Discussion Papers In Economics
And Business

Age Gap in Voter Turnout and Size of Government Debt

Ryo Arawatari and Tetsuo Ono
Discussion Paper 16-24-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAge Gap in Voter Turnout and Size of Government Debt

Ryo Arawatari and Tetsuo Ono
Discussion Paper 16-24-Rev.

February 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAge Gap in Voter Turnout and Size of Government
Debt∗
Ryo Arawatari†

Tetsuo Ono‡

Nagoya University

Osaka University

Abstract
We consider a cross-country diﬀerence of age gap in voter turnout and its impact on fiscal policymaking in a multi-country, overlapping-generations model. We
present conflict over fiscal policy between successive generations (i.e., the young
and elderly). We show that higher turnout of the elderly in voting may have a nonmonotone eﬀect on the size of government debt, depending on voters’ inter-temporalelasticity of substitution of public expenditure.
Keywords: fiscal policy; voter turnout; public debt; probabilistic voting; small
open economies.
JEL Classification: D70; E62; H63∗

Ryo Arawatari acknowledges financial support from Grant-in-Aid for Scientific Research (C) from
the Ministry of Education, Culture, Sports, Science and Technology of Japan No. 17K03620. Tetsuo
Ono acknowledges financial support from Grant-in-Aid for Scientific Research (C) from the Ministry of
Education, Culture, Sports, Science and Technology of Japan No. 15K03509.
†
Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601,
Japan. E-mail: arawatari@soec.nagoya-u.ac.jp‡
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Email: tono@econ.osaka-u.ac.jp

1

Introduction

The Organisation for Economic Co-operation and Development (OECD, 2007) reportsthat the voter turnout of young people is, on average, 20 points lower than that of individuals aged 65 years and over. In particular, it indicates a considerable diﬀerencein the age gap in voter turnout among OECD countries (see Table 1). This trend isalso reported by Smets (2012), who surveys national election data for the following 10
countries: Canada, Denmark, Finland, Germany, Great Britain, Italy, the Netherlands,
Norway, Sweden, and the United States. She defines the age gap as the turnout diﬀerence
between younger (aged ≤ 35 years) and older voters (aged > 35 years) and reports that inthe 2000s, the gap was around or above 20% in Canada, Finland, Norway, Great Britain,
and the United States, whereas it was around or below 10% in the other five countries.
[Table 1 is here.]
The age gap in voter turnout could aﬀect fiscal policy formation (Salavov, 2006). In
particular, it should pressurize politicians to shift the fiscal burden from the present generation to future generations. This pressure incentivizes the government to finance a partof government expenditure through the issue of public bonds. This in turn may increase
the debt-to-gross domestic product (GDP) ratio in the long run (Song, Storesletten, andZilibotti, 2012). However, this prediction does not seem to fit well with observations inOECD countries. Figure 1 depicts the correlation between the age gap in voter turnout
and government debt-to-GDP ratio. The age gap is measured by the percentage-point
diﬀerence in voting rates between older voters (aged>55 years) and younger voters (aged
16–35 years). No remarkable correlation could be observed from the data when the twooutliers, Great Britain and Japan, were omitted from the sample.
[Figure 1 is here.]
Against this background, the present study provides a simple dynamic politico-economic
model that enables us to explain various relationships between the age gap in voter turnout
1and debt-to-GDP ratio observed in OECD countries. For the analysis, we use the model ofSong, Storesletten, and Zilibotti (2012), who propose a dynamic politico-economic modelof public debt. Their model assumes that the world economy consists of a unit mass of
small open economies and that each country is populated by overlapping generations wholive for two periods, that is, the working young and retired old, both of whom benefit frompublic goods provision financed by tax and public bond issues. The authors employ theprobabilistic voting method developed by Lindbeck and Weibull (1987) to demonstratefiscal policymaking. Under this voting system, the government in each period chooses
fiscal policy that maximizes the weighted average utility of voters (i.e., the young andold). We extend the framework of Song, Storesletten, and Zilibotti (2012) by assuming
cross-country diﬀerences in the rates of abstention from voting. In particular, we assume
that voter turnout rates diﬀer between the young and old as well as among countries.
Within this framework, we consider an increased turnout rate (i.e., a lower abstentionrate) of the old. A higher turnout rate of the old is associated with a larger weight on
the old in the political objective. This induces the government to respond more strongly
to the demand of the retired old. Given that the retired old do not have a tax burden
but benefit from public goods, the government responds to the request from the old byraising the tax rate and issuing more public bonds. Therefore, a larger age gap in voter
turnout works to increase public goods provision and thus raise the tax and public debt
burdens on the working young.
Higher public goods provision today makes the young prefer higher public goods intheir old age because they want to smooth public goods consumption over their life cycles.
The government responds to this demand of the young by restraining the issue of publicbonds and compensates for the loss of revenue by raising the tax rate. This in turn lowers
debt repayment costs, increases the budget in the next period, and thereby enables thegovernment to increase public goods provision in the next period. Therefore, the higher
voter turnout of the old definitely increases the tax rate, while it creates two opposing
eﬀects on public bond issues.

2

We show that the negative eﬀect on public bond issues depends on the magnitude of
the inter-temporal elasticity of substitution of public goods. A lower elasticity implies lessdemand for future public goods provision by the young. In other words, the governmentis less inclined to control public bond issues as elasticity decreases. Our analysis showsthat there is a critical value of elasticity, such that the negative eﬀect of the voter turnout
rate of the old on public bond issues is outweighed by its positive eﬀect when elasticity
is below the critical value. In addition, the analysis shows that there is another criticalvalue of elasticity such that the negative eﬀect outweighs the positive eﬀect. Furthermore,
the two opposing eﬀects balance each other at some voter turnout rate of the old whenelasticity lies between the two critical values. This case could be viewed as representing
the non-monotone relationship between the age gap in voter turnout and debt-to-GDP
ratio.
We further analyze the eﬀect of the age gap in voter turnout on public bond issues
by focusing on the parameter representing the preferences for public goods, as in Song,Storesletten, and Zilibotti (2012). The preferences may aﬀect the two opposing eﬀects
of the voter turnout rate of the old on public bond issues in the following ways. First,
greater preferences for public goods increase the weight on the old’s utility of public
goods, thereby strengthening the positive eﬀect on public bond issues. Second, greater
preferences for public goods increase the weights on the young’s utility of public goodsin their youth and old age. However, greater preferences for public goods do not aﬀect
the negative eﬀect on public bond issues because the weights on the two periods of life
increase to the same degree. Therefore, the positive eﬀect on public bond issues becomeslarger as the preferences for public goods increase.
Our results contribute to the literature in the following two ways. First, the present
study extends the model of Song, Storesletten, and Zilibotti (2012), who examine therelationship between the political power of the old and debt-to-GDP-ratio. They showthat an increase in the relative political power of older agents, caused mainly by population
aging, leads to a higher steady-state level of public debt. Their model prediction, however,3does not seem to fit well with the observation in OECD countries. The present study
instead focuses on the age gap in voter turnout and shows various relationships between
this age gap and the size of government debt, which may fit well with the evidence inOECD countries.1
Second, the present study is related to the political economy of population aging
and government expenditure. As the population ages, the median voter becomes older
and hence is more willing to support larger government expenditure. Several studies
support this view from the political economy perspective (e.g., Gradstein and Kaganovich,
2004; Bassetto, 2008; Gonzalez-Eiras and Niepelt, 2008, 2012). However, an eﬀect inthe opposite direction could be expected because aging reduces the willingness of theworking generation to accept larger government expenditure (Razin, Sadka, and Swagel,
2002; Galasso and Profeta, 2007). These studies assume a balanced government budget,and thus leave the issue of government debt unresolved. The present study focuses on
government debt rather than government expenditure as a measure of government size andshows a possible negative eﬀect of the age gap in voter turnout on the size of government.
The rest of this paper is organized as follows. Section 2 presents the model. Section
3 introduces the political mechanism of fiscal policy formation and demonstrates fiscal
policy in voting. Section 4 characterizes a steady-state equilibrium and shows how thecross-country debt distribution is aﬀected by the age gap in voter turnout, preferences
for public goods, and inter-temporal elasticity of substitution of public goods. Section 5
provides concluding remarks.

2

Model

Our model is based on that developed by Song, Storesletten, and Zilibotti (2012). Timeis discrete and is denoted by t = 0, 1, 2, · · · . The world economy consists of a unit massof small open countries. Each country is populated by overlapping generations of agents
1

The present study is also related to Röhrs (2016), who focuses on the absence of commitment invoting as a source of ineﬃciency in a closed economy. However, it diﬀers from hers in that we show theimportance of the age gap in voter turnout for explaining the cross-country diﬀerence in fiscal policy.

4

who live for two periods: they work in the first period and retire in the second period.
There is no population growth and each generation has a unit mass. We assume thatcountries are identical except for voter turnout rates in order to shed light on their rolein shaping fiscal policy.

2.1

Utility Maximization

Each agent is assumed to receive utility from private consumption and publicly provided
goods. The utility of a type-i young agent in country j = [0, 1] born in period t is}
{ o(cyj,t )1−σ − 1(cj,t+1 )1−σ − 1(gj,t )1−σ − 1(gj,t+1 )1−σ − 1yUj,t =+θ·+β·+ θλ ·
,
1−σ

1−σ

1−σ

1−σ

where cyj,t is consumption during youth, coj,t+1 is consumption during old age, gj,t is public
goods provision in period t, β ∈ (0, 1] is a discount factor, and θ(> 0) and θλ(> 0) capture
the preference weights on public goods for the young and old, respectively. In particular,
the parameter λ captures the relative strength of the preference for public goods of the oldagents. The parameter σ(> 0) is an inverse of the inter-temporal elasticity of substitution.
A higher σ implies a lower elasticity (de la Croix and Michel, 2002, p.11).
Each young agent supplies one unit of labor inelastically and earns exogenous wage w.
The individual budget constraints of type-i agents during youth and old age are given by

cyj,t + sj,t ≤ (1 − τj,t ) · w,
coj,t+1 ≤ Rsj,t ,where sj,t represents savings, τj,t is an income tax rate in period t, and R is the (endogenous) world interest rate. Following Song, Storesletten, and Zilibotti (2012), we focus on
stationary equilibria and thus, we characterize an allocation of each country as a functionof a constant R.
We solve the utility-maximization problem and obtain the following consumption and5savings functions:cyj,t =
coj,t+1 =sj,t =

1
1 + β (βR)(1−σ)/σβ (βR)(1−σ)/σ1 + β (βR)(1−σ)/σβ (βR)(1−σ)/σ1 + β (βR)(1−σ)/σ· w · (1 − τj,t ) ,· R · w · (1 − τj,t ) ,· w · (1 − τj,t ) .

