GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 228

Marital Status and Derived Pension Rights:
A Political Economy Model of Public Pensions with
Borrowing Constraints

Tetsuo Ono

November 2011; revised May 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Marital Status and Derived Pension Rights: A
Political Economy Model of Public Pensions with
Borrowing Constraints
Tetsuo Ono
Osaka University
May 11, 2012

Abstract
This paper develops an overlapping-generation model featuring four types ofhouseholds: single female, single male, one-breadwinner couple and two-breadwinnercouple. The paper considers majority voting over public pension in the presence of
derived pension rights for one-breadwinner couples. In an economy with a low intertemporal elasticity of substitution, borrowing-constrained one-breadwinner couples may prefer a lower tax rate than do other types of households, although the former attain a higher bene…t-to-cost ratio of public pension than do others. Changes
in the gender wage gap, the level of derived pension rights, and the fraction of twobreadwinner couples produce an inverse U-shaped relationship between the relevant
variable and the tax rate.
Keywords: Borrowing constraint; Marital status; Gender wage gap; Derived pension rights; Political economy
JEL Classi…cation: D72, H55, J12

Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Tel.: +81-6-6850-6111; Fax: +81-6-6850-5256. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

Most OECD (Organization for Economic Co-operation and Development) countries o¤er
pension bene…ts for non-working spouses and divorcees. The bene…ts, called derived pension rights, include (i) survivors’ bene…ts for widows; (ii) bene…ts for divorced spouses;and (iii) spousal bene…ts as a supplement to a worker’s bene…t (Choi, 2006). Thesebene…ts imply that derived pension rights have an intra-generational redistribution component from working singles and two-breadwinner couples to one-breadwinner couples.
Thus, recent pension reforms in many OECD countries that attempt to link contributions
and bene…ts more closely (OECD, 2011) may provoke an intra-generational con‡ict overpension policy.
Despite the con‡ict among singles and couples caused by derived pension rights, there
are few studies focusing on these rights in the political-economic literature. Exceptions
are the works of Leroux and Pestieau (2011) and Leroux, Pestieau, and Racionero (2011).
Leroux and Pestieau (2011) consider an economy composed of couples who maximize
the joint lifetime utility of a husband and a wife. A husband always works regardlessof his labor productivity, while a wife chooses whether or not to work depending on her
reservation wage. Under this framework, Leroux and Pestieau (2011) demonstrate aninteraction between a wife’s labor supply decision and pension policy preferences, andthey show that a pension system with derived pension rights is likely to emerge as avoting equilibrium outcome.
Leroux, Pestieau, and Racionero (2011) assume that the degree of derived pension
rights is …xed. Instead, they allow for the presence of single males and females andexamine how the degree of derived pension rights a¤ects tax burden policy preferences
for public pensions and thus a resulting pension system via voting. Their results are as
follows: (1) a reduction of derived pension rights results in a smaller tax burden for public
pension, and (2) an increase in the share of two-breadwinner couples have two opposinge¤ects on the pension burden, where the net e¤ect may be positive or negative depending
on other economic factors.
The results in Leroux, Pestieau, and Racionero (2011) provide signi…cant policy predictions for public pensions. However, their results heavily depend on the following two
assumptions: quasi-linear utility and no borrowing constraints. The …rst assumption,
which is often adopted in the political-economic analyses of social security (see, for example, Conde-Ruiz and Galasso, 2003, 2004, 2005; Borck, 2007), makes the analysis tractable,
but draws attention away from the considerable e¤ect of the intertemporal elasticity of
substitution on a household’s decisions concerning saving and voting (Casamatta et al.,
2000; Conde-Ruiz and Profeta, 2007; Cremer et al., 2007; Arawatari and Ono, 2011). The
second assumption allows for borrowing against future pension bene…ts, which is di¢ cult1

to support from the empirical viewpoint (Diamond and Hausman, 1984; Mulligan andSala-i-Martin, 1999). The aim of this paper is to relax these two assumptions in theframework of Leroux, Pestieau, and Racionero (2011) and to provide new insight into
derived pension rights from a political-economic viewpoint.
For analytical purposes, we will extend the framework of Leroux, Pestieau, and Racionero(2011) in the following two ways. First, the preferences of each household are represented
by a utility function with a constant intertemporal elasticity of substitution. Second, eachhousehold is unable to borrow against its future pension bene…ts. Under this extended
framework, we show the following two results. First, in an economy where an intertemporal elasticity of substitution is below one, one-breadwinner couples who bene…t frompublic pensions may prefer a lower, rather than higher, tax rate than single females who
owe net burden, because of the presence of borrowing constraints. Borrowing-constrained
one-breadwinner couples want to choose a low tax rate to keep their after-tax income levelas high as possible. There is then a voting equilibrium, much like an ends-against-themiddle equilibrium, in which single females, along with the old, form a coalition against
the others.
Secondly, when the intertemporal elasticity of substitution is below one, an inverse
U-shaped relationship is created between the relevant variable and the tax rate due to thegender wage gap, the level of derided pension rights, and the ratio of double-breadwinnercouples. Near the maximum of the inverse U-shaped curve, the decisive voter is borrowingunconstrained on one side and borrowing-constrained on the other side. This two-tonede¤ect, as well as the ends-against-the-middle equilibrium, both of which were not shownin Leroux, Pestieau, and Racionero (2011), are derived by the presence of a borrowing
constraint associated with a low intertemporal elasticity of substitution.
The organization of this paper is as follows. Section 2 describes the economic environment. Section 3 demonstrates the utility maximization of singles, one-breadwinnercouples, and two-breadwinner couples. Section 4 presents the political institution andpension policy preferences of the young and the old. Section 5 characterizes the politicalequilibrium. Section 6 performs a comparative statics analysis and shows how genderwage gap, derived pension rights, and the share of two-breadwinner couples a¤ect theequilibrium pension policy. Section 7 provides concluding remarks. Proofs are provided
in the Appendix.

2

The Economic Environment

Consider a discrete time economy in which time is denoted by t = 0; 1; 2;. The economy

is comprised of overlapping generations of individuals, each of whom lives for two de…ned
periods: youth and old age. Each generation is composed of a continuum of agents.
2

Speci…cally, in each generation, there are males and females; the size of each gender
population is normalized to unity. Thus, the total population size of each generation is
two.
Each generation consists of four di¤erent categories of households: single males, single
females, one-breadwinner couples, and two-breadwinner couples. The fraction of singles
is 1' 2 (0; 1), and the fraction of couples is '. Among couples, a fraction 1

2 (0; 1)

are to be considered one-breadwinner couples, and the remaining fraction is composedof two breadwinners. Therefore, in each generation, the population size of each type ofhousehold is as follows: the number of single males is 1
is 1'; the number of single females'; the number of one-breadwinner couples is 2'(1); and the number of two-breadwinner couples is 2' . The fraction of each type of household is …xed over time.
For simplicity, marriage decisions are not factored into the analysis.
Each agent is endowed with one unit of labor in youth and retires in old age. Males
supply labor regardless of their marital status, while only females who are single or belong
to two-breadwinner couples supply labor. Females who belong to one-breadwinner couples
do not supply labor; instead, they devote their time to home production and leisure, bothof which are assumed not to have an e¤ect on utility or household income.
In this economy, there are two types of heterogeneity between males and females: wage
and longevity. These types are characterized by the parameter pairs (wm ;and (wf ; f ) for females, such that
(wm ; m ) = (w; );(wf ; f ) = ( w; 1);m) for males

2 (0; 1);2 (0; 1);

where wi (i = f; m) represents the wage, and i represents the probability of surviving
in old age. The term 2 (0; 1) represents the gender wage gap, and the term 2 (0; 1)

represents the longevity di¤erence between males and females. It is assumed that females
have a longer life span than males but obtain a lower wage.

