Chukyo University Institute of Economics
Discussion Paper Series
March 2024

No. 2307
Old-age support policy effects on economic growth and fertility
Akira Yakita

Old-age support policy effects on economic growth and fertility
Akira Yakita
Nagoya City University
Abstract
Salient features of major economies around the world, specifically in high-income
countries, are graying population and declining fertility. Incorporating the quantity–
quality tradeoff of children into an overlapping generations model, this paper presents ananalysis of the effects of formal old-age support on fertility and economic growth. Formal
old-age support provision is assumed to involve management costs and labor costs. Whenthe management costs are sufficiently small, i.e., cost-efficient, formal old-age support
raises the balanced economic growth rate, involving smaller tax burden and freeing
individual time from family old-age support, but it lowers fertility. The effect on the
lifetime utility of individuals is indeterminate. By contrast, when management costs are
high, the increased formal old-age support deters economic growth through a negative
income effect caused by the greater tax burdens, but it also lowers the fertility rate. In thiscase, lifetime utility becomes lower.
Keywords: fertility, human capital, old-age support, quantity-quality tradeoff
JEL Classification: H51, J14, J22, J24, O40

Correspondence: Yakita, A.: Professor Emeritus, Graduate School of Economics, Nagoya
City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya 467-0802, Japan
E-mail: yakita@ econ.nagoya-cu.ac.jp
ORCID: 0000-0002-8966-9602
1

1. Introduction
Data for life expectancy at birth for 1950–2021, expressed in logarithmic values, aredepicted for World Bank income groups of countries in Fig. 1. They increased for all
countries during the period. Specifically, the growth rate in low-income countries afterlate 1990s is fairly high. 1 Although life expectancy after 2020 decreased because of theCovid-19 pandemic, long-term trends are positive for all country groups.
Data of the elderly dependency ratios, as expressed in logarithmic values, are depicted
for World Bank income groups of countries in Fig. 2. The ratio is defined as the population
size of people aged 70 and older relative to the population size of people aged 20–69 incountries. In contrast to longevity, the growth rate of the elderly dependency ratio has
been higher for high-income countries. Therefore, in high-income countries, the young
working generation must support more aged people per young worker now and perhaps
in the future. Although the elderly dependency ratio remains low in middle-income
countries, the growth rates of the ratio have become higher recently. Such might be thecase even for low-income countries. This low dependence ratio might reflect high fertilityof these countries. Therefore, in these countries, the needs for old-age support are
expected to become great in the near future.

4.4
4.2
4
3.8
3.6

High-income countries

Figure 1

Middle-income countries

2019

2016

2013

2010

2007

2004

2001

1998

1995

1992

1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

1950

3.4

Low-income countries

Life expectancy at birth of both sexes (in logarithm)

Source: United Nations: World Population Prospects 2022
https://population.un.org/wpp/Download/Standard/Mortality/

Because life expectancy is measured in logarithmic terms, the slopes of the loci reflect
their respective growth rates.
1

2

Mayhew (2011) reports that the gap separating life expectancy and healthy life
expectancy has expanded in the past decade. Moreover, it is expected to expand in the
future. The Ministry of Health, Labour and Welfare in Japan (2022) reports that, in Japan,
the gaps between life expectancy and healthy life expectancy were 8.73(= 81.41 − 72.68)
years for men and 12.06(= 87.45 − 75.38) years for women in 2019, whereas those were
8.67(
= 78.07 − 69.40) years and 12.28(= 84.93 − 72.65) years in 2001. Extended
longevity might engender a longer old-age period during which individuals need to becare-dependent. In that case, young adult children might have to provide old-age care for
their parents in addition to caring for their children simultaneously. If individuals arealtruistic toward their children, then they might also invest in children’s human capitalthrough education. Human capital accumulation boosts economic growth, increasing thechildren’s affordability. Therefore, extensions of life expectancy might make the time
allocation of young adult children between working and caring for both parents andchildren severer.

