Discussion Papers In Economics
And Business

Pensions, Education, and Growth: A Positive Analysis
Tetsuo Ono and Yuki Uchida
Discussion Paper 14-37-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPensions, Education, and Growth: A Positive Analysis
Tetsuo Ono and Yuki Uchida
Discussion Paper 14-37-Rev.2

March 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPensions, Education, and Growth: A Positive
Analysis
Tetsuo Ono

Yuki Uchida

Osaka University

Osaka University

Abstract
This study presents an overlapping generations model to capture the nature of
the competition between generations regarding two redistribution policies, public
education and public pensions. From a political economy viewpoint, we investigate
the e¤ects of population aging on these policies and economic growth. We show that
greater longevity results in a higher pension-to-GDP ratio. However, an increase inlongevity produces an initial increase followed by a decrease in the public educationto-GDP ratio. This, in turn, results in a hump-shaped pattern of the growth rate.
Keywords: economic growth; population aging; public education; public pensions
JEL Classi…cation: D78, E24, H55

Tetsuo Ono: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp. Yuki Uchida: Graduate Schoolof Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail:pge001uy@student.econ.osaka-u.ac.jp. Ono acknowledges …nancial support from Grant-in-Aid for Scienti…c Research (C) from the Ministry of Education, Culture, Sports, Science and Technology of Japan
No. 15K03509. Uchida acknowledges …nancial support from Grant-in-Aid for JSPS Fellows of the Japan
Society for the Promotion of Science No. 15J01375.

1

Introduction

Redistribution policy preferences are dictated largely by age. Middle-aged workers whoare parents have altruistic concern for their children and are more likely to support public
education because they can bene…t from highly educated children. This makes expenditure on education more attractive for these parents. On the other hand, retired oldpeople are less likely to support public education because they cannot obtain the bene…tsof education directly. Instead, they support public pension expenditure that compensates
for loss of earnings in their retirement. The age-dependent di¤erence in policy preferencessuggests that the recent trend of aging in developed countries, which increases the political power of the old, may induce governments to shift expenditure from education topensions.
This prediction on pensions is in line with the observation in developed countries.
Figure 1 suggests that the public pension spending-to-GDP ratio is positively correlatedwith the share of the population aged 65 years and above. However, the aforementioned prediction on public education does not …t the observation: the public education
spending-to-GDP ratio shows a hump-shaped pattern in response to the share of the aged
population. In addition, the growth rate shows a hump-shaped pattern, which seems
to be induced by the hump-shaped public education spending pattern. Therefore, there
should be another mechanism for the non-linear relationship between aging and public
education. The aim of this study to present a political economy model that provides theprediction …tting the observation in Figure 1.
[Figure 1 here.]
For analysis, this study presents a politico-economic model to characterize the natureof the competition between generations, and in addition, examines the e¤ect of population
aging on redistribution policy and economic growth. To capture generational con‡ict, themodel economy contains an in…nite sequence of overlapping generations in which eachis comprised of many identical individuals who live over three periods, namely, young,middle, and old ages. The middle-aged individuals are faced with uncertain lifetimes andare endowed with altruism toward children (i.e., the young). The middle-aged and old individuals participate in voting, but policy disagreements between them arise owing to thelonger planning horizon of the middle-aged individuals. In addition, the model contains
physical and human capital accumulation through savings and educational investment,
which enables us to demonstrate the e¤ect of redistribution policy on economic growth.
Within this framework, we consider probabilistic voting a la Lindbeck and Weibull(1987). In each period, the middle-aged and old individuals participate in voting; theyoung are not enfranchised. The government in power maximizes a political objective
1

function of the weighted sum of the utilities of the middle-aged and old population, taking
account of the impact on the middle-aged population’s economic decisions (for applicationsof the probabilistic voting for overlapping-generations models, see, e.g., Grossman andHelpman, 1998; Hassler et al., 2005; and Song, Storesletten, and Zilibotti, 2012).
In this voting environment, the redistribution preferences of the middle-aged population are state dependent. In other words, successive generations are linked throughphysical and human capital accumulation. The state dependence of policy preferences
gives the middle-aged voters the means to in‡uence the policy outcome when they areold. Their incentive to manipulate policy preferences stems from their desire to obtainlarge pension bene…ts in old age. We capture this forward-looking behavior of the middleaged voters by focusing on Markov-perfect political equilibria, in which voters condition
their strategies on payo¤-relevant state variables (i.e., physical and human capital in thepresent model).
The Markov-perfect political equilibrium is a¤ected by the three factors representing
population aging: uncertain lifetimes capturing individual longevity, the political power ofthe old that re‡ects their share in voting, and the population growth rate. The politicalpower of the old and the population growth rate have de…nite e¤ects on pensions andeducation. With greater political power of the old and a lower population growth rate,
the pension-to-GDP ratio increases, but the education-to-GDP ratio decreases. In otherwords, population aging stemming from a larger share of the old in voting and a lower
population growth rate shifts the allocation of tax revenue from education for the young
to pensions for the old.
In addition, longevity has a de…nite e¤ect on pensions; greater longevity results ina higher pension-to-GDP ratio. However, the e¤ect of longevity on public education is
not straightforward. Greater longevity implies a larger weight on the utility of the oldand the utility of the middle-aged population for their consumption in old age. This
incentivizes the government to shift its spending from education to pensions to improvetheir utility. However, at the same time, greater longevity incentivizes the government
to shift the allocation of spending from pensions to education because greater longevity
implies a larger weight of middle-aged voters’altruistic utility from the human capital of
their children. These opposing e¤ects produce an initial increase followed by a decreasein the education-to-GDP ratio and this, in turn, results in a hump-shaped pattern of thegrowth rate. The model predictions described thus far are in line with the observationsof developed countries demonstrated in Figure 1.
We obtain our results by assuming perfect annuity markets where individuals can purchase private pensions. However, in the real world, some countries have limited or noaccess to private annuity markets. To account for this possibility, we consider an alternative case without an annuity market, and show that the growth rate increases with rising2

longevity. This result re‡ects data from some Eastern European countries, such as Czech
Republic, Estonia, and Hungary, with low degrees of private annuitization (OECD, 2014),
as demonstrated in Panel (c) of Figure 1. Therefore, we may well conclude that the overall trend in Organisation for Economic Co-operation and Development (OECD) countries
shows a non-linear relationship between longevity and growth, but some countries show
a positive relationship due to limited access to private annuities.
The remainder of this paper is organized as follows. We …rst present a literature review
in Subsection 1.1. Thereafter, Section 2 presents the model and characterizes economic
equilibrium. Section 3 describes political equilibrium. Section 4 investigates how thethree aging factors a¤ect government expenditure on education and pensions. Section 5analyzes the e¤ects of the aging factors on economic growth. Section 6 undertakes theanalysis under an alternative scenario where annuity markets are absent, and compares
the result here to that in Section 5. Section 7 provides concluding remarks.

1.1

Literature Review

The present study is related to the literature on the political economy of public educationand pensions by Bearse, Glomm, and Janeba (2001), Soares (2006), Iturbe-Ormaetxe andValera (2012), Kaganovich and Meier (2012), Kaganovich and Zilcha (2012), and Naito(2012). A common feature of these studies is that the two-dimensional voting aspect
is reduced to one dimension for simplicity of analysis. In other words, they consider
a vote over public education for a given pension bene…t, or a vote over the allocationof tax revenue for a given tax rate. Therefore, these studies do not indicate how thesize of the government (i.e., the tax rate) and the allocation of government spending
between education and pensions are determined jointly through voting in the presence ofgenerational con‡ict.1
This problem is resolved by introducing two-dimensional voting (Rangel, 2003; Levy,
2005; Poutvaara, 2006; and Arawatari and Ono, 2014). However, these studies abstractfrom physical and/or human capital formation, and thus, show nothing about the interaction between policy and capital formation. Capital formation is introduced by Kemnitz(2000), Gradstein and Kaganovich (2004), Holz-Eakin, Lovely, and Tosun (2004), Tosun(2008), and Bernasconi and Profeta (2012). These studies assume myopic voting, in which
the current voters take future policy as given. In other words, the forward-looking decisions of voters are absent in the analysis of these studies. Therefore, they abstract from thefeedback mechanism between current and future redistribution policies through physicaland/or human capital accumulation, which plays a crucial role in shaping redistribution
1

An alternative to the political economy approach is the normative approach (e.g., Boldrin and Montes,2005), which takes the two spending programs as given and focuses on their role as a means to support
the complete market allocation.

