Discussion Papers In Economics
And BusinessOld-age Social Security vs. Forward
Intergenerational Public Goods Provision
Ryo Arawatari and Tetsuo Ono
Discussion Paper 11-26-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOld-age Social Security vs. Forward
Intergenerational Public Goods Provision
Ryo Arawatari and Tetsuo Ono
Discussion Paper 11-26-Rev.2

August 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOld-age Social Security vs. Forward
Intergenerational Public Goods Provision
Ryo Arawatariy

Tetsuo Onoz

Nagoya University

Osaka University

August 1, 2013

Abstract
This paper introduces an overlapping-generations model with earnings heterogeneity and borrowing constraints. The labor income tax and the allocation oftax revenue between social security and forward intergenerational public goods are
determined in a bidimensional majoritarian voting game played by successive generations. The political equilibrium is characterized by an ends-against-the-middle
equilibrium where low- and high-income individuals form a coalition in favor of alower tax rate and less social security while middle-income individuals favor a higher
tax rate and greater social security. Government spending then shifts from social
security to public goods provision if higher wage inequality is associated with theborrowing constraint and a high elasticity of marginal utility of youthful consumption.
Keywords: Borrowing constraint; Old-age social security; Forward intergenerational public goods; Ends-against-the-middle equilibrium; Wage inequality
JEL Classi…cation: H41; H55; D72

The authors would like to thank the Editor of this journal and three anonymous referees for theirvaluable comments and suggestions. Financial support of the Nomura Foundation is gratefully acknowledged.
yGraduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601,
Japan. E-mail: arawatari@soec.nagoya-u.ac.jpzCorresponding Author: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

1.1

Background and Motivation

Almost all Organization for Economic Cooperation and Development (OECD) countries
have experienced some increase in wage inequality over the past few decades. Standard
political economy theory suggests that higher wage inequality results in greater social
security as the decisive voter becomes less aﬄuent as wage inequality increases (Romer,
1975; Roberts, 1977; Meltzer and Richard, 1981). Given this theoretical prediction, it isnatural to expect that higher wage inequality leads to a larger volume of social security.
Empirical evidence, however, does not necessarily support the above-mentioned theoretical prediction. For instance, OECD cross-country data shows that social security is
negatively correlated with wage inequality (for example, Gottschalk and Smeeding, 1997;
Chen and Song, 2009). In particular, the United Kingdom and the United States feature higher wage inequality and a smaller volume of social security whereas the Nordic
countries display lower wage inequality and a larger volume of social security.
Several researchers have attempted to provide political economy models that explain
this negative correlation (Benabou, 2000; Rodriguez, 2004; Chen and Song, 2009). In
particular, some studies suggest that the presence of borrowing constraints is a key todemonstrate said negative correlation (Casamatta, Cremer and Pestieau, 2000; Bellettini
and Berti Ceroni, 2007; Conde-Ruiz and Profeta, 2007; Cremer et al., 2007; Arawatari
and Ono, 2013). However, they focus on a single policy issue, that is, social security that
beneﬁts the old at the expense of the young, and abstract away other policy issues that
beneﬁt the young. Because of this limitation, they fail to show how inequality within ageneration aﬀects political choices between conﬂicting expenditure demands by diﬀerent
generations.
The conﬂict over the distribution of government spending between diﬀerent generations is considered from several viewpoints: income redistribution within and acrossgenerations (Conde-Ruiz and Galasso, 2005) and redistribution for the elderly vs. public
goods provision for the young (Rangel, 2003; Levy, 2005; Bernasconi and Profeta, 2012).
However, all these studies demonstrate the intergenerational conﬂict in the absence ofborrowing constraint. In other words, they say nothing about how inequality within ageneration aﬀects the distribution of tax revenue across generations in the presence ofborrowing constraint that produces a negative correlation between social security andinequality.
To resolve the limitation of the previous studies, we develop a model including borrowing constraints to demonstrate the empirically ﬁtted correlation between inequality andthe volume of old-age social security, and then add public goods provision that beneﬁts
1

the young into the model as alternative government spending.1 Within this extended
framework, we aim to provide theoretical predictions about the intergenerational distribution of tax revenue aﬀected by wage inequality in the presence of borrowing constraint.
In this respect, the present paper extends the two branches of literature mentioned above
and ﬁlls a gap between them.

1.2

Analysis and Results

For the purpose of this analysis, we introduce an overlapping-generation economy withstorage technology. In this economy, young workers are of three income categories: low,
middle and high. Because they are not permitted to borrow in youth as a result ofimperfect ﬁnancial markets, lower-income individuals are more likely to be borrowing
constrained. In youth, individuals decide how much to save and how much to consume.
In old age, they retire and consume the return from saving and a pay-as-you-go (PAYG)
social security beneﬁt. Young workers pay a ﬁxed proportion of their labor income to
the government, and the tax revenue is divided into PAYG social security payments andpublic goods such as pure science and environmental maintenance. The former is enjoyedby the old. The latter, called forward intergenerational public goods, take a one-period
lag to mature and thus beneﬁt only the young.
The tax rate and the distribution of tax revenue between social security and public
goods provision are determined in a bidimensional majoritarian voting game played by
the young and the old. Voters cast a ballot on the labor income tax, which ﬁnances social
security and public goods provision, and over the allocation of tax revenue between social
security and public goods provision. Under this type of voting game, the existence ofa Condorcet winner of the majority voting game is not necessarily guaranteed becauseof the multidimensionality of the issue space. To deal with this problem, we utilize theconcept of a structure-induced equilibrium (Shepsle, 1979) with the notion of a once-andfor-all voting, which is applied to an overlapping generations framework by Conde-Ruizand Galasso (2003, 2005).
Based on the above-mentioned concept of equilibrium, we consider the voting behaviorof each type of individual. The preferences of the old are identical across all types ofindividuals because they owe no tax burden and receive the same level of social security
beneﬁt. In contrast, the preferences of the young depend on their income type because
the tax burden diﬀers across the board. In particular, the key to their preferences are theborrowing constraint and the elasticity of marginal utility of youthful consumption.
1

The public good in the present model does not satisfy the non-rivalry property; per capita public
spending for the young decreases as the number of the young increases. The good is classified as an impure
public good in a strict sense. However, in the following, we call it as ”a public good” for simplicity ofdescription.

2

To understand the role of these two factors, consider ﬁrst the case where a certain
type of individual is borrowing unconstrained. A reduction in his wage decreases his
marginal cost of taxation, thereby giving him an incentive to choose a higher tax rate.
However, when he is borrowing unconstrained, there is an additional eﬀect which works
in the opposite direction. A decrease in consumption by one unit, which is caused by arise in the tax rate, lowers the utility of consumption. This eﬀect represents the marginalloss of utility, which becomes larger as his wage is decreased. Therefore, the additionalloss of utility works to increase the marginal cost of taxation in response to a reduction
in one’s wage.
Which eﬀect outweighs the other depends on the magnitude of the elasticity of marginalutility of consumption, denoted by σ. When σ ≤ 1, the former eﬀect outweighs the latter.
Regardless of borrowing status, one’s marginal cost of taxation decreases as his wage isdecreased. Therefore, he prefers a higher tax rate in response to a reduction in his wage
irrespective of whether he is borrowing constrained or not. However, when σ > 1, thelatter eﬀect outweighs the former when one is borrowing constrained. The borrowingconstrained individual prefers a higher tax rate in response to a reduction in his wage.
The result in the case of σ > 1 implies that there is a V -shaped relationship between
the wage and the marginal cost of taxation. For a high-wage case where an individual isborrowing-unconstrained, a reduction in his wage decreases the marginal cost of taxationand thus increases his preferred tax rate. The opposite holds true for a low-wage casewhere he is borrowing-constrained: a borrowing-constrained individual prefers a lower tax
rate in response to a reduction in his wage. The latter case entails a situation where aborrowing-constrained low-income individual prefers a lower tax rate than a borrowingunconstrained middle-income individual. There is then an ends-against-the-middle equilibrium where low-income and high-income individuals form a coalition in favor of a low
tax rate, and where middle-income individuals favor a high tax rate.
Given the characterization of political equilibrium, we investigate how the tax rate andthe distribution of tax revenue are altered in response to changes in wage inequality. In
particular, we consider a mean-preserving reduction of the decisive voter’s wage in order
to compare two groups of countries with similar per capita income levels but diﬀerent
levels of income inequality. We show that the standard theoretical result holds when theelasticity is below unity: an increase in wage inequality leads to a higher tax rate. We also
show that a larger tax rate is associated with a larger fraction of old-age social security
and a smaller fraction of forward intergenerational public goods provision in government
expenditure.
However, when the elasticity is above unity, the mean-preserving reduction of the decisive voter’s wage creates an inverse V -shaped relationship between the decisive voter’s

3

wage and the fraction of social security in government expenditure. The negative correlation arises when the decisive voter’s wage is high and thus he is borrowing unconstrained,
while the positive correlation arises when his wage is low and thus he is borrowing constrained. In particular, the latter case predicts that a higher level of inequality results
in a lower tax rate, a smaller fraction of social security and a larger fraction of forward
intergenerational public goods in government expenditure.
In the current framework, the negative correlation between inequality and the shareof social security in government expenditure arises only in the equilibrium where thefollowing two conditions hold: (i) the elasticity of marginal utility of youthful consumption
is above unity; and (ii) the decisive voter is borrowing constrained. When one of theconditions fails to hold, the economy displays a positive correlation between inequality andthe share of social security. Therefore, our analysis suggests that these factors are the keys
to demonstrate the above-mentioned inverse V-shaped relationships. These relationships
still hold even if we relax the assumption of the utility function.
The organization of this paper is as follows. Section 2 introduces the model andcharacterizes the economic equilibrium. Section 3 develops the political system, introduces
the equilibrium concept of the voting game and demonstrates the voting behavior of eachindividual. Section 4 characterizes the political equilibrium. Section 5 examines how wage
inequality aﬀects the tax rate and the allocation of tax revenue between social security
and forward intergenerational public goods. Section 6 brieﬂy undertakes the analysis
under a generalized utility function. Section 7 provides concluding remarks. Proofs of the
propositions are provided in the appendix.

