed countries have experienced a shift towards a progressively older population over the past several decades. This demographic
change has induced an increase in the political power of the elderly, and social security
expenditure for them is expected to increase (Casey et al., 2003). This may imply areduction in spending on the young (e.g., public education) because of government budgetary constraints (Poterba, 1997, 1998; Fernandez and Rogerson, 2001; Harris, Evans,
and Schwab, 2001).
Such predictions are not always possible when altruistic concern for one’s oﬀspringis considered (Cattaneo and Wolter, 2009). Greater longevity means that parents canenjoy the human capital of their children longer in later life, making public and private
investment in education more attractive for the working population. In addition, theeﬀects of the increased political power of the elderly on social security expenditure is notstraightforward. A greater social security burden on the working population discourages
them from investing in education, which may result in a smaller tax base and a lower
level of social security beneﬁts in the long run. Furthermore, all these eﬀects interact
with each other. The following question arises: how does a conﬂict of interest between
generations aﬀect political decision-making on social security and public education (andhence, human capital accumulation) in the long run? This study aims to answer thisquestion from a political economy viewpoint.
For analysis, this paper presents an overlapping-generations model with uncertain
lifetimes and altruism towards children. In each generation, there are identical individuals
who live for at most three periods: young, middle, and old. An individual produces one
oﬀspring during the middle period and dies at the end of the middle period with someprobability. A middle-age individual is endowed with a stock of human capital that alsodeﬁnes his/her labor capacity. He/she sets the allocation of disposable income between
his/her personal consumption and investment in furthering the child’s education. An
individual who lives throughout old age can receive and consume social security beneﬁts.
The level of an oﬀspring’s human capital is determined by public education and theparents’ human capital and private investment. Social security and public education are
ﬁnanced using taxation on the middle.
Within this framework, we consider a probabilistic voting (à la Lindbeck and Weibull,1987) in which in each period, the middle and the old participate in voting. Here, thegovernment in power maximizes a political objective function of the weighted sum of theutilities of the middle and the old (see, e.g., Grossman and Helpman (1998), Hassler et al.
(2005), and Song, Storesletten and Zilibotti (2012) for applications for the overlapping
generations models). In particular, this paper restricts its attention to Markov-perfect1

equilibria, where voters condition their strategies on payoﬀ-relevant state variables (i.e.,
human capital in the present model). This concept of equilibrium captures the forwardlooking behavior of the middle, who expect the beneﬁt of social security when they reachold age.
The present model demonstrates two types of political equilibria: the presence ofprivate investment in education and the absence of public education, and vice versa. In
both equilibria, social security is provided to old individuals. It is shown that the former
equilibrium is realized if the eﬃciency of public education is relatively low compared tothat of private education; otherwise, the latter equilibrium is observed.
Voters’ preferences are aﬀected by longevity. In particular, longevity has eﬀects on
the political determination of social security through the following three factors: (i) the
weight on the utility of old-age social security that beneﬁts the old; (ii) the tax burdenof the middle to ﬁnance the current old-age social security payments; and (iii) the sumof the weights on the utilities of the oﬀspring’s human capital and old-age social security
that are expected to beneﬁt the current middle in their old age. The ﬁrst factor works toincrease old-age social security, whereas the second factor works to reduce it. The eﬀect
created by the ﬁrst factor is oﬀset by that created by the second factor. Therefore, there
remains an eﬀect created by the third factor, which includes the forward-looking behaviorof agents.
The third factor implies a negative eﬀect of longevity on old-age social security. This
somewhat counterintuitive result arises as follows. Greater longevity implies a larger
weight on the utility of old-age social security that the middle receive in their old age.
To maintain a certain level of future social security, the middle must sustain the humancapital level of their oﬀspring and thus invest privately and/or publicly in education.
Then, the middle have an incentive to reduce the tax burden for old-age social security payments today and instead increase private and/or public investment in education.
Therefore, greater longevity results in a lower level of old-age social security.
Longevity also aﬀects public education, but its eﬀect is non-monotone. In a politicalequilibrium with public education, greater longevity results in a higher level of public
education spending. However, a further increase in longevity induces voters to preferold-age social security to public education and thus to leave educational investment to
the private sector. In other words, a further increase in longevity puts the economy intoa state with no public education. Overall, greater longevity produces a non-monotone
eﬀect on public education spending. This non-monotone eﬀect is peculiar to the present
model that includes two alternatives for educational investments.
This study employs a Markov-perfect equilibrium to demonstrate the forward-looking
behavior of voters. In order to examine the role of this equilibrium concept, we investigate2

an alternative concept of equilibrium, that is, myopic voting, which is often employed in
the literature (see, e.g., Holtz-Eakin, Lovely, and Tosun, 2004; Tosun, 2008; Gradstein
and Kaganovich, 2004; Boldrin, 2005). In this voting scheme, voters take future policy asgiven. Under this alternative concept, we ﬁnd that the model fails to demonstrate a political equilibrium with the presence of public education. In other words, the model shows
only a political equilibrium with the absence of public education, which is empirically an
implausible scenario. The result suggests that the Markov-perfect equilibrium is key todemonstrating the two types of political equilibria with empirically relevant properties.
The remainder of this paper is organized as follows. We ﬁrst present a literature
review. Thereafter, Section 2 presents the model. Section 3 demonstrates individual
decision-making on education and then characterizes an economic equilibrium. Section 4
demonstrates a period-t political equilibrium. Section 5 investigates how longevity aﬀects
education and social security policies. Section 6 shows the existence and stability of asteady-state political equilibrium. Section 7 undertakes the analysis using an alternative
assumption of myopic voting. Section 8 concludes.

