Discussion Papers In Economics
And Business

Marital Status and Derived Pension Rights:
A Political Economy Model of Public Pensions
with Borrowing Constraints
Tetsuo Ono
Discussion Paper 11-32-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMarital Status and Derived Pension Rights:
A Political Economy Model of Public Pensions
with Borrowing Constraints
Tetsuo Ono
Discussion Paper 11-32-Rev.2

January 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMarital Status and Derived Pension Rights: A
Political Economy Model of Public Pensions with
Borrowing Constraints
Tetsuo Onoy
Osaka University
January 2013

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

The author would like to thank two anonymous referees for their valuable comments and suggestions
which greatly improved the paper. The author also thanks participants of the Osaka Workshop on
Economics of Institutions and Organizations. This research was supported by the Grant-in-Aid for
Scienti…c Research (C) from the Japan Society for the Promotion of Science (No. 24530346).
yGraduate 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; Leroux
and Pestieau, 2011). These bene…ts imply that derived pension rights have an intragenerational 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 anintra-generational con‡ict over pension 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 (2012) and Leroux, Pestieau, and Racionero (2011).
Leroux and Pestieau (2012) 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 (2012) 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 tothe gender wage gap, the level of derived pension rights, and the ratio of two-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 Environment1

2

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.
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 total population ofeach generation is divided as follows: (1') single females, ' females in couples whereof them are workers and 1of them are non-workers, (1 ') single males and 'males in couples. The allocation of households 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 ; m ) for malesand (wf ;f) for females, such that
(wm ; m ) = (w; );(wf ; f ) = ( w; 1);

where wi (i = f; m) represents the wage, andi

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

represents the probability of surviving in

old age. The term 2 (0; 1) represents the gender wage gap; an increase in implies a
reduction of the gender wage gap. 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 loss of e¢ ciency
in taxation by assuming convex costs for 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) [w + w(11') + w' ] ;Since the framework is based on that in Leroux, Pestieau, and Racionero (2011), a part of description
in this section follows Leroux, Pestieau, and Racionero (2011).

3

where the terms w, w(1'), 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.
Let p denote pension bene…ts for contributors; let p denote pension bene…ts for noncontributors, where 2 [0; 1] represents the level of derived pension rights. The total
pension payments are

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 becomes
p = w ( ) (1
where
()

(1)

);

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 7.2

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. An old agent does not make any economic
decisions because his/her saving is predetermined during 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:
Uj =(cj )11

1+

2(dj )1
1

1;

The paper makes several assumptions: the fraction of each type of household is …xed; the pension
bene…ts are equal for all contributors; and the derived pension rights are …xed. We employ these assumptions to keep the comparability with the result in Leroux, Pestieau, and Racionero (2011), and to shedlight on the roles of borrowing constraint and an intertemporal elasticity of substitution, both of whichwere abstracted away in Leroux, Pestieau, and Racionero (2011).

4

where cj is consumption in youth, dj is consumption in old age,

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.3
Type-j’s (j = f; m) individual budget constraints in youth and in old age are respectively given by,
c j + sj
dj
where sj is saving,

(1

)wj ;

sj +

j

p;

is the income tax 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 = . For thetractability of analysis, we assume that the gross rate of interest is equal to one. In
addition, we assume borrowing constraints, that is, sj

0. This constraint precludes

the possibility of borrowing when young against future pension bene…ts (Diamond andHausman, 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 = ( )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
(
"
#)
1=
j
(
)
p
sj = max 0;(1)wj; j = f; m:
(2)
1=
1+( )
( )1=
The saving function (2) and the government budget constraint (1) imply that there isa critical rate of tax such that
( )1=
f
f
;
s >0, <^
()
m

s >0,

m

<^

( )1=
:
()

The critical rate for single males, ^m , is higher than that for single females, ^f , 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) :
(
1
j
1
1
Vs>0
[(1
)wj + j w ( ) (1
)]
1=
j
1
1+(
)
V =
1
1
j
1
Vs=0
((1
)wj )
+ 1 f j w ( ) (1
)g
1
3

1+
1
1+
1

if

< ^j ;if^j :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.

5

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 + )=(1unrelated to the political decision on taxes.

3.2), summarizes the parameters

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(dj )1
1
Uj = 2+; j = c1; c2:
1
1
Under this speci…cation, spouses play cooperatively and share their resources over theirlifecycle.4
A couple chooses consumption and saving to maximize the household utility subject
to the budget constraints in youth and old age:2cj + sj
2dj
The borrowing constraint is sj(1

sj + ( +

)wj ;
j

)p:

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

c1

=

and

c2

= 1.

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, sj , the pension bene…t paid to the husband, p, and that to the wife,jp.

By taking the borrowing constraint into account, the saving function of the type-j
couple becomes
(
"
#)
1=
j
(
)
(
+
)p
sj = max 0;(1)wj; j = c1; c2
(3)
1=
1+( )
( )1=
The saving function (3) and the government budget constraint (1) imply that there is a4

An alternative model of a household is to assume that “members of a family cannot cooperate because
they cannot communicate with one another, and so the best that they can do is to behave according to
the de…nition of a Nash equilibrium” (Ermisch, 2003, p.22). This alternative modeling may produce a
result di¤erent from that achieved in the current model. However, we adopt the unitary model to keep
the comparability with the result in Leroux, Pestieau and Racionero (2011). The author would like tothank one of the referees for pointing out an alternative option.

6

critical tax rate for couples such that
c1

s >0,
c2

s >0,

c1

( )1=
;
( + ) ()

c2

(1 + ) ( )1=:( + 1) ( )<^<^With 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 couple’s indirect utility function:
j

V =

j
Vs>0
if
j
Vs=0 if< ^j ;^j ;(1)wj + ( +112where:
j
Vs>0

j
Vs=0

1
1
1
1

1
2

1
1=

1+( )

1
2

(1

)wj

!
1

+

j

)w ( ) (1( +j)w ( ) (1)

2 (1 + )
;
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 rate is 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, wheret 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
7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

4.1

oj

, is

oj

= 1=2 8j = f; m; c1; c2.

