ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E13-3

Pension and the family
byHikaru Ogawa
Mizuki Komura

Revised version as of October 2013

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Pension and the family

Hikaru Ogawa*・Mizuki Komura**

Abstract. The effects of pension policies on fertility have been examined in the overlapping
generations (OLG) model of unitary household in which no heterogeneity exists between thewife and the husband. This paper departs from the OLG model and focuses on the maritalbargaining arising from the heterogeneity in a couple in a non-unitary model. Specifically,
this paper examines how the pension policy affects the endogenous fertility of a bargaining
couple who have different lifespans. The analysis finds out a new channel of pension policy
on fertility decisions: an increase in pension size affects fertility not only via the changes incurrent and future income, but through a change in marital bargaining power. This channel
leads a plausible argument that an increase in a pay-as-you-go (PAYG) pension further
accelerates a decline in fertility through the empowerment of women.

Keywords Pension・Fertility・Longevity・Marital bargaining
JEL Classification

H55・J12・J13* H. Ogawa. Graduate School of Economics, Nagoya University, Furocho, Chikusaku,
Nagoya 464-8601, Japan, e-mail: ogawa@soec.nagoya-u.ac.jp.
** M. Komura. Graduate School of Economics, Nagoya University, Furocho, Chikusaku,
Nagoya 464-8601, Japan, e-mail: komura.mizuki@h.mbox.nagoya-u.ac.jp.

11.

Introduction

Acceleration in demographic aging has caused many developed countries to reform their
existing pension systems. Against a background of this policy concern, the mutual
dependence relationship between fertility and public pension has come under intense study(Cigno, 1993; Zhang and Zhang, 1998; Wigger, 1999; Yakita, 2001; Groezen et al., 2003;
Groezen and Meijdam, 2008; Hirazawa and Yakita, 2009). Economists are intrigued with the
pay-as-you-go (PAYG) pension system as an inter-generational redistribution device thatinvolves the intra-generational redistribution effect. 1

If the PAYG pension system isgenerous to be equally beneficial to all individuals, it induces the redistribution among
heterogeneous households. Some studies deal with heterogeneity among households, butthey draw the household as a single decision unit based on unitary model, and do notconsider heterogeneity within the household. This paper adopts a different approach. By
focusing on the marital bargaining arising from the heterogeneity in a couple, we describe
the intra-generational redistribution effect of the pension policy and how its expansion
affects the balance of power between husband and wife, and the fertility of the couple
having different lifespans.
Heterogeneity within a household can be factors that bring about intra-generational
redistribution. For example, an expansion of a generous pension system means an implicit
income transfer from the shorter longevity spouse to the spouse with longer longevity.
Recent studies based on the non-unitary model have considered this intra-redistribution

1

Sinn (2004) considers redistribution from households with children to those without,because children can be insurance devices for households who cannot have children in the
PAYG pension scheme. Cremer et al. (2008) consider redistribution under both funded andunfunded pension systems in the presence of different abilities in raising children amonghouseholds. Hirazawa et al. (2013) focus on redistribution among households with different
contributions as a result of different childcare schedules. Heterogeneity among households
is discussed in the many studies on public policies. Bommier et al. (2011a, 2011b) consider
the problem of redistribution among households with different longevities. Cremer et al.
(2004) examine how the redistribution effect of implicit tax imposed on postponed retirement
affects households' retirement activities in the presence of different productivities and healthstatuses. Cremer et al. (2010) consider a trade-off between redistribution due toheterogeneous productivities and redistribution caused by heterogeneous longevity (whichis positively correlated with productivities).
2effect in the retirement period.2 Theoretically, Browning (2000) showed that a generous
pension system involves redistribution from husbands to wives following the fact that womentend to live longer, and that it results in increased savings, which is a favorable household
allocation for wives. 3

An empirical work by Duflo (2003) also confirms theintra-generational redistribution effect of an expansion of a generous pension in South Africa,
which is more likely to be beneficial for wives because they live longer. Although they shed
new light on the redistribution effect of public policies within the families in the retirement
period, the long-run effects of policies on household fertility decisions are not examined. The
purpose of this paper is to explore the effects of the PAYG pension system on fertility, taking
into account gender differences in longevity and its effect on redistribution within thehousehold.
Our model has three features. The first is that the household makes a decision throughintra-household bargaining.4

Homogeneous couples look like the picture of happiness, but

the reality proves different. It is known that wives tend to be younger than their husbands,
and that they also tend to live longer than their husbands do.5