1.1

Literature Review

The present work can be seen as integrating two bodies of literature. The ﬁrst is concernedwith public education as a means of redistribution and its possible eﬀects on humancapital accumulation in political economy models of public education. These models,
however, do not include private educational investment as a choice for individuals (Glomm
and Ravikumar, 1995, 2001; Glomm, 2004; Gradstein and Kaganovich, 2004; Boldrin,
2005; Palivos and Varvarigos, 2013). Several studies overcome this issue by comparing an
economy with public education to one with private education (Glomm and Ravikumar,
1992; Saint-Paul and Verdier, 1993; Gradstein and Justman, 1997; de la Croix and Doepke,2004) or by considering parents’ choice of public and/or private education to maximize
their altruistic utility (Stiglitz, 1974; Epple and Romano, 1996; Gradstein and Justman,
1996; Hoyt and Lee, 1998; Glomm and Ravikumar, 1998; Cardak, 2004; Bearse, Glomm,
and Patterson, 2005; de la Croix and Doepke, 2009; Kunze, 2014).
Most of these studies capture public education as public expenditure on children ﬁnanced by taxation on adults. In other words, they focus on a forward intergenerationaltransfer from parents to their children. However, in the real world, there is another intergenerational transfer that works in the opposite direction: income transfer from the young
to the elderly, such as social security. The present study focuses on this alternative public
spending and investigates how an intergenerational conﬂict over these two policy issues
aﬀects human capital accumulation and the allocation of government spending on public
education and social security through individual decision-making regarding education. In
3particular, the paper sheds light on the role of longevity on political decision-making.
The second body of literature focuses on two-issue voting in the presence of intergenerational conﬂict, such as two types of income redistribution. Examples include redistribution within a generation and between diﬀerent generations (Conde-Ruiz and Galasso,
2005; Bassetto, 2008; Gonzalez-Eiras and Niepelt, 2008); public goods provision and social security (Creedy and Moslehi, 2009; Creedy, Li and Moslehi, 2011); public capital andsocial security (Konrad, 1995; Bellettini and Berti Ceroni, 1999); and medicare services
and social security (Bethencourt and Galasso, 2008). In particular, the present paper isconcerned with works on public education that beneﬁts the young and social security that
beneﬁts the elderly (Bearse, Glomm, and Janeba, 2001; Soares, 2006; Iturbe-Ormaetxe
and Valera, 2012; Kaganovich and Meier, 2012; Kaganovich and Zilcha, 2012; Naito,2012). However, these studies assume either a vote over public education for a givensocial security beneﬁt or over the allocation of tax revenue for a given tax rate. In otherwords, the two-dimensional voting aspect is reduced to one dimension. Therefore, thesestudies do not indicate how the size of the government (i.e., the tax rate) and the allocation of government spending between diﬀerent generations are jointly determined throughvoting in the presence of an intergenerational conﬂict.
The present study, therefore, attempts to integrate both literatures in an analyticalframework. Some recent works that share these concerns are Kemnitz (2000), Rangel(2003), Levy (2005), Poutvaara (2006), Bernasconi and Profeta (2012), Gonzalez-Eirasand Niepelt (2012) and Lancia and Russo (2013). However, these diﬀer from the present
study in that (1) there is no private education as an alternative choice (Kemnitz, 2000;
Gonzalez-Eiras and Niepelt, 2012; Lancia and Russo, 2013) or no human capital accumulation (Rangel, 2003; Levy, 2005), (2) there is a focus on the intragenerational conﬂict
rather than the intergenerational one (Bernasconi and Profeta, 2012), and (3) there isno analysis on the eﬀect of increasing longevity on the allocation of government spending(Poutvaara, 2006). By contrast, this paper demonstrates how an intergenerational conﬂict
over two policy issues (i.e., public education and social security) aﬀects human capital
accumulation and the allocation of government spending in the presence of private education as an alternative to public education. The paper then shows that private education
as an alternative to public education and forward-looking behavior of voters are key todemonstrating the two types of political equilibria.

2

The Model

We consider a discrete-time overlapping-generations model that starts at time 0. Individuals live for at most three periods: young, middle, and old. An individual dies at the endof his/her middle age with probability 1 − p and lives throughout old age with probability4p ∈ (0, 1). A higher p means greater longevity, which is interpreted as population aging.1
Young and middle individuals are identical within each generation, and each middle individual produces one oﬀspring.2 There is no population growth, and the number of young
individuals is assumed to be constant. Young individuals are economically inactive exceptthat they consume education.
Consider a middle individual at time t (i.e., generation t). He/she is endowed witha stock of human capital ht , which also deﬁnes his/her labor capacity. Given the labor
income tax τt , a middle individual sets his/her allocation of disposable income ((1 − τt )ht )between consumption in the middle period (cmt ) and private investment in his/her child’s
further education (zt ) subject to the budget constraint:cm
t + zt ≤ (1 − τt )ht .
In old age, an individual receives social security bt+1 and consumes it. The budget constraint in old age is
cot+1 ≤ bt+1 ,where cot+1 denotes consumption in old age. We assume that there is no means of storing
private goods for old-age consumption.3 Figure 1 depicts the structure of the model.
[Figure 1 here.]
The level of the oﬀspring’s human capital, ht+1 , is determined by the parent’s humancapital, ht , material private input, zt , and public expenditure on education, et . The humancapital production function is assumed to have the following form:
ht+1 = A · (ht )η · (zt + (1 − γ)et )δ ,where A, η and δ are constant and satisfy A > 0, η ∈ (0, 1), γ ∈ (0, 1), δ ∈ (0, 1), andη + δ ∈ (0, 1). The function implies that private input zt is a perfect substitute for the
publicly provided input et .4
1

There are two aspects of population aging: an increase in longevity, which is basically out of thecontrol of individuals, and a decline in fertility rates, which is the outcome of individual decision-making.
The current study focuses on the former aspect to examine the eﬀect of an exogenous change in the
demographic structure on education and social security policies via voting.
2
Given the assumption of identical individuals within a generation, we ignore intragenerational politicalconﬂict in the present study. Instead, we focus on the conﬂict between generations.
3This assumption is unusual in the literature, but it enables us to demonstrate the intergenerational
conﬂict over the two policy issues (i.e., public education for the young and social security for the old) ina tractable manner.
4
In general, private education serves as a substitute and a complement to public education (Glomm
and Kaganovich, 2003; Bearse, Glomm, and Patterson, 2005). The latter role is not included in thepresent analysis.