Individuals contribute to the pension system during youth and receive a pension bene…ts in old age. Following the convention in the literature, we present the e¢ ciency lossof taxation by assuming convex costs of collecting taxes (see, for example, Casamatta,
Cremer, and Pestieau, 2000; Bellettini and Berti Ceroni, 2007; Borck, 2007; Cremer etal., 2007). The actual tax revenue from the young is therefore given by(1
where the terms w, w(1) [w + w(1') + w' ] ;'), and w' in the square brackets correspond to the con-tributions by males, single females, and females who belong to two-breadwinner couples,
respectively. The term (1) is the distortionary factor. The assumption of distortionary

taxation is made solely to ensure an interior solution to preferred tax rates and otherwise
plays no role.
3Let p denote pension bene…ts for contributors; let p denote pension bene…ts for noncontributors, where
pension payments are

2 [0; 1] represents the level of derived pension rights. The total
p [ + (1') + ' + '(1)] :The pension bene…t for males is p , rather than p, because their length of life in old age
is assumed to be 2 [0; 1).
Under the assumption of a balanced budget, the government budget constraint be-comes
p = w ( ) (1

(1)

);

where

1 + (1 ' + ' ):+ 1 ' + ' + '(1)
The tax rate is determined via majority voting, whereas the degree of derived pension
rights is assumed to be …xed at the constitutional level. Voting over will be discussed()

in Section 7.

3

Economic Decisions

Let j = f; m; c1, and c2 denote single females, single males, one-breadwinner couples,
and two-breadwinner couples, respectively. In this section, we illustrate the economic
decisions on saving by singles and couples. Any old agent makes no economic decisionbecause his/her saving is predetermined in youth.

3.1

Singles

Each single agent is assumed to receive utility from private consumption. The lifetime
utility function of a type-j (j = f; m) single young agent is speci…ed by:(cj )1
1(dj )11+;
11where cj is consumption in youth, dj is consumption in old age,Uj =2 (0; 1) is a discountfactor, and (> 0) is the inverse of the intertemporal elasticity of substitution. A lower
1= implies a lower intertemporal elasticity of substitution.1
Type-j’s (j = f; m) individual budget constraints in youth and in old age are given
by, respectively,
c j + sj
dj

(1
Rsj +

1

)wj ;
j

p;

For j = m, the second-period utility might be more appropriately written asf(dj )11g=(1)
since is interpreted as the probability of surviving to the second period of life. Following Borck (2007),
we assume away the e¤ect of on the second-period utility for the tractability of analysis.

4where sj is saving,is the income tax rate, R(> 1) is the gross interest rate, and p is the

per capita pension bene…t. If j = f , then wf = w and f = 1; if j = m, then wm = w
and m = . Throughout this paper, we assume borrowing constraints, that is, sj
0.
This constraint precludes the possibility of borrowing when young against future pension
bene…ts (Diamond and Hausman, 1984; Mulligan and Sala-i-Martin, 1999).
A type-j young agent maximizes his/her utility subject to his/her budget constraint
and the borrowing constraint. When sj > 0, the …rst-order condition for an interior
solution is dj = ( R)1= cj . This condition determines an interior solution of saving by a
type-j agent. By taking the borrowing constraint into account, the saving function of a
type-j agent becomes(

sj = max 0;( R)1=
1=

R + ( R)

"

j

)wj

(1

p

( R)1=#); j = f; m:(2)

The saving function (2) and the government budget constraint (1) imply that there isa critical rate of tax such that
f

s >0,
m

s >0,

f

<^

m

<^

( R)1=;()( R)1=:()

The critical rate for single females, ^f , is lower than that for single males, ^m , because
single males obtain higher wages and live shorter than do single females.
With the saving function and the private and government budget constraints, we canobtain the consumption functions of a type-j (= f; m) agent in youth and in old age.
We use the functions to obtain the indirect utility function of type-j singles, denoted by
V j (j = f; m) :
8
h
i1
j
< j
1+
Vs>0 1 1 R+( RR)1=(1)wj + wR ( ) (1)
if
< ^j ;
j
1
V =
1
1
1+
1
:
Vj
((1
)wj )
+
f j w ( ) (1
)g
if
^j :
s=0

1

1

1

j
The function Vs>0(j = f; m) denotes the indirect utility of a type-j young householdjwhen it saves some portion of income, and Vs=0
denotes the indirect utility when it isfaced with a borrowing constraint and saves nothing. The term in square brackets injrepresents the lifetime income; the …rst and the second terms on thethe equation of Vs>0jright-hand side in the equation for Vs=0
represent the utilities of consumption in youth andin old age, respectively; the constant term, (1 + )=(1), summarizes the parameters

unrelated to the political decision on taxes.

5

3.2

Couples

We next consider consumption decisions by couples. Following Leroux, Pestieau, andRacionero (2011), we adopt the unitary model of a household that has only one set ofpreferences:(cj )1
1

Uj = 2

1(dj )1
1+

1; j = c1; c2:

Under this speci…cation, spouses play cooperatively and share their resources over theirlifecycle.
A couple chooses consumption and saving to maximize the household utility subject
to the budget constraints in youth and old age:2cj + sj
2dj
and the borrowing constraint, sj
c2
= 1.

)wj ;

(1

Rsj + ( +j)p;0, where wc1 = w, wc2 = (1 + )w;

c1

=

and

In the …rst period of life, a husband and/or wife work and earn the after-tax wage
income, (1)wj . A couple consumes a part of the after-tax wage and saves the restfor old-age consumption. In the second period of life, the couple obtains the return fromsavings, Rsj , the pension bene…t paid to the husband, p, and that to the wife, j p.
By taking the borrowing constraint into account, the saving function of the type-j
couple becomes(

sj = max 0;( R)1=
1=

R + ( R)

"

)wj

(1

( +

j

)p

( R)1=#); j = c1; c2(3)

The saving function (3) and the government budget constraint (1) imply that there is a
critical tax rate for couples such that
c1

s >0,

<^

( R)1=;( + ) ()

sc2 > 0 ,< ^c2

(1 + ) ( R)1=:
( + 1) ( )c1With the saving function and the private and government budget constraints, we canobtain the consumption functions in youth and in old age. We use these functions toderive a type-j couples’indirect utility function:
Vj =

j
Vs>0
if
j
Vs=0 if6< ^j ;^j ;where:
j
Vs>0

j
Vs=0

1
1
1
1

1
2
1
2

R
1=

R + ( R)
(1

)wj

1

!
+

)wj +

(1
1
2

1

j

( +

)w ( )
(1
R

)

( +

j

)

)w ( ) (11

2 (1 + );1

1

2 (1 + )1

j
The function Vs>0
denotes the indirect utility of a type-j couple when it saves in youth,jand Vs=0
denotes the indirect utility when it is faced with a borrowing constraint andsaves nothing in youth. The interpretation of each term in these equations follows thato¤ered for singles.

4

The Political Institution and Policy Preferences

The tax rateis determined by individuals through a political process of majority voting.

Elections take place every period and all living individuals, both young and old, cast aballot over . The tax preferences of young individuals are represented by the indirect
utility functions presented in the previous section. The tax preferences of old agents are
determined by the size of the pension because their saving when young is predeterminedand has no critical e¤ect on voting behavior. Every individual has zero mass, and thus
no individual vote can change the outcome of the election. We thus assume individuals
vote sincerely.
The majority voting game is intrinsically dynamic because it describes the interaction
among successive generations. To address this feature, we assume commitment, or inother words, once-and-for-all-voting. Here, voters determine the constant sequence ofthe parameters: t = t+1 = for all t, where t denotes the tax rate in period t (see,for example, Casamatta, Cremer and Pestieau, 2000; Cond-Ruiz and Profeta, 2007).
We can view the full commitment solution as the solution that includes intergenerational
interaction because the full commitment solution can be supported as the subgame perfectequilibrium in repeated voting (see, for example, Conde-Ruiz and Galasso, 2003, 2005;
Poutvaara, 2006).
Given the stationary environment, the current model presents a static voting game.
Therefore, the median voter theorem can be applied to the voting game. To …nd thevoting equilibrium, we need to show that tax preferences of voters are single-peaked. As
for the tax preferences of old voters, their objective is to maximize their pension bene…ts
regardless of their marriage status, labor supply, and saving. Although the bene…t levels
di¤er between old agents, the factor related to political decision is common to all old
agents and is speci…ed by the La¤er curve (1). Thus, the tax preferences of the old
are single-peaked; their preferred tax rate, denoted by oj , is oj = 1=2 8j = f; m; c1; c2.
74.1