Ignoring irrelevant terms, we can express the indirect utility function of a type-i young
agent as follows:[{}σ1(1−σ)/σ· 1 + β (βR)· {w · (1 − τj,t )}1−σ
1−σ]+θ · (gj,t )1−σ + βλθ · (gj,t+1 )1−σ .

yVj,t= V y (τj,t , gj,t , gj,t+1 ) ≡(1)The first term in the square brackets on the right-hand side denotes the utility of consumption during youth and old age, the second term denotes the utility of public goods
during youth, and the third term denotes the utility of public goods during old age.
The indirect utility function of old agents in period t is expressed asoVj,t= V o (gj,t ) ≡1· λθ · (gj,t )1−σ ,
1−σ

(2)

where the irrelevant terms are omitted from the expression. Old agents each have the sameindirect utility function regardless of their type because their savings during youth arepredetermined and an individual’s utility function is assumed to be additively separable.

2.2

Government Budget Constraint

Government bonds are traded in international asset markets. Given inherited debt foryoung agents, bj,t , the government of country j in period t chooses an income tax rate,τj,t , public goods expenditure, gj,t , and a new issue of government bonds, bj,t+1 , subject

6

to the following government budget constraint:

bj,t+1 = gj,t + Rbj,t − τj,t w.

(3)where the revenue shortfall, gj,t +Rbj,t −τj,t w, could be covered by issuing new government
bonds, bj,t+1 .
Governments are committed to not repudiating debt. Thus, sovereign debt cannot
exceed the present value of the maximum feasible tax revenue (i.e., the natural debt
limit). In an environment with a constant interest rate and exogenous wages, the tax
revenue in period t is maximized at τj,t = 1. Therefore, the natural debt limit, denotedby b̄, is identical across countries, and is given by

bj,t+1 ≤

∞
∑
w
s=1

Rs

=

w
≡ b̄ for all j.
R−1

Hereafter, we omit time index t and use recursive notation with x′ denoting next periodx.

3

Politico-Economic Equilibrium

This section introduces the political mechanism of fiscal policy formation in the presenceof an age gap in voter turnout.

3.1

Probabilistic Voting with Abstention

The political mechanism in the present model is based on the probabilistic voting developed by Lindbeck and Weibull (1987). In this voting scheme, there is electoral competition
between two oﬃce-seeking candidates, denoted by, for example, A and B. Each candidate
announces a set of fiscal policies subject to the government budget constraint and thenatural debt limit to maximize his/her probability of winning the election. As we demonstrate in Appendix A.1, the two candidates’ platforms converge in equilibrium to the same7

fiscal policy that maximizes the weighted-average utility of voters. Formally, the politicalobjective function in country j is given byΩ(τj , gj , b′j ; ω(pj ), R) = ωj V o (gj ) + (1 − ωj )V y (τj , gj , b′j ),where ωj presents a relative political weight on the old in country j.
The present study diﬀers from standard probabilistic voting in that some individuals
abstain from voting. In particular, voter turnout rates diﬀer between the young and old.
For example, among the young’s (old’s) votes in country j, proportion qjy (qjo ) participatesin voting, while proportion 1 − qjy (1 − qjo ) abstains from voting. Therefore, the relative
political weight of the old, denoted by ωj , is specified as
ωj ≡

qjo
pj
,
y , i.e., ω (pj ) ≡oqj + qj
1 + pjwhere pj ≡ qjo /qjy denotes a relative turnout of the old. A higher relative turnout of theold implies a larger relative political weight on them. The weights on the young and old
are equal if there is no abstention.2
The current fiscal policy aﬀects the future fiscal policy through its eﬀect on debt
accumulation. Each candidate takes this eﬀect into account when shaping his/her fiscal
policy. To capture such an inter-temporal eﬀect, we here use the concept of stationary
Markov-perfect equilibrium (SMPE) described by Song, Storesletten, and Zilibotti (2012),
which is defined as follows:
Definition 1.

An SMPE comprises an interest rate R, a stationary debt distri-bution {bj }j , a debt rule b′j = B(bj ; ω(pj ), R) , a government expenditure rulegj = G(bj ; ω(pj ), R) , and a tax rule τj = T (bj ; ω(pj ), R), such that the following
two conditions hold:2

One way to endogenize the voter turnout rates is to assume a rational voting choice (see, for example,
Dhillon and Peralta, 2002). However, the present study assumes exogenous voter turnout rates because
the focus is on the eﬀect of the diﬀerence in the voter turnout rates on fiscal policy rather than themechanism behind the diﬀerence.

8

(i) ⟨B(bj ; ω(pj ), R), G(bj ; ω(pj ), R), T (bj ; ω(pj ), R)⟩ = arg max{τj ,gj ,b′j }Ω(τj , gj , b′j ; ω(pj ), R),and the government’s budget constraint and natural debt limit
are satisfied:B(bj ; ω(pj ), w, R) = G(bj ; ω(pj ), R) + Rbj − T (bj ; ω(pj ), R)w,
B(bj ; ω(pj ), R) ≤

w
≡ b̄.
R−1

(ii) the world asset market clears,∫∫
sj dj =jwhere b′j = B(bj ; ω(pj ), R) and sj =b′j dj,

j

1−σ
σ
1−σ
1+β(βR) σ· w · (1 − τj ).

β(βR)

The timing of events is as follows. (i) The two candidates, A and B, simultaneously
and non-cooperatively announce their electoral platforms. (ii) The election is held. (iii)
The elected candidate implements his/her announced policy platform. (iv) Given thepolicy platform, each individual sets savings and consumption from the viewpoint ofutility maximization. (v) Finally, international financial markets clear.

3.2

Voting on Fiscal Policy

Hereafter, we take the market-clearing interest rate as given for a while, and we solve themodel by backward induction. Since we have solved the utility-maximization problem
already, we are now ready to find the equilibrium fiscal policy. Given R, the first-orderconditions with respect to gj and τj are
−σ

(gj )

1−σ

∂G(b′j ; ω(pj ), w, R)
,
∂b′j

(4)

1
,
gj

(5)

= −f (ω(pj )) · (gj′ )−σ ·1 + β(βR) σ=
1 − τj(θβλf (ω(pj ))

) σ1

9

·w·

wheref (ω(pj )) ≡(1 − ω(pj ))βλ,(1 − ω(pj )) + ω(pj )λ(6)and gj = G(bj ; ω(pj ), w, R), gj′ = G(b′j ; ω(pj ), w, R), τj = T (bj ; ω(pj ), w, R), and b′j =gj + Rbj − τj w ≡ B(bj ; ω(pj ), w, R). The derivation of (4) and (5) is given in Appendix
A.2.
Condition (4) is a generalized Euler equation for public goods provision. The government chooses public goods provision to equate the marginal benefit on the left-handside to the marginal cost on the right-hand side. The term ∂G(b′j ; ω(pj ), w, R)/∂b′j on theright-hand side captures the disciplining eﬀect exercised by the young voters; this is qualitatively similar to that demonstrated in Song, Storesletten, and Zilibotti (2012). Thiseﬀect implies that the young agents’ concern for future public goods provision restrains
government expenditure and thus, prevents the government from running up too much
public bond issues. In particular, the term f (ω(pj )) on the right-hand side of (4) indicatesthat the disciplining eﬀect decreases as the political power of the old increases.
Condition (5) states that the government chooses the tax rate to equate the marginal
cost of taxation to its marginal benefit. A higher tax rate lowers the consumption of theyoung. The left-hand side of (5) presents this negative eﬀect of taxation. On the other
hand, a higher tax rate increases tax revenue. This incentivizes the government to reduce
current public bond issues, which, in turn, increases future public goods provision. The
right-hand side of (5) represents this positive eﬀect of taxation. We should note that, as
observed on the right-hand side, the relative turnout of the old, pj , aﬀects the shape ofthe fiscal policy through the decision on the tax rate. This point is investigated further
in the following analysis.
Based on the guess-and-verify method, we find policy functions that satisfy conditions(4) and (5), as demonstrated in the following lemma.

10Lemma 1. Given R and bj , country j’s policy functions in SMPE are given byG(bj ; ω(pj ), R) = γj∗ · (b̄ − bj ),} { f (ω(p )) } σ1 1{
1−σj
T (bj ; ω(pj ), R) = 1 − 1 + β(βR) σ ·· · γj∗ · (b̄ − bj ),
βλθ
w
{
}1
B(bj ; ω(pj ), R) = b̄ − f (ω(pj ))γj∗ σ · (b̄ − bj ),where γj∗ (> 0) satisfies following condition:[{} { f (ω(p )) } σ1
1−σ
j
∗
σ
R − γj · 1 + 1 + β(βR)
·
βλθ

]

{
}1
= f (ω(pj ))γj∗ σ .

(7)

Proof. See Appendix A.3.
The government expenditure on public goods is linearly related to the stock of public
debt. In particular, higher public debt is associated with lower public goods provision,
because debt accumulation increases debt repayment costs and thus, reduces the available
resources for the government. To compensate for this loss, the government increases thetax rate and public bond issues. This incentivizes the future government to issue more
public bonds to repay the increased debt. This might result in debt accumulation up to
the natural debt limit. We formally investigate this possibility in the next section.

