Chukyo University Institute of Economics
Discussion Paper Series
July 2023

No. 2301
Old-age support policy and fertility
with strategic bequest motives

Akira Yakita

Old-age support policy and fertility with strategic bequest motives
Akira Yakita
Nagoya City University

Abstract
This paper presents an analysis of the effects of public old-age support on individuals’
fertility decisions and the long-term equilibrium by assuming strategic bequest motivesof individuals in a discrete-time overlapping generations model. An increased wage tax
for financing public old-age support reduces the opportunity cost of child-rearing time,
although it decreases after-tax income. The increased public support tends to decrease thefamily old-age support time. Thereby, the net effect of public support on fertility is
positive when the rationality constraint is binding. Nevertheless, the tax increase lowers
the per-worker capital. Therefore, the net effect of public support on the long-term
lifetime utility of an individual is ambiguous. It is also shown that intergenerational
exchange based on strategic bequest motives might lead to excessively high fertility ratecompared to social optimum.
JEL Classification: D13, J13, J14, J22
Keywords: fertility, old-age support, rationality constraint, strategic bequest motives

Correspondence: Akira Yakita, Professor Emeritus, Graduate School of Economics,
Nagoya City University, Yamanohata 1, Mizuho-cho, Mizuho-ku, Nagoya 467-0802,
Japan.
1

1. Introduction
Most advanced economies have experienced extended longevity and declining fertility
in past decades. Longer life expectancy increases the possibility of long-term care needsof individuals and entire countries (Mayhew, 2011). Because the health conditions ofindividuals during the old age period vary among individuals, some countries have public
long-term care support systems to supplement or substitute informal supports, e.g., Japan,
Germany, Korea, Luxembourg, Austria, Canada, the UK, and the US. 1
This paper presents an analysis of effects of public old-age support on fertility rates by
incorporating endogenous fertility decisions of individuals into the dynamic generalequilibrium model based on strategic bequest motives. With strategic bequest motives, aparent obtains additional (marginal) utility and also extra old-age support from an extra
child at the cost of rearing time and strategic bequests given to the child. Few studies in
the literature describe works on fertility analyses using such a dynamic model based on
strategic bequest motives.
As population aging progresses in countries, many theoretical studies of effects ofelderly long-term care on family labor supply decisions have been presented (Pestieau
and Sato, 2008; Cremer and Roeder, 2013; Ponthiere, 2014; Cremer et al., 2017; Yakita,
2020a). Many empirical analyses of family caregiving effects on the market labor supplyof family members have also been presented (van Houtven et al., 2013; Skira, 2015). A
vast literature related to Japan has presented analyses of the Japanese Long-Term Care
Insurance system on the market labor supply of families (Tamiya et al., 2011; Sugawara
and Nakamura, 2014; Yamada and Shimizutani, 2015; Oshio and Usui, 2017; Niimi,
2017; Fu et al., 2017). Most such studies specifically examine the female labor supply
because long-term care is often provided by women. The results are mixed. Nevertheless,these theoretical and empirical studies do not consider fertility decisions of families
simultaneously.
Since works by Becker, e.g., Becker and Barro (1988), the analysis of fertility decisions
have been described in many reports (e.g., Galor and Weil, 1996; de la Croix and Doepke,
2003; Apps and Rees, 2004). Nevertheless, few reports describe the effects of long-termcare on the fertility decisions made by families. Recently, Yakita (2023a) presents ananalysis of the long-term effects on fertility rate in an overlapping generations model inwhich children are altruistic toward their parents. He demonstrates that increases in public
long-term care provision might not raise the fertility rate, depending on the efficiency of

1

Canada, the UK, and the US have means-tested public programs.
2

public care provision. 2 By contrast, this study considers such an issue in a dynamic
general equilibrium model including many generations, assuming strategic bequest
motives of individuals.
As described herein, the author also considers broader old-age support services, suchas inclusion of chore assistances, rather than a narrowly defined long-term care services
such as ADL and IADL services. 3 In Japan, for example, the ratio of individuals in needof long-term care among people aged 75 and older is 32.1%; the ratio of those aged 85
and older is 60.6% (Japanese Ministry of Health, Labour and Welfare, 2020). 4 The life
expectancy at birth was 81.47 years for men and 87.57 years for women in Japan in 2021
(Japanese Ministry of Health, Labour and Welfare, 2021). 5 Therefore, one might inferthat most elderly people are supported by family members or by public programs, even if
they do not even need narrowly-defined long-term services such as IADL services. 6 To
simplify the arguments, for these analyses we assume that all elderly people receive oldage support both informally from family members and formally from governments.
Herein, we consider only old-age support in kind. 7
The main result is that public old-age support financed through wage taxes raises thefertility rate when the rationality constraint is binding. Under the rationality condition,
the cost of family old-age support for young adult children is exactly compensated bybequests from elderly parents. In addition, the positive effect of lower opportunity costsof child rearing outweighs the negative income effects of tax increases. Moreover, that
Yakita (2022b, 2023b) examines the fertility decisions of parents in a model assumingintergenerational exchange game between children and parents and demonstrates that anincrease in the probability of becoming dependent or in the bargaining power of children
might lower the fertility rate. Nevertheless, the models are essentially a two-generation
model.
3
Van Houtven et al. (2013 p.244) define chore care as cares including household chores,
errands, and transportation.
4
MinistryofHealth,LabourandWelfareofJapan(2020)
https://www.mhlw.go.jp/toukei/saikin/hw/life/life21/dl/life21-02.pdf.
5
MinistryofHealth,LabourandWelfareofJapan(2022)
https://www.mhlw.go.jp/toukei/saikin/hw/life/tdfk20/dl/tdfk20-05.pdf.
6
Most municipalities pay for medical service costs of elderly persons in a universal
health insurance system in Japan, although elderly patients cover some costs.
7
Miyazawa (2010), incorporating human capital accumulation as a growth engine,compares the growth effects of public old-age support in cash and in kind. He
demonstrates that transfers in kind promote growth to a great degree. Pensions asintergenerational cash transfers have been studied widely, as reported in the literature (e.g.,
Cipriani and Fioroni, 2023; Tamai, 2023). Nishimura and Zhang (1992) and Yakita (2001),
for example, examines the fertility effect of social security.
2