3

policies.
The feedback mechanism is demonstrated by Beauchemin (1998), Forni (2005), Bassetto (2008), Gonzalez-Eiras and Niepelt (2008, 2012), Song (2011), Bishnu and Wang(2014), Chen and Song (2014), and Ono (2015). In particular, the present study is closely
related to Lancia and Russo (2015), who analyze the politics of public education andpensions in overlapping generations models. Lancia and Russo (2015) show that greater
political power of the old in voting increases pension spending but decreases education
spending. However, the cross-country evidence shows that the education-to-GDP ratioshows a hump-shaped pattern in response to the share of the old in the population (seeFigure 1). The present study shows that altruism toward children and uncertain lifetimes,
which are absent in the model of Lancia and Russo (2015), are the keys to demonstratesuch a hump-shaped pattern of education spending. In addition, this pattern induces
a non-linear relationship between aging and economic growth, which is in line with theempirical evidence (e.g., An and Jeon, 2006; Kunze, 2014; and Panel (c) in Figure 1).
Ludwig, Schelkle, and Vogel (2012) and Heer and Irmen (2014) also show the potential
for a non-linear relationship. An aging population decreases the share of the working-age
population, which in turn increases the capital-labor ratio, lowers the rates of return on
capital, and thus creates a disincentive to save. Endogenous human capital adjustment(Ludwig, Schelkle, and Vogel, 2012) or …rms’incentive to invest in innovation that a¤ect
total factor productivity (Heer and Irmen, 2014) could mitigate this negative impact .
The present study instead focuses on the political power of the elderly that a¤ects growth
rates through their in‡uence on public education and pension policies.
Apart from the abovementioned studies, the present study is related to Lambrecht,
Michel, and Vidal (2005) and Kunze (2014), who investigate the growth e¤ect of redistribution policy in overlapping generations models, in which altruistic parents …nance theeducation of their children. However, both of these studies focus on a single policy issue:
public pensions in the case of Lambrecht, Michel, and Vidal (2005), and public educationin the case of Kunze (2014). The present study di¤ers from theirs in that we consider
the two policy issues, investigate how they are shaped by population aging, and in turn,
analyze how they a¤ect economic growth.

2

Model

The model is based on that presented by Lambrecht, Michel, and Vidal (2005) and Kunze(2014). The economy starts at period 0 and consists of overlapping generations. Individuals are identical within a generation, live at most for three periods, namely, young,middle, and old ages. Each middle-aged individual gives birth to 1 + n children. The
middle-aged population for the period-t is Nt , and the population grows at a constant4

rate n(>1) : Nt+1 = (1 + n)Nt .

Individuals are faced with uncertain lifetimes in the third period of life. The probabilityof living in old age is 2 [0; 1]. This is idiosyncratic for all individuals and is constantacross periods.

2.1

Individuals

The economic behavior of individuals over their life cycles is as follows. During youngage, individuals make no economic decisions; they receive public education …nanced by
the government. In middle age, individuals work, receive market wages, and make tax
payments. They use after-tax income for consumption and savings. In old age, individuals
are retired. They receive the returns from savings and pension bene…ts, and consume both.
Consider an individual born in period t 1. In period t, he is middle aged and isendowed with ht units of human capital. He supplies it inelastically in the labor market,
and obtains the wage wt ht , where wt is the wage rate per e¢ ciency unit of labor inperiod-t. After paying the tax t wt ht where t 2 (0; 1) is the period-t income tax rate,
the individual distributes his after-tax income into consumption, ct , and savings held as
an annuity and invested in physical capital, st . Therefore, the period-t budget constraint
for the middle becomes as follows:c t + st(1t )wt ht :The period-t + 1 budget constraint in old age is
dt+1

Rt+1

st + pt+1 ;where dt+1 is consumption in old age, Rt+1 (> 0) is the gross return from investment
in capital, Rt+1 st =is the return from savings, and pt+1 is the pension bene…t. If an

individual dies at the end of the middle age, his annuitized wealth is transferred to theindividuals who live throughout old age via annuity markets. Therefore, the return fromsaving becomes Rt+1 = under the assumption of perfect annuity markets. The case of noannuity market will be investigated in Section 6.
A period-t middle-aged individual cares about his children’s per capita human capitalin period t + 1; ht+1 . This is a function of the government spending on public education,
xt , and the parent’s human capital, ht . In particular, ht+1 is formulated by the following
equation:ht+1 = D (xt ) (ht )1
where D(> 0) is a scale factor, and;

2 (0; 1) denotes the elasticity of education technology

with respect to public education spending.

5It should be noted that private investment in education may also contribute to the formation of human capital. For example, parents’time (Glomm and Ravikumar, 1995, 2001,
2003; Glomm and Kaganovich, 2008) or spending (Glomm, 2004; Lambrecht, Michel, andVidal, 2005; Kunze, 2014) devoted to education may work as a complement to public
education. In the present study, we abstract private education from the analysis to simplify the presentation of the model and to focus on the con‡ict between generations withrespect to public spending on education and pensions. In a former version of this paper(Ono and Uchida, 2014), we assume private education spending as a complement to public
education, and obtain qualitatively similar results to those in the present version of thepaper.
We assume that parents are altruistic toward their children and are concerned about
the disposable income of their children in middle age, (1 t+1 )wt+1 ht+1 . The preferencesof an individual born in period t 1 are speci…ed by the following expected utility functionof the logarithmic form:
Ut = ln ct + ln dt+1 + ln(1t+1 )wt+1 ht+1 ;where (> 0) denotes the intergenerational degree of altruism. We substitute the budget
constraints and human capital production function into the utility function to write thefollowing unconstrained maximization problem:
max ln [(1fst gt )wt htst ] + ln

Rt+1

st + pt+1 + ln(1t+1 )wt+1 D (xt )(ht )1:

By solving the problem, we obtain the following savings and consumption functions:
st =

1+
1
ct =
1+

pt+1
;
Rt+1
pt+1
:
t )wt ht +
Rt+1

(1

(1)

t )wt ht

(1

(2)

The savings function in (1) states that a higher level of after-tax wage, (1t )wt , implies
higher savings, whereas a higher level of pension, pt+1 , implies lower savings. The consumption function in (2) states that a higher lifetime income, (1t )wt ht +pt+1 =Rt+1 ,results in larger spending on consumption.

2.2

Firms

In each period, there is a continuum of identical …rms that are perfectly competitive pro…tmaximizers. According to Cobb–Douglas technology, they produce a …nal good Yt using
two inputs, aggregate physical capital Kt and aggregate human capital Ht
aggregate output is given by
Yt = A (Kt ) (Ht )1
6,

Nt ht . The

where A(> 0) is a scale parameter and

2 (0; 1) denotes the capital share.

Let kt Kt =Ht denote the ratio of physical to human capital. The …rst-order conditions for pro…t maximization with respect to Ht and Kt are as follows:
wt = (1)A (kt ) ;

(3)

t =

1

(4)

A (kt );

where wt and t are the wage of labor and the rental price of capital, respectively. The
conditions state that …rms hire human and physical capital until the marginal products
are equal to the factor prices.

2.3

Government Budget Constraint

Pensions and public education are …nanced by the tax on labor income. The aggregate
tax revenue is t wt ht Nt , while the aggregate expenditure is the spending on public pensions, Nt 1 pt , plus the spending on public education, Nt+1 xt . Therefore, the government
budget constraint in period t is
1+n

pt + (1 + n)xt =t wt ht :The left-hand side shows the expenditure on pensions, pt =(1 + n), and public education,(1 + n)xt , and the right-hand side shows the revenue from taxing labor income.