4

Steady State

Here, we focus on a steady-state equilibrium in which the relevant terms are constantacross periods, and we assume it is stable. Our task is to determine the world interest ratethat satisfies Definition 1(ii) and the associated debt distribution that satisfies Definition
1(i). However, an analytical solution is not available, except for the case of 1/σ = 2,
as demonstrated in the next subsection. Therefore, we employ the following analysis
strategy. First, in Subsection 4.1, we pin down the world interest rate, R, for a given
world distribution of debt. Next, in Subsection 4.2, we determine the world distribution11

of debt for a given R. Finally, in Subsection 4.3, we solve for R and the debt distribution
simultaneously.

4.1

World Interest Rate

For the purpose of analysis, we rewrite the law of motion of debt in Lemma 1 as follows:()
b̄ − b′j = ϕ∗j · b̄ − bj ,

where
{
}1
ϕ∗j = ϕ∗ (ω(pj ), R) ≡ f (ω(pj ))γj∗ σ , j ∈ [0, 1].

(8)

We define ϕ̄ as
ϕ̄ ≡ max {ϕ∗ (ω(pj ), R)}j∈[0,1] .
Same as the model in the main text, the steady-state equilibrium interest rate, R∗ ,satisfies ϕ̄ = 1.Substituting ϕ̄ = 1 into (7) enables us to express the condition that
determines the equilibrium interest rate as follows:][} { f (ω(p ∗ )) } σ1
{
1−σ
1
j
R∗ = 1 +,· 1 + 1 + β(βR∗ ) σ ·f (ω(pj ∗ ))
βλθ

(9)

where j ∗ denotes a country that accumulates debt below b̄ and satisfies ϕ̄ = 1. Other
countries accumulate debt up to b̄ and attain ϕ̄ < 1. We should note here that we takethe distribution of debt as given, and thus, country j ∗ is not specified at this moment.
The following lemma establishes the condition for the existence and uniqueness of thesteady-state equilibrium interest rate, R∗ , for a given distribution of debt in the world
economy.
Lemma 2. Given the distribution of debt, there is a unique steady-state equilibrium interest rate, R∗ ≥ 1, satisfying (9) if 1/σ ∈ (0, 2) or if 1/σ = 2 and θ > f (ωj∗ )1/2 /λ.
Proof. See Appendix A.4.
12

The assumption about 1/σ is supported by the empirical studies of Kydland andPrescott (1982), Hall (1988), Campbell and Mankiw (1989), Browning, Hansen, andHeckman (1999), and Campbell (1999), who report 1/σ ∈ (0, 1], and those of Hansen
and Singleton (1982), Attanasio and Weber (1989), and Vissing-Jorgensen and Attanasio (2003), who report 1/σ ∈ (1, 2]. Following these estimations, hereafter, we assume1/σ ∈ (0, 2], which is also employed by Song, Storesletten and Zilibotti (2012). While
this assumption narrows the scope of our model, it enables us to solve the model in a
tractable way. When σ = 1/2, we add the assumption of θ > f (ωj ∗ )1/2 /λ to ensure thatthe gross interest rate R∗ lies within the range (1, ∞).

4.2

International Diﬀerences in Voter Turnout and Public Debt
Distribution

The policy function B(bj ; ω(pj ), R) in Lemma 1 suggests that the relative political weighton the old has an eﬀect on the public debt distribution through ϕ∗j . To isolate its eﬀect
from the interest rate eﬀect, we take R as given, and investigate the eﬀect of ω(pj ) on ϕ∗j .
Lemma 3. Given R, the following result holds:






∂ϕ∗j
≤0
∂ω(pj )

if Ψ(R, ω(pj )) ≥ σ1 − 1,∂ϕ∗j>0
∂ω(pj )

otherwise,whereΨ(R, ω(pj )) ≡1
1 + β (βR)(1−σ)/σ

{
·

βλθ
f (ω(pj ))

}1/σ
> 0.

Proof. See Appendix A.5.
[Figure 2 is here.]Let ωmax and ωmin denote the maximum and minimum values of ω(pj ) (0 < ωmin <ωmax < 1), respectively. Figure 2 illustrates the relationship between ω(pj ) and ϕj within
the range [ωmin , ωmax ]. In particular, panel (a) illustrates the case of 1/σ − 1 ≤ Ψ(R, ωmin )13

where ∂ϕj /∂ω(pj ) < 0 holds for any ω(pj ) ∈ [ωmin , ωmax ]. This case implies that countries
with the lowest ω(pj ) (ω(pj ) = ωmin ) have the highest ϕj (ϕj = ϕ̄ = 1). Panel (c)demonstrates the case of Ψ(R, ωmax ) ≤ 1/σ − 1 where ∂ϕj /∂ωj > 0 holds for any ω(pj ) ∈[ωmin , ωmax ]. In this case, countries with the highest ω(pj ) (ω(pj ) = ωmax ) have thehighest ϕj (ϕj = ϕ̄ = 1).
Panel (b) depicts the intermediate case between the cases of panels (a) and (c), whereΨ(R, ωmin ) < 1/σ − 1 < Ψ(R, ωmax ). There is a threshold level of ω(pj ) denoted by ωe (R),
which satisfies

Ψ(R, ω
e (R)) =1−1
σ[
1 − 1θ ·

1−σ
·
σ

{

}]σ

(10)

1 + β (βR)(1−σ)/σ[{}]σ ∈ (ωmin , ωmax )⇔ ωe (R) =
(1−σ)/σ1 − λ − 1θ · 1−σ
·
1
+
β
(βR)
σ
In this case, ∂ϕj /∂ωj ≥ 0 holds when ω(pj ) ∈ [ωmin , ωe (R)], whereas ∂ϕj /∂ωj < 0 holds
when ω(pj ) ∈ (e
ω (R), ωmax ]. Therefore, countries with ω(pj ) = ωe (R) have the highest ϕjwhere ϕj = ϕ̄ = 1.
The result described so far is summarized in the following proposition.
Proposition 1. Given the world interest rate, R, the steady-state distribution of public
debt is as follows.
(i) When 1/σ − 1 ≤ Ψ(R, ωmin ), countries with ω(pj ) = ωmin accumulate public
debt below b̄, whereas other countries accumulate public debt up to b̄.
(ii) When Ψ(R, ωmin ) < 1 /σ − 1 < Ψ(R, ωmax ), countries with ω(pj ) = ωe (R) ∈
(ωmin , ωmax ) accumulate public debt below b̄, whereas other countries accumulate public debt up to b̄.
(iii) When Ψ(R, ωmax ) ≤ 1/σ − 1, countries with ω(pj ) = ωmax accumulate public
debt below b̄, whereas other countries accumulate public debt up to b̄.
The result in Proposition 1 suggests that the relative turnout of the old, pj , and14

the inter-temporal elasticity of substitution, 1/σ, aﬀect the steady-state distribution of
public debt in the world economy. To understand their impacts, we first consider therole of the relative voter turnout of the old in shaping fiscal policy. Higher turnout of
the old increases their political influence in policymaking. This in turn incentivizes thegovernment to respond more strongly to the request from the old. Because the current
public goods provision is the only concern for the old, the government increases it byraising the tax rate as well as by increasing public bond issues.
However, higher public goods provision today makes the young prefer higher public
goods in the future because they want to smooth the utility of public goods across periods
from the viewpoint of utility maximization. In addition, the government responds to thispreference of the young by restraining the issue of public bonds, and compensates forthe loss of the revenue by raising the tax rate. Therefore, higher voter turnout of theold definitely increases the tax rate, while it creates two opposing eﬀects on public bondissues.
The negative eﬀect on public bond issues depends on the magnitude of the intertemporal elasticity of substitution, represented by 1/σ. A lower 1/σ implies a lower intertemporal elasticity of substitution, thereby providing less incentive for the government
to control public bond issues. In other words, the negative eﬀect becomes less eﬀective
as the inter-temporal elasticity of substitution decreases. In particular, when 1/σ is low
such that 1/σ − 1 ≤ Ψ (R, ωmin ), the negative eﬀect is outweighed by the positive eﬀect.
As a result, we obtain an equilibrium in which all the countries except the country withthe lowest voter turnout of the old accumulate debt up to the natural debt limit, as we
describe in Proposition 1(i) (see panel (a) of Figure 2).
The result is reversed when 1/σ is high such that 1/σ−1 ≥ Ψ (R, ωmax ) (see panel (c) of
Figure 2). In this case, the negative eﬀect outweighs the positive eﬀect because of the highinter-temporal elasticity of substitution. As a result, we obtain an equilibrium in which
all countries except the country with the highest voter turnout of the old accumulate
debt up to the natural debt limit (Proposition 1(iii)). Finally, when the elasticity lies
15

betweenΨ (R, ωmin ) and Ψ (R, ωmax ), there is a critical value of pj , defined by ω (pj ) = ω̃(R)such that the two opposing eﬀects balance each other. Below (above) the critical value,
the positive eﬀect outweighs (is outweighed by) the negative eﬀect, thereby resulting in an
equilibrium in which all the countries except the country with ω (pj ) = ω̃(R) accumulates
debt up to the limit (see panel (b) of Figure 2).

4.3

Stationary Markov-perfect Equilibrium

Based on the result established thus far, we are now ready to demonstrate the SMPE. We
first solve the model analytically for the case of 1/σ = 2, and then, we solve it numericallyfor the case of 1/σ ∈ (0, 2).
We first consider the case of 1/σ = 2. For the characterization of the equilibrium, we
focus on θ, representing the preference of public goods, and introduce the following two
threshold values of θ :
((
)
)
]1/2
1 [
· 2f (ωmax ) + 1 + β 2 /β 2 · (f (ωmax ))2
,
λ
((
)
)
]1/2
1 [
θ ≡ · 2f (ωmin ) + 1 + β 2 /β 2 · (f (ωmin ))2,λθ≡where θ < θ holds. Condition (9) yields the following world interest rate when 1/σ = 2:
1 + (1 + β ) ·2R∗ = 1 +{f (ω(pj )) − β 2 ·f (ω(pj ))
βλθ

{

}2

f (ω(pj ))
βλθ

}2 .