3

study found that the policy lowers the capital stock per worker in non-health consumption
goods production. Consequently, public old-age support would not necessarily improve
the steady-state lifetime utility of individuals.
The remainder of this paper is the following. An overlapping generations model withstrategic bequest motives of individuals is introduced in the next section. 8 Section 3
describes an examination of the model dynamics. Section 4 presents comparative static
results with respect to old-age support provision. Section 5 compares the results with thesocial optimum. The objective of a social planner is the individual lifetime utility in steadystates. The last section concludes the paper.

2. ModelWe consider a discrete-time overlapping generations model. For our purposes, we
assume unisex individuals. Individuals are homogeneous and live for three periods:
childhood, a working period and old-age retirement period. The lifetime is certain, andthe length of each period is normalized to unity. An individual is fed by a parent during
his childhood. The individual then supplies labor to the labor market, consumes some of
the wage income, and rears children in the second working period. The working
individual also provides family old-age support for the parent in exchange for bequests,
i.e., with strategic bequest motives. 9 A government provides public old-age support,
produced using labor, and finances it through wage taxes. Non-health consumption goods
are produced under Cobb Douglas technology using capital and labor.
2.1 Individuals
While consuming part of the sum of personal wage income and a bequest from a parent,
the individual saves the remainder for his old-age retirement in the second period. He
allocates the fruits of savings between non-health consumption and makes bequests to hischildren in exchange for old-age support from them in the third period. The lifetime
optimization problem of an individual can be split into two stages: the first is the
maximization of utility during the old-age period, the second is the maximization of the
lifetime utility in the second working period. We first consider the choice of a strategic
bequest in the third period of individual’s life and then consider the choice of
The strategic bequest motives are called the exchange motives for old-age caregiving.
This setting is similar to the one which Miyazawa (2010) presents as a case of old-age
support in kind. Bernheim et al. (1985) provide evidence that strongly suggests thatbequests are used as compensation for services rendered by beneficiaries.
894

consumption-savings plan in the second period in turn.
An old-age individual in period t + 1 allocates the fruit of his lifecycle savings
between old-age non-health consumption and bequests to maximize old-age utility. The
old-age utility function is assumed
as u2t +1 c12−t +γ1[nt ( zt +1 + ε ztG+1 )]γ , where c2t +1 is=
non-health consumption during the old-age period, nt stands for the number of hischildren, zt +1 represents old-age support per child for a parent, and ztG+1 denotes public
old-age support provided by government in per-child terms. Parameter ε > 0 represents
the degree of cost-efficiency of public old-age support time relative to family support.10
The total elderly support per child is given as zt +1 + ε ztG+1 in terms of family support.
Letting bt +1 be the bequest per child, the rationality constraint can be written as(1)
bt +1 ≥ (1 − τ ) wt +1 zt +1 ,where τ ∈ (0,1) is the wage tax rate and where wt +1 denotes the wage rate for periodt + 1 . No child provides family old-age support unless that child receives a bequest greaterthan or equal to the opportunity cost of support provision. 11 Letting I t +1 be the fruit of
the child’s lifecycle savings from the working period, the budget constraint in the old-age
retirement period is given as 12(2)=
I t +1 c2t +1 + nt bt +1 .
The optimization problem of the old-age individual is to choose non-health consumption
c2t +1 and old-age support (demand) zt +1 to maximize the old-age utility. Because of
maximization by the individual, it is natural to consider equality for the rationality
constraint (1). Therefore, the choice of support demand zt +1 can be regarded as that ofbequests to children bt +1 . Inserting the equal constraint into the budget constraint, wehave
I t +1 = c2t +1 + nt (1 − τ ) wt +1 zt +1 .

(3)

The first-order conditions for old-age utility maximization are
10

Although public support is expected to be more efficient than family support, i.e.,ε ≥ 1 , one can have ε < 1 when the public support provision needs goods inputs otherthan labor or when X-inefficiency exists.
Coefficient of zt +1 in constraint (1), (1 − τ ) wt +1 , corresponds to the equilibrium priceof the private market of old-age support in a model of exchange motives for caring, whichis considered by Klimaviciute et al. (2017). In equilibrium bt +1 is the amount paid forzt +1 .
11

12

We assume that parents regard their children as identical, giving them bequests equally.
5(1 − γ )c2−tγ+1ntγ ( zt +1 + ε ztG+1 )γ − µ =,
0 and

(4a)

0.
γ c12−t +γ1ntγ ( zt +1 + ε ztG+1 )γ −1 − µ nt (1 − τ ) wt +1 =(4b)Variable µ is the Lagrange multiplier attached to constraint (3). From (4) we obtain1− γ
c2t +1 = nt (1 − τ ) wt +1 ( zt +1 + ε ztG+1 ) .