2.4

Economic Equilibrium

The market clearing condition for capital is Kt+1 = Nt st , which expresses the equality of
total savings by the middle-aged population in period t; Nt st , to the stock of aggregate
physical capital at the beginning of period t + 1, Kt+1 . With the use of kt+1 Kt+1 =Ht+1
and ht+1 = Ht+1 =Nt+1 , we can rewrite the condition as (Kt+1 =Ht+1 ) (Ht+1 =Nt+1 )(Nt+1 =Nt ) = st ; or
(1 + n)kt+1 ht+1 = st :The economic equilibrium in the present model is de…ned as follows.
De…nition 1. Given a sequence of policies, f t ; xt ; pt g1
t=0 , an economic equilibrium is a1
sequence of allocations fct ; dt ; st ; kt+1 ; ht+1 g1
t=0 and prices f t ; wt ; Rt gt=0 with the ini-tial conditions k0 (> 0) and h0 (> 0) such that (i) given (wt ; Rt+1 ; t ; xt ; pt+1 ) ; cyt ; cot+1 ; stsolves the utility maximization problem; (ii) given (wt ; t ), kt solves the pro…t maximization problem of a …rm; (iii) given (wt ; ht ; kt ) ; ( t ; xt ; pt ) satis…es the government budget constraint; (iv)t

= Rt holds; and (v) the capital market clears:(1 + n)kt+1 ht+1 = st .
7

In economic equilibrium, the indirect utility function of the middle-aged populationin period t; VtM , and that of the old in period t, Vto , can be expressed as functions ofgovernment policy and physical and human capital as follows:
V M (kt ; ht ; t ; kt+1 ; ht+1 ; t+1 ; pt+1 ) = ln

1
1+

+ ln

(1

t )(1

A (kt+1 )1+ ln(1V o (kt ; ht ; t ; pt ) = ln
+ ln(1pt+1
A (kt+1 )!

1(1 + n)kt+1 ht+1 + pt+1

t+1 )(1A (kt ))A (kt ) ht +)A (kt+1 ) ht+1 ;
!

1

(5)

(1 + n)kt ht + pt

t )(1

(6)

)A (kt ) ht ;where some irrelevant terms are omitted from the expressions. The …rst and second termsin (5) correspond to the utility of consumption in middle and old ages, respectively, andthe third term shows the utility from the disposable income of their children. The …rst
term in (6) corresponds to the utility of old-age consumption and the second term shows
the utility from the disposable income of their children.

3

Political Equilibrium

The present study assumes probabilistic voting developed by Lindbeck and Weibull (1987)
for demonstrating the political mechanism. In each period, the government in power
maximizes a political objective. Formally, the political objective function in period t isgiven byt = !

V o (kt ; ht ; t ; pt ) + (1 + n)V M (kt ; ht ; t ; kt+1 ; ht+1 ; t+1 ; pt+1 ) ;where ! (> 0) and 1 + n are the relative weights of old-age and middle-age agents,
respectively. In particular, the parameter !(> 0) represents the political power of theold-age agents. An explicit microfoundation for this modeling is explained in Persson
and Tabellini (2000, Chapter 3) and Acemoglu and Robinson (2005, Appendix). The
government problem in period t is to maximize
constraint, given the state variables, kt and ht .

tsubject to the government budget

In this study, we restrict our attention to a Markov-perfect equilibrium. In the present
framework, Markov perfectness implies that outcomes depend only on the payo¤-relevant
state variables, that is, physical and human capital, k and h, respectively. Therefore, theexpected levels of tax and public pension for the next period, t+1 and pt+1 , are given byfunctions of the next period stock of physical and human capital,
8

t+1 = T (kt+1 ; ht+1 ) andpt+1 = P (kt+1 ; ht+1 ), respectively. By using recursive notation with z 0 denoting the next
period z, we can de…ne a Markov-perfect political equilibrium as follows.
De…nition 2.
A Markov-perfect political equilibrium is a set of functions, hT; X; P i, where T :<++ <++ ! [0; 1] is a tax rule,= T (k; h), X : <++ <++ ! <++ is a
public education expenditure rule, x = X(k; h), and P : <++ <++ ! <++ is a
public pension expenditure rule, p = P (k; h), such that the following conditions are
satis…ed:(i) the capital market clears,(1 + n)k 0 h0 =(1

1+

T (k; h)) (1)A (k) h(ii) given k and h, hT (k; h); X(k; h); P (k; h)i = arg max

P (k 0 ; h0 );A (k 0 ) 1(7)(k; h; ; x; p; p0 ) subject to

p0 = P (k 0 ; h0 ); the capital market clearing condition in (7), the government budget

constraint,
P (k; h) + (1 + n)X(k; h) = T (k; h)(1)A (k) h;
1+nand the human capital production function, h0 = D(X(k; h)) (h)1 ; wherede…ned by(k; h; ; x; p; p0 )is

! V o (k; h; ; p) + (1 + n)V M (k; h; ; k 0 ; h0 ; 0 ; p0 ).

In order to obtain a set of functions in De…nition 2, we conjecture the following functions:p0 = P A (k 0 ) h0 ;x0 = X A (k 0 ) h0 ;(8)where P (> 0) and X(> 0) are constant parameters. By using this conjecture and theconstraints in De…nition 2(ii), we can obtain the political objective function as follows:!
A (k) 1
= ! ln(1 + n)kh + p
(9)
+ f!

+ (1 + n) (1 + ( + ) )g ln (1+ (1 + n) ( + ) (1)A (k) h1+np(1 + n)x) ln x;where the terms unrelated to policy are omitted from the expression. The derivation of(9) is provided in Appendix A.1.
We solve the problem of maximizing. The …rst-order conditions with respect to p

and x are
p:

!
A(k)

1

f!
(1

+ (1 + n) (1 + ( + ) )g 1+n;)A (k) h 1+n p (1 + n)x(1 + n)kh + p
f! + (1 + n) (1 + ( + ) )g (1 + n)
(1 + n) ( + ) (1x:=(1)A (k) h 1+n p (1 + n)x
x
9

)

:

A strict inequality holds in the …rst condition if p = 0. By using these conditions, we
obtain the following result.
Proposition 1. There is a Markov-perfect political equilibrium distinguished by p > 0
if ! >; and p = 0 otherwise, whereis de…ned by) + f!

! + (1 + n) ( + ) (1+ (1 + n) (1 + ( + ) )g :The corresponding policy functions are as follows:
Pp>0 A (k) h; Xp>0 A (k) h0; Xp=0 A (k) h

if ! >
if !

( + ) (1( + ) (1!

(P (k; h); X(k; h)) =

;
;

where
Pp>0

!

; Xp>0); and Xp=0

1+n

)2

:

Proof. See Appendix A.2.
To understand the result in Proposition 1, recall the political objective function,!
A (k) 1= |{z}! ln(1 + n)kh + p
( 1)

8
>
<

9
>
=
+ ! + (1 + n) (1 + ( + ) ) ln (1{z}>>:|{z} |;( 2)

)A (k) h1+np(1 + n)x( 3)+ (1 + n) ( + ) (1
|
{z
( 4)

) ln x:}(10)