2

The Economic Environment

Consider a discrete time economy where time is denoted by t = 0, 1, 2 · · · . The economy ismade up of overlapping generations of individuals, each of whom lives two periods: youth
and old age. The size of a generation born in period t, called generation t, is denotedby Nt . Population grows at a constant rate n > 0 : Nt+1 = (1 + n)Nt for all t ≥ 0.
Within each generation, there are three types of agents according to ability to work, low,
middle and high (j = L, M, H), whose proportions are respectively ρL , ρM and ρH , where∑ jjj ρ = 1 and ρ satisﬁes the following assumption.
n> ρj > 2(1+n), j = L, M, H.
Assumption 1. 1+(1+n)
2(1+n)

The ﬁrst inequality of Assumption 1, {1 + (1 + n)} /{2(1 + n)} > ρj , states that the
proportion of type-k (k = L, M, H) young individuals, ρj (1 + n), must be less than halfof the population, (2 + n)/2. Otherwise, the type-k young individual becomes a decisive
voter regardless of the preferences of others. The second inequality of Assumption 1,
4

ρj > n/{2(1 + n)}, ensures that a young individual who prefers the highest tax rate
among young individuals becomes the decisive voter.
To understand the argument stemming from the second inequality condition in Assumption 1, consider ﬁrst the preferences of the old. As we explain below, the old choosea higher tax rate than young individual because they bear no tax burden but beneﬁt fromtaxation via social security; the tax burden when young is viewed as a sunk cost for theold. In addition, the old have the same preferences over the policy because they beneﬁt
from the same social security.
Next, consider the preferences of the young. Suppose that a type-k (k = L, M or H)
prefers the highest tax rate. When the young and the old participate in voting, the sumof the type-k young and the old is given by Nt ρk + Nt−1 , which is greater than half ofthe population in period t, (Nt + Nt−1 )/2, under the assumption of ρk > n/2(1 + n). Thisimplies that the decisive voter becomes the old or the type-k young. However, the oldcannot become the decisive voter because the population size of the old is smaller thanthat of the young under the assumption of n > 0. Therefore, the type-k young individual
becomes the decisive voter. Figure 1 provides an example of preferences over the tax rate.
[Figure 1 about here.]
To assess the empirical plausibility of Assumption 1, let us suppose a generation to be
30 years in length. Assumption 1 becomes:
1 + (1.0063)30(1.0063)30 − 1j>ρ >,2(1.0063)302(1.0063)30
that is,0.91414 > ρj > 0.085861,
where 1 + n = (1.0063)30 comes from the data by OECD (2013): the average annual
population growth rate is 0.63% in sample OECD countries in 2010.2

2.1

Individuals

Each individual is assumed to receive utility from private consumption and publicly provided goods. The utility function of a type-j young individual in period t is speciﬁed
by:
[
]
1−σ
(cyj
−1
(gt )1−σ − 1(gt+1 )1−σ − 1
j
oj
t )
Ut =
+η
+ β · ct+1 + η
,
1−σ
1−σ
1−σ
2

The assumption requires that the fraction of each type of the young must be more than 8.5% andless than 91.4% from the empirical viewpoint. This requirement is satisfied by the evidence reported by
Ichino, Karabarbounis and Moretti (2011). They use the data from the World Value Surveys, and reportthat on average, 33%, 37% and 30% of the population are classified as poor, middle and rich, respectively,
for twelve OECD countries.

5oj
where cyj
t is consumption in youth, ct+1 is consumption in old age, gt is per capita public

goods in period t, η(> 0) is the parameter representing the preference for public goods,β ∈ (0, 1] is the discount factor, and σ(> 0) is the elasticity of marginal utility of youthful
consumption or publicly provided goods. A lower σ implies a lower elasticity.
Following the literature (Conde-Ruiz and Galasso, 2005; Borck, 2007; Bethencourt andGalasso, 2008; Leroux, Pestieau and Racionero, 2011), we assume a quasi-linear utility
function for analytical tractability. In Section 6, we will brieﬂy investigate the case where1−σ
the utility of old-age consumption is given by β{(coj− 1}/(1 − σ) and show that the
t+1 )main result is not qualitatively unchanged under this alternative utility function.
Each individual works in his youth and retires in old age. The wage income is related
to working ability. The wage of a type-j individual is given by wj (j = H, M, L), where
wj is constant over time and wL < wM < wH . The average of the wage is denoted byw̄ ≡ ρL wL + ρM wM + ρH wH .
Type-j’s individual budget constraints in youth and old age are given respectively by:jjcyj
t + st ≤ (1 − τt )w ,j
coj
t+1 ≤ Rst + bt+1 ,where sjt is saving, τt is the income tax rate in period t, R is the gross interest rate,
and bt+1 is the per capita social security beneﬁt in old age. We impose the restriction ofnonnegative savings as:
sjt ≥ 0.
This rules out the possibility of borrowing in youth against future social security beneﬁts(Diamond and Hausman, 1984; Conde-Ruiz and Profeta, 2007).
We assume that the economy is dynamically eﬃcient.
Assumption 2. R ≥ 1 + n.
The assumption implies that the rate of return from social security is lower than theprivate rate of return from saving. Nevertheless, low- and middle-income individuals may
have an incentive to support this inferior system of intertemporal resource reallocation.
This is because the current social security system involves an intragenerational redistribution component that transfers resources from the high to the low and the middle.
We also assume that (i) the interest rate is exogenous, and (ii) each individual receives
the same amount of old age social security beneﬁts regardless of contributions in their
youth. The ﬁrst assumption abstracts away the general equilibrium eﬀect via the interest
rate investigated by, for example, Cooley and Soares (1999) and Boldrin and Rustichini(2000). However, this simpliﬁcation enables us to demonstrate more simply the analytical6

solution of the model. The second assumption abstracts away from the choice of social security systems (for example, Bismarckian vs. Beveridgean) as analyzed by Borck (2007),
Conde-Ruiz and Profeta (2007) and Cremer et al. (2007). We adopt the second assumption to concentrate on the role of the borrowing constraint in the political determinationof social security and public goods provision.
The representative type-j young individual maximizes his utility subject to the budget
constraints and the restriction of nonnegative saving. When sjt > 0, the ﬁrst-order condi−σtion for an interior solution is (cyj
= βR, and thus deﬁnes the optimal saving decisiont )jof a type-j individual given by st = (1 − τt )wj − (βR)−1/σ . By taking the borrowing
constraint into account, the saving function of a type-j individual is:{}
sjt = max 0, (1 − τt )wj − (βR)−1/σ .

(1)Eq. (1) indicates that the saving decision depends on the current tax rate τt , but is
independent of the future tax rate τt+1 and the proportion of tax revenues devoted tosocial security in old age, denoted by λt+1 . This property comes from the assumptionof a linear utility function of old-age consumption. Because of this property, we easily
demonstrate the joint political determination of the tax rate τ and the proportion λ.
The saving function (1) implies that there is a critical rate of tax such that:sjt > 0 ⇔ τt < τ̂ (wj ) ≡ 1 −1.
(βR)1/σ wj(2)

A type-j individual chooses positive savings when the tax is below the critical rate. However, when the tax is above the critical rate, a type-j individual faces a borrowing constraint and can save nothing in youth. The critical rate of tax is higher when the wage
income is larger because, given a tax rate common to all types of individuals, a morecompetent individual receives a higher level of disposable income.

2.2

The Government

In each period, the government collects tax revenue from the young by imposing an
income tax. Following the conventions in the literature, we present the eﬃciency loss oftaxation by assuming convex costs of collecting taxes (for example, Casamatta, Cremer,
and Pestieau, 2000; Bellettini and Berti Ceroni, 2007; Cremer et al., 2007). Therefore,
the actual tax revenue is given by (1 − τt )τt (ρL wL + ρM wM + ρH wH ) = (1 − τt )τt w̄, where
the term (1 − τt ) is the distortionary factor. The assumption of distortionary taxation issolely to ensure an interior solution to preferred tax rates and otherwise plays no role.
The government uses the tax revenue for old-age social security payments along withforward intergenerational public goods such as environmental preservation and pure science. The proportion λt ∈ [0, 1] of tax revenue is devoted to old-age social security beneﬁts
7and the remainder (1 − λt ) is devoted to forward intergenerational public goods provision.
The old-age social security is then an intergenerational transfer from the young to the old
within a period. The budget constraint is λt Nt (1 − τt )τt w̄ = Nt−1 bt . The per capita social
security beneﬁt in period t, bt , is given by:
bt = (1 + n)λt (1 − τt )τt w̄.
The formation of public goods requires investment one period ahead of time. This
assumption reﬂects the idea that pure science and investment in the environment do
not obtain immediate results. Importantly, the current young generation can enjoy theoutcomes of any investment in the future, while the current old generation cannot enjoy it
while they are still alive. The budget constraint is (1−λt )Nt (1−τt )τt w̄ = (Nt +Nt+1 )gt+1 .
The per capita public goods provision in period t + 1, gt+1 , is given by:
gt+1 =