The difference in lifespanleads couples to bargain over the saving in the young because the wife wants to havegreater wealth at the retirement stage (Lundberg and Ward-Batts, 2000; Lundberg et al.,2003), which may reduce the demand for private consumption and the number of children in2

The distribution effect of public policy among family members in the young period istheoretically considered by Lundberg and Pollak (1993). Komura (2013a, 2013b)
theoretically examines the intra-family distribution effect of policy shifts from in-cash childsupport to in-kind child support, and the shift of unit of income taxation from household toindividual, respectively. Lundberg et al. (1997) found a significant redistribution effectcaused by the shift of the child allowance recipient from fathers to mothers in the UK.
3
Aura (2005) also uses the American legislation change of 1984 in favor of wives, who arelikely to be widowed because of their higher life expectancies, to show that this implicit
income transfer leads the household wealth portfolio to reflect the wife's intentions more.
4
In contrast to our setting of family bargaining, Glazer (2008) showed that couple's
non-cooperative strategic interactions result in inefficient household savings, and that thesocial security system can improve welfare because it forces them to secure savings.
Grossbard-Shechtman and Pereira (2013) also explores the effects of marital status on
individual saving behavior, rather than household saving behavior, as responses to them ina non-cooperative game, treating the distribution problem of marriage.
5
The United Nations (2000), based on 236 countries, reported that husbands are older thantheir wives in all but one country. This husband-wife age gap tends to be larger indeveloping countries, especially African nations, but smaller in developed countries.
3the young period. To gain insight into the declining birthrate, we must find the channel ofpolicy effects by accounting for the endogenous marital relationship of heterogeneous
spouses. The second feature is considering the effect of the pension system on fertility in a
family bargaining model in which the balance of power within the young couple is affected bysocial norms or peer pressure.6

In our model, the bargaining power depends on the

difference between the average lifetime income of men and women in the economy, andhence, the bargaining positions of marriage are affected by the PAYG pension system. This
reflects empirical evidence that social security affects the balance of power in a couple(Duflo, 2003). The third feature of our model is that fertility is determined endogenously.
Most studies based on life-cycle models of a household with multiple decision units focus onhousehold wealth or behaviors for the retirement period, with little interest in fertility. Here,
we formulate a model of family bargaining in which fertility is endogenous under the PAYG
pension system.
This study reveals a new channel of pension policy on fertility decisions; an increase inpension size affects fertility not only via the changes in current and future income, but also
through a change in marital bargaining power. Specifically, the study presents a plausible
argument that an increase in the PAYG pension further accelerates a decline in fertility
compared to the unitary model, in which the bargaining power of the couple is not of interest.
Increasing a generous pension system induces intra-household redistribution between
spouses with different longevities, as well as the inter-generational income redistribution
between young and old generations which is discussed in the conventional unitary model.
Since this redistribution from short-living husbands to long-living wives alters the balance ofpower of couples through the changes in their relative expected lifetime incomes, thischange favors wives who expect that they will live longer than their husbands will. Thechange in the balance of power within the couple affects their decision on the number ofchildren they have, because the wife has the longer lifespan, so that she has a larger
6

This concept is based on Grossbard-Shechtman (1984) and Lundberg and Pollak (1993),
both of which suggested that marriage relations are determined in a marriage market thatreflects cultures or social norms.
4incentive for saving by reducing expenditure in the young period.
This paper is organized as follows. Section 2 presents a model of two family memberswith different longevity. Section 3 explores the equilibrium of our model. Section 4 carries out
our policy analysis and Section 5 concludes the paper.

2.

Model

Consider a small open economy, comprising one representative household and agovernment. The household consists of two individuals (i  f , m) , where fand mdenote the female (wife) and male (husband), respectively. Each individual lives for at mosttwo periods: young and old. Although everyone can certainly live through the young period,
there is uncertainty in old age. For the sake of simplicity, we assume that the individual iseither alive or dead at the beginning of the old period. The probability that individual isurvives in the old period is denoted by i . To incorporate the gender difference inlongevity, we assume that the wife has a longer lifespan than the husband:  f   m . In theyoung period, individuals get married with the partner j ( j  i ), raise their children, andearn an income by supplying their time in the labor market. After paying tax, they allocate
their collective earnings among their private consumptions and saving for retirement. In theretirement period, they consume by making use of the pension benefit and the return fromtheir savings. The government employs the PAYG pension scheme for income distribution
from young to old generations. It imposes a tax on each household in the young period tofinance the pension benefit for the elderly living in the same period.