γ(5)

By inserting c2t +1 from (5) into budget constraint (3), one obtains
zt +1 + ε ztG+1= γ [ I t +1 + nt (1 − τ ) wt +1ε ztG+1 ] / [nt (1 − τ ) wt +1 ] .

(6)

From (5) and (6) we obtain the indirect old-age utility of
1 − γ 1−γ
u2t +1 = () [nt (1 − τ ) wt +1 ]1−γ ( zt +1 + ε ztG+1 )1−γ ntγ ( zt +1 + ε ztG+1 )γγ= Qt +1[ I t +1 + nt (1 − τ ) wt +1ε ztG+1 ] ≡ u2*t +1 ,(7)

where Qt +1 =(1 − γ )1−γ γ γ [(1 − τ ) wt +1 ]−γ . Because the rationality constraint is satisfied
with equality, each child provides old-age support zt +1 to his parent in exchange forbequest bt +1 . Income I t +1 and the number of children nt are already determined in
the prior period; public old-age support ztG+1 and tax rate τ are determined by thegovernment; market prices (i.e., wage rate wt +1 and interest rate rt +1 ) are given for
individuals.
Next, we consider the optimization of working individuals. Each individual chooses
the time allocation among market labor supply, child rearing time, and family old-age
support time. Each individual also allocates the sum of wage income and bequest received
between non-health consumption during the working period and lifecycle savings. Market
labor consists of that employed in non-health consumption goods production and that
employed in the public old-age support provision sector. For convenience, we denote theformer by lt and the latter by ztG . We assume that labor in public sector is compensatedusing the same wage rate wt as in the non-health consumption goods production sector
during period t because a cost-minimizing government must pay the same wage rate toemploy labor in the public support provision sector. Assuming that the child-rearing time
per child is constant at φ > 0 , then the time constraint of a working individual is writtenas6lt + ztG + zt + φ nt =1,

(8)where lt + ztG is the market labor supply and where zt + φ nt is the labor supply forfamily production of family old-age support and childcare.
The budget constraint of a working individual in period t is given asIbt + (1 − τ ) wt (lt + ztG ) = c1t + t +1 .
1 + rt +1(9)

Therein, c1t denotes non-health consumption during the working period. Because therationality constraint is satisfied with equality for each generation, we havebt= (1 − τ ) wt zt . Therefore, budget constraint (10) can be rewritten as(1 − τ ) wt (1 − φ nt ) =c1t +I t +1.
1 + rt +1(10)

The lifetime utility function is assumed as U =ln c1t + σ ln nt + β ln u2*t +1 . From (7) andbecause Qt +1 is given for the individual, the objective function of the utility maximizing
individual can be written as
U = ln c1t + σ ln nt + β ln[ I t +1 + nt (1 − τ ) wt +1ε ztG+1 ] .

(11)The optimization problem of the individual is to choose non-health consumption c1t , thenumber of children nt , and lifecycle savings I t +1 / (1 + rt +1 ) ≡ st to maximize lifetime
utility (11). The first-order conditions for lifetime utility maximization are
10,− λt =
c1t

(12)

λt
β
0 , and−=
I t +1 + nt (1 − τ ) wt +1ε ztG+1 1 + rt +1(13)

β (1 − τ ) wt +1ε ztG+1
σ+− λt (1 − τ ) wtφ =0.
nt I t +1 + nt (1 − τ ) wt +1ε ztG+1(14)

Variable λt is the Lagrange multiplier attached to constraint (10). Conditions (12) and(13) give the optimal intertemporal allocation between the working-period consumption
and the retired-period consumption, including old-age support services. The first term on
the right-hand side of (14) represents the marginal utility of having an extra child. The
second term is the marginal utility of old-age support from the child net of bequest (costs)
given to the individual. The third term stands for the rearing costs of the child in terms ofutility.
7

From (12) and (13) we obtain
c1t =I t +1 + nt (1 − τ ) wt +1ε ztG+1.
β (1 + rt +1 )Using (10) and (15) and defining Gt +1 =
c=
1t

(15)
nt (1 − τ ) wt +1ε ztG+1, we have
1 + rt +1

1[(1 − τ ) wt (1 − φ nt ) + Gt +1 ] .
1+ β

(16)

Herein, Gt +1 is the discounted present value of public old-age support in terms of family
old-age support. Using the definition of lifecycle savings, we obtain1[ β (1 − τ ) wt (1 − φ nt ) − Gt +1 ] .
st
=
1+ β

(17)

We also have
=
nt

Gt +1
σ
[
].
+
φ 1 + β + σ (1 − τ ) wt
1

(18)

A greater discounted present value of public old-age support engenders a greater numberof children. It is noteworthy that equation (15)–(18) give implicit solutions for hon-healthconsumption, savings, and the number of children. They will be obtained in the next
section.
2.2 Government
Government provides public old-age support to old-age individuals, employing laborfrom the labor market at the market wage rate. The government old-age support provision
is financed by a wage income tax, balancing the budget in each period. We assume thatthe government expends tax revenues only for old-age support. By maintaining abalanced budget, the government budget constraint is given asτ wt (lt + ztG ) Nt =wt ztG Nt .