Policy Preferences of the Young

Next, let us consider the preferences of the young. To show that the preferences of ayoung agent who belongs to a type-j (j = f; m; c1; m2) household are single peaked,j=@ 2 < 0 and
we should note that the following three properties hold. First, @ 2 Vs>0jjj
@ 2 Vs=0=@ 2 < 0 hold; that is, Vs>0and Vs=0
are single peaked. Second, the indirect utility

V j of a young agent in a type-j household is continuous atj
j = f; m; c1; c2. Third, the slope of Vs>0
at
j
@Vs>0
@

j

=^

j
@Vs=0
=
@
j

j
= ^j : Vs>0=^jj
= Vs>0j
is equivalent to that of Vs=0
at

=^j
j

;

=^:

; j = f; m; c1; c2:=^j=^

The details of the calculation are given in Appendix A.1. The three properties imply thatV j has a unique local maximum. In what follows, we derive the conditions that determine
the tax rates preferred by four types of households.
First, consider a preferred tax rate by a young single female agent. She choosesfthat satis…es @Vs>0=@= 0 when she is borrowing-unconstrained; andthat satis…es

f
@Vs=0
=@

= 0 when she is borrowing-constrained. After some calculation, we can …ndthat the preferred tax rate by single females satis…es:(
Rif< ^f ;()f
1
LHS 1 2 = RHS(4)1( ) if^f :()The terms LHS and RHS f represent the marginal e¢ ciency loss of taxation and themarginal cost-to-bene…t ratio of taxation, respectively. Single females choose the tax ratethat balances these terms. In the next subsection, we provide an interpretation of theratio and explain its role in the determination of a preferred tax rate by each type of
agent.
As in the case of a single female, we …nd that a preferred tax rate by single males
satis…es:
LHS

1

2 = RHS

m

(

1

1
()

1
R
()
1

a preferred tax rate by one-breadwinner couples satis…es:(
1R( + ) ()c1
1
LHS 1 2 = RHS
1
1

( )

if

< ^m ;

if

^m ;

if

< ^c1 ;

( )

if

^c1 ;

and a preferred tax rate by two-breadwinner couples satis…es:(1+R( +1) ( )c2
1
LHS 1 2 = RHS11+( )( +1) ( )if

< ^c2 ;if^c2 :( + ) ()

8

(5)

(6)

(7)

4.2

The Role of Marginal Cost-to-bene…t Ratio of Taxation

The conditions (4), (5), (6), and (7), determine the preferred tax rates by single females,
single males, one-breadwinner couples, and two-breadwinner couples, respectively. Each
type of young household chooses its preferred tax rate to equate the marginal e¢ ciency
loss of taxation, represented by the left-hand side, to the marginal cost-to-bene…t ratio of
taxation, represented by the right-hand side; each household prefers a lower tax rate as
the ratio becomes higher.
To understand the reasoning behind the abovementioned property, let us consider the
single female’s condition, (4), as an example. To understand the role of the ratio, we
focus on the parameter , representing the gender wage gap, for illustrative purposes.
There are two opposing e¤ects of an increase in on the ratio. First, given a tax rate, anincrease in

imposes a further tax burden. This burden gives a single female an incentive

to choose a lower tax rate, resulting in a negative e¤ect on the preferred tax rate. Second,an increase in augments wage income for single females, and thus pension bene…ts in oldage, because the total wages on which the tax is levied are increased. This augmentation
gives a young single female an incentive to choose a higher tax rate, resulting in a positivee¤ect on the preferred tax rate via the term ( ).
When a young single female is borrowing-unconstrained, she can reallocate income
freely across periods. Because of this intertemporal reallocation of income, the positivee¤ect is compensated for by the negative e¤ect regardless of the degree of intertemporalelasticity of substitution. Therefore, an increase in

results in a higher marginal cost-to-bene…t ratio and thus a lower preferred tax rate when a young single female is borrowingunconstrained. The result holds regardless of 1= because the objective for a borrowingunconstrained household is to maximize lifetime income, which is independent of 1= .
When a single female is borrowing-constrained, the positive e¤ect is not necessarily
compensated for by the negative one. The borrowing-constrained single female wants to
consume more when young, but her demand is restricted by the borrowing constraint. In
this situation, the borrowing-constrained individual attaches a large weight to the utility
gain of an increase in her wage. This e¤ect might lead to a situation in which the positivee¤ect overcomes the negative one, resulting in a lower, rather than higher, marginal costto-bene…t ratio and thus a higher preferred tax rate in response to an increase in .
Which e¤ect outweighs the other depends on the degree of intertemporal elasticity of
substitution. A lower elasticity implies a stronger incentive for single females to smoothconsumption over periods. Because of this incentive, borrowing-constrained single females
attach a smaller weight to the positive e¤ect on youthful consumption via a decrease inher preferred tax rate as the elasticity becomes lower. In other words, the negative e¤ect
on the preferred tax rate is more likely to be overcome by the positive one as the elasticity
9becomes lower.
The net e¤ect depends on the degree of the intertemporal elasticity of substitution.
When the elasticity is high, that is, 1=
1, the net e¤ect on the tax is negative. An
increase in results in a lower preferred tax rate by borrowing-constrained single females.
In contrast, when the elasticity is low, that is, 1= < 1, the net e¤ect becomes positive.
An increase inleads to a higher preferred tax rate.

In concluding this section, we note that the tax rates preferred by the young are lower
than those preferred by the old who choose = 1=2. The result of this phenomenon isthat the decisive voter with respect tobelongs to the young generation because thepopulation size of the young is larger than that of the old given the death of some males
in early life. Given this result, we focus on young agents’preferences overthe determination of in majority voting in the next section.

5and consider

Political Equilibrium

This section characterizes the political equilibrium of the majority voting. In what follows,
an “agent”implies a “young agent”if not otherwise speci…ed.

5.1

Political Environment

We impose the following assumption to proceed with the analysis.
no11Assumption 1: (i) 1 < (1 + ) and (ii) max 4(1< ' < 1+2 .
;) 4
The …rst assumption, 1 < (1 + ), enables us to illustrate a variety of cases. Under
this assumption, there are two critical values of :
=

1
1+

;

1

:

With these two critical values, the order of critical tax rates that determine the status ofsaving becomes as follows:
Case (a) ^f < ^c2 ^c1 < ^m if
Case (b) ^f ^c1 < ^c2 < ^m if
Case (c) ^c1 ^f < ^c2 < ^m if

0
1
1+
1

<
<

1
1+
1

;
;

(8)

1:

Case (c) is available if and only if Assumption 1(i) holds. We will investigate who becomesa decisive voter for each case.
The second assumption, max

n

1
4(1

;1
) 4

o

< ' < 1+2 , ensures that one who prefers

the highest tax rate among the young becomes the decisive voter. To understand theargument stemming from Assumption 1(ii), suppose that a young agent who belongs to a
10

type-j (j = f; m; c1; c2) household prefers the highest tax rate among the young. He/she
becomes a decisive voter if the size of the old plus the size of young agents who belong
to the type-j household are more than a half of the population. Based on the argument
above, a young single female or male becomes a decisive voter if (1') + (1 + ) >(3 + )=2; a young agent who belongs to a one-breadwinner couple becomes a decisive
voter if 2'(1 )+(1+ ) > (3+ )=2; and a young agent who belongs to a two-breadwinnercouple becomes a decisive voter if 2' + (1 + ) > (3 + )=2.
For each condition, the …rst term on the left-hand side shows the number of young
agents who prefer the highest tax rate among the young, the second term, 1+ , shows thenumber of old agents, and the term on the right-hand side, (3 + )=2, shows a half of the
population size. The above-mentioned three conditions are summarized as in Assumption
1(ii).
Hereafter, we will focus on the parameter , which represents the inverse of the intertemporal elasticity of substitution, and consider two cases separately: a high elasticity
case (1=1) and a low elasticity case (1= < 1). We adopt this classi…cation because

the order of preferences for the tax rate depends critically on the degree of elasticity.
Because the former case includes the case of Leroux, Pestieau, and Racionero (2011) as a
special one, we leave it to Appendix A.6. In what follows here, we will focus exclusively
on the latter case.

