GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.213

Old-age Social Security vs. Forward
Intergenerational Public Goods Provision

Ryo Arawatari and Tetsuo Ono

August 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanOld-age Social Security vs. Forward
Intergenerational Public Goods Provision
Ryo Arawatariy

Tetsuo Onoz

Nagoya University

Osaka University

August 19, 2011

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; Old-age social security; Forward intergenerational public goods; Ends-against-the-middle equilibrium; Wage inequality
JEL Classi…cation: H41; H55; D72

Arawatari acknowledges the …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 forYoung Scientists (B) (No. 20730208).
y
Graduate 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.jp1

1

Introduction

The unfunded old-age social security program bene…ts the retired old at the expense of
the current working young and thus creates an intergenerational con‡ict over government
expenditure. The voting models of Browning (1975) and Boadway and Wildasin (1989)
have precisely this ‡avor. In addition to this intergenerational con‡ict, the unfundedsocial security program also creates intragenerational con‡ict when income inequality
within a generation is taken into account. Some previous studies introduce this inequality
to emphasize the intragenerational redistribution component built in many social security
systems (for example, Casamatta, Cremer, and Pestieau, 2000; Tabellini, 2000; CondeRuiz and Galasso, 2005; Galasso and Profeta, 2007; Koethenbuerger, Poutvaara andProfeta, 2008). They show that the redistribution from the rich to the poor via social
security plays a crucial role in the voting game because social security becomes appealing
to the low-income young.
The studies on the political economy of social security have been expanded in various
directions for the past decade (see the surveys by Galasso and Profeta, 2002; de Walque,2005; Borck, 2007b). In particular, the following two directions are of concern in thepresent study. The …rst is the introduction of borrowing constraints into the analysis andthe second is the voting over multidimensional choice spaces.
As for the …rst direction, some previous studies exclude the presence of the constraints
by assuming no consumption in youth (for example, Conde-Ruiz and Galasso, 2003, 2005;
Bethencourt and Galasso, 2008) or by implicitly assuming that saving does not hit theborrowing constraint (for example, Galasso and Profeta, 2007). Given no restriction on
saving, lower-income agents prefer a larger size of social security, which coincides with thestandard result in the literature of the political economy of redistribution (Romer, 1975;
Roberts, 1977; Meltzer and Richard, 1981). The result implies that higher inequality
associated with a reduction of the decisive voter’s income results in a larger volume ofsocial security.
This theoretical prediction, however, is not necessarily supported by the empiricalevidence: OECD cross-country data show that the size of social security is negatively
correlated with inequality (Gottschalk and Smeeding, 1997; Chen and Song, 2009). Somestudies suggest that the presence of borrowing constraints resolves this issue (Casamatta,
Cremer, and Pestieau, 2000; Bellettini and Berti Ceroni, 2007; Conde-Ruiz and Profeta,
2007; Cremer et al., 2007). In particular, borrowing-constrained agents, who cannot reallocate resources freely across periods, attach a large weight to the utility loss of income
reduction in youth, and thus prefer a lower tax burden for social security …nance in response to a reduction of their income, provided their interest-rate elasticity of consumption
is low (Arawatari and Ono, 2011).
2

The second direction, with which we are concerned here, is a two-dimensional voting
including intergenerational as well as intragenerational con‡icts. Under a balanced government budget constraint, the unfunded old-age social security bene…ts are provided at theexpense of other government programs such as intragenerational redistribution (CondeRuiz and Galasso, 2005), public goods provision that bene…ts only the young (Levy,
2005) and medical services for the elderly (Bethencourt and Galasso, 2008). Bernasconi
and Profeta (2011) focus on two competing programs, public education and redistribution, and examine an intergenerational consequence of these two programs in the political
economy. Society must choose via voting both the size of government (i.e., a tax) andhow to allocate its resources between two competing programs. 1
In this paper, we particularly focus on public goods, such as pure science and environmental maintenance, called forward intergenerational public goods in this study, whichtake a long time to mature and thus bene…t only the young. Levy (2005) analyzed thisissue in a two-party system. She focused on lump-sum redistribution and public education that bene…ts only the young, and showed that an increase in income inequality
may decrease the size of redistribution when the majority is the young, which …ts theempirical evidence. However, the predictions about the share of redistribution and forward intergenerational public goods (i.e., public education) in government expenditure
are abstracted in her analysis because no provision of forward intergenerational public
goods arises in the young-majority equilibrium because of the speci…cation of the model.
The aim of this study is to combine the abovementioned two strands of literature thathave received attention in recent years. In particular, we analyze how income inequalitya¤ects the size of government as well as the shares of old-age social security and forward
intergenerational public goods in government expenditure, which has not been fully investigated in the previous studies. Therefore, the current paper sheds light on the roleof borrowing constraints in the two-dimensional voting over old-age social security andforward intergenerational public goods provision.
For this purpose, we introduce an overlapping-generations model with heterogeneous
agents. In this economy, young workers are of three types of income: low, middle, andhigh. Because they are not permitted to borrow in youth, as a result of imperfect …nancial
markets, lower-income individuals are more likely to be borrowing constrained. Youngworkers then pay a …xed proportion of their labor income to the government, and the tax
1

Besides the abovementioned studies, there is an emerging literature on two-dimensional voting, particularly focusing on voting over payroll tax and the pension system. This literature is classi…ed into
two categories according to the concept of equilibrium. The …rst includes the studies based on sequential
voting (for example, Casamatta, Cremer, and Pestieau, 2000, 2005; Cremer et al., 2007); and the second
includes the studies based on Shepsle’s (1979) notion of structure-induced Nash equilibrium voting (forexample, Conde-Ruiz and Galasso, 2003; Conde-Ruiz and Profeta, 2007). As explained further below,
the current paper falls into the second category.

3

revenue is divided into social security payments from which the old can bene…t 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 theyoung and the old. Voters cast ballots over the labor income tax, which …nances 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 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 have no tax burden, receive the same level of social security
bene…t, and cannot capture the bene…t of forward intergenerational public goods. Instead,they prefer the tax rate that attains the top of the La¤er 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 di¤ers across the types of income. In particular, the key
factors for 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 e¤ects: a negative e¤ect that results in lower after-tax income,and thus the utility loss of taxation in youth, and a positive e¤ect that produces greatersocial security bene…t, 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.
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 di¢ cult.
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 favora high tax rate.
Given the characterization of political equilibrium, we investigate how the shares ofsocial security and forward intergenerational public goods in government expenditure are4altered 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 in government expenditure when
the interest-rate elasticity is low. The positive correlation arises when the decisive voter’s
wage is high, and thus he/she is borrowing unconstrained, while the negative correlationarises 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 the borrowing constraint are the key factors needed todemonstrate the negative correlation between wage inequality and the share (or size) ofsocial 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 a¤ects the tax rate and the allocation of tax revenue between social security andforward intergenerational public goods. Section 6 brie‡y undertakes the analysis under ageneralized framework. 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 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 highP(j = L; M; H), whose proportions are respectively L ; M and H , where j j = 1 andjsatis…es the following assumption.
Assumption 1.