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
we should note that the following three properties hold. First, @ 2 Vs>0=@ 2 < 0 andjjj@ 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

=^

j
@Vs=0
@

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 maximum. In what follows, we derive the conditions that determine thetax rates preferred by four types of households.
First, consider a preferred tax rate by a young single female agent. Suppose that she
fprefers a positive tax rate: > 0. She chooses that satis…es @Vs>0=@ = 0 when she
is borrowing-unconstrained; andfthat satis…es @Vs=0=@= 0 when she is borrowing-

constrained. After some calculation, we can …nd that the preferred tax rate by single
females satis…es:
LHS

1

2 = RHS

f

(

()
1

1
()

( )

if

< ^f ;

if

^f :

(4)

The terms LHS and RHS f represent the marginal e¢ ciency loss of taxation and themarginal cost-to-bene…t ratio of redistribution, respectively. Single females choose the tax
rate that balances these terms.
Condition (4) determines the preferred tax rate by single females provided that they
prefer a positive tax rate. However, single females may want to prefer no taxation and8thus no redistribution via pension when the marginal cost of redistribution is larger thanf
= 0, that is, when @Vs>0=@

the marginal bene…t of redistribution at

0 (,
=0

(1
)= '(1)) holds. Therefore, the preferred tax rate by single females,
denoted by f , is summarized as:
f

=0
if
f
(> 0) satis…es (4)

1
'(1

;

)

otherwise.

m=@ j =00 always
Following the same procedure, we can immediately …nd that @Vs>0mm
5
holds. The preferred tax rate by single males, denoted by, becomes= 0:

The next task is to …nd preferred tax rates by couples. As in the case of a single female,
we …rst suppose that couples prefer a positive tax rate and compute their preferred tax
rate as follows. A preferred tax rate by one-breadwinner couples satis…es:(1
if< ^c1 ;( + ) ()c1
1
LHS 1 2 = RHS
11( ) if^c1 ;( + ) ()
and a preferred tax rate by two-breadwinner couples satis…es:
(
1+
LHS

1

2 = RHS( +1) ( )11+
( +1) ( )

c2

1

( )

if

< ^c2 ;

if

^c2 :

(5)

(6)

Taking into account of the case where couples prefer no redistribution, we …nd that apreferred tax rate by one-breadwinner couples, denoted byc1, satis…es:

1
;
=0
if
1+
c1
(> 0) satis…es (5) otherwise;c1

and a preferred tax rate by two-breadwinner couples, denoted byc2, satis…es:

1
=0
if
;
1+
c2
(> 0) satis…es (6) otherwise.
c2

Two remarks are in order. First, single males always prefer no taxation and thus
no redistribution because they always pay more than they receive. Second, the thresholdvalue of the derived pension rights for one-breadwinner couples coincides with that for twobreadwinner couples. However, the order of preferences of the one-breadwinner couples
is the total opposite to that of the two-breadwinner couples because the one-breadwinner
5

After some calculation, we obtain:
m
@Vs>0
@

where LHS < 0 < RHS.

=0

0,1

1
,(
|
()

1) (1 '(1
{z
LHS

9

))
}

'(1
) ;
| {z }
RHS

couples obtain more bene…ts whereas the two-breadwinner couples obtain fewer bene…ts
as the degree of derived pension right becomes higher.
The above-mentioned two properties are qualitatively identical to those demonstrated
by Leroux, Pestieau and Racionero (2011). However, by taking account of borrowing
constraint and an intertemporal elasticity of substitution, we can …nd non-monotonee¤ects of gender wage gap, derived pension rights and the fraction of two-breadwinnercouples on the equilibrium tax, which were not observed in Leroux, Pestieau and Racionero(2011). This point will be investigated in the following sections. Before going to the next
section, we would like to consider more in detail the roles of borrowing constraint and anintertemporal elasticity of substitution in the determination of a preferred tax rates.

4.2

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

The result in Section 4.1 presents the preferred tax rates by single females, single males,
one-breadwinner couples, and two-breadwinner couples, respectively. The intuition forthe corner solution is described in Section 4.1. To understand the reasoning behind theinterior solution, let us consider the single female’s condition, (4), as an example. In
particular, we focus on the parametersand , representing the gender wage gap andthe degree of derived pension rights, respectively. They are observed on the right-handside of (4); they a¤ect the marginal cost-to-bene…t ratio of redistribution.
There are two opposing e¤ects on the ratio via the terms and . First, given a tax
rate, an increase in(i.e., an increase in females’ wage) imposes a further tax burden

on a single female via the termon the numerator of the right-hand side in (4); andan increase in (i.e., an increase in the degree of derived pension rights) produces fewer
bene…ts for a single female via the term ( ). Greater burden and fewer bene…ts give her
an incentive to choose a lower tax rate and thus a smaller size of redistribution, resulting
in a negative e¤ect on the preferred tax rate. Second, an increase inaugments wage

income for single females, and thus pension bene…ts in old age, 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 positive e¤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, increases inandresult in a higher marginalcost-to-bene…t ratio of redistribution and thus a lower preferred tax rate when a young
single female is borrowing-unconstrained. The result holds regardless of 1= because the10objective for a borrowing-unconstrained household is to maximize lifetime income, whichis 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 of redistribution and thus a higher preferred tax rate in response to anincrease inor .