Accompanied by the result in Proposition 1, this equilibrium interest rate leads to thefollowing result:
Proposition 2. Suppose that 1/σ = 2 and θ > f (ωmax )1/2 /λ hold. The world distribu-

16

tion of debt is












bj < b̄ for ωj = ωmin and bj = b̄ for ωj ̸= ωminbj < b̄ for ωj = ωe (R∗ ) and bj = b̄ for ωj ̸= ωe (R∗ ) if
bj < b̄ for ωj = ωmax and bj = b̄ for ωj ̸= ωmax)[
θ ∈ θ, ∞ ,( )θ ∈ θ, θ ,

if

if

]
(
θ ∈ f (ωmax )1/2 /λ, θ ,where ωe (R∗ ) satisfies∗ωe (R ) =

1−

β+

1−λ−

√ 2

β +θ 2 λ2 (1+β 2 )
θ2 λ

√ 2

β+

β +θ 2 λ2 (1+β 2 )θ2 λ.

The corresponding world interest rate is
1+(1+β 2 )·{f (ωmin )/βλθ}2


1
+
if

f (ωmin )−β 2 ·{f (ωmin )/βλθ}2


1/2
2
2
2
2
1+{1+θ λ (1+β )/β }
R∗ =
1+2·
if
θ 2 λ2
22 1 + 1+(1+β )·{f (ωmax )/βλθ} 2 iff (ω)−β 2 ·{f (ω)/βλθ}
max

max

[
)
θ ∈ θ, ∞ ,( )θ ∈ θ, θ ,](θ ∈ f (ωmax )1/2 /λ, θ .

Proof. See Appendix A.6.
The result in Proposition 2 suggests that the parameter θ, representing the preferences
for public goods, also aﬀects the world distribution of debt. To understand the eﬀect of
θ in detail, recall the two opposing eﬀects of the relative voter turnout of the old, ω,on public bond issues, which are described in the previous section. The preferences for
public goods might or might not aﬀect these two opposing eﬀects of ω in the following
ways. First, a larger θ increases the weight on the old’s utility of public goods, thereby
strengthening the positive eﬀect of ω on public bond issues. Second, a larger θ increases
the weights on the young’s utility of public goods in their youth and old age. Because theweights on the two periods of life increase to the same degree, an increase in θ does not
aﬀect the young’s incentive for consumption smoothing behavior. This implies that thenegative eﬀect of ω is not aﬀected by an increase in θ.
Consequently, the positive eﬀect of ω increases as θ increases. In particular, when θis high such that θ ≥ θ, the positive eﬀect of ω outweighs its negative eﬀect. As a result,17

we obtain an equilibrium in which all countries except the country with the lowest voter
turnout of the old, ωj = ωmin , accumulate public debt up to the natural debt limit. The
result is reversed when θ is low such that θ ≤ θ. In this case, the negative eﬀect of ω
outweighs its positive eﬀect, and thus, we obtain an equilibrium in which all countries
except the country with the highest turnout of the old, ωj = ωmax , accumulate public
debt up to the natural debt limit. Finally, when θ lies between θ and θ, there is a criticalvalue of ωj , defined by ωj = ωe (R∗ ), such that the two opposing eﬀects balance each other.
Figure 3 illustrates three possible cases shown in Proposition 2.
[Figure 3 is here.]
The results in Propositions 1 and 2 suggest that the country with lower voter turnoutof the old is more likely to accumulate public debt below the natural debt limit and to
become a lender as 1/σ becomes lower and θ becomes higher. To confirm this, we solve
the model numerically and show that the statement holds for 1/σ ∈ (0, 2). Figure 4 showsthat the country with lowest voter turnout of the old, denoted by ωmin , accumulates public
debt below the natural debt limit and becomes a lender when 1/σ is low and θ is high.
On the other hand, the country with the highest voter turnout of the old, denoted byωmax , accumulates public debt below the natural debt limit and becomes a lender when1/σ is high and θ is low. Therefore, the present study suggests that the age gap in voter
turnout is a key determinant of the distribution of public debt in the world economy, andthat the impact of the age gap heavily depends on the preferences for public goods as wellas the inter-temporal elasticity of substitution.
[Figure 4 is here.]5

Conclusion

In this study, we investigate the eﬀect of the age gap in voter turnout on fiscal policymaking and show that this age gap has two opposing eﬀects on public bond issues. First,
18the higher voter turnout of the old increases their political influence. This incentivizes
the government to increase public goods provision today by issuing more public bonds
because the old do not bear the cost of future debt repayment. Second, higher public
goods provision today makes the young prefer higher public goods provision in the futurefrom the viewpoint of public goods consumption smoothing. Furthermore, the government responds to this request by the young by restraining the issue of public bonds today.
In particular, this negative eﬀect on public bond issues becomes less eﬀective as the intertemporal elasticity of substitution decreases. The net eﬀect of the two opposing eﬀects
depends on the magnitude of the inter-temporal elasticity of substitution. This is themain cause of the non-monotone relationship between the age gap and size of governmentdebt.
We demonstrate the above result by adopting several assumptions that make theanalysis tractable. In particular, we assume inelastic labor supply and thus we ignore thelabor–leisure choice by households. While this assumption enables us to derive an analytical solution, it results in an equilibrium that overly portrays the cross-country diﬀerence:
all countries except one are distinguished by debt up to the natural debt limit. We might
resolve this problem by introducing the labor–leisure choice of households; however, here,
we retain the assumption of inelastic labor supply to simplify the presentation.
We also assume no cross-country diﬀerence in population growth rates but instead
focus on the age gap in voter turnout. Introducing the diﬀerence in population growth
rates is expected to produce a similar eﬀect as the diﬀerence in age gaps because therelative political weights to the young and elderly vary as population growth rates change.
However, this prediction is not fulfilled, as demonstrated in Appendix B. We show that ahigher population growth rate is associated with a higher level of public debt. Therefore,
the diﬀerence in abstention from voting, rather than the diﬀerence in population growth
rates, is crucial for explaining the non-monotone eﬀect of voter turnout on the size ofgovernment debt.

19

A

ProofsA.1

Derivation of the Political Objective Function Ω (·)

The description of probabilistic voting follows that in Persson and Tabellini (2000). The
population consists of two distinct groups, J ∈ {y, o}, representing the young and old,
respectively. Since there is no population growth, the population share of each group is
1/2. An electoral competition takes place between two oﬃce-seeking candidates, denotedby A and B. Each candidate announces a set of fiscal policies {b′j , τj , gj }, subject to thegovernment budget constraint, b′j = gj + Rbj − τj w, and to the natural debt limit, b′j ≤ b̄.
At the time of the election, voters base their voting decisions on economic policy
announcements as well as the two candidates’ ideologies. Specifically, voter k in group J
prefers candidate A if
V J (τj,A , gj,A , b′j,A ) > V J (τj,B , gj,B , b′j,B ) + σ kJ + δ,
where V J is the indirect utility of group J, {τj,A , gj,A , b′j,A } and {τj,B , gj,B , b′j,B } are policy
sets that are announced by candidates A and B, respectively. Within each group, voters
diﬀer according to their ideology toward the two candidates. The parameter σ kJ , whichmeasures voter k’s individual ideological bias toward candidate B, is an individual-specificparameter that can take a negative or positive value. A positive value of σ kJ implies that
voter k is biased in favor of party B, whereas voters with σ kJ = 0 are ideologicallyneutral, that is, they care only about economic policy. We assume that this parameterhas a uniform distribution on [−1/(2ϕ), 1/(2ϕ)]. In addition, voters’ decisions are aﬀectedby the candidates’ average popularity. The parameter δ measures the average (relative)
popularity of candidate B in the population as a whole, and has a uniform distribution
on [−1/(2ψ), 1/(2ψ)].
The timing of events is as follows. (i) Candidates A and B simultaneously and noncooperatively announce their electoral platforms: {τj,A , gj,A , b′j,A } and {τj,B , gj,B , b′j,B }.

20

(ii) The actual value of δ is realized. (iii) Elections are held. (iv) The elected candidate
implements his/her announced policy platform.
In the present framework, some voters abstain from voting. The voter turnout rate ofgroup J is denoted by qjJ ∈ (0, 1), J ∈ {y, o}, which is independent from their ideologicalbias, σ kJ . For example, among the young in country j, proportion 1 − qjy participates invoting, whereas proportion qjy abstains from voting.
A swing voter is one whose ideological bias, given the candidates’ platform, makes
him/her indiﬀerent between the two parties. The swing voter’s ideology in group J(J = y, o), σ J , satisfies
σ J = V J (τj,A , gj,A , b′j,A ) − V J (τj,B , gj,B , b′j,B ) − δ.

