Multiple Equilibria between Fertility Rates and Pension Levels Based on the Target Level of Government Debt

doi:10.4236/tel.2011.13020

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Theoretical Economics Letters, 2011, 1, 95-98

doi:10.4236/tel.2011.13020 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

E-mail: [email protected], [email protected]

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

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.

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)



.. 1

st cznw







, (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.

wThe first-order conditions of the utility maximization

problem yield:



11 1

cc w





 







21211

ccr w

 





 





(4)



nn w







 





(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:

ttt t

BrBNpNw









where t is the government debt in period t, and

is the population born in period t-1.

GDP per capita is constant in our model. Thus, we

consider the debt-per-capita dynamics4. These dynamics

are represented as:

12,

rpw

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:

rpw

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.

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:



nAAB









,







(8)



nAAB









,







(9)

where 1

Awwr















and



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 n

Let us examine the comparative statistics:

nb

, 0

nb

, 0



 and 0





(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.

[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.