5.2

An Economy with 1= < 1

To determine the decisive voter over when 1= < 1, we recall the conditions (4), (5), (6),and (7), that determine the preferred tax rates by four types of households. The graphsof these conditions when 1= < 1 are illustrated in Figure 1. The main feature is thatRHS j and RHS k (j 6= k) intersect at some tax ratej

2 (0; 1=2). In an economy with

1= < 1, the slope of RHS becomes steeper the lower the marginal cost-to-bene…t ratio
becomes when a type-j household is borrowing-constrained. To investigate in detail theproperty of the political equilibrium when 1= < 1, we hereafter consider in turn threecases, (a), (b), and (c), classi…ed according to the level of derived pension right as in (8),and then summarize the results of the three cases to provide a global characterization of
the political equilibrium.
[Figure 1 about here.]
5.2.1

Low Level of Derived Pension Rights: 0

1
1+

Panel (a) of Figure 1 illustrates the conditions (4), (5), (6), and (7) that determine thepreferred tax rates of the four types of households when the level of derived pension
11

rights is low such that

2 [0; (1)=(1 + )]. As depicted in the …gure, there are three

critical values of , ~f;c2 2 ^f ; ^c2 , ~c2;c1 2 (^c2 ; ^c1 ), and ~c1;m 2 (^c1 ; ^m ), such thatRHS f and RHS c2 intersect at = ~f;c2 , RHS c2 and RHS c1 cross at = ~c2;c1 , andRHS c1 and RHS m intersect at
f;c2

~

c1;m

~

1

1

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

c2;c1

; ~

()

1

R

= ~c1;m . By direct calculation, we obtain1+( + 1) ( )

1

;

1

1

()

1

:

( + ) ()The tax rate preferred by a type-j household is determined by the crossing point of
LHS and RHS j . Given the assumption of household distribution in Assumption 1(ii),the decisive voter overis the one who prefers the highest tax rate among the young

households. Based on the illustration in Panel (a) of Figure 1, the decisive voter overwhen 1= < 1 and 2 0; 11+
is determined as follows.
Lemma 1. Suppose that 1=

2 0; 11+< 1 andhold. There exists a uniqueequilibrium of the voting game with

2 (0; 1=2). The decisive voter over(j) a type-f single female agent if()

1(1+ )R
+1

2

1

( )

is

+ (1++1)R ;(ii) a type-c2 agent who belongs to a two-breadwinner couple, otherwise.
Proof. See Appendix A.2.
The result in Lemma 1 indicates that (i) a type-m or type-c1 agent never becomesa decisive voter; and (ii) the decisive voter is a type-f single female agent if the gender1

wage gap is small such that ( )2(1+ )R+11( )+ (1++1)R ; otherwise, the decisive

voter is a type-c2 agent who belongs to a two-breadwinner couple.
To understand the mechanism behind the …rst result, suppose that a type-m or typec1 agent is a decisive voter. Based on the observation in Panel (a) of Figure 1, theequilibrium tax rate must be high so that> ~c2;c1 . For this high tax rate, the marginalcost-to-bene…t ratios of taxation for type-m and type-c1 agents, which are increasing in
the tax rate, are always larger than the marginal e¢ ciency loss of taxation, 1 2 , which
is decreasing in the tax rate. The marginal ratio and e¢ ciency loss of taxation are neverbalanced for> ~c2;c1 for type-m and type-c1 agents. Therefore, neither of the agents

becomes a decisive voter.
Next, to understand the mechanism behind the second result, recall the condition that
produces an equilibrium in which a type-f single female agent becomes a decisive voter
in Lemma 1. The condition is rewritten as
1
1

'(1

1+
'(1

)(1
+1

)
)

1

2 (R )

1( + 1)
12

1

1

1+

+ R;

(9)

where the left-hand side is decreasing in, and the right-hand side is increasing in.

Therefore, the condition (9) states that a type-f agent is more likely to become a decisive
voter when is higher.
Single females owe greater tax burden as
higher). However, an increase inbecomes higher (i.e., as their wage becomes

results in a lower marginal cost-to-bene…t ratio andthus a higher preferred tax rate by single females when 1=< 1 as demonstrated in

Subsection 4.2. Therefore, type-f agents prefer a higher tax rate than type-c2 agents
when is high such that (9) holds.
5.2.2

Medium Level of Derived Pension Right: 11+

1<

Panel (b) of Figure 1 illustrates the conditions (4), (5), (6), and (7) that determine thepreferred tax rates by the four types of households when the level of derived pension rightis medium such that 2 ((1)=(1 + ); (1)= ]. As depicted in the …gure, there
f;c1
f
c1
c1;c2
are three critical values of , ~2 ^ ;^ , ~2 (^c1 ; ^c2 ), and ~c2;m 2 (^c2 ; ^m ),such that RHS f and RHS c1 intersect atand RHS c2 and RHS m intersect at
f;c1

~

c2;m

~

R
( + ) ()
1R()

= ~f;c1 , RHS c1 and RHS c2 cross at= ~c2;m . By direct calculation, we obtain
1

1

R (1 + )( + 1) ( )

c1;c2

; ~

()
1+
( + 1) ( )= ~c1;c2 ,111( + ) ();

1:

As in the previous case, we can characterize the political equilibrium for the case of
1= < 1 and

2

1
1+

;1

as follows.

Lemma 2. Suppose that 1= < 1 and

2

1
1+

;1

hold. There exists a uniqueequilibrium of the voting game with

2 (0; 1=2). The decisive voter over(j) a type-f single female agent if

()

1

2

R
+

( )

1

+

R
+

is

;

(ii) a type-c1 agent who belongs to a one-breadwinner couple otherwise.
Proof. See Appendix A.3.
The main di¤erence from the previous case is that a type-c2 agent cannot be a decisive
voter; instead, a type-c1 agent can be a decisive voter under a certain condition. A changeof the decisive voter is due to that, in the current case, a type-c1 agent attains a higher
pension bene…t because of a higher level of derived pension right relative to the previous
case. This relationship results in a higher marginal cost-to-bene…t ratio of taxation for a
type-c1 agent compared to a type-c2 agent. Therefore, a type-c1 agent prefers a higher
tax rate than does a type-c2 agent.
13

In the current case, the decisive voter is a type-f or a type-c1 agent. The former
becomes a decisive voter if the condition in Lemma 2 holds. The left-hand side of thecondition is decreasing in , while the right-hand side is increasing in . An interpretation
similar to that of the previous case applies to the current case. For a higher , a type-f ,single female agent is more likely to become a decisive voter.
High Level of Derived Pension Right: 1

5.2.3

<

1

Panel (c) of Figure 1 illustrates the conditions (4), (5), (6), and (7) that determine thepreferred tax rates by four types of households when the level of derived pension rights
is high such that 2 ((1)= ; 1]. As depicted in the …gure, there are three criticalvalues of , ~c1;f 2 ^c1 ; ^f , ~f;c2 2 ^f ; ^c2 , and ~c2;m 2 (^c2 ; ^m ), such that RHS c1 and

RHS f intersect at= ~c1;f , RHS f and RHS c2 intersect at

RHS m intersect at= ~c2;m . By direct calculation, we obtain~~(1 + )R( + 1) ( )

f;c2

; ~

( + ) ()
1

R

c2;m

1

1

R
()

c1;f

()

1+
( + 1) ( )= ~f;c2 , and RHS c2 and

1

1

;

()

1

:

As in the previous two cases, we can characterize the political equilibrium for the caseof 1= < 1 and

2 ((1)= ; 1] as follows:

Lemma 3. Suppose that 1=< 1 andequilibrium of the voting game with2

1

; 1 hold. There exists a unique

2 (0; 1=2). The decisive voter over(j) a type-c1 agent who belongs to a one-breadwinner couple if()is2( R )

1

1

1
+

+ R;

(ii) a type-f single female agent otherwise.
Proof. See Appendix A.4.
A set of possible decisive voters is unchanged from case (b): the decisive voter is a
type-f or a type-c1 agent depending on . However, the qualitative impact of theseparameters on the determination of the decisive voter is completely reversed. In case (b),a type-f agent is more likely to be a decisive voter for a higher . In contrast, she is more
likely to be a decisive voter for a lower

in the current case.