Which e¤ect outweighs the other depends on the degree of an 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 positive e¤ect
on the preferred tax rate is more likely to overcome the negative one as the elasticity
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 or 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 inorleads 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: max 4(1 ) ; 4< ' < 1+2 .
11Assumption 1 ensures that one who prefers the highest tax rate among the young
becomes a decisive voter. To understand the argument stemming from Assumption 1,
recall that (i) young agents are ranked in terms of their preferred tax rates; (ii) all the old
prefer = 1=2 which is higher than any other preferred tax rates by the young. However,
the old cannot be majority voters because the number of the old population, 2 , is lessthan that of the young population, 2:Given the abovementioned argument, we can …nd the decisive voter from the young. In
particular, by imposing Assumption 1, we can determine the identity of the decisive voter
from an agent who prefers the highest tax rate among the young. For example, supposethat a single female prefers the highest tax rate among the young. She can be a decisive
voter if the number of single females plus the old is larger than half of the population, thatis, if (1 ') + (1 + ) > (3 + )=2 , ' < (1 + )=2: Following the same argument, we cansay that an agent who belongs to a one-breadwinner couple becomes a decisive voter if2'(1)+(1+ ) > (3+ )=2 , ' > (1)=4(1); and an agent who belongs to a two-breadwinner couple becomes a decisive voter if 2' +(1+ ) > (3+ )=2 , ' > (1The three conditions are summarized as in Assumption 1.6)=4 :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.5. In what follows here, we will focus exclusively
on the latter case.

5.2

An Economy with 1= < 1

To determine the decisive voter overwhen 1= < 1, we recall the conditions demon-

strated in Section 4.1 that determine the preferred tax rates by four types of households. Because= (1)=(1 + ) is the threshold value of the derived pension
rights common to one-breadwinner and two-breadwinner couples, we consider two cases,
2 [0; (15.2.1)=(1 + )] and2 ((1)=(1 + ); 1] in turn.

Low Level of Derived Pension Rights: 0

1
1+

First, we consider the case of a low level of derived pension rights. In this case, single
males and one-breadwinner couples prefer no taxation, whereas single females and twobreadwinner couples prefer taxation. Figure 1 illustrates the conditions (4) and (6) that6

A single male cannot be a decisive voter because he always prefers no taxation.

12

determine the preferred tax rates by single females and two-breadwinner couples, respectively. As depicted in the …gure, there is a critical value of , ~f;c2 2 ^f ; ^c2 , such thatRHS f and RHS c2 intersect at= ~f;c2 . By direct calculation, we obtain:(1 + )( + 1) ( )

f;c2

~

1

1

:

()

[Figure 1 about here.]
The tax rate preferred by a type-j (j = f; c2) household is determined by the crossing
point of LHS and RHS j . Given the assumption of household distribution in Assumption
1, the decisive voter over is the one who prefers the highest tax rate among the young
households. Based on the illustration in Figure 1, the decisive voter over when 1= < 1and

2 0; 11+is determined as follows.

hold. There exists a unique
Lemma 1. Suppose that 1= < 1 and
2 0; 11+equilibrium of the voting game with 2 (0; 1=2). The decisive voter over is
1(i) a type-f single female agent if

()

(1+ )
+1

2

1

( )

+ 1++1 ;(ii) a type-c2 agent who belongs to a two-breadwinner couple, otherwise.
Proof. See Appendix A2.
To understand the mechanism behind the 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( )

1

where the left-hand side is decreasing in( + 1)

1

1

1+

+ 1;

, and the right-hand side is increasing in(7).

Therefore, the condition (7) states that a type-f agent is more likely to become a decisive
voter when is higher (that is, when the gender wage gap is smaller).
The intuition behind the condition (7) is as follows: For a low degree of derived pension
rights such that

2 [0; (1)=(1 + )], single females expect a high level of pension

bene…ts in old age, which gives them a disincentive to save. Thus, single females would
be borrowing constrained for a low degree of derived pension rights. In this situation,
they owe greater tax burden as becomes higher (i.e., as their wage becomes higher).
However, when 1= < 1, an increase in

results in a lower marginal cost-to-bene…t ratioof redistribution and thus a higher preferred tax rate by borrowing-constrained single
females as demonstrated in Section 4.2. Therefore, type-f agents prefer a higher tax rate
than type-c2 agents when is high such that (7) holds.
13

High Level of Derived Pension Right: 11+

5.2.2

<

1

Next, let us consider the case of a high level of derived pension rights. In this case,single males and two-breadwinner couples prefer no taxation, whereas single females andone-breadwinner couples prefer taxation. Figure 2 illustrates the conditions (4) and (5)that determine the preferred tax rates by single females and one-breadwinner couples,
respectively.
[Figure 2 about here.]
The current case could be divided at most into the following three sub-cases:n (2-a) a ocase of

2

1
1+

; min

1

(see Panel (a)); (2-b) a case of

;1

1

2

provided that 1 < (1 + ) holds (see Panel (b)); and (2-c) a case ofPanel (c)) provided thatf;c1values of the tax, ~

1
(1

)

2

;1
i )
; 1 (see)1'(1< ' holds. As depicted in the …gure, there are critical

for the case (2-a), and ~c1;f for the case (2-b), de…ned by:
1

1

f;c1

1

( + ) ()

;~

()

1

1

c1;f

~

1
'(1

; min
h

()

( + ) ()

respectively. As in the previous subsection, we can characterize the political equilibrium
for the current case as follows.
Lemma 2. Suppose that 1=< 1 andequilibrium of the voting game with

2

1
1+

; 1 hold. There exists a unique

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

1
1+

; min

1

1

;1

and

()

2

+

( )

1

+

1
;
+

or if:
2

1; 1 ; 1 < (1 + ) and( ) > 2(

)

1

1
+

1

+ ;

