Social security tends to be unsustainable in nature. It reduces individuals'' demand for children as a measure to support their lifestyle during old age, which in turn undermines the financial basis of social security. Using a simple overlapping generations model with endogenous fertility and income transfer from children to parents, we discuss the maximum size of a pay-as-you-go social security program that can prevent a cumulative reduction of fertility and make a program sustainable. We also show that a child-care allowance raises the maximum size of the program and raises an individual''s lifetime utility.social security, fertility, intergenerational income transfer

Masaya Yasuoka

Takashi Oshio

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 2 
  
Maximum size of social security in a model of endogenous fertility 
  
 
 
Takashi Oshio  
Hitotsubashi University 
Masaya Yasuoka  
The University of Kitakyushu
Abstract 
Social security tends to be unsustainable in nature. It reduces individuals'' demand for children as a measure to support 
their lifestyle during old age, which in turn undermines the financial basis of social security. Using a simple overlapping 
generations model with endogenous fertility and income transfer from children to parents, we discuss the maximum 
size of a pay-as-you-go social security program that can prevent a cumulative reduction of fertility and make a 
program sustainable. We also show that a child-care allowance raises the maximum size of the program and raises an 
individual''s lifetime utility.
Research for this paper was supported financially by a Grant-in-Aid for Scientific Research (No.19530237) and the University of Kitakyushu. 
The authors are thankful to Hisakazu Kato, Motohiro Sato, Toshiyuki Uemura, Akira Yakita, and Shinji Yamashige and seminar participants at 
the seminar held at Hitotsubashi University. Nevertheless, any remaining errors are the authors'' responsibility. 
Citation: Takashi Oshio and Masaya Yasuoka, (2009) ''Maximum size of social security in a model of endogenous fertility'', Economics 
Bulletin, Vol. 29 no.2 pp. 644-654. 
Submitted: Apr 18 2009.   Published: April 25, 2009. 
 
     2 
 
1.  Introduction 
 
Declining fertility puts strong pressures on the sustainability of social security. Most 
advanced countries have instituted pay-as-you-go (PAYG) social security programs, 
which rely heavily on contributions from young and future generations. Because a 
decreasing number of children is most likely to make programs less sustainable, many 
policymakers now call for child-care support, which is expected to prevent fertility 
from declining further. Indeed, various studies have examined the effectiveness of 
child-care support to mitigate the negative impact of low fertility on social welfare 
(Groezen, Leers, and Meijdam, 2003; Fenge and Meier, 2005; Hirazawa and Yakita, 
2008).  
However, social security tends to be unsustainable or even self-destructive in 
nature. The old-age security hypothesis, which treats children as capital goods for the 
material support during old age, implies that social security reduces demand for 
children (Zhang and Nishimura, 1993). This is also the case if we interpret a PAYG 
program as insurance against not having children (Sinn, 2004). Social security 
provides older individuals with financial support, at least partially substituting children. 
A reduced motive for having children reduces fertility and renders the financial base of 
social security vulnerable.   
The negative feedback loop between social security and fertility, which is inherent 
in social security, must not be ignored, especially if sustainability of social security 
confronts an imminent risk under conditions of declining fertility. In this study, we 
explicitly address the risk of a cumulative reduction in fertility and discuss how to 
prevent social security from collapsing, exclusively examining the role of children as 
capital goods for support during old age.   
To this end, we explore a simple overlapping-generations model with endogenous 
fertility and income transfer from children to parents. Incorporating the old-age gift 
into the model of endogenous fertility, Zhang and Zhang (1998) and Wigger (1999) 
show that social security programs, if small sized, can stimulate per capita income 
growth, but not otherwise. We extend their analysis to examine explicitly the 
maximum size of social security that can prevent fertility from cumulatively declining 3 
 
and prevent social security from collapsing.  Moreover, we show that social security 
reduces utility. 
The remainder of this paper is constructed as follows. Section 2 presents a basic 
model and discusses the benchmark state that exists before introducing social security. 
Section 3 introduces a PAYG social security program and examines the dynamics of 
fertility and necessary conditions to make social security sustainable. Section 4 
concludes. 
 