(19)

Variable Nt stands for the number of workers in period t . The left-hand side represents
the tax revenues. The right-hand side is the labor cost of public old-age support
provision. 13 From (19) it follows that τ wt lt= (1 − τ ) wt ztG in per-worker terms.

Public old-age support for an elderly person in period t , ztG , is expressed in per
adult-child terms. Therefore, each elderly person receives public support
138

2.3 Non-health consumption goods production
The production function of non-health consumption goods is assumed as
Yt = Ktα L1t−α ( 0 < α < 1 ).

(20)

In that equation, Yt is the aggregate output; Kt and Lt represent the aggregate capitaland labor in period t . The production function can be rewritten as yt = ktα lt1−α in perworker terms, where yt = Yt / Nt , kt = Kt / Nt and lt = Lt / Nt . Assuming perfectlycompetitive factor markets, we have
(1 − α )ktα lt−α =(1 − α ) yt / lt , and
wt =

(21)

−1 1−α=lt
α yt / kt .
1 + rt α ktα=(22)

The marginal productivity of each factor is equal to the factor price.
2.4 Capital market
The clearing condition of capital market is given as
Kt +1 = st Nt or, in per worker terms, nt kt +1 = st .

(23)

3. Dynamics and Long-Term Equilibrium
In this section we study the dynamics of the economy. First, we consider temporary
equilibrium; and then we examine the dynamics of the system.
3.1 Temporary equilibrium
We first analyze a temporary equilibrium in a period, period t . From the government
budget constraint and from the definition of Gt +1 , we have Gt +1 = εθ nt kt +1 , whereθ= [(1 − α ) / α ]τ is the wage tax burden relative to capital income (hereinafter, it iscalled the tax parameter) .
Using (17) andnt kt=+1 εθβ (1 − τ ) wt (1 − φ nt ) / (1 + β + εθ ) . Therefore, we have
Gt +1 =εθβ (1 − τ ) wt (1 − φ nt ).
1 + β + εθ

(23),

we

obtain(24)

From (12), (16), and (24), we obtainnt −1 ztG = Nt ztG / Nt −1 from the next generation of their children.
9

1
=

λt

1 + εθ
(1 − τ ) wt (1 − φ nt ) .
1 + β + εθ(25)(24) andfrom (25) into condition
1/ λt
λt Gt +1 + σ = λt (1 − τ ) wtφ nt , which is obtained from (13) and (14), one can obtain(26)(1 − φ nt )[εθβ + σ (1 + εθ )] = (1 + β + εθ )φ nt .

Inserting

Gt +1

from

Therefore, the fertility rate in period t is given asσ (1 + εθ ) + εθβ.
nt =φ (1 + εθ )(1 + β + σ )](27)

The right-hand side of (27) does not depend on the period. The fertility rate in each period
is constant, i.e., nt = n for all t , when the wage tax rate τ and θ are kept constant.
It is noteworthy that even when parents are non-altruistic toward their children, i.e.,
σ = 0 , parents want to have children merely for exchange motives. 14
From (6), (23) and definition st ≡ I t +1 / (1 + rt +1 ) , we have
(1 − τ ) wt +1[ zt +1 + ε ztG+1 (1 − γ )] = γ (1 + rt +1 )kt +1 .

(28a)

From (21), (22), and from the government budget constraint (19), it follows thatzt +1 = {[γ − (1 − γ )εθ ] / θ ]ztG+1 .

(28b)

The government budget constraint (19) is rewritten as
1 − α (1 + θ ) G
zt +1 .

(29)αθwhere (21) and (22) are used. Inserting these into the time constraint, one obtains
1 − α (1 + θ ) γ − (1 − γ )εθ + θ G[]zt +1 =1−φn ,+
αθθfrom which one has the following:αθ
1 + β + εθ.
(30)
ztG+1 =
1 − α (1 − γ )(1 + εθ ) (1 + εθ )(1 + β + σ )lt +1 =

From the time constraint (8), we obtain

14

This case corresponds to a model of children as investment goods, described by
Cochrane (1975). Adult children provide ‘future consumption’ of old-age support forparents. Even in this case, young adults care for both their children and elderly parents,
i.e., sandwich caregivers. Nevertheless, a case of α = 0 is apparently unrealistic because
one has nt = 0 without public old-age support. Although public long-term care systems
vary among countries, countries such as Germany, Japan, France, Korea and theNetherlands introduced or reformed public long-term care system around 2000.
10

zt =

1 + β + εθα [γ − (1 − γ )εθ ].
1 − α (1 − γ )(1 + εθ ) (1 + εθ )(1 + β + σ )(31)