The function indicates that the political power of the old (!), longevity ( ), and the
population growth rate (n) a¤ect the provisions of public pensions and education throughthe four factors represented by the terms (*1), (*2), (*3), and (*4). The other terms,
includingand n, have no critical impact on political decisions because the e¤ects throughthese terms cancel each other out.
The term (*1) implies that greater political power of the old and greater longevity
imply a larger weight of the utility of consumption for the old. This incentivizes thegovernment to allocate tax revenue more to public pensions for the old and less to public
education for the young. The term (*2) implies that greater power of the old and greater
longevity signal a larger weight of the old’s utility from their children’s disposable income.
To improve this utility, the government cuts the tax burden of the middle-aged population,
thereby resulting in lower levels of public pensions and education.
The term (*3) is the weight of the utility of the middle-aged agents for their consumption. The two factors included in this term, and n, produce opposite e¤ects on public
10spending. Greater longevity implies a larger weight of the middle-aged agents’utility of
their old-age consumption. This gives the government an incentive to save more for theirold-age consumption by cutting current pension and education expenditures. However, alower population growth rate works in the opposite direction because it implies a smaller
weight of the middle-aged agents’utility of their consumption.
Finally, the term (*4) is the weight of the utility of the middle-aged agents from theirold-age consumption, denoted by (1+n) ( + ) . Greater longevity provides an incentive
for the government to increase pension bene…ts for the middle-aged agents in their old age
and to increase the human capital level of their children. The government can realize thesetwo purposes by shifting the allocation of tax revenue from pensions for the old to public
education for the children. This shift expands the tax base in the future by improving
human capital of the children, and thus, increases pension bene…ts for the middle-aged
agents in their old age. A lower population growth rate works in the opposite direction
because the weight (1 + n) ( + ) decreases as the population growth rate decreases.
Based on the argument thus far, we now consider how the condition for p > 0; given
by ! > ; is a¤ected by the three aging factors, !; ; and n. The condition is rewritten
as follows:1! > (1 + n) ( + ) (1
|{z}
|
{z
( 1)

( 4)

8
>
<

9
>
=
) + ! + (1 + n) (1 + ( + ) ) ;
} >
{z
}>
:|{z} |
;
( 2)

( 3)

where the terms (*1), (*2), (*3), and (*4) correspond to those in the political objectivefunction. The condition states that the e¤ect of the population growth rate is straightforward. The terms (*3) and (*4), including n, indicate that public pensions are more
likely to be provided in political equilibrium if the population growth rate is lower. A
lower population growth rate implies smaller weights to the utility of the middle-aged
agents for their old-age consumption and to the utility of these agents for the disposable
income of their children. Given a lower weight for the utility of the middle-aged agents,
the government shifts its spending from education to pensions for the current old.
The e¤ects of longevity ( ) and the political power of the old (!) on pension provision
are not straightforward. First, greater longevity has a positive e¤ect on pensions throughthe term (*1), while it has a negative e¤ect on pensions through the terms (*2), (*3),
and (*4). Second, greater political power of the old has a positive e¤ect on pensions
through the term (*1), while it has a negative e¤ect on pensions through the term (*2).
Greater longevity and greater political power of the old have two competing e¤ects on
pensions, but the condition implies that the public pensions are more likely to be provided
in political equilibrium as longevity and the political power of the old increase.

11

4

Pensions and Education

The result established in Section 3 indicates that pensions and education are a¤ected by
the three aging factors, , !, and n. To consider their e¤ects on pensions and education,
we focus on the pension-to-GDP ratio,

Nt 1 pt =Yt , and the education-to-GDP ratio,

Nt+1 xt =Yt , and analyze the e¤ects of increases inand ! and a decrease in n on theseratios.
The following proposition demonstrates the e¤ects of ; !; and n on the pension-toGDP ratio.
Proposition 2. Suppose that ! >holds: a Markov-perfect political equilibrium

exists with p > 0. The pension-to-GDP ratio, Nt 1 pt =Yt , increases with greater longevity,
greater political power of the old, and a lower population growth rate: @ ( Nt 1 pt =Yt ) =@ >0; @ ( Nt 1 pt =Yt ) =@! > 0; and @ ( Nt 1 pt =Yt ) =@n < 0.
Proof. See Appendix A.3.
To con…rm the statement in Proposition 2, we compute the pension-to-GDP ratio as
follows:
2
8
0
193 1
>
>
=7
<
Nt 1 pt 6
1BC6(1 + n) ( + )(1) + @! + (1 + n)(1 + ( + ) )A 7
= 41 +
5|{z} |
|
{z
}
{z
} >
>
Yt
!
;
:
|{z}
( 2)
( 4)
( 3)
( 1)

(11)

The derivation of Eq. (11) is given in Appendix A.3.

The terms (*1), (*2), (*3), and (*4) in the above expression correspond to those in thepolitical objective function in Eq. (10). We can apply the interpretation for the conditionof p > 0 to the result in Proposition 2 in the following way. First, the population growthrate has an e¤ect on the ratio through the terms (*3) and (*4). As described in Section 3,
both terms imply a positive e¤ect on pension provision when the population growth rate
decreases. Second, the political power of the old has two competing e¤ects on the ratio
through the terms (*1) and (*2); and longevity also has two competing e¤ects on the ratio
through the terms (*1), (*2), (*3), and (*4). However, when p > 0, the positive e¤ectoutweighs the negative e¤ect, as demonstrated in Section 3. Therefore, greater longevity
and greater political power of the old lead to a higher pension-to-GDP ratio.
Given the result in Proposition 2 and the government budgetary constraint, it isnatural to conjecture that the education-to-GDP ratio, Nt+1 xt =Yt , decreases as

and !

increase and as n decreases. The following proposition shows that the conjecture is true
with regard to ! and n, but it is not necessarily true with regard to .
Proposition 3. The education-to-GDP ratio decreases with greater political power ofthe old and a lower population growth rate: @ (Nt+1 xt =Yt ) =@! < 0 and @ (Nt+1 xt =Yt ) =@n >12:

!( )2 ; increases

0. With greater longevity, the education-to-GDP ratio decreases if 1+n
if !(1 + )1 + n, and shows a hump-shaped pattern if !( )2 < 1 + n < !(1 + ) .

Proof. See Appendix A.4.
To con…rm the statement in Proposition 3, we …rst compute the Nt+1 xt =Yt ratio whenp > 0 and p = 0, as follows:
2

8
93 1
( 3)
( 2)
<
}|
{=
z}|{
z}|{ z6! + ! + (1 + n) (1 + ( + ) ) 7
6
:
;7
7
6
Nt+1 xt
7
6= 61 +7 ;7
Yt p>0 6(1 + n) ( + )(1
)
7
6
{z
}
|
5
4
( 4)

Nt+1 xt= (1Yt p=0

( 1)

2

( 3)

( 2)

3 1

}|
{
z}|{ z6! + (1 + n) (1 + ( + ) ) 7
76) 61 +
7(1 + n) ( + )(1
)
5
4
{z
}
|

:

( 4)

The terms (*1), (*2), (*3), and (*4) in the expressions correspond to those in the politicalobjective function in Eq. (10).
The e¤ects of the political power of the old (!) are as follows. Greater political powerof the old has a negative e¤ect on education spending via the term (*1), representing theweight on the utility of old agents for their consumption, and the term (*2), representing
the weight on the utility of old agents from the disposable income of their children. These
terms give the government incentive to shift the allocation of tax revenue from education
to pensions and to cut the tax burden of the children of old agents. Therefore, theeducation-to-GDP ratio decreases as the political power of the old increases.
A lower population growth rate has two competing e¤ects on the education-to-GDP
ratio. The positive e¤ect comes from the term (*3), implying a smaller weight on theutility of middle-aged agents for their consumption, whereas the negative e¤ect comesfrom the term (*4), implying a smaller weight on the utility of the middle-aged agentsfrom the disposable income of their children. The result suggests that for both cases ofp > 0 and p = 0, the negative e¤ect outweighs the positive one. Therefore, a decrease in
the population growth rate results in a decline in the education-to-GDP ratio.
Greater longevity a¤ects the ratio in the following ways. The …rst e¤ect involves
the term (*4), representing the weight of the utility of the middle-aged agents from thedisposable income of their children. When longevity increases, the government shifts the
allocation of tax revenue from pensions to education in order to increase the income of thechildren. This positive e¤ect is o¤set partly by the negative e¤ect of longevity represented
by the term (*3). This term works to decrease the tax burden of the middle-aged agentsby cutting the expenditure on education.
13