2.1 Household
Individual i in a household gains utility from consumptions in the young and old periods, aswell as the number of children. The expected utility function of individual i born at period t ,who belongs to generation t , is assumed to be:

EU it  ln nt  ln ct  i ln d t 1 ,(1)

where n t is the number of children of each gender who belong to the couple in generation5

t , which means that one unit of n t corresponds to a pair of son and daughter.7 In Eq.1,
c t and d t 1 are the couple's consumption of private goods in the young and old periods,
respectively.

We assume that the wife and the husband consume the same amount ofprivate goods. Our main results do not change if we remove this assumption. Even though
their preferences are identical, there exists a gender gap in longevity within their expectedutilities. The household welfare function is the sum of the weighted utilities of the spouses8:

Vt   t EU ft  (1   t ) EU mt ,(2)

where  t  0,1 represents the bargaining power of the wife. Following Chiappori (1988,
1992) and Apps and Rees (1988), we assume that household members can always achieve
an efficient allocation based on certain distributional rules within the household. Here,  t
can be interpreted as the distribution rule in our model.9
Each individual is endowed with one unit of time in their young period, and supplies
their time in market and domestic work. Childcare activities are the domestic production, so
the husband and/or wife commit time to the upbringing of the children. The fixed time forparental attention per child is denoted by z and the time spent on market work byiindividual i is denoted by Lt .

Although the results remain intact, based on the averagesof the observations, we assume that husband's wage is higher than the wife's: wm  w f .
Furthermore, we assume the parental times of the wife and the husband are completely
7This approach was also used in the theoretical models of Galor and Weil (1996), Abio et al.
(2004), Doepke and Tertilt (2009), and de la Crox and Donckt (2010). Abio et al. examined
the relationship between social security and fertility in the OLG model of two-earnerhousehold, but they treated household as decision unit and do not deal with the maritalbargaining.
8
Eq.1 is a simple expected utility function at the beginning of youth period. When
individuals enter into the retirement period, he (she) knows whether oneself and its partner
die or alive. There are four possible cases; both die, both alive, the husband dies but thewife alive, and the wife dies but the husband alive. However, individuals only know the
probability for survival before the fact (at the beginning of youth period). Since all decisions
are made at the beginning of youth period, the utility function is specified by Eq.1. Specifying
fm
fm
private consumption by separating c t into ct and ct , d t 1 into d t 1 and d t 1 , does
not affect our main results.
9
The classic formulation of intra-household bargaining was studied in the Nash bargaining
model (Manser and Brown, 1980; McElroy and Horney, 1981). They regarded the statusesof single or divorced as threat-points. The threat-points of the Lundberg and Pollak (1993)
bargaining model are statuses of non-cooperative equilibrium. Lundberg and Pollak (1996)
give an excellent survey on intra-family bargaining.
6

substitutable. Thus, it is the most efficient for a couple if the wife takes care of their children,and the husband's endowed time is spent solely on market work: Lt 1  znt andf

Lmt  1 . 10 Note that, in this paper, we assume that there is no substitution between
domestic and market childcare.
In the young period, the couple chooses the number of children they have, n t , andallocates the disposable income among their own private goods consumption, c t , and theirsavings for retirement, s t . The budget constraint of the household in the young period isgiven by:ct  w f znt  st  w f  wm  (3)where  denotes the lump-sum tax imposed on the household.11
Here, w f  wm   is assumed in the following analysis. In the retirement period, thehousehold members enjoy their private goods consumption, financed by the return fromtheir savings and the pension benefit. The budget constraint of the household in theretirement period is:

d t 1  Rst  ( f  m ) Pt 1 ,(4)

where Pt 1 is the pension benefit to each individual and

R

1 r
 f  m   f m(5)

is the return rate for the savings after one period, while r is the interest rate. Here, Eq.5
implies that the savings in the young period are returned to the household, as long as one
member of the couple survives. In Eq.4, the term ( f  m ) Pt 1 implies that the moreprobability that the household member i lives during the second period, the more likely
he/she can obtain the total pension benefit.
Given the bargaining power,  t , the household maximizes Eq.2 subject to Eq.3 and10

Easing the assumption of complete substitution does not affect our main results as longas the wife is better at caring for the children.
11
In order to make the analysis simpler and clearer, we assume the government employs
the lump-sum tax. In the case of income taxation, the model includes the substitution effectdue to a change in the opportunity cost of raising children and additional redistribution effectfrom husbands to wives through different labor income, but our main remark on theredistribution effect caused through gender difference in longevities remains intact.
7