2.3

1(1 − λt )(1 − τt )τt w̄.
2+n

The Economic Equilibrium

We deﬁne the economic equilibrium as follows.
Deﬁnition 1. For a given sequence of tax rates and social security shares in government expenditure, {τt , λt }∞t=0 , an economic equilibrium is a sequence of allocations,yj oj j t=0,··· ,∞{ct , ct , st }j=L,M,H with the initial condition sj0 (j = L, M, H), such that (i) in every
period, a type-j individual maximizes his utility subject to the budget constraints
and the nonnegativity constraint of saving, (ii) the social security budget and the
public goods budget are balanced in every period, and (iii) the goods market clears
every period.
From (1) and the private and government budget constraints, the consumption functions of a type-j individual in youth and old age are given respectively by:{(βR)−1/σ
if τt < τ̂ (wj )
yj
ct =
j
(1 − τt )w
if τt ≥ τ̂ (wj ){R{(1 − τt )wj − (βR)−1/σ } + (1 + n)λt+1 (1 − τt+1 )τt+1 w̄ if τt < τ̂ (wj )oj
ct+1 =(1 + n)λt+1 (1 − τt+1 )τt+1 w̄
if τt ≥ τ̂ (wj ).
Because of the assumption of a quasi-linear utility function, the consumption in youth is
type-independent and constant over time when the tax is below the critical rate.
The utility level obtained by individuals in economic equilibrium is represented bytheir indirect utility functions. We use the above-mentioned consumption functions to
obtain an indirect utility function of a type-j young individual:{ yj
Vt,s>0 if τt < τ̂ (wj )
yj
Vt =
(3)
yj
Vt,s=0if τt ≥ τ̂ (wj ),8where:y,jVt,s>0≡((βR)−1/σ

)1−σ

1−σ

−1

[ {
}
]
−1/σ
j
+ β R (1 − τt )w − (βR)
+ (1 + n)λt+1 (1 − τt+1 )τt+1 w̄
{
}
( 1
)1−σ
(1
−
λ
)(1
−
τ
)τ
w̄
−
1
(gt )1−σ − 1ttt+η+ β 2+n
,
1−σ
1−σ

1−σ
((1 − τt )wj )−1y,j+ β(1 + n)λt+1 (1 − τt+1 )τt+1 w̄
Vt,s=0 ≡

1−σ

{

+η

(gt )1−σ − 1
+β
1−σ

( 1

2+n

(1 − λt )(1 − τt )τt w̄
1−σ

)1−σ

−1

}
.

y,j
Vt,s>0denotes the indirect utility of a type-j young individual when he saves a portiony,jof his income, and Vt,s=0
denotes the indirect utility when he is faced with a borrowing

constraint and saves nothing. For each indirect utility function, the ﬁrst term on theright-hand side shows the utility of consumption in youth, the second term shows theutility of consumption in old age and the third term shows the utility of public goods inold age. The public goods provision in period t, gt , is omitted in the above expression
because it is predetermined in period t − 1 and thus, is politically irrelevant in period t.
For a type-j old individual in period t, the indirect utility function is:(gt )1−σ − 1
,
(4)
1−σ
where the ﬁrst-term on the right-hand side shows the social security beneﬁts. The term
Rst−1 , representing the return from saving, is omitted in this expression because it ispredetermined in period t − 1. Old individuals have the same indirect utility function
regardless of their type because their savings in youth are predetermined and the level ofVto,j ≡ (1 + n)λt (1 − τt )τt w̄ + η

public goods they enjoy is predetermined one period in advance. Therefore, old individuals
have the same preferences for the tax rate, τ , and the share of social security, λ.

3

The Political Institution and Voting

The tax rate τ and the proportion λ are determined by individuals through a politicalprocess of majoritarian voting. Elections take place every period and all young and old
individuals cast a ballot over τ, the income tax, and λ, the share of social security ingovernment expenditure. Individual preferences over the two issues are represented by
the indirect utility functions at Eqs. (3) and (4) for the young and the old, respectively.
Every individual has zero mass and thus, no individual vote can change the outcome ofthe election. Therefore, we assume individuals vote sincerely.
This majoritarian voting game has two signiﬁcant characteristics. First, the issue
space is bidimensional (τ and λ), and thus, the Nash equilibrium of a majoritarian voting
9

game may fail to exist. To deal with this feature, we use the concept of issue-by-issue
voting, or structure-induced equilibrium, as formalized by Shepsle (1979) and applied by
Conde-Ruiz and Galasso (2003, 2005) for the framework of overlapping generations.
Second, the game is intrinsically dynamic because it describes the interaction among
successive generations. To deal with this feature, we assume once-and-for-all voting (see,for example, Casamatta, Cremer, and Pestieau, 2000; Conde-Ruiz and Profeta, 2007).
That is, period-t young individuals vote over the current and future taxes (τt and τt+1 )and the allocation (λt and λt+1 ) given the static expectation that successive generations
will make the same choice with their current choice persistent over time: τt = τt+1 = τ and
λt = λt+1 = λ for all t. The previous literature has shown that this type of behavior can besupported as a sub-game perfect equilibrium in a repeated voting if voters expect that any
deviation from this static behavior will be punished by future generations (Conde-Ruizand Galasso, 2003, 2005).3
Because of the above-mentioned assumption, the current model presents a static voting
game. Therefore, the result in Shepsle (1979) can be applied to obtain the suﬃcientconditions for the existence of a structure-induced equilibrium. In particular, if preferences
are single peaked along every dimension of the issue space, a suﬃcient condition for (τ ∗ , λ∗ )to be an equilibrium of the voting game is that τ ∗ represents the outcome of majority
voting over the jurisdiction τ when the other dimension is ﬁxed at its level λ∗ , and viceversa.
Preferences of the old are immediately shown to be single peaked along every dimension because they are given by Vto,j ≡ (1 + n)λt (1 − τt )τt w̄ and satisfy ∂ 2 V o,j /∂τ 2 < 0and ∂ 2 V o,j /∂λ2 = 0. The preferences of the young are also single peaked along every
dimension. Nevertheless, the proof of this argument is not straightforward because thepreferences of the young are kinked at the critical rate τ̂ (wj ). The formal proof is given
in Appendix 8.1.
In what follows, we demonstrate preferences of the old and the young over policy.

3.1

Preferences of the Old Over Policy

The old choose τ to maximize V o,j in (4) given λ, and λ to maximize V o,j in (4) given τ .
Their preferred tax rate and the share of social security are respectively given by:τ oj =

1and λoj = 1 for all j.
2Maximization is realized when the tax rate is set to attain the top of the Laﬀer curve,(1 − τ )τ . The old prefer to use the maximized tax revenue exclusively for social security
3

The authors would like to thank one of the referees for pointing this out.

10

because they cannot beneﬁt from the investment in public goods that take a one-period
lag in formation.

3.2

Preferences of the Young Over the Tax Rate

Consider ﬁrst the preferences of the young over τ . A type-j young individual chooses τ toy,jy,jmaximize Vs>0
when he is borrowing unconstrained; he chooses τ to maximize Vs=0
when
he is borrowing constrained. Thus, the tax rate τ chosen by the type-j young individual
satisﬁes the following ﬁrst-order condition:
{
}−σ
1
1
β(1 + n)λ(1 − 2τ )w̄ + βη
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
(5)
2+n
2+n
{
βRwj
if τ < τ̂ (wj )=(1 − τ )−σ (wj )1−σ if τ ≥ τ̂ (wj )The ﬁrst term on the left-hand side is the marginal beneﬁt of social security, the second
term on the left-hand side is the marginal beneﬁt of public goods provision, and the righthand side shows the marginal cost of taxation. The marginal cost is given by βRwj whena type-j young individual is borrowing unconstrained; and it is given by (1 − τ )−σ (wj )1−σ
when he is borrowing constrained. The type-j young individual chooses the tax rate toequate marginal beneﬁts and costs of taxation from the viewpoint of utility maximization.
Condition (5) shows that the marginal beneﬁt of taxation depends on the average wage
rather than each individual’s wage; but the marginal cost depends on each individual’s
wage. In particular, a rise in type-i’s wage increases his marginal cost of taxation when heis borrowing unconstrained. Therefore, a rise in his wage decreases his preferred tax rate
as long as he is borrowing unconstrained. However, when he is borrowing-constrained, arise in his wage may or may not increase his marginal cost of taxation depending on theelasticity of the marginal utility of youthful consumption, denoted by σ. The elasticity
could have a crucial role in the determination of the preferred tax rate by a borrowingconstrained individual.
To understand the role of the elasticity in a borrowing-constrained case more precisely,
consider the marginal cost of taxation for the type-i borrowing-constrained individual,
denoted by M CT j :M CT j ≡ (1 − τ )−σ (wj )1−σ = wj · (cyj )−σ ,where the second equality comes from cyj = (1−τ )wj . This expression shows that type-j’s
wage has two opposing eﬀects on M CT j . The ﬁrst eﬀect is expressed by the term wj .
An increase in the tax rate by one unit entails an increase in the tax burden by wj units.
In other words, a lower-income individual pays less tax than a higher-income individual11

in response to a marginal increase in the tax rate. Therefore, the term works to decrease
the marginal cost of taxation in response to a reduction in one’s wage.
The second eﬀect is expressed by the term (cyj )−σ . This term shows a marginal utilityof consumption: a decrease in consumption by one unit, which is caused by a rise in
the tax rate, leads to a decrease of utility of consumption by (cyj )−σ units. Given thatcyj = (1 − τ )wj , this term becomes smaller as one’s wage is decreased. Therefore, theterm works to increase the marginal cost of taxation in response to a reduction of one’s
wage.
Which eﬀect outweighs the other is likely to depend on the magnitude of the elasticity
of marginal utility of consumption. When σ ≤ 1, the former eﬀect outweighs the latter.
Regardless of borrowing status, one’s marginal cost of taxation decreases as his wage isreduced (Panel (a) of Figure 2). However, the latter eﬀect becomes more dominant as σ isincreased. In particular, when σ > 1, the latter eﬀect outweighs the former. There is thena V -shaped relationship between the wage and the marginal cost of taxation (Panel (b)of Figure 2). The latter case gives an insight into the mechanism of an inverse V -shaped
relationship between inequality and policy variable, which will be investigated in detailin Section 5.
[Figure 2 about here.]