j> n=f2(1 + n)g; j = L; M; H:

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, consider …rst the preferences of the old. As we explain below, the oldchoose a higher tax rate than any young individual because they bear no tax burden but
bene…t from taxation via social security; the tax burden when young is viewed as a sunkcost for the old. In addition, the old have the same preferences over the policy because
they bene…t from the same social security.
5

Next, consider the preferences of the young. 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 andthe old participate in voting, the sum of the type-k young and the old is given by Nt k +Nt 1 , which is greater than half of the population in period t; (Nt + Nt 1 )=2; under theassumption of k > n=2(1+n). This implies that the decisive voter becomes the old or the
type-k young. However, the old cannot become the decisive voter because the population
size of the old is smaller than that of the young under the assumption of n > 0. Therefore,
the type-k young individual becomes the decisive voter. Figure 1 provides an example ofpreferences 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 speci…edby:(gt )11(gt+1 )11
(cyj )1
1
+
+
coj
+
;
Utj = t
t+1
1
1
1
oj
where cyj
t is consumption in youth, ct+1 is consumption in old age, gt is per capita public
goods in period t; (> 0) is the parameter representing the preference for public goods,
2 (0; 1] is the discount factor, and (> 0) is the inverse of the elasticity of young-age
consumption with respect to the interest rate. A lower 1= implies a lower interest-rateelasticity of young-age consumption.
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 brie‡y investigate the case where the1utility of old-age consumption is given by f(coj1g=(1) and show that the
t+1 )main result is not qualitatively unchanged under this alternative utility function.
Each individual works in his/her youth and retires in old age. The wage income isrelated 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
L L
by w
w + M wM + H wH :Type-j’s individual budget constraints in youth and old age are given respectively by:jcyj
t + st

coj
t+1

(1

j
t )w ;

Rsjt + bt+1 ;6

where sjt is saving, t is the income tax rate in period t; R is the gross interest rate,
and bt+1 is the per capita social security bene…t 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 bene…ts(Diamond and Hausman, 1984; Conde-Ruiz and Profeta, 2007).
We assume that the economy is dynamically e¢ cient.
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 bene…ts regardless of contributions in their
youth. The …rst assumption abstracts away the general equilibrium e¤ect via the interest
rate investigated by, for example, Cooley and Soares (1999) and Boldrin and Rustichini(2000). However, this simpli…cation enables us to demonstrate more simply 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 role of the borrowing constraint in the political
determination of social security and public goods provision.
The representative type-j young individual maximizes his/her utility subject to thebudget constraints and the restriction of nonnegative saving. When sjt > 0; the …rstorder condition for an interior solution is (cyj
= R, and thus de…nes the optimal
t )jsaving decision of a type-j individual given by sjt = (1
( R) 1= . By taking thet )wborrowing constraint into account, the saving function of a type-j individual is:
sjt = max 0; (1

t )w

j

( R) 1=:(1)

Eq. (1) indicates that the saving decision depends on the current tax rate t ; but is
independent of the future tax rate t+1 and the proportion of tax revenues devoted tosocial security in old age, denoted by t+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 and the proportion .
7

The saving function (1) implies that there is a critical rate of tax such that:sjt > 0 ,jt < ^(w )1

1:( R)1= wj(2)

A type-j individual chooses positive saving 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 higher
ability 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 convention in the literature, we present the e¢ ciency loss of taxation 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
L L
revenue is given by (1w + M wM + H wH ) = (1t) t(t ) t w, where the term
(1t ) is the distortionary factor. The assumption of distortionary taxation is solely toensure 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 t 2 [0; 1] of tax revenue is devoted to old-age social security bene…ts
and the remainder (1t ) 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 t Nt (1 t ) t w = Nt 1 bt . The per capita social
security bene…t in period t, bt , is given by:
bt = (1 + n) t (1t ) t w.

The formation of public goods requires investment one period ahead of time. This
assumption re‡ects 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 (1t )Nt (1t ) t w = (Nt + Nt+1 )gt+1 . The percapita pubic goods provision in period t + 1, gt+1 , is given by:
gt+1 =

1
(1
2+n

t )(1

8

t ) t w:

2.3

The Economic Equilibrium

We de…ne the economic equilibrium as follows.
De…nition 1. For a given sequence of tax rates and social security shares in government expenditure, f t ; t g1
t=0 , an economic equilibrium is a sequence of allocations,yj oj j t=0; ;1
fct ; ct ; st gj=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 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:
cyj
t =
coj
t+1 =

(

(

( R) 1=
j
(1
t )w

if
if

j
Rf(1
t )w
(1 + n) t+1 (1( R) 1= g + (1 + n) t+1 (1t+1 ) t+1 w

t+1 ) t+1 w

jt < ^(w )^(wj )t

if t < ^(wj )if t ^(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 abovementioned consumption functions to
obtain an indirect utility function of a type-j young individual:
Vtyj =(yj
Vt;s>0
if t < ^(wj )yjVt;s=0
if t ^(wj );(3)

where:y;j
Vt;s>0y;jVt;s=0( R) 1=

1

1

1

((1

j 1
t )w )

1

h n
R (1
(
(gt )1
+
1

+

1

t )w

1

j

+

( R) 1=1(1
2+n+ (1 + n) t+1 (1t+1 ) t+1 w(
1(1(gt )1
1
2+n
+
+
1
9

o

+ (1 + n) t+1 (1
t )(1
1

t) tw

t )(1

t) tw

1

1

1

t+1 ) t+1 w

1

)

;

1

)

:

i

y;j
Vt;s>0denotes the indirect utility of a type-j young individual when he/she saves somey;jdenotes the indirect utility when he/she is faced
portion of his/her income, and Vt;s=0
with a borrowing constraint and saves nothing. For each indirect utility function, the …rst
term on the right-hand side shows the utility of consumption in youth, the second termshows the utility of consumption in old age and the third term shows the utility of public
goods in old age. The public goods provision in period t, gt , is predetermined in periodt 1.
For a type-j old individual in period t; the indirect utility function is:Vto;j(1 + n) t (1t) tw +(gt )1
1

1

;

(4)

where the …rst-term on the right-hand side shows the social security bene…ts. Old individuals have the 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,, and the share of social security, .