The right-hand sides of (29), (30), and (31) do not depend on the period. Therefore, for agiven tax rate, not only public old-age support, but also the labor employment in nonhealth consumption goods sector, and family old-age support time are constant throughall periods, i.e., ztG = z G , lt = l and zt = z for all t . For family and public old-age
support to be positive, we must have 1 − α (1 + θ ) > 0 and γ − εθ (1 − γ ) > 0 . We assume
that these two conditions are satisfied in this paper. 15
3.2 Dynamics of the economy
From (17), (23), and Gt +1 = εθ nt kt +1 , we obtain the rule of motion of capital per
worker as
kt +1 =β (1 − τ )(1 − φ n)(1 − α )l −α αkt .
(1 + β + εθ )n(32)

Because the coefficient of ktα is positively constant and because 0 < α < 1 , the capitalper worker has stable dynamics. In addition, the long-term equilibrium steady state is
unique. From (32) the long-term steady state capital per worker is given as

β (1 − τ )(1 − φ n)(1 − α )l −α 1/(1−α ),](1 + β + εθ )nwhere=
τ [α / (1 − α )]θ .
k =[

(33)

4. Effects of Public Old-Age Support Policy
In this section, we analyze the effects of public old-age support policy changes on thesteady state. The next subsection presents analyses of the effects. Subsection 4.2 provides
a numerical example.
4.1 Policy effects
The effect on the fertility rate is ascertained by differentiating (27) with respect to thetax parameter θ as

The latter condition is the same as the condition for family old-age support to bepositive in Miyazawa (2010).
15

11

dn
1
εβ
=
> 0.
dθ φ (1 + β + σ ) (1 + εθ ) 2(34)

Therefore, we obtain the following result.
Proposition 1 We assume that individuals have strategic bequest motives and that therationality constrain is binding. Then, increases in the tax for public old-age support
always raise the fertility rate.
The result can be interpreted as follows. When public and family old-age support aresubstitutes, an increase in public old-age support will reduce family support. It might
increase the number of children and the market labor supply by freeing adult-children’stime from providing family old-age support. Nevertheless, with strategic exchanges
between elderly parents and young adult children, children’s old-age support is provided
correspondingly to the bequest received (see the rationality constraint with equality). Thatis, the intergenerational transfers in families, i.e., bequests, are neutralized by family oldage support in this model. 16 Therefore, the after-tax wage rate does not affect thedecisions related to family old-age support provision. However, the fertility decision isaffected by the after-tax wage rate, i.e., the opportunity cost of rearing children. Decreases
in the after-tax wage rate lower the opportunity cost of child rearing and therefore raises
the fertility rate more than offsetting the negative income effect of the tax increases. 17
Therefore, public old-age support financed by wage taxes increases the fertility rate. The
lowered wage rate might also reduce the market labor supply.
It is noteworthy that the result does not depend on whether public old-age support iscost-efficient relative to family support. That is in contrast to the result reported by Yakita(2023a), which assumes altruistic elderly care supply for parents. Yakita (2023a)
demonstrated that an expansion of public long-term care lowers fertility when public
long-term care provision is less cost-efficient than family care. In fact, that can be thecase even when public long-term care is more cost-efficient. His result is based on theassumption that public long-term care provision needs goods inputs other than labor. As
described in this paper, by contrast, public old-age support negatively affects fertility
regardless of whether public old-age support is more cost-efficient, even when produced
16

It holds true al far as bequests or family old-age supports are not constrained to corner
solutions.
17
The dominance of price effect is supported by the empirical studies of Jones and Tertilt(2008) and Jones et al. (2010), who find that fertility and wage rates are negatively related
in most countries.
12

using labor alone.
Next, we present the analyses of policy effects on the time allocation of working
individuals. The effect on public old-age support is demonstrated by differentiating (29)with respect to θ as

dz G
dθ

1
α [1 − α (1 − γ )](1 + β + εθ ) εβ[].
−
(1 + εθ )(1 + β + σ ) [1 − α (1 − γ )(1 + εθ )]2
1 + εθ

(35)

Although the first term in brackets on the right-hand side is positive, the second term isnegative from (34). Therefore, the sign of the effect is not determined a priori. However,
one can reasonably consider that an increase in the tax rate will increase public old-age
support. Otherwise, no government have an incentive to introduce the tax for old-age
support provision. Therefore, we assume that the sign of (35) is positive in the following.
The policy effect on family old-age support is derived from (31) as
α (α − 1)ε (1 − γ )(1 + β + εθ )dz
1=[dθ [1 − α (1 − γ )(1 + εθ )](1 + εθ )(1 + β + σ )1 − α (1 − γ )(1 + εθ )−εβα [γ − (1 − γ )εθ ]φ]< 0.
1 + εθ

(36)

Increases in the tax rate decrease family old-age support. This result can be interpreted asfollows. Tax increases reduce the disposable income of workers. The negative income
effect leads to decreases in family support even if bequests from the parent are fixed. If
public old-age support increases, then this increase enforces the effect. In fact,
improvements in the cost-efficiency of public old-age support provision reduce family
support, i.e., dz / d ε < 0 . That freed-up time might be allocated to rearing children andmarket labor supply, i.e., dn / d ε > 0 from (27). 18 However, summing up above
comparative static analyses, the net policy effects on the market labor supply l andl + z G are indeterminate. Therefore, we have the following proposition.

Proposition 2 The effects of old-age support tax changes on the total old-age support,family and public, and the market labor supply are indeterminate a priori.
Finally, the policy effect on capital per worker in the non-health consumption
production is derived from (33) as

Assuming altruistic family long-term care, Yakita (2023a) demonstrates that
improvements in the cost efficiency of public long-term care can increase the fertility
rate.