3.3

Preferences of the Young Over the Share of Social Security

Next, consider the preferences of the young over λ. The ﬁrst derivative of V y,j with respecty,jy,j/∂λ. Direct calculation/∂λ = ∂Vs=0
to λ is independent of the status of saving: ∂Vs>0
leads to:
y,j
y,j
∂Vs>0
∂Vs=0
=
∂λ
∂λ

= β(1 + n)(1 − τ )τ w̄ − βη{

1(1 − λ)(1 − τ )τ w̄
2+n

}−σ

1
(1 − τ )τ w̄,2+nwhere the ﬁrst term on the right-hand side shows the marginal increase in the beneﬁt ofsocial security given by an increase in the share of social security, and the second termis the marginal loss of utility of public goods given by a decrease in the share of public
goods provision. The share of social security, λ, is chosen to balance the marginal beneﬁt
and loss in terms of utility.
A noteworthy feature of the current model is that the preferred share by the young is
type-independent. This is because (i) all types of young individuals enjoy the same levelof forward intergenerational public goods, (ii) the utility of forward intergenerational
public goods is separable from the utility of private goods, and (iii) the utility of old-age
consumption is speciﬁed by a linear utility function. In Section 6, we employ a more12

generalized utility function and show that the preferred share becomes type-dependent
but the main result still holds.
Another noteworthy feature is that for a low tax rate, the share λ could be zero: all
tax revenue goes to public goods provision. When the tax revenue is low, the marginalutility of public goods is high even when all tax revenue is devoted to it. However, themarginal utility of old-age consumption is always constant because of a quasi-linear utility
function.4 Therefore, choosing λ = 0 is optimal for a young individual from the viewpointof utility maximization when the tax rate is below the critical rate. The corner solution
is not peculiar to the model with a quasi-linear utility function. As we demonstrate in
Section 6, the qualitatively similar result also holds under a generalized utility function.
Based on the above argument, the preferred share of the young becomes:{
0
if τ ∈ [0, τ ]
(
)1/σ
( 1]
λ=
η
2+n
1 − (1−τifτ∈τ, 2 ,)τ w̄ (2+n)(1+n)√

where:
τ≡1−

1 − 4(2+n)w̄(η(2+n)(1+n)

(6)

)1/σ

.
2
Figure 3 illustrates the graphs of (6). To proceed the analysis we make the following
assumption.
[Figure 3 about here.]
Assumption 3.
)1/ση
;(2 + n)(1 + n)
2(√(4(2 + n)(i) 1 >w̄(ii) wL >
1/σ

(βR)

(

1+

1 − 4(2+n)w̄η(2+n)(1+n)

)1/σ

).

The ﬁrst assumption implies that the preferred share in (6), which attains the highest
value at τ = 1/2, takes a positive value at τ = 1/2. Therefore, the assumption ensures thata political equilibrium exists with λ > 0 for a range of [τ , 1/2); otherwise, λ = 0 holds forany τ ∈ [0, 1/2), implying a trivial outcome of no provision of old-age social security. The
range of τ is limited to (0, 1/2) because the preferred tax rate by the old is equal to 1/2 and( )that by the young is less than 1/2. The second assumption is equivalent to τ < τ̂ wL .
This assumption enables us to demonstrate cases of borrowing-unconstrained as well as
borrowing-constrained type-L individuals in the presence of social security, λ > 0.
4

We would like to thank one of the referees for suggesting this interpretation.

13

4

The Political Equilibrium

The previous section analyzed the voting behavior of each type of individual along thetwo dimensions of the issue space, τ and λ. Given that preferences are single peaked foreach issue, we now apply Shepsle’s (1979) result and characterize the structure-induced
equilibrium of the game.
The structure-induced equilibrium outcome is found as follows. First, we determine
the decisive voter over λ and calculate his most preferred share, denoted by λdec (τ ), asa function of the tax rate τ , where the superscript “dec” indicates the decisive voter.
Second, we determine the decisive voter over τ and calculate his most preferred tax rate,
denoted by τ dec (λ), as a function of the share parameter λ. Finally, we ﬁnd the point
where these reaction functions λdec (τ ) and τ dec (λ) cross. This point corresponds to thestructure-induced equilibrium outcome of the voting game.
Consider the political determination of λ. The decisive voter over λ is a young individual because (i) the population size of the young is larger than that of the old, and (ii)all young individuals have the same preferences for λ regardless of their type. Therefore,
from (6), the decisive voter’s reaction function λdec (τ ) is given by:{0
if τ ∈ [0, τ ]
(
)1/σ
(
]
(7)
λdec (τ ) =η
2+nif τ ∈ τ , 12 .
1 − (1−τ)τ w̄ (2+n)(1+n)
Next, consider the political determination of τ . The decisive voter over τ belongs to
the young generation because (i) young individuals choose lower tax rates than the old,and (ii) the population size of the young is larger than that of the old. In particular,
to determine the type of decisive voter, we focus on the parameter σ representing theelasticity of marginal utility of youthful consumption and consider two cases separately:a low elasticity (σ ≤ 1 in Subsection 4.1) and a high elasticity (σ > 1 in Subsection 4.2).
We adopt the above classiﬁcation because the order of preferences for the tax ratecritically depends on the degree of elasticity. For the case of σ ≤ 1, a lower-income young
individual prefers a higher tax rate. However, for the case of σ > 1, a low-income young
individual may prefer a lower tax rate than middle-income (or middle- and high-income)individuals. For each case, we show the existence and uniqueness of a structure-induced
equilibrium of the voting game and explain the mechanism underlying the result.

4.1

The Case of a Low Elasticity (σ ≤ 1)

To determine the type of decisive voter over τ in the case of σ ≤ 1, we consider the
preferred tax rate of a type-j young individual given by (5). Figure 4 illustrates thecondition (5) that determines the preferred tax rate by a type-j young (j = L, M, H)
14

individual. The left-hand side of (5), denoted by LHS, is decreasing in τ and is independent of the type of young individual. In contrast, the right-hand side of (5), denoted by
RHS j , is nondecreasing in τ , and dependent on the type of young individual and featured
by RHS H ≥ RHS M ≥ RHS L , where an equality holds if and only if σ = 1. The kink
point of τ = τ̂ (wj ) implies that a type-j young individual can save part of his income
if τ < τ̂ (wj ) and nothing if τ ≥ τ̂ (wj ). It is immediately observed from Figure 4 thatgiven λ, a lower-income young individual prefers a higher tax rate: τ yH < τ yM < τ yL forall λ ∈ [0, 1], where τ yj (j = L, M, H) denotes the preferred tax rate of a type-j young
individual.
[Figure 4 about here.]
Given the assumption of demographic structure (Assumption 1) and the fact that
τ yH < τ yM < τ yL < τ oj , the decisive voter over τ is the one who prefers the highest tax
rate among young individuals, that is, a type-L young individual. Therefore, the reaction
function of τ, τ dec (λ), is implicitly given by (5) with j = L. To ﬁnd the crossing point of
the two reaction functions, λdec (τ ) and τ dec (λ), we substitute (7) into (5) with j = L to
obtain:y(τ ; w̄, n) = z(τ ; wL ),where:{y(τ ; w̄, n) ={z(τ ; wL ) ={ 1 }1−σ 1−2τw̄
if τ ∈ [0, τ ]βη 2+n((1−τ )τ )σβ(1 + n)(1 − 2τ )w̄
if τ ∈ (τ , 1/2]
βRwL(wL )1−σ(1−τ )σif τ < τ̂ (wL )if τ ≥ τ̂ (wL ).

The function y(τ ; w̄, n) represents the marginal beneﬁt of taxation including the politically determined λ which is adjusted to a change in τ ; and the function z(τ ; wL ) represents
the marginal cost of taxation for a type-L individual. Solving y(τ ; w̄, n) = z(τ ; wL ) for τ
leads to the tax rate in a structure-induced equilibrium of the voting game. The corresponding λ is obtained by substituting the equilibrium τ into the reaction function λdecin (7).
Proposition 1. Suppose that σ ≤ 1 holds. There exists a unique structure-induced
equilibrium of the voting game such that the decisive voter over τ is a type-L young
individual.
Proof. See Appendix 8.2.
There are two possible cases of the equilibrium. The ﬁrst is the case where the wageof type-L individuals is high such that they can save part of their income in youth for15

future consumption. In this case, the equilibrium tax rate represented by the crossing
point of y(τ ; w̄, n) and z(τ ; wL ) is below the critical rate of τ̂ (wL ). The second is the casewhere the wage of type-L individuals is low such that they save nothing in their youth.
The equilibrium tax rate is given above the critical rate of τ̂ (wL ).