2.  Before introducing social security 
 
We consider a simple overlapping generations model, in which individuals live in two 
life periods, respectively, when they are young and old. Individuals treat children 
solely as capital goods for material support during their old age; there are no altruistic 
motives
1. We start with the case in which no social security program exists. Each 
individual maximizes lifetime utility: 
() ( ) . 1 0 , ln 1 ln , 2 1 2 1 < < − + = = γ γ γ c c c c u u  (1) 
Therein, c1 and c2 respectively signify consumption in young and old age periods. The 
budget constraints are given as 
( ) [] , 1 1 s w n c c − − − = θ  
() , 1 1 1 2 n w s r c + + + + = θ  
for each life stage, where s, w, r, n, θ, and c(n) represent savings, wages, the interest 
rate, the number of children, gifts to parents, and the cost function of childrearing. The 
suffix “+1” indicates one period ahead. Both θ and c(n) are defined in terms of the ratio 
to wage. When young, an individual earns wage income, bears some children, and 
gives some pecuniary or material gifts to their old parents. When old, individuals rely 
on their own savings and gifts from their children. No bequest exists
2. We also assume 
for simplicity that individuals perfectly foresee w+1 and r+1. 
As for the old-age gift ratio θ, individuals choose its optimal value to maximize 
their lifetime utility, assuming that their children will make the same choice as their 
                                                  
1    This setup is in contrast to that of many preceding analyses which have interpreted children as consumption 
goods—that is, they have included the number of children in an individual’s utility function—and/or incorporated 
altruistic motives.   
2    We can discuss (non-altruistic) income transfer from parents to children by replacing θ with –θ, while keeping 
the main results unchanged. 4 
 
own if other variables remain unchanged. We assume that old parents take the value of 
the gift received from their children as given, even if it differs from what they expected 
to receive from their children. In the equilibrium, each generation calculates the 
optimal gift such that each generation gives the same fraction of their own wage 
income and no generation has an incentive to change the size of the gift (see Zhang and 
Zhang, 1998). 
The cost function of childrearing is specified as 
() , 0 , > = c cn n c
ε  (2) 
where  ε is the elasticity of the cost of childrearing with respect to the number of 
children. Moreover, we assume ε>1. 
The first-order conditions for utility maximization are given as 
() , 1 2 1 1 u r u + + =  
() , 2 1 1 u w wu n c + = ′ θ  
2 1 1 nu w wu + =  
with respect to saving, the number of children, and the gift ratio, respectively, where 
ui=∂u/∂ci. From these three conditions, we have 
()
, 1 1 1
1 1
+
+ + = − = −
′
r
w
n w
w n c
w θ
                ( 3 )   
which means that the rates of return from childrearing, the old-age gift, and saving are 
all equalized in utility maximization. This condition (3) is reduced to 
* *
* 1 1
) (
r n
n c
= − = −
′
θ
                   ( 4 )  
in the steady state, where n
* and r
* are the steady-state number of children and interest 
rate.
3 
If (3) holds, then (i) the lifetime budget constraint is reduced to 
() [] , 1
1 1
2
1 w n c
r
c
c − =
+
+
+
 (5) 
(ii) the old-age gift ratio is given as 
() , n c ε θ =  (6) 
using (3), and (iii) the optimal saving is calculated as 
                                                  
3  This resultant fertility is determined solely by the interest rate, giving basically the same result as that of Becker 
and Barro (1988), who incorporate altruistic bequests in the model of endogenous fertility. 5 
 