18

13

dk
εk
=
−
< 0.
1 + β + εθ
dθ

(37)

Per-worker capital decreases with the tax rate. Because the policy effects on market labor
employment are ambiguous, the effects on the steady-state wage rate and interest rate arealso ambiguous. Therefore, the effects on non-health consumption during working andold-age periods and on lifetime utility are indeterminate a priori. The following
proposition holds.
Proposition 3 Increases in the tax rate lowers the steady-state level of per worker capitalstock.
4.2 Numerical example
In this subsection, we present numerical results obtained by assuming model
parameters. According to de la Croix and Doepke (2003), we set the utility weight of
120=
0.299 . 19 We further
children as σ = 0.271 and the discount factor =as β 0.99

assume here that one period lasts 30 years. The income share of capital is assumed to be
α = 0.3 , as usual in the macroeconomics literature. The scale parameter for non-healthconsumption goods is set to unity for this example. Although the burden rate for old-age
support in kind varies from country to country, we set τ = 0.10 as a benchmark rate
because, for example, the social security benefits-GDP ratio was 23.6%, about half ofwhich was of pension benefits in Japan in 2019 20. In this case, we have θ = 0.23 . The
contribution ratio of old-age support in the old-age utility is assumed to be γ = 0.864 .
Liu et al. (2023) show that an increase in Long-term Care Insurance (LTCI) increases
non-health consumption by 0.157%. The elasticity of substitution between non-healthconsumption and old-age support might be nearly zero (e.g., ADL), whereas the demandfor old-age support depends on its price (Sano et al., 2022). 21 We consider here that themagnitude of the marginal effect approximates an elasticity. The cost-efficiency of public

De la Croix and Doepke (2003) also assume the same utility weight on human capitalof children.
20
MinistryofHealth,LabourandWelfareofJapan(https://www.mhlw.go.jp/stf/newpage_21509.html: cited on 4 July 2023). Japanese LongTerm Care Insurance does not provide cash transfers.
21
Tax change effects are qualitatively unaltered for a smaller γ . It is noteworthy herethat changes in γ do not affect the steady state fertility rate because the right-hand sideof (27) does not include γ .
1914

old-age support is set as ε = 0.5 . Yakita (2023) calculates it from empirical works forJapanese long-term care industry such as Aya (2014). 22 Per-child rearing time is assumed
variously in the literature. We set the variable to generate the fertility rate for unity, i.e.,φ = 0.19 . This value is approximately equal to 0.15, which is assumed by de la Croix andDoepke (2003). The parameter set engenders the equilibrium fertility rate of about unity.
With these parameters, we calculate the steady state values of endogenous variables ofthe model. For comparative statics, we consider variations of τ by 0.01 from thebenchmark case. The results are demonstrated in Table 1.
As shown in Proposition 1, a tax rise for old-age support increases the fertility rate.
The capital stock per worker and family support decrease. With the assumed parameters,
the tax rise increases public old-age support, as expected by the policymaker. The market
labor l + z G decrease with the tax rise. Although old-age non-health consumption
decreases, the total old-age support z + ε z G increases. Consequently, the net effect on
the old-age utility u2* is positive. Nevertheless, the negative effect on the workingperiod non-health consumption through negative income effect overwhelms the positiveeffect on fertility, consequently lowering the lifetime utility U . 23,24
Table 1 Simulation results

τ

0.09
0.10
0.11

zG
0.053
0.059
0.065

n

z

l

k

u2 *

U

1.004
1.013
1.022

0.21600.21530.2145

0.540
0.533
0.526

0.05530.05320.0512

0.162
0.165
0.168

−2.129−2.141−2.153

5. Social optimum
As described earlier, it was demonstrated that tax rises for public old-age support
always increase the fertility rate, although they lower the per-worker capital stock. In thisThe relative cost efficiency of public long-term care is calculated as 1/1.55 by Yakita(2023) when the labor productivities of public and family elderly care are equal. Because
elderly people might prefer broader old-age support by family members to public support,
we assume a lower value.
23
Effects on non-health consumption during the working and retired periods are notpresented in Table 1.
24
Sensitivity analyses ensure that these qualitative results hold for wider ranges ofparameters.
22

15

section, we present evaluation of the policy of public old-age support by comparing thedecentralized long-term equilibrium under public old-age support policy with the social
optimum. In doing so, the role of strategic bequest motives in the dynamic allocation
might be clarified.
The long-term equilibrium is achieved as a steady state. Therefore, we consider thesocial optimum as the steady state which maximizes the lifetime utility of an individual
by controlling the resource allocation centrally. For our purposes, we designate the total
old-age support per worker as Z .
The social optimization problem of the social planner can be formalized asMax

c1,c2 ,n, Z ,k ,lU = ln c1 + σ ln n + β (1 − γ ) ln c2 + βγ ln(nZ )csubject to k α l1−α − c1 − 2 − nk =0 , andn
1 − l − Z − φ n =0 .