(ii) a type-c1 agent who belongs to a one-breadwinner couple otherwise.
Proof. See Appendix A3.
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. Thechange of 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 rights relative tothe previous case. This relationship results in a lower marginal cost-to-bene…t ratio of14

redistribution 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.
The two conditions in Lemma 2(i) implies that for a lower , a single female agent
prefers a higher tax rate; and she is more likely to be a decisive voter (see Appendix A3).
This result is qualitatively di¤erent from that in the previous, low-derived pension case:for a higher , a single female agent prefers higher tax rate; and she is more likely to bea decisive voter. The di¤erence between the two cases comes from the saving behaviora¤ected by the degree of derived pension rights.
To understand the role of derived pension rights, suppose that the degree of derived
pension rights is higher than the critical value (1)=(1 + ). Under this situation,

single females are less likely to be borrowing constrained because they expect a lower levelof pension bene…ts, which gives them an incentive to save more for old-age consumption.
In this situation, a decrease in

results in a decrease of the cost-to-bene…t ratio ofredistribution for unconstrained single females; and this gives them an incentive to choosea higher tax rate. Therefore, when is higher than the critical value (1single female agent is more likely to be a decisive voter for a lower .

)=(1 + ); a

Alternatively, suppose that the degree of derived pension rights is lower than thecritical value (1)=(1 + ). Under this assumption, single females are more likely to

be borrowing constrained because they expect a higher level of pension bene…ts and thus…nd less need to save for old-age consumption. In this situation, an increase in results
in an increase of the cost-to-bene…t ratio of redistribution for constrained single females;
but this gives them an incentive to choose a higher, rather than a lower, tax rate because
single females are borrowing constrained and are faced with a low intertemporal elasticity
of substitution. Therefore, when is lower than the critical value (1)=(1 + ), a
single female agent is more likely to be a decisive voter for a higher .
5.2.3

A Decisive Voter When 1= < 1

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 over15is(i) a type-f single female agent if:
1
2 0;
1+
2
2

1
1+

and
; min

1

1

()

2

1

(1 + )
+1

1

( )

+

1+
; or
+1
1

;1

and

; 1 ; 1 < (1 + ) and()2( ) > 2(

+
)

1

( )

1
; or
+

1

1
+

1

+
+ ;

(ii) a type-c2 agent who belongs to a two-breadwinner couple if:
1
2 0;
1+

and

()>2

(1 + )
+1

1

( )

1

+

1+
;
+1

(iii) a type-c1 agent who belongs to a one-breadwinner couple otherwise.
Proof. The proof is immediate from Lemmas 1 and 2.
Proposition 1 presents the determination of the identity of the decisive voter in majority voting. A type-f single female agent becomes a decisive voter if the degree of derived
pension rights is low and the gender wage gap is small as demonstrated in the …rst condition of the statement (i); or if the degree of derived pension right is high and the genderwage gap is large as demonstrated in the second and third conditions of the statement(i). Interpretations for these results are discussed in the previous subsections.
A type-c2 agent who belongs to a two-breadwinner couple becomes a decisive voter if
the degree of derived pension right is low such that the …rst condition in the statement(ii) holds; and the bene…t-to-burden ratio of public pension, denoted by ( ), is high suchthat the second condition in the statement (ii) holds. Because ( ) is increasing in , thelatter 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.
Finally, a type-c1 agent who belongs to a one-breadwinner couple becomes a decisive
voter if the conditions in statements (i) and (ii) of Proposition 1 fail to hold. In particular,
all types except type-c1 agents prefer no taxation and thus no redistribution via pension
1
holds (see Panel (c) of Figure 2). This condition implies that the genderwhen > '(1)
wage gap is small, the fraction of one-breadwinner couples is large, and the longevity ofmen is high. All these factors imply greater bene…ts via pension compared to the costof taxation for type-c1 agents. Therefore, type-c1 agents have an incentive to choose
taxation on working agents although all the other types of agents …nd it optimal tochoose no taxation.
The e¤ect of the derived pension rights on the identity of the decisive voter is qualitatively equivalent to that demonstrated in Leroux, Pestieau and Racionero (2011). However, the e¤ect of gender wage gap on the identity of the decisive voter is di¤erent from that16

in Leroux, Pestieau and Racionero (2011). As the gender wage gap becomes smaller, themarginal cost-to-bene…t ratio of redistribution for single females becomes larger. However,they prefer a larger size of redistribution as long as they are borrowing constrained with a
low intertemporal elasticity of substitution (Lemma 1). This counterintuitive result was
not demonstrated in Leroux, Pestieau and Racionero (2011) who assume a quasi-linear
utility and no borrowing constraint. This di¤erent e¤ect is further investigated in thenext section.

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 thetax rate changes in response to recent trends in developed economies: a reduction of thegender wage gap, a reduction of derived pension rights, and an increase in the fraction of
two-breadwinner couples. The aim of the analysis is to explore the roles of the borrowing
constraint and the intertemporal elasticity of substitution on the determination of the
tax rate, which were not demonstrated in Leroux, Pestieau and Racionero (2011). The
analysis also aims to discuss policy implications of the result.

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 inlocally produces an inverse U-shaped relationship between

and

)

j

around ^ .

Proof. See Appendix A.4.
To understand the mechanism behind the result in Proposition 2, we …rst note that
there is a critical value offor a type-j agent, denoted byj(j = f; c1; c2):

jis thejthat makes a type-j agent choose = ^ (see Figure 3). Around this critical value, there
is a change in the pattern of saving as discussed in the previous sections. In particular,
for j = f and c2, an agent is borrowing constrained for
unconstrained for

>

j

j

; and he/she is borrowing(see Panel (a) of Figure 3). For j = c1, an agent is borrowingjunconstrained for; and he/she is borrowing constrained for > j (see Panel (b)of Figure 3). The detail of the mechanism behind the change of saving pattern around jis shown in Appendix A.4.1.
[Figure 3 about here.]17