4.2

The Case of a High Elasticity (σ > 1)

Next, consider the case of a high elasticity such that σ > 1. The decisive voter over λis equivalent to that in the previous case; the reaction function λdec (τ ) is given by (7).
However, the decisive voter over τ may diﬀer from the previous case; the order of preferred
tax rates may change depending on the value of λ.
To determine the decisive voter over τ , we recall the condition (5) that determines thetax rate preferred by a type-j young individual for a given λ. The graphs of (5) for thecase of σ > 1 are illustrated in Figure 5. The main diﬀerence from the previous case isthat RHS i and RHS j (i ̸= j) cross at tax rate τ ∈ (0, 1/2). This is because when a type-j
individual is borrowing constrained, the slope of RHS j becomes steeper as the elasticity σ
increases. There are two critical values of τ , τ̃ LM and τ̃ M H , such that RHS L and RHS M
cross at τ = τ̃ LM and RHS M and RHS H cross at τ = τ̃ M H . By direct calculation, we
obtain:( M 1−σ )1/σ
( L 1−σ )1/σ
(w )
(w )
MH
LM
and τ̃
≡1−
,
τ̃
≡1−
M
βRw
βRwH
where τ̂ (wL ) < τ̃ LM < τ̂ (wM ) < τ̃ M H < τ̂ (wH ) (see Figure 5). The derivation of τ̃ LM
and τ̃ M H is given in Appendix 8.3.
[Figure 5 about here.]
The tax rate preferred by a type-j young individual is determined by the crossing pointof LHS and RHS of (5). RHS is independent of λ while LHS is strictly increasing in
λ. The tax rate preferred by a type-j young individual depends on the size of λ. Overall,
he prefers a higher tax rate when λ is higher.
The order of tax rates preferred by the three types of individuals is changed by thesize of λ, as illustrated in Figure 5. First, when λ is low such that LHS of (5) crosses
RHS of (5) with j = L within the range (0, τ̃ LM ], the tax rates preferred by the young
are ordered by τ yH < τ yM < τ yL , where τ yj (j = L, M, H) denotes the preferred tax rateby type-j young: the type-L young individual becomes the decisive voter. Second, when
λ attains a middle value such that LHS of (5) crosses RHS of (5) with j = M within therange (τ̃ LM , τ̃ M H ], the tax rates preferred by the young are ordered by τ yH < τ yL < τ yMor τ yL ≤ τ yH < τ yM : the decisive voter in this case is the type-M young individual.
Finally, when λ is high such that LHS of (5) crosses RHS of (5) with j = H within the16range [τ̃ M H , 1/2], the tax rates preferred by the young are ordered by τ yL < τ yM < τ yH :the decisive voter becomes the type-H young individual.
Given the abovementioned feature, the reaction function of τ , τ = τ dec (λ), is now
implicitly given by:{}1−σ1
1 − 2τLM
H
β(1 + n)λ(1 − 2τ )w̄ + (1 + β)η
(1 − λ)w̄
σ = z̃(τ ; w , w , w ), (8)2+n((1 − τ )τ )where z̃(τ ; wL , wM , wH ) ≡ minj {z(τ, wj )}. The graph of the function z̃ is illustrated by
the bold curve in Figure 5.
We substitute the reaction function of λdec (τ ), given by (7), into the left-hand side of(8) to obtain the condition that determines the equilibrium tax rate:y(τ ; w̄, n) = z̃(τ ; wL , wM , wH ),where y(·) has been already deﬁned in the previous subsection. Figure 6 illustrates thegraphs of y(τ ; w̄, n) and z̃(τ ; wL , wM , wH ). Solving y(τ ; w̄, n) = z̃(τ ; wL , wM , wH ) forτ leads to the tax rate in a structure-induced equilibrium of the voting game. The
corresponding λ is obtained by substituting the equilibrium τ into the reaction function
λdec in (7).
[Figure 6 about here.]
Proposition 2. Suppose that σ > 1 holds. There exists a unique structure-induced
equilibrium of the voting game such that the decisive voter over τ is:{}1/σ(i) a type-L individual if −1 + 2 (wL )1−σ /βRwM≤ RwM /(1 + n)w̄;(ii) a type-M individual otherwise.
Proof. See Appendix 8.4.
A noteworthy feature of Proposition 2 is that under certain conditions, the middleincome individuals prefer a higher tax rate than the low-income individuals. In particular,
if the condition in statement (ii) of Proposition 2 holds, there exists an equilibrium, like anends-against-the-middle equilibrium, where the low- and high-income young individuals
form a coalition in favor of a low tax rate and the middle-income individual favoring ahigh tax rate becomes the decisive voter (see Figure 6).
The key factors in Proposition 2 are the borrowing constraints and the elasticity of
marginal utility of youthful consumption. To understand the roles of these two factors,
consider the case where the low-income individuals are faced with a borrowing constraint.
Here, they wish to consume more in their youth, but cannot because of the borrowing
constraint. In this situation, a higher tax rate produces two opposing eﬀects: a negative17

eﬀect that results in lower after-tax income and thus, the utility loss of taxation in youth,
and a positive eﬀect that produces higher social security beneﬁt and thus, the utility gain
in old age.
The net impact of taxation depends on the elasticity of marginal utility of youthful
consumption. When the elasticity is low such that σ < 1, the positive eﬀect outweighs thenegative eﬀect. The low-income individual then chooses the highest tax rate among theyoung and thus becomes the decisive voter. In contrast, when the elasticity is high suchthat σ > 1, the negative eﬀect may outweigh the positive eﬀect for low-income individuals.
They choose a lower tax rate than the middle-income individuals, and this results in an
equilibrium where the middle-income individual becomes the decisive voter.
Figure 7 illustrates the conditions that determine the decisive voter and his status ofsaving in a wL − wM space. From the ﬁgure, we ﬁnd that the decisive voter is a type-L
individual when wage incomes levels of the two types of individuals are high such thatthe pair (wL , wM ) is set within the area marked by (j = L) in Figure 7. The order ofpreferred tax rates is the reverse of the wage rates. However, the order is changed when
the wage income level of the type-L individual is suﬃciently low such that (wL , wM ) is
set within the area marked by (j = M ) in Figure 7. The decisive voter becomes thetype-M young individual. The equilibrium is featured by the situation that resembles theends-against-the-middle equilibrium.
[Figure 7 about here.]5Eﬀects of Inequality on Policy

Given the characterization of the political equilibrium in Section 4, we now investigate
how the tax rate and the share of social security in government expenditure change in
response to a change in inequality. In particular, we consider a mean-preserving reductionof the decisive voter’s wage in order to compare two groups of countries with similar per
capita income levels but diﬀerent levels of income inequality. For this purpose, we suppose
a reduction of the type-L’s (or type-M ’s) wage associated with an increase in type-H’s
wage when the decisive voter is a type-L (or type-M ) individual.
We focus on a non-trivial equilibrium with λ > 0 to observe the marginal eﬀect
on the share of social security in government expenditure. Under Assumption 3(ii), we
can demonstrate cases of borrowing-unconstrained as well as borrowing-constrained typeL individuals in the presence of social security, λ > 0. Otherwise, the type-L young
individual is always borrowing constrained when λ > 0.
Proposition 3. Consider a political equilibrium with λ > 0.
18

(i) In an economy with σ ≤ 1 where the decisive voter is a type-L young individual, thetax rate and the share of social security are nondecreasing in response to a meanpreserving reduction of wL .
(ii) In an economy with σ > 1 where the decisive voter is a type-j ( j = L or M )young individual, a mean-preserving change in the decisive voter’s wage ( wj ) locally
produces inverse V-shaped relationships between wj and the tax rate ( τ ) and between
wj and the share of social security ( λ).
Proof. See Appendix 8.5.
Figure 8 illustrates the eﬀects of a mean-preserving change in the decisive voter’s wage
on the equilibrium tax rate when a type-L individual is the decisive voter. Panel (a) isfor the case of σ ≤ 1; Panel (b) is for the case of σ > 1. Proposition 3 states that if theelasticity is low such that σ ≤ 1, there is, in general, a monotone relationship between
the decisive voter’s wage and his preferred tax rate: the decisive voter prefers a higher
tax rate as he becomes poorer. However, when the elasticity is high such that σ > 1,such a monotone relationship no longer holds. Once the decisive voter’s wage falls below
the threshold level that changes his status from unconstrained to constrained, he prefers
a lower tax rate as he becomes poorer, as demonstrated in Subsection 3.2. Thus, there
is an inverse V-shaped relationship between the decisive voter’s wage and the preferred
tax rate around the threshold level of wage, as illustrated in panel (c) of Figure 8. Givena positive correlation between the tax and the share of social security, there is also an
inverse V-shaped relationship between the decisive voter’s wage and the share of social
security in government expenditure.
[Figure 8 about here.]
Two remarks are in order. First, a positive correlation between the tax and the shareof social security arises even if we assume a representative individual and thus, no wage
inequality within a generation. This statement is easily conﬁrmed by looking at thepreferred share of social security by the young, Eq. (7). However, when the assumptionof wage inequality is dropped, we cannot investigate the eﬀect of wage inequality on the
distribution of government expenditure between diﬀerent generations, which is the main
objective of this paper.
Second, in the current framework, the negative correlation between inequality andthe share of social security in government expenditure arises only in the equilibrium
where the following two conditions hold: (i) the elasticity of marginal utility of youthful
consumption, σ, is above unity; and (ii) the decisive voter is borrowing constrained. Whenone of the conditions fails to hold, the economy displays a positive correlation between
19inequality and the share of social security. Therefore, our analysis suggests that thesefactors are the keys in demonstrating the above-mentioned inverse V-shaped relationships.
These relationships still hold even if the assumption of a quasi-linear utility function is
dropped, as we brieﬂy demonstrate in the next section.