( ) [] () [] ( ) ( ) [ ]w n c w n c w n c s ε γ γ γ θ + − − − = − − − − = 1 1 1 1 . (7) 
The wage income and the interest rate are derived from the competitive firms’ 
profit maximization. Assuming that the production function is given as 
, 1 0 , < < = α
α k y  
where k is the capital–labor ratio and that capital stock fully depreciates in one life 
period, then we have 
() . 1 , 1
1 − = + − =
α α α α k r k w  (8) 
The market equilibrium for capital (and for goods) is expressed as   
. 1 n
s
k = +  (9) 
Then, combining (3), (7), (8), and (9) yields the fertility equation: 
() () ( )
() ( ) ε γ α
α γ α
+ − −
− − −
=
1 1
1 1
n c , (10) 
assuming (1-α)(1-γ)>α. Normalizing the number of children before introducing social 
security as unity, we have the equation shown below. 
() ()
() ( ) ε γ α
α γ α
+ − −
− − −
=
1 1
1 1
c  (11) 
 
3.  Introducing social security 
 
This section introduces a PAYG social security program, by which a young individual 
pays the social security tax of t×100 percent of wages and an older individual receives 
the benefit with a replacement ratio of β×100 percent of the wage paid to the young 
individual. Therefore, the lifetime budget constraints are given as 
( ) [] , 1 1 s w t n c c − − − − = θ  
() ( ) 1 1 2 1 + + + + + = w n s r c β θ . 
Because the number of children, the old-age gift ratio, and savings are adjusted in 
the same way as before introducing social security, condition (3) holds here again. 
Consequently, the lifetime budget constraint is expressed as 
() , 1
1 1
2
1 w
n
t n c
r
c
c ⎥ ⎦
⎤
⎢ ⎣
⎡ + − − =
+
+
+
β
 (12) 6 
 
and the optimal saving is given as 
() ( ) w
n
t n c s ⎥ ⎦
⎤
⎢ ⎣
⎡ − − − + − − − =
γβ
γ ε γ γ ) 1 ( 1 1 . (13) 
Meanwhile, the budget constraint for the government is given as 
1 −
=
n
t
β
, (14) 
where the government first sets up the replacement ratio β; then it adjusts the tax rate t 
to balance the PAYG social security program at each period, taking the observed 
number of the current young individuals n-1 as given. 
Then, combining (3), (8), (9), (13), and (14) yields the dynamic equation of 
fertility: 
() () ( )
() ( ) ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛ −
+
+ −
−
+ − −
− − −
=
−1
1
1 1 1
1 1
n n
n c
γ γ
ε γ
β
ε γ α
α γ α
. 
Normalizing the number of children before introducing social security as unity and 
using (11), we have 
,
1
1
1 A n n
n
β γ γ ε
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛ −
+ − =
−
 (15) 
where 
() ()
0
1
1 1
>
−
− − −
≡
α
α γ α
A . 
From (15), we confirm that the number of children continues to decline after 
introduction of social security. Introducing social security reduces the demand for 
children as capital goods used for material support during old age and correspondingly 
reduces the cost of child rearing, which also engenders a reduction in the old-age gift 
(see (6)). Consequently, individuals can increase saving, which accelerates capital 
accumulation and reduces the interest rate. This brings a reduction in the rate of return 
from child rearing (see (3)) and engenders a further reduction in fertility. Under this 
adjustment, old parents depend less on the gifts from their children than before 
introducing social security because they receive social security benefits. 
Next, we consider the maximum size of social security that can prevent a 
cumulative reduction in fertility and make social security sustainable. From (15), the 
equation which solves the steady-state number of children, n
*, is expressed as 7 
 