Proposition 2 says that starting from a low value of , borrowing-constrained type-for type-c2 agent prefers a higher tax rate as becomes larger (that is, as the genderwage gap becomes narrower). However, he/she becomes borrowing-unconstrained once
approachesj; and he/she prefers a lower tax rate as

is further increased. Type-j’s(j = f; c2) preferences over the tax qualitatively changes at = j . As for a type-c1
agent, he/she is borrowing unconstrained for below his/her critical value c1 . For thisrange of , he/she prefers a higher tax rate as
borrowing constrained whenbecomes larger. However, he/she becomesis beyond the critical value

c1. He/she turns to choose alower tax rate in response to a further increase in
The result described so far could be understood in the following way. We …rst consider the e¤ect of an increase inon the marginal cost-to-bene…t ratio of redistribution,

denoted by RHS j (j = f; c1; c2), when an agent is borrowing-unconstrained. After some
calculation, we obtain@RHS f@RHS c1@RHS c2> 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 redistribution for type-f
and type-c2 agents, whereas it decreases the ratio for type-c1 agents. The di¤erence isdue to the 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 marginal cost-to-bene…t
ratio of redistribution and thus a lower preferred tax rate for type-f and type-c2 agents,
whereas it results in a lower marginal cost-to-bene…t ratio of redistribution and thus ahigher 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 Section 4.2. A reduction of the gender wage
gap results in lower RHS f and RHS c2 and thus in higher preferred tax rates by 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 relationship between
and j around the critical value of the tax, ^j , that divides the status of saving (seePanel (c) of Figure 3).

18

6.2

A Reduction of Derived Pension Rights

Next, we consider the e¤ect of a reduction of derived pension rights on the determinationof 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.4.
To understand the result in Proposition 3, we …rst note that there is a critical valuejof ;, for a type-j agent:jis the= ^j . Aroundthat makes a type-j agent choose

this critical value, there is a change in the patterns of saving. In particular, for j = f; c2;
jan agent is borrowing constrained for; he/she is borrowing unconstrained for>j: For j = c1; an agent is borrowing unconstrained forconstrained for

j

<

j

; he/she is borrowing:

Given the saving pattern of each agent, we …rst consider the e¤ect of the derived
pension rights on the marginal cost-to-bene…t ratio of redistribution when an agent isborrowing unconstrained:( 1)@RHS f@RHS c1@RHS c2< 0; ( 1)> 0; ( 1)< 0 if sj > 0 (j = f; c1; c2)@@@

We multiply the derivatives by ( 1) to demonstrate the qualitative e¤ect of a decrease in.
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 marginal cost-to-bene…t ratio of redistribution, and thus raises their preferred tax rate. Therefore, type-f and type-c2 agents
prefer a higher tax rates in response to a reduction of the degree of derived pension rights
as long as

is above the critical valuej(j = f; c2). In contrast, such a reduction

decreases the pension bene…ts for type-c1 agents, raises their cost-to-bene…t ratio of redistribution and thus lowers their preferred tax rate. Type-c1 agent prefers a lower tax
rate in response to a reduction of as long as is below a critical value c1 .
The above result is reversed when an agent is borrowing-constrained as demonstrated
in the case of a change in. Therefore, the two opposing e¤ects result in an inverse

U-shaped relationship between and j around the critical value of ;tax rate is maximized and is given by = ^j at = j :j. The preferred

The result in Proposition 3 di¤ers from that demonstrated in Leroux, Pestieau andRacionero (2011). They numerically showed that a reduction of derived pension rights(i.e., a decrease in ) results in decreases in the tax rate and pension size for plausible set
19

of parameters. Their monotone result comes from the assumption of quasi-linear utility
and no borrowing constraint. The current paper takes away these assumptions and showsa non-monotone e¤ect of the derived pension rights on the equilibrium tax.

6.3

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.4.
To understand the result in Proposition 4, let us …rst consider how the bene…t-toburden ratio of public pension, denoted by ( ), is a¤ected by the parameter :1 + (1 ' + ' )+ 1 ' + ' + '(1()
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 ( ). On the other hand, an increase in implies a larger size of twobreadwinner couples and a smaller size of one-breadwinner couples. Because the levelof pension rights is larger for working females than for non-working females, an increase
in results in a decrease of per capita pension bene…ts. This e¤ect is observed in the
denominator of ( ).
The former positive e¤ect one¤ect on( ) is dependent of( ) is independent of , whereas the latter negative. Speci…cally, the negative e¤ect becomes larger as

becomes smaller. Therefore, the negative e¤ect overcomes the positive one when the levelof derived pension rights is low such that < (1)=(1 + ): that is, @ ( )=@ < 0holds if

< (1)=(1 + ); the opposite holds if@ ( )=@> 0 holds if

> (1

> (1

)=(1 + ): that is,)=(1 + ). The two opposing e¤ects are o¤set eachother if = (1)=(1 + ).
In what follows, we consider the case of< (1)=(1 + ) and the case of>)=(1 + ) in turn. First, for a low degree of derived pension rights such that

(1
< (1

)=(1 + ), type j = f and j = c2 agents prefer a positive tax rate. A

low degree of derived pension rights for non-working females produces a large size of percapita pension bene…ts for working females. In addition, a low (i.e., a low share of20

two-breadwinner couples) implies a high level of ( ) (i.e., a high bene…t-to-burden ratioof public pension) and thus a high level of per capita public pension. These two e¤ects
give types j = f and j = c2 agents a disincentive to save for their old-age consumption.
Thus, they are borrowing constrained for
for

j

>

)=(1 + ) and

< (1

<

j

. However,, a level of per capita pension bene…ts becomes low; and this negative e¤ect

on the size of pension overcomes the positive e¤ect produced by a low degree of derived
pension rights. Therefore, types j = f and j = c2 agents are borrowing unconstrained
for

< (1

)=(1 + ) and

>

j

.