There are two additional negative e¤ects of longevity on public education; they are
represented by the terms (*1) and (*2). These e¤ects are enhanced as the political powerof the old increases. That is, the sum of the negative e¤ects by the terms (*1), (*2), and(*3) outweighs the positive e¤ect by the term (*4) when the political power of the old isabove a critical level. However, we should note that the e¤ect via the term (*1) appears
only when p > 0. Therefore, the critical values of political power that balance the twocompeting e¤ects di¤er between the two cases, p > 0 and p = 0: This di¤erence is a sourceof the hump-shaped pattern of the education-to-GDP ratio.
[Figure 2 here.]
Figure 2 illustrates a numerical example of the hump-shaped pattern. We …x the shareof capital at = 1=3, following Song, Storesletten, and Zilibotti (2012) and Lancia andRusso (2015). Each period lasts 30 years; this assumption is standard in quantitative
analyses of the two-period overlapping-generations model (see, for example, GonzalezEiras and Niepelt, 2008; Lancia and Russo, 2015). The middle-age period is from 30 to
59 years old, while the old-age period is from 60 to 89 years old. Following Lancia andRusso (2015), we assume an annual gross population growth rate of 1:006; which is theOECD average rate during 1995 2009: This assumption implies that the gross population
growth rate for 30 years is (1:006)30 ' 1:197:

Following the literature (Gonzalez-Eiras and Niepelt, 2008; Lancia and Russo, 2015),
the political power of the elderly, denoted by !, is set equal to the per-capita in‡uence ofthe middle-aged and the elderly. We set ! = 2:3 to indicate that the middle-aged and theelderly have approximately the same per-capita in‡uence when

= 0:52; that is, when

individuals live for 74:6 years on average. The remaining two parameters, and , areset at = 0:61 and = 0:259, respectively, to satisfy the following two requirements: (i)the education-to-GDP ratio is around 5:4%, which is the OECD average for 1995-2009(Lancia and Russo, 2015), and (ii) the condition of ! ( )2 < 1 + n < !(1 + ) has a
hump-shaped pattern in the education-to-GDP ratio in the present framework.

5

Economic GrowthBased on the result established thus far, we derive the growth rate of the economy, andinvestigate how it is a¤ected by population aging. For the presentation of the analysis,
consider per capita output, yt , which is de…ned by yt
growth rate of per capita output isy0
A(k 0 ) h0=;yA(k) h14

Yt =Nt = A(kt ) ht : Then, thewhere x0 denotes the next period x(= k; h; y). In the steady state with k 0 = k, thegrowth rate of per capita output, y 0 =y, is equal to the growth rate of human capital, h0 =h.
Therefore, in what follows, we focus on the steady-state growth rate of human capital.
The analysis proceeds as follows. First, we consider the cases of p > 0 and p = 0,
separately. We show that the physical-to-human capital ratio, k = K=H, stably converges
to a unique steady state for each case, and that at the steady state, the growth rate ofhuman capital remains constant across periods. Second, we undertake numerical analysis
to investigate the overall e¤ects of increases in ! andand a decrease in n on the growthrate at the steady state.

5.1

Steady-state Growth Rate

Recall the capital market-clearing condition in De…nition 2. With the use of the policy
functions derived in Proposition 1, we reformulate it as follows:
k0 =

k

P

(1

)A (k) h1+np(1 + n)x (x)(h) (1

)

(12)

;

where
k

The term1+n
D (1 + n) + 1+

PP

k ( ) is constant and dependent on P , where P

o:is the coe¢ cient of the policy

function, p = P A(k) h, and is presented in Proposition 1. The derivation of theseequations and the de…nition of k ( ) are provided in Appendix A.1.
Suppose that p > 0. The policy functions are given by
P (k; h) = Pp>0 A (k) h;X(k; h) = Xp>0 A (k) h:
Substituting these functions into the physical capital formation function in (12), we obtainthe law of motion of physical capital when p > 0, as follows:
k0 =

k

Pp>0

(1

)

1+n

Pp>0

(1 + n)Xp>0

Xp>0

(A)(1

)

(k) (1): (13)The equation implies that a unique and non-trivial steady state exists and that for any
initial condition k > 0, the sequence of k stably converges to the unique steady state.
From (13), we compute the steady-state level of k when p > 0, denoted by kp>0 , as follows:
kp>0 =

"

1
k

Pp>0

(1

)

1+n

Pp>0

(1 + n)Xp>0

Xp>0

(1

(A)

)

#1=f1
(14)

15

(1

)g

:

Using kp>0 in (14) and the policy functions in Proposition 1, we write the law of motionof human capital when p > 0 as h0 jp>0 = D Xp>0 A kp>0 h(h)1 ; or
h0
=D
h p>0

Xp>0 A kp>0

(15)

:

The equation shows that the growth rate is constant across periods at the steady state.
Following the same procedure, we obtain the growth rate when p = 0 as follows:
h0
=D
h p=0

Xp=0 A kp=0

;

(16)

where kp=0 is a unique and stable steady-state level of k when p = 0.

5.2

Numerical Analysis

To investigate the growth e¤ect of increases in ! and and a decrease in n, we undertake
numerical analysis. We follow the assumption introduced in Section 4: each generation
lasts for 30 years; the population growth rate for 30 years is (1:006)30= 1=3,= 0:61; and1 ' 0:197; and= 0:259: In addition, we normalize A as A = 1, and set D = 2:77

to obtain an empirically plausible result of the growth rates.
Within these assumptions, we undertake the numerical analysis and obtain the following results, as illustrated in Figure 3. First, an increase in ! leads to a decrease in thegrowth rate, as depicted in Panel (a). Second, a decrease in n leads to an increase in thegrowth rate, as depicted in Panel (b). Third, an increase in leads to a hump-shaped
pattern when !( )2 < 1 + n < !(1 + ) , as depicted in Panel (c). Therefore, the threeparameters, !; , and n, representing population aging, have di¤erent e¤ects on economic
growth.
[Figure 3 here.]
To understand the mechanism behind the result, recall the growth rate of humancapital in Eqs. (15) and (16). The growth rate is a¤ected by !; , and n through the two
factors, that is, the policy function of public education represented by Xp>0 and Xp=0 ,and the steady-state capital, represented by kp>0 and kp=0 . The …rst factor is crucial indetermining the growth rate because parameters related to population aging, !; , and n;
directly a¤ect the growth rate via the term Xp>0 or Xp=0 . Therefore, hereafter, we focus
on Xp>0 and Xp=0 to interpret the result.

16

Recall Xp>0 and Xp=0 in Proposition 1, which are reformulated as follows:

Xp>0 =( + ) (1)}|{( + ) (1
)
=
;
!
+
(1
+
n)
+
!
+
(1
+
n)(1
+
(
+
)
)
|{z} | {z } |{z} |
{z
}
( 1)

Xp=0 =( + ) (1
!

)2

( 4a)

z

( 4b)

z

( 2)

( 3)

( 4a)

}|
{
( + ) (1)2=;
(1 + n) + ! + (1 + n)(1 + ( + ) )| {z } |{z} |
{z
}
( 2)

( 4b)

( 3)

where the terms (*1), (*2), and (*3) correspond to those in the political objective function
in Eq. (10), and the terms (*4a) and (*4b) correspond to the term (*4) in Eq. (10). The
terms (*1) and (*2) show a negative e¤ect of increased political power of the old on
education spending, and thus, on the growth rate of human capital; the terms (*3) and(*4b) show a positive e¤ect of a decreased population growth rate on education spending
and the growth rate of human capital. Therefore, the e¤ect on the growth rate is de…nitewhen ! and n.2
The e¤ect ofon the growth rate is not straightforward. The terms (*1), (*2), and(*3) show a negative e¤ect of longevity on education spending, and thus, on economic
growth, while the term (*4a) shows a positive e¤ect. However, the negative e¤ect via
the term (*1) appears only when the public pension is provided, because the term (*1)
is irrelevant for political decision making when there is no provision of public pension.
That is, the negative e¤ect ofwhen p > 0 is larger than that when p = 0. Becauseof this di¤erence, the negative e¤ect outweighs the positive e¤ect when p > 0, while thenegative e¤ect is outweighed by the positive e¤ect when p = 0. This is the source of the
hump-shaped pattern of the growth rate a¤ected by longevity.

