This paper analyzes the effects of child allowance on households without a child with respect to the pay-as-you-go public pension system. We demonstrate that the child allowance can improve the utilities of households without a child through an increased pension benefits when the rate of households raising the number of children is sufficiently low.

Masatoshi Jinno

English

Research Papers in Economics

Is it beneficial for households without children to subsidy the
cost of rearing children to increase pension benefits? 
Masatoshi Jinno
Nagoya university
Abstract
This paper analyzes the effects of child allowance on households without a child with respect
to the pay-as-you-go public pension system. We demonstrate that the child allowance can
improve the utilities of households without a child through an increased pension benefits
when the rate of households raising the number of children is sufficiently low.
Citation: Jinno, Masatoshi, (2008) "Is it beneficial for households without children to subsidy the cost of rearing children to
increase pension benefits?." Economics Bulletin, Vol. 8, No. 9 pp. 1-7
Submitted: August 29, 2008.  Accepted: October 1, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume8/EB-08H50015A.pdf1. Introduction 
In recent times, the aging of the population combined with the diminishing number of children has 
become a serious problem faced by advanced countries. What causes a decline in the number of 
children? Cigno (1993) and Groezen, Leers, and Meijdam (G-L-M) (2003) show that there are two 
opposite externalities that children have under an economy where the pay-as-you-go pension system 
exists. An additional child increases the future output, and thus, the pension is beneficial; however, it 
also causes a decline in the capital-labor ratio. Households that do not consider these externalities 
tend to have fewer than the optimal number of children. G-L-M and Groezen and Meijdam (2008) 
present the necessity of subsidizing the cost of child rearing in order to attain the command 
optimum. 
Households, while not considering the social importance of having a child in an economy where 
the pay-as-you-go pension system exists, tend to have fewer children or no children because the 
pay-as-you-go public pension system provides insurance for old age. This free-riding behavior on 
the public pension system results in the decline in the number of children. G-L-M and Groezen and 
Meijdam both stress the need for child allowance; however, they assume that households are 
representative and homogeneous. If all households had children, their utilities would rise relatively 
easily since an increase in the rate of child allowance decreases the cost of child rearing and the 
burden of households directly decreases and the child allowance indirectly increases the pension 
benefits. 
However, there are some households who will not have any children even if child allowance 
decreases the cost of child rearing. We will therefore analyze whether the utilities of such households 
rise or not since the child allowance increasing the tax rate only becomes a direct burden for them, 
even if it does indirectly increase pension benefit. We will present the condition that indirect benefits 
from the child allowance system exceed direct burdens for households without a child.  
The remainder of this paper is organized as follows. The model is presented in section 2, and 
section 3 summarizes the discussion and presents issues for further study. 
 
2. The Model 
This section develops an overlapping generational model wherein two types of household coexist. 
The first type has positive savings and zero children and the second type has positive savings and 
some children. The utility of the former type depends on lifetime consumption only while that of the 
latter type depends on life time consumption and number of children. Households are characterized 
by their preferences to having children. The type of household is determined by its own nature. The 
duration of a household can be categorized into three periods: youth, middle age, and old age. We 
term the current generation as the t
th generation; households are born in period t – 1, they work, save, 
and have some or no children in period t, and retire to live on saving returns and pensions in period t 
+ 1. During middle age, households inelastically supply one unit of labor. The government adopts 
the pay-as-you-go public pension system and the child allowance system. 
We assume that there are only two types of typical household in an economy: (1) those that have 
positive savings and zero children and (2) those that have positive savings and some children. The   3 
   
