Appendix A: Numerical example for a tax cut
In this Appendix, we calculate the time paths of the key variables by specifying the
parameters of the model for the case of a tax cut.
In order to demonstrate the
qualitative properties of the model, we assume an extreme case of parameters:
  0.25
,
 1  0.5 ,
z  0.15 ,
  0.5 , ( 1 , 2 )  (0.4,0) and
A  6.5 ,
( N 1 , N 2 )  (1,1) in the initial steady state. One period consists of 25 years. With
these
parameters
1.208  1.008 24
we
have
 2  0.130 ,
1
2
(k 
, k
)  (0.302,0.764)
,
(n1 , n 2 )  (1.208,1.208)
,
W
R
 3.391( 1.052 24 )
where
and
2
(u1 , u
)  (0.404,0.427) in the initial steady state in which the relative population
weights of the two economies remain constant.
Then, we suppose that the tax rate in country 1 decreases from 0.4 to 0.25 in
period 5, say.
Then, the fertility rate in country 1 falls from 1.208 to
1.114( 1.005 24 ) , whereas that in country 2 remains unchanged. In period 5, the tax
rate change induces reallocation of existing capital between the countries; that is, a
certain amount of capital moves from country 2 to country 1. These capital flows
increase the wage rate in country 1 and thereby the savings of workers in the period,
while they decrease the wage rate and per worker savings in country 2.
As shown in
the text, since the savings rate is higher in country 1, the increased wage rate in
country 1 increases the aggregate savings in the world, offsetting the decrease in
savings of country 2. The increased aggregate savings is allocated between countries
through arbitrage (see Fig. A1). Thus, the wage rates in both countries rise in period 6
and as do savings in both countries (see Fig. A2).
However, as time passes, the
decreased fertility in country 1 reduces the aggregate world-wide savings since the
savings per worker in country 1 is greater. Therefore, the capital labor ratios of both
countries begin to decline. The declining capital labor ratios lower the wage rates and
thereby savings. In this example, the capital labor ratio of country 1 will be lower than
the initial ratio for 8 periods after the policy change. The world-wide after-tax rate of
return to capital moves to the opposite direction of the capital labor ratio in country 2,
as depicted in Fig. A3.
The retired generations in both countries in the period of policy change benefit
from the rise in the world-wide return to capital. But after the period the change in
the utility moves following the capital labor ratio. The time paths of the lifetime utility
in both countries are demonstrated in Fig. A4. In the new steady state, we have
~ ~
~W
1 ~2
(k 1 , k 2 )  (0.084,0.141) , R
 8.580( 1.094 24 ) and (u~
, u )  (0.169,0.107) .
Appendix B: Transfers to both working and retired generations
As mentioned in Section 5.2, it is difficult to trace the transitional time path to the
steady state when the tax revenue is rebated to the retired generation.
In this
appendix, assuming the existence and stability of the steady state, we examine the
long-term effect of an increase in life expectancy in country 1 in the presence of income
transfers to the retired generation.
1
Denoting the lump-sum transfer to the retired agent by B , the budget constraint of
agents in country 1 in the second period becomes
R1s1  B1  d 1 .
(A1)
The optimal plans of agents are then
s1 
1
1    1
n1 
( w1  T 1 ) 

z (1     1 )
(1 
1 
B1
1     1 R1
T 1  B 1 / R1
w1
,
(A2)
).
The government budget constraint becomes
(A3)
 1r1K 1  T 1 N 1  B1 N 11 or, in per
worker terms,
 1r1k 1 (1  zn1 )  T 1  B1 / n1
(A4)
and where subscript 1 denotes the value in the previous period. For expositional
purpose, we assume the following allocation of the tax revenue:
T  
1
where
1
1
(1  zn )k 1 f k1 and
B  R (1   )(
1
1
1
1
1
)(1  zn1 )n1k 1
(A5)
 is the distribution ratio to the working generations ( 0    1 ). Using (6)
and (7) in the text and inserting (A5) into (A3), we obtain
1
n 

1
z 1   
1
{1 
 1 /(1   1 )
f
1
 k 1 f k1
(1  zn1 )k 1[ (1   1 ) f k1  (1   )n1 ]} .
(A6)
1
2
1
In this case, the steady state is characterized by triplet ( k , k , n ) satisfying (A6),
(11) and (12), where per worker savings is obtained by inserting (A5) into (A2) as
s1 
1
{ f 1  [1   1 (1  zn1 )]k 1 f k1 }
1   1

where
s1k 
1
1   
1
1 
1
1    1
1
s1 
  1 z 1
1   
1
(1   )(1  zn1 )n1k 1
(A7)
1
{ 1 (1  zn1 ) f k1  [1   1 (1  zn1 )]k 1 f kk
}

s1n 
1
k 1 f k1 
(1  zn1 )k 1
1    1
1 
1
1    1
1
(1   )[ 1 /(1   1 )]
1   
[ 1 f k1  (1   )n1
1
1
(1   )(1  zn1 )n1
(1   )(1  2 zn1 )k 1  0 .
1 
(1   1 ) 2
].
From (A6), (11) and (12), we obtain