Eq.4. Solving the optimization problem, we have the following demand functions:

ct 

1
It ,
2   t  f  (1   t )md t 1 

nt 

st 



R  t  f  (1   t ) m

(6)

I ,

2   t  f  (1   t ) m

(7)

t

1
It ,
w f z 2   t  f  (1   t )m









(8)

R  t  f  (1   t ) m w f  wm     2 f   m Pt 1

R 2   t  f  (1   t ) m



,

(9)

where I t  w f  wm    ( f  m ) Pt 1 R is the net lifetime income for the couple. The
1demand functions show that, as the bargaining power of the wife rises, the private goods
consumption in the young period and fertility fall, while the savings, and thus, theconsumption in the old period, increases: ct  t  0 , nt  t  0 , st  t  0 andd t 1  t  0 . Intuitively, there is a conflict between spouses in terms of lifetime goals
because the wife thinks she will outlive her husband, while the husband believes the
opposite. Hence, the wife wants to save the more money for retirement than her husband. If
the wife's bargaining position becomes more favorable, the household outcomes are more
likely to reflect her intentions. Consequently, the private goods consumption in retirement
age increases by reducing consumption in the young period, as well as the number ofchildren they have.

2.2 PAYG Pension
The government operates the PAYG pension system. It imposes a tax on each household in
the young period so that it can finance the pension benefit for people living in the old period.
The government's budget constraint in per household terms is given by:nt  ( f  m ) Pt 1 .

(10)

2.3 Bargaining Power8

We assume that the power balance of young couples is shaped within the marriage market
and is affected by social norms or peer pressure (Grossbard-Shechtman, 1984, 1993, 2013;
Lundberg and Pollak, 1993; Komura, 2013a, 2013b).12

They anticipate the balance ofpower within the marriage by observing the behaviors and economic relations of theirparents. Specifically, the balance of power of the couple in generation t depends on the
difference in their average resources of men and women in generation t  1 , including theincome from the pension benefits of men and women in the preceding generation: t   w f Ltf1  wm Lmt1   ( f  m ) Pt R 1  ,(11)where    0 is assumed. For simplicity of notation, we assume    0 . In Eq.11, the term

w f Ltf1  wm Lmt1 represents the gap in labor income, and the term  ( f  m ) Pt R 1
represents the gap in the expected pension benefits that affect the bargaining power of thecouple.

  0,1 captures the degree of how the pension policy (or income in theretirement period) affects the balance of power within the marital relationship. If   1 , thegap in the expected pension benefits and the gap in labor income have the same effect on
the bargaining power. Therefore, the determinant of power for the couple simply becomes
the difference between the expected lifetime incomes of men and women. In contrast,  0 reduces our model to that of Komura (2013a, 2013b) essentially, in which thepension policy has no impact on the bargaining power.
Recalling that the husband spends his time solely in the labor market, Eq. 11 can be
rewritten as: t   w f (1  znt 1 )  wm   ( f  m ) Pt R 1 ,(12)where nt 1 stands for the average number of children per household in generation t  1 .
Note that the wife's bargaining power decreases as the average number of children in thesociety increases,  t nt 1   w f z  0 . This implies that having children by couples in
the previous generation weakens the wife's say in the next generation, because a reductionin her earning is expected from peer pressure. It is also worth mentioning that the expansion
12

Using data of European countries and Japan, Feyrer et al. (2008) pointed out thatwomen's status is affected not only by common economic factors but also by thelongstanding cultural and social factors.
9

of a public pension, which mainly aims to transfer income from young to old, plays a role in
the income transfer from husbands to wives in our model, and hence, it increases the wife's
bargaining power,  t Pt    ( f  m ) R

3.

1

0.