Based on the abovementioned argument, we can now show the e¤ect ofon thepreferred tax rate by a type-j (j = f; c2) agent when < (1)=(1 + ). First, supposejthat > : a type-j (j = f; c2) agent is borrowing unconstrained. In this situation, ahigherresults in a lower size of ( ), and thus a larger marginal cost-to-bene…t ratio ofredistribution, RHS j (j = f; c2) in Eqs. (4) and (6). A type-j (j = f; c2) agents prefers
a lower tax rate as becomes larger for the range of > j . However, the opposite result
holds for the range of < j because he/she is borrowing constrained. Therefore, aroundj, there arises an inverse U-shaped relationship between

and the preferred tax rate by

the decisive voter.
Next, consider the case of a high degree of derived pension rights such that>(1)=(1 + ): type-f and type-c1 agents prefer a positive tax rate. A high degreeof derived pension rights gives a high level of per capita pension bene…ts for type-c1
agents. However, while it gives a low level of per capita pension bene…ts for type-f
agents, this negative e¤ect on pension size is compensated for by the e¤ect of via ( )because a higher results in a larger size of ( ) and thus a larger size of per capita
pension, @ ( )=@> 0. Therefore, for a highsuch that>j, a type-j (j = f; c1)

agent obtains a large sized pension bene…t; and this gives him/her a disincentive to savefor old-age consumption, thereby resulting in being borrowing constrained for > j .
Because @ ( )=@ > 0 holds, a larger results in a lower marginal cost-to-bene…t ratioof redistribution for a constrained type-j agent and thus leads to a lower preferred tax
rate by him/her. The opposite result holds for<jbecause a type-j (j = f; c1) agent

is borrowing unconstrained due to a low size of pension bene…ts. A larger results ina lower marginal cost-to-bene…t ratio of redistribution for a type-j unconstrained agent
and thus results in a higher preferred tax rate.

6.4

Discussion

We have analyzed the e¤ects of changes in (gender wage gap), (the degree of derived
pension rights) and (the fraction of two-breadwinner couples) on the equilibrium tax
rate, and shown the inverse U-shaped relationship between the tax rate and parameters.
21

Based on the result demonstrated so far, in this subsection we brie‡y discuss the following
three issues: (i) the e¤ect of longevity gap between males and females on the equilibriumtax rate, (ii) the e¤ect of a decisive voter’s switch on the equilibrium tax rate, and (iii)
policy implications of the result.
The current framework assumes that females live two periods, but males die at the endof youth with probability 1. The parameter

represents the longevity di¤erence: asmaller implies a larger longevity di¤erence between males and females. From equations(4), (5) and (6), we can immediately …nd that a smaller (i.e., a larger longevity gap)
results in a lower marginal cost-to-bene…t ratio of redistribution for type-f and type-c2
agents, while it results in a higher marginal cost-to-bene…t ratio of redistribution for typec1 agents. Given this result, we can apply the analysis and result established so far andconclude that type-f and type-c2 agent prefer a higher tax rate in response to an increasein longevity gap when they are borrowing unconstrained, whereas they prefer a lower tax
rate when they are borrowing constrained. The opposite result holds for type-c1 agents.
Therefore, a change in longevity di¤erence also produces an inverse U-shaped e¤ect on
the equilibrium tax.
The result in this section, combined with the result in Section 5, suggests that theequilibrium tax would show two peaks in response to changes in parameters. For illustrative purpose, assume < (1)=2: the result in Lemma 1 holds for any 2 [0; 1],
and the decisive voter is a type-f or type-c2 agent depending on the value of . First, we
take a su¢ ciently low value ofsuch that the decisive voter is a borrowing-constrained

type-c2 agent. His/her preferred tax rate is increased in response to an increase in(i.e., a reduction of gender wage gap), and peaks at c2 . A further increase in makes
him/her to be borrowing-unconstrained; and to prefer a lower tax rate (Proposition 2).
When the value ofapproaches beyond the critical level, the decisive voter changes froma borrowing-unconstrained type-c2 agent to a borrowing-constrained type-f agent. The
same argument holds for a type-f agent: her preferred tax rate attains a peak at f
(Proposition 2). Therefore, there are two peaks of the equilibrium tax rates in responseto changes in .
Finally, we discuss the policy implications of the results. Many developed countries
have been faced with increasing burden of public pension for the past decades. It has beenargued that population aging is one of the most causes for this increasing burden. The
local comparative statics analysis and result in this section provide an alternative view:
decreases in the gender wage gap, the reduction of derived pension rights, and an increasein the fraction of two-breadwinner couples, observed in developed countries for the past
decades, also provide a political incentive to increase the tax burden of public pension.
However, our analysis and result suggest that further increases in these parameters may
result in a decrease or a non-monotone change in tax burden. This possibility could be22

further investigated in future research.

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,
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.

23

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 maximum over the whole range ofthat V f is single peaked over .

and thusy;jy;jwith respect to are
and Vs=0
The …rst and the second derivatives of Vs>0
!
f
@Vs>0
1
=
[(1
) w + w ( ) (1)]
f w + w ( )(1 2 )g ;@1 + ( )1=!
f
@ 2 Vs>011=( ) [(1) w + w ( ) (1)]
f w + w ( )(1
1=
@ 2
1+( )
!
1
[(1
) w + w ( ) (1)]( 2)w ( )+1 + ( )1=

< 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

1 + ( )1=
1

=^f

( )1=
()

!

1

( w)1
24

( )1=
()

!1

1+

( w)1

()

+ 2 ( )1=

1+
;
1

:

2 )g2

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 maximum with respect to over the whole range of . Speci…cally, V f isfff
maximized at = arg max Vs>0
if arg max Vs>0
< ^f ; it is maximized at = arg max Vs=0
otherwise.

A.2

Proof of Lemma 1

From Panel (a) of Figure 1, a type-f agent becomes a decisive voter if and only if thefollowing condition holds:
1+:( + 1) ( )2~f;c2

1This condition is rewritten as the one in Lemma 1(i). Otherwise, a type-c2 agent becomesa decisive voter (see Panel (b) of Figure 1).