6

Role of Annuity Markets

The analysis thus far has assumed perfect annuity markets. However, in the real world,
some countries have limited or no access to annuity markets. We expect this to in‡uence
individual economic and political decisions, which in turn a¤ect long-run economic growth.
To explore this possibility, this section modi…es the model by assuming no annuity market,
and investigates how the results di¤er in this alternative scenario.
2

We should note that a lower population growth rate results in a lower aggregate education spendingto-GDP ratio, as demonstrated in Section 4, whereas it results in larger per capita spending on education,
as demonstrated here. The di¤erence arises because Xp>0 (or Xp=0 ) is multiplied by the gross population
growth rate, 1 + n, when we compute the aggregate education spending. The factor 1 + n works in anegative direction and outweighs the positive e¤ect through Xp>0 (or Xp=0 ).

17

For this purpose, we demonstrate the no-annuity-market case in the following way.
The budget constraints of the middle-aged and the elderly are now given by
c t + stdt+1

(1t )wt ht + bt ;Rt+1 st + pt+1 ;where bt is the per capita accidental bequest. If an individual dies at the end of middle
age, his/her unannuitized wealth, Nt (1)Rt+1 st ; is distributed to his/her o¤spring as

an accidental bequest: Nt+1 bt+1 = Nt (1)Rt+1 st ; orbt+1 =

1
Rt+1 st :
1+n

(17)

We solve the utility maximization problem with the above budget constraints andobtain the saving and consumption functions. Then we substitute these, the government
budget constraints, and (17) into the utility function of the middle-aged, and make aconjecture related to the policy functions as in Section 3 to obtain the following indirect
utility function for the middle-aged:
V M = f1 + ( + ) g ln (1+ ( + ) (1where the term (1)A (k) h + (1

) A (k) h1+np(1 + n)x) ln x;) A (k) h peculiar to the modi…ed model represents the accidental

bequest. The indirect utility function of the elderly is also obtained as
V o = ln [(1 + n) A (k) h + p] + ln (1)A (k) h1+np(1 + n)x :Following the same procedure as in the previous sections, we seek the policy functions
that maximize the political objective,= ! V o + (1 + n)V M : However, we are unable toobtain analytical solutions of the policy functions because of the presence of the additionalterm (1) A (k) h. To resolve this di¢ culty, we solve the maximization problem
numerically. We adopt the parameter values introduced in the previous sections, and use
the numerically obtained policy functions to compute the steady-state level of capital andthe rate of per capita human capital growth. Figure 4 illustrates how longevity a¤ects
the pension-to-GDP ratio and the steady-state growth rate.
[Figure 4 here.]
The …gure shows that the no annuity market case di¤ers from the case with perfectannuity markets in three respects. First, public pensions are provided for any probabilityof living to old age,

2 [0; 1]. No annuity markets implies a lower return from savings18

than when there are perfect annuity markets. This strengthens the elderly population’s
incentives to expand public pension provision. Second, the pension-to-GDP ratio exhibits
a hump-shaped pattern as longevity increases. Children receive a smaller bequest as
parents’ longevity increases, implying a negative income e¤ect that gives the middleaaged an incentive to prefer a smaller tax burden and thus a lower level of public pension
provision. This negative e¤ect on pension dominates the positive e¤ect demonstrated in
Proposition 2 when longevity is above a threshold level. Third, the negative e¤ect on
pensions incentivizes individuals to save more for the retirement period. This leads toan increase in the steady-state level of capital. This positive e¤ect on capital dominates
the negative growth e¤ect through the policy function of public education presented in
Section 5.
The result in this section indicates that the presence (or absence) of annuity markets
is a key to the e¤ect of longevity on economic growth. As demonstrated in the previous
section, an increase in longevity could reduce the growth rate when individuals are able
to purchase private annuity contracts. However, the result in this section shows thatrising longevity de…nitely increases the growth rate when private annuity contracts areunavailable. To check the empirical relevance of the con‡icting predictions, we again lookat the observation in Panel (c) of Figure 1. In particular, we focus on four countries
with a high proportion of the population aged 65 years and above: the Czech Republic,
Estonia, Hungary, and the United Kingdom. The United Kingdom has GDP per capita
growth below 5%, while the Czech Republic, Estonia, and Hungary have growth rates
above 10%. There is a signi…cant di¤erence in the growth rates among these countries,
although they share a similar demographic structure.
The divergence between the United Kingdom and the three Eastern European countries could be explained by the degree of private annuitization. In fact, the OECD (2014,
Factbook) reports that the ratio of private expenditure on pension to GDP, which captures the degree of private annuitization, was 1.6% in 2010 for the United Kingdom. Thisis above the OECD average of 0.9%. On the other hand, the ratio was 0.5% for the
Czech Republic, 0% for Estonia, and 0.2% for Hungary. These …gures are stable for 20072012. Therefore, we may well conclude that the overall trend in OECD countries showsa non-linear relationship between longevity and economic growth, though some countries
experience continued economic growth due to limited access to private annuities.3
3

Japan also exhibits a high share of population aged 65 years and above, though is not included in
the discussion above because of the lack of data related to private annuitization.

19

7

Summary and Conclusion

How does the con‡ict of interest between generations a¤ect the two redistribution policies,
namely, public education for the young and public pensions for the old? In turn, howdoes the con‡ict a¤ect economic growth? The present study attempted to answer thesequestions from a political economy viewpoint.
We considered three factors representing population aging: longevity, the politicalpower of the old, and the population growth rate. We showed that with greater politicalpower of the old and a lower population growth rate, the pension-to-GDP ratio increases
but the education-to-GDP ratio decreases. In addition, greater longevity results in ahigher pension-to-GDP ratio.
We demonstrated that the e¤ect of longevity on education spending is complex.
Greater longevity increases the weight of the utility of the elderly. This incentivizes thegovernment to shift spending allocations from education to pensions. However, greater
longevity also increases the weight of middle-aged agents’ utility from their children’s
human capital. This gives the government an incentive to increase public education
spending. These opposing e¤ects produce an initial increase followed by a decrease in the
education-to-GDP ratio, which in turn results in a hump-shaped pattern in the growthrate. This model prediction generally …ts the cross-country empirical evidence in developed countries. We also demonstrated that greater longevity always increases the growthrate when there is no private annuity market, and found that this result …ts the data fromsome Eastern European countries with limited access to private pensions.
The result established in the present study have policy implications related to aging
and economic growth. First, a decline in the population growth rate and an increase inlongevity have di¤erent e¤ects on economic growth. In particular, a decline in the population growth rate de…nitely increases per-capita growth, while an increase in longevity
has a non-monotone growth e¤ect. Therefore, policymakers should focus on increasing
life expectancy and its associated costs rather than falling birth rates. Second, diminishing the political power of the elderly can increase economic growth. One way to realize
this is to lower the minimum voting age. Japan, with the highest life expectancy amongdeveloped countries, has recently lowered the minimum voting age from 20 to 18. Thisis expected to strengthen the political power of the young, and to increase the economic
growth rate in the long run.

20

A

ProofsA.1

Derivation of (9)To derive (9), we …rst use the government budget constraint to replace in the indirect
utility functions by x and p. Then, we substitute the conjectures in (8) into the politicalobjective function. Finally, we replace k 0 and h0 with k; h; x and p, respectively, byusing the capital market clearing condition and the human capital production function.
In what follows, we provide the details of the calculation step by step.
Step 1.
Recall the government budget constraint in De…nition 2(ii), which is rewritten as
follows:
1=(1)A (k) h 1+n p (1 + n)x:(1)A (k) h

Plugging this into the indirect utility functions in (5) and (6), we obtain
V M = ln

1
1+

(1

1

A (k 0 )

+ ln

p
1+n
!

(1 + n)k 0 h0 + p0
1

A (k)

V o = ln

)A (k) h(1 + n)kh + p!