(38)
(39)

Constraint (37) is the resource constraint, also, (38) is the time constraint per individual.
Derivation of the solution is set aside in the Appendix. From the first-order conditions for
maximization, we obtain the optimal resource allocation as shown below.
c1
1−α
,
(40a)
=
y 1 + β (1 − γ )c2 / n (1 − α ) β (1 − γ )
, and
=
y
1 + β (1 − γ )

(40b)

nk
=α .
y

(40c)

From (40c) we can immediately obtain=
α y / k[ α=k α −1l1−α ] n . This is the Golden Rulecondition for capital accumulation.
Using (38)–(40), we obtain the following solution:(1 − α )(σ + β ) − α [1 + β (1 − γ )],nso =φ{(1 − α )(1 + σ + 2 β ) − α [1 + β (1 − γ )]}
lso =

(41)

(1 − α )[1 + β (1 − γ )], and(1 − α )(1 + σ + 2 β ) − α [1 + β (1 − γ )](42)

βγ (1 − α ).
(1 − α )(1 + σ + 2 β ) − α [1 + β (1 − γ )](43)

Z so =

Subscript so designates the optimal value of variables in the Golden Rule optimum.25

25

With the parameters assumed in the preceding section, the social optimum fertility rate
16

For these values to be non-negative, the denominators on the right-hand sides of theseequations must be positive. We assume that these conditions are satisfied. For expositional
purposes, we also assume that the social optimum fertility rate is positive, i.e., nso > 0 .26
Comparing these optimum solutions with those obtained under decentralization withpublic old-age support policy, we can infer the following: First, the tax rate which
achieves the optimum fertility rate is obtainable from (27) and (41), i.e.,τ [α / (1 − α )]θ satisfying the following condition.
=θε =(1 + β ){β (1 − α ) − α [1 + β (1 − γ )]}.
(1 − α ) β (σ + β ) + α [1 + β (1 − γ )](44)

Two remarks must be made: First, tax rate θ is inversely proportion to the relative costefficiency of public old-age support ε . When the public support efficiency is higher, thetax rate is lower by comparison. 27 Nevertheless, we cannot rule out the possibility thatthe decentralized fertility rate with zero wage tax, i.e., with no old-age support policy, canbe higher than the social optimum fertility rate. To illustrate this point, we set τ = 0 in
(27). Then, the decentralized fertility rate without policies is
nss τ =0 =σ.
φ (1 + β + σ )(27’)

From (41) and (27’), we can demonstrate that the difference between two rates
nso − nss τ =0 as(1 + β ){(1 − α ) β − α [1 + β (1 − γ )]}nso − nss τ =0 =.
φ{(1 − α )(1 + σ + 2β ) − α [1 + β (1 − γ )]}(1 + β + σ )(45)

The sign of expression (44) cannot be ascertained a priori. If it is negative, i.e., if
is nso = 0.460 . Although the optimum rate is too low to maintain the population size inthe steady state, some strategic bequest motives are observed, e.g., in Horioka et al (2018).
Therefore, the equilibrium fertility rate would be higher than the social optimum even
without public old-age support. The optimum level of lifetime utility is U so = −1.085 .
This finding implies that if government also aims to maintain, or even increase, thepopulation size, then the optimal policy would be to provide sufficient public old-age
support, lowering the lifetime utility. By contrast, if, for example, α = 0.2 , other things
being equal, then the social optimum fertility rate nso would be greater than unity.
26
Nevertheless, we cannot rule out the possibility of nso ≤ 0 a priori.
The optimal tax rate might be zero because of the non-negative constraint. For theparameters assumed in a numerical example, the right-hand side of (44) is −0.309 .
Therefore, we must have a corner solution τ = 0 as the optimal policy in this case.
Nevertheless, when α = 0.2 with other parameters unchanged, we have=
θε 0.1182 > 0 and hence τ = 0.045 .
27

17

nso − nss τ =0 < 0 , then the public old-age support provision fails to achieve the optimum
fertility rate because increases in τ always raise the fertility rate, as condition (34)indicates. 28
Second, even if a wage tax τ achieves the optimum fertility rate, the tax rate might
not achieve the overall resource allocation efficiency. 29 The tax rate which equalizes lss
in (30), with lso in (42) perhaps not equalizing nso with nss .
Summing up the above arguments, we obtain the following proposition.
Proposition 4 Suppose that individuals have strategic bequest motives. Then, old-age
support policy alone might not lead the decentralized market equilibrium to the social
optimum.
In our model, there are two potential sources of inefficiency, i.e., finite lifetimes andstrategic behaviors. Nevertheless, the numerical example implies that strategic bequest
motives might keep fertility rate higher. We have only old-age support financed by wage
taxes. Dynamic efficiency in the sense of a golden rule might require multiple policy
measures.