A.3

Proof of Lemma 2

Suppose that

1
1+

2

; min 1; 1

holds (see Panel (a) of Figure 2). A type-ff;c1

agent becomes a decisive voter if 1
1

2

+

1

( )

+

Next, assume 1

1
+

2~

1=( + ) ( ) holds, that is, if

holds.

< 1 (i.e., 1 < (1 + ) ) and2n; min 1;

1

1

1

)

o

hold (see2~c1;f > = ( ) holds, that is,

Panel (b)). A type-f agent becomes a decisive voter if 1
1
if ( ) > 2 ( )++voter becomes a type-c1 agent.

1
'(1

()

holds. When these conditions fail to hold, the decisive

The conditions demonstrated above imply that a single female agent is more likely to
be a decisive voter for a lower . To con…rm this statement, let us rewrite the conditions
in Lemma 2(i) as follows:
1

()

2

+

f1 '(1,f( + 1)( ) > 2(

)

1

( )

1

+

)g [( + )
(1
)'(1
1
+

1
+
(1
)= ]
)g ( + )

1

2

( )

+

1

;

(8)

1

+

f + '(1))g,> 2(( + 1) (1
)'(1
)
1

25

1=

)

1
+

(

1)=

(9)

The left-hand side of (8) is increasing in whereas the right-hand side of (8) is decreasing in ; (8) is more likely to hold for a lower . The left-hand side of (9) is decreasing
in

whereas the right-hand side of (9) is increasing in ; (9) is more likely to hold for alower .

A.4

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 redistribution for type-f; c1 and c2 agents, respectively:
see equations (4), (5) and (6). 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
c1

f

c2

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

]
]
@ RHS
@ RHS
R 0;
@
@

c1

]
@ RHS
R 0;
@

(10)
(11)

c2

R 0 if and only if

Q

1
1+

(12)

:

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
consider the e¤ects ofand;around the critical value ^j de…ned in Section 3, we

calculate the e¤ects of these parameters on ^j as follows:
@ ^f
@ ^c1
@ ^c2
> 0;
< 0;
> 0;
@
@
@
@ ^c1
@ ^c2
@ ^f
> 0;
< 0;
> 0;
@
@
@
@ ^c1
@ ^c2
@ ^f
R 0;
R 0;
R 0 if and only if
@
@
@
A.4.1

The e¤ect of

(13)
(14)
Q

1
1+

(15)

:

on the equilibrium tax rate: Proof of Proposition 2

Consider …rst the equilibrium where the decisive voter is a type-f agent. Suppose thatis initially given such that type-f ’s preferred tax rate is= ^f . We denotefas thefthat makes a type-f young agent choose = ^ .
f] in (10) and the property of ^f in (13), we …nd a positive
With the property of RHS
real number "(> 0) aroundfsuch that the type-f agent is borrowing-constrained for
26

2

( f "; f ) and borrowing-unconstrained for 2 [ f ; f +"). This statement is con…rmed
by looking at Panel (c) of Figure 3. For a low , a preferred tax rate by a type-f agent
is determined by the crossing point of LHS

1

2 and RHS f = ( = ( ))1( ) = ;

showing that she is borrowing constrained. For a high , it is determined by the crossing
point of LHS 1 2 and RHS f = = ( ); showing that she is borrowing unconstrained.
Therefore, the equilibrium tax rate satis…es the following condition:(
11( ) for2 ( f "; f ];f()
LHS 1 2 = RHS
for
2 f; f + " :()Given the properties in (10) and (13), we can illustrate the e¤ects of an increase in
on RHS f and ^f as in Panel (a) of Figure 3. The illustration leads to the following
result:

@ f
@
@ f
@

> 0 for< 0 for

The result shows that an increase inbetween and f around = ^f .

2( f
2 f;

"; f );f+" :locally produces an inverse U-shaped relationship

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 (10) and (13).
Next, consider the equilibrium where the decisive voter is a type-c1 agent. Suppose
thatis initially given such that type-c1’s preferred tax rate is= ^c1 . We denotec1

as the that makes a type-c1 young agent choose = ^c1 . Because the properties ofc1f] in (10) and ^c1 in (13) are opposite to those of RHS] and ^f ; the saving pattern of
RHS
type-c1 agent aroundc1

is also opposite to that of type-f agents around…nd a positive real number "(> 0) around2 ( c1";

2 = RHSc1

unconstrained for

c1

c1

1

) and borrowing-constrained for1: That is, wesuch that the type-c1 agent is borrowing-

equilibrium tax rate satis…es the following condition:
(
1
LHS

f

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

( )2 [ c1 ;c1for

2 ( c1";for

2 [ c1 ;c1+ "). The

c1

);

+ "):

Given the properties in (11) and (14), we obtain the following result:
@ c1
>0
@
@ c1
<0
@

The result shows that an increase inbetweenandc1

aroundforfor

2 ( c12 ( c1 ;"; c1 );
c1+ ") :

locally produces an inverse U-shaped relationship

c1

=^ .

27

A.4.2The e¤ect ofon the equilibrium tax rate: Proof of Proposition 3

Suppose that the decisive voter is a type-j (j = f; c1; c2) agent. Suppose that is initiallygiven such that type-j 0 s preferred tax rate is = ^j . We denote j as the that makes
a type-j agent choose= ^j .

