Discussion Papers In Economics
And Business

The Political Economy of Social Security and Public
Goods Provision in a Borrowing-constrained Economy
Ryo Arawatari and Tetsuo Ono
Discussion Paper 09-38-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Political Economy of Social Security and Public
Goods Provision in a Borrowing-constrained Economy
Ryo Arawatari and Tetsuo Ono
Discussion Paper 09-38-Rev.

August 2010

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe Political Economy of Social Security and Public
Goods Provision in a Borrowing-constrained
EconomyRyo Arawatariyand Tetsuo Onoz
August 23, 2010

Abstract
This paper introduces an overlapping-generations model with earnings heterogeneity and borrowing constraints. The labor income tax and the allocation oftax revenue across 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 a
low tax rate and less social security while middle-income individuals favor a hightax 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 low interest-rate elasticity of consumption.
Keywords: Borrowing constraint; Social security; Public goods provision; Endsagainst-the-middle equilibrium; Wage inequality
JEL Classi…cation: H41; H55; D72

Arawatari acknowledges …nancial support of the Japan Society for the Promotion of Science (JSPS)
through a Grant-in-Aid for Young Scientists (Start-up) (No. 21830046). Ono acknowledges the …nancial
support of the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Young
Scientists (B) (No. 20730208).
yFaculty of Economics, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano 390-8621 Japan. E-mail:ryo_a@shinshu-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.jp1

1

Introduction

Almost all 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 a larger volume of redistribution as the decisive voter becomes less well-oﬀ
as wage inequality increases (Romer, 1975; Roberts, 1977; Meltzer and Richard, 1981;
Alesina and Rodrik, 1994; Benabou, 1996; Persson and Tabellini, 1994). This prediction
still holds when the focus is on intergenerational redistribution, such as pay-as-you-go(PAYG) social security. That is, the intragenerational redistribution component of the
PAYG social security system makes this program palatable to low-income young individuals (Conde-Ruiz and Galasso, 2003, 2005; Bethencourt and Galasso, 2008).
The empirical evidence, however, does not necessarily support the abovementioned
theoretical predictions. OECD cross-country data show that the volume of redistribution
is negatively correlated with wage inequality (Gottschalk and Smeeding, 1997; Chen andSong, 2009). For instance, the United Kingdom and the United States feature higher
wage inequality and smaller redistribution whereas the Nordic countries display lower
wage inequality and larger redistribution. In fact, Pineda and Rodriguez (2006) found astrong negative correlation between redistribution and the share of capital in GDP where
the share is considered an indicator of income inequality.
Several theories have been provided to explain the negative correlation between inequality and redistribution. Examples include political bias toward the rich (Benabou,
2000); the prospect of upward mobility by low-income agents (Benabou and Ok, 2001;
Arawatari and Ono, 2009); and lobbying by rich capitalists (Rodriguez, 2004). Thesestudies, however, abstract away forward intergenerational public goods provision (forexample, public education, environmental maintenance, pure science and other public
services for future generations) as alternative public spending and thus say nothing about
how the intergenerational conﬂict over the composition of public spending is aﬀected by
the increase in intragenerational inequality.
In sum, most contributions addressing the negative correlation focus on a single policy
issue. A notable exception is Levy (2005), who develops a two-dimensional political
economy model with endogenous formation of parities. In essence, Levy (2005) considers
political conﬂicts over income redistribution that beneﬁts the poor and public educationthat beneﬁts the young. One of the striking results is that higher income inequality may
decrease, rather than increase, the tax rate in the young-majority equilibrium. However,her analysis ignores predictions about the share of redistribution and public education ingovernment because no provision of public education arises in equilibrium.1
1

In Levy (2005), public provision of education arises only in the old-majority equilibrium. However,inequality and redistribution have a positive correlation in that equilibrium, and this does not readily fit

2

The present paper contributes to this literature by predicting how the shares of social security and forward intergenerational public goods in government expenditure areaﬀected by inequality within a generation. For this purpose, we introduce an overlappinggenerations model with heterogeneous agents. In this economy, young workers are of three
types of income: low, middle and high. Because they are not permitted to borrow in youth
as a result of imperfect ﬁnancial markets, lower-income individuals are more likely to beborrowing constrained. Young workers then pay a ﬁxed proportion of their labor income
to the government, and the tax revenue is divided into PAYG social security paymentsfrom which the old can beneﬁt and public goods provision that the old cannot capture.
The tax rate and the allocation of tax revenue between social security and public
goods provision are determined in a two-dimensional majoritarian voting game played by
the young and the old. Voters cast a ballot over the labor income tax, which ﬁnances
social security and public goods provision, and over the allocation of tax revenue betweensocial security and public goods provision. Under this type of voting game, the existenceof a 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 once-andfor-all voting, which is applied to an overlapping-generations framework by Conde-Ruizand Galasso (2003, 2005).
Based on the abovementioned 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, receive the same level of social security
beneﬁt, and cannot capture the beneﬁt of forward intergenerational public goods. Instead,they prefer the tax rate that attains the top of the Laﬀer curve and full use of the revenue
for social security. In contrast, the preferences of the young depend on their income
type because the tax burden diﬀers across the types of income. In particular, the key
factors to their preferences are the borrowing constraint and the interest-rate elasticity of
consumption.
To understand the role of these two factors, consider the case where only low-income
individuals are faced with the borrowing constraint. 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 negative 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 of taxation in old age.
When the interest-rate elasticity is high, substitution across periods is easy: the utility
loss of taxation in youth is compensated for by the utility gain of taxation in old age.
with the available empirical evidence.

3

Therefore, low-income individuals choose a higher tax rate than middle- and high-income
individuals. However, when the interest-rate elasticity is low, compensation is not necessarily available for low-income individuals because substitution across periods is diﬃcult.
Low-income individuals then prefer a lower tax rate than middle-income individuals. This
results in an ends-against-the-middle equilibrium where low- and high-income individuals
form a coalition in favor of a low tax rate while middle-income individuals favor a hightax rate.
Given the characterization of political equilibrium, we investigate how the shares ofsocial security and forward intergenerational public goods in government expenditure arealtered in response to changes in wage inequality. We show that the mean-preserving
reduction of the decisive voter’s wage creates an inverse U-shaped relationship between
the decisive voter’s wage and the share of social security when the interest-rate elasticity islow. The positive correlation arises when the decisive voter’s wage is high and thus he/she
is borrowing unconstrained, while the negative correlation arises when his/her wage is low
and thus he/she is borrowing constrained. The same relation also holds between wage
inequality and the size of social security. Therefore, the interest-rate elasticity and theborrowing constraint are the key factors needed to demonstrate the negative correlation
between wage inequality and the share (or size) of social security.
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, the allocation of tax revenue and the levels of social
security and public goods provision via the political process. Section 6 brieﬂy undertakes
the analysis under a generalized framework. Section 7 provides some concluding remarks.