Equilibrium

3.1 Dynamics
Using Eqs.8, 10 and 12, the dynamics of bargaining power can be obtained as: t 1    w f  wm R( w f  wm   ) A,w f z 2   t  f  (1   t )m R   





(13)

where

A  zw f  ( f  m ).
R ( f   m )(14)

Differentiation gives:13
2
 t 1 ( f  m ) z nt w f A, t
w f  wm  

(15)

2 2
2 3
 2 t 1 2 z w f ( f  m ) nt  A

.
 t2
( w f  wm   ) 2(16)

If  is sufficiently small, the sign of Eq.14 tends to be positive, A  0 . This corresponds tothe case in which the pension policy is inactive, or   0 . In contrast, when the pension
policy is active and  is sufficiently large, Eq. 14 is likely to take a negative sign, A  0 .
Supposing that  t 1   0  0 , Eqs. 15 and 16 reveal that the bargaining power convergestmonotonically (cyclically) to the steady-state if A  ()0 .14[Figures 1(a) and 1(b) are here]

Behind the dynamics of the bargaining power, the dynamics of fertility can be derived in
13
14

See Appendix A.
The stability of the steady-state is ensured by assuming  t 1  t  1 .
10

a similar way to show that the fertility converges to the steady state monotonically(cyclically) if A  ()0 .

3.2 Steady state
Using Eqs. 8, 10 and 12, the steady-state value of the bargaining power and the fertility
satisfy: n( f  m ) w f zn  w f  wm  , ( f  m ) R   n

R ( w f  wm   )

w f z 2   f  (1   )m R  

.

(17)

(18)

To plot combinations ( , n) that satisfy Eqs. 17 and 18 we first reveal how the fertility
affects the bargaining power in the steady state equilibrium. From Eq. 17 and    0 , we
have:


  A,
n

(19)

 2
0.
n 2

(20)

In Eq.19, if  is sufficiently small, then A  0 , the effect of fertility on the bargaining power
is negative,  n  0 .This implies that having another child reduces women's bargaining
position since the wife is forced into child rearing, reducing her income from the labor market.
Having a child as a factor against women's bargaining power is captured by the first term in
Eq. 14. If   0 , the sign of  n depends on the relative magnitude of two terms. The
second term in Eq. 14 captures the positive effect of having a child on the wife's bargaining
power, owing to the PAYG pension system. The more children there are in society, the more
the pension benefit increases for the elderly. Because the wife is more likely to survive in theretirement period, her expected benefit from the PAYG pension system is higher than that ofher husband's. As the weight of pension benefit in the bargaining power increases, owing toan increase in fertility, the wife becomes invulnerable, strengthening her power within the11couple.

[Figures 2(a), 2(b), 3(a) and 3(b) are here] (n) in Figures 2(a) and 3(a) represents Eq. 17 when the pension policy is inactive,i.e.,  is sufficiently small to cause the sign of Eq. 19 negative. In contrast,  (n) in
Figures 2(b) and 3(b) represents the alternative case, in which the pension policy is active,  0 leading the sign of Eq. 19 to be positive.
We next study how fertility is related to the bargaining power in the steady-state. From
Eq. 18, differentiating n with respect to  gives:15

n 2 zw f ( f  m )
n

 0,

w f  wm  

(21)

2

 nzw f ( f  m ) 
 2n
n

2

  0.
 2
 w f  wm   

(22)

Eq. 18 is illustrated as n( ) in Figures 2 and 3, showing that the number of children the
couple has decreases as the woman's bargaining power increases. This is simply because
the wife lives longer than the husband does. As women are likely to live longer, they want tosave more money for future consumption rather than spending it on raising children. In suchsituation, therefore, a rise in women's bargaining power leads to a fall in fertility rate,
reflecting their intentions in household decisions.

4.

Effects of Pension Policy

In this section, we examine the effects of the changes in the size of a PAYG social security
system on fertility and women's bargaining power, focusing on the stable steady-state
equilibrium of

E1 in Figures 2(a) and 3(a), and E5 in Figures 2(b) and 3(b).

First, we come back to the traditional argument in which the bargaining power is fixed15

See Appendix A.
12(    ).

Using Eq. 18, this can be confirmed by differentiating n with respect to  ;16dnn( R  n )

.
d 
R( w f  wm   )(23)Here, Eq. 23 shows that, given bargaining power  , an increase in the tax rate reduces(increases) the fertility if R  ()n .
An increase in the size of the pension policy, represented by  , affects fertility via thechange in the lifetime full income in two ways: (i) the reduction in the disposable income of
the working period decreases the number of children, and (ii) the increase in the pension
benefit allows the household to reduce their savings for the old period and increases theirfertility. In other words, the comparison between