(A.1)

Recall that q J denotes the voting rate of group J; this is independent from their ideologicalbias, σ kJ . Candidate A’s actual vote share is



)
)
(
(

1
1
1 
 o

y
o
y
×  qj ϕ · σ +ΓA ≡ y+ qj ϕ · σ +oqj + qj 
2ϕ
2ϕ 
|
{z
} |
{z
}
( old
)
∑
1
ϕ
J
J
·
q · σ +.
= y
qj + qjo J j
2ϕ

young

(A.2)

Using Eqs. (A.1) and (A.2), we compute candidate A’s probability of winning the election
as follows:[]
1
P = Probδ Γ ≥
2
[
]
(
)
∑
qjy + qjo
1
J
J
′
J
′
= Probδ
qj · V (τj,A , gj,A , bj,A ) − V (τj,B , gj,B , bj,B ) − δ +
≥
2ϕ
2ϕ
J
]
[
∑ {
}
1
J
′
J
′
J
)
≥δ
)
−
V
(τ
,
g
,
b
= Probδ y
V
(τ
,
g
,
b
·
q
j,B
j,B
j,A
j,A
j,B
j,A
j
qj + qjo J
A

A

=

∑ qjJ
}
{
1
+ψ
· V J (τj,A , gj,A , b′j,A ) − V J (τj,B , gj,B , b′j,B ) .
yo2
qj + qjJ
21

Candidate B wins the election with probability P B = 1 − P A ,PB = 1 − PA
=

∑ qjJ
{
}
1
+ψ
· V J (τj,B , gj,B , b′j,B ) − V J (τj,A , gj,A , b′j,A ) .
yo2
qj + qjJ

Each candidate chooses a policy platform to maximize his/her probability of winningthe election, given the other candidate’s policy platform. The two candidates’ choices
satisfy the following equations:∗∗, gj,A, b′∗(τj,Aj,A ) = arg

∗
(τj,B
,

∗
gj,B
,

b′∗
j,B ) = argmax ′τj,A ,gj,A ,bj,Amax

τj,B ,gj,B ,b′j,B

∑
J

}
{
qjJ
∗
∗
, gj,B, b′∗· V J (τj,A , gj,A , b′j,A ) − V J (τj,B
y
j,B )
o
qj + qj

J

{ J
}
qjJ
′
J
∗
∗
′∗
·
V
(τ
,
g
,
b
)
−
V
(τ
,
g
,
b
)
.
j,B
j,B
y
j,B
j,A
j,A
j,A
qj + qjo∑

The expressions suggest that the two candidates’ platforms converge in equilibrium to thesame fiscal policy that maximizes the weighted-average utility of each group,
∗
∗
∗
∗
′∗
(τj,A
, gj,A, b′∗j,A ) = (τj,B , gj,B , bj,B )= arg maxτj ,gj ,b′j ω(pj )V o (gj ) + (1 − ω(pj ))V y (τj , gj , b′j ),where pj ≡ qjo /qjy and ω(pj ) ≡ pj /1+pj . This is the objective function given in Subsection
3.1.
■

22

A.2

Derivation of Eqs. (4) and (5)We substitute Eq. (B.4) in Eq. (B.5) and reformulate the political objective function as
follows:1Ω(τj , gj , gj′ ; ω(pj ), w, R) = ω(pj ) ·· λθ · (gj )1−σ
1−σ
]
[{
}σ
1−σ
1
+ (1 − ω(pj )) ·· 1 + β(βR) σ· {w(1 − τj )}1−σ + θ · (gj )1−σ + βλθ · (gj′ )1−σ .
1−σ
(A.3)
We diﬀerentiate (A.3) with respect to gj and τj and obtain{(1 − ω(pj )) + ω(pj )λ}θ(gj )−σ + (1 − ω(pj ))βλθ(gj′ )−σ ·{}σ1−σ(1 − ω(pj )) 1 + β(βR) σ(−1)(1 − τj )−σ w1−σ∂gj′ ∂b′j·= 0,∂b′j ∂gj+ (1 − ω(pj ))βλθ(gj′ )−σ ·∂gj′ ∂b′j·= 0.
∂b′j ∂τj

(A.4)

(A.5)

The government budget constraint in (3) implies ∂b′j /∂gj = 1. We use this condition
to rewrite (A.4) as in (4). In addition, we rewrite (A.5) as}σ{
1−σ
1 + β(βR) σ(1 − τj )σ

∂gj′
·w
∂b′j
1
(1 − ω(pj )) + ω(pj )λ
=θ·
·w·
.
1 − ω(pj )(gj )σ· w1−σ = (−1)βλθ(gj′ )−σwhere we derive the first equality by using ∂b′j /∂τj = −w, and the second equality byusing (4). By rearranging the terms, we obtain Eq. (5).
■

23

A.3

Proof of Lemma 1

Recall the first-order conditions with respect to gj and τj , given by (4) and (5), respectively. We substitute (5) into the government budget constraint and obtain[]} { f (ω(p )) } σ1 1{1−σjb′j = gj + Rbj − w · 1 − 1 + β(βR) σ ·· · gj ,
βλθ
w
or

{
} { f (ω(p )) } σ11−σjb′j = gj + Rbj − (R − 1)b̄ + 1 + β(βR) σ ·· gj ,βλθ(A.6)

where we derive the second expression by using b̄ ≡ w/(R − 1).
To find the solution satisfying (4), (5), and (A.6), we estimate()
gj′ = γj · b̄ − b′j ,(A.7)

where γj is an undetermined coeﬃcient. We substitute (A.7) into (4) and obtain
(

(
) )σ
γj · b̄ − b′j= γj f (ω(pj )),gjorb′j = b̄ − (γj ) σ −1 · {f (ω(pj ))} σ · gj .
1

1

Plugging this expression into (A.6) and rearranging the terms, we obtain()
R · b̄ − bj
.
gj =
{
} {
}1
1
1
1−σ
f (ω(pj )) σ−1σσ
· {f (ω(pj ))}1 + 1 + β(βR) σ ·+ (γj )βλθOur estimate is verified if, for a given R, γj satisfies the following condition:

γj =

R
,
{
}
{
}1
1
1
1−σ
f (ω(pj )) σ−1
1 + 1 + β(βR) σ ·+ (γj ) σ · {f (ω(pj ))} σ
βλθ

24

or

[

{
} { f (ω(p )) } σ11−σj
R − γj · 1 + 1 + β(βR) σ ·βλθ]= {γj f (ω(pj ))}(A.8)

The left-hand side of (A.8), denoted by LHS, is decreasing in γj with LHS|γj =0 = R
and limγj →∞ LHS = −∞. The right-hand side of (A.8), denoted by RHS, is an increasing
function of γj with RHS|γj =0 = 0 and limγj →∞ RHS = ∞. Therefore, there is a uniqueγj (> 0) satisfying (A.8). We obtain the corresponding tax and debt policy functions by()
substituting gj = γj · b̄ − bj into (5) and (A.6), respectively.
■

A.4

Proof of Lemma 2

First, suppose that 1/σ ≤ 1 holds. We denote the right-hand side of (9) by RHS(R∗ ) with
RHS(1) > 1. Moreover, RHS(R∗ ) is decreasing in R∗ and satisfies limR∗ →∞ RHS(R∗ ) ∈(1, +∞). Therefore, when 1/σ ≤ 1, there is a unique R∗ ∈ (1, +∞) satisfying Eq. (9).
Next, suppose that 1/σ ∈ (1, 2) holds. The first and second-order diﬀerentials of
RHS(·) with respect to R∗ are
(
) σ1
1
1
−
ω
(1 − ωj ) + ωj λ∂RHS(R∗ )j=· (β) σ ·∂R∗
(1 − ωj ) βλθ{(1 − ωj ) + ωj λ}
(
)
1
1
×
− 1 · (R∗ ) σ −2 > 0,
σ
(
) σ1
1
1
−
ω
∂ 2 RHS(R∗ )(1 − ωj ) + ωj λj=· (β) σ ·∂R∗2(1 − ωj ) βλθ{(1 − ωj ) + ωj λ}
) (
)
(
1
1
−1 ·
− 2 · (R∗ )1/σ−3 < 0,×σσ
where limR∗ →+∞ ∂RHS(R∗ )/∂R∗ = 0 holds when 1/σ ∈ (1, 2). Therefore, there is aunique R∗ satisfying Eq. (9) when 1/σ ∈ (1, 2).
Finally, suppose that 1/σ = 2 holds. Then, we can derive R∗ directly by solving (9)25for R∗ as follows:
[
{
}2 ]
f
(ω
)
1
j
· 1 + {1 + β(βR∗ )} ·,R∗ = 1 +f (ωj )
βλθ
or

{

}2
}2
{
f (ωj )f (ωj )
2
1 + f (ωj ) +1 + (1 + β ) ·
βλθ
βλθ
∗
R =
}2 = 1 +
}2 ,
{
{
f
(ω
)
f
(ω
)
j
j
f (ωj ) − β 2 ·f (ωj ) − β 2 ·
βλθ
βλθ

(A.9)

where R∗ > 1 holds if and only if θ > {f (ωj )}1/2 /λ.

A.5

Proof of Lemma 3

Recall that γj∗ satisfies (7). We diﬀerentiate this with respect to γj∗ and ω(pj ) and obtain∂γj∗∂ω(pj ){
} ( )1/σ( ∗ )1/σ11γj{f (ω (pj ))}1/σ−1 f ′ (ω (pj )){f (ω (pj ))}1/σ−1 f ′ (ω (pj )) + γj∗ 1 + β (βR)1/σ−1
θβλ
σ
[
=−
{
}(
)1/σ ]( ∗ )1/σ−1f (ω(pj ))
1/σ
1/σ−1
1
γj
{f (ω (pj ))} + 1 + 1 + β (βR)
σ
θβλ
1
σ

)1/σ
{
}(
( ∗ )1/σ−1f (ω(pj ))γj{f (ω (pj ))}1/σ + 1 + β (βR)1/σ−1θβλ[= −γj∗ {f (ω(pj ))}−1 f ′ (ω(pj )) ·{}()1/σ ] ,( ∗ )1/σ−1f (ω(pj ))
1/σ
1/σ−1
γj
{f (ω (pj ))} + σ 1 + 1 + β (βR)θβλ(A.10)
where f ′ (ω(pj )) < 0 holds from (6).