5

The assumption γ ∈ (0, 1) implies that public education is less eﬃcient than private
education is. In other words, the rate of return to investment in public education islower than that in private education (Gradstein and Justman, 1996). This assumption
reﬂects the fact that public education provides standardized (rather than individualized)
education to each child. Because of this uniformity of public school education, eachchild is unable to receive the type of education suitable for his or her needs. The lackof individualized education programs in public education results in a lower return toinvestment compared to private education.
An individual in generation t derives utility from consumption in middle and old ages,ocmt and ct+1 , respectively, and from his/her child’s anticipated future income, ht+1 . We
assume that parents do not care about the welfare of their children and only about their
human capital. Generation t’s preferences are speciﬁed by the following expected lifetime

utility function:

{
}
o
ln cm
t + p · θ ln ht+1 + (1 − θ) ln ct+1 ,where θ ∈ (0, 1) and 1−θ denote the relative weights on the utility of the child’s anticipatedfuture income and of old-age consumption, respectively. A middle individual in generationt chooses cm
t and zt to maximize his/her expected lifetime utility subject to the budget
constraints in the middle and old ages given (1 − τt )ht , bt+1 , and et .
In each period, the government raises tax revenues to ﬁnance the provision of uniform public schooling for all children, et , and social security, bt . The government budget
constraint is given byτt ht = et + pbt ,where τt ht is tax revenue from the middle, et is public expenditure on education, and pbt
is expenditure on social security (here, the expenditure on social security, bt , is multipliedby p because the number of old individuals is p in each period).
The timing of events in period t is as follows. First, middle and old individuals voteon the tax rate (τt ) and on expenditures on public education and social security (et andbt , respectively). Second, each middle individual sets an allocation of disposable income
between consumption and private education subject to his/her budget constraints. We
solve the model using backward induction in the following two sections.

3

Economic Equilibrium

This section demonstrates a middle individual’s decision on consumption and private
investment in education and its consequence for utility and human capital accumulation.
Thus, we ﬁrst deﬁne the economic equilibrium as the outcome of a middle individual’sutility-maximizing behavior.
6

Deﬁnition 1. Given a sequence of tax rates and the sizes of redistribution and public education, {τt , et , bt }∞t=0 , an economic equilibrium is a sequence of allocations,∞mo{ct , zt , ct , ht+1 }t=0 , with an initial condition h0 (> 0) such that (i) in each period,
a middle individual maximizes his/her lifetime utility subject to the budget constraints, non-negativity constraint of investment in private education, and the human capital production function, and (ii) the government budget is balanced in eachperiod.
The problem of a middle individual in period t is as follows:max

cm
t ,zt ∈[0,(1−τt )ht ]

{
}
o
ln cm
t + p θ ln ht+1 + (1 − θ) ln ct+1

subject tocm
t + zt ≤ (1 − τt )ht ,cot+1 ≤ bt+1 ,
ht+1 = A · (ht )η · (zt + (1 − γ)et )δ ,given ht , τt , bt+1 and et ,where the ﬁrst and second constraints are the budget constraints in middle and old ages,
respectively, and the third constraint is the human capital production function.
Solving the utility maximization problem leads to the following private education
decision:{}1
zt = max 0,[pθδ(1 − τt )ht − (1 − γ)et ] .
(1)
1 + pθδ
Equation (1) indicates that the investment decision depends on an individual’s humancapital ht and on government policy variables τt and et . In particular, a middle individual
chooses to invest privately in education if his/her human capital is high, the tax rate islow, and/or the level of public education is low. Otherwise, he/she chooses no private
investment in education and spends disposable income for his/her own consumption.
Therefore, the consumption function in middle age is given by}{1m[(1 − τt )ht + (1 − γ)et ] ,ct = min (1 − τt )ht ,1 + pθδ
where the ﬁrst and second arguments in the brackets correspond to consumption whenz = 0 and z > 0, respectively.
Equation (1) indicates that a middle individual invests in education if and only ifpθδ(1 − τt )ht − (1 − γ)et > 0. Using the government budget constraint, the condition isrewritten as follows:()
1−γ
z t > 0 ⇔ ht >+ 1 · et + pbt .
pθδ7This condition states that lower levels of public education and social security beneﬁts
produce a larger income eﬀect, thereby giving a middle individual incentive to invest
more in education.
Considering the condition zt > 0, we can write the indirect utility function of a middle
individual as follows:

(
)
1−γ
m
 Vt,z>0ifh+1
et + pbt ,
t >
( pθδ
)
Vtm =
m
 Vt,z=0
if ht ≤ 1−γ+ 1 et + pbt ,pθδmm
are the indirect utility functions of a period-t middle-aged individual
and Vz=0
where Vz>0born in the previous period (i.e., period t − 1) when zt > 0 and zt = 0, respectively. These
are given bymVt,z>0= (1 + pθδ) ln (ht − γet − pbt ) + p(1 − θ) ln bt+1
[
]
1
pθδ
+ ln
+ pθδ ln
+ pθ ln A + pθη ln ht ;1 + pθδ1 + pθδmVt,z=0= ln(ht − et − pbt ) + pθδ ln et + p(1 − θ) ln bt+1

+ [pθ ln A + pθδ ln(1 − γ) + pθη ln ht ] .

(2)

(3)

The old do not make an economic decision. They receive social security beneﬁts andconsume them. In addition, they receive the utility of their oﬀspring’s human-capitallevel. The indirect utility of an old individual alive at time t is as follows:
Vto = (1 − θ) ln bt + θ ln ht ,(4)where the ﬁrst and second terms on the right-hand side represent the utilities of social
security and the oﬀspring’s human capital, respectively.
Using the government budget constraint and the condition zt > 0, the human capital
production equation is given by the following:

]δ
[
pθδ
η

(h
−
γe
−
pb
)
H
(h
,
e
,
b
)
≡
A(h
)

t
t
t
z>0 t t t
t

1+pθδ

(
)


1−γ
if ht > pθδ + 1 et + pbt ;ht+1 = H(ht , et , bt ) =δ
Hz=0 (ht , et ) ≡ A(ht()η [(1 − γ)e
t]

)



if ht ≤ 1−γ+ 1 et + pbt .
pθδ

(5)

Equation (5) implies that there exists a critical level of human capital, ((1 − γ) /pθδ + 1) et +pbt , that determines educational investment behavior. For a given set of policy variables,e and b, human capital accumulates according to the ﬁrst equation in (5) when the stockof human capital is above the critical level. In addition, it evolves according to the second
equation in (5) when the stock of human capital is below the critical level. The critical level depends on public education and social security, both of which are determinedthrough voting (as demonstrated in the next section).
84