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 denoted by
Nt . Population grows at a constant rate n > 0 : Nt+1 = (1 + n)Nt for all t ≥ 0. Withineach generation, there are three types of agents according to ability, low, middle and high∑(j = L, M, H), whose proportions are respectively ρL , ρM and ρH , where j ρj = 1 andρj satisﬁes the following assumption.
Assumption 1. ρj > n/{2(1 + n)}, j = L, M, H.

4

Assumption 1 ensures that a young individual who prefers the highest tax rate among
young individuals becomes the decisive voter. To understand the argument stemmingfrom Assumption 1, suppose that a type-k (k = L, M or H) prefers the highest tax rate.
As explained below, all the old have the same preferences over policies and choose a higher
tax rate than any young agent. When the young and the old participate in voting, thesum of 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.]

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 given 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

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 inverse of the elasticity of young-age
consumption with respect to the interest rate. Following the literature (Conde-Ruiz andGalasso, 2005; Borck, 2007; Bethencourt and Galasso, 2008), we assume a quasi-linear
utility function for analytical tractability.2
Each individual works in 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 by, respectively:jjcyj
t + st ≤ (1 − τt )w ,j
coj
t+1 ≤ Rst + bt+1 ,2

In Section 6, we briefly analyze the case where the utility of old-age consumption is given by
1−σ
β(coj
/(1−σ). We will show that the main results are qualitatively unchanged under this alternativet+1 )assumption.

5

where sjt is saving, τt is the income tax rate in period t, R is the gross interest rate andbt+1 is the per capita social security beneﬁt in old age. We impose the restriction ofnonnegative savings, that is:
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 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 more simply demonstrate the analyticalsolution of the model. The second assumption abstracts away from the choice of social
security systems (for example, Bismarckian vs. Beveridgean) analyzed by, for example,
Borck (2007), Conde-Ruiz and Profeta (2007) and Cremer et al. (2007). We adopt thesecond assumption to concentrate on the allocation of tax revenue over social security andpublic goods provision as the main focus of the paper.
The representative type-j young individual maximizes his/her utility subject to thebudget constraints and the restriction of nonnegative saving. When sjt > 0, the ﬁrst−σorder condition for an interior solution is (cyj
= βR and thus deﬁnes the optimal
t )jsaving decision of a type-j individual given by st = (1 − τt )wj − (βR)−1/σ . By taking theborrowing 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 assumption ofa linear utility function of old-age consumption. Because of this property, we can easily
demonstrate the joint political determination of the tax rate τ and the proportion λ.
We should note that Conde-Ruiz and Galasso (2003, 2005) and Bethencourt andGalasso (2008) cut the link between the current saving decision and future policy variables by assuming no consumption in youth; that is, households save all of their incomein their youth and consume it in their old age. In contrast, the current paper introduces
the saving decisions of households because our focus is on the borrowing constraint.
The saving function (1) implies that there is a critical rate of tax such that:sjt ≷ 0 ⇔ τt ≶ τ̂ (wj ) ≡ 1 −61.
(βR)1/σ wj(2)

Figure 2 illustrates the relation between savings and the tax rate for each type of anindividual. A type-j individual chooses positive saving when the tax is below the criticalrate. 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 higher
ability individual receives a higher level of disposable income.
[Figure 2 about here.]

2.2

The Government

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

2.3

1+n(1 − λt )τt (1 − τt )w̄.
2+n

The Economic Equilibrium

We deﬁne the economic equilibrium as follows:7

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/her utility subject to the budget
constraints and the nonnegativity constraint of saving, (ii) the social security budget and the public goods budget are balanced 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 by, respectively:
{
cyj
t =
{
coj
t+1 =

(βR)−1/σ(1 − τt )wj

if τt < τ̂ (wj )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 )(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 can be represented bytheir indirect utility functions. We can use the above-mentioned consumption functions
to obtain an indirect utility function of a type-j young individual:{
Vtyj =yj
Vt,s>0if τt < τ̂ (wj )yjVt,s=0if τt ≥ τ̂ (wj ),(3)

where:(
y,j
Vt,s>0≡(βR)−1/σ

)1−σ

1−σ

1−σ

y,j
Vt,s=0≡−1((1 − τt )wj )
1−σ

−1

[ {
}
]
+ β R (1 − τt )wj − (βR)−1/σ + (1 + n)λt+1 (1 − τt+1 )τt+1 w̄
( 1+n
)1−σ
(1
−
λ
)(1
−
τ
)τ
w̄
−1
t
t
t
,
+ βη 2+n1−σ+ β(1 + n)λt+1 (1 − τt+1 )τt+1 w̄
( 1+n
)1−σ
(1
−
λ
)(1
−
τ
)τ
w̄
−1
t
t
t
+ βη 2+n.
1−σy,jVt,s>0denotes the indirect utility of a type-j young individual when he/she saves somey,jportion of his/her income, and Vt,s=0
denotes the indirect utility when he/she is facedwith a borrowing constraint and saves nothing. For each indirect utility function, theﬁrst term on the right-hand side shows the utility of consumption in youth, the second8term shows the utility of consumption in old age and the third term shows the utility of
public goods in old age. Variables unrelated to political decisions are removed from theindirect utility functions.
For a type-j old individual in period t, the indirect utility function is:
Vto,j ≡ (1 + n)λt (1 − τt )τt w̄,(4)where the right-hand side shows the PAYG social security beneﬁts. Old individuals havethe same indirect utility function regardless of their type because their saving in youth
is predetermined and the level of public goods they enjoy is predetermined one period in
advance. Therefore, old individuals have the same preferences for the tax rate, τ , and theshare of PAYG 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 individuals alive,
both young and old, cast a ballot over τ, the income tax, and λ, the share of social security
in government 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. We thus 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 equilibria of a majoritarian voting
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 the second feature, we assume full commitment, i.e.,
once-and-for-all voting. That is, voters determine the constant sequence of the parameters:
τt = τt+1 = τ and λt = λt+1 = λ for all t as in Casamatta, Cremer and Pestieau (2000)
and Conde-Ruiz and Profeta (2007). We can view the full commitment solution as thesolution including intergenerational interaction because the full commitment solution canbe supported as the subgame perfect equilibrium (see, for example, Conde-Ruiz andGalasso, 2003, 2005).
Given the stationary environment, the current model presents a static voting game.
Therefore, the result in Shepsle (1979) can be applied to obtain the suﬃcient conditions
for the existence of a structure-induced equilibrium. In particular, if preferences are9single peaked along every dimension of the issue space, a suﬃcient condition for (τ ∗ , λ∗ )to be an equilibrium of the voting game with full commitment is that τ ∗ represents theoutcome of majority voting over the jurisdiction τ when the other dimension is ﬁxed at
its level λ∗ , and vice versa. Taking second derivatives of Vtyj and Vto,j with respect to τand λ, we immediately ﬁnd that ∂ 2 V o,j /∂τ 2 < 0, ∂ 2 V o,j /∂λ2 < 0, ∂ 2 V y,j /∂τ 2 < 0, and∂ 2 V y,j /∂λ2 < 0, showing that preferences are single peaked.
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 rate of tax and the share of PAYG social security are respectively givenby:1τ oj = and λoj = 1 for all j.
2Maximization is realized when the tax rate is set to attain the top of the Laﬀer curve,τ (1 − τ ), and the maximized tax revenue is used exclusively for PAYG social security.
In what follows, we sequentially investigate the preferred tax and share of government
expenditure by the borrowing-unconstrained and borrowing-constrained young.