26With the use of (8) and (A.10), we obtain
{
}
) 1−σ
∂ϕ∗j
∂γj∗
1 (
∗ σ
′
∗
= · f (ω(pj )) · γj
· f (ω(pj )) · γj + f (ω(pj )) ·∂ω(pj )σ∂ω(pj )) 1−σ
1 (= · f (ω(pj )) · γj∗ σ · f ′ (ω(pj )) · γj∗
σ

)1/σ
{
}(
( ∗ )1/σ−1f (ω(pj ))
1/σ
1/σ−1
γj
{f (ω (pj ))} + 1 + β (βR)


θβλ
.
[
]
×
1
−
{
}
(
)

1/σ 
( ∗ )1/σ−1f (ω(pj ))
1/σ
1/σ−1
γj
{f (ω (pj ))} + σ 1 + 1 + β (βR)θβλ(A.11)The expression in (A.11) leads to the following condition:[]{} ( f (ω(p )) ) σ1{
} ( f (ω(p )) ) σ1
∂ϕ∗j
j
j
1/σ−1
1/σ−1
⋛ 0 ⇔ σ 1 + 1 + β (βR)·
⋚ 1 + β (βR)·
∂ω(pj )
θβλ
θβλ
[
]1/σ
θ · (1 − ω (pj )) + ω (pj ) λ
1
1−σ
⇔
·
⋚
.
1/σ−1
1 − ω (pj )σ
1 + β (βR)■

A.6

Proof of Proposition 2(i) Suppose that ϕ∗ (ωmin , R∗ ) = ϕ̄ = 1. Eq. (A.9) leads to the following equilibrium
interest rate:{}2
f (ωmin )1 + (1 + β ) ·
βλθ
∗
R =1+
{
}2 (> 1).
f
(ω
)
min
f (ωmin ) − β 2 ·
βλθ
2

27

(A.12)

As demonstrated in Proposition 1(i), ϕ∗ (ωmin , R∗ ) = ϕ̄ = 1 holds when 1/σ − 1 ≤Ψ(R∗ , ωmin ) is satisfied. We substitute 1/σ = 2 and (A.12) into this condition and obtain}2βλθf (ωmin ){}2 
f (ωmin )2
1 + (1 + β ) · { βλθ }2βλθ2
⇔ 1 + β 1 +{}2  ≤f (ωmin )f (ωmin ) f (ωmin ) − β 2 ·
βλθ
{
}2
{
}4
f (ωmin )f (ωmin )2}2{+ (1 + β ) ·f (ωmin )
βλθ
βλθ
2
2
⇔ (1 + β ) ·
+β ·
≤1
{
}2
βλθ
f
(ω
)
min
f (ωmin ) − β 2 ·
βλθ
{
}2
{
}4
f (ωmin )f (ωmin )2
2
2⇔ (1 + β ) ·· f (ωmin ) − (1 + β )β ·
βλθ
βλθ
{
}2
{
}4
{
}2
f (ωmin )f (ωmin )f (ωmin )
2
2
2
2
+β
+ (1 + β )β ·≤ f (ωmin ) − β ·
βλθ
βλθ
βλθ
{
}2
{
}2
{
}2
f (ωmin )f (ωmin )f (ωmin )2
2
2⇔ (1 + β ) ·· f (ωmin ) + β≤ f (ωmin ) − β ·
βλθ
βλθ
βλθ

1
1 ≤ Ψ(R , ωmin ) =·1 + β 2 R∗
{∗

⇔ (1 + β 2 ) · {f (ωmin )}3 + β 2 · {f (ωmin )}2 ≤ β 2 λ2 θ2 f (ωmin ) − β 2 · {f (ωmin )}2⇔ (1 + β 2 ) · {f (ωmin )}2 + 2β 2 f (ωmin ) ≤ β 2 λ2 θ2
((
)
)
]1/2
1 [
⇔ θ ≥ θ ≡ · 2f (ωmin ) + 1 + β 2 /β 2 · (f (ωmin ))2.
λ(A.13)

Therefore, when the condition in (A.13) is satisfied, countries with ω(pj ) = ωmin accumulate public debt below b̄.
(ii) Suppose that ϕ∗ (eω (R∗ ), R∗ ) = ϕ̄ = 1. Eq. (A.9) leads to the following equilibrium
interest rate:

{

}2
f (e
ω (R∗ ))
1 + (1 + β ) ·
βλθ
∗
R =1+
{
}2 (> 1).
∗
f
(e
ω
(R
))
f (e
ω (R∗ )) − β 2 ·βλθ
2(A.14)

When 1/σ = 2, (10) and (6) are reformulated asωe (R) =

1 − 1θ · (1 + β 2 R)

1/2

1 − λ − 1θ · (1 + β 2 R)1/2
28,(A.15)

and
βθλ

f (e
ω (R)) =(1 + β 2 R)1/2,(A.16)

respectively.
Substituting (A.16) into (A.14) leads to
∗

R =1+

1 + (1 + β 2 ) 1+β12 R∗βθλ− β 2 1+β12 R∗(1+β 2 R∗ )1/2.

This yields two solutions for R∗ . The solution satisfying R∗ > 1 is taken as equilibrium
interest rate
R∗ = 1 + 2 ·

1+

√
2 2
2)
1 + θ λ β(1+β2θ 2 λ2> 1.

(A.17)

Proposition 1(ii) shows that ϕ∗ (eω (R∗ ), R∗ ) = ϕ̄ = 1 holds if Ψ(R∗ , ωmin ) < 1/σ −1 < Ψ(R∗ , ωmax ) is satisfied. With 1/σ = 2 and (A.17), the first inequality condition,
Ψ(R∗ , ωmin ) < 1/σ − 1, is reformulated as
1·
1 > Ψ(R , ωmin ) =1 + β 2 R∗

{

∗

βθλ
f (ωmin )}2,or⇔ θ<θ≡

((
)
)
]1/2
1 [
· 2f (ωmin ) + 1 + β 2 /β 2 · (f (ωmin ))2.
λThe second inequality condition, 1/σ − 1 < Ψ(R∗ , ωmax ), is reformulated as
1
1 < Ψ(R , ωmax ) =·1 + β 2 R∗
∗

{

βθλ
f (ωmax )

}2
,

or
θ>θ≡

((
)
)
]1/2
1 [
.
· 2f (ωmax ) + 1 + β 2 /β 2 · (f (ωmax ))2λ29

By substituting (A.17) into ωe (R∗ ), we obtain the corresponding value of ωe (R∗ ) as√
β 2 + θ2 λ2 (1 + β 2 )
1−
2
√ θ λ
ω
e (R∗ ) =,β + β 2 + θ2 λ2 (1 + β 2 )
1−λ−
θ2 λ
β+

(iii) Suppose that ϕ∗max = ϕ̄ = 1. Eq. (A.9) leads to{}2
f (ωmax )1 + (1 + β ) ·
βλθ
∗
R =1+
{
}2 (> 1).
f (ωmax )2
f (ωmax ) − β ·βλθ2(A.18)

Proposition 1(iii) shows that ϕ∗max = ϕ̄ = 1 holds when Ψ(R∗ , ωmax ) ≤ 1/σ − 1 is satisfied.
We substitute 1/σ = 2 and (A.18) into this condition and obtain}2βλθf (ωmax )
{
}2 
f
(ω
)
max
1 + (1 + β 2 ) · { βλθ }2
βλθ


2
⇔ 1 + β 1 +{}2  ≤f (ωmax )f (ωmax ) f (ωmax ) − β 2 ·
βλθ
[
((
)
)
]1/2
1
.
⇔ θ ≤ θ ≡ · 2f (ωmax ) + 1 + β 2 /β 2 · (f (ωmax ))2λ1·
1 ≥ Ψ(R , ωmax ) =1 + β 2 R∗{∗

(A.19)

Therefore, if (A.19) is satisfied, countries with ω(pj ) = ωmin accumulate public debt below
bmin < b̄. Finally, from (A.18), R∗ > 1 holds if and only if θ > {f (ωmax )}1/2 /λ.
■B

An Alternative Setting: Cross-Country Diﬀerencein Population Growth Rates

In this appendix, we assume away the cross-country diﬀerence in the age gap in voter
turnout. Instead, we assume a cross-country diﬀerence in population growth rates. Under30

this alternative assumption, we show that a higher population growth rate is associatedwith a higher level of public debt. The model presented below is based on that developed
by Arawatari (2018).

B.1

Model

We retain the notation used in the main text. The present model diﬀers from the one in
the main text in that there is a cross-country diﬀerence in population growth rates. The
population growth rate of country j is denoted by nj . Let us assume that nj is drawn
from a finite-valued set, nj ∈ {n, n1 , n2 , · · · , n̄} ≡ ΓN , n > −1. The size of the population
born in period t is denoted by Nj,t : Nj,t+1 = (1 + nj )Nj,t .
B.1.1

Utility Maximization

The utility-maximization problem of young agents is given bymax

cyt ,cot+1{ o}(cyj,t )1−σ − 1(cj,t+1 )1−σ − 1(gj,t )1−σ − 1(gj,t+1 )1−σ − 1yUj,t =
+θ·
+β·
+θ·
,
1−σ

1−σ

1−σ

1−σ

s.t. cyj,t + sj,t ≤ (1 − τj,t ) · w,
coj,t+1 ≤ Rsj,t ,where gj,t is public goods provision per capita. To simplify the analysis, we assume λ = 1.
Solving the problem leads to the following consumption and savings functions:cyj,t =11−σ

1 + β(βR) σ· w · (1 − τj,t ),(B.1)· R · w · (1 − τj,t ),(B.2)· w · (1 − τj,t ).

(B.3)

1−σ

coj,t+1 =β(βR) σ1−σ1 + β(βR) σ1−σsj,t =

β(βR) σ1−σ1 + β(βR) σ

Ignoring the irrelevant terms, we can express the indirect utility functions of type-i

31young agents and old agents as follows:

y
Vj,t
=

[{
}σ
]
1−σ
1
· 1 + β(βR) σ· {w · (1 − τj,t )}1−σ + θ · (gj,t )1−σ + βθ · (gj,t+1 )1−σ
1−σ

= V y (τj,t , gj,t , gj,t+1 ),
o
Vj,t
=

(B.4)

1
· θ · (gj,t )1−σ
1−σ

= V o (gj,t ).

B.1.2

(B.5)

Government Budget Constraint

The government budget constraint is given by

bj,t+1 =

gj,t
Rbj,t
τj,t w+−.
1 + nj 1 + nj
2 + njA lower population growth rate results in a smaller tax base in the future. Therefore, thenatural debt limit, denoted by b̄j , diﬀers across countries. Country i’s limit, b̄j , is defined
as follows:(Nj,t+1 + Nj,t ) · bj,t+1 ≤∞∑Nj,t+s · w
s=1

B.2

Rs

⇔ bj,t+1 ≤1 + njw·≡ b̄j .
2 + nj R − (1 + nj )Politico-Economic Equilibrium

This section explains the political mechanism of fiscal policy formation in the presence ofa cross-country diﬀerence in population growth rates.