Political Equilibrium

Public education e and old-age social security b are determined by individuals through apolitical process. Elections occur every period, and all middle and old individuals cast aballot on e and b. Individuals’ preferences for the two policy issues are represented by theindirect utility functions in Eqs. (2) and (3) for the middle and by those in Eq. (4) for
the old.
The issue space is bi-dimensional, and thus, a majoritarian voting game equilibrium
may not exist. To resolve this problem, the present paper assumes probabilistic votingà la Lindbeck and Weibull (1987) in the demonstration of the political mechanisms. In
each period, the middle and the old participate in voting, and the government in power
maximizes a political objective function that reﬂects the preferences of the middle andthe old. Formally, the political objective function in period t is given by the following:Ωt = pVto + Vtm ,where p (attached to the utility of the old, Vto ) is the relative weight of the old measured asa percentage of the population in the economy. The government’s problem is maximizingΩt subject to the human capital production function, Eq. (5), given ht .5
This study restricts its attention to a Markov-perfect equilibrium as described in
Krusell, Quadrini, and Rios-Rull (1997) and applied to a political economy analysis (e.g.,
Grossman and Helpman, 1998; Hassler et al., 2003, 2005; Hassler, Storesletten, and Zilibotti, 2007; Gonzalez-Eiras and Niepelt, 2008, 2012; Song, 2011, 2012; Song, Storesletten,
and Zilibotti, 2012). Voters condition their strategies only on payoﬀ-relevant state variables. In the current framework, human capital h is the payoﬀ-relevant state variable.
Therefore, the expected level of social security for the next period included in the utility
function of the middle, Vtm , is given by bt+1 = B̃(ht+1 ) : ℜ++ → ℜ+ . We can now deﬁne
a period-t political equilibrium as follows:
Deﬁnition 2. A period-t political equilibrium is a pair of functions, {B, E}, where B
and E are two policy rules, bt = B(ht ) : ℜ++ → ℜ+ and et = E(ht ) : ℜ++ → ℜ+ ,respectively, such that (i) B and E are solutions to the government’s problem for agiven expectation of bt+1 = B̃(ht+1 ) and (ii) B̃ = B holds.
Next, we characterize a period-t political equilibrium, that is, a voting outcome in someperiod t. For this purpose, we seek the voters’ preferred policies when zt > 0 (Section 4.1)
and zt = 0 (Section 4.2). In Section 4.3, we summarize the two cases to characterize a
5

An explicit microfoundation for this model is explained in Persson and Tabellini (2000, Chapter 3)
and Acemoglu and Robinson (2005, Appendix). Song, Storesletten, and Zilibotti (2012, Appendix B)
show the process of deriving the political objective function in an overlapping generations framework.

9period-t political equilibrium. Section 4.4 examines how the period-t political equilibriumoutcome is aﬀected by the increased longevity of agents. In Section 4.5, we review thelong-run consequences and show the existence and stability of a steady-state politicalequilibrium.

4.1

Voters’ Preferred Policy When zt > 0

Suppose that a middle individual in period t invests a portion of his/her disposable income
in education: zt > 0. Because the preferences are speciﬁed by the logarithmic utility
function, we conjecture a linear policy function of social security in period t + 1 : bt+1 =B0 · ht+1 , where B0 ∈ (0, ∞) is a constant parameter. Under this conjecture, the social
security beneﬁt is given by the following:bt+1 = B0 AHz (ht , et , bt )
)δ
(
pθδ
η
(ht − γet − pbt )δ ,= B0 A (ht )1 + pθδ(6)

where the second line is derived using the equation of human capital production (5).
Under the assumption of zt > 0 and the expectation of bt+1 in (6), the objectivefunction of the period-t government becomesΩt,z>0 = p(1 − θ) ln bt + (1 + pδ) ln (ht − γet − pbt ) ,where unrelated terms are omitted from the expression. The ﬁrst term on the righthand side, (1 − θ) ln bt , denotes the utility of social security beneﬁts for the period-t
old weighted by the number of old, p. The second term denotes the expected utility ofthe period-t middle. In particular, it is the sum of the utilities of their consumption((1 + pθδ) ln (ht − γet − pbt )) and social security beneﬁts they receive in their old age,p(1 − θ)δ ln (ht − γet − pbt ). The latter beneﬁt is speciﬁc to the model considering theMarkov-perfect political equilibrium.
The problem of the government in period t is choosing a pair of (bt , et ) that maximizeΩt,z>0 . The ﬁrst-order conditions with respect to et and bt are−γ≤ 0,
ht − γet − pbt
p(1 − θ)pbt := (1 + pδ).
btht − γet − pbt

et : (1 + pδ)These conditions lead toet = 0 and bt =

1−θ
ht .
1 + p ((1 − θ) + δ)

(7)

Therefore, B = B̃ holds if B0 = (1 − θ) · (1 + p ((1 − θ) + δ))−1 . That is, function (7)constitutes a period-t political equilibrium as long as B0 = (1 − θ) · (1 + p ((1 − θ) + δ))−1 .
10We substitute the solution in (7) into the condition zt > 0 in (5) and ﬁnd that thecondition holds for any ht > 0.6 The result established thus far is summarized as follows.
Lemma 1. For any ht > 0, there is a solution to the period-t government problem
distinguished by zt > 0, et = 0 and bt = (1 − θ) · (1 + p ((1 − θ) + δ))−1 · ht .
The result in Lemma 1 implies that given the expectation that the middle invest privately in education, the government ﬁnds it optimal to invest nothing in public educationand to use all tax revenue for old-age social security for any level of ht . Faced with thisgovernment policy, the middle choose to invest privately in education to maximize utility. Zero public investment in education, which seems to be an extreme result, is partly
because of the speciﬁcations of the utility and human capital production functions. However, it may be viewed as demonstrating an economy where government spending is infavor of old-age social security.

4.2

Voters’ Preferred Policy When zt = 0

Alternatively, suppose that a middle individual in period t privately invests nothing in
education: zt = 0. Following the same procedure as in the case of zt > 0, we conjecture
a linear function of social security in period t + 1 : bt+1 = B1 · ht+1 where B1 ∈ (0, ∞) isa constant parameter. Under this conjecture, the social security beneﬁt is given by
bt+1 = B1 A(ht )η (1 − γ)δ (et )δ .
The objective function of the period−t government becomesΩt,z=0 = p(1 − θ) ln bt + ln (ht − et − pbt ) + pδ ln et ,where unrelated terms are omitted from the expression. The ﬁrst term on the right-handside, (1 − θ) ln bt , denotes the utility of social security beneﬁts for period-t old weighted by
the number of old, p. The second term denotes the utility of consumption for the period-t
middle, and the third term is part of the expected utility of old-age social security for theperiod-t middle.
The problem for the government in period t is choosing a pair of (bt , et ) that maximizeΩt,z=0 . The ﬁrst-order conditions with respect to bt and et are1pδ= ,ht − et − pbtetp(1 − θ)
p
bt :
=
.
bt
ht − et − pbt

et :

6

The condition zt > 0 becomes
zt > 0 ⇔ ht >()
p(1 − θ)1−γ
+ 1 et + pbt ⇔ 1 >.
pθδ
1 + p ((1 − θ) + δ)

The last inequality holds for any set of parameters.