R and n means whether the present

value of leaving  as disposable income in the young period is larger or smaller than thepresent value of the pension benefit as a return of tax payments. The well-known
Aaron-Samuelson condition states that future generations benefit from the PAYG pension
system as an inter-generational transfer when the economy is dynamically inefficient,1  r  n (Samuelson, 1958; Aaron, 1966). While the interest factor is adjusted by thelongevity in our model, there is strong support for the government operating the PAYG
pension system, and that examining the effects of pension policy is relevant if R  n . WhenR  n , the positive effects of a pension expansion on lifetime income overwhelms thenegative effects. As a result, an increase in lifetime income induces a rise in the number ofchildren, as children are normal goods in our model.
On the other hand, if the economy is dynamically efficient, 1  r  n , the introduction of
the lump-sum financed PAYG pension scheme basically loses its theoretical foundation.
However, as summarized by Groezen et al. (2003), the political incentives may promote theintroduction of an unfunded pension scheme or, a drastic policy reform is often difficult froma practical standpoint, even if the environment surrounding the economy becomes against a
PAYG pension. If the PAYG pension system is operated under the condition thatR  1  r  n , and the government increases the size of the pension policy, the negative
16

See Appendix A.
13

effect on fertility of a decrease in disposable income in the young period outweighs the
positive effect of an increase in pension benefit, which results in a fall in fertility rate.
In Figures 2(a) and 2(b), the policy effects on the fertility, given  , are depicted by theshift of the n( ) curve.

When R  n , the sign of Eq. 23 becomes positive, which shifts

the curve from n( ) to n ( ) . In contrast, if R  n , Eq. 23 becomes negative, and the1

curve n( ) shifts left (see Figures 3(a) and 3(b)).
The important feature of our model is that the pension policy influences the bargaining
power in the couple, which also has an impact on the fertility.

From Eq. 17, given n( n  n ), we find that, as the pension policy increases, so does the woman's bargaining
power:

 n( f   m )
d
 0.

( f   m ) R
d n

(24)

This indicates that an increase in the size of a pension policy, represented by  , induces anupward shift of the  (n) curve in Figures 2 and 3. This is simply because the increase inpension works to the longer-living woman's advantage, increasing her bargaining power. In
this case, as  is larger, the pension benefit is appreciated in determining the balance ofpower between men and women, so that the change in the pension policy affects significantly. Similarly, as the gender gap in life expectancy  f  m is larger, the gap in the
expected pension benefit between men and women widens, resulting in a significant changein  . In such situations, an increase in  induces a relatively large upward shift in  (n) .
The steady-state equilibrium, n*and  , satisfy Eqs. 17 and 18. We first explain the*effects of an increase in  on ( n ,  ) when R  n , i.e.,**dn d    0 in Eq. 23.

Figures 2(a) and 2(b) show the cases in which, given other variables, the effect of anincrease in pension size on fertility is modest, while an increase in the pension has a great
impact on bargaining power. This case tends to take place when

 and  f  m arelarge as explained above. In Figure 2(a), for instance, if we assume that the bargaining
power is an exogenous parameter, then the increase in the size of the pension increases
fertility along the course from

E1 to E3 . However, as the bargaining power is endogenous14

in our model, the woman's bargaining power is strengthened by the increase in the size ofthe pension, moving the stable equilibrium from

E1 to E4 . This shows that a rise in 

induces a fall in fertility. The same holds for Figure 2(b). If the bargaining power isexogenous, the increase in  raises fertility, along the course from E5 to E6 . However, it
decreases fertility by shifting the equilibrium from E5 to E7 as the bargaining power of thewife increases. Consequently, the fertility rate in the economy may fall if the bargaining
power is determined endogenously.
The case of R  n , i.e., dn d    0 , can be interpreted in a similar fashion, by
making use of Figures 3(a) and 3(b). Both figures show that fertility decreases not only
through the decrease in lifetime net income but also through the increase in the bargaining
power of the wife, if the pension policy changes the balance of power significantly.
Some empirical studies with unitary models support the case in which the pension
expansion by an increase in  causes a decline in fertility (Cigno and Rosati, 1996; Boldin
et al., 2005). Our model shows that the fertility rate falls not only because of the negative
income effect (i.e., the leftward shift of the n( ) curve) but also the negative bargaining
effect (i.e., the upward shift of the  (n) curve). If there is no heterogeneity within ahousehold,  f  m , the pension policy has no influence on the bargaining power (see Eq.
24), and thus the effects of the pension policy are essentially the same as that the traditional
unitary model found. This is because the spouses' lifetime objects are identical, so that they
do not need to negotiate household allocations. In the real economy with gender differences,
if policy makers ignore the bargaining power effect, the negative effects of a pension reformon fertility could be biased or could reverse the sign of the impact estimated initially.
Using Eqs. 17 and 18, the graphical analysis mentioned above can be formally restatedas follows:dn*  n n   1 n 
   n   n   ,d     (25)d *    n  1     n n  ,d   n  