6

A Generalized Utility Function

At this point, we have conducted an analysis assuming a quasi-linear utility function
where the utility of old-age consumption is given by βcoj
t+1 . This speciﬁcation enables us
to illustratively show the existence and uniqueness of the political equilibrium. However,the speciﬁcation also results in (i) a saving decision unaﬀected by social security; and (ii)type-independent preferences over the share of social security. We introduce a generalized
utility function of old-age consumption to resolve these problems.
The main result of this section is that most of the previous results still hold true
under the alternative utility function. That is, under a certain condition, there exists anequilibrium, like an ends-against-the-middle equilibrium, when the elasticity of marginalutility of youthful consumption is above unity and the decisive voter is borrowing constrained. In this equilibrium, a mean-preserving spread of income inequality results in alower equilibrium tax rate and a lower share of social security in government expenditure.
For the purpose of analysis, we assume the following utility function:[]oj 1−σyj 1−σ
1−σ
1−σ
(c
)
−
1
(g
)
−
1
(g
)
−
1
(c
)
−
1
t
t+1
t+1
+η
+β
+η
.
Utj = t
1−σ
1−σ
1−σ
1−σ
The main diﬀerence from the previous model is that the utility of old-age consumption is1−σgiven by β{(coj− 1}/(1 − σ) rather than βcoj
t+1 )t+1 . The maximization of their lifetime
utility under the budget constraints leads to the following saving function:{[]}(βR)1/σbt+1jj
st = max 0,(1 − τt )w −.
(βR)1/σ + R(βR)1/σSaving now depends on the social security beneﬁt bt+1 that gives individuals a disincentive
to save. We hereafter drop the time subscript because our focus is on the time-invariant
policy.
We substitute the government budget constraint for social security b = (1 + n)λτ (1 −τ )w̄ into the above saving function to obtain the following condition that determines thesaving behavior of a type-j individual:wj (βR)1/σ·> λτ .
w̄
1+nThis inequality condition states that a type-j individual is borrowing unconstrained if his
wage is high, the tax burden is low, and/or the share of social security in government
sj > 0 ⇔expenditure is also low.
20

With the saving function and the government budget constraints, we give the consumption functions of a type-j individual in youth and old age as follows:
[
]
{
1/σ
j
(1+n)λ(1−τ )τ w̄
R
j
(1
−
τ
)w
+
if ww̄ · (βR)
> λτ
yj
R
1+n
(βR)1/σ +R
ct =
1/σ
j
(1 − τ )wj
if ww̄ · (βR)
≤ λτ
1+n
]
{ (βR)1/σ R [
1/σ
j
)τ w̄
(1 − τ )wj + (1+n)λ(1−τ
if ww̄ · (βR)
> λτ
oj
R
1+n
(βR)1/σ +R
ct+1 =
1/σj(1 + n)λ(1 − τ )τ w̄
if ww̄ · (βR)
≤ λτ.
1+n
Unlike the previous case, the consumption in youth is now type-dependent and is linearly
related to lifetime income when individuals are borrowing unconstrained.
After some calculation, we can obtain indirect utility functions of type-j young andold individuals as follows:
{
1/σ
j
y,j
Vs>0
if ww̄ · (βR)
> λτ
yj
1+n
V =
y,j
wj (βR)1/σ
Vs=0 if w̄ · 1+n ≤ λτ
{
1/σ
j
o,j
Vs>0
if ww̄ · (βR)
> λτ
oj
1+n
V =
o,j
wj (βR)1/σ
Vs=0 if w̄ · 1+n ≤ λτ
where:]1−σ(1 + n)λ(1 − τ )τ w̄
η(1 − τ )w +
+
(g)−σ
1/σ
(βR) + R
R
1−σ
[
]1−σ
1
βη
(1 − λ)(1 − τ )τ w̄
;+1−σ 2+n
)1−σ
1 (
β
η
y,j
Vs=0
≡
(1 − τ )wj+[(1 + n)λ(1 − τ )τ w̄]1−σ +
(g)−σ
1−σ
1−σ
1−σ
[
]1−σ
1
βη
+
(1 − λ)(1 − τ )τ w̄
;1−σ 2+n
]1−σ
1 [ j
o,j
;
Vs>0
≡
Rs−1 + (1 + n)λ(1 − τ )τ w̄
1−σ
[
]1−σ
βη
1+n
o,j
Vs=0 ≡(1 − λ)(1 − τ )τ w̄
.
1−σ 2+n

y,j
Vs>0
≡

1
1−σ

(

R

)−σ [

j

The terms unrelated to political decisions are omitted from the above expressions. We
can show that these preferences satisfy single-peaked properties by following the samemanner as in the case of a quasi-linear utility function.
The policy preferences of the old are the same as for quasi-linear utility. That is,regardless of type and saving behavior, the old wish to maximize the tax revenue fromthe young and use it exclusively for social security: τ oj = 1/2 and λoj = 1 hold for all j.
Accordingly, generalization of the utility function does not aﬀect the policy preferencesof the old.
21

We next consider the policy preferences of the young. Given λ, the preferred tax rateof a type-j young individual satisﬁes the following ﬁrst-order condition with respect to τ :
{
1/σ
j
yj
RHSs>0if τ < τ ∗ (wj , λ) ≡ ww̄ · (βR)
· λ1
y
yj
1+n
LHS = RHS ≡(9)yj
RHSs=0if τ ≥ τ ∗ (wj , λ),where:[]−σ
1
1
LHS ≡ βη(1 − λ)τ w̄
(1 − λ)(1 − 2τ )w̄,
2+n
2+n
(
)−σ [
]−σ [
]
(1 + n)λτ w̄
(1 + n)λ(1 − 2τ )w̄
Ryjjj
RHSs>0 ≡
w +
w −
, and
(βR)1/σ + R
RR( )1−σyj
RHSs=0≡ wj− β [(1 + n)λτ w̄]−σ (1 + n)λ(1 − 2τ )w̄.
y

LHS y represents the marginal beneﬁt of taxation in terms of the utility of public goods.
This beneﬁt is common to the three types of young agents because of the nature of public
goods. RHS yj represents the marginal cost of taxation plus the marginal beneﬁt of social
security in terms of the utility of consumption. The sum of these costs and beneﬁts diﬀers
among individuals. In particular, the following properties hold (see Appendix 8.6 for the
proof):{

RHS yL ≤ RHS yM ≤ RHS yH if σ ≤ 1yL
yM
yH
RHSs=0> RHSs=0> RHSs=0
if σ > 1,(10)where an equality in the ﬁrst line holds if and only if σ = 1 and s = 0. Similar to theyjprevious model, the order of RHSs=0(j = L, M, H) critically depends on the degree ofσ.5
Panel (a) of Figure 9 illustrates the graph of (9) when σ ≤ 1 holds. The crossing pointof LHS y and RHS yj determines the tax rate preferred by a type-j young individual.
The ﬁgure shows that a lower-income young individual prefers a higher tax rate. Under
the demographic structure assumption given in Assumption 1, a type-L young individual
becomes the decisive voter over τ . That is, the ends-against-the-middle equilibrium neverarises when the elasticity is low such that σ ≤ 1.
[Figure 9 about here.]
Panel (b) of Figure 9 illustrates the graph of (9) when σ > 1 holds. A noteworthy
feature is that lower-income young individuals prefer a lower tax rate when they areborrowing constrained. In particular, there may arise an equilibrium where the low- andthe high-income young individuals form a coalition against the middle, as illustrated inPanel (b) of Figure 9. Therefore, the high elasticity and the borrowing constraint remain
the keys to the existence of the ends-against-the-middle equilibrium.
5

yj
If 1/σ < 1, the order of RHSs>0(j = L, M, H) is ambiguous.

22

The determination of the share of social security λ is slightly diﬀerent from that in
the previous quasi-linear utility case. The preferred share of a type-j young is given by:0ifτ ≤ τ̃ (wj )


j
1/σ

βηR
1
R
 2+n
−( (2+n)(1+n)·w) · (βR)1/σ
+R τ w̄
if τ̃ (wj ) < τ < τ ∗ (wj )
1/σ
βηR
1+n
1
λyj =
(11)
+
(
)
2+n
(2+n)(1+n)(βR)1/σ +R


1

2+n

 1
if
τ ∗ (wj ) ≤ τ
1/σ
η
+
(1+n)
2+n ( (2+n)(1+n) )
where

(

)1/σ
βηR
R
(2 + n)wj
τ̃ (w ) ≡··;(2 + n)(1 + n)(βR)1/σ + R
w̄
}
{
(
)1/σ
1
η
(2 + n)wj+(1 + n).
τ ∗ (wj ) ≡ (βR)1/σ ·2+n(2 + n)(1 + n)
(1 + n)w̄
j

The ﬁrst two lines of the right-hand side in (11) represent the choice of λ when a type-j
young is borrowing unconstrained; the third line represents the choice of λ when he isborrowing constrained. The derivation of (11) is given in Appendix 8.6.
When the tax burden is low, such that τ ≤ τ̃ (wj ), a type-j young individual can savemuch for his old-age consumption and thus, ﬁnds it unnecessary to use tax revenue forsocial security: λ = 0. However, when the tax is above τ̃ (wj ), a type-j young individualﬁnds it optimal to oﬀset part of their tax-induced consumption loss with a social security
beneﬁt. In particular, a lower income agent prefers a higher share of social security.
A type-j young individual is borrowing constrained when the tax rate is high such thatτ ≥ τ ∗ (wj ). Borrowing-constrained individuals choose the same share of social security
regardless of their type. This is because they have the same level of old-age consumptionthat is equal to the lump-sum pension beneﬁt. They then choose that share to equate
the marginal utilities of old-age consumption and public goods, both of which are typeindependent. This result is diﬀerent from that under a quasi-linear utility function.
Panel (c) of Figure 9 illustrates the reaction function of λ for each type of an individual.
The ﬁgure shows that λyH (τ ) ≤ λyM (τ ) ≤ λyL (τ ) < λo holds for any τ . Thus, under thedemographic structure in Assumption 1, a type-L individual agent becomes the decisive
voter. We can derive the political equilibrium tax rate by substituting λ = λyL into thedecisive voter’s ﬁrst-order condition with respect to τ .
Given a brief characterization of the political equilibrium, we now compare the income
inequality eﬀects between the current and former models. In particular, we focus on thesituation where the decisive voters over τ and λ are borrowing constrained. The decisive
voter’s choice of λ in the current framework is given by:
λ

dec

=
1
+
2+n

(

1
2+n
η
(2+n)(1+n)