11

These conditions lead to
et =

pδ
1−θ
ht and bt =ht .
1 + p ((1 − θ) + δ)
1 + p ((1 − θ) + δ)

(8)Therefore, B = B̃ if B1 = (1 − θ) · (1 + p ((1 − θ) + δ))−1 .
Plugging the solution in (8) into the condition zt = 0, we ﬁnd that zt = 0 holds if andonly if θ ≤ 1 − γ ⇔ γ ≤ 1 − θ holds. Therefore, we obtain the following result.
Lemma 2. Suppose that γ ≤ 1−θ holds. Then, for any ht > 0, there is a solution to theperiod-t government problem distinguished by zt = 0, et = pδ ·(1+p ((1 − θ) + δ))−1 ·ht and bt = (1 − θ) · (1 + p ((1 − θ) + δ))−1 · ht .
The result in Lemma 2 states that given the expectation that the middle privately
invest nothing into education, the government ﬁnds it optimal to use a part of the tax
revenue for public education as long as the eﬃciency of public education is suﬃciently highthat γ ≤ 1 − θ. Here, a lower γ implies a higher eﬃciency of public education. Faced withthis policy, the middle chooses to privately invest nothing into education to maximize
utility. Compared to the result in Lemma 1, the result in Lemma 2 demonstrates an
economy with government spending in favor of public education.

4.3

Period-t Political Equilibrium

The results in Lemmas 1 and 2 suggest that there is a unique solution distinguished byz > 0 if γ > 1 − θ; furthermore, there may exist two solutions distinguished by z > 0
and z = 0 if γ ≤ 1 − θ. The case γ ≤ 1 − θ implies that there are two local maxima forthe political objective function. The government can choose the better of the two local
maxima by controlling policies.
To select the equilibrium for the case γ ≤ 1 − θ, we substitute the policy functions in
Eq. (7) when z > 0 and in Eq. (8) when z = 0 into the political objective functions Ωz>0and Ωz=0 , respectively. For a given ht , we have the following relation:mmΩt,z>0 R Ωt,z=0 ⇔ Vt,z>0
R Vt,z=>0,mm
are deﬁned in (2) and (3), respectively. This relation holds
and Vt,z=>0
where Vt,z>0
because the old-age social security functions are equivalent in both cases (z > 0 andz = 0) for a given ht . Thus, Vto when z > 0 is equal to Vto when z = 0.

12

Using a direct calculation, we obtain7mmVt,z>0R Vt,z=>0⇔ γ R 1 − θ · ϕ(p),where ϕ(p) is deﬁned as(ϕ(p) ≡

1 + pδ
1 + pθδ(9))(1+pδ)/pδ> 1,

and 1 − θ > 1 − θ · ϕ(p) holds. Therefore, we can conclude that the government chooses
the policy in Lemma 1 if γ ∈ (1 − θ · ϕ(p), 1), that in Lemma 2 if γ ∈ (0, 1 − θ · ϕ(p)),
and is indiﬀerent between the two if γ = 1 − θ · ϕ(p). The result is summarized in thefollowing proposition.
Proposition 1 (Period-t political equilibrium)(i) If γ > 1 − θ · ϕ(p), there exists a unique period-t political equilibrium with zt > 0 and
et = 0.
(ii) If γ = 1 − θ · ϕ(p), there are two period-t political equilibria: one is distinguished by
zt > 0 and et = 0, and the other is distinguished by zt = 0 and et > 0.
(iii) If γ < 1 − θ · ϕ(p), there exists a unique period-t political equilibrium with zt = 0
and et > 0.
Figure 2 illustrates the conditions γ R 1 − θ and γ R 1 − θ · ϕ(p) in a θ − γ space. The
condition γ > 1 − θ · ϕ(p) indicates that given γ, obtaining the equilibrium with zt > 0 is
positively correlated with the size of θ. Greater parental interest in the child’s education
gives the parents a stronger motivation for providing education. Given the property thatprivate education is a perfect substitute for public education, parents ﬁnd it optimal toprivately provide education and to use tax revenue for old-age social security instead offor public education. Therefore, the economy realizes an equilibrium where education isprovided in private-sector institutions when γ > 1 − θ · ϕ(p) holds.
7

Using this direct calculation, we havemm
Vt,z>0R Vt,z=>0⇔ (1 + pδ) ln(

1 + pδ
+ pδ ln θ R pδ ln(1 − γ)
1 + pθδ)(1+pδ)1 + pδpδ⇔ ln+ ln (θ) R ln(1 − γ)pδ
1 + pθδ)(1+pδ)(1 + pδpδ⇔(θ) R (1 − γ)pδ
1 + pθδ()(1+pδ)/pδ
1 + pδ
⇔θ R 1 − γ.
1 + pθδ
Using the deﬁnition of ϕ(p) in the text, we obtain (9).