(1 + n)x ++ ln (1+ ln (1p0
A (k 0 )1)A (k 0 ) h0
)A (k) h

1+n

1+n

p

p0

(1 + n)x0 ;(1 + n)x :(18)

Step 2.
We substitute the conjecture of the policy functions in (8) into V M to obtain
V M = ln

1
1+

+ ln

(1
A (k 0 ))A (k) h

1+n

1

p

(1 + n)x +!

P A (k 0 ) h0
A (k 0 ) 1(1 + n)k 0 h0 + P A (k 0 ) h0

+ ln (1)A (k 0 ) h0

= ln (1)A (k) h

1+n
1+n

p

P A (k 0 ) h0

(1 + n)XA (k 0 ) h0

(1 + n)x +P k 0 h0 +( + ) ln k 0 + ( + ) ln h0 ;(19)

where the terms unrelated to the political decision are omitted from the expression.
Step 3.
To replace k 0 and h0 in (19) with k; h; p and x; we …rst recall the human capital

21

production function, h0 = D (x) (h)1 ; and the capital market clearing condition,(1 + n)k 0 h0 =
=

1+
1+

p0
R0

(1

) wh

(1

)A (k) h1+np(1 + n)x

P A (k 0 ) h0;
A (k 0 ) 1

where the …rst line comes from the capital market clearing condition, (1 + n)k 0 h0 = s withthe saving function in (1), and the second line comes from the pro…t maximization conditions in (3) and (4) and the conjecture of the policy function p0 in (8). After rearranging
the terms, we rewrite the abovementioned expression as follows:
k 0 h0 =1+

We substitute h0 = D (x) (h)1
k0 =
where

k

k

P

(1

(1

P

(1 + n) + 1+

)A (k) h1+np(1 + n)x :(x)(h) (1(20)

into (20) to obtain)A (k) h1+np(1 + n)x

)

;

(21)

P is de…ned by:k1+n
D (1 + n) + 1+P

By using (20) and (21), we rewrite V M in (19) as
V M = f1 + ( + ) g ln (1+ ( + ) (1)A (k) h

P

o:

1+n

p

(1 + n)x(22)) ln x;where constant terms are omitted from the expression. With V o in (18) and V M in (22),we write the political objective function

A.2= ! V o + (1 + n)V M as expressed in (9).

Proof of Proposition 1

Suppose that public pensions are provided in the next period, p0 > 0. Given that preferences are speci…ed by the logarithmic utility function, we conjecture linear policy functionsof public education and public pensions for the next period, x0 = Xp>0 A (k 0 ) h0 and
p0 = Pp>0 A (k 0 ) h0 ; respectively, where Xp>0 (> 0) and Pp>0 (> 0) are policy function
parameters when p > 0. Under this conjecture, the solution to the problem becomes:X(k; h) =P (k; h) =( + ) (1!

) A (k) h;

A (k) h;1+n22

whereis de…ned in Proposition 1. The solution for P (k; h) indicates that P (k; h) > 0

holds if and only if ! >. When ! >holds, the abovementioned solution
constitutes a Markov-perfect political equilibrium if(( + ) (1
);
Xp>0
(23)
!
:
Pp>0
1+n

Alternatively, suppose that p0 = 0; that is, public pensions are not provided in thenext period. Consider the estimation of policy functions as x0 = Xp=0 A (k 0 ) h0 and
p0 = Pp=0 A (k 0 ) h0 , where Xp=0 and Pp=0 are policy function parameters when p = 0.
The solution to the problem becomes:( + ) (1)2
A (k) h;!
P (k; h) = 0 if !
:X(k; h) =This solution constitutes a Markov-perfect political equilibrium if
Xp=0

)2

( + ) (1!

and Pp=0(24)

0.

The tax rates for p > 0 and p = 0 are obtained by substituting the corresponding
policy functions X and P into the government budget constraint.

A.3

Proof of Proposition 2

The pension-to-GDP ratio is
Nt 1 pt
Nt 1 pt(!
==
Yty t NtA(kt ) ht (1 + n))

A(kt ) ht =

1+n

1
=!

;

where the …rst equality comes from yt = Yt =Nt and the second equality comes from thepolicy function of pt presented in Proposition 1. Using the de…nition of in Proposition
1, we rewrite this expression as
Nt 1 pt=
Yt
1 + !1

1) + f!

[(1 + n) ( + ) (1;+ (1 + n)(1 + ( + ) )g]that is,Nt 1 pt=
Yt
1 + (1 + n) ! 1 +(1

1) + + (1 + n) !1

This equation states that Nt 1 pt =Yt is increasing in

23

1

+ 1+

:

and ! and is decreasing in n.

A.4

Proof of Proposition 3

Let Nt+1 xt =Yt jp>0 and Nt+1 xt =Yt jp=0 denote the public education-to-GDP ratio whenp > 0 and p = 0, respectively. Using the policy function of xt in Proposition 1, they are
expressed as follows:
Nt+1 xt
1;= (1 + n)Xp>0 =Yt p>01 + p>0
Nt+1 xt
1;= (1 + n)Xp=0 =Yt p=01 + p=0
where
p>0

p=0

The termsp>0 and! + f! + (1 + n) (1 + ( + ) )g;
(1 + n) ( + )(1)! + (1 + n) (1 + ( + ) ):(1 + n) ( + )(1)

p=0 are increasing in ! and decreasing in n.

Therefore, the ratio

decreases as ! increases and as n decreases for both cases of p > 0 and p = 0.
To determine the e¤ect oftoon the ratio, we di¤erentiate

p>0 and

p=0 with respect

and obtain
@ p>0
=
@
(1 + n) (@ p=0=@(1 + n) (1+ )2 (11+ )2 (1))[!(1 + )!( )2(1 + n)] ;(1 + n) :These expressions indicate that the following holds:!
!
Nt+1 xt
Nt+1 xt=@
0 and @=@ > 0 if !(1 + )(1 + n);@Yt p>0Yt p=0!
!
Nt+1 xt
Nt+1 xt@=@ < 0 and @=@ > 0 if !( )2 < (1 + n) < !(1 + ) ;Yt p>0Yt p=0!
!
Nt+1 xt
Nt+1 xt@=@ < 0 and @=@0 if (1 + n) !( )2 :Yt p>0Yt p=024

References[1] Acemoglu, D., and Robinson, J., 2005. Economic Origins of Dictatorship and Democracy.
Cambridge University Press, Cambridge, United Kingdom.
[2] An, C.B., and Jeon, S.H., 2006. Demographic change and economic growth: An inverted
U-shape relationship. Economics Letters 92, 447–454.
[3] Arawatari, R., and Ono, T., 2014. Old-age social security vs. forward intergenerational
public goods provision. Japanese Economic Review 65, 282–315.
[4] Bassetto, M., 2008. Political economy of taxation in an overlapping-generations economy.
Review of Economic Dynamics 11, 18–43.
[5] Bearse, P., Glomm, G., and Janeba, E., 2001. Composition of government budget, nonsingle peakedness, and majority voting. Journal of Public Economic Theory 3, 471–481.
[6] Beauchemin, K.R., 1998. Intergenerational politics, …scal policy and productivity. Review
of Economic Dynamics 1, 835–858.
[7] Bernasconi, M., and Profeta, P., 2012. Public education and redistribution when talents
are mismatched. European Economic Review 56, 84–96.
[8] Bishnu, M., and Wang, M., 2014. The political intergenerational welfare state. Iowa State
University Department of Economics Working Paper No. 14014.
[9] Boldrin, M., and Montes, A., 2005. The intergenerational state education and pensions.
Review of Economic Studies 72, 651–664.
[10] Chen, K., and Song, Z., 2014. Markovian social security in unequal societies. Scandinavian
Journal of Economics 116, 982–1011.
[11] Forni, L., 2005. Social security as Markov equilibrium in OLG models. Review of Economic
Dynamics 8, 178–194.
[12] Glomm, G., 2004. Inequality, majority voting and the redistributive e¤ects of public education funding Paci…c Economic Review 9, 93–101.
[13] Glomm, G., and Kaganovich, M., 2008. Social security, public education and the growthinequality relationship. European Economic Review 52, 1009–1034.
[14] Glomm, G., and Ravikumar, B., 1995. Endogenous public policy and multiple equilibira.
European Journal of Political Economy 11, 653–662.
[15] Glomm, G., and Ravikumar, B., 2001. Human capital accumulation and endogenous public
expenditure. Canadian Journal of Economics 34, 807–826.
[16] Glomm, G., and Ravikumar, B., 2003. Public education and income inequality. European
Journal of Political Economy 19, 289–300.
[17] Gonzalez-Eiras, M., and Niepelt, D., 2008. The future of social security. Journal of Monetary
Economics 55, 197–218.
[18] Gonzalez-Eiras, M., and Niepelt, D., 2012. Aging, government budgets, retirement, andgrowth. European Economic Review 56, 97–115.
[19] Gradstein M., and Kaganovich, M., 2004. Aging population and education …nance. Journalof Public Economics 88, 2469–2485.
[20] Grossman, G., and Helpman, E., 1998. Intergenerational redistribution with short-lived
governments. Economic Journal 108, 1299–1329.