3

The Political Institution and Voting

The tax rate and the proportion 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 ; the income tax, and ; 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. Thus, we assume individuals vote sincerely.
This majoritarian voting game has two signi…cant characteristics. First, the issue
space is bidimensional ( and ), and thus the Nash equilibrium 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:
and t = t+1 = for all t as in Casamatta, Cremer, and Pestieau (2000)
t = t+1 =and Conde-Ruiz and Profeta (2007). We can view the full commitment solution as thesolution including intergenerational interaction because the full commitment solution can
10be 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 su¢ cient conditions
for the existence of a structure-induced equilibrium. In particular, if preferences are single
peaked along every dimension of the issue space, a su¢ cient condition for ( ; ) to bean equilibrium of the voting game with full commitment is that represents the outcomeof majority voting over the jurisdiction when the other dimension is …xed at its level, and vice versa.
Preferences of the old are immediately shown to be single peaked along every dimension because they are given by Vto;j (1 + n) t (1 t ) t w and satisfy @ 2 V o;j =@ 2 < 0 and@ 2 V o;j =@ 2 = 0. The preferences of the young are also single peaked along every dimension. However, the proof of this argument is not straightforward because the preferencesof the young are kinked at the critical rate ^(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 to maximize V o;j in (4) given , and to maximize V o;j in (4) given .
Their preferred rate of tax and the share of social security are respectively given by:
oj

=

1
and
2

oj

= 1 for all j:

Maximization is realized when the tax rate is set to attain the top of the La¤er curve,(1) . The maximized tax revenue is used exclusively for social security because theold cannot bene…t from the investment in public goods that take a one-period lag information.

3.2

Preferences of the Young Over the Tax Rate

Consider …rst the preferences of the young over . A type-j young individual choosesy;jto maximize Vs>0
when he/she is borrowing unconstrained; he/she chooses to maximizey;jVs=0 when he/she is borrowing constrained. Thus, the tax rate chosen by the type-j

11young individual satis…es the following …rst-order condition:(1 + n) (1
(

=

(1

1
(1
2+n

2 )w +Rwj) (wj )1

if
if

)(1

) w

1
(1
2+n

)(1

2 )w

(5)

< ^(wj )^(wj )The …rst term on the left-hand side is the marginal bene…t of social security, the second
term is the marginal bene…t of public goods provision, and the right-hand side shows themarginal cost of taxation. The marginal cost is given by Rwj when the type-j young
individual is borrowing unconstrained; and it is given by (1) (wj )1 when he/she
is borrowing constrained. The type-j young individual chooses the tax rate to equate
marginal bene…ts and costs of taxation from the viewpoint of utility maximization.
Preferences of the young over are a¤ected by the wage income. In order to understandthe e¤ect, we …rst assume that a type-j young individual is borrowing unconstrained:sj > 0. Given a tax rate , a reduction of wj decreases the after-tax income in youth,
but has no e¤ect on the consumption in youth because cyj = ( R) 1= if sj > 0. The
negative income e¤ect is then absorbed by saving because of the assumption of a quasilinear utility function. In other words, a reduction of wj only has an e¤ect on consumption
in old age. A young individual then wishes to o¤set the loss of saving by increasing thesocial security bene…ts. Therefore, a reduction of the wage gives a young individual an
incentive to choose a higher tax rate when he/she is borrowing unconstrained.
Next, suppose that a type-j young individual is borrowing constrained: sj = 0. The
equilibrium tax rate satis…es y( ; w; n) = (wj )1 =(1) , where the right-hand sideis the marginal utility loss of taxation when the type-j young individual is borrowing
constrained (see Eq. (5)). A marginal reduction of wj a¤ects the marginal utility loss oftaxation such that @(wj )1 =(1) =@wj = (1)(wj ) =(1) ? 0 , 1= ? 1.
j
When 1= > 1; a reduction of w leads to a decrease in the marginal utility loss, thereby,
giving the decisive voter an incentive to increase the tax rate. However, when 1= < 1holds, a reduction of wj leads to the opposite e¤ect. Therefore, if 1= < 1, a reduction of
the wage gives a young individual an incentive to choose a lower tax rate when he/she isborrowing constrained.
In order to understand intuitively the result for the case of a borrowing-constrained
young individual, we consider the following two opposing e¤ects of a reduction in wage
on the preferred tax rate. First, given a tax rate, a decrease in the wage reduces thetax burden and thus raises a bene…t-to-burden ratio. This gives a young individual an
incentive to choose a higher tax rate: this is a positive e¤ect on the preferred tax rate.
Second, a decrease in the wage reduces the disposable income and thus the consumption
12

level when young. This gives a young individual an incentive to choose a lower tax ratefrom the viewpoint of maintaining the utility of consumption: this is a negative e¤ect on
the preferred tax rate.
When a young individual is borrowing constrained, the positive e¤ect does not necessarily overcome the negative one. The borrowing-constrained individual wants to consume
more when young, but his/her demand is restricted by the borrowing constraint. Under
this situation, the borrowing-constrained individual attaches a large weight to the utility
loss of a reduction in his/her wage. This might lead to a situation where the negativee¤ect overcomes the positive e¤ect, which results in a lower preferred tax rate in response
to a wage reduction.
Which e¤ect outweighs the other depends on the degree of the interest-rate elasticity.
A lower elasticity results in a smaller change of consumption in response to a change
in the interest rate. In other words, a lower elasticity implies a stronger incentive for ayoung individual to smooth consumption across periods. Because of this incentive, theborrowing-constrained individual attaches a larger weight to the negative e¤ect on youngage consumption as the interest-rate elasticity becomes lower. Therefore, the borrowingconstrained individual prefers a lower tax rate as his/her wage becomes lower when theinterest-rate elasticity is low enough so that 1= < 1.

3.3

Preferences of the Young Over the Share of Social Security

Next, consider the preferences of the young over . The …rst derivative of V y;j with respecty;jy;jto is independent of the status of saving: @Vs>0=@ = @Vs=0=@ . Direct calculation
leads to:
y;j
y;j
@Vs>0
@Vs=0
=
@
@

= (1 + n)(1

) w

1
(1
2+n

)(1

) w

1
(1
2+n

) w;

where the …rst term on the right-hand side shows the marginal increase in the bene…t 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, , is chosen to balance the marginal bene…tand 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
13

consumption is speci…ed by a linear utility function.
Another noteworthy feature is that the share could be zero. In order to understandthe mechanism behind the result of “a corner solution”, we …rst assume an interior solutiony;jof and set @Vs>0=@ = 0: By rearranging the terms, we obtain:(1 + n)w =

w
2+n

1

(1

1
)(1

;