1
2 1
1
1 1 t t t t c s r       K ,                (5) 
where  t T  is the lump-sum tax, 
i
t s  the savings,  1  t r  the interest rate , and  1  t K  the public 
pension benefit  which is the same for all households. T he middle-age budget constraint  of the 
second type becomes 
      t t t t t t t b n s c T w W I        1 1
2 2 2
1            (4b) 
where  t b  is the cost of rearing a child , and  t W  is a subsidy rate per child.  The last-age budget 
constraint is the same as equation (5).  Equations (4a) and (4b) express the disposable income when 
middle age is divided among consumption, savings, and child rearing (in the case of  second type 
households). Equation (5) expresses the return on middle -age savings and the pension benefit s are 
spent on old-age consumpti on. 
     The government runs the public pension sy stem denoted by the following budget constraint per 
household, 
  1 1     t t t w nI K ,                 (6) 
If households consider the investment  aspect of having children, that is, the public pension 
premiums they pay , equation (6) should be included in the  maximization problem. However, 
because we assume that children only  bring happiness to households and that the effect of changing 
the number of children  is limited to the extent where it can be ignored with respect to  the public 
pension benefit,  this paper does not include equation ( 6) in the problem  (see, for example , Groezen 
et al. 2003).   
The government lowers  the cost of child rearing by  a subsidy of 
c
t t bW  per child. To finance the 
child allowance  or the subsidy for savings , the government imposes the lump -sum tax  t T . The 
budget constraint per household for this system is   
  t t t t b n T W   .                  (7) 
In each period t,  the Cobb-Douglas technology  is employed for production  using two inputs : 
physical capital  t K  and labor  t N ; 
E D
t t t N K Y     such that we have   t t t K Y r / 1 D     and 
t t t N Y w / E   . Perfect competition in the factor markets ensures that , in equilibrium , the return 
from savings and th e wage per labor  unit are equal to the respective marginal products.  In each 
period, the capital stock  is the result of  the households’ savings in the preceding period . 
  t t
i
t t N s s K    ¦ 1 .                 (8)   4 
The capital stock lasts only for one period and attains zero scrap value in the subsequent period. The 
initial capital stock ( 0 K ), which belongs to  1  N   households, is given to those who are old in 
period 0. Each of them own  1 0 1 /     N K s . Using the perfect competition in the factor markets 
and equations (2), (7) and (8), the following relation is formed: 
  1 1 1      t
t
t
t w
s
n
r
E
D .                (9) 
A household born in period t – 1 maximizes utility (3) subject to the budget constraints (4) and 
(5). The household’s optimal choices are characterized by the first-order condition: 
   
0
1
2
1 2
1
1  


 
i
t
t
i
t c
r a
c
a
               (10a) 
   
0
1
2
3
2
1
1   
 
t t
t t
n
a
c
b a W
.              (10b) 
Equation (10a) indicates that households equate the marginal rate of substitution between current 
and future consu mptions to the rate of return from s avings. This is common for all households.  The 
optimal condition ( 10b) can be interpreted as follows. The marginal disutility of children, resulting 
from a decrease in consumption during middle age, is equal to the direc t marginal benefit of having 
children. This applies to  second type households  only. 
     By substituting  equations (3) and (7) into equations ( 10), we have the optimal savings and  the 
optimal number of children  as follows: 
      
   1 2 1
1 1 1 2 1
1
1 1

 
 
   
 
t
t t t t
t r a a
a T w r a
s
K I
,          (11a) 
        
  1
1 3 1 1 2 2
1
1 1

 

    
 
t
t t t t
t r
a a T w r a
s
K I
,        (11b) 
      
  b r
T w r
a n
t
t t t t
t W
K I
 
   
 

 
1 1
1 1
1
1 1
3
2 .          (12) 
   By using equations (1), (6), (9), (11) and (12), the optimal average savings and number of children 
become 
    ^ ` t t t T w
z
z a
s     I
D
1
2
1 2 , and            (13a) 
 
    ^ ` t t t T w a p
bz
z z a
n  


  I
W
EI
1 ˆ
1
3
2
2 1 2 ,           (13b)   5 
where      p p a a z ˆ 1 ˆ 2 1 1    {  and      2 1 3 2 1 2 ˆ a a p a a a z    { D EI . According to 
equation (13), the capital -labor ratio, the wage rate and the interest rate  in period t+1 depend on the 
subsidy rate of child allowance in period  t: 
 
  2 1 2 3
1 2
1 ˆ
1
z z a a p
z ba
k
t
t 

   EI
W D
,              (14) 
   
D E 1 1     t t k w                 (15) 
    1
1
1 1   

 
D D t t k r                 (16) 
We assume that the government imposes a  lump-sum tax  t T  on households in order to 
subsidize the cost of  child rearing. By substituting  equation (13b) into equation (7), the lump-sum tax 
rate becomes 
        2 2 1 2 3
2 1 2
3 1 ˆ
1 ˆ
z z z a p a
z z a
w p a T
t t
t t t W W EI
EI
I W
  