B.2.1

Probabilistic Voting with Abstention

Recall that the sizes of the young and old in country j are Nj,t and Nj,t−1 , respectively.
Consider two political candidates, A and B, as in Appendix A.1. Candidate A’s actual

32vote share is









)


)
(
(
1
1
1
o
ΓA =
· Nj,t−1 · ϕ · σ ++ Nj,t · ϕ · σ y +Nj,t + Nj,t−1 
2ϕ
2ϕ 



{z
} |
{z
}
|

young
old
)
(
)}
{(
ϕ
1
1
y
o
=
+ (1 + nj ) · σ +.
·σ +2 + nj
2ϕ
2ϕ

(B.6)

Therefore, we can compute candidate A’s probability of winning the election as follows:[]
1
P = Probδ Γ ≥
2
[
1
1
= +ψ·
· {V o (gj,A ) − V o (gj,B )}2
2 + nj]}
1 + nj { y′y′+· V (τj,A , gj,A , bj,A ) − V (τj,B , gj,B , bj,B ) .
2 + njA

A

Candidate B wins the election with probability P B = 1 − P A ,PB = 1 − PA

[
1
1
= +ψ·
· {V o (gj,B ) − V o (gj,A )}2
2 + nj]}
1 + nj { y′y′· V (τj,B , gj,B , bj,B ) − V (τj,A , gj,A , bj,A ) .
+2 + njThese expressions suggest that the two candidates’ platforms converge in equilibrium
to the same fiscal policy that maximizes the weighted average utility of each group,
o
y
′
′∗
∗
∗
∗
∗
, b′∗
, gj,A(τj,Aj,A ) = (τj,B , gj,B , bj,B ) = arg max ω(nj )V (g) + (1 − ω(nj ))V (τ, g, b ).
τ,g,b′

where ω(nj ) ≡ 1/(2 + nj ), denoting the relative political weight of the old, is a decreasing
function of the population growth rate. The weights on the young and old are equal if
there is no population growth.
Definition B1. An SMPE comprises an interest rate R, a stationary debt distribution33

{bj }j , a debt rule b′j = B(bj ; nj , R), a government expenditure rule gj = G(bj ; nj , R),
and a tax rule τj = T (bj ; nj , R), such that the following two conditions hold:(i) ⟨B(bj ; nj , R), G(bj ; nj , R), T (bj ; nj , R)⟩ = arg max{τj ,gj ,b′j }Ω(τj , gj , b′j ; nj , R), and the government’s budget constraint and natural debt limit aresatisfied:G(bj ; nj , R)
Rbj
T (bj ; nj , R)w+−,
1 + nj
1 + nj2 + nj
1 + njw
B(bj ; nj , R) ≤·≡ b̄j .
2 + nj R − (1 + nj )B(bj ; nj , R) =(ii) the world asset market clears,∫∫

b′j dj,sj dj =

where b′j = B(bj ; nj , R) and sj =
B.2.2

j

j

1−σ
σ
1−σ
1+β(βR) σ· w · (1 − T (bj ; nj , R)).

β(βR)

Voting on Fiscal Policy

Hereafter, we take the market-clearing interest rate as given and solve the model by
backward induction. Since we have solved the utility-maximization problem already, we
are now ready to find the equilibrium fiscal policy. Given R, the first-order conditions
with respect to gj and τj are( ′ )σgj∂G(b′j ; nj , R)1
1 + nj
=−
·
·β·
,
gj
1 + nj 2 + nj
∂b′j

(B.7)

1−σ

1
1 + β(βR) σ
1= θσ · w · ,1 − τjgj(B.8)

where gj = G(bj ; nj , R), gj′ = G(b′j ; nj , R), τj = T (bj ; nj , R), and b′j = gj + Rbj − τj w ≡B(bj ; nj , R).
The population growth rate aﬀects the generalized Euler equation for public goods
provision in Eq. (B.7) via the two terms, 1/(1 + nj ) and (1 + nj )/(2 + nj ). The term34

1/(1 + nj ) indicates that a higher population growth rate weakens the disciplining eﬀect
expressed by the term ∂G(b′j ; nj , R)/∂b′j . Population growth implies an expansion of thetax base in the next period and thus reduces the per capita fiscal burden for the next
generation. This incentivizes current politicians to issue more public debt. The term
(1 + nj )/(2 + nj ) is the share of the young in the population, indicating their relative
political power in voting. A higher population growth rate implies the greater politicalpower of the young and thus strengthens the disciplining eﬀect. Hence, population growth
has two opposing eﬀects on the disciplining eﬀect (i.e., public debt issuance), but theformer eﬀect outweighs the latter one in the present framework.
Lemma B1. Given R and bj , country j’s policy functions in the SMPE are given byG(bj ; nj , R) = (1 + nj ) · γj∗ · (b̄j − bj ),
{
}
1−σ
1
1 + nj ∗σT (bj ; nj , R) = 1 − 1 + β(βR)· θ− σ ·· γj · (b̄j − bj ),w() σ1
1 + nj∗ σ1·β· (b̄j − bj ),B(bj ; nj , R) = b̄j − (γj ) ·2 + njwhere γj∗ (> 0) satisfies the following condition:
]
(
[
) σ1
}
1−σ
1
1
1 + njR
1 + nj {−∗∗− γj · 1 +· 1 + β(βR) σ · θ σ = (γj ) σ ··β. (B.9)1 + nj2 + nj2 + nj
Proof.
We substitute (B.8) into the government budget constraint and obtain
b′j =

[
]
{
}
1−σ
1
gj
Rbj
w
1
−
+
−
· 1 − 1 + β(βR) σ · θ σ · · gj ,1 + nj 1 + nj
2 + nj
w

or
b′j =

{
}
1−σ
1
gj
Rbj
R − (1 + nj )1+−· b̄j +
· 1 + β(βR) σ · θ− σ · gj , (B.10)
1 + nj 1 + nj
1 + nj2 + njwhere we derive the second expression by using w/(2 + nj ) = {R − (1 + nj )}b̄j /(1 + nj ).
35

To find the solution satisfying (B.7), (B.8), and (B.10), we estimate()
gj′ = (1 + nj ) · γj · b̄j − b′j ,(B.11)

where γj is an undetermined coeﬃcient. We substitute (B.11) into (B.7) and obtainb′j = b̄j −11· (γj ) σ −1 ·1 + nj(

1 + nj·β
2 + nj) σ1· gj .

By plugging this expression into (B.10) and rearranging the terms, we obtaingj =R · (b̄j − bj ){}() σ1 .
1
1−σ
1+nj
1+nj
−1
− σ1
σ
σ
1 + 2+nj · 1 + β(βR)· θ + (γj )· 2+nj · β

Our estimate is verified if, for a given R, γj satisfies the following condition:(1 + nj ) · γj =R}{(
) σ1 ,
1
1−σ
1
1+nj
1+nj
−1
−
σ
1 + 2+nj · 1 + β(βR) σ · θ σ + (γj )· 2+nj · β

or
]
[
(
) σ1
}
1
1−σ
1
R
1 + nj
1 + nj {−σ− γj · 1 +· 1 + β(βR) σ · θ σ = γj·β.
1 + nj2 + nj2 + nj(B.12)The left-hand side of (B.12), denoted by LHS, is decreasing in γj with LHS|γj =0 =R/(1 + nj ) and limγj →∞ LHS = −∞. The right-hand side of (B.12), denoted by RHS,
is an increasing function of γj with RHS|γj =0 = 0 and limγj →∞ RHS = ∞. Therefore,
there is a unique γj (> 0) satisfying (B.12). We obtain the corresponding tax and debt)(
policy functions by substituting gj = γj · b̄ − bj into (B.8) and (B.10), respectively.
■

36

B.3

Steady State

B.3.1

World Interest Rate

For the analysis, we rewrite the law of motion of debt in Lemma 1 as follows:()
b̄j − b′j = ϕ∗j · b̄j − bj ,

where

(
1
ϕ∗j = ϕ∗ (nj , R) ≡ (γj∗ ) σ

1 + nj·β
2 + nj) σ1, j ∈ [0, 1].

(B.13)We define ϕ̄ as
ϕ̄ ≡ max {ϕ∗ (nj , R)}j∈[0,1] .
Following the procedure in the main text, we determine the steady-state world interest
rate as follows. First, there is no country with ϕ∗ (nj , R) > 1. Otherwise, some countries
would accumulate ever-increasing surpluses, while the other counties could accumulate
debt up to b̄ at most. This condition prevents the international asset market from clearing.
Second, there is no R such that ϕ̄ < 1. Otherwise, all countries would accumulate public
debt up to b̄, which prevents the international asset market from clearing. Thus, thesteady-state equilibrium interest rate, R∗ , satisfies ϕ̄ = 1.
Substituting ϕ̄ = 1 into (B.9) enables us to express the condition that determines theequilibrium interest rate as follows:
]
[
}
1
R∗
2 + nj ∗1 + nj ∗ {
∗ 1−σ
−
=1+
· 1+
· 1 + β(βR ) σ · θ σ ,1 + nj ∗(1 + nj ∗ )β
2 + nj ∗(B.14)

where j ∗ denotes a country that accumulates debt below b̄ and satisfies ϕ̄ = 1. Other
countries accumulate debt up to b̄ and attain ϕ̄ < 1. We take the distribution of debt asgiven, and thus country j ∗ is not specified at this moment.
Hereafter, we assume 1/σ ∈ (0, 2) as in the main analysis. The following lemma
establishes the condition for the existence and uniqueness of the steady-state equilibrium37interest rate, R∗ , for a given distribution of debt in the world economy.
Lemma B2. Assume 1/σ ∈ (0, 2). Given the distribution of debt, there is a uniquesteady-state equilibrium interest rate, R∗ ≥ 1, satisfying (B.14).
Proof.
First, suppose that 1/σ ≤ 1 holds. We denote the right-hand side of (B.14) by
RHS(R∗ ) with RHS(1) > 1/(1 + nj ∗ ). Moreover, RHS(R∗ ) is decreasing in R∗ andsatisfies limR∗ →∞ RHS(R∗ ) ∈ (1, +∞). Therefore, when 1/σ ≤ 1, there is a uniqueR∗ ∈ (1, +∞) satisfying Eq. (B.14).
Next, suppose that 1/σ ∈ (1, 2) holds. The first- and second-order diﬀerentials of
RHS(·) with respect to R∗ are
1−2σ
1
1∂RHS(R∗ )1−σ= β σ −1 θ− σ ·· (R∗ ) σ > 0,
∗
∂R
σ
1
1
∂ 2 RHS(R∗ )1− σ 1 − 2σ
−1 − σ∗ 1−2σ
σ
=
β
θ
·
·
·
(R
) σ < 0,∂(R∗ )2σσ∂RHS(R∗ )
lim
= 0.
R∗ →∞
∂R∗