)

where the left-hand side shows the marginal bene…t of an increase in social security bene…ts, and the right-hand side shows the marginal utility loss of a decrease in public goods
provision. The left-hand side of the above equation is independent of , while the righthand side depends on .
The marginal loss of taxation, represented by the right-hand side, increases as the tax
rate decreases. There then exists a critical rate of the tax, denoted by 2 (0; 1=2), suchthat for 2 [0; ], the marginal loss is greater than the marginal bene…t regardless of the
choice of 2 [0; 1]. Choosing = 0 is optimal for a type-j young individual from theviewpoint of utility maximization when is below . The corner solution is not peculiar
to the model with a quasi-linear utility function. The qualitatively similar result also
holds under a generalized utility function, which is demonstrated in Section 6.
Based on the above argument, the preferred share of the young becomes:=

where:

8
<

: 1

0
2+n
(1 ) w

1

r

1

if

2 [0; ]if21=(2+n)(1+n)

4(2+n)w; 21 ;(6)

1=(2+n)(1+n):
2Figure 2 illustrates the graph of (6). The range of is limited to (0; 1=2) because thepreferred tax rate by the old is equal to 1=2 and that by the young is less than 1=2. Whenthe tax rate is below the critical rate , the tax revenue is too small, so that the marginal
bene…t of raising the share of social security is lower than the marginal cost of reducing
the share of public goods provision for any 2 [0; 1]. Therefore, the young choose no
expenditure to social security, = 0; if.
[Figure 2 about here.]
When the tax rate is above the critical rate , the tax revenue is su¢ cient, so that
there is a share 2 (0; 1) that balances the marginal bene…ts and costs. The optimal
share for the young increases as the tax rate increases. The optimal share attains its
highest value at = 1=2, where the tax revenue is maximized.
14

4

The Political Equilibrium

The previous section analyzed the voting behavior of each type of individual along thetwo dimensions of the issue space, and . Given that preferences are single peaked foreach issue, we now apply Shepsle’s (1979) result and characterize the structure-induced
equilibrium of the game. To proceed with the analysis, we impose the following assumption.
Assumption 3.
1=

4(2 + n)(i) 1 >w;

(2 + n)(1 + n)r

11(ii) wL > wL

4(2+n)w

2( R)1= (2+n)w

1=(2+n)(1+n)1=:(2+n)(1+n)

The …rst assumption implies that the preferred share in (6), which attains the highest
value at = 1=2; takes a positive value at = 1=2. Therefore, the assumption ensuresthat a political equilibrium exists with > 0 for a range of [ ; 1=2); otherwise, = 0 holds
for any 2 [0; 1=2); implying a trivial outcome of no provision of PAYG social security.
The second assumption enables us to demonstrate cases of borrowing-unconstrained aswell as borrowing-constrained type-L individuals in the presence of social security, > 0.
Otherwise, the type-L young individual is always borrowing constrained when > 0.
The structure-induced equilibrium outcome is found as follows. First, we determine
the decisive voter over and calculate his/her most preferred share, denoted by dec ( ),as a function of the tax rate , where the superscript “dec” indicates the decisive voter.
Second, we determine the decisive voter over and calculate his/her most preferred tax
rate, denoted by dec ( ), as a function of the share parameter . Finally, we …nd thepoint where these reaction functions dec ( ) and dec ( ) cross. This point corresponds tothe structure-induced equilibrium outcome of the voting game.
Consider the political determination of . The decisive voter over 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 regardless of their type. Therefore,
from (6), the decisive voter’s reaction function dec ( ) is given by:

dec

( )=

8
<

: 1

0
2+n
(1 ) wif

2 [0; ]if21=(2+n)(1+n)

15; 12 :(7)

Next, consider the political determination of . The decisive voter over 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, todetermine the type of the decisive voter, we focus on the parameter representing theinverse of the interest-rate elasticity of young-age consumption and consider two cases
separately: a high elasticity (1=1 in Subsection 4.1) and a low elasticity (1= < 1 in
Subsection 4.2).
We adopt the above classi…cation because the order of preferences for the tax ratecritically depends on the degree of interest-rate elasticity. For the case of 1=
1, alower-income young individual prefers a higher tax rate. However, for the case of 1= < 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=

1)

To determine the type of the decisive voter over in the case of 1=
1, we consider
the preferred tax rate of a type-j young individual given by (5). Figure 3 illustrates thecondition (5) that determines the preferred tax rate by a type-j young (j = L; M; H)
individual. The left-hand side of (5), denoted by LHS, is decreasing in and is independent of the type of young individual. In contrast, the right-hand side of (5), denoted by
RHS j , is nondecreasing in , 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 = 1. The kink
point of = ^(wj ) implies that a type-j young individual can save part of his/her income
if < ^(wj ) and nothing if^(wj ). It is immediately observed from Figure 3 thatgiven , a lower-income young individual prefers a higher tax rate: yH < yM < yL for
all 2 [0; 1], where yj (j = L; M; H) denotes the preferred tax rate of a type-j young
individual.
[Figure 3 about here.]
Given the assumption of demographic structure (Assumption 1) and the fact that< yM < yL < oj , the decisive voter over is the one who prefers the highest tax
rate among young individuals, that is, a type-L young individual. Therefore, the reaction
function of ; dec ( ); is implicitly given by (5) with j = L: To …nd the crossing point of
the two reaction functions, dec ( ) and dec ( ); we substitute (7) into (5) with j = L toyH

16

obtain:y( ; w; n) = z( ; wL );where:y( ; w; n) =Lz( ; w ) =

(

(

1
1
w
2+n

1 2
((1 ) )

(1 + n)(1

2 )w

RwL
(wL )1(1 )if< ^(wL )if^(wL ):if
if

2 [0; ]

2 ( ; 1=2]

Solving y( ; w; n) = z( ; wL ) for leads to the tax rate in a structure-induced equilibriumof the voting game. The corresponding is obtained by substituting the equilibrium
into the reaction function dec in (7).
Proposition 1. Suppose that 1=
1 holds. There exists a unique structure-induced
equilibrium of the voting game such that the decisive voter over is a type-L young
individual.
Proof. See Appendix 8.2.
There are two possible cases of the equilibrium. The …rst is the case where the wageof type-L 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 of y( ; w; n) and z( ; wL ) is below the critical rate of ^(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 ^(wL ).