The di¤erence between the two cases is due to the di¤erent levels of derived pension
right. The level of derived pension right in the current case is higher than that in case(b). This di¤erence implies that whenis high enough so that both types of agents areborrowing-unconstrained, the marginal cost-to-bene…t ratio of taxation for type-c1 agents14

is lower than that for type-f agents. Therefore, type-c1 agents prefer the highest tax
rate among the young and become the decisive voter if is high such that the condition
in Lemma 3 holds. In contrast, when

is low enough so that both types of agents areborrowing-constrained, the order of the ratios is reversed because of a low intertemporalelasticity of substitution. The ratio for type-f agents becomes lower than that for the
type-c1 agents; the former prefer the highest tax rate among the young and becomedecisive voters.
A Decisive Voter When 1= < 1

5.2.4

The results established so far are summarized as follows.
Proposition 1. Suppose that 1= < 1 holds. There exists a unique equilibrium of thevoting game with

2 (0; 1=2). The decisive voter overis(i) a type-c1 agent who belongs to a one-breadwinner couple if
1() > 2

R
+

( )

1

+

R
+

; or if

1

1
+

2

1

; 1 and1(1+ )R
+1

1
1+

;1

and

()

2( R )

2 0; 11+

and

1

+ R;

(ii) a type-c2 agent who belongs to a two-breadwinner couple if
2

2

( )

1

()>

+ (1++1)R ;(iii) a type-f single female agent otherwise.
Proof. It is immediate from Lemmas 1-3.
The fraction of two-breadwinner couples, denoted by, has a crucial e¤ect on thedetermination of a decisive voter when 1= < 1. The result in Proposition 1 states thatfor an economy with 1= < 1, a type-c2 agent who belongs to a two-breadwinner couple
becomes a decisive voter if (a) the level of derived pension right is set at a low level, and(b) the bene…t-burden ratio of public pensions in the economy, denoted by ( ), is high.
Because ( ) is increasing in , the latter condition implies that a type-c2 agent is more
likely to be a decisive voter when the fraction of type-c2 agents is larger in the economy.

6

Gender Wage Gap, Derived Pension Rights andthe Fraction of Two-breadwinner Couples

Given the characterization of the political equilibrium in Section 5, we investigate how
the tax rate changes in response to recent trends in developed economies: a reduction of
the gender wage gap, a reduction of derived pension rights, and an increase in the fraction15

of two-breadwinner couples. The aim of the analysis is to predict the direction of futurechange in the tax rate in response to these trends. The analysis also aims to explore theroles of the borrowing constraint and the intertemporal elasticity of substitution on thedetermination of the tax rate.

6.1

A Reduction of the Gender Wage Gap

First, we investigate the e¤ect of a reduction of the gender wage gap on the determinationof the tax rate.
Proposition 2: In an economy with 1= < 1 where the decisive voter is a type-j( j = f; c1, or c2) agent, a reduction of the gender wage gap (i.e., an increase in )locally produces an inverse U-shaped relationship between

andaround ^j .

Proof. See Appendix A.5.
To understand the mechanism behind the result in Proposition 2, we …rst consider thee¤ect of an increase in on the marginal cost-to-bene…t ratio of taxation, denoted by
RHS j (j = f; c1; c2), when an agent is borrowing-unconstrained. After some calculation,
we obtain@RHS c1@RHS c2@RHS f> 0;< 0;> 0 if sj > 0 (j = f; c1; c2).
@@@
When an agent is borrowing-unconstrained, a reduction of the gender wage gap (i.e.,
an increase in ) increases the marginal cost-to-bene…t ratio of taxation for type-f andtype-c2 agents, whereas it decreases the ratio for type-c1 agents. The di¤erence is due tothe fact that single females and two-breadwinner couples owe an additional tax burden
when there is an increase in females’ wages, whereas one-breadwinner couples owe no
additional burden. Because increased tax revenue is returned to all types of agents as
lump-sum pension bene…ts, single females and two-breadwinner couples pay more thanthey receive, whereas one-breadwinner couples pay nothing but receive additional bene…ts.
Therefore, a reduction of the gender wage gap results in a higher cost-to-bene…t ratio of
taxation and thus a lower preferred tax rate for type-f and type-c2 agents, whereas it
results in a lower cost-to-bene…t ratio and thus a higher preferred tax rate for type-c1
agents.
The opposite result holds when an agent is borrowing-constrained because the e¤ectof on RHS j is reversed as demonstrated in Subsection 4.2. A reduction of the genderwage gap results in a lower RHS j (j = f; c2) and thus in a higher preferred tax rateby type-f and type-c2 agents, whereas it results in a higher RHS c1 and thus in a lower
preferred tax rate by type-c1 agents. Therefore, there is an inverse U-shaped relationshipbetween and(see Figure 2).

jaround the critical value of the tax, ^j , that divides the status of saving
16

[Figure 2 about here.]

6.2

A Reduction of Derived Pension Rights

Next, we will consider the e¤ect of a reduction of derived pension rights on the determination of the tax rate.
Proposition 3: In an economy with 1=< 1 where the decisive voter is a type-j( j = f; c1, or c2) agent, a reduction of derived pension rights (i.e., a decrease in) produces an inverse U-shaped relationship between

around ^j .

andProof. See Appendix A.5.
To understand the result in Proposition 3, we …rst consider the e¤ect of the derived
pension rights on the marginal cost-to-bene…t ratio of taxation when an agent is borrowing
unconstrained:@RHS c1@RHS c2@RHS f< 0; ( 1)> 0; ( 1)< 0 if sj > 0 (j = f; c1; c2)
( 1)
@
@
@
We multiply the derivatives by ( 1) to demonstrate the qualitative e¤ect of a decreasein . A reduction of derived pension rights (i.e., a decrease in ) increases the pension
bene…ts for type-f and type-c2 agents, lowers the cost-to-bene…t ratio of taxation, andthus raises their preferred tax rate. In contrast, such a reduction decreases the pension
bene…ts for type-c1 agents, raises their cost-to-bene…t ratio of taxation and thus lowers
their preferred tax rate. However, the opposite result holds when an agent is borrowingconstrained as demonstrated in the previous subsection. Therefore, the two opposinge¤ects result in an inverse U-shaped relationship between
value of the tax, ^j .

6.3

and

j

around the critical

An Increase in the Fraction of Two-breadwinner Couples

Finally, we examine the e¤ect of the share of two-breadwinner couples on the equilibriumtax rate.
Proposition 4: In an economy with 1=< 1 where the decisive voter is a type-j( j = f; c1, or c2) agent, an increase in the fraction of two-breadwinner couples(i.e., an increase in ) (a) locally produces an inverse U-shaped relationship between
andifaround ^j if

= (1

6= (1

)=(1 + ); (b) has no e¤ect on the equilibrium tax)=(1 + ).