13The condition γ > 1−θ ·ϕ(p) also indicates that given θ, the equilibrium with zt > 0 is
positively correlated with γ, that is, it is negatively correlated with the relative eﬃciencyof public education. As previously mentioned, public education is a perfect substitute forprivate education, but the former is less eﬃcient and thus more costly than the latter.
However, for the middle-aged voters, public education acts as a collective mechanism toincrease the average human capital of subsequent generations and their old-age pension
beneﬁts. Middle-aged voters examine the costs and beneﬁts of public education and ﬁndthat costs are larger (smaller) than beneﬁts when the eﬃciency of public education is
below (above) the critical value, denoted by θ · ϕ(p). Eventually, middle-aged voterschoose private education rather than public education when γ is suﬃciently high that
1 − γ < θ · ϕ(p) (i.e., γ > 1 − θ · ϕ(p)). Furthermore, they choose public education when
γ is suﬃciently low that 1 − γ > θ · ϕ(p) (i.e., γ < 1 − θ · ϕ(p)).
[Figure 2 here.]
The situation described thus far is somewhat extreme, because it lacks either public
or private education. However, this situation demonstrates two types of states in the realworld: states with a high and low share of private education in the total education expenditure, respectively. The equilibrium with z > 0 (z = 0) can be viewed as demonstrating
the former (latter) state. To examine the plausibility of these situations, we review dataon OECD countries (OECD, 2013, Education at a glance 2013). Table 1 shows the relative
proportions of private and public expenditure on educational institutions for all levels of
education in 2010. Panel (a) presents the list of countries with an over-25 % share ofprivate education; Panel (b) presents the list of countries with an under-10% share.
[Table 1 here.]
The evidence suggests that the group of countries with high shares of private education
includes Anglo-Saxon countries (Australia, the United Kingdom, and the United States)
and East-Asian countries (Japan and Korea)8 . The group of countries with low shares
includes European countries (Austria, Belgium, Denmark, Estonia, Finland, Iceland, Ireland, Italy, Portugal, and Sweden). One interpretation of this evidence is that countries
in the former group features low eﬃciency of public education and/or larger concern ofparents regarding the education of their children. Alternatively, countries in the lattergroup feature high eﬃciency of public education. In other words, the degree of eﬃciencyof public education and parental motivation for education tend to aﬀect decision making
on education and the choice of redistribution policies. Furthermore, they are keys to
8

An exception is Chile with a 42.1% share. The share is 24.2% in Canada, which can plausibly be
included in the group of Anglo-Saxon countries.

14

explaining cross-country diﬀerence in the composition of education expenditures. This isa testable implication of the theory, which should be conﬁrmed by empirical evaluationin future work.

5Eﬀects of Longevity

The analysis and results thus far suggest that longevity, represented by the parameterp, aﬀects the equilibrium policies and characterizations demonstrated in Section 4. To
investigate the longevity eﬀects in more detail, we ﬁrst consider the eﬀect of an increasein p on the equilibrium policies e and b for the cases z > 0 and z = 0 in Proposition
2. Then, we provide an interpretation of the results, examine the eﬀects of an increasein p on the threshold condition in (9), and show the overall eﬀect of longevity on public
education spending around the threshold condition. Finally, we compare our results toprevious results and ﬁndings.
The following proposition summarizes the eﬀect of greater longevity on the equilibrium
policies.
Proposition 2. Consider a period-t political equilibrium.
(i) Greater longevity results in a lower level of old-age social security.
(ii) Suppose that z = 0 holds. Greater longevity results in a higher level of public education spending.
To understand the mechanism behind the result, we recall the political objective function when z > 0:Ωt,z>0 = p (1 − θ) ln bt + (1 + pδ ) ln ht − γet − pbt  .
|{z}
|{z}
|{z}
(a.i)

(a.iii)(a.ii)

Longevity aﬀects the determinants of old-age social security using the following threefactors: (a.i) the weight on the utility of old-age social security that beneﬁts the current
old; (a.ii) the tax burden of the middle required to ﬁnance the current old-age social
security payments; and (a.iii) the sum of the weights on the utilities of consumption,
the oﬀspring’s human capital, and old-age social security expected to beneﬁt the current
middle in their old age. In the present speciﬁcation of the model, the eﬀect created by
the ﬁrst factor is oﬀset by that created by the second one. Therefore, the eﬀect producedby the third factor remains.
The third factor includes the discipline eﬀect exercised by the middle voters. Greater
longevity implies a larger weight on the utility of old-age social security that they will
15

receive in their old age. To maintain a certain level of social security beneﬁts, they need tosustain the human capital level of their oﬀspring and thus need to invest privately and/or
publicly in education. Therefore, the middle have an incentive to reduce the tax burden
for old-age social security and to increase private and/or public investment in educationin response to an increase in longevity. Therefore, greater longevity results in a lower
level of old-age social security.
To conﬁrm that a similar result is obtained for the case z = 0, recall the politicalobjective function when z = 0:Ωt,z=0 = p(1 − θ) ln bt + ln ht − et − pbt  + pδ ln et .
|{z}
|{z}
| {z }(b.ii)(b.i)(b.iii)

Similar to the case z > 0, longevity has eﬀects on the determination of old-age social
security and public education through the following three factors: (b.i) the weight on
the utility of old-age social security that beneﬁts the old; (b.ii) the tax burden on the
middle to ﬁnance the current old-age social security payments; and (b.iii) the sum of theweights on the utilities of the oﬀspring’s human capital and old-age social security that is
expected to beneﬁt the current middle in their old age. As in the case z > 0, we ﬁnd thatthe eﬀect created by the ﬁrst factor is oﬀset by that created by the second factor. The
remaining factor, that is, the third factor, shows that greater longevity means a larger
weight on public education. Therefore, we can conclude that greater longevity results ina larger level of public education spending and thus a lower level of old-age social security
for the case z = 0.
This analysis and its results suggest a monotone eﬀect of longevity on old-age social
security. However, the eﬀect on public education is not straightforward, because public
education spending increases with longevity when z = 0; however, there is no public
education spending when z > 0. To understand the overall eﬀect, we must investigate
how an increase in p aﬀects the threshold condition γ = 1 − θ · ϕ(p) in (9).
To see the eﬀect of p on the threshold condition, we show the conditions for the threevalues of p in Figure 2 using a graph. The corresponding values of p are p = 0.2, 0.6, and0.99. The ﬁgure suggests that the equilibrium with z > 0 is more likely to be obtainedwith high longevity. A higher p implies a larger weight on old-age social security in thepolitical objective function. Therefore, voters tend to prefer old-age social security to
public education and to leave educational investment to the private sector as longevity
increases.
Combining the results in Proposition 2 with the numerical results demonstrated in
Figure 2, we obtain the following implication of longevity for public education. For a low
p such that the middle privately invest nothing in education, an increase in p results in a16higher level of public education spending. However, a further increase in p results in no
spending on public education, because the equilibrium allocation is changed from the one
distinguished by z = 0 and e > 0 (as in Lemma 2) to that distinguished by z > 0 and
e = 0 (as in Lemma 1) in response to an increase in longevity. That is, greater longevity
produces a non-monotone eﬀect on public education. This non-monotone eﬀect is peculiar
to the present model that includes two alternatives for educational investment.
The positive eﬀect of aging on public education shown in Proposition 2(ii) is in linewith the theoretical predictions in the literature. Gradstein and Kaganovich (2004) show
an overall positive impact of increasing longevity on public education funding. Levy
(2005) shows a relatively high level of per-capita public provision of education when theyoung are a minority in the population. In addition, the negative eﬀect of aging on social
security shown in Proposition 2(i) is in line with the prediction in the model by Razin,
Sadka, and Swagel (2002).
These theoretical predictions are somewhat counterintuitive, and one might expectthat aging voters show less support for public education that does not directly beneﬁt theold and place more emphasis on social security. However, empirical investigations suggestthat the ﬁndings are mixed regarding the eﬀect of aging on public education (see Cattaneo
and Wolter, 2009, for a literature survey). In addition, Razin, Sadka, and Swagel (2002)
show that aging has a negative impact on social security in the United States and in some
European countries. The predictions of the present model can therefore provide a possible
explanation for these counterintuitive ﬁndings.