Therefore, there is a unique R∗ ∈ (1, +∞) satisfying Eq. (B.14) when 1/σ ∈ (1, 2).
■

B.4

International Diﬀerences in the Population Growth Rateand Public Debt Distribution

The policy function B(bj ; nj , R) in Lemma 1 suggests that the relative political weight on
the old influences the public debt distribution through ϕ∗j . To isolate its eﬀect from theinterest rate eﬀect, we take R as given and investigate the eﬀect of nj on ϕ∗j . We obtainthe following proposition.
Proposition B1. Countries with the lowest population growth rate, nj = n, accumulate
public debt below b̄, whereas the other countries accumulate public debt up to b̄.
38Proof.
Recall that γj∗ satisfies (B.12). We diﬀerentiate this with respect to γj∗ and nj andobtain{} 1( ∗ ) σ1 1 1 ( 1+nj ) σ1 −1 1
1−σ
1
R
∗
σ
σ
∗
1
+
β
(βR)
γ
β
+
γ
θ− σ + (1+n222jj (2+nj )∂γjσ 2+nj(2+nj )
j)
=−
.
{
} 1
( ∗ ) σ1 −1 ( 1+nj ) σ1
1−σ
∂nj
1+nj
1
−σ
σ
γj
β + 1 + 2+nj 1 + β (βR)θσ
2+nj
(B.15)
With the use of (B.13) and (B.15), we obtain
(
) σ1
(
)1
( ∗ ) σ1 1 1 1 + nj σ −1
∂ϕ∗j1 ∗ 1 −1 ∂γj∗ 1 + nj
1
= (γj ) σβ
+ γj β σ∂nj
σ∂nj 2 + nj
σ 2 + nj
(2 + nj )2() σ1 { ∗}∂γjγj∗1 ∗ 1 −1 1 + nj
= (γj ) σβ·+
σ
2 + nj∂nj (1 + nj ) (2 + nj )
(
) σ1
γj∗
1 ∗ 1 −1 1 + nj
σ
= (γj )β·σ
2 + nj(1 + nj ) (2 + nj )


(
{
) σ1
} 1
1−σ
1
1+nj
1+nj
2+nj

1
−1
−
∗
σ


(γj ) σβ + 2+nj 1 + β (βR)θ σ + γ ∗ (1+nj ) R 
σ
2+nj
j
× 1−
{
(
) σ1
} 1
1−σ


1+nj


1
∗ σ1 −1 1+nj
σ
1
+
β
(βR)
(γ
)
β
+
1
+
θ− σ
j
σ
2+nj
2+nj
(
) σ1
γj∗
1 ∗ 1 −1 1 + nj
= (γj ) σβ·
σ
2 + nj(1 + nj ) (2 + nj )2+n1 − γ ∗ (1+nj j ) R
j
×
(
{
} 1.
) σ1
1−σ
1
1+nj
1
∗ σ −1 1+nj
(γj )β + 1 + 2+nj 1 + β (βR) σ θ− σσ
2+nj
From (B.9), we obtain2 + nj·R∗γj (1 + nj )

(
) σ1
} 1
{
1−σ
1
1 + nj−∗−1> 1.
= (2 + nj ) + (1 + nj ) 1 + β(βR) σ θ σ + (2 + nj )(γj ) σβ
2 + njTherefore, we obtain ∂ϕ∗j /∂nj < 0.
Recall that countries with the highest ϕ∗j accumulate debt below b̄ and satisfy ϕ̄ = 1(Section B.1). With ∂ϕ∗j /∂nj < 0, we can conclude that countries with nj = n accumulate
public debt below b̄, whereas the other countries accumulate public debt up to b̄. This39

finding implies that a higher population growth rate is associated with a higher level of
public debt.
■

40

References[1] Arawatari, R., 2018. A political economy model of population growth rate and thesize of government debt. Keizai Kagaku (The Economic Science), forthcoming. (inJapanese)[2] Attanasio, O., and Weber, G., 1989. Intertemporal substitution, risk aversion andthe Euler equation for consumption. Economic Journal 99, 59–73.
[3] Bassetto, M., 2008. Political economy of taxation in an overlapping-generations economy. Review of Economic Dynamics 11, 18–43.
[4] Browning, M., Hansen, L.P., and Heckman, J., 1999. Micro data and general equilibrium models. In: Taylor, J., and Woodford, M. (eds.), Handbook of Macroeconomics,
Volume 1A. North-Holland, Amsterdam.
[5] Campbell, J., 1999. Asset prices, consumption and the business cycle. In: Taylor, J.,
and Woodford, M. (eds.), Handbook of Macroeconomics, Volume 1C. North-Holland,
Amsterdam.
[6] Campbell, J.Y., and Mankiw, N.G., 1989. Consumption, income, and interest rate:
reinterpreting the time series evidence. NBER Macroeconomics Annual. MIT Press,
Cambridge, MA.
[7] de la Croix, D., and Michel, P., 2002. A Theory of Economic Growth: Dynamics andPolicy in Overlapping Generations. Cambridge University Press, Cambridge, UK.
[8] Dhillon, A., and Peralta, S., 2002. Economic theories of voter turnout. Economic
Journal 112, F332–F352.
[9] Galasso, V., and Profeta, P., 2007. How does ageing aﬀect the welfare state? European Journal of Political Economy 23, 554–563.
[10] Gonzalez-Eiras, M., and Niepelt, D., 2008. The future of social security. Journal of
Monetary Economics 55, 197–218.
41

[11] Gonzalez-Eiras, M., and Niepelt, D., 2012. Aging, government budgets, retirement,
and growth. European Economic Review 56, 97–115.
[12] Gradstein M., and Kaganovich, M., 2004. Aging population and education finance.
Journal of Public Economics 88, 2469–2485.
[13] Hall, R., 1988. Intertemporal substitution in consumption. Journal of Political Economy 96, 339–357.
[14] Hansen, L., and Singleton, K., 1982, Generalized instrumental variables estimationof nonlinear rational expectations models. Econometrica 50, 1269–1286.
[15] Kydland, F., and Prescott, E., 1982. Time to build and aggregate fluctuations. Econometrica 50, 1345–1370.
[16] Lindbeck, A., and Weibull, J., 1987. Balanced-budget redistribution as the outcomeof political competition. Public Choice 52, 273–297.
[17] Organisation for Economic Co-operation and Development (OECD), 2007. Society
at a Glance 2006. OECD Social Indicators, OECD Publishing, Paris.
[18] Persson, T., and Tabellini, G., 2000. Political Economics: Explaining Economic Policy. MIT Press, Cambridge, MA.
[19] Razin, A., Sadka, E., and Swagel, P., 2002. The aging population and the size of thewelfare state. Journal of Political Economy 110, 900–918.
[20] Röhrs, S., 2016. Public debt in a political economy. Macroeconomic Dynamics 20,
1282–1312.
[21] Salavov, S. N., 2006. Age bias in fiscal policy: why does the political process favor
the elderly? Topics in Theoretical Economics 6, 1–37.
[22] Smets, K., 2012. A widening generational divide? The age gap in voter turnout
through time and space, Journal of Elections, Public Opinion and Parties 22, 407–430.
42

[23] 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.
[24] Vissing-Jorgensen, A., and Attanasio, O., 2003. Stock market participation, intertemporal substitution and risk aversion, American Economic Review 93, 383–391.

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aaa
Country
aaa
ITA
BEL
AUS
NLD
DEU
SWE
HUN
DNK
ESP
CZE
NOR
CAN
NZL

aaa
Age gap invoter turnout
−3.51
−1.49
−0.35
1.18
2.11
2.64
4.62
5.74
7.17
11.06
12.29
13.12
13.71

aaa
Country
aaa
USA
FRA
IRL
PRT
POL
SVN
ISR
CHE
FIN
KOR
JPN
GBR
Averageaaa
Age gap invoter turnout
14.37
15.75
17.82
18.61
19.17
19.55
19.76
19.87
20.33
22.79
25.20
38.18
12.79

Table 1: The percentage-point diﬀerence in voting rates between those aged +55 years
and those aged 16–35 years among OECD countries. A higher percentage diﬀerence invoting rates implies greater political power of old citizens. Data source: OECD Society
at a Glance 2011.

44

Percentage point difference in voting rates (2001−2008)
0
10
20
30
40

GBR

JPN
KOR
FIN CHEISR
SVN POL
PRT
IRL
FRA
USA
NZL
CAN
NOR
CZE
ESP
DNK
SWE
AUS

NLD

HUN
DEU
BEL
ITA

0

50
100
150
200
General government gross debt (Percent of GDP, 2009)250

Figure 1: This scatter plot shows the relationship between the percentage-point diﬀerence in voting rates between those aged +55 years and those aged 16–35 years and thegovernment debt-to-GDP ratio. A higher percentage diﬀerence in voting rates implies
greater political power of old citizens. Data source: OECD Society at a Glance 2011,
International Monetary Fund, World Economic Outlook Database, April 201245

Figure 2: This figure shows the relationship between ω(pj ) and ϕj . Panel (a) shows thecase of (1 − σ)/σ ≤ Ψ(R, ωmin ); panel (b) shows the case of Ψ(R, ωmin ) < (1 − σ)/σ <Ψ(R, ωmax ); and panel (c) shows the case of Ψ(R, ωmax ) ≤ (1 − σ)/σ.

46

Figure 3: The equilibrium pattern under σ = 1/2.

47

Figure 4: A numerical example of the equilibrium patterns.

48