3

2.5

2

1.5

High-income countries

Figure 2

Middle-income countries

2019

2016

2013

2010

2007

2004

2001

1998

1995

1992

1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

1950

1

Low-income countries

Elderly dependency ratio (70+/20–69) (in logarithm)

Source: United Nations: World Population Prospects 2022
https://population.un.org/wpp/Download/Standard/Population/

For each, whereas the number of children and the amount of care for them are fairly
predictable, the risks related to the parents’ health condition are unpredictable because
health conditions during old age vary from person to person. What degree of care becomes
necessary and how long the care is necessary are all uncertain. The risk of caring for3

parents might be severer for each family member than caring for children. In fact, many
developed countries directly provide formal or public elderly long-term care services. 2
Nevertheless, individuals come to make decisions related to family or informal old-age
support and on child care simultaneously during the young adulthood period. 3 These
young-adult working generations are called sandwich generations.
As pointed out by Carney (2023), few theoretical studies examine sandwich
generations. Therefore, few studies have assessed their simultaneous decisions aboutchild care and elderly care. 4 Nevertheless, among others, Suh (2016) empirically reports
for 2012 that about half of people aged between 47–59 cared for older parents aged 65
and older and simultaneously for children under age 18 or provided financial support tochildren older than 18 in the US. Yamashita and Soma (2020) find, based on a 2012–2018
sample survey of parents who have children of university student age and younger, thatabout 30% of Japanese people have experienced caring for both parents and children. The
analyses described in this paper are intended to fill the gap.
Miyazawa (2010) shows that public old-age support promotes economic growth morethan public cash transfers do because the valuation of the formal old-age support
generates an additional benefit of parental education. Nevertheless, he does not consider
the fertility decisions of individuals. d’Albis et al. (2018) report that a rise in humancapital pushes the economy toward fewer births. Moreover, when the human capital level
becomes sufficiently high, it engenders the so-called modern regime of the decline infertility associated with birth postponement. However, in that study, they do not consider
family elderly care provision. Yakita (2023) presents an analysis of effects of increases informal long-term care provision and of increases in the labor productivity on fertility
when formal elderly long-term care provision involves management costs and labor
employment costs. He demonstrates that increases in long-term care provision lower
fertility when formal long-term care is less cost-efficient than informal care. Nevertheless,
2

They are Japan, Germany, Korea, Luxembourg, Austria, Canada, and the UK. Those
listed as the last three countries have means-tested public programs. Public long-term care
support systems established in the US provide supplemental or substitute informal care.
3
Klimaviciute et al. (2017) describe motivations of three main types for family longterm care provision: altruism, exchange, and family norm (forced altruism). This study
presumes the last motive. Those hypotheses are tested by Alessie et al. (2014) andKlimaviciute et al. (2017) among others. The empirical results are mixed.
4
For instance, van Houtven et al. (2013) and Skira (2015) present empirical analyses ofthe effects of formal long-term care provision on the labor supply of working generationand the wage rates. Many theoretical studies have examined elderly long-term care, e.g.,
investigations by Pestieau and Sato (2008), Cremer and Roeder (2013), Ponthiere (2014),
Cremer et al. (2017), and Yakita (2020).
4

he does not consider parental education of the children or human capital accumulation.
This paper extends Yakita’s (2023) analyses by also considering the quantity–quality
tradeoff of children to present an analysis of the effects of formal old-age support in kindon fertility and economic growth of the economy through these sandwich generation’s
decisions. 5
The salient findings are the following. The effects of formal old-age support provisionon labor employment in consumption goods production, the physical capital–humancapital ratio, and the balanced economic growth rate all depend on the magnitude ofmanagement costs other than labor costs in formal old-age support provision. We assume
that formal old-age support is financed by lump-sum taxes. The effect on fertility is
negative. The more cost-inefficient is formal old-age support, the decline of the fertility
rate is greater. If the management cost factor of formal old-age support provision is
sufficiently small (large), then the balanced growth rate and the physical capital–humancapital ratio becomes higher (lower) with greater formal old-age support provision. These
effects derive from the fact that the management costs affect worker behaviors throughthe negative income effect caused by changes in the tax burden. The effects on lifetime
utility of individuals might be positive if the management costs are sufficiently small, andindeterminate otherwise.
The next section introduces the model used for the analyses. Section 3 analyzes thelong-term equilibrium of the model. Section 4 presents an analysis of the effects of oldage support on the long-term equilibrium. The last section concludes the paper.