4.2

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

Next, consider the case of a low interest-rate elasticity such that 1= < 1. The decisive
voter over is equivalent to that in the previous case; the reaction function dec ( ) is
given by (7). However, the decisive voter over may di¤er from the previous case; theorder of preferred tax rates may change depending on the value of .
To determine the decisive voter over , we recall the condition (5) that determines
the tax rate preferred by a type-j young individual for a given . The graphs of (5) forthe case of 1= < 1 are illustrated in Figure 4. The main di¤erence from the previous
case is that RHS i and RHS j (i 6= j) cross at some tax rate 2 (0; 1=2). This is because
when a type-j individual is borrowing constrained, the slope of RHS j becomes steeper
as the elasticity 1= decreases. There are two critical values of , ~LM and ~M H , such17

that RHS L and RHS H cross at
direct calculation, we obtain:
LM~= ~LM and RHS M and RHS H cross at(wL )1
RwM

1

1=
MH

and ~

(wM )1RwH

1= ~M H . By

1=;where ^(wL ) < ~LM < ^(wM ) < ~LM < ^(wH ) (see Figure 4). The derivation of ~LM and~M H is given in Appendix 8.3.
[Figure 4 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 while LHS is strictly increasing in. The tax rate preferred by a type-j young individual depends on the size of . Overall,he/she prefers a higher tax rate when is higher.
The order of tax rates preferred by the three types of agents is changed by the size of, as illustrated in Figure 4. First, when is low such that LHS of (5) crosses RHS of(5) with j = L within the range (0; ~LM ], the tax rates preferred by the young are orderedby yH < yM < yL , where 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 attains
a middle value such that LHS of (5) crosses RHS of (5) with j = M within the range(~LM ; ~M H ], the tax rates preferred by the young are ordered by yH < yL < yM or
yL
yH< yM : the decisive voter in this case is the type-M young individual. Finally,
when is high such that LHS of (5) crosses RHS of (5) with j = H within the range[~M H ; 1=2], the tax rates preferred by the young are ordered by yL < yM < yH : thedecisive voter becomes the type-H young individual.
Given the abovementioned feature, the reaction function of , = dec ( ), is now
implicitly given by:(1 + n) (12 )w + (1 + )where:

L

M

1

1
(1
2+n

H

z~( ; w ; w ; w )

1
((1

)w

8
>
RwL
>
>
>
(wL )1>>> (1 )
>
>
< RwM
M 1

(w )
>
>
(1 )
>
>
>
>
RwH
>
>
>
: (wH )1
(1

)

if

2
= z~( ; wL ; wM ; wH ); (8)) )< ^(wL )if ^(wL )~LM

if ~LM << ^(wM )if ^(wM )~M H

if ~M H << ^(wH )if ^(wH ):The graph of the function z~ is illustrated by the bold curve in Figure 4.
18

We substitute the reaction function of dec ( ), given by (7), into the left-hand side of(8) to obtain the condition that determines the equilibrium tax rate:y( ; w; n) = z~( ; wL ; wM ; wH );where y( ) has been already de…ned in the previous subsection. Figure 5 illustrates thegraphs of y( ; w; n) and z~( ; wL ; wM ; wH ). Solving y( ; w; n) = z~( ; wL ; wM ; wH ) for
leads to the tax rate in a structure-induced equilibrium of the voting game. The
corresponding is obtained by substituting the equilibrium into the reaction functiondecin (7).
[Figure 5 about here.]
Proposition 2. Suppose that 1= < 1 holds. There exists a unique structure-induced
equilibrium of the voting game such that the decisive voter over is:(i) a type-L individual if

1 + 2 (wL )1

= RwM

1=

RwM =(1 + n)w;(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 5).
The key factors in Proposition 2 are the borrowing constraints and the interest-rateelasticity of consumption. To understand the roles of these two factors, consider the casewhere 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 thissituation, a higher tax rate produces two opposing e¤ects: a negative e¤ect that results inlower after-tax income and thus the utility loss of taxation in youth, and a positive e¤ectthat produces higher social security bene…t and thus the utility gain in old age.
The net impact of taxation depends on the interest-rate elasticity. When the elasticity
is high such that 1= > 1, the positive e¤ect overcomes the negative e¤ect. The lowincome individual then chooses the highest tax rate among the young and thus becomes
the decisive voter. In contrast, when the elasticity is low such that 1= < 1, the negativee¤ect may dominate the positive e¤ect for low-income individuals. They choose a lower
19

tax rate than the middle-income individuals, and this results in an equilibrium where themiddle-income individual becomes the decisive voter.
Figure 6 illustrates the conditions that determine the decisive voter and his/her statusof saving in a wL wM space. From the …gure, we …nd that the decisive voter is a type-L
individual provided the wage income levels of the two types of individuals are high suchthat the pair (wL ; wM ) is set within the area marked by (j = L) in Figure 6. 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 su¢ ciently low such that (wL ; wM ) is
set within the area marked by (j = M ) in Figure 6. The decisive voter becomes thetype-M young individual. The equilibrium is featured by the situation that resembles theends-against-the-middle equilibrium.
[Figure 6 about here.]5E¤ects 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 ( ) change in response to a change ininequality. 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
di¤erent levels of income inequality. We focus on the nontrivial equilibrium with > 0
to observe the marginal e¤ect on the share of social security in government expenditure.
Proposition 3. Consider a political equilibrium with

> 0.

(i) In an economy with 1=
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 1= < 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 U-shaped relationships between wj and the tax rate ( ) and between
wj and the share of social security ( ).
Proof. See Appendix 8.5.
Figure 7 illustrates the e¤ects of a mean-preserving change in the decisive voter’s wage
on the equilibrium tax rate. Panel (a) is for the case of 1=
1; Panel (b) is for thecase of 1= < 1. Proposition 3 states that if the interest-rate elasticity is high such that1=
1, there is, in general, a monotone relationship between the decisive voter’s wage
20

and his/her preferred tax rate: the decisive voter prefers a higher tax rate as he/she
becomes poorer. However, when the elasticity is low such that 1= < 1, such a monotone
relationship no longer holds. Once the decisive voter’s wage falls below the threshold levelthat changes his/her status from unconstrained to constrained, he/she prefers a lower tax
rate as he/she becomes poorer, as demonstrated in Subsection 3.2. Thus, there is aninverse U-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 7. Givena positive correlation between the tax and the share of social security, there is also an
inverse U-shaped relationship between the decisive voter’s wage and the share of social
security in government expenditure.
[Figure 7 about here.]
We should note that, in the current framework, the negative correlation arises only inthe 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 theconditions fails to hold, the economy displays a positive correlation between inequality
and the share of social security. Therefore, our analysis suggests that these factors are
the key to demonstrate the abovementioned inverse U-shaped relationships. These relationships also hold even if the assumption of a quasi-linear utility function is dropped, as
we demonstrate in the next section.

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 coj
t+1 . This speci…cation enables
us to illustrate the existence and uniqueness of the political equilibrium. However, thespeci…cation also results in (i) a saving decision una¤ected 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.
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 interest-rate elasticity
is low and the decisive voter is borrowing constrained. In this equilibrium, a meanpreserving spread of income inequality results in a lower equilibrium tax rate and a lower
share of social security in government expenditure.