Proof. See Appendix A.5.
17

The result in Proposition 4 can be understood by focusing on the e¤ects ofon themarginal cost-to-bene…t ratio of taxation, denoted by RHS j . When agents are borrowing
unconstrained, the e¤ects are summarized as follows:@RHS f@RHS c1@RHS c2
R 0;
R 0;
R 0 if and only if
@
@
@

Q

1
1+

:

The marginal cost-to-bene…t ratio of taxation, RHS j (j = f; c1; c2), is a¤ected by anincrease in via the bene…t-to-burden ratio of public pension, denoted by ( ), in thefollowing two ways. On one hand, an increase in

results in an increase of tax revenue

from the working females and thus in an increase per capita pension bene…t. This e¤ect
is observed in the numerator of ( ) in (1), which yields a negative e¤ect on RHS j . On
the other hand, an increase in implies a larger size of two-breadwinner couples and asmaller size of one-breadwinner couples. Because the level of pension rights is larger forworking females than for non-working females, an increase in

results in a smaller per

capita pension bene…ts. This e¤ect is observed in the denominator of ( ) in (1), whichyields a positive e¤ect on RHS j .
As observed in (1), the former e¤ect is independent of , whereas the latter e¤ect
is dependent of . Speci…cally, the latter e¤ect becomes larger as

becomes smaller.

Therefore, the latter e¤ect overcomes the former one when the level of derived pension
rights is low such that < (1)=(1 + ). That is, RHS j is increased by an increase
in

if

< (1

)=(1 + ). The opposite result holds if

opposing e¤ects are o¤set by each other if

= (1

> (1

)=(1 + ). The two)=(1+ ). These e¤ects are reversed

when an agent is borrowing-constrained as demonstrated in Subsection 4.2. Therefore,
there is an inverse U-shaped relationship between and j around the critical value ofthe tax, ^j .

7

Concluding Remarks

This paper developed an overlapping-generation model based on that of Leroux, Pestieau
and Racionero (2011). The model includes four types of households: single female, single male, one-breadwinner couple and two-breadwinner couple. The paper introduced aborrowing constraint into their model and generalized the model by assuming a utility
function with a constant intertemporal elasticity of substitution. Under this generalized
framework, we consider majority voting over public pension policy in the presence of
derived pension rights, and investigate how the borrowing constraint and intertemporalelasticity of substitution a¤ect the preferences of each household over pension and theresulting equilibrium pension policy.
The paper showed the following two results. First, in an economy where an intertemporal elasticity of substitution is below one, one-breadwinner couples may prefer a lower,
18

rather than higher, tax rate than do single females because of the presence of borrowing
constraints. There is an equilibrium, much like an ends-against-the-middle equilibrium,
where the old and single females form a coalition against the others.
Second, the gender wage gap, the level of derived pension rights, and the fractionof two-breadwinner couples create an inverse U-shaped relationship between the relevant
variable and the tax rate when the intertemporal elasticity of substitution is below one.
This two-toned e¤ect was derived via a borrowing constraint associated with a low intertemporal elasticity of substitution.
Throughout the analysis, we assumed that the degree of derived pension rights is…xed. This assumption can be relaxed by assuming a structure-induced Nash equilibriumof voting (for example, Conde-Ruiz and Galasso, 2003; 2005; Casamatta, Cremer andPestieau, 2005; Conde-Ruiz and Profeta, 2007; Bethencourt and Galasso, 2008). In thisvoting equilibrium, one-breadwinner couples prefer a full derived pension right, whereas
others prefer no right. Thus, the full derived pension right is realized if the number of
one-breadwinner couples is larger than a half of the population; no derived pension rightis realized otherwise. However, in the real world, the degree of the derived pension rightis set between these two extreme solutions. There is a need to add an institutional feature
to demonstrate a more realistic situation: this task is left as future work.

19

A

Appendix

A.1

Single-peaked Preferences of the YoungIn this appendix, we prove that preferences of a type-f young agent are single peaked.
The proof applies to other types of young agents.
f
f
are single
and Vs=0
The proof proceeds as follows. First, we show that both Vs>0ffff
peaked over . Then we demonstrate that @Vs>0=@ = @Vs=0=@ and Vs>0
= Vs=0hold

at = ^f , implying that V f has a unique local maximum over the whole range ofthus that V f is single peaked over .

and

y;j
y;j
with respect to are
and Vs=0
The …rst and the second derivatives of Vs>0
!
f
@Vs>0
R
w ()
w ()
=
(1
) w+
(1
)
w+
(1 2 ) ;1=@
R
R
R + ( R)!
1f
@ 2 Vs>0
R
w ()
w ()
(1
)
(1
=
(
)
(1
)
w
+
w
+
@ 2
R
R
R + ( R)1=
!
w ()
w ()
R
(1
)
( 2)
(1
) w+
+
1=
R
R
R + ( R)

< 0;
f
@Vs=0

= [(1
) w]
( w) +[w ( ) (1@f@ 2 Vs=01= ( ) [(1
) w]
( w)2
2
@
1
[w ( ) (1)](w ( ))2 (12[w ( ) (1)])]

w ( ) (1

2 );

2 )2

w ()

< 0:
f
f
The functions Vs>0and Vs=0
are single peaked overbecause the second derivatives arenegative.
ffff
Next, we show that @Vs>0=@ = @Vs=0=@ and Vs>0
= Vs=0hold at= ^f . By direct

calculation, we have:
y;j
Vs>0

=^f

y;j
= Vs=0

=
y;j
@Vs>0
@

1
1

y;j
@Vs=0
=
@
f

=^

=

1

=^f

R + ( R)1=
R=^f

( R)1=
()

!

1

( w)1
20

( R)1=
()

!1

1+

( w)1

1+
;
1

()
2
+ ( R)1=
R
R

:

2

2 )

y;j
y;j
With this result and the single-peakedness of Vs>0and Vs=0
over , we can conclude thatV f has a unique local maximum with respect to over the whole range of . Speci…cally,ff
V f is maximized at= arg max Vs>0
if arg max Vs>0
< ^f ; it is maximized at=fotherwise.
arg max Vs=0

A.2

Proof of Lemma 1

1st step. We …rst show that a type-j (j = c1; m) agent cannot be a decisive voter. To
show this, suppose that an agent who belongs to a type c1 or type m household is a
decisive voter. From Panel (a) of Figure 1, it must be that LHS > RHS c2 atthat is,R
1 2~c2;c1 >:( + ) ()
By rearranging this condition, we obtain:
12R( + ) (),( + )

1

1

1+( + 1) ( )R+( ) > 2( + )>1= ~c2;c1 ;R( + ) ()

1+
+1

1

+ R:

(10)

The left-hand side of (10), ( + ) ( ), is less than one under the current assumption of
2 0; 11+.2 Conversely, the right-hand side is greater than one under the assumptionof R > 1. Therefore, the condition (10) fails to hold, implying that an agent who belongs
to a type j (j = c1; m) household never becomes a decisive voter.
2nd step. Given the result in the …rst step, a candidate for the decisive voter is anagent who belongs to a type f or type c2 household. From Panel (a) of Figure 1, a
type-f agent becomes a decisive voter if and only if
12~f;c2(1 + )R;( + 1) ( )that is, if and only if()

2(1 + )R
+1

1

( )

+

(1 + )R:+1'(1)] 7 1

1

Otherwise, a type-c2 agent becomes a decisive voter.

2

The condition ( + ) ( ) ? 1 is rewritten as( + ) ()?1
, [( 1)(1) + (1 + )] [1

where the left-hand side in hthe second
i line is negative because ( 1)(11current assumption of 2 0; 1+ . Thus, we obtain ( + ) ( ) < 1.

21

) + (1 + ) < 0 holds by theA.3

Proof of Lemma 2

1st step. We …rst show that a type-j (j = c2; m) household cannot be a decisive voter.
To show this, suppose that a type j(= c2; m) be a decisive voter. From Panel (b) of
Figure 1, it must be that LHS > RHS c1 at
2~c1;c2 >1= ~c1;c2 , i.e.,
R(1 + ):
( + 1) ( )By rearranging this condition, we obtain( + 1) ( )
+1
>
2
1+
1+

R (1 + )
+1

1

1
+

1

+ R:

(11)

The left-hand side of (11) is less than one under the current assumption of21;1, while the right-hand side is greater than one under the assumption of R > 1.
1+
Therefore, (11) fails to hold, implying that a type j (j = c2; m) agent never becomes adecisive voter.
2nd step.
Given the result in the …rst step, a decisive voter is an agent who belongs to a type-for type-c1 household. From Panel (b) of Figure 1, a type-f agent becomes a decisive voter
if and only if
12~f;c1

2

R
+

or
()

R
;
( + ) ()
1

( )

1

R
:
+

+

Otherwise, a type c1 agent becomes a decisive voter.