2. ModelWe consider an overlapping generations model populated by individuals with three life
periods. Individuals are unisex and are homogeneous, except for their ages. The time
horizon of the economy is infinite. Letting the population size of the working generationin period t be Nt , then the population of their children is Nt +1 = Nt nt , where nt
denotes the number of children per working individual in period t , i.e., the fertility rate
in period t . Each individual receives educational expenditures by the parent during the5

We consider elderly support in kind, which requires some time input. Becker et al.
(2016) point out that many elderly parents do not rely much on pecuniary support fromtheir children in countries with government-financed social security systems. Fu et al.
(2017) point out that the fact the introduction of Japanese Long-Term Care Insurance
exerted positive spillovers on the caregivers’ labor force participation can be explainedby the benefits in kind, unlike other countries such as Germany and Korea, whose systems
have cash allowances.
5

first childhood period. During the second adulthood period, the child allocates the timeendowment between market labor supply and care time for the parent and for the children.
With human capital accumulated during childhood, the individual earns wage income bysupplying labor in the labor market. After paying taxes, the individual allocates the earned
wage income among adulthood consumption, education payments for children, andsavings for the individual’s own old age. The second adulthood period consumption
includes the individual’s children’s consumption. The young adulthood generation is
employed by the consumption goods production sector and the government. Consumption
goods are produced under constant-returns-to-scale production technology using physicalcapital and effective labor. The human capital of a child in period t + 1 is produced byparental education expenditure and the average human capital stock of the parent’s
generation.
As described in this paper, especially emphasizing effects of formal old-age support
on fertility and human capital accumulation, we assume that all old-age individuals
receive a minimum level of old-age support z from the children and the government. 6
The minimum level of support is assumed to remain constant along growth paths becausesuch support is considered necessary for life irrespective of income levels. Financed bytaxes on the working generation, agencies of the government provide formal old-age
support z G (< z ) . Children are altruistic toward their parents in the sense that they areforced to provide informal old-age support for parent (Cremer and Roeder, 2017), making
up for the difference z − z G . We assume that children are responsible for their parents’
health during old age by family norm. 7 In this paper, we assume that old-age support

6

The old-age support under consideration herein is conceptually wider than either ADL(e.g., eating, bathing, and changing clothes) or IADL (e.g., shopping and handling phone
calls). Every elderly individual described herein is assumed to need the support in a broad
sense. In Japan, only 33 percent of care-need certificated persons (as support-required(Yo-Shien) levels and as severe-care (Yo-Kaigo) levels in the Japanese Long-Term Care
Insurance system) were cared by long-term care facilities in 2020 (Ministry of Health,Labour and Welfare of Japan, 2020).
7
Barigozzi et al. (2020) describe the responsibility of children as a fact. Because ofhomogeneous individuals, assuming that individuals provide old-age support byemploying labor privately, the results might not be altered fundamentally. Chakrabarti etal. (1993) report that even though parents are uncertain about their children’s altruism,
the optimal investment in their children’s human capital is the efficient one. We do notconsider exchange motives for old-age support in this paper.
6

provision is time intensive both in formal old-age support and informal old-age support. 8
We also assume that time productivity in both provision sectors are equal to µ for thesake of emphasis of management costs in formal old-age support provision. 9 Old-age
for formal old-age support andsupport production functions are z G Nt −1 = µ LGt
( z − z G ) Nt −1 =µ Lt for informal old-age support. Herein, z G Nt −1 and ( z − z G ) Nt −1

respectively represent the total old-age support provided by the government and byfamilies: LGt and Lt are the total support time employed in the formal old-age support

sector and the total informal support time of working generation, respectively, during
period t . We also assume that formal old-age support provision involves management
costs in addition to labor cost.
2.1. Individuals
Letting the time endowment of each in a period be unity, the time constraint of arepresentative worker in period t is then given as
1 − (ltG + lt ) = φ nt + lt .

(1)

Herein, φ denotes the per-child rearing time, which is assumed to be constant. In
addition, lt represents the labor time employed in the consumption goods production
sector, ltG denotes labor time employed by government, and lt stands for the informal
old-age support time for the worker’s parent at home. 10 Therefore, the market labor timeof the worker in period t is ltG + lt , whereas ltG + lt stands for total old-age support
time, both formal and informal.
We assume that labor in the public sector is compensated using the same wage rate wt

Miyazawa (2010) also uses such an assumption. The opportunity cost of informal oldage support time at home for each is the wage rate per labor time wt ht . Therefore, theopportunity cost of old-age support might change along with economic growth paths
although the minimum old-age support level remains constant.
9
Yakita (2023) considers a case in which they are different.
10
According to OECD (2005), informal long-term care provided by children differs bynation: 24% in Australia, 28% in Germany, 55% in Korea, 60% in Japan, 43% in the UK,
and 41% in the USA.
87

as in the consumption goods production sector because a cost-minimizing government
must pay the same wage rate to employ labor in the public support provision sector.
Therefore, each chooses the total market labor time supply

ltG + lt

without

distinguishing between being employed by the government and by the consumption
goods production sector.
The budget constraint of a worker in period t is expressed as
wt ht (lt + ltG ) − Tt = c1t + nt et + st .