. 1 *
*
An
n
β ε − =  (16) 
To consider the solutions of this equation graphically, Figure 1 depicts curves of 
f(n
*)=n
*ε and g(n
*)=1-β/(An
*). This figure suggests that an overly high value of  β 
engenders no steady-state solution of n
* because, thereby, the g (n
*) curve is shifted 
downward and located below the f (n
*) curve. The maximum value of β, denoted by β+, 
is such that makes the two curves mutually tangent. Considering f (n
*)=g(n
*) and 
f ’(n
*)=g’(n
*), we calculate 
()
( ) A
ε ε ε ε β
/ 1 1
+ −
+ + = , 
which engenders n=(1+ε)
-1/ε. Simple calculations show that β+ is an increasing function 
of ε and a decreasing function of α and γ. The number of children continues falling 
cumulatively and the social security program collapses if β exceeds β+. Consequently, 
we can state that A PAYG social security program should be maintained within a 
certain limited size to prevent a cumulative reduction of fertility and a collapse of the 
program. 
To graphically illustrate this conclusion, let us tentatively assume γ=0.5, α=0.25, 
and ε=2. Normalizing the number of children before introducing social security as 
unity, we have c=0.0667 from (11) and A=0.0167. Then, the dynamics of fertility, (15), 
is expressed as 
        ,
1 1
3 1
1
β ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
+ − =
− n n
n  
and the maximum size of social security is calculated as β+=0.0642. Figure 2 portrays 
the dynamics of fertility, with the n=n(n-1) curve and the 45-degree line along which 
steady states must lie, for three difference replacement ratios, β=0.05, 0.0642(=β+), and 
0.07. If β=0.05, the n=n(n-1) curve crosses the 45-degree line at n=0.787 and n=0.319, 
which correspond to stable and unstable state solutions, respectively. Assuming that the 
economy starts at n=1, the number of children will fall to and stabilize at 0.787. When 
β is raised to 0.0642, the curve becomes tangent with the 45-degree line and yields the 
only stable number of children of n=0.579. When β is 0.07, which is greater than 
0.0642, the curve does not cross the 45-degree line, suggesting that the number of 
children falls cumulatively to zero. 
Another interesting question to be addressed is whether introducing social security 8 
 
raises an individual’s lifetime utility. We concentrate on the steady state, in which the 
lifetime budget constraint (12) is reduced to (5), the same as before introducing social 
security. In addition, an individual’s adjustment equalizes the rate of return from the 
old-age gift to that from saving; that is, n
*-1=r
*, which makes the net rate of return 
from PAYG social security equal to zero. Therefore, social security affects the lifetime 
budget and utility entirely through its impact on fertility. Because the level of utility in 
the steady state, u
*, is given as 
( ) ( ) [ ] () ( ) ( ) [ ]
() [] (), ln 1 1 ln ln
1 ln 1 1 ln
* * *
* * * * * *
n n c w
w n c n w n c u
γ
γ γ
− + − + =
− − + − =
 
we have 
()
() β
γ
β d
dn
n n c
n c
dn
dw
w d
du
*
* *
*
*
*
*
* 1
1
1
⎥
⎦
⎤
⎢
⎣
⎡ −
+
−
′
− = .        ( 1 7 )  
We can show dn
*/dβ<0 as long as social security stays sustainable (β≤β+) from (16) 
and also [    ] >0 in the RHS of (17), as long as 
() ( )
() ( )
,
1 1
1 1
/ 1
*
ε
α ε γ α
ε γ α
⎥
⎦
⎤
⎢
⎣
⎡
− + − −
+ − −
< n         ( 1 8 )  
using dw
*/dn
*= -(α/n
*)
1/(1-α)<0, c’(n
*)=cεn
*ε-1>0 and (11). We start with n
*=1. Therefore, 
(18) holds and so we have du
*/dβ<0. Consequently, we can state that a PAYG social 
security program reduces an individual’s lifetime utility in the presence of income 
transfer from children to their parents.       
      This conclusion is consistent with a conventional view that a PAYG social security 
program reduces lifetime utility under declining fertility. It is noteworthy, however, 
that the negative impact of social security on utility is not caused by a reduction of 
lifetime income in our model. Indeed, social security raises lifetime income because it 
makes individuals reduce the number of children and increase saving, which in turn 
accelerates capital accumulation and raises per-capita income. At the same time, 
however, it reduces the interest rate, which directly implies a higher cost of old-age 
consumption. The negative sign of eq. (17) indicates that this negative effect dominates 
positive effects from lower fertility—that is, an increase in lifetime income and a 
reduction in the childrearing cost—and reduces net lifetime utility. 
 