Under the abovementioned situation, suppose that an increase inproduces an inverse U-shaped relationship between
This assumption implies that a decrease in aroundU-shaped relationship betweenjaroundjlocally

and type-j’s preferred tax rate.
also locally produces an inverse

and type-j’s preferred tax rate. Therefore, it is su¢ cient

to show the e¤ect of an increase inon 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.
A.4.3

The e¤ect ofSuppose thaton the equilibrium tax rate: Proof of Proposition 4)=(1 + ) holds. The decisive voter is a type-f or type-c2 agentj] and ^j (j = f; c2) are qualitatively similar to those(Lemma 1). The e¤ects of on RHS< (1j] and ^j (j = f; c2). We can apply the analysis and result in Proposition 2
on RHSofto the current 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> (1qualitatively similar to those of]
on RHSc1and ^c1 . We can apply the analysis andresult in Proposition 2 to the current case.
Finally, suppose that = (1)=(1 + ) holds. The parameter has no e¤ect onjj]
RHS and ^ . A change in has no e¤ect on the equilibrium tax rate.

A.5

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

Figure 4 illustrates the conditions that determine the preferred tax rates by agents who
prefer taxation for the case of 1=

1. From the observation in Figure 4, 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 panels (c) and (d)).
[Figure 4 about here.]28

Proposition A1. Suppose that 1=
voting game with

1 holds. There exists a unique equilibrium of the

2 (0; 1=2). The decisive voter over1(i) a type-f , single female agent if

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, an agent
who belongs to a two-breadwinner couple cannot become a decisive voter. Such agent’s
marginal cost-to-bene…t ratio of redistribution in terms of utility is always higher than
the ratios of the other two types of households. This result implies that two-breadwinnercouples prefer a lower tax rate than do other two types of young agents.
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 that)= . Then, the marginal cost-to-bene…t ratio of redistribution in terms of utility(1

for single females is lower than that for one-breadwinner couples because the former owe
less tax 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.5.1

)= .
Comparative Statics Analysis

Consider the political equilibrium in an economy with 1=

1. Suppose that(1)= 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
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 (10) - (12), we obtain the following result:With the property of RHS
@ f
@
and

Next, suppose that

> (1

0;

@ f
@

0;

@ f
Q0,
@

Q

1
1+

)=

:

holds: the decisive voter is a type-c1 agent(Proposition A1). The optimality condition for a type-c1 agent, given by (5), indicates29]that a higher RHS
c1

s = 0:

c1

results in a lower preferred tax rate except the case of 1= = 1 and(]With the property of RHS

@ c1
]
@ RHS

c1

c1

= 0 if 1= = 1 and sc1 = 0;
@ c1
c1 < 0 otherwise.
]@ RHS

in (10) - (12) and the assumption of> (1)= , weobtain the following result:
@ c1
@

0;

@ c1
@

0;

@ c1
@

0:

The result established in this appendix indicates monotone e¤ects of the parameters ;andon the equilibrium tax rate.

30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 andredistributive policies. European Economic Review 51, 625–645.
[3] Bethencourt, C., and Galasso, V., 2008. Political complements in the welfare state: Healthcare 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 arecorrelated. 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. Journalof Public Economics 88, 1849-1869.
[10] Conde-Ruiz, J.I., and Galasso, V., 2005. Positive arithmetic of the welfare state. Journalof 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, Journalof Public Economics 23, 81–114.
[14] Ermisch, J., 2003. An Economic Analysis of the Family. Princeton University Press, Princeton, Mew Jersey.
[15] Leroux, M.L., and Pestieau, P., 2012. The political economy of derived pension rights.
International Tax and Public Finance 19, 753–776.
[16] Leroux, M.L., Pestieau, P., and Racionero, M., 2011. Voting on pensions: Sex and marriage.
European Journal of Political Economy 27, 281–296.
[17] Mulligan, C.B., Sala-i-Martin, X., 1999. Social security in theory and practice (I): facts andpolitical theories. NBER Working Papers 7118, National Bureau of Economic Research.
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Journal of Population Economics 19, 345–365.

31

Panel (a)(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2
τ
̂
̂
τ ,

Panel (b)(1 + )( + 1) (∙)(∙)

LHS ≡ 1 − 2
τ
̂
̂
τ ,

Figure 1.

The figure illustrates the case of γ ∈ (0, (1 − απ)/(1 + α) . In Panel (a), a

type-f agent is a decisive voter; in Panel (b), a type-c2 agent is a decisive voter.

32

Panel (a)1( + ) (∙)

LHS ≡ 1 − 2
(∙)

̂

τ ,

Panel (b)

τ
̂

(∙)

LHS ≡ 1 − 2

1( + ) (∙)

τ
̂

τ

̂

,

Panel (c)1
( + ) (∙)

LHS ≡ 1 − 2τ̂Figure 2. Panels (a), (b) and (c) illustrate cases of γ ∈
,

(

)

,1

and γ ∈()

, 1 , respectively.

33

,

1,

, γ∈

Panel (a)ττ (j = f, c2)αConstrained

Unconstrainedα ( = , 2)

Panel (b)τταUnconstrained αPanel (c)Constrainedwith a low α

with a high α

LHS ≡ 1 − 2̂τFigure 3. Panel (a) illustrates the saving pattern and the corresponding preferred tax
rate by a type-j = f, c2 agent; Panel (b) illustrates the saving pattern and thecorresponding preferred tax rate by a type-c1 agent; Panel (c) illustrates changes in thepreferred tax rate by a type-j agent in response to an increase in α.
34

Panel (a)Panel (b)(1 + )( + 1) (∙)

1( + ) (∙)

LHS ≡ 1 − 2(∙)

LHS ≡ 1 − 2

(∙)

τ

τ
̂

̂
̂

̂

Panel (c)Panel (d)(∙)
1
( + ) (∙)

1( + ) (∙)

LHS ≡ 1 − 2

LHS ≡ 1 − 2

τ
̂

τ

̂
̂

Figure 4. Panel (a) is the case of γ ∈ 0,
Panel (c) is the case of γ ∈

,

(

)

; Panel (b) is the case of γ ∈; and Panel (d) is the case of γ ∈

35

;

,

(

)

,1 .