23

)1/σ

,
(1 + n)which is independent of τ . We substitute this into the ﬁrst-order condition with respectto τ , (9), for the case of τ ≥ τ ∗ (wj , λ) and obtain the following condition that determines
the equilibrium tax rate when the decisive voter is borrowing constrained:(() )1−σ( j )1−σ
1 − λdec w̄
= βηw(τ )−σ (1 − 2τ )2+n(() )1−σ ( dec )1−σ
+ β (1 + n) 1 − λdec w̄λ(τ )−σ (1 − 2τ ),where the left-hand side shows the marginal cost of taxation, the ﬁrst term on the righthand side shows the marginal beneﬁt of public goods, and the second term on the righthand side shows the marginal beneﬁt of social security. Given τ , the right-hand side isindependent of wj whereas the left-hand side is decreasing (increasing) in wj if σ > (<)1.
Thus, a mean-preserving reduction of the decisive voter’s wage decreases (increases) theequilibrium tax rate if the elasticity is high (low) such that σ > (<)1. This result isqualitatively equivalent to that in the quasi-linear utility function model.

7

Conclusion

How does wage inequality aﬀect the distribution of tax revenue between social security
and forward intergenerational public goods provision in the presence of borrowing constraints? This paper develops a political economy model that addresses this question.
Two features are crucial to our analysis and results: the elasticity of marginal utilityof youthful consumption and the borrowing constraint. These features derive an endsagainst-the-middle equilibrium where low- and high-income individuals form a coalitionin favor of a low tax rate and middle-income individuals favor a high tax rate. In addition,higher wage inequality results in a lower level of social security and a lower share of social
security (i.e., a higher share of public goods provision) in government expenditure when
the decisive voter is borrowing constrained and the elasticity is above unity.
To obtain these results, we simplify the analysis by adopting a quasi-linear utility
function. Because of this simpliﬁcation, we can remove the link between saving and theallocation of tax revenue between social security and public goods provision. However,as shown in Section 6, we demonstrate that the main result is qualitatively unchanged
under a generalized utility function. Thus, our analysis and result are almost robust to
the assumption of a quasi-linear utility function.6
6

The result established in this paper may fail to hold when we introduce intra-generational redistribution, i.e., income redistribution within young agents, into the model. Borrowing-constrained agents may
prefer a higher, rather than a lower, tax rate in response to a reduction of their wage. We would like tothank one of the reviewers for pointing this out.
We exclude this possibility from the analysis because, as surveyed in Introduction, the empirical evidence shows the negative correlation between old-age social security and wage inequality in the presence

24

8

Appendix

8.1

Single-peakedness of Preferences

8.1.1

Single-peakedness of preferences over τy,jy,jThe proof proceeds as follows. First, we show that both Vs>0and Vs=0
are single peaked
y,j
y,j
y,j
y,j
over τ . Then, we demonstrate that ∂Vs>0/∂τ = ∂Vs=0/∂τ and Vs>0
= Vs=0hold atjy,jτ = τ̂ (w ), implying that V has a unique local maximum over the whole range of τ andthus that V y,j is single peaked over τ .
y,jy,jThe ﬁrst and the second derivatives of Vs>0and Vs=0with respect to τ are:
y,j
∂Vs>0
= −βRwj + β(1 + n)λ(1 − 2τ )w̄∂τ()−σ(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
+ βη
·
;
2+n
2+n
(
)−σ
y,j
∂ 2 Vs>0(1 − λ)(1 − τ )τ w̄
(1 − λ)w̄= (−2)β(1 + n)λw̄ + (−2)βη
2
∂τ
2+n
2+n
(
)−σ−1 {
}2
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
+ βη(−σ)
2+n
2+n
< 0;
y,j
∂Vs=0
= (−1)(1 − τ )−σ (wj )1−σ + β(1 + n)λ(1 − 2τ )w̄∂τ()−σ(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
+ βη
·
;
2+n
2+n
y,j
∂ 2 Vs=0= (−σ)(wj )1−σ (1 − τ )−σ−1 + (−2)β(1 + n)λw̄
∂τ 2
(
)−σ
(1 − λ)(1 − τ )τ w̄
(1 − λ)w̄+ (−2)βη2+n2+n()−σ−1 {
}2
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
+ βη(−σ)
2+n
2+n
< 0.
y,j
y,j
Vs>0
and Vs=0
are single peaked over τ because the second derivatives are negative.
y,jy,jNext, we show that ∂Vs>0/∂τ = ∂Vs=0/∂τ at τ = τ̂ (wj ). By direct calculation, we
have:
y,j
y,j
∂Vs>0
∂Vs=0
R
⇔ −βRwj R (−1)(wj )1−σ (1 − τ )−σ .
∂τ
∂τ
j
j
τ =τ̂ (w )τ =τ̂ (w )of intra-generational redistribution; and because our aim of this paper is to consider the allocation of government spending between old-age social security and forward intergenerational public goods provisionin an empirically plausible situation.

25

At τ = τ̂ (wj ) ≡ 1 − 1/ (βR)1/σ wj , the right-hand side of the above condition is rewrittenas:(−1)(wj )1−σ (1 − τ̂ (wj ))−σ = −βRwj ,y,jy,j/∂τ = ∂Vs=0/∂τ at τ = τ̂ (wj ).
implying that ∂Vs>0y,jy,jFinally, we show that Vs>0
= Vs=0hold at τ = τ̂ (wj ). By direct calculation, we have:y,jy,jR Vs=0Vs>0τ =τ̂ (wj )τ =τ̂ (wj )
(
)1−σ
−1/σ
{
} ((1 − τ )wj )1−σ − 1
(βR)
−1
−1/σ
j
⇐⇒
+ βR (1 − τ )w − (βR)
=
.
1−σ
1−σ

At τ = τ̂ (wj ) ≡ 1 − 1/ (βR)1/σ wj , the left-hand and right-hand sides of the above
condition are reduced to, respectively:
(
)1−σ
−1/σ
(βR)
−1
LHS = RHS =
,
1−σ
y,j
y,j
implying that Vs>0
= Vs=0hold at τ = τ̂ (wj ).

8.1.2

Single-peakedness of preferences over λ

Before proceeding to the proof, we note that the status of saving is independent of λbecause of the assumption of a quasi-linear utility function. Thus, it is suﬃcient to showy,jy,jthat ∂ 2 Vs>0/∂λ2 < 0 and ∂ 2 Vs=0/∂λ2 < 0 for the proof.
y,jy,jThe ﬁrst and the second derivatives of Vs>0and Vs=0with respect to λ are:
(
)−σ
y,j
y,j
∂Vs>0
(1 − λ)(1 − τ )τ w̄
∂Vs=0(1 − τ )τ w̄
== β(1 + n)(1 − τ )τ w̄ + (−1)βη
·
;
∂λ
∂λ
2+n
2+n
(
)−σ−1 (
)2y,jy,j
∂ 2 Vs>0(1 − λ)(1 − τ )τ w̄
(1 − τ )τ w̄
∂ 2 Vs=0== (−σ)
·
< 0.
∂λ2
∂λ2
2+n
2+n

8.2

Proof of Proposition 1

As shown in the text, when σ ≤ 1, the decisive voter is a type-L individual and his
preferred tax rate satisﬁes y(τ ; w̄, n) = z(τ ; wL ). The functions y(τ ; w̄, n) and z(τ ; wL )have the following properties: ∂y(τ ; w̄, n)/∂τ < 0, limτ →0 y(τ ; w̄, n) = ∞, y(1/2; w̄, n) =0, ∂z(τ ; wL )/∂τ ≥ 0, z(0; wL ) = max{βRwL , (wL )1−σ } < ∞, and z(1/2; wL ) ∈ (0, ∞).
These properties indicate that there exists a unique τ ∈ (0, 1/2) that satisﬁes y(τ ; w̄, n) =z(τ ; wL ).



26

8.3

The Derivation of τ̃ LM and τ̃ M H

The derivation of τ̃ LM is as follows. For the range of (τ̂ (wL ), τ̂ (wM )), the right-hand sideof (5), denoted by RHS j , is given by:{
RHS L = (wL )1−σ /(1 − τ )σ for j = L
j
RHS =RHS M = βRwM
for j = M.
RHS L < RHS M holds at τ = τ̂ (wL ); RHS L > RHS M holds at τ = τ̂ (wM ). Thus, there
exists a unique τ , denoted by τ̃ LM ∈ (τ̂ (wL ), τ̂ (wM )), that satisﬁes RHS L = RHS M
because RHS L is continuous and strictly increasing in τ whereas RHS M is independentof τ . We can derive τ̃ LM by solving (wL )1−σ /(1 − τ )σ = βRwM for τ .
Similarly, the tax rate that satisﬁes RHS M = RHS H for the range of (τ̂ (wM ), τ̂ (wH ))is derived by solving (wM )1−σ /(1 − τ )σ = βRwH for τ . The solution is denoted by τ̃ M H .