(2)

In that equation, c1t stands for working-period consumption, including consumption by
the individual’s children, ht stands for the individual’s human capital stock, et denotes
educational expenditure per child, and st expresses his lifecycle savings for retirement
period in period t . Additionally, wt stands for the wage rate per efficient labor in periodt , and Tt is the tax payment in period t . The budget constraint of the individual’s

retirement is given as(3)
c2t +1= (1 + rt +1 ) st ,where c2t +1 represents retirement-period consumption, apart from old-age support
services, and where rt +1 denotes the interest rate in period t + 1 .
The human capital of a child ht +1 is assumed to be produced under the nested Cobb–
Douglas production technology as 11
ht +1 = σ etδ (htβ ht1− β )1−δ .

( σ > 0 , δ , β ∈ (0,1) )(4)Herein, ht denotes the average human capital stock of his parent’s generation. Assuming
homogeneous individuals, we have ht = ht .
The lifetime utility of an individual is assumed as the following log linear function:(5)
U t =ln c1t + ε ln nt + λ ln ht +1 + ρ ln c2t +1 .
The working generation is altruistic toward both children and parents. For expositionalease, we omit the utility from the minimum old-age support z , which is provided bygovernment and his children. Parameter ρ ∈ (0,1) is the discount factor for utility fromconsumption during old age, ε > 0 is the utility weight on the number of children, and
λ > 0 is the utility weight on the human capital per child. 12
11

As described by de la Croix and Michel (2002), other functions with constant returns
to scale can be considered for sustainable endogenous growth.
12
Because the utility weight on working-period consumption is set as equal to unity, ε8We obtain the following optimal plans from the first-order conditions for lifetime
utility maximization as 13
1[ wt ht (1 − lt ) − Tt ] ,
=
c1t
(6)
1+ ε + ρ=ntTε − λδ
[(1 − lt ) − t ] ,φ (1 + ε + ρ )
wt ht

(7)

λδ
w h , andε − λδ t tρ=st[ wt ht (1 − lt ) − Tt ] .
1+ ε + ρ

(8)

et =

(9)

Herein, the individual devotes time to old-age support for parents to a level such that theparent receives total old-age support of the minimum level by the family norm. For
expositional simplicity, we do not explicitly include the utility of the parent from the total
old-age support in defining individual lifetime utility (5). As might be apparent from (7)and (8), both the number of children and education expenditure per child increase withthe wage rate per labor-timer for a given tax payment. However, it is noteworthy thatparental education expenditures are unaffected by the tax burden.
2.2. Formal old-age support
The costs of providing formal old-age support are assumed to be wt ht (1 + M ) LG
t , where
M denotes a management cost factor. A government pays its employees a wage rateequal to the market wage rate to employ them in the labor market. The management costs
include training and re-training costs of formal old-age carers in response to increases in
the number of care workers, maintenance of the quality of old-age support, and also thecosts of nursing facilities. 14 The plausibility of the existence of such a management costis suggested in Yakita (2023). The sum of these costs, except the wage payments, isdesignated as the management cost for these analyses. For simplicity, we measure themanagement cost in terms of labor costs. The provision cost is assumed to be financed byand λ might be assumed as less than unity.
13
We assume that each worker has more than one child, i.e., the marginal benefit of
having a child is more valuable than the child’s life performance. In other words, we
assume that condition ε − λδ > 0 holds true.
14
The costs might include the net costs of care-system digitalization (e.g., remotemonitoring systems). Such systems are expected to reduce labor-time inputs or to improvetime efficiency, reducing the cost of formal old-age support provision. Nevertheless, we
do not consider investment decisions on such systems in this paper.
9

lump-sum taxes on the working generation. This assumption might clarify the effects ofmanagement costs of formal old-age support provision. The budget constraint of thegovernment in period t can be written as=Tt wt ht ( z G / µ nt −1 )(1 + M )(10)

in per-worker terms.
2.3. Consumption goods production
The production function of consumption goods is assumed as
Yt = Ktα L1t−α ,( α ∈ (0,1) )(11)

where Yt , Kt , and Lt = ht lt Nt respectively represent the aggregate output, aggregate
physical capital, and aggregate effective labor in period t . Assuming perfectlycompetitive markets, the following conditions hold as
wt =(1 − α ) Ktα L−t α =(1 − α )ktα (ht lt ) −α , andα −1 1−α=
1 + rt α K=
Lt
α ktα −1 (ht lt )1−α ,
t