3.1

Voting by the Borrowing-unconstrained Youngy,jThe borrowing-unconstrained young choose τ to maximize Vs>0in (3) given λ, and λ toy,jmaximize Vs>0 in (3) given τ . First, we consider the choice of τ . The ﬁrst derivative ofy,jVs>0with respect to τ is given by:
y,j
∂Vs>0
= −βRwj + β(1 + n)λ(1 − 2τ )w̄
∂τ
}−σ
{
1+n
1+n
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄,
+ βη
2+n
2+n

where the ﬁrst term on the right-hand side shows a marginal decrease in old-age consumption, the second term shows a marginal increase in social security beneﬁts and the third
term shows a marginal utility beneﬁt of an increase in public goods provision produced
by an increase in tax revenue.
The borrowing-unconstrained young choose the τ that balances marginal costs andy,jbeneﬁts in terms of utility, that is, that attains ∂Vs>0/∂τ = 0, which is equivalent to:βRwj = β(1 + n)λ(1 − 2τ )w̄
{
}−σ
1+n
1+n
+ βη
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄.
2+n
2+n

(5)

The left-hand side of (5) is constant whereas the right-hand side of (5), denoted by RHS,
is featured by limτ →0 RHS = +∞ and RHS|τ =1/2 = 0. Therefore, the preferred tax rate10

by the borrowing-unconstrained young is set within the range (0, 1/2) as illustrated inPanel (a) of Figure 3.
[Figure 3 about here.]y,jNext, we consider the choice of λ. The ﬁrst derivative of Vs>0
with respect to λ is
given by:y,j∂Vs>0
= β(1 + n)(1 − τ )τ w̄ − βη
∂λ

{

1+n
(1 − λ)(1 − τ )τ w̄
2+n

}−σ
·

1+n
(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 PAYG social security, λ, is chosen to balance the marginal
beneﬁt and loss in terms of utility. However, under a certain condition, the share λ could
be zero because the marginal loss of public goods in terms of utility may overcome themarginal beneﬁt of PAYG social security at λ = 0.
Based on the above argument, the preferred share of the borrowing-unconstrainedy,jy,jyoung satisﬁes ∂Vs>0/∂λ = 0 when λ > 0 and ∂Vs>0/∂λ ≤ 0 when λ = 0; that is:
{
λ=

0
2+n
1 − (1+n)(1−τ)τ w̄

where:
τ≡1−√( η )1/σ
2+n

4(2+n)
1 − (1+n)w̄

if τ ∈ [0, τ ](]if τ ∈ τ , 12 ,(6)( η )1/σ2+n.
2Panel (c) of Figure 3 illustrates the graph of (6). 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. When the tax rate is below the critical rate τ , the tax revenue is too small, so that themarginal beneﬁt of raising the share of PAYG social security is lower than the marginal
cost of reducing the share of public goods provision for any λ ∈ [0, 1]. Therefore, theborrowing-unconstrained young choose no expenditure to social security, λ = 0, if τ ≤ τ .
When the tax rate is above the critical rate τ , the tax revenue is suﬃcient, so that
there is a share λ ∈ (0, 1) that balances the marginal beneﬁts and costs. The optimal
share for the borrowing-unconstrained young increases as the tax rate increases. The
optimal share attains its highest value at τ = 1/2, where the tax revenue is maximized.

11

3.2

Voting by the Borrowing-constrained Youngy,jThe borrowing-constrained young choose τ to maximize Vs=0in (3) given λ, and λ toy,jmaximize Vs=0 in (3) given τ . First, we consider the choice of τ . The ﬁrst derivative ofy,j
Vs=0with respect to τ is:
y,j
∂Vs=0
= −((1 − τ )wj )−σ wj + β(1 + n)λ(1 − 2τ )w̄
∂τ
{
}−σ
1+n
1+n
+ βη
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄,2+n2+nwhere the ﬁrst term on the right-hand side shows the marginal cost of a decrease indisposable income in youth, the second term shows the marginal increase in social security
beneﬁts, and the third term shows the marginal utility beneﬁt of an increase in public
goods provision produced by an increase in tax revenue.
The borrowing-constrained young choose τ that balances the marginal costs and beny,jeﬁts in terms of utility. That is, they choose τ that attains ∂Vs=0/∂τ = 0, which isequivalent to:(wj )1−σ= β(1 + n)λ(1 − 2τ )w̄
(1 − τ )σ
{
}−σ
1+n
1+n
+ βη
(1 − λ)(1 − τ )τ w̄
·(1 − λ)(1 − 2τ )w̄.
2+n
2+n

(7)

Panel (b) of Figure 3 illustrates the determination of τ that maximizes the utility of theborrowing-constrained young.
y,jNext, we consider the choice of λ. The ﬁrst derivative of Vs=0
with respect to λ is
given by:y,j∂Vs=0
= β(1 + n)(1 − τ )τ w̄
∂λ
{
}−σ
1+n
1+n
− βη
(1 − λ)(1 − τ )τ w̄
·(1 − τ )τ w̄,
2+n
2+n
y,j
which is equivalent to that of the young without borrowing constraints: ∂Vs=0
/∂λ =
y,j
∂Vs>0 /∂λ. This equality stems from the quasi-linear utility function. An increase in theshare λ creates a marginal utility beneﬁt irrespective of whether the borrowing constraint
is binding. Therefore, the preferred share of the borrowing-constrained young is given by:

{
λ=

0
2+n
1 − (1+n)(1−τ)τ w̄( η )1/σ
2+n

12

if τ ∈ [0, τ ](]if τ ∈ τ , 12 ,(8)

which is equivalent to that of the borrowing-unconstrained young.3

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
for each issue, we can now apply Shepsle’s (1979) result and characterize the structureinduced equilibrium of the game. To proceed with the analysis, we impose the following
assumption.
4(2+n)
Assumption 2: 1 > (1+n)
w̄

( η )1/σ
2+n.

This assumption implies that the preferred share in (6) and (8), which attains thehighest value at τ = 1/2, takes a positive value at τ = 1/2. Therefore, the assumption
ensures that a political equilibrium exists with λ > 0 for a certain range of τ ; otherwise,
λ = 0 holds for any τ ∈ [0, 1], implying a trivial outcome of no provision of PAYG social
security.
The structure-induced equilibrium outcome is found as follows. First, we determine
the decisive voter over λ and calculate his/her most preferred share, denoted by λdec (τ ),as a 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/her most preferred tax
rate, denoted by τ dec (λ), as a function of the share parameter λ. Finally, we ﬁnd thepoint where these reaction functions λdec (τ ) and τ dec (λ) cross. This point corresponds tothe structure-induced equilibrium outcome of the voting game.
First, 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, as shownin (6) and (8). Therefore, from (6) and (8), the decisive voter’s reaction function λdec (τ )is given by:{0
if τ ∈ [0, τ ]
dec
(
(
)
]
λ (τ ) =
(9)
1/σ
η
2+n
1 − (1+n)(1−τ
if τ ∈ τ , 12 .
)τ w̄ 2+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 oldand (ii) the population size of the young is larger than that of the old. In particular, todetermine the type of the decisive voter, we focus on the parameter σ representing theinverse of the interest-rate elasticity of young-age consumption and consider two cases
3

The equivalence stems from the assumption of a quasi-linear utility function of old-age consumption.
In Section 6, we show how the preferences over λ change when this assumption is removed.