8.4

Proof of Proposition 2

8.4.1

Existence and uniqueness of the equilibrium

As shown in the text, when σ > 1, the decisive voter’s preferred tax rate satisﬁesy(τ ; w̄, n) = z̃(τ ; wL , wM , wH ). The functions y(τ ; w̄, n) and z̃(τ ; wL , wM , wH ) have thefollowing properties:∂y(τ ; w̄, n)/∂τ < 0,lim y(τ ; w̄, n) = ∞,

τ →0y(1/2; w̄, n) = 0,
∂ z̃(τ ; wL , wM , wH )/∂τ ≥ 0,z̃(0; wL , wM , wH ) = max{βRwL , (wL )1−σ } < ∞,z̃(1/2; wL , wM , wH ) ∈ (0, ∞).
These properties indicate that there exists a unique τ ∈ (0, 1/2) satisfying y(τ ; w̄, n) =z̃(τ ; wL , wM , wH ).
8.4.2

The determination of the decisive voter

Suppose that the type-H young individual is the decisive voter. He/she is borrowing unconstrained under Assumption 3(ii). Then, from Figure 6, it must hold thaty(τ̃ M H ; w̄, n) = β(1 + n)(1 − 2τ̃ M H )w̄ > z̃(τ̃ M H ; wL , wM , wH ) = βRwH at τ = τ̃ M H ,that is:(1 + n)(1 − 2τ̃ M H )w̄ > RwH .
This condition never holds under the assumptions of R ≥ 1 + n (Assumption 2) and wH >w̄. Therefore, the decisive voter is a type-L or type-M young individual. From Figure 6,27the type-L young individual becomes the decisive voter if β(1 + n)(1 − 2τ̃ LM )w̄ ≤ βRwM ,that is, if:( L 1−σ )1/σ
(w )R
−1 + 2 ·
≤
wM .
M
βRw
(1 + n)w̄
Otherwise, the decisive voter is a type-M young individual.

8.5



Proof of Proposition 3(i) For the case of σ ≤ 1, the decisive voter is a type-L young individual and theequilibrium tax rate satisﬁes y(τ ; w̄, n) = z(τ ; wL ), as shown in Subsection 4.1. Whenthe mean-preserving change in wL is considered, y(τ ; w̄, n) is unchanged while z(τ ; wL )is nonincreasing with reductions of wL . Therefore, the equilibrium tax rate satisfyingy(τ ; w̄, n) = z(τ ; wL ) is nondecreasing in response to a mean-preserving reduction of wL .
Given that λdec (τ ) is increasing in τ for τ ∈ (0, 1/2), λ is also nondecreasing in response
to a mean-preserving decrease in wL .
(ii) For the case of σ > 1, the decisive voter is a type-j (j = L or M ) individual
depending on parameter values, as shown in Proposition 2. To simplify the presentation,
suppose that a type-L individual is the decisive voter. Note that the following argument
applies for the case where a type-M is the decisive voter.
Assume that the equilibrium tax rate is given by τ equil = τ̂ (wL ): a type-L individual
is indiﬀerent between saving and not saving. Under this situation, the decisive voter’s
wage wL satisﬁes β(1 + n)(1 − 2τ̂ (wL ))w̄ = βRwL , or:R(wL )2 + (1 + n)w̄wL − 2(1 + n)w̄= 0.
(βR)1/σSolving this equation for wL , we obtain:√−(1 + n)w̄ + {(1 + n)w̄}2 + 8R(1 + n)w̄/(βR)1/σ
LLw = ŵ ≡.
2R
Therefore, the equilibrium tax rate is given by τ equil = τ̂ (wL ) when a type-L individualwith wL = ŵL is the decisive voter.
We now consider a mean-preserving change of wL around ŵL . As shown in Subsection
4.2, the equilibrium tax rate satisﬁes y(τ ; w̄, n) = z̃(τ ; wL , wM , wH ) if σ > 1. In particular,
around wL = ŵL , there exists a positive real number ε such that the equilibrium tax ratesatisﬁes the following condition:{
βRwL for wL ∈ (ŵL − ε, ŵL ],L
M
Hz̃(τ ; w , w , w ) =(wL )1−σfor wL ∈ [ŵL , ŵL + ε).
(1−τ )σWe focus on the range (ŵL − ε, ŵL + ε) and consider a mean-preserving change of
wL around ŵL . The right-hand side of the above equation is increasing in wL within the28

range (ŵL − ε, ŵL ) and decreasing in wL within the range (ŵL , ŵL + ε). This property
implies that the equilibrium tax rate attains the highest value at wL = ŵL within therange (ŵL − ε, ŵL + ε). Therefore, there is an inverse V-shaped relationship between thedecisive voter’s wage and the equilibrium tax rate around wL = ŵL . Given λdec (τ ) is
increasing in τ for τ ∈ (0, 1/2), there is also an inverse V-shaped relationship between thedecisive voter’s wage and the equilibrium share of social security around wL = ŵL .


8.6

Supplementary Explanation for Section 6

8.6.1

Derivation of (10)yj
In order to establish that condition (10) holds, we ﬁrst investigate the property of RHSs>0.
yj
The ﬁrst derivative of RHSs>0
with respect to wj leads to:()−σ
yj∂RHSs>0R·(βR)1/σ + R∂wj()−σ−1 []
(1 + n)λτ w̄
(1 + n)λw̄
j
j
= w +
· (1 − σ)w +{σ(1 − 2τ ) + τ } ,RRyj
where the term {σ(1 − 2τ ) + τ } is positive provided that τ > 1/2. Thus, ∂RHSs>0/∂wj >yL
yM
yH
0 holds if σ ≤ 1: this implies that RHSs>0< RHSs>0< RHSs>0
if σ ≤ 1.
yj
Next, we investigate the property of RHSs=0 . Direct calculation leads to:( )1−σ ( M )1−σ
yM
yL
⇔ wL
R w
;
R RHSs=0
RHSs=0
(
)
(
)
1−σ
1−σ
yM
yH
RHSs=0R RHSs=0
⇔ wM
R wH
.

Therefore, we obtain:yL
yM
yM
RHSs=0R RHSs=0
R RHSs=0⇐⇒ 1/σ Q 1.

An equality holds if and only if σ = 1.
8.6.2

Derivation of (11)Suppose ﬁrst that the type-j young agent is borrowing unconstrained. The ﬁrst-ordery,jcondition for the maximization of Vs>0
with respect to λ is given by:
(
)1/σ
j
βηR
R
1
y,j
− (2+n)(1+n)· (βR)1/σ
·w
2+n
∂Vs>0
+R τ w̄yj= 0 ⇔ λ = λs>0 ≡(< 1).
(
)1/σ
∂λ
βηR
1
1+n
+
2+n
(2+n)(1+n)(βR)1/σ +R

Taking into account the corner solution τ = 0, we obtain:


(
)1/σ
βηR

R
wj 
1
 2+n
− (2+n)(1+n)· (βR)1/σ +R · τ w̄ 
yj
λs>0 ≡ max 0,
.
(
)1/σ


βηR
1+n
1


+ (2+n)(1+n)2+n(βR)1/σ +R
29

The preferred share λ is increasing in τ and is positive if and only if τ = τ ∗ (wj ).
y,j
∂Vs>0
= 0 ⇔ λ = λyj
s=0 ≡
∂λ

1
+
2+n

(

1
2+n
η
(2+n)(1+n))1/σ(< 1),(1 + n)yj
yj
where λyj
s=0 is constant and independent of τ . The equality holds between λs>0 and λs=0
at τ = τ ∗ (wj ). Therefore, the preferred share λ by a type-j young agent is given as (11).

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32

Figure 1: This ﬁgure illustrates an example of the tax rates preferred by the old and the
young. In this example, a type-L young individual becomes a decisive voter.

33

Figure 2: Marginal cost of taxation. Panel (a) is the case of σ ≤ 1; Panel (b) is the caseof σ > 1.

34

Figure 3: This ﬁgure illustrates the share of social security (λ) preferred by the young in
response to a change in the tax rate (τ ).

35

Figure 4: The tax rates preferred by the three types of young individuals in the case ofσ ≤ 1.

36

Figure 5: The tax rate preferred by a type-j young individual in the case of σ > 1. The
bold curve illustrates the graph of ze in (8).

37Figure 6: The determination of the tax rate in the case of σ > 1. The ﬁgure illustrates
the case where the decisive voter is a type-M young individual.

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4

3

2

1

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4

3

2

1

0

4

3

2

1

0

Figure 7: The conditions that determine the decisive voter and his status of saving in awL − wM space. To illustrate the ﬁgure, we assume a generation to be 30 years in length.
Our selection of parameters is 1 + n = (1.01)30 , β = (0.98)30 , R = (1.015)30 and σ = 1.2.
We illustrate the set of parameters for the following three cases: w = 2.0 (panel (a)),
w = 3.0 (panel (b)), and w = 4.0 (panel (c)). The triangular area surrounded by thevertical axis, the 45-degree line and wM = w̄ line covers the set of wages (wL , wM ) relevant
for the analysis. The wage of the middle, wM , is assumed to be below the average: thisassumption reﬂects a typical right-skewed income distribution employed in the literature.
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Figure 8: Panels (a) and (b) depict the graphs of the equation y = z that determines
the equilibrium tax rate when the decisive voter is a type-L individual for the casesof σ < 1 and σ > 1, respectively. The three graphs of the function z are associatedto the three levels of type-L′ s wage income, wL , wL′ and wL′′ where wL > wL′ > wL′′ .
Panel (c) illustrates the relation between the decisive voter’s (i.e., type-L’s) wage and theequilibrium tax rate around the critical value of type-L’s wage, ŵL , in the case of σ > 1.

40

Figure 9: Panels (a) and (b) illustrate the condition expressing the preferred tax rates by
the young in cases of σ ≤ 1 and σ > 1, respectively. Panel (c) illustrates the condition of
the preferred shares of social security by three types of young individuals.

41