(12)
(13)

where kt = Kt / Nt stands for physical capital per worker. The factor prices are,respectively, equal to the marginal productivity of the factor.

3. Dynamic System and the Long-Term Equilibrium
As described in this section, we first analyze the short-term market equilibrium in thenext subsection and then the long-term equilibrium in Subsection 3.2.
3.1. Market equilibrium in the short term
The market-clearing condition in the labor market in period t is given as=(1 − lt − φ nt ) N( Nt lt ) + LG
t
t ,

(14)

where the left-hand side is the labor time supply of the working generation, and where
the right-hand side represents the sum of the labor time demand of the consumption goods
G
production sector and that of the formal old-age support provision sector, i.e., LG
t = N t lt .

The total informal old-age support time in period t is Lt = Nt lt .
10

The difference between the minimum old-age support and the formal old-age support
is met by family old-age support provision, i.e., zNt −1 − µ LG
µ Lt , where the secondt =term on the left-hand side represents formal old-age support provision by governmentz G Nt −1 .
The equilibrium condition in the capital market can be expressed as
st Nt = Kt +1 , or, in per-worker terms, st = nt kt +1 .

(15)

The left-hand side of (15) denotes the savings of the period- t working generation. The
right-hand side is capital stock in the consumption goods production sector in period t + 1 .
We assume here that the capital stock depreciates completely after one-period use inproduction.
3.2. Dynamic system
The state variables in this dynamic model are the fertility rate nt , human capital per
worker ht , and physical capital per worker kt . The dynamic system can be described by
the three equations of these variables.
By inserting the government budget (10) and informal old-age support production
function ( z − z G ) Nt −1 =µ lt Nt into (7), we obtain the following rule of motion in thefertility rate asnt=

z + Mz Gε − λδ(1 −).
φ (1 + ε + ρ )µ nt −1(16)

From the human capital production function (4) and by inserting (12) into (8), we obtainφλδ δ kt
(17)
] [( ) / lt ]αδ ht .
ht +1 = σ [ε − λδ
ht
Substituting (10) and (12) into (9) gives the following equation:st=z + Mz G kt α −αρ (1 − α )(1 −)( ) l h .
1+ ε + ρµ nt −1 ht t t(18)

Inserting (16) and (18) into (15), we have the dynamic equation of per worker physicalcapital asφρ (1 − α ) kt α −α(19)( ) l h.
kt +1 =
ε − λδ ht t t
The dynamic system of this model consists of the three equations (16), (17), and (19). In
11the system, the labor employed in the consumption goods production sector is described
as
lt
=

zG
z
1 + ρ + λδ−[(1 + ρ + λδ ) − (ε − λδ ) M],z
1 + ε + ρ µ nt −1 (1 + ε + ρ )(20)

where we use the time constraint (1) and old-age support production functions. The labor
supply of the working generation implicitly reflects the old-age support time, informal
and formal, and child-rearing time. The old-age support time in turn depends on the ratioof the population size of working generation to the population size of the retired
generation or the fertility rate of the retired generation.
By defining a new variable υt = kt / ht and thereby simplifying the analyses of thelong-term equilibrium of the system, equations (17) and (19) can be united into thefollowing dynamic equation:

υt +1 =

φ1−δ ρ (1 − α )(υt / lt )α (1−δ ) .
1−δδσ (λδ ) (ε − λδ )(21)

Therefore, the dynamic system becomes a set of two equations of the fertility rate nt
and the physical capital–human capital ratio υt , (16) and (21).
As might be readily apparent, equation (16) is a difference equation only of the fertility
rate nt . Assuming the existence of stable long-term fertility rate n for analytical
purposes, the fertility rate satisfies the following condition. 15

z + Mz Gε − λδ=n(1 −)φ (1 + ε + ρ )
µn

(16’)