21

For the purpose of analysis, we assume the following utility function:(cyj )1
Utj = t1

1(gt )1
+
1

1

"

+

1
(coj
t+1 )
1

1

(gt+1 )1
+
1

1

#

:

The main di¤erence from the previous model is that the utility of old-age consumption is
1
1g=(1) rather than coj
given by f(cojt+1 . The maximization of their lifetime
t+1 )utility under the budget constraints leads to the following saving function:
sjt = max 0;( R)1=(1( R)1= + R

t )w

bt+1
( R)1=

j:

Saving now depends on the social security bene…t bt+1 that gives individuals a disincentive
to save. Hereafter, we 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) (1)w into the above saving function to obtain the following condition that determines thesaving behavior of a type-j individual:
sj > 0 ,wjw( R)1=>
1+n.

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

coj
t+1 =

8
<

R
( R)1= +R

h

(1

: (1
)wj
8
h
< ( R)1= R (1
(

R)1=

+R

: (1 + n) (1)wj+ (1+n) R(1)wj + (1+n) R(1
) w

) w

) w

i
i

j

if ww

wj

if w

j

if ww
wj

if w

( R)1=
1+n( R)1=
1+n( R)1=
1+n( R)1=
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 obtain the indirect utility functions of type-j young and22old individuals as:
yj

V
V

oj

=
=

(

(

y;j
if
Vs>0
y;j
Vs=0
if
o;j
Vs>0
if

wj
w
wj
w

( R)1=
1+n( R)1=
1+n

wj
w
wj
w

( R)1=
1+n( R)1=
1+n

o;j
Vs=0

if

(1

)wj +

>
>
;

where:
1

y;j
Vs>0

+

( R)1= + R

1

1
(1
2+n
(1

1
+

o;j
Vs=0

1

1

y;j
Vs=0

o;j
Vs>0

R

1

)(1

)wj

1

+
)(1

1

1+n
(1
2+n

1

) w

1

+

1

(g)

1

)(1

) w

1

+

(g)

1

+

1

(g)

(1 + )(1 + )
;
1

) w

Rsj 1 + (1 + n) (1) w]1[(1 + n) (11) w(1 + )(1 + )
;
1

) w

1

1
(1
2+n

1

1

(1 + n) (1
R

+

1

(g)

(1 + );
1(1 + ).
1o;jo;jThe term (g) =(1) in the functions of Vs>0and Vs=0
is predetermined. We canshow that these preferences satisfy single-peaked properties in the same manner as in thecase of a quasi-linear utility function.
The policy preferences of the old are the same as for the 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: oj = 1=2 and oj = 1 hold for all j:
Accordingly, generalization of the utility function does not a¤ect the policy preferencesof the old.
We next consider the policy preferences of the young. Given , the preferred tax rateof a type-j young individual satis…es the following …rst-order condition with respect to :y

LHS = RHS

yj

(

yj
RHSs>0ifyj
RHSs=0 if

23<(wj ; )
(wj ; )wjw( R)1=
1+n

1

;

(9)

where:
1
(1
2+n

LHS y
yj
RHSs>0yj
RHSs=0

R
(

R)1=

wj

1

1
(1
2+n

) w

+R

wj +

[(1 + n)w]

(1 + n)
R

)(1

2 )w;

w

wj

(1 + n) (1(1 + n) (1
R

2 )w

; and

2 )w:

LHS y represents the marginal bene…t of taxation in terms of the utility of public goods.
This bene…t is common to the three types of young agents because of the nature of public
goods. RHS yj represents the marginal loss of taxation plus the marginal bene…t of social
security in terms of the utility of consumption. The sum of these losses and bene…ts di¤ers
across individuals. In particular, the following properties hold (see Appendix 8.6 for the
proof):(
RHS yL RHS yM RHS yH if 1=
1
(10)
yL
yM
yH
RHSs=0 > RHSs=0 > RHSs=0 if 1= < 1;
where an equality in the …rst line holds if and only if 1= = 1 and s = 0. Similar to theyjprevious model, the order of RHSs=0(j = L; M; H) critically depends on the degree of2
the interest-rate elasticity 1= .
Panel (a) of Figure 8 illustrates the graph of (9) when 1=
1 holds. The crossingyyj
point of LHS and RHS determines the tax rate preferred by a type-j young individual.
The …gure 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 . That is, the ends-against-the-middle equilibrium neverarises when the interest-rate elasticity is high such that 1=
1.
[Figure 8 about here.]
Panel (b) of Figure 8 illustrates the graph of (9) when 1= < 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 lowand the high-income young individuals form a coalition against the middle, as illustrated
in Panel (b) of Figure 8. Therefore, the low interest-rate elasticity and the borrowing
constraint remain the keys to the existence of the ends-against-the-middle equilibrium.
The determination of the share of social security is slightly di¤erent from that in2yj(j = L; M; H) is ambiguous.
If 1= < 1, the order of RHSs>024

the previous quasi-linear utility case. The preferred share of a type-j young is given by:yj=

where:

8
>
>
>
>
>
<
>
>
>
>
>
:

1
2+n

1=

R
( (2+n)(1+n)
)

~(w )
(wj )

R
wj
( R)1= +R w

1=
R
1+n
1
+ (2+n)(1+n)
2+n( R)1= +R
1
2+n
1=
1
(1+n)
+ (2+n)(1+n)
2+n

(

(

if ~(wj ) <

<

if

(wj )

)

)

1=

j

~(wj )

if

0R(2 + n)(1 + n)(1( R)1=+
2+n(2 + n)wj;( R)1= + R
w

(wj )

(11)

;

R

1=

(2 + n)(1 + n))

(1 + n)(2 + n)wj:(1 + n)w

The …rst two lines of the right-hand side in (11) represent the choice of when a type-j
young is borrowing unconstrained; the third line represents the choice of when he/she
is borrowing constrained. The derivation of (11) is given in Appendix 8.5.
When the tax burden is low such that~(wj ), a type-j young individual can savemuch for his/her old-age consumption and thus …nds it unnecessary to use tax revenue forsocial security: = 0. However, when the tax is above ~(wj ), a type-j young individual…nds it optimal to o¤set part of their tax-induced consumption loss with a social security
bene…t. 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 that(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 bene…t. 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 di¤erent from that under a quasi-linear utility function.
Panel (c) of Figure 9 illustrates the reaction function of for each type of individual.
yMyL
The …gure shows that yH ( )( )( ) < o holds for any . 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 = yL into thedecisive voter’s …rst-order condition with respect to .
Given a brief characterization of the political equilibrium, we now compare the income
inequality e¤ects between the current and former models. In particular, we focus on thesituation where the decisive voters over and are borrowing constrained. Thus, the25

decisive voter’s choice of , in the current framework, is given by:
dec

1
2+n

=
1
+
2+n

;

1=
(2+n)(1+n)(1 + n)which is independent of . We substitute this into the …rst-order condition with respect
to , (9), for the case of(wj ; ) and obtain the following condition that determines
the equilibrium tax rate when the decisive voter is borrowing constrained:

w

j 1

=
+

1

dec

2+n
(1 + n) 1

w

!1

dec

w

( )

(1

2 )

1

dec 1

( )

(1

2 );

where the left-hand side shows the marginal cost of taxation, the …rst term on the righthand side shows the marginal bene…t of public goods, and the second term on the righthand side shows the marginal bene…t of social security. Given , the right-hand side isindependent of wj whereas the left-hand side is decreasing (increasing) in wj if 1= < (>)1.
Thus, a mean-preserving reduction of the decisive voter’s wage decreases (increases) theequilibrium tax rate if the interest-rate elasticity is low (high) such that 1= < (>)1. This
result is qualitatively equivalent to that in the quasi-linear utility function model.