13

separately: a high elasticity (1/σ ≥ 1 in Subsection 4.1) and a low elasticity (1/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 interest-rate elasticity. For the case of 1/σ ≥ 1, alower-income young individual prefers a higher tax rate. However, for the case of 1/σ < 1,a low-income young individual may prefer a lower tax rate than middle-income (or middleand high-income) individuals. For each case, we show the existence and uniqueness of astructure-induced equilibrium of the voting game and explain the mechanism underlying
the result.

4.1

The Case of a High Interest-rate Elasticity of Consumption(1/σ ≥ 1)

To determine the type of a decisive voter over τ in the case of 1/σ ≥ 1, we consider the
preferred tax rate of a type-j young in the following way. When τ < τ̂ (wj ), he/she saves
part of his/her income, and his/her preference for τ follows (5). When τ ≥ τ̂ (wj ), he/shesaves nothing and his/her preference over τ follows (7). From (5) and (7), given λ, thepreferred tax rate by the type-j young satisﬁes the following condition:{

1+n(1 − λ)w̄2+n{z

β(1 + n)λ(1 − 2τ )w̄ + βη
|
LHS
{
j
βRw
if τ < τ̂ (wj )=(wj )1−σ
if τ ≥ τ̂ (wj ),(1−τ )σ
{z
}
|

}1−σ

1 − 2τ
((1 − τ )τ )σ
}
(10)

RHS j

where LHS and RHS j show marginal beneﬁt and loss of taxation in terms of utility,
respectively.
Figure 4 illustrates the condition (10) that determines the preferred tax rate by a
type-j young (j = L, M, H) individual. The left-hand side of (10), denoted by LHS,
is decreasing in τ and is independent of the type of young individual. In contrast, theright-hand side of (10), denoted by RHS j , is nondecreasing in τ , and dependent on thetype 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 cansave part of his/her income if τ < τ̂ (wj ) and nothing if τ ≥ τ̂ (wj ). It is immediately
obvious from Figure 4 that given λ, a lower-income young individual prefers a higher tax
rate: τ yH < τ yM < τ yL for all λ ∈ [0, 1], where τ yj (j = L, M, H) denotes the preferred
tax rate of a type-j young individual.
14[Figure 4 about here.]
Given the assumption of demographic structure (Assumption 1) and the fact 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 (10) with j = L. To ﬁnd the crossing point of
the two reaction functions, λdec (τ ) and τ dec (λ), we substitute (9) into (10) with j = L to
obtain:y(τ ; w̄, n) = z(τ ; wL ),where:{y(τ ; w̄, n) ={z(τ ; wL ) =(1 + β)η

{ 1+n }1−σ 1−2τw̄
if τ ≤ τ
2+n((1−τ )τ )σβ(1 + n)(1 − 2τ )w̄
βRwL

if τ < τ̂ (wL )(wL )1−σ

if τ ≥ τ̂ (wL ).

(1−τ )σif τ > τ

Solving y(τ ; w̄, n) = z(τ ; wL ) for τ leads to the tax rate in a structure-induced equilibriumof the voting game. The corresponding λ is obtained by substituting the equilibrium τ
into the reaction function λdec in (9).
Proposition 1. Suppose that 1/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 8.1.
Figure 5 illustrates two examples. Panel (a) depicts the case where the wage of typeL individuals is high such that they can save part of their income in youth for future
consumption. In this case, the equilibrium tax rate represented by the crossing point ofy(τ ; w̄, n) and z(τ ; wL ) is below the critical rate of τ̂ (wL ). In contrast, panel (b) illustrates
the case where 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 ).
[Figure 5 about here.]

4.2

The Case of a Low Interest-rate Elasticity of Consumption(1/σ < 1)

Next, consider the case of a low interest-rate elasticity such that 1/σ < 1. The decisive
voter over λ is equivalent to that in the previous case: that is, the reaction function
15

λdec (τ ) is given by (9). However, the decisive voter over τ may diﬀer from the previous
case: that is, the order of preferred tax rates may change depending on the value of λ.
To determine the decisive voter over τ , let us recall the condition (10) that determines
the tax rate preferred by a type-j young individual for a given λ. The graphs of (10) for
j = L, M, H are illustrated in Figure 6. The main diﬀerence from the previous case isthat RHS i and RHS j (i ̸= j) cross at some tax rate τ ∈ (0, 1/2). This is because whena type-j individual is borrowing constrained, the slope of RHS j becomes steeper as theelasticity 1/σ becomes lower. Thus, there are two critical values of τ , τ̃ LM and τ̃ M H , suchthat RHS L and RHS H cross at τ = τ̃ LM and RHS M and RHS H cross at τ = τ̃ M H . By
direct calculation, we obtain:
(
τ̃

LM

≡1−

(wL )1−σ
βRwM

)1/σ

(
and τ̃

MH≡1−(wM )1−σ
βRwH

)1/σ
,

where τ̂ (wL ) < τ̃ LM < τ̂ (wM ) < τ̃ LM < τ̂ (wH ) (see Figure 6). The derivation of τ̃ LM andτ̃ M H is given in Appendix 8.2.
[Figure 6 about here.]
The tax rate preferred by a type-j young is determined by the crossing point of
LHS and RHS of (10). RHS is independent of λ while LHS is strictly increasingin λ. Thus, the tax rate preferred by a type-j young depends on the size of λ. Overall,he/she prefers a higher tax rate when λ is higher.
The order of tax rates preferred by the three types of agents is changed by the size of λas illustrated in Figure 6. First, when λ is low such that LHS of (10) crosses RHS of (10)with j = L within the range (0, τ̃ LM ], the tax rates preferred by the young are orderedby τ yH < τ yM < τ yL , where τ yj (j = L, M, H) denotes the preferred tax rate by type-j
young: the type-L young individual becomes the decisive voter. Second, when λ attains
a middle value such that LHS of (10) crosses RHS of (10) with j = M within the range(τ̃ LM , τ̃ M H ], the tax rates preferred by the young are ordered by τ yH < τ yL < τ yM orτ 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 (10) crosses RHS of (10) with j = H within the range[τ̃ M H , 1/2], the tax rates preferred by the young are ordered by τ yL < τ yM < τ yH : thedecisive 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:{β(1+n)λ(1−2τ )w̄+(1+β)η