6

Steady-state Equilibrium

Having established the period-t political equilibrium, we now investigate the law of motionof human capital in the political equilibrium. In particular, we examine the existence andstability of a steady-state political equilibrium, where ht = ht+1 holds. Thus, we computethe human capital production equation by substituting the policy when zt > 0 (7) into
ht+1 = Hz>0 (·) in (5) and that when zt = 0 (8) into ht+1 = Hz=0 (·) in (5). Thus, we
obtain

)δ (
)δ
(

1+pδ
 Hz>0 (ht ) ≡ A pθδ(ht )η+δ if zt > 0;1+pθδ(1−θ)+δ()δ(10)
ht+1 =(1−γ)pδ
Hz=0 (ht ) ≡ A (1−θ)+δ(ht )η+δ if zt = 0,
where Hz>0 and Hz=0 are strictly increasing and strictly concave in h with Hz>0 (0) =′′(·) = 0.
(·) = 0, and limh→∞ Hz=00, Hz=0 (0) = 0, limh→∞ Hz>0
Given the result in Proposition 1, we can write the law of motion of human capital as17

follows:



Hz>0 (ht )if γ > 1 − θ · ϕ(p){Hz>0 (ht ), Hz=0 (ht )} if γ = 1 − θ · ϕ(p)ht+1 =

Hz=0 (ht )if γ < 1 − θ · ϕ(p).

Using this motion, we obtain the following result regarding the existence and stability ofthe steady-state political equilibrium.
Proposition 3.
(i) If γ > 1 − θ · ϕ(p), there exists a unique and stable steady-state political equilibriumwith z > 0 and e = 0.
(ii) If γ = 1 − θ · ϕ(p), there are multiple steady-state political equilibria: the steady statewith z > 0 and e = 0 and that with z = 0 and e > 0.
(iii) If γ < 1 − θ · ϕ(p), there exists a unique and stable steady-state political equilibriumwith z = 0 and e > 0.
The result in Proposition 3 implies that there is a unique political equilibrium paththat stably converges to the steady state with z > 0 and e = 0 when γ > 1 − θ · ϕ(p). In
addition, there is a unique political equilibrium path that stably converges to the steadystate with z = 0 and e > 0 when γ < 1 − θ · ϕ(p). Multiple steady-state equilibria arise
only when γ = 1 − θ · ϕ(p). In this case, the economy either experiences a monotone
convergence to either state or oscillates between the two equilibria, depending on the
choice of the government in each period.
Figure 3 illustrates two numerical examples of the human capital production equation
when γ ≤ 1−θ. In this situation, there are two solutions to the government problem. The
government selects one of solution to maximize the objective. In each panel, a solid curve(dashed curve) presents an equation of human capital production realized (not realized)
in equilibrium in accordance with the government’s selected (non-selected) policy.
[Figure 3 here.]In the present model, no switch occurs between the state with z > 0 and e = 0 andthat with z = 0 and e > 0 along the equilibrium path for most sets of parameters. We
observe a switch between the two states only when γ = 1 − θ · ϕ(p) holds. Therefore,
a question arises as to whether it is possible to show a switch between the two statesalong an equilibrium path for a larger set of parameters. To answer this question, we
can assume that γ depends on the human capital level, ht . For example, assume that
γ is decreasing in ht : the return on investment from public education increases as thehuman capital level increases. This assumption implies that productivity (or eﬃciency)18

of public education increases as the teachers’ level in public schools (represented by thehuman capital level in the economy) increases.
Under this assumption, we can ﬁnd a critical value of ht , denoted by ĥ. For ht < ĥ, thecondition γ > 1 − θ · ϕ(p) is satisﬁed: there exists a unique equilibrium path with z > 0.
For ht > ĥ, the condition γ < 1−θ·ϕ(p) is satisﬁed: there exists a unique equilibrium pathwith z = 0. Therefore, we can predict that the economy attains a political equilibrium
distinguished by z > 0 and e = 0 when ht is below the critical value and one distinguishedby z = 0 and e > 0 when ht is above the critical value. A switch from the state withz > 0 and e = 0 to that with z = 0 and e > 0 may occur along the equilibrium path.
However, the plausibility of this prediction should be tested in an empirical manner, which
is beyond the scope of the present study.
Another assumption may demonstrate the switch between the two states. For example,
it is plausible to assume that longevity, represented by parameter p, is increasing in
the human capital level, ht (see, e.g., Castello-Climent and Domenech, 2008). However,this assumption does not satisfy the Markov property for a given conjecture of a linear
social security function, bt+1 = B̃(ht+1 ) = B0 · ht+1 , where B0 is a constant parameter.
That is, the period-t social security function, bt = B(ht ), becomes bt = (1 − θ) · (1 +p(ht ) · ((1 − θ) + δ))−1 · ht ., which is a nonlinear function of ht : B̃(·) is not equal to
B(·). Therefore, the assumption that p is dependent on ht is not available in the present
framework.