7

Conclusion

How does wage inequality a¤ect the allocation of tax revenue between social security andforward 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 interest-rate elasticity of consumption and theborrowing constraint. These features derive an ends-against-the-middle equilibrium where
low- and high-income individuals form a coalition in favor of a low tax rate and middleincome individuals favor a high tax rate. In addition, higher wage inequality results in alower level of social security and a lower share of social security in government expenditurewhen the decisive voter is borrowing constrained and the interest-rate elasticity is low.
To obtain these results, we simplify the analysis by adopting a quasi-linear utility
function. Because of this simpli…cation, we can remove the link between saving and theallocation of tax revenue between social security and public goods provision. However, as
shown in our analysis, the e¤ect of the interest-rate elasticity of consumption appears in
the political equilibrium in spite of this simpli…cation. In addition, Section 6 demonstrates
that the main results are qualitatively unchanged under a generalized utility function.
26

Thus, our analysis is almost robust to the assumption of a quasi-linear utility function.

27

8

Appendix

8.1

Single-peakedness of Preferences

8.1.1

Single-peakedness of preferences overy;jy;jare single peaked
and Vs=0
The proof proceeds as follows. First, we show that both Vs>0
y;j
y;j
y;j
y;j
over . Then, we demonstrate that @Vs>0 =@ = @Vs=0 =@ and Vs>0 = Vs=0
hold atjy;j= ^(w ), implying that V has a unique local maximum over the whole range of andthus that V y;j is single peaked over .
y;jy;jwith respect to are:and Vs=0
The …rst and the second derivatives of Vs>0
y;j
@Vs>0
=
@

Rwj + (1 + n) (1
(1

+

2 )w

)(1
) w
2+n

y;j
@ 2 Vs>0= ( 2) (1 + n) w + ( 2)
@ 2

+

(

)

(1

)(1 2 )w
;
2+n

(1

)(1
) w
2+n
1

(1

(1

)(1
) w
2+n

)

(wj )1+ (1 + n) (1

)(1
) w
2+n

(1

(1
)w
2+n

)(1 2 )w
2+n

2

< 0;
y;j
@Vs=0
= ( 1)(1@(1+y;j@ 2 Vs=0=(@ 2)(wj )1+ ( 2)
+

(

)

(1

)

1

2 )w

)(1 2 )w;
2+n

+ ( 2) (1 + n) w

(1

)(1
) w
2+n

(1
)w
2+n

(1

)(1
) w
2+n

1

(1

)(1 2 )w
2+n

2

< 0:
y;j
y;j
are single peaked over because the second derivatives are negative.
Vs>0
and Vs=0y;jy;jNext, we show that @Vs>0=@ = @Vs=0=@ at = ^(wj ). By direct calculation, we
have:
y;j
y;j
@Vs>0
@Vs=0
R
,
Rwj R ( 1)(wj )1 (1
) :
@
@
j
j
=^(w )At= ^(wj )1=^(w )

1= ( R)1= wj , the right-hand side of the above condition is rewritten

28

as:
( 1)(wj )1(1^(wj ))

=

Rwj ;

y;j
y;j
implying that @Vs>0=@ = @Vs=0=@ at = ^(wj ).
y;jy;jFinally, we show that Vs>0
= Vs=0hold at = ^(wj ). By direct calculation, we have:
y;j
Vs>0

=^(wj )y;j
R Vs=0=^(wj )( R) 1=
()

1

1

1

n
+ R (1

)w

j

( R)

1=

o

=

((1

1

)wj )
1

1

At
= ^(wj )1 1= ( R)1= wj , the left-hand and right-hand sides of the above
condition are reduced to, respectively:
1( R) 1=
LHS = RHS =y;jy;jimplying that Vs>0
= Vs=0hold at

8.1.2

1
;

1

= ^(wj ).

Single-peakedness of preferences over

Before proceeding to the proof, we note that the status of saving is independent of
because of the assumption of a quasi-linear utility function. Thus, it is su¢ cient to showy;jy;jthat @ 2 Vs>0=@ 2 < 0 and @ 2 Vs=0=@ 2 < 0 for the proof.
y;jy;jThe …rst and the second derivatives of Vs>0and Vs=0with respect to are:
y;j
y;j
@Vs>0
@Vs=0
=
= (1 + n)(1@@

) w + ( 1)y;jy;j
@ 2 Vs>0@ 2 Vs=0
=
=(
@ 2
@ 2

)(1
) w
2+n

8.2

)

(1

(1
1

)(1
) w
2+n
(1

) w
2+n

(1

) w
;
2+n

2

< 0.

Proof of Proposition 1

As shown in the text, when 1=
1, the decisive voter is a type-L individual and his/her
preferred tax rate satis…es y( ; w; n) = z( ; wL ). The functions y( ; w; n) and z( ; wL )have the following properties: @y( ; w; n)=@ < 0; lim !0 y( ; w; n) = 1; y(1=2; w; n) =0; @z( ; wL )=@0; z(0; wL ) = maxf RwL ; (wL )1 g < 1; and z(1=2; wL ) 2 (0; 1).
These properties indicate that there exists a unique 2 (0; 1=2) that satis…es y( ; w; n) =z( ; wL ).

29

:

8.3

The Derivation of ~LM and ~M H

The derivation of ~LM is as follows. For the range of (^(wL ); ^(wM )), the right-hand sideof (5), denoted by RHS j , is given by:
RHS j =(

RHS L = (wL )1 =(1RHS M = RwM)

for j = L
for j = M:

RHS L < RHS M holds at = ^(wL ); RHS L > RHS M holds at = ^(wM ). Thus, there
exists a unique , denoted by ~LM 2 (^(wL ); ^(wM )), that satis…es RHS L = RHS M
because RHS L is continuous and strictly increasing in whereas RHS M is independentof . We can derive ~LM by solving (wL )1 =(1) = RwM for .
Similarly, the tax rate that satis…es RHS M = RHS H for the range of (^(wM ); ^(wH ))is derived by solving (wM )1 =(1) = RwH for . The solution is denoted by ~M H .