   .       (17) 
We investigate the effect of child allowance through the pension system on the first type of 
household. Substituting equations (11a), (14)~(17) into the utility function leads to the indirect utility 
function: 
      ¸ ¸
¹
·
¨ ¨
©
§


  t t t W
a a z
z z a
a a c c U
2 1 2
2 1 2
1 1
1
2
1
1 ln ,
EI
 
          ¸
¸
¹
·
¨
¨
©
§
¸ ¸
¹
·
¨ ¨
©
§
 ¸ ¸
¹
·
¨ ¨
©
§




t W
z z a p a
bz
z
a
a a
z z a
a
1
2 1 2 3
1
2
2
2 1
2 1 2
2 ˆ
ln
D
D
EI
E
D
E
EI
  (18) 
where    
    t t
t t
t z z z a p a
z w
W
W W EI
I W
  
 
{
1 ˆ
1 1
2 2 1 2 3
2 . Differentiating it with respect to  t W   and 
evaluating it with  0   t W , we obtain 
        ^ `   > @p a a a a a a a a
z d
c c dU
t t
t t ˆ 2
1 ,
3
2
2 1
2
1
2
1 2 1 3
2 0
2
2
2
1    

 
 
D D EI
W
W
 
    ^ ` 2 1 1 2
2
1
1
a a a a
z
    D EI D .           (19) 
The first term in equation (18) denotes the direct effect of an increase in the lump-sum tax. The 
second term denotes the indirect effect of an increase in child allowance in the pension system. Child   6 
allowance results in an increase in the number of children. Further, it increases the number of 
workers in the next period but decreases the wage rate. An increase in the number of workers 
implies a rise in those supporting the system, which in turn implies a rise in pension benefits. A 
decline in the wage rate implies a decrease in the contributions because the premium rate is constant. 
Thus, an increase in child allowance has ambiguous effects on the second type of household in the 
pension system. 
The sign of equation (19) depends on  p ˆ  
 
0
,
0
2
2
2
1

!
  t t
t t
d
c c dU
W W
, 
   if     
    
»
¼
º
«
¬
ª
   
 
»
¼
º
«
¬
ª 
!

EI D EI D
D EI D
1 2
1
ˆ
2
2 2 1 1
2 1
3
2
a a a a
a a
a
a a
p .    (20) 
Equation (20) implies that unless the rate of households with some children is small, the utility of 
households without a child does not rise even if there is child allowance. The higher the rate of 
households with children, the higher the burden of child allowance. Thus, child allowance does not 
increase the utility of households without a child through an increase in pension benefits when the 
rate of households with children is sufficiently high. 
 
Proposition 
Child allowance can improve the utility of households without a child through the pay-as-you-go 
pension system in an economy where the rate of households with children is sufficiently small. 
 
3. Conclusions and Remaining Issues 
This paper assumes the case that there are households without a child and analyzes the effects of the 
child allowance system on such households through an increase in pension benefits. With regard to 
the returns from an uncontrolled pension system, this paper shows that child allowance increases the 
utility of households without a child when the rate of households with some children is sufficiently 
small because of increased pension returns. 
When considering an endogenous fertility rate, the relationship between the cost of raising a child 
and that of education is also very important. Thus, it is important to take into account the relationship 
between the number of children, savings, the pension system, and investment in education. It 
remains to be seen whether child allowance or a subsidy for education investment more effectively 
improves the utility of households without a child through increased pension benefits.  
 
Endnotes 
 
１  An increase in child allowance offers households without a child an incentive to have one because it 
lowers the cost of raising a child. This aspect of child allowance is of some importance when the 
government considers the optimal child allowance rate; however, this has not been considered in this   7 
                                                                                                                                            
paper. This paper focuses on whether child allowance improves the utility of the last household that has 
no child even if the cost of child rearing is lowered. 

Is it beneficial for households without children to subsidy the cost of rearing children to increase pension benefits?

http://www.accessecon.com/pubs/EB/2008/Volume8/EB-08H50015A.pdf

Is it beneficial for households without children to subsidy the cost of rearing children to increase pension benefits?

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