1+n(1 − λ)w̄
2+n

}1−σ

16

·

1 − 2τ
= z̃(τ ; wL , wM , wH ), (11)((1 − τ )τ )σwhere:βRwL if τ < τ̂ (wL )(wL )1−σif τ̂ (wL ) ≤ τ ≤ τ̃ LM(1−τ )σ βRwM if τ̃ LM < τ < τ̂ (wM )L
M
Hz̃(τ ; w , w , w ) ≡(wM )1−σ
if τ̂ (wM ) ≤ τ ≤ τ̃ M H
(1−τ )σβRwH if τ̃ M H < τ < τ̂ (wH ) (wH )1−σ
if τ̂ (wH ) ≤ τ < 21 .
(1−τ )σ

The graph of the function z̃ is illustrated by the bold curve in Figure 6.
We substitute the reaction function of λdec (τ ) given by (9) into the left-hand side of(11) to obtain the condition that determines the equilibrium tax rate:y(τ ; w̄, n) = z̃(τ ; wL , wM , wH ).
Figure 7 illustrates the graphs of y(τ ; w̄, n) and z̃(τ ; wL , wM , wH ). Solving y(τ ; w̄, n) =z̃(τ ; wL , wM , wH ) for τ leads to the tax rate in a structure-induced equilibrium of thevoting game. The corresponding λ is obtained by substituting the equilibrium τ into thereaction function λdec in (9).
[Figure 7 about here.]
Proposition 2. Suppose that 1/σ < 1 holds. There exists a unique structure-induced
equilibrium of the voting game such that the decisive voter over τ is:(i) a type-L individual if y(τ̃ LM ; w̄) ≤ βRwM ;(ii) a type-M individual if βRwM < y(τ̃ LM ; w̄) and y(τ̃ M H ; w̄) ≤ βRwH ; and(iii) a type-H individual if βRwH < y(τ̃ M H ; w̄).
Proof. See Appendix 8.3.
A noteworthy feature of the result in Proposition 2 is that under certain conditions,the middle-income (or middle- and high-income) individuals prefer a higher tax rate thanthe low-income individuals. In particular, if the condition in statement (ii) in Proposition
2 holds, there exists an ends-against-the-middle equilibrium where the low- and highincome young individuals form a coalition in favor of a low tax rate and the middle-income
individual favoring a high tax rate becomes the decisive voter (see Figure 7). In contrast,
if the condition in statement (iii) holds, the low- and middle-income individuals form a
coalition and the high-income young individual becomes the decisive voter.
The key factors in the result in Proposition 2 are the borrowing constraints and theinterest-rate elasticity of consumption. To understand the roles of these two factors,
17

consider 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 negative eﬀect that
results in lower after-tax income and thus the utility loss of taxation in youth, and apositive eﬀect that produces higher social security beneﬁt and thus the utility gain in oldage.
The net impact of taxation depends on the interest-rate elasticity. When the elasticity
is high such that 1/σ > 1, the positive eﬀect overcomes the negative eﬀect. The lowincome individual then chooses the highest tax rate among the young and thus becomes
the decisive voter. When 1/σ = 1, both eﬀects perfectly cancel out. The three types ofyoung then choose the same tax rate. Finally, when the elasticity is low such that 1/σ < 1,
the negative eﬀect may dominate the positive eﬀect. In particular, if βRwM < y(τ̃ LM ; w̄)and y(τ̃ M H ; w̄) ≤ βRwH hold as in Proposition 2(ii), the negative eﬀect dominates the
positive eﬀect for low-income individuals. They choose a lower tax rate than the middleincome individuals, and this results in an equilibrium where the middle-income individualis the decisive voter.

5Eﬀects of Income Inequality on Policy

Given the characterization of the political equilibrium in Section 4, we wish to investigate
how the tax rate (τ ), the share of social security (λ), the level of social security beneﬁt(b) and public goods provision (g) change in response to changes in income inequality. In
particular, we consider a mean-preserving reduction of the decisive voters’ wage in order
to compare two groups of countries with similar per capita income levels but diﬀerent
levels of income inequality. We focus on the nontrivial equilibrium with λ > 0 to observe
the marginal eﬀect on the share of social security in government expenditure.
Proposition 3. Consider a political equilibrium with λ > 0.
(i) In an economy with 1/σ ≥ 1 where the decisive voter is a type-L young individual,a mean-preserving reduction of wL increases the tax rate ( τ ), the share of social
security ( λ) and the level of social security benefit ( b), and keeps the level of public
goods ( g) constant.
(ii) In an economy with 1/σ < 1 where the decisive voter is a type-j ( j = L, M, H)
young individual, a mean-preserving change in the decisive voter’s wage ( wj ) locally
produces inverse U-shaped relationships between wj and the tax rate ( τ ), between wj
and the share of social security ( λ), and between wj and the level of social security