A.4

Proof of Lemma 3

1st step.
We …rst show that a type j(= c2; m) agent cannot be a decisive voter. To show this,
suppose that a type j(= c2; m) agent is a decisive voter. From Panel (c) of Figure 1, it
must be that LHS > RHS f at

= ~f;c2 , i.e.,
12~f;c2 >(1 + )R;( + 1) ( )which is rewritten as( + 1) ( )>21+(1 + )R( + 1) ( )
22

1

( )

1

+1
+ R:
1+

(12)

The left-hand side of (12) is less than one under the assumption of21; 1 , while

the right-hand side is greater than one under the assumption of R > 1. Therefore, (12)fails to hold, implying that a type j(= c2; m) agent never becomes a decisive voter.
2nd step.
Given the result in the …rst step, a decisive voter is a type j (= c1; f ) agent. A
type-c1 agent becomes a decisive voter if and only if
1

R
;
()

2~c1;for()2( R )

1

1
+

1

+ R:

Otherwise, a type f agent is a decisive voter.

A.5

Proof of Propositions 2-4

First, we focus on the terms = ( ); 1=( + ) ( ) and (1 + )=( + 1) ( ) that a¤ect themarginal cost-to-bene…t ratios of taxation for type-f; c1 and c2 agents, respectively: seeequations (4), (6) and (7). We denote these terms as:
]
RHS

f

()

]
; RHS

c1

1
( + ) ()

]
; RHS

c2

1+
.
( + 1) ( )j] (j = f; c1; c2):
After some calculation, we obtain the following properties of RHS
f

c1

c2

]
]
]
@ RHS
@ RHS
@ RHS
> 0;
< 0;
> 0;
@
@
@
f
c1
c2
]
]
]
@ RHS
@ RHS
@ RHS
> 0;
< 0;
> 0;
@
@
@
f

]
]
@ RHS
@ RHS
R 0;
@
@

c1

In what follows, we denote

]
@ RHS
R 0;
@
j

(13)
(14)

c2

R 0 if and only if

Q

1
1+

:

(15)

(j = f; c1; c2) as the tax rate preferred by a type-j

agent.
A.5.1

The case of 1= < 1

Consider the political equilibrium in an economy with 1=< 1. The decisive voter inthe current case is a type-f , type-c1 or type-c2 agent (Proposition 1). Because we here

23

consider the e¤ects of;around the critical value ^j de…ned in Section 3, weandcalculate the e¤ects of these parameters on ^j as follows:
@ ^f
@ ^c1
@ ^c2
> 0;
< 0;
> 0;
@
@
@
@ ^f
@ ^c1
@ ^c2
> 0;
< 0;
> 0;
@
@
@
@ ^f
@ ^c1
@ ^c2
R 0;
R 0;
R 0 if and only if@@@(i) The e¤ect of

(16)
(17)
Q

1
1+

(18)

:

on the equilibrium tax rate: Proof of Proposition 2 Consider…rst the equilibrium where the decisive voter is a type-f agent. Suppose that is initiallygiven such that type-f ’s preferred tax rate is = ^f . We denote f as the that makes
a type-f young agent choose = ^f .
f
] in (13) and the property of ^f in (16), we …nd a positive
With the property of RHS
real number "(> 0) around f such that the type-f agent is borrowing-constrained for
2 ( f "; f ) and borrowing-unconstrained for 2 [ f ; f + "). The equilibrium tax
rate satis…es the following condition:
(
LHS

1

2 = RHS

f

1

1

for
for

( )

()
()

R

2( f
2 f;

"; f ];f+" :Given the properties in (13) and (16), we can illustrate the e¤ects of an increase in
on RHS f and ^f as in Figure 2. The illustration leads to the following result:
@ f
@
@ f
@

> 0 for< 0 for

The result shows that an increase inbetweenandfaround2( f
2 f;"; f );f+" :locally produces an inverse U-shaped relationship

f

=^ .

The analysis and result apply to the equilibrium in which the decisive voter is a typej] and ^j are qualitatively similar between thec2 agent because the e¤ects of on RHS
two types of agents, as demonstrated in (13) and (16).
Next, consider the equilibrium where the decisive voter is a type-c1 agent. Supposethat is initially given such that type-c1’s preferred tax rate is = ^c1 . We denote c1
as the that makes a type-c1 young agent choose = ^c1 .
c1] in (13) and the property of ^c1 in (16), we …nd a positive
With the property of RHS
real number "(> 0) aroundc1

c1

such that the type-c1 agent is borrowing-unconstrainedc1

for 2 ("; ) and borrowing-constrained fortax rate satis…es the following condition:(
1R( + ) ()c1
1
LHS 1 2 = RHS
11( + ) ()

24

2 [ c1 ;( )c1+ "). The equilibriumfor

2 ( c1";for

2 [ c1 ;

c1

c1

);

+ "):

Given the properties in (14) and (17), we obtain the following result:
@ c1
>0
@
@ c1
<0
@

The result shows that an increase inbetweenandc1

around2 ( c12 ( c1 ;forfor"; c1 );
c1+ ") :

locally produces an inverse U-shaped relationship

c1

=^ .

on the equilibrium tax rate: Proof of Proposition 3 Suppose(ii) The e¤ect ofthat the decisive voter is a type-j (j = f; c1; c2) agent. Suppose thatsuch that type-j 0 s preferred tax rate is= ^j . We denotejas theis initially giventhat makes a type-jjagent choose = ^ .
Under the abovementioned situation, suppose that an increase inaroundjlocally

produces an inverse U-shaped relationship between

and type-j’s preferred tax rate. Thisjlocally produces an inverse U-shaped

assumption implies that a decrease inaroundrelationship between and type-j’s preferred tax rate. Therefore, it is su¢ cient to show
the e¤ect of an increase in on the preferred tax rate by the decisive voter.
The analysis of the e¤ect of applies to the current analysis because the e¤ects ofjj] and ^j are qualitatively similar to those of on RHS] and ^j . Therefore, weon RHSobtain the result described in Proposition 3.
(iii) The e¤ect ofthaton the equilibrium tax rate: Proof of Proposition 4 Suppose)=(1 + ) holds. The decisive voter is a type-f or type-c2 agent (Lemmaj] and ^j (j = f; c2) are qualitatively similar to those of on1). The e¤ects of on RHS< (1j] and ^j (j = f; c2). We can apply the analysis and result in Proposition 2 to theRHScurrent case.
Next, suppose that)=(1 + ) holds. The decisive voter is a type-f orj] and ^j (j = f; c1) area type-c1 agent (Lemmas 2 and 3). The e¤ects of on RHS> (1]qualitatively similar to those of on RHS
result in Proposition 2 to the current case.

c1and ^c1 . We can apply the analysis andFinally, suppose that = (1)=(1 + ) holds. The parameter has no e¤ect onjj] and ^ . A change in has no e¤ect on the equilibrium tax rate.
RHS

A.6

The Case of a High Intertemporal Elasticity of Substitution: 1=
1

Figure 3 illustrates the conditions (4), (5), (6), and (7) that determine the preferred
tax rates by single females, single males, one-breadwinner couples, and two-breadwinner25

couples, respectively, for the case of 1=

1. Speci…cally, panels (a), (b), and (c) cor-

respond to the cases (a), (b), and (c) in (8), respectively. The left-hand side of eachcondition, denoted by LHS, is decreasing in , independent of the type, and featured
with LHSj =0 = 1 and LHSj =1=2 = 0. In contrast, the right-hand side of each condition, denoted by RHS j (j = f; m; c1 and c2), is non-decreasing in , dependent on thetype of a household, and characterized byCase (a): RHS f
Case (b): RHS f
Case (c): RHS c1

RHS c2
RHS c1
RHS f

RHS c1
RHS c2
RHS c2

RHS m if
RHS m if
RHS m if

0
1
1+
1

<
<

1
1+
1

;
1;

where equality holds if and only if 1= = 1 and sj = 0. The kink point ofthat a type-j household can save part of its income if;= ^j implies< ^j and nothing if^j .

