Theoretical Economics Letters, 2011, 1, 95-98
doi:10.4236/tel.[phone] Published Online November 2011 (http://www.SciRP.org/journal/tel)
Multiple Equilibria between Fertility Rates and Pension
Levels Based on the Target Level of Government Debt
Hideki Nakamura1, Masaya Yasuoka2
1Faculty of Economics, Osaka City Un iversity, Osaka, Japan
2Faculty of Economics and Business Administration, the University of Kitakyushu, Kitakyushu, Japan
Received August 1, 2011; revised August 26, 2011; accepted September 2, 20 11
Abstract
We show that when the government has a target debt level, multiple equilibria exist in the relationship be-
tween fertility rates and pension levels. One is associated with a high fertility rate and a high pension level.
The other is associated with a low fertility rate and a low pension level. If the government fails to provide
adequate security for individuals during their retirement years, it would result in a failure of coordination
between the government and individuals.
Keywords: Fertility, Pensions, Target Level of Government Debt, Multiple Equilibria
1. Introduction
The fertility rate in Japan has been decreasing, while the
ratio of elderly people to the total population has been
increasing. In addition, the Japanese government has
attempted to reduce the number of issued government
bonds by implementing the Reform of Fiscal Structure
Law in 1997, because the government debt in Japan had
increased significantly during the 1990s. This led to con-
siderable controversy over the sustainability of the pay-as-
you-go (PAYG) social security scheme in an aging society
with fewer children and a large public debt1.
Should the government reduce the pension level in
order not to increase the government debt as the pen-
sionable proportion of the population rises? This paper
assesses the possible relationship between the level of
pension and the fertility rate when the government has a
target for restricting the level of government debt. Pen-
sion levels affect lifetime income, which in turn affect
the fertility rate. We show that given income taxes or
consumption taxes, there exist multiple equilibria between
pension levels and fertility rates. If the government
chooses to set pensions at a low level, th e fertility rate of
those dependent on such pensions decreases during their
child-rearing years. A low pension level means that indi-
viduals cannot count on a reasonable income during their
retirement years. Thus, they would seek to have fewer
children during their younger years so as to reduce their
household living expenses. The government then cannot
increase the pension level becau se of the low level of tax
revenue. On the other hand, if the government chooses to
fix pensions at a higher level, this would tend to increase
the fertility rate, as the prospect of larger pensions en-
ables individuals to consider having more children. The
government, in due course, obtains more tax revenue,
because a higher fertility rate eventually expands the size
of the taxpaying labor force relative to the pensionable
group. Thus, the government should provide adequate
security for individuals during their retirement years not
to cause a failure of coordination between the govern-
ment and individuals2.
We must emphasize the following points. Assuming a
balanced government budget, Zhang [1], Groezen, Leers,
and Meijdam [2], Groezen and Meijdam [3], and Hi-
razawa and Yakita [4] examined the effects of the PAYG
social security scheme on fertility and welfare. As in
their models, our model also considers endogenous fer-
tility, in that children are assumed to be consumption
goods. By introduc ing a target level of government debt,
Futagami, Iwaisako, and Ohdoi [5] showed that multiple
equilibria exist in the relationship between productive
government spending and economic growth. Considering
1Member countries of the European Union must maintain a ratio o
f
government debt to GDP less than 60% in line with the Maastricht
Treaty. In addition, a decline in the fertility rate and growing public
debt are also evident in South Korea and Germany.
2In Japan and some European developed countries, including Italy,
Germany, France, and Sweden, the correlation between the fertility rate
and the ratio of pension benefit to labor income is high.

H. NAKAMURA ET AL.
96
1t
a situation in which population aging causes a heavy
burden of social security payments, Ono [6] showed that
the effects of pension reform and population aging on
capital accumulation are different between potential two
equilibria. This paper shows that multiple equilibria exist
in the relationship between the pension level and the fer-
tility rate because the existence of a target level for gov-
ernment debt yields a complementary relationship be-
tween the government and individuals through the pen-
sion system.
2. Model
We consider a small , open, overlapping-generations model.
Individuals live in three periods. In the first period, the
cost of child rearing is p aid for by parents. In the second
period, during which individuals are working, they de-
cide how much to consume, how much to save, and how
many children to have. They have to pay income taxes
during this time3. In the third period, they receive the
interest payments and a pension and consume their in-
comes. The government collects income taxes and issues
bonds used for providing pensions while maintaining a
target level of government debt. Firms are assumed to be
perfectly competitive. The interest rate which is exoge-
nously given is assumed to be constant. The wage rate
also remains constant.
2.1 Individuals
The utility maximization problem for an individual born
in period t-1 is expressed as:

121 12
,,
max lnln1ln
tt ttt
nc cnc c
 


, (1)

21
1
.. 1
t
tt
cp
st cznw
r



r
, (2)
where 0

, 0

, and 1




c. t is the number of children born in period t, 1t and 21t are the
consumption levels in periods t and t+1, respectively,
nc

is the income tax rate, is the pension level, is the
cost of child rearing, is the gross interest rate, and
is the wage rate.
p
rz
wThe first-order conditions of the utility maximization
problem yield:

11 1 tp
cc w
r


 



21211 tp
ccr w
r
 


 


,
(4)

1.
tp
nn w
zr



 
,


(3)


 (5)

The relationships between consumption in periods t
and t+1 and the fertility rate in period t are linear be-
cause of the log -linear function of utility. Th e number of
children is proportionate to their parents’ lifetime in-
come.
2.2. The Government
The government finances the pension system by two
methods: levying income taxes and issuing bonds. Its
budget constraint can be written as:
11
,
ttt t
BrBNpNw




where t is the government debt in period t, and
is the population born in period t-1.
Bt N
GDP per capita is constant in our model. Thus, we
consider the debt-per-capita dynamics4. These dynamics
are represented as:
12, tt
rpw
bb
nnn