18

benefit ( b) if the equilibrium tax rate is initially given by τ equil = τ̂ (wj ); and keeps
the level of public goods ( g) constant.
Proof. See Appendix 8.4.
Proposition 3 states that if the interest-rate elasticity is high such that 1/σ ≥ 1, there
is a monotone relationship between the decisive voter’s wage and his/her preferred tax
rate. The decisive voter then prefers a higher tax rate as he/she becomes poorer. However,when the elasticity is low such that 1/σ < 1, such a monotone relationship no longer holds.
Once the decisive voter’s wage falls below the critical level that changes his/her statusfrom unconstrained to constrained, he/she prefers a lower tax rate as he/she becomes
poorer. There is then an inverse U-shaped relationship between the decisive voter’s wage
and the preferred tax rate around the critical wage.
To understand the result intuitively, let us assume that the decisive voter is a typeL individual regardless of the value of σ, and consider cases where the decisive voter
is borrowing unconstrained and constrained, respectively. The following argument also
holds for the case of 1/σ < 1 where the decisive voter is a type-M or type-H individual.
Figure 8 illustrates the eﬀects of a mean-preserving change in the decisive voter’s wage
on the equilibrium tax rate. Panel (a) is for the case of 1/σ ≥ 1; Panel (b) is for the caseof 1/σ < 1.
[Figure 8 about here.]
To proceed with the analysis, we ﬁrst assume that the decisive voter is borrowing
unconstrained: sL > 0. Given a tax rate τ , a reduction of wL decreases the after-tax
income of the decisive voter in youth, but has no eﬀect on the consumption in youth
because cyj = (βR)−1/σ if sL > 0. The negative income eﬀect is then absorbed by saving
because of the assumption of a quasi-linear utility function. In other words, a reduction of
wL only has an eﬀect on consumption in old age. The decisive voter then wishes to oﬀset
the loss of saving by increasing the PAYG social security beneﬁts. Therefore, a meanpreserving reduction of the decisive voter’s wage gives him/her an incentive to choose ahigher tax rate when he/she is borrowing unconstrained.
Next, suppose that the decisive voter is borrowing constrained: sL = 0: the equilibriumtax rate satisﬁes y(τ ; w̄, n) = (wL )1−σ /(1 − τ )σ , where the right-hand side is the marginalutility loss of taxation when the decisive voter is borrowing constrained. A marginal
reduction of wL aﬀects the marginal utility loss of taxation such that ∂(wL )1−σ /(1 −τ )σ /∂wL = (1 − σ)(wL )−σ /(1 − τ )σ ≷ 0 ⇔ 1/σ ≷ 1. When 1/σ ≥ 1, a reduction of
wL leads to a decrease in the marginal utility loss, thereby giving the decisive voter an
incentive to increase the tax rate. However, when 1/σ < 1 holds, a reduction of wLleads to the opposite eﬀect. Therefore, if 1/σ < 1, a mean-preserving reduction of the19decisive voter’s wage gives him/her an incentive to choose a lower tax rate when he/she
is borrowing constrained.
The share of social security (λ) changes in the same direction as τ because the decisive
voter’s reaction function λdec (τ ), given by (9), is increasing in τ as long as τ ∈ (0, 1/2).
The social security beneﬁt (b) also changes in the same direction as τ because b is given byb = (1 + n)λτ (1 − τ ) from the government budget constraint. However, the level of public
goods (g), given by g = (1 + n)(1 − λ)τ (1 − τ )w̄/(2 + n), remains unchanged because anincrease (decrease) in the tax revenue, τ (1 − τ )w̄, is oﬀset by a decrease (increase) in theshare of public goods provision, (1 − λ).
The result established in Proposition 3 implies that an economy with a low interest-rateelasticity (i.e., the case of 1/σ < 1) exhibits an inverse U-shaped relationship between
the decisive voter’s wage and the level of social security, as illustrated in Panel (c) of
Figure 8. When the decisive voter (i.e., a type-L individual in the current assumption) isborrowing unconstrained, a lower wL results in a higher tax rate, a higher share of social
security and thus a higher level of social security beneﬁt. However, when the decisive
voter is borrowing constrained, a lower wL results in a lower tax rate, a lower share ofsocial security and thus a lower level of social security beneﬁt.
Empirical Implications
As mentioned in Section 1, standard political economy theory suggests a positive
correlation between inequality and social security: higher inequality results in greater
redistribution. However, cross-country data do not necessarily support this prediction.
For example, Gottschalk and Smeeding (1997) and Chen and Song (2009) ﬁnd a negative
correlation between wage inequality and social security: put diﬀerently, countries withsmaller earnings inequality have, on average, greater social security as a percentage ofGDP. For example, the United Kingdom and the United States feature high income
inequality, a low tax rate and a low level of social security beneﬁts. In contrast, many
continental European and Nordic countries feature low income inequality, a high tax rate
and a high level of social security beneﬁts.
In the current framework, the negative correlation arises only in the equilibrium where
the following two conditions hold: (i) the interest-rate elasticity of consumption is low
and (ii) the decisive voter is borrowing constrained. When one of the conditions fails tohold, the economy displays a positive correlation between income inequality and the tax
rate. This is inconsistent with the empirical evidence. Therefore, our analysis suggeststhat these factors are the key to explaining cross-country diﬀerences in income inequality,
tax rates and the share of social security.

20

6

A Generalized Utility Function

To this point, we have conducted the analysis by 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 nonlinear
utility function of old-age consumption to resolve these problems. We show that most of
the previous results still hold true under this alternative utility function. That is, undera certain condition, there exists the same ends-against-the-middle equilibrium when theinterest-rate elasticity is low and the decisive voter is borrowing constrained. In thisequilibrium, a mean-preserving spread of income inequality results in a lower equilibriumtax rate and a lower share of social security in government expenditure.
For the purpose of analysis, we assume the following utility function:(cyj)1−σ − 1(gt )1−σ − 1jUt = t
+η
+β
1−σ

1−σ

[

]
1−σ
(coj
−1
(gt+1 )1−σ − 1
t+1 )
+η
.
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). We maximize their lifetime utility under the budget
t+1 )constraints and obtain the following saving function:
sjt = max

{
0,

[
]}
(βR)1/σbt+1
j(1 − τt )w −.
(βR)1/σ + R(βR)1/σSaving now depends on the social security beneﬁt bt+1 that gives agents 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:
sj > 0 ⇔wj (βR)1/σ·> λτ .
w̄
1+nThis inequality condition states that a type-j individual is borrowing unconstrained if
his/her wage is high, the tax burden is low, and/or the share of social security in government expenditure is also low.
With the saving function and the government budget constraints, we give the con-21

sumption functions of a type-j individual in youth and old age as follows:
cyj
t =

coj
t+1 =




[
]
(1+n)λ(1−τ )τ w̄
j(1 − τ )w +R(βR)1/σ +RR1/σif ww̄j · (βR)
> λτ
1+n

 (1 − τ )wj[] (βR)1/σ R (1 − τ )wj + (1+n)λ(1−τ )τ w̄

≤ λτ
if ww̄ · (βR)
1+n

 (1 + n)λ(1 − τ )τ w̄

if ww̄ · (βR)
≤ λτ.
1+n

(βR)1/σ +R

R

j

1/σ

j

1/σ

j

1/σ

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 agents are borrowing unconstrained.
After some calculation, we can obtain indirect utility functions of type-j young andold individuals as follows:

[
]1−σ
[ 1+n
]1−σ
(1+n)λ(1−τ )τ w̄
y,j
ϕ
βη
j

V
≡
(1
−
τ
)w
+
+
(1
−
λ)(1
−
τ
)τ
w̄

s>0

1−σ
R
1−σ 2+n
1/σj
if ww̄ · (βR)
> λτ
yj
1+n
V =
1−σ
j
[
]1−σ

((1−τ )w )
1−σ
y,j
β
βη
1+n

+
[(1
+
n)λ(1
−
τ
)τ
w̄]
+
(1
−
λ)(1
−
τ
)τ
w̄
 Vs=0 ≡1−σ1−σ1−σ 2+n
1/σj
if ww̄ · (βR)
≤ λτ
1+n
{
]
[
1/σ
1−σ
j
y,j
1
if ww̄ · (βR)≡ 1−σRsj−1 + (1 + n)λ(1 − τ )τ w̄
> λτ
Vs>0
oj
1+n
V =
[ 1+n
]1−σ
y,j
βη
wj (βR)1/σ
Vs=0 ≡ 1−σ 2+n (1 − λ)(1 − τ )τ w̄
if w̄ · 1+n ≤ λτ,
where variables unrelated to political decisions are dropped from V yj and V oj .
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.
We next consider the policy preferences of the young. The preferred tax rate of a
type-j young individual satisﬁes the following ﬁrst-order condition with respect to τ :{y

LHS = RHS

yj

≡

1/σ

yj
RHSs>0if τ < τ ∗ (wj , λ) ≡ ww̄ · (βR)
· λ1
1+n
yj
RHSs=0if τ ≥ τ ∗ (wj , λ)
,
j

(12)

where:
]−σ
1+n
1+n
·
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄,LHS ≡ βη
2+n
2+n
]−σ [
]
[
(1 + n)λ(1 − 2τ )w̄
(1 + n)λ(1 − τ )τ w̄
yj
j
j
· w −
RHSs>0 ≡ ϕ (1 − τ )w +
, and
R
R
( )1−σyj
RHSs=0≡ (1 − τ )−σ wj− β [(1 + n)λ(1 − τ )τ w̄]−σ (1 + n)λ(1 − 2τ )w̄.
[

y

22

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 natureof public goods. RHS yj represents the marginal loss of taxation plus the marginal beneﬁt
of social security in terms of the utility of consumption. However, the sum of theselosses and beneﬁts diﬀers across agents. In particular, the following properties hold (seeAppendix 8.5 for the proof):{

RHS yL < RHS yM < RHS yH if 1/σ ≥ 1yL
yM
yH
RHSs=0> RHSs=0> RHSs=0
if 1/σ < 1.