[Figure 3 about here.]
The crossing point of LHS and RHS j determines the tax rate preferred by a young
agent who belongs to a type-j household. Under Assumption 1(ii), the decisive voter is theone who prefers the highest tax rate among the young. From the observation in Figure 3,
we can conclude that the decisive voter is a single female (j = f ) agent if(1)=(see panels (a) and (b)); this voter is an agent who belongs to a one-breadwinner couple(j = c1) if > (1)= (see panel (c)).
Proposition A1. Suppose that 1=
voting game with

1 holds. There exists a unique equilibrium of the

2 (0; 1=2). The decisive voter over(i) a type-f , single female agent if

1

is

;

(ii) a type-c1 agent who belongs to a one-breadwinner couple otherwise.
The result established in Proposition A1 has the following two features. First, a single
male or an agent who belongs to a two-breadwinner couple does not become a decisive
voter. Such agents’ marginal cost-to-bene…t ratios of taxation in terms of utility arealways higher than those of the other two types of households. This result implies thatsingle males and two-breadwinner couples prefer a lower tax rate than do other two typesof young agents. The decisive voter therefore is a single female or an agent who belongs
to a one-breadwinner couple.
Second, which household becomes a decisive voter depends on ; andthat represent

the gender wage gap, life expectancy of men, and the fraction of derived pension rights,
respectively. Suppose that the gender wage gap is high (i.e.,

is low), the life expectancy

of men ( ) is low, and the level of derived pension rights ( ) is low such that26)= . Then, the marginal cost-to-bene…t ratio of taxation in terms of utility for(1single females is lower than that for one-breadwinner couples because the former owe lesstax burden whereas the latter receive lower pension bene…ts. Therefore, single females
prefer a higher tax rate than do one-breadwinner couples and thus become decisive voters
if

(1

A.6.1

)= .
Comparative Statics Analysis

Consider the political equilibrium in an economy with 1=

1. Suppose that)= holds: the decisive voter is a type-f agent (Proposition A1). The optimalityf] results in a lower
condition for a type-f agent, given by (4), indicates that a higher RHS(1

preferred tax rate except for the case of 1= = 1 and sf = 0:
( @ f
= 1 and sf = 0;
f = 0 if 1=
]
@ RHS

@ f
f
]
@ RHS

< 0 otherwise.

f] in (13) - (15), we obtain the following result:With the property of RHS
0;

@ f
@

0;

@ f
Q0,
@

Q

1
1+

@ f
@
and

:

Next, suppose that> (1)= holds: the decisive voter is a type-c1 agent(Proposition A1). The optimality condition for a type-c1 agent, given by (6), indicatesc1] results in a lower preferred tax rate except the case of 1= = 1 andthat a higher RHS
sc1 = 0:(]With the property of RHS@ c1= 1 and sc1 = 0;c1 = 0 if 1=
]
@ RHS
@ c1
c1 < 0 otherwise.
]
@ RHS
c1

in (13) - (15) and the assumption ofobtain the following result:
@ c1
@

0;

@ c1
@

27

0;

@ c1
@

0:

> (1

)= , weReferences[1] Arawatari, R., and Ono, T., 2011. The political economy of social security
in a borrowing-constrained economy. Journal of Public Economic Theory, forthcoming. Available at http://www.accessecon.com/pubs/JPET/tempPDF/JPET-1000157_0_0_2144890_temp.pdf[2] Bellettini, G., and Berti Ceroni, C., 2007. Income distribution, borrowing constraints
and redistributive policies. European Economic Review 51, 625–645.
[3] Bethencourt, C., and Galasso, V., 2008. Political complements in the welfare state:
Health care and social security. Journal of Public Economics 92, 609–632.
[4] Borck, R., 2007. On the choice of public pensions when income and life expectancy
are correlated. Journal of Public Economic Theory 9, 711–725.
[5] Casamatta, G., Cremer, H., and Pestieau, P., 2000. The political economy of social
security. Scandinavian Journal of Economics 102, 503–522.
[6] Casamatta, G., Cremer, H., and Pestieau, P., 2005. Voting on pensions with endogenous retirement age. International Tax and Public Finance 12, 7–28.
[7] Choi, J., 2006. The role of derived rights for old-age income security of women.
OECD social, employment, and migration working papers No. 43. Available at
http://www.oecd.org/dataoecd/3/55/37817844.pdf[8] Conde-Ruiz, J.I., and Galasso, V., 2003. Early retirement. Review of Economic Dynamics 6, 12–36.
[9] Conde-Ruiz, J.I., and Galasso, V., 2004. The macroeconomics of early retirement.
Journal of Public Economics 88, 1849-1869.
[10] Conde-Ruiz, J.I., and Galasso, V., 2005. Positive arithmetic of the welfare state.
Journal of Public Economics 89, 933–955.
[11] Conde-Ruiz, J.I., and Profeta, P., 2007. The redistributive design of social security
systems. Economic Journal 117, 686–712.
[12] Cremer, H., De Donder, P., Maldonado, D., and Pestieau, P., 2007. Voting over typeand generosity of a pension system when some individuals are myopic. Journal of
Public Economics 91, 2041–2061.
[13] Diamond, P.A., Hausman, J.A., 1984. Individual retirement and savings behavior,
Journal of Public Economics 23, 81–114.
28

[14] Leroux, M.L., and Pestieau, P., 2011. The political economy of derived pension rights.
International Tax and Public Finance: DOI 10.1007/s10797-011-9205-9.
[15] Leroux, M.L., Pestieau, P., and Racionero, M., 2011. Voting on pensions: Sex andmarriage. European Journal of Political Economy 27, 281–296.
[16] Mulligan, C.B., Sala-i-Martin, X., 1999. Social security in theory and practice (I):facts and political theories. NBER Working Papers 7118, National Bureau of Economic Research.
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G20 Countries. OECD, Paris.
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29

Panel (a). 0 ≤ γ ≤(∙)( + ) (∙)
(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2̂̂τ ,Panel (b).

,

τ

,

τ

τ
̂

̂

<γ≤

(∙)
(1 + )( + 1) (∙)( + ) (∙)

LHS ≡ 1 − 2(∙)̂
Panel (c).

<

̂

τ ,

̂
τ

τ

,

,

τ
̂

≤1

(∙)
(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2( + ) (∙)

̂
̂
τ

,

̂
τ ,

Figure 1. The preferred tax rates when

̂
τ

τ

,

< 1. Panel (a) is the case of

0 ≤ γ ≤ (1 − απ)/(1 + α); panel (b) is the case of (1 − απ)/(1 + α) <γ ≤ (1 − απ)/α; and panel (c) is the case of (1 − απ)/α < γ ≤ 1.
30

LHS ≡ 1 − 2̂τFigure 2. The figure illustrates the effect of an increase in α on theequilibrium tax rate in the case of 1/σ < 1 with a decisive voter j = f.

31

Panel (a). 0 ≤ γ ≤(∙)( + ) (∙)
(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2Panel (b).

τ
̂

̂

̂
̂

<γ≤

(∙)
(1 + )( + 1) (∙)( + ) (∙)

LHS ≡ 1 − 2

(∙)

̂
̂

Panel (c).

<

̂
̂

τ

≤1

(∙)
(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2( + ) (∙)

̂
̂

̂
̂

τ

Figure 3. The preferred tax rates when 1/σ ≥ 1. Panel (a) is the case of
0 ≤ γ ≤ (1 − απ)/(1 + α); panel (b) is the case of (1 − απ)/(1 + α) < γ ≤ (1 −απ)/α; and panel (c) is the case of (1 − απ)/α < γ ≤ 1.
32