 (6)
where ttt bBN

, which is public debt per capita, and
11tttt
NNNN n


.
The government is assumed to have a target level of
debt per capita, which is represented by b>0. By con-
trolling the income tax rate and the pension level, the
government tries to satisfy b which is represented as:
2.
rpw
bb
nnn

 (7)
When the government follows (6), a negative primary
balance implies that r/n<1 must hold for convergence.
3. Multiple Equilibria
Figure 1 shows the relationship between the fertility rate
and the pension level represented by (5) and (7). Given
b and

, (5) implies a linear relationship. A rise in the
pension level causes an increase in the fertility rate, be-
cause it increases the level of lifetime income. Since
higher pensions provide individuals with added security
after their retirement, it allows them to have more chil-
dren. On the other hand, (7) represents a nonlinear rela-
tionship between the fertility rate and the pension level.
When the fertility rate is low, the government can pay
only low pensions. It is difficult to maintain the target
level of government debt because of the low level of tax
revenue. A higher pension level becomes possible with a
3It is widely argued in Japan that consumption taxes should be in-
creased because of growing public debt. Assuming consumption taxes
obtains the same results.
4If we assumed technological progress exogenously, we would con-
sider the dynamics of ratio of public debt to national income. How-
ever, our res ults would remain intact.

97 H. NAKAMURA ET AL.
Figure1. Relationship between fertility rate and pension
level.
higher fertility rate because of the higher level of tax
revenue. It is easier for the government to maintain its
target level of debt.
Using (5) and (7), the fertility rate in equilibrium can
be represented as:

1
12
2
24
l
nAAB




,



(8)

1
12
2
24
h
nAAB




,



(9)
where 1
1
zb
Awwr







and

1
b
Brw

. We
assume that and . 0A240AB
There exist multiple equilibria between the fertility
rate and the pension level. Equatio n. (7) implies l and
h which correspond with l and h, respectively.
When the pension level represented by l is low, the
fertility rate represented by l is low. On the other
hand, when the pension level represented by h is high,
the fertility rate represented by h is high. If the gov-
ernment can make young individuals believe they will
receive the large pension, they will have more children.
The government can then provide higher pensions.
However, if the government fails to make them believe
this, they will choose to have a lower number of children.
This will eventually force the government to provide the
low pension level to maintain the budget. Thus, if the
government provides a lower pension level, it would
cause a failure of coordination between the government
and individuals. In addition, although the level of labor
income is identical in both equilibria, the income level
during the retirement with the h igh fertility rate is higher
than that with the low fertility rate. This is because a
higher pension level is possible.
p
p n
n
np
p
n
Let us examine the comparative statistics:
0
l
nb , 0
h
nb , 0
l
n

 and 0
h
n


(10)
First, an increase in the target debt level implies that
the line representing (7) shown in Figure 1 shifts down-
ward, because the burden of debt on the government
budget decreases. Consequently, an increase in the target
debt level then tends to raise the low fertility rate and
lower the high fertility rate. If the government sets a low
target level to avoid incurring a large debt in an aging
society with fewer children, it will reduce the already
low fertility rate. Note that given the tax rate, a high tar-
get level would make it difficult for the government to
maintain its budget, and the equilibria may disappear.
Next, an increase in the income tax rate implies that
when the line representing (5) in Figure 1 shifts down-
ward, the line representing (7) also shifts downward.
Here, although an increase in the tax rate may induce
individuals to have fewer children, it also allows the
government to fund higher pensions. In our model, the
latter effect dominates the former. Consequently, al-
though an increase in the tax rate decreases the high fer-
tility rate, it increases the low fertility rate.
4. Conclusions
When the government has a target level of debt, there
exist multiple equilibria between the fertility rate and the
pension level because of a strategic complementary rela-
tionship between the government and individuals. If the
government cut the level of pensions so as not to in-
crease the government debt, it would take away relief for
individuals’ old age, and, thereby, it would cause a fail-
ure of coordination between the government and indi-
viduals.
5. References
[1] J. Zhang, “Social Security and Endogenous Growth,”
Journal of Public Economics, Vol. 58, No. 2, 1995, pp.
185-213. doi:10.1016/0047-2727(94)01473-2
[2] B. V. Groezen, T. Leers, L. Meijdam, “Social Security
and Endogenous Fertility: Pension and Child Allowances
as Siamese Twins,” Journal of Public Economics, Vol. 87,
No. 2, 2003, pp. 233-251.
doi:10.1016/S0047-2727(01)00134-7
[3] B. V. Groezen, L. Meijdam, “Growing Old and Staying
Young: Population Policy in an Aging Closed Economy,”
Journal of Population Economics, Vol. 21, No. 3, 2008,
pp. 573-588. doi:10.1007/s00148-006-0067-x
[4] M. Hirazawa, A. Yakita, “Fertility, Child Care Outside
the Home, and Pay-as-You-Go Social Security,” Journal
of Population Economics, Vol. 22, No. 3, 2009, pp. 565-
583. doi:10.1007/s00148-007-0153-8
[5] K. Futagami, T. Iwaisako, R. Ohdoi, “Debt Policy Rule,
Productive Government Spending, and Multiple Growth
Paths,” Macroeconomic Dynamics, Vol. 12, No. 4, 2008,
pp. 445-462. doi:10.1017/S1365100508070235

H. NAKAMURA ET AL.
98
[6] T. Ono, “Social Security Policy with Public Debt in an
Aging Economy,” Journal of Population Economics, Vol. 16, No. 2, 2003, pp. 363-387.