(26)

15



where   1  n  n  0 and  km ( k  m ) denotes the elasticity of k ( n, )with respect to m(  , n,  ) .17
The first term in Eq.25 represents the effects of an increase in  on fertility throughchanges in lifetime full income, which is regarded as an inter-generational distribution effectby conventional studies. The sign can be positive or negative, depending on the relative
magnitude between

R and n as in Eq.23. The second term in Eq.25 stands for the effectcaused by a change in the power balance within a household, and can be interpreted as an
intra-generational distribution effect. As other studies on intra-generational distribution
effects, this effect is caused by heterogeneity among individuals in the same generation, butthat within the family. Because the pension expansion by an increase in  leads womenwith higher longevity to a more favorable position, the sign of   is positive. On theother hand, n  is negative because women want to reduce the number of children sothat they can ensure resources in the retirement period, taking into account their higher
probability to survive. Thus, overall, the second term is negative. The sign of the total effectof the change in  on n , therefore, is not determined a priori. If the negative effect of an*increase in pension size on fertility due to a change in bargaining power exceeds the
positive effect of the increase in  , then fertility is decreased by an increase in  .
Now, we turn to the total effects of a change in  on  . The first term in Eq.26*

captures the direct positive effect of a change in  on bargaining power by changing the
gap of the pension benefit between wife and husband. The second term in Eq.26 represents
the indirect effect through a change in fertility caused by an increase in  . The sign of n depends on the relative magnitude of the effects of n on women's bargaining
power as a result of reduced labor income and an increased pension benefit, as comparedto that of their husbands. Consequently, the sign of the overall effect is determined by these
direct and indirect effects.

5.

Conclusion

17

See Appendix B.
16

The family is the key constitutional unit of human society. It offers comfort, security, and aplace to grow. However, the role and the structure of family change according to the
influence of many factors, such as changing lifestyles and increasing personal mobility.
Public policy is also a significant factor affecting the family shape. The decision on having
children must be affected by the system of childcare leave, educational costs, and various
family policies. The systems of taxation and social security are also factors that influence theway couples work and the balance of power within the family. This paper can be placed as avariant on a line that examines the intra-familial structure, focusing on the balance of power
and the number of children in the family.
There has been intense research into the effects of pension policy on the fertility. Most
studies approach this matter using an overlapping generation (OLG) model with a unitary
household, assuming no interaction between wife and husband. These studies are
successful in analyzing long-run macroeconomic steady-state outcomes. At the same time,these standard approaches rely on some strong assumptions. In particular, they assume aunitary household with no heterogeneity in preferences or the lifespans of wives andhusbands. This means they assume that a couple never bargains over household resource
allocation, such as the number of children they have and/or the amount they save for the
future. This paper approaches from different perspective than the orthodox homogenous
couple, to explore the effects of a public pension on the household resource allocation by
the bargaining couple with heterogeneity.
Following the trend of analyses on the heterogeneous couples, we also consider asetting with heterogeneity in the lifespan between husband and wife. The wife tends to live
longer than her husband, causing incentives for them to bargain over the amount of saving
they do and the number of children they have. The bargaining power between wife andhusband is endogenously determined in the social level based on their relative average
lifetime income, including pension benefit. This is affected critically by a pension reform,
which may in turn influence endogenous fertility. To demonstrate our results simply, we
consider a small open economy characterized by an exogenous interest rate and wages.

17

The interest rate is the same for both the husband and wife, but the wage rate is not. Men
tend to get a higher wage rate than the women, therefore the women often decide against
participating in labor market.
Using this setup, we find out a new channel of pension policy on fertility decisions. An
increase in pension size affects not only via the changes in current and future incomes, but it
affects the fertility through the change in marital bargaining power. The conventional OLG
literature has always seen marital bargaining power as fixed, which meant that the increasein the pension benefit of the old accompanied by the tax increase in the young simply
changes the lifetime income. As a result, this change in lifetime income caused by anincrease in the pension size affects the household fertility behaviors. In our model, however,the development of a pension alters the marital relationship defined by the gender gap inlifetime incomes, because the wife lives longer and is expected to gain higher amount ofpension benefit. The change in the balance of power within the heterogeneous couple
affects their saving behavior as well as their fertility. This results in the PAYG pension
accelerating the falling birthrate, in contrast to the case of homogenous couples.
In closing this paper, we briefly discuss the decision of the pension size in majority
voting and its optimality. Suppose an economy where the size of pension system isdetermined by voting at every period. Every individual living in t period can vote for thedecisions. It is easily expected that the individuals who enter the old period prefer toincrease the tax rate as high as possible, because they realize that they surely survive thewhole period once they enter the old period. On the other hand, individuals in the young
period would choose the optimal level of pension size for themselves  i , which maximize*