(13)

yj
Similar to the previous model, the order of RHSs=0(j = L, M, H) critically depends on4
the degree of interest-rate elasticity 1/σ.
Panel (a) of Figure 9 illustrates the graph of (12) when 1/σ ≥ 1 holds. The crossing
point of LHS y and RHS yj determines the tax rate preferred by a type-j young agent. The
ﬁgure shows that a lower-income young individual prefers a higher tax rate. Under thedemographic structure assumption given in Assumption 1, a type-L young agent becomes
the decisive voter over τ . That is, the-ends-against-the-middle equilibrium never arises
when the interest-rate elasticity is high such that 1/σ ≥ 1.

[Figure 9 about here.]
Panel (b) of Figure 9 illustrates the graph of (12) when 1/σ < 1 holds. A noteworthy
feature is that lower-income young prefer a lower tax rate when agents are borrowing
constrained. In particular, there may arise an equilibrium where the low- and the highincome young agents form a coalition against the middle, as illustrated in Panel (b) of
Figure 9. Therefore, the low interest-rate elasticity and the borrowing constraint remain
the key to the existence of the ends-against-the-middle equilibrium.
The political determination of the share of social security λ is slightly diﬀerent fromthat in the previous quasi-linear utility case. The preferred share of a type-j young isgiven by:

λyj =

4









0

if

1/(2+n)−(βηR/(2+n))1/σ (wj /τ w̄(1+n))1/(2+n)+(βηR/(2+n))1/σ /R
1/(2+n)1/(2+n)+(η/(2+n))1/σ

if τ̃ (wj ) < τ < τ̂ (wj )ifyjIf 1/σ < 1, the order of RHSs>0(j = L, M, H) is ambiguous.

23τ ≤ τ̃ (wj )τ̂ (wj ) ≤ τ(14)where()1/σwj (2 + n);w̄(1 + n){()1/σ }
j
1
η
w
(2
+
n)
(βR)1/σ+.
τ̂ (wj ) ≡w̄(1 + n)2+n2+nτ̃ (w ) ≡jβηR
(2 + n)·The ﬁrst two lines of the right-hand side in (14) represent the choice of λ when a type-j
young is borrowing unconstrained; the third line represents the choice of λ when he/she
is borrowing constrained. The derivation of (14) is given in Appendix 8.5.
When the tax burden is low, such that τ ≤ τ̃ (wj ), a type-j young individual can savemuch for his/her 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 suchthat τ ≥ τ̂ (wj ). Borrowing-constrained agents 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.
Panel (c) of Figure 9 illustrates the reaction function of λ for each type of an agent.
The ﬁgure shows that λyH (τ ) ≤ λyM (τ ) ≤ λyL (τ ) < λo holds for any τ . Thus, under
the demographic structure in Assumption 1, a type-L young 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. Thus, thedecisive voter’s choice of λ in the current framework is given by:
λdec =

1
1 + (2 + n)(σ−1)/σ (η)1/σ,

which is independent of τ . We substitute this into (12) for the case of τ ≥ τ ∗ (wj , λ) andobtain the following condition that determines the equilibrium tax rate when the decisive24

voter is borrowing constrained:
[

]−σ
1+n
1+n
βη
(1 − λ)(1 − τ )τ w̄
(1 − λ)(1 − 2τ )w̄
2+n2+n( )1−σ− β [(1 + n)λ(1 − τ )τ w̄]−σ (1 + n)λ(1 − 2τ )w̄,= (1 − τ )−σ wj
where λ is now constant. Given τ , the left-hand side is independent of wj whereas theright-hand side is decreasing (increasing) in wj if 1/σ < (>)1. Thus, a mean-preserving
reduction of the decisive voter’s wage decreases (increases) the equilibrium tax rate if theinterest-rate elasticity is low (high) such that 1/σ < (>)1. This result is qualitatively
equivalent to that in the quasi-linear utility function model.
The corresponding levels of social security and public goods are given by b = (1 +n)λτ (1 − τ )w̄ and g = ((1 + n)/(2 + n))(1 − λ)τ (1 − τ )w̄. Because λ is unchanged whileτ is decreased (increased) by the mean-preserving reduction of wj if 1/σ < (>)1, b and g
are decreased (increased) if the interest-rate elasticity is low (high) such that 1/σ < (>)1.
The main departure from the result in the previous model is that public goods provision
is now nonneutral with respect to the mean-preserving expansion of income inequality.

7

Conclusion

Why is wage inequality negatively correlated with the level of social security? How does
wage inequality aﬀect the allocation of tax revenue between social security and forward
intergenerational public goods provision? This paper develops a political economy modelthat responds to both these important questions.
Two features are crucial to our analysis and results: the interest-rate elasticity of
consumption and the borrowing constraint. These features derive an ends-against-themiddle equilibrium where low- and high-income individuals form a coalition in favor ofa 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 in government expenditure when the decisive voter is borrowing constrained andthe interest-rate elasticity is low.
The negative correlation between inequality and social security we obtain appears to ﬁt
the available empirical evidence. This correlation arises only in the equilibrium where theabovementioned conditions are eﬀective for the decisive voter. When one of the conditions
fails to hold, the economy displays a positive, rather than negative, correlation between
inequality and social security. Thus, our analysis suggests that the two conditions play a
key role in explaining cross-country diﬀerences in inequality and social security.
To obtain these results, we simplify the analysis by adopting a quasi-linear utility
25

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,
and as shown in our analysis, the eﬀect of the interest-rate elasticity of consumption on
the political determination of the tax rate and the allocation of tax revenue remains. In
addition, Section 6 demonstrates that the main results are qualitatively unchanged undera generalized utility function except for the level of public goods. Thus, our analysis isalmost robust to the assumption of a quasi-linear utility function.

26

8

Appendix

8.1

Proof of Proposition 1

As shown in the text, when 1/σ ≥ 1, the decisive voter is a type-L individual and his/her
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 →∞ 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 ).


8.2

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 (10), denoted by RHS j , is given by:{
RHS j =

RHS L = (wL )1−σ /(1 − τ )σ for j = L
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 .
By the same token, 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 isdenoted by τ̃ M H .

8.3

Proof of Proposition 2

As shown in the text, when 1/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 the27following properties:∂y(τ ; w̄, n)/∂τ < 0,lim y(τ ; w̄, n) = ∞,τ →∞y(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 ). The condition for the determination of the decisive voter over τ canbe found from Figure 7.