6. Concluding Remarks
For a dynamic general equilibrium model in which old-age support is provided bychildren to parents based on strategic bequest motives, an analysis of fertility decisionsof individuals is presented. The public old-age support provision increases the fertility
rate and lowers the capital stock per worker, thereby engendering ambiguous effects on
the lifetime utility of individuals. Comparing the decentralized long-term equilibriumwith the golden rule social optimum, we can infer that public old-age support provision
might not be socially desirable under strategic bequest behaviors of individuals. Even
when no public old-age support is provided, strategic bequest motives of individuals
might engender a fertility rate higher than the social optimum rate.
28

For the parameters in the numerical example, the value of nso − nss τ =0 is −0.858

when τ = 0 . When α = 0.2 with other parameters unchanged, the difference becomes
0.183 > 0 .
29
There are multiple tax rates which equalize the decentralized steady-state employmentof (29) and the social optimum employment of (41) in the non-health consumption goods
sector.
18

Further research directions can be inferred. First, as Horioka et al. (2018) demonstrate,individuals might have both altruistic and strategic bequest motives simultaneously.30
The analyses can be extended to such a case. Although the effect of public old-age support
on fertility does not depend on the cost-efficiency of public support in this study, Yakita(2023a) demonstrates that public long-term care provision lowers fertility if the public
provision is inefficient. Second, we have not considered any child policy. If old-age
support affects child policy, then that must be considered in a model simultaneously. Third,
whether an old individual becomes dependent and the degree to which dependency exists
can be expected to vary among individuals. Such uncertainty must be considered. Many
works such as those reported by Pestieau and Sato (2008) and by Cremer and Roeder(2013) introduce such uncertainty. Fourth, we have assumed only child-rearing time cost.
This assumption simplifies the analyses together with Cobb Douglas utility functions.
Goods costs of rearing and educating children might be significant. Consideration of thesecosts makes the fertility decisions dependent on the relative magnitudes of the wage rate
and the goods price (Becker and Barro, 1988).
Finally, and more importantly, we have assumed that steady-state old-age support,family plus public, is determined as an interior solution. Nevertheless, given the level of
public support, individuals might want to make the level of family support negative. In
other words, the public old-age support is too great relative to old-age non-healthconsumption. If the level of public support is merely equal to the necessary level for oldage individuals to remain alive, then the chosen fertility rate might be a corner solution
to their utility maximization. In this case, the negative income effect of tax increases
becomes significant for individuals, consequently affecting the fertility rate negatively.
Therefore, in such a situation, together with positive effects of the opportunity cost change,
the net effect of a tax increase on the fertility rate might be indeterminate a priori. This
presents an interesting case for future research in this area.

Declaration
The author declares that he has no competing financial interest or personal relationshipthat might influence the work reported in this paper.
Acknowledgments
30

Klimaviciute et al. (2014) demonstrate, using SHARE data, that long-term caring isdriven by altruism or by family norm, whereas Alessie et al. (2014) show that exchange
motive is important in intergenerational transfers.
19

An earlier version of this paper was read at Nagoya Macroeconomics Workshop. The
author thanks participants for their insightful comments and suggestions. Financial
support from the Japan Society for the Promotion of Science (KAKENHI Grant No.
22K001544) is gratefully acknowledged.

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Appendix: Social optimum
Assuming Lagrange multipliers attached to constraints (38) and (39) respectively as
q1 and q2 , the first-order conditions for the optimum are given as1− q1 =0,c1

β (1 − γ )c2

σ + βγ
n

βγ
Z

− q1

(A1)
1
=
0,
n

(A2)

c
− q1 (− 22 + k ) − q2φ =
0,
n

(A3)

0,
− q2 =(A4)

q1 (α k α −1l1−α − n) =0 , and(A5)

q1 (1 − α )k α l −α − q2 =
0.

(A6)

Condition (A5) stipulates that the marginal productivity of capital equals the population
growth rate, i.e., the Golden Rule for capital accumulation. Inserting (A1), (A5), and (A6)
into the resource constraint (38), one obtains
1
1−α
.
(A7)
=
yq1 1 + β (1 − γ )Together with (A1), we have
c1
1−α
.
=
y 1 + β (1 − γ )(A8)From (A2) and (A7), we havec2 / n (1 − α ) β (1 − γ ).
=y
1 + β (1 − γ )(A9)Conditions (A8) and (A9) are the same with (40a) and (40b) in the text. Inserting (A5)and (A9) into (A3), and using (A7), we obtain23

σ +β −α [1 + β (1 − γ )]q2φ n .
=
1−α

(A10)

We also have, from (A6) and (A7),l=[1 + β (1 − γ )] / q2 .

(A11)

Therefore, inserting (A11) and (A4) into the time constraint (39), we obtain(A12)
q2 (1 − φ n) =1+ β .
Eliminating q2 from (A10) and (A12), one can have the following equation.
(σ + β )(1 − α ) − α [1 + β (1 − γ )]
φn
.
=
1−φn
(1 − α )(1 + β )Therefore, from (A13), we obtain the social optimum fertility rate as(σ + β )(1 − α ) − α [1 + β (1 − γ )].
n=φ{(1 + σ + 2β )(1 − α ) − α [1 + β (1 − γ )]}From (A11) and (A12), we have
(1 − α )[1 + β (1 − γ )].
l=
(1 + σ + 2 β )(1 − α ) − α [1 + β (1 − γ )]Inserting (A14) and (A15) into the time constraint, we also haveβγ (1 − α ).
Z=(1 + σ + 2 β )(1 − α ) − α [1 + β (1 − γ )]

(A13)

(A14)

(A15)

(A16)

Solutions in (A14)–(A16) are identical with those in (41)–(43) of the text.
We also obtain, using (A5),k = (n / α )1/ (α −1) l ,(A17)from (A1) and (A7),(1 − α k α l1−α,c1 =1 + β (1 − γ )(A18)and, from (A8) and (A9),=c2 c1β (1 − γ )n ,(A19)

where n and l are given by (A14) and (A15).

24