25

[21] Hassler, J., Krusell, P., Storesletten, K., and Zilibotti, F., 2005. The dynamics of government. Journal of Monetary Economics 52, 1331–1358.
[22] Heer, B., and Irmen, A., 2014. Population, pensions, and endogenous economic growth.
Journal of Economic Dynamics and Control 46, 50-72.
[23] Holtz-Eakin, D., Lovely, M.E., and Tosun, M.S., 2004. Generational con‡ict, …scal policy,
and economic growth. Journal of Macroeconomics 26, 1–23.
[24] Iturbe-Ormaetxe, I., and Valera, G., 2012. Social security reform and the support for public
education. Journal of Population Economics 25, 609–634.
[25] Kaganovich, M., and Meier, V., 2012. Social security systems, human capital, and growth
in a small open economy. Journal of Public Economic Theory 14, 573–600.
[26] Kaganovich, M., Zilcha, I., 2012. Pay-as-you-go or funded social security? A general equilibrium comparison. Journal of Economic Dynamics and Control 36, 455–467.
[27] Kemnitz, A., 2000. Social security, public education, and growth in a representative democracy. Journal of Population Economics 13, 443–462.
[28] Kunze, L., 2014. Life expectancy and economic growth. Journal of Macroeconomics 39,
54–65.
[29] Lambrecht, S., Michel, P., and Vidal, J.P., 2005. Public pensions and growth. European
Economic Review 49, 1261–1281.
[30] Lancia, F., and Russo, A., 2015. Public education and pensions in democracy:
A political economy theory. http://www.sv.uio.no/econ/english/research/unpublishedworks/working-papers/2015/memo012015.html (July 11, 2015).
[31] Levy, G., 2005. The politics of public provision of education. Quarterly Journal of Economics 120, 1507–1534.
[32] Lindbeck, A.J., and Weibull, W., 1987. Balanced-budget redistribution as the outcome ofpolitical competition. Public Choice 52, 273–297.
[33] Ludwig, A., Schelkle, T., and Vogel, E., 2012. Demographic change, human capital andwelfare. Review of Economic Dynamics 15, 94-107.
[34] Naito, K., 2012. Two-sided intergenerational transfer policy and economic development: A
politico-economic approach. Journal of Economic Dynamics and Control 36, 1340–1348.
[35] OECD, 2014. OECD Factbook 2014: Economic, Environmental and Social Statistics.
http://www.oecd-ilibrary.org/economics/oecd-factbook-2014_factbook-2014-en (February
20, 2016).
[36] Ono, T., 2015. Public education and social security: A political economy approach. Economics of Governance 16, 1–25.
[37] Ono, T., and Uchida, Y., 2014. Pensions, education, and growth: A positive analysis.
https://ideas.repec.org/p/osk/wpaper/1437.html (July 11, 2015).
[38] Persson, T., and Tabellini, G., 2000. Political Economics: Explaining Economic Policy,
MIT Press, Cambridge, Massachusetts.
[39] Poutvaara, P., 2006. On the political economy of social security and public education.
Journal of Population Economics 19, 345–365.
[40] Rangel, A., 2003. Forward and backward intergenerational goods: Why is social security
good for the environment? American Economic Review 93, 813–834.

26

[41] Soares, J., 2006. A dynamic general equilibrium analysis of the political economy of public
education. Journal of Population Economics 19, 367–389.
[42] Song, Z., 2011. The dynamics of inequality and social security in general equilibrium. Review
of Economic Dynamics 14, 613–635.
[43] Song, Z., Storesletten, K., and Zilibotti, F., 2012. Rotten parents and disciplined children:
A politico-economic theory of public expenditure and debt. Econometrica 80, 2785–2803.
[44] Tosun, M.S., 2008. Endogenous …scal policy and capital market transmissions in the presence of demographic shocks. Journal of Economic Dynamics and Control 32, 2031–2060.

27

Public spending on education (% of GDP, Average 00−10)
2
3
4
5
6
7
8
9

Public spending on pension (% of GDP, average 05−09)
0
5
10
15

ITA
FRA
AUT
POL

GRC

PRT

DEU

SVN
HUN BEL
FIN
ESP
SWE

CZE
LUX
CHE

SVKUSA

TUR

DNK
GBR
CHL

ISR
IRL

NZL

JPN

EST

NLD
NOR
CAN
AUS

MEX

5

KOR

ISL

10
15
Population ages 65 and above (% of toal, average 05−09)

20

DNK
ISL
SWE

NOR
NZL

ISR

IRL

MEX

FIN

NLD
USA
POL
CAN
AUS

BEL

FRA
SVN AUT
CHE
EST
PRT
HUN
GBR
ESP

KOR
SVK

LUX

CHL

DEU
ITA

CZE
GRC

JPN

TUR

5

10
15
Population ages 65 and above (% of total, Average 00−10)

(b)
EST
CZE
SVK
HUN

POL
TUR

CHL

NZLAUS

KOR

IRL

ISR
MEX

SVN NORLUXCAN NLDGRC

ESP
CHE PRT
BEL
FINAUTDNKFRA SWE ITA
DEU

GBR
ISL USA
JPN

0

GDP per capita growth rate (Average 00−10)
5
10
15

(a)

20

5

10
15
Population ages 65 and above (% of total, 00)

20

(c)

Figure 1: Panel (a): public pension spending-to-GDP ratio and population aged 65 years
and above, 2005–2009. Panel (b): public education spending-to-GDP ratio and population
aged 65 years and above, 2000–2010. Panel (c): per capita growth rate and population
aged 65 years and above. Source: World Bank, 2015, World Bank Indicators; OECD, social expenditure database (https://data.oecd.org/socialexp/pension-spending.htm, July11, 2015).

28

0.09
P>0
P=0

0.085

public education-to-GDP ratio

0.08
0.075
0.07
0.065
0.06
0.055
0.05
0.045
0.04
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2: Education spending-to-GDP ratio and the longevity parameter π.

29

1.1
P>0
P=0

1.4

P>0
P=0

1.08

1.35
1.06
1.3
1.04
1.25
1.02
1.2
1
1.15
0.98
1.1
0.96

1.05
1

0.94

0.95

0.92

0.9

0.9
0

0.5

1

1.5

2

2.5

3

3.5

0

0.1

0.2

(a)

0.3

0.4

0.5

0.6

(b)
1.1
1.05
1
0.95
0.9
0.85
0.8
P>0
P=0

0.75
0.7
0.65
0.6
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c)

Figure 3: Panel (a): per capita human capital growth rate and ω. Panel (b): per capita
human capital growth rate and n. Panel (c): per capita human capital growth rate and
π.

30

0.14

1.1

public pension-to-GDP ratio

P=0
P>0

0.12

1

0.1

0.9

0.08

0.8

0.06

0.7

0.04

0.6

0.02

0.5

0

P=0
P>0

0.4
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

(a)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 4: Panel (a): public pension-to-GDP ratio and π. Panel (b): per capita humancapital growth rate and π.

31