8.4

Proof of Proposition 2

8.4.1

Existence and uniqueness of the equilibrium

As shown in the text, when 1= < 1, the decisive voter’s preferred tax rate satis…esy( ; w; n) = z~( ; wL ; wM ; wH ). The functions y( ; w; n) and z~( ; wL ; wM ; wH ) have thefollowing properties:@y( ; w; n)=@ < 0;lim y( ; w; n) = 1;!0y(1=2; w; n) = 0;
@ z~( ; wL ; wM ; wH )=@0;z~(0; wL ; wM ; wH ) = maxf RwL ; (wL )1z~(1=2; wL ; wM ; wH ) 2 (0; 1):
These properties indicate that there exists a uniquez~( ; wL ; wM ; wH ).
8.4.2

g < 1;

2 (0; 1=2) satisfying y( ; w; n) =The determination of the decisive voter

Suppose that the type-H young individual is the decisive voter. Then, from Figure 5, it
must hold that y( ; w; n) > z~( ; wL ; wM ; wH ) at = ~M H ; that is:(1 + n)(12~M H )w > RwH :30

This condition never holds under the assumptions of R 1 + n (Assumption 2) and wH >w. Therefore, the decisive voter is a type-L or type-M young individual. From Figure 5,
the type-L young individual becomes the decisive voter if (1 + n)(1 2~LM )w
RwM ;that is, if:
1=(wL )1
R
1+2
wM :
M
Rw
(1 + n)w
Otherwise, the decisive voter is a type-M young individual.

8.5

Proof of Proposition 3(i) For the case of 1=
1, the decisive voter is a type-L young individual and theequilibrium tax rate satis…es y( ; w; n) = z( ; wL ), as shown in Subsection 4.1. Whenthe mean-preserving change in wL is considered, y( ; w; n) is unchanged while z( ; wL )is nonincreasing with reductions of wL . Therefore, the equilibrium tax rate satisfyingy( ; w; n) = z( ; wL ) is nondecreasing in response to a mean-preserving reduction of wL .
Given that dec ( ) is increasing in for 2 (0; 1=2), is also nondecreasing in response
to a mean-preserving decrease in wL .
(ii) For the case of 1= < 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 equil = ^(wL ): a type-L individual
is indi¤erent between saving and not saving. Under this situation, the decisive voter’s
wage wL satis…es (1 + n)(1 2^(wL ))w = RwL , or:R(wL )2 + (1 + n)wwL

2(1 + n)w
= 0:( R)1=

Solving this equation for wL , we obtain:Lw = w^

L(1 + n)w +pf(1 + n)wg2 + 8R(1 + n)w=( R)1=:
2R

Therefore, the equilibrium tax rate is given by equil = ^(wL ) when a type-L individualwith wL = w^ L is the decisive voter.
We now consider a mean-preserving change of wL around w^ L . As shown in Subsection
4.2, the equilibrium tax rate satis…es y( ; w; n) = z~( ; wL ; wM ; wH ) if 1= < 1. In particular, around wL = w^ L , there exists a positive real number " such that the equilibrium

31

tax rate satis…es the following condition:z~( ; wL ; wM ; wH ) =(RwL(wL )1(1

for wL 2 (w^ L"; w^ L ];for wL 2 [w
^ L ; w^ L + "):)We focus on the range (w^ L "; w^ L + ") and consider a mean-preserving change of
wL around w^ L . The right-hand side of the above equation is increasing in wL within therange (w^ L "; w^ L ) and decreasing in wL within the range (w^ L ; w^ L + "): This property
implies that the equilibrium tax rate attains the highest value at wL = w^ L within therange (w^ L "; w^ L + "). Therefore, there is an inverse U-shaped relationship between thedecisive voter’s wage and the equilibrium tax rate around wL = w^ L . Given dec ( ) is
increasing in for 2 (0; 1=2), there is also an inverse U-shaped relationship between thedecisive voter’s wage and the equilibrium share of social security around wL = w^ L .

8.6

Supplementary Explanation for Section 6

8.6.1

Derivation of (10)yj
To establish that condition (10) holds, we …rst investigate the property of RHSs>0. Theyj…rst derivative of RHSs>0 with respect to wj leads to:R( R)1= + R=(1 + n)w +Rjyj@RHSs>0
@wj

w

1

(1

)wj +

(1 + n) w
f (1R2 )+ g ;yj
where the term f (1 2 ) + g is positive provided that > 1=2. Thus, @RHSs>0=@wj >yL
yM
yH
0 holds if
1: this implies that RHSs>0< RHSs>0< RHSs>0
if1.
yj
Next, we investigate the property of RHSs=0 . Direct calculation leads to:yLyM
RHSs=0R RHSs=0
, wL

yM
yH
, w
RHSs=0R RHSs=0

1

M 1

R wM
R w

1

H 1

;
:

Therefore, we obtain:yM
yL
yM() 1= Q 1:
RHSs=0R RHSs=0
R RHSs=0

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

32

8.6.2Derivation of (11)Suppose …rst that the type-j young agent is borrowing unconstrained. The …rst-ordery;jwith respect to is given by:condition for the maximization of Vs>0
y;j
@Vs>0
=0,
@

=

yj
s>0

1
2+n

1
+
2+n

Taking into account the corner solution

yjs>0

The preferred shareR(2+n)(1+n)1=R(2+n)(1+n)

wjw

R( R)1= +R1=(< 1):

1+n( R)1= +R= 0, we obtain:

8
9
1=
j >
R
>
R
1
w
< 2+n
=
(2+n)(1+n)
w( R)1= +R
max 0;
.
1=
>
>
R
1+n
1
:
;
+ (2+n)(1+n)
2+n( R)1= +R

is increasing in

y;j
@Vs>0
=0,
@

=

and is positive if and only if
1
2+n

yj
s=0
1
+
2+n

(wj ):(< 1);

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

33

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36

Figure 1: This …gure 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.

37

Figure 2: This …gure illustrates the share of social security ( ) preferred by the young.

38

Figure 3: The tax rates preferred by the three types of young individuals in the case of
1=
1.

39

Figure 4: The tax rate preferred by a type-j young individual in the case of 1= < 1. The
bold curve illustrates the graph of ze in (8).

40Figure 5: The determination of the tax rate in the case of 1= < 1. The …gure illustrates
the case where the decisive voter is a type-M young individual.

41

Figure 6: A decisive voter and his/her status of saving.

42

Figure 7: The e¤ect of a mean-preserving reduction of the decisive voter’s wage on the tax
rate. Panel (a) illustrates the case of 1=
1. Panel (b) illustrates the case of 1= < 1.
Panel (c) illustrates the relation between the decisive voter’s wage and the equilibriumtax rate in the case of 1= < 1.

43

Figure 8: Panels (a) and (b) illustrate the preferred tax rates in cases where 1=
1= < 1, respectively. Panel (c) illustrates the preferred shares of social security.

44

1 and