7

Myopic Voting

Thus far, we have focused on the Markov-perfect political equilibrium of a voting gameover two policy issues, public education for the young and social security for the old. Inorder to examine the role of this equilibrium concept, this section introduces an alternative
political equilibrium concept adopted by many studies: myopic voting, where voters today
take future policy as given (e.g., Gradstein and Kaganovich, 2004; Holz-Eakin, Lovely,
and Tosun, 2004; Boldrin, 2005; Tosun, 2008; Kaganovich and Meier, 2012). Under thisalternative voting that does not include the forward-looking behavior of agents, we ﬁndthat the model demonstrates only one type of equilibrium with z > 0 and e = 0.
Under the myopic voting assumption, the objective of the period−t government is tochoose (et ,bt ) to maximize Ωt , which is deﬁned by()
 Ωz>0 ≡ p ln bt + (1 + pθδ) ln(ht − γet − pbt ) if ht > 1−γ + 1 et + pbt ;
( pθδ
)
Ωt =
 Ωz=0 ≡ p ln bt + ln(ht − et − pbt ) + pθδ ln et if ht > 1−γ + 1 et + pbt .
pθδThe function Ωt does not include the term pβ ln bt+1 , representing the utility of social
security beneﬁts in the next period as a politically relevant variable, because myopic voters
19today (i.e., in period t) take the next-period policy as given. In other words, we do notinclude the intertemporal eﬀect of current policy on next-period social security throughthe human capital accumulation equation from the analysis. Therefore, the equation ofhuman capital accumulation that appeared as a constraint in the previous sections is notconsidered in the current problem formulation.
Following the same procedure as in Section 4, we consider the cases z > 0 and z = 0.
First, suppose that z > 0 holds. The problem of the period-t government is as follows:
max p ln bt + (1 + pθδ) ln(ht − γet − pbt ).
bt ,etThe solution to this problem is((et , bt ) = 0,

)
1
ht .
1 + p(1 + θδ)Plugging in the solution to the condition zt > 0, we have
(
)
1−γ
1
ht >
+1 ·0+pht .
pθδ
1 + p(1 + θδ)This condition holds for any ht : there always exists a political equilibrium with z > 0.
Next, suppose that z = 0 holds. The problem of the period-t government is
max p ln bt + ln(ht − et − pbt ) + pθδ ln et .
bt ,etThe solution to the problem is((et , bt ) =

)
pθδ
1
ht ,
ht .
1 + p(1 + θδ) 1 + p(1 + θδ)We substitute the solution into the condition z = 0 and obtainzt = 0 ⇔ 0 ≤ −γ,
which never holds. Therefore, we can conclude that
Proposition 4. Suppose that voters are myopic in the sense that they take future policy
as given. Then, there exists a unique political equilibrium distinguished by z > 0
and e = 0.
Why is it impossible to have an equilibrium with no private investment, z = 0? In
order to answer this question, we recall the condition z = 0 in (5):
(
)
1−γ
ht ≤
+ 1 et + pbt .
pθδ20

For a given h, this condition requires high levels of public education and social security,
both of which give individuals a disincentive to privately invest in education. However,
myopic voting results in a high level of social security but a low level of public education.
Moreover, the former eﬀect is outweighed by the latter, and thus, the condition z = 0
fails to hold for any h. The model fails to show an economy distinguished by suﬃcientpublic support for education, which is an implausible scenario from an empirical pointof view. The result in this subsection therefore suggests that there is a need to capture
the forward-looking behavior of agents, which is peculiar to the Markov-perfect politicalequilibrium when we consider the two-dimensional voting on public education and old-age
social security.

8

Conclusion

This paper developed an overlapping-generations model with altruism toward children
and considers voting over two policy issues, public education for the young and social
security for the old. In the model, there is a conﬂict of interest between generations overthese policy issues. The analysis shows that the two types of political equilibria exist in
the model: one with private (but not public) education and the other with public (but not
private) education. In both equilibria, social security is provided to all old individuals.
One of the two equilibria is selected by the government to maximize its objective.
This study ﬁrst shows that longevity critically aﬀects this selection. In addition, greater
longevity results in a lower level of old-age social security in both equilibria and a higher
level of spending on public education in the equilibrium with public education. These
results imply that longevity produces a non-monotone eﬀect on the provision of public
education.
This study also shows the role of the forward-looking behavior assumption of agents,
which is peculiar to the Markov-perfect political equilibrium. When we remove this assumption and alternatively assume myopic behavior, the model fails to show an economy
distinguished by the presence of public education. In other words, the economy attains
only an equilibrium distinguished by the absence of public education, which is empirically an implausible scenario. Therefore, the present analysis sheds light on the role ofthe Markov-perfect equilibrium in demonstrating two types of equilibria reﬂecting crosscountry diﬀerences in public education and social security.

21

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Period t

Period t+1

Generation t-1

Private budget

Generation t
Government budget
Private budget

Generation t+1
Human capital production
A+

Figure 1: The structure of the model.

25

Period t+2

1
0.9

p=0.2
p=0.6
p=0.99

0.8
0.7

gamma

0.6
0.5
0.4
0.3
0.2
0.1
0

0

0.2

0.4

theta

0.6

0.8

1

(
Figure 2: The dashed line presents γ = 1−θ. The curves present γ = 1−θfor three values of p : p = 0.2, 0.6 and 0.99.

26

1+pδ
1+pθδ

)(1+pδ)/(pδ)

Panel (a)Panel (b)Table 1: Relative proportions of public and private expenditure on educational institutions
for all levels of education (2010 year). Panel (a) is the list of countries where the share ofprivate education is more than 25% ; Panel (b) is the list of countries where the share ofprivate education is below 10%.
Source: OECD (2013) Education at a glance 2013: OECD indicators, OECD Publishing.

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1
0.9
0.8
0.7

h(t+1)

0.6
0.5
0.4
0.3
0.2
0.1
0

0

0.2

0.4

0

0.2

0.4

h(t)

0.6

0.8

1

0.6

0.8

1

1
0.9
0.8
0.7

h(t+1)

0.6
0.5
0.4
0.3
0.2
0.1
0

h(t)

Figure 3: In each panel, a solid curve presents an equation of human capital production
which is realized in equilibrium in accordance with a selected policy by the government;
and a dashed curve presents one which is not realized in equilibrium because the corresponding policy is not selected by the government. The parameter values are set at
A = 2.0, p = 0.8, θ = 0.5, δ = 0.4 and η = 0.3. Panel (a) with γ = 0.3 illustrates the resultin Proposition 3(i); and Panel (b) with γ = 0.1 illustrates the result in Proposition 3(iii).

28