their own expected indirect utility functions Vt  ln ct ( )  ln nt ( )  i ln d t 1 ( ). Becausei f  m and the pension policy in our model is more beneficial for the longer-living
individuals, women prefer the larger pension size,  f   m . In sum, the preferred tax rate**

(pension size) of the elderly, the young women and the young men are the highest possiblelevel,  f and  m , respectively. Since the relative voting power (population) is  f  m , n**

and n , for decision branch of highest possible level,  f and  m , the political equilibrium as
*

18

*

the Condorcet winner is  f . However, if we define the social welfare function as the sum of*

expected life-time utilities of a man and a woman which are weighted equally as Vt  Vt ,fm

it is obvious that the tax rate in the political equilibrium is larger than the social optimallevel.18

Acknowledgement
This research has been supported by 2013 Kampo-Zaidan Grants.

Appendices
Appendix A
Derivation of Eqs 15 and 16.
From Eq. 13, we have2
 t 1  R ( w f  wm   ) Aw f z ( f   m ).
 tw f z 2   t  f  (1   t )m R   2





(A1)

From Eqs. 8 and 10, we have

nt 



R ( w f  wm   )

w f z 2   t  f  (1   t )m R  

.

(A2)

Substituting Eq.A2 into Eq. A1, we have Eq. 15.
The differentiation of Eq. A1 gives
2 2 23
 2 t 1 2( f  m ) w f z  R ( w f  wm   ) A

.
 t2
w f z 2   t  f  (1   t )m R   3





(A3)

Substituting Eq.A2 into Eq.A3, we obtain Eq.16.

Derivation of Eqs. 21 and 22.
From Eq. 18, we have18

See Leroux et al. (2011) for similar analysis of political equilibrium with marital statuses
and gender difference in longevity and productivity.
19

w f zR 2 ( f  m )( w f  wm   )
dn

 0,
d
w f z 2   f  (1   )m R   2





2 2
2 3
d 2 n 2 w f z ( f  m ) R ( w f  wm   )

 0.
d 2
w f z 2   f  (1   )m R   3





Eq. 18 is rewritten as

2   f  (1   )m 

R( w f  wm   )  n
nw f zR 0,which is used to derive Eqs. 21 and 22.

Derivation of Eq. 23.
From Eq. 18, we have

( w f  wm ) R  2   f  (1   )m zw f R 2dn.
d d 0
w f z 2   f  (1   )m R   2



Substituting Eq. 18 into this equation, we have Eq. 23.

Appendix B
Derivation of Eqs. 25 and 26.
Fertility and bargaining power in the steady-state are given by total differentiation:

dn* 



n * n
d 
d ,



 *

dn  d * 
d .
n


Using these equations, we have n    dn*   n  
 1
d ,


  n
1  d *    where the determinant of the coefficient matrix is 1  (n  )( n) , which is reasonable
to assume positive. Solving this, we have Eqs. 25 and 26.

20

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24

Figure 1(a). Steady-state bargaining power; A  0 .

Figure 1(b). Steady-state bargaining power; A  0 .

25

Figure 2(a). Effect of increase in  when R  n and the pension policy is inactive, A  0 .
Note. An increase in the size of pension policy shifts a stable equilibrium from E1 to E4 .

E2 is an unstable equilibrium.

26

Figure 2(b). Effect of increase in  when R  n and the pension policy is active, A  0 .
Note. An increase in the size of pension policy shifts a stable equilibrium from E5 to E7 .

27

Figure 3(a). Effect of increase in  when R  n and the pension policy is inactive, A  0 .
Note. An increase in the size of pension policy shifts a stable equilibrium from E1 to E4 .

E2 is an unstable equilibrium.

28

Figure 3(b). Effect of increase in  when R  n and the pension policy is active, A  0 .
Note. An increase in the size of pension policy shifts a stable equilibrium from E5 to E7 .

29