 9 
 
4. Conclusion 
 
We discussed how to make a pay-as-you-go social security sustainable, based on a 
simple overlapping-generations model with endogenous fertility and an old-age gift 
from children to parents. We confirmed that a PAYG social security program should 
not be too large, because of the risk that a large program leads to a cumulative 
reduction in fertility. To make the program sustainable, we should contain its size 
within a certain limit, as determined by parameters related to individual utility, 
production, and the cost of childrearing functions. 
We also showed that a PAYG social security program reduces an individual’s 
lifetime utility. In response to an introduction of social security, individuals reduce the 
number of children and the old-age gift to their parents, which raises saving and raises 
per-capita income. However, the lowered interest rate raises the cost of old-age 
consumption and reduces the net lifetime utility by more than offsetting the positive 
effects of lower fertility. 
These results hold even if we consider the opposite direction of intergenerational 
transfer, i.e., bequests from old parents to children, as far as an individual’s utility does 
not include an altruistic aspect. Individuals increase income transfer to their children to 
offset its impact on lifetime income in response to an introduction of social security. 
Consequently, social security affects an individual’s utility entirely through fertility, 
just as in the case of the old-age gift. 
Including another aspect of children, especially their role as consumption goods, 
and altruistic motives will most likely engender mixed results. However, the inherent 
unsustainability of old-age social security cannot be alleviated completely so long as 
social security programs at least partially substitute intergenerational income transfer. 
 10 
 
References 
Becker, G.S. and R. Barro (1988) “A Reformation of the economic theory of fertility” 
Quarterly Journal of Economics, 103, 1-25. 
Fenge R. and V. Meier (2005) “Pension and fertility incentives” Canadian Journal of 
Economics, 38(1), 28-48. 
Groezen B. van, T. Leers and L. Meijdam (2003) “Social security and endogenous 
fertility: pension and child allowances as Siamese twins” Journal of Public 
Economics, 87, 233-251. 
Hirazawa M. and A. Yakita (2008) “Fertility, child care outside the home, and 
pay-as-you-go social security” Journal of Population Economics, forthcoming. 
Sinn H.-W. (2004) “The pay-as-you-go pension system as fertility insurance and an 
enforcement device” Journal of Public Economics, 88, 1335-1357. 
Wigger B. U. (1999) “Pay-as-you-go financed public pensions in a model of 
endogenous growth and fertility” Journal of Population Economics, 12, 625-640. 
Zhang, J. and K. Nishimura (1993) “The old-age hypothesis revisited” Journal of 
Development Economics, 41, 191-202. 
Zhang U. and J. Zhang (1998) “Social security, intergenerational transfer, and 
endogenous growth” Canadian Journal of Economics, 31, 1225-1241. 11 
 
Figure 1. Solving the steady-state number of children 
-2
-1
0
1
f(n*), g(n*)
n*
f(n*)=n*ε
g (n*)=1-β/(An*)  for β<β+
g (n*)=1-β/(An*)  for β=β+
g (n*)=1-β/(An*)  for β>β+
 
 
Figure 2. Dynamics of fertility under social security 
0.0 
0.2 
0.4 
0.6 
0.8 
1.0 
1.2 
0.0 0.2 0.4 0.6 0.8 1.0 1.2 
n
n-1
β=0.05
β=0.0642
β=0.07
 
Note: γ=0.5, α=0.25, and ε=2 are assumed. 

A Reformation of the economic theory of fertility”

Fertility, child care outside the home, and pay-as-you-go social security”

Pay-as-you-go financed public pensions in a model of endogenous growth and fertility”

Pension and fertility incentives”

Social security and endogenous fertility: pension and child allowances as Siamese twins”

Social security, intergenerational transfer, and endogenous growth”

The old-age hypothesis revisited”

The pay-as-you-go pension system as fertility insurance and an enforcement device”

Maximum size of social security in a model of endogenous fertility

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Maximum size of social security in a model of endogenous fertility

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