 Hk

1
 s  (1  zn1 )n1
 k
 (1   1 ) f 1
kk

0
 dk 1 / d 1   H  
 


1
1 1  2
1
s n  (1  2 zn )k
dk / d    s1 

 1
 dn / d 1   f k1 
0


Hn
s k2  (1  zn 2 )n 2
2
 f kk
(A8)
where
Hk 
[ 1 /(1   1 )](1  zn1 )
(1     1 )( f 1  k 1 f k1 )
{(
1
(k 1 ) 2 f kk
f
1
 k 1 f k1
1
 1)[ (1   1 ) f k1  (1   )n1 ]   (1   1 )k 1 f kk
} (A9)
Hn 
z


 1 /(1   1 )
(1     1 )( f 1  k 1 f k1 )
{zk 1[ (1   1 ) f k1  (1   )n1 ]  (1   )(1  zn1 )k 1}
H 

(1  zn1 )k 1
1
z 1    1 f 1  k1 f 1
k
[f k1  (1   )n1 (1   1 )  2 ]  0
(A10)
(A11)
Under the assumption of the Cobb-Douglas production function, we have
Hk 
[ 1 /(1   1 )](1  zn1 )
(1     )( f
1
1
 k 1 f k1 )
[(1   )(1   )n1   (1   1 )]  0
(A12)
and, from (A11), we obtain
H n  0 if (1  
1
) f k1
1   1  2 zn1
.


z
(A13)
Letting H be the determinant of the coefficient matrix on the right-hand side of (A8),
we have
2 1
H   H k f kk
[sn  (1  2 zn1 )k 1 ]
2
1
 H n {[ s1k  (1  zn1 )n1 ] f kk
 [sk2  (1  zn 2 )n 2 ](1   1 ) f kk
}.
(A14)
Assuming that the world economy is in a steady state (where ski  (1  zn i )n i  0 for
i  1,2 ), we have H  0 if H n  0 and/or if the absolute value of H k ( 0) is
sufficiently great. From (A13), a sufficient condition for H n  0 is that  is
sufficiently great, i.e.,
  (1  2 zn1 ) /[1  2 zn1  (1   1 ) f k1z] .
For expositional
purpose, we assume H  0 .1 From (A8), we obtain
dn1
d
1
2
 H 1H k {s1 f kk
 [ sk2 )  (1  zn 2 )n 2 ] f k1}
2
1
 H 1H {[ s1k  (1  zn1 )n1 ] f kk
 [sk2  (1  zn 2 )n 2 ](1   1 ) f kk
} . (A15)
We cannot rule out the possibility of indeterminacy of a long-term equilibrium with
respect to the distribution ratio between the working and retired generations in the
1
sense that there exists
present study.
  (0,1) satisfying H ( )  0 . We assume it away in the
The second term on the right-hand side of (A15) is positive. On the other hand, as can
be
seen
from
(A12),
if

is
sufficiently
great,
i.e.,
if
  [n1 (1   ) /(1   1 )] /[  1 f k1  n1 (1   ) /(1   1 ) 2 ] , s1 is positive although the value
of H k becomes smaller. Although the sign of the first term on the right-hand side is
 is sufficiently
1
1
ambiguous, we cannot rule out the possibility of dn / d  0 if
1 1
1
1 2
1 1
1
great, i.e., if   max( 1 /{ f k /[ n (1   ) /(1   ) ]  1}, 1 /[1  (1   ) f k z /(1  2 zn )]) .
Setting, for example,
  0.25
,
z  0.15 ,  1   2 (  1 )  0.25 , n1  1.115( 1.004 24 ) ,
 1  0.5
and
1 /[1  (1   1 ) f k1 z /(1  2 zn1 )]  0.341
for
  1 .2
f k1  11.440( 1.107 24 )
and
,
we
have
1 /{ 1 f k1[n1 (1   ) /(1   1 ) 2 ]  1}  0.302
For example, the ratio of social security in the general account expenditure
in Japan was 0.295 in 2010, where the social security spending includes those on relief
works, health and hygiene measures and so on, i.e., spending for young generations.
Thus, supposing that the distribution ratio to the retired generation is about 30%, i.e.,
1    0.3 , we may say that dn1 / d 1  0 is not implausible even taking transfers to
the retired into consideration.
2
Parameters are those used in the numerical example. We can show that even when
   2  0.40 , the values of these equations are ceteris paribus 0.556 and 0.597 ,
1
respectively. It should be noted, however, that when 1    0 , n will be smaller
1
1
and f k will be greater.
Fig. A1 Capital labor ratios
0.9
0.8
0.7
0.6
0.5
k1
0.4
k2
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
period
Fig. A2 Savings
0.9
0.8
0.7
0.6
0.5
s1
0.4
s2
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
period
Fig. A3 World-wide after-tax rate of return to capital
6
5
4
3
Rw
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
period
Fig. A4 Utilities
0.5
0.45
0.4
0.35
0.3
u1
0.25
u2
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
period