8.4

Proof of Proposition 3(i) For the case of 1/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
decreased by a reduction of wL . Therefore, a mean-preserving reduction of wL results in
an increase in the equilibrium tax rate that satisﬁes y(τ ; w̄, n) = z(τ ; wL ). Given λdec (τ )is increasing in τ for τ ∈ (0, 1/2), a mean-preserving decrease in wL results in an increasein λ.
From the government budget constraints for social security (b) and public goods provision (g) in Subsection 2.2, b and g are given by:b = (1 + n)λτ (1 − τ )w̄,1+ng=
(1 − λ)τ (1 − τ )w̄ =2+n((1 + β)η
β(2 + n)

(15)
)1/σ
,

(16)

where the second equality in the equation (16) is obtained by substituting the decisive
voter’s reaction function λdec (τ ) in (9) into the equation. (15) and (16) imply that b isincreased by an increase in τ and λ, thereby implying a negative eﬀect of wL on b; and g
is independent of τ and λ, thereby implying no eﬀect of wL .
(ii) For the case of 1/σ < 1, the decisive voter in this case is a type-j (j = L, M or H)
individual depending on parameter values as shown in Subsection 4.2. To simplify thepresentation, suppose that a type-L individual is a decisive voter. Note that the following
argument applies for the case where a type-j (j = M or H) is a decisive voter.
Assume that the equilibrium tax rate is given by τ equil = τ̂ (wL ): a type-L individual28

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:
w = ŵ ≡L

L−(1 + n)w̄ +√{(1 + n)w̄}2 + 8R(1 + n)w̄/(βR)1/σ.
2R

Therefore, the equilibrium tax rate is given by τ equil = τ̂ (wL ) when a type-L individualwith wL = ŵL is a decisive voter.
Let us 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 1/σ < 1. In particular, around wL = ŵL , there exists a positive real number ε such that the equilibriumtax rate satisﬁes the following condition:{z̃(τ ; wL , wM , wH ) =βRwL

for wL ∈ (ŵL − ε, ŵL ],(wL )1−σ(1−τ )σfor wL ∈ [ŵL , ŵL + ε).

We 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 therange (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 U-shaped relationship between thedecisive voters’ 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 U-shaped relationship between thedecisive voters’ wage and the equilibrium share of social security around wL = ŵL . Given
the result for τ and λ, the result for b and g arises from (15) and (16). 

8.5

Supplementary Explanation for Section 6

8.5.1

Derivation of (13)In order to establish that (13) holds, we focus on the relationship between RHS yL
and RHS yM . The result immediately applies to the relationship between RHS yM and
RHS yH .

29

yL
yM
Let us ﬁrst compare RHSs>0and RHSs>0. We obtain:yL
yM
RHSs>0≶ RHSs>0]−σ [][(1 + n)λ(1 − 2τ )w̄
(1 + n)λ(1 − τ )τ w̄
L
L
· w −
⇔ (1 − τ )w +
R
R
[
]−σ [
]
(1 + n)λ(1 − τ )τ w̄
(1 + n)λ(1 − 2τ )w̄
MM≶ (1 − τ )w +
· w −
R
R
]
[
(1+n)λ(1−τ)τ w̄ σ(1 − τ )wL +wL − (1 + n)λ(1 − 2τ )w̄/R
R
⇔ M
.
(17)
≶
(1+n)λ(1−τ )τ w̄
Mw − (1 + n)λ(1 − 2τ )w̄/R(1 − τ )w +R

The left-hand side of (17) is smaller than wL /wM whereas the the term in the large
parentheses on the right-hand side is larger than wL /wM ; that is:
wL − (1 + n)λ(1 − 2τ )w̄/R
wL<;
wM − (1 + n)λ(1 − 2τ )w̄/RwM
)τ w̄
(1 − τ )wL + (1+n)λ(1−τwL(1 − τ )wL
R
.
=
<
wM
(1 − τ )wM(1 − τ )wM + (1+n)λ(1−τ )τ w̄
R

Therefore, we have:yL
yM
RHSs>0< RHSs>0
if

wL
≤
wM

(

wL
wM

(

)σ
⇔

wL
wM

)1−σ
≤ 1.

(18)

yM
yL
holds if σ ≤ 1, i.e., if 1/σ ≥ 1.
< RHSs>0(18) indicates that RHSs>0yL
yM
We next compare RHSs=0 and RHSs=0. Direct comparison leads to:( )1−σ ( M )1−σyL
yM
RHSs=0≶ RHSs=0⇔ wL≶ w.
Therefore, we can conclude:yL
yM
RHSs=0≤ RHSs=0
if σ ≤ 1 ⇔ 1/σ ≥ 1,
yL
yM
RHSs=0> RHSs=0
if σ > 1 ⇔ 1/σ < 1.

8.5.2

Derivation of (14)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:
y,j
∂Vs>0
1/(2 + n) − (βηR/(2 + n))1/σ (wj /τ w̄(1 + n))= 0 ⇔ λ = λyj≡(< 1).
s>0∂λ1/(2 + n) + (βηR/(2 + n))1/σ /R30

The preferred share λ is increasing in τ and is positive if and only if τ > τ̃ (wj ).
Next, suppose that the type-j young agent is borrowing constrained. The ﬁrst-ordery,jcondition for the maximization of Vs=0
with respect to λ leads to:
y,j
∂Vs>0
1/(2 + n)= 0 ⇔ λ = λyj(< 1).
s=0 ≡∂λ1/(2 + n) + (η/(2 + n))1/σ

We should note that λyj
s=0 is constant and independent of τ . The equality holds betweenyjyj
λs>0 and λs=0 at τ = τ̂ (wj ). Therefore, the preferred share λ by a type-j young agent isgiven as (14).

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33

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.

34

Figure 2: The relation between savings and the tax rate for each type of individual.

35

Figure 3: Panel (a) illustrates the determination of τ that maximizes the utility of theborrowing-unconstrained young. Panel (b) illustrates the determination of τ that maximizes the utility of the borrowing-constrained young. Panel (c) illustrates the share ofsocial security preferred by the young.

36Figure 4: The tax rates preferred by the three types of young individuals in the case of1/σ ≥ 1.

37

Figure 5: The ﬁgure illustrates the determination of the tax rate in the case of 1/σ ≥ 1.
Panel (a) is for the case of sL > 0 and λ > 0. Panel (b) is for the case of sL = 0 and
λ > 0.

38

Figure 6: The tax rate preferred by a type-j young individual in the case of 1/σ < 1. The
bold curve illustrates the graph of ze in (11).

39

Figure 7: The determination of the tax rate in the case of 1/σ < 1. The ﬁgure illustrates
the case where the decisive voter is a type-M young individual.

40Figure 8: The eﬀect of a mean-preserving reduction of the decisive voter’s wage on the tax
rate. Panel (a) illustrates the case of 1/σ ≥ 1. Panel (b) illustrates the case of 1/σ < 1.
Panel (c) illustrates the relation between the decisive voter’s wage and the equilibriumtax rate in the case of 1/σ < 1.

41

³

Figure 9: Panels (a) and (b) illustrate the preferred tax rates in cases where 1/σ ≥ 1 and1/σ < 1, respectively. Panel (c) illustrates the preferred shares of social security.

42