From (20), when n remains constant, lt also becomes constant, i.e., l . In this case,equation (21) becomes a difference equation of υt . Because 0 < α (1 − δ ) < 1 , a stablesteady-state solution υ exists as

15

Equation (16) is a quadratic function in n . If the discriminant conditionµ (ε − λδ ) − 4φ (1 + ε + ρ )( z + Mz G ) > 0 holds, then a stable solution is obtained. Thiscondition gives the maximum level of management factor M max for n to be a realdnt
ε − λδ z + Mz G=
∈ (0,1) .
value. The stability conditionis given as
dnt −1 φ (1 + ε + ρ ) µ n 2121

φ1−δ ρ (1 − α )υ =[l −α (1−δ ) ]1−α (1−δ ) .
δ
1−δσ (λδ ) (ε − λδ )(22)

The growth rates of human capital and of physical capital become equal in the long term,
where the growth rate is given from (17) as
ht +1
φλδ δ
(23)
] (υ / l )αδ ,= σ[
ε − λδ
ht
where from (22), we haveφ (1 − α ) 1−δ1)1−α (1−δ )
−
ε
λδ
.
]
=[
l
σ (λδ )δ l

υ

(

(24)

Therefore, the human capital growth rate gives the balanced growth rate, which can be
described as=γ ( ht +1 / h=t − 1 kt +1 / kt − 1) .
From the above findings, we might have two comparative statics results. First, thebalanced growth rate γ is higher when the per-child rearing time φ is greater. From(16’), it is apparent that an increase of the per-child rearing time decreases the stablefertility rate. The lower fertility rate in turn decreases the employment of labor time in the
consumption goods production (see (20)). Therefore, together with (22) and (24), theincrease in the per-child rearing time increases the physical capital–human capital ratio.
Consequently, the increased per-child rearing time raises the wage rate and thereby
education expenditure through the income effect (see (8)). This result reflects a quantity–
quality trade off of children.
Second, higher efficiency in human capital production σ also engenders a higher
growth rate for the following reason. From (4), the increased production efficiency boosts
human capital accumulation for a given parental education expenditure. Greater amountsof human capital per worker lower the physical capital–human capital ratio (see (22)),lowering the wage rate (see (12)). Although the low wage rate works to decrease parental
education expenditure (see (8)), the positive effect of the higher human capital production
efficiency dominates the negative effects of the low wage rate, consequently increasing
the growth rate (see (23)). The steady-state effects of other parameters are indeterminate
a priori.

4. Formal Old-Age Support ProgramThis section presents an analysis of the effects of an increase in formal old-age support
provision by the government. The policy change engenders two immediate effects on the13economy: one is decreased informal old-age support of families because old-age support
provision remains minimal; the other is an increased tax burden to finance a greater
formal support. We are concerned with the long-term effects on the economy in thefollowing.
The long-term effect on the fertility rate is obtained from (16’) as
dn
M
=
−
<0.
(25)
G
dz
φ (1 + ε + ρ ) z + Mz Gµ n[−]ε − λδ
µ n2
The denominator on the right-hand side of (25) is positive from the stability condition of
the long-term fertility rate. The increased formal old-age support lowers the long-term
fertility rate. We have the following lemma:
Lemma 1 If formal old-age support provision involves management costs, then an
increase in formal old-age support provision lowers the long-term fertility rate.
In this model setting, formal old-age support is less cost-efficient than informal old-age
support because of the existence of management costs. The increased tax burden brought
about by an increase in formal old-age support lowers the number of children of theworking generation through a negative income effect. This result is consistent with thatobtained by Yakita (2023). 16
Next, the effect of an increase in formal old-age support on the labor of theconsumption goods production sector is obtained from (20) as

dl
=
dz G

z G dnz G dn
+
ρ
+
λδ
−
)
(1
)(
)
z
n dz Gn dz G z[M −].
µ n(1 + ε + ρ )z G dn z G(ε − λδ )(1 −)n dz G(ε − λδ )(1 −(26)

From (25), both the coefficient of the bracketed terms and the second term in the bracket
on the right-hand side of (26) are positive. Therefore, we obtain the following results.

z G dn)n dz G z[≡ H ] .
z G dn z G(ε − λδ )(1 −)n dz G> (1 + ρ + λδ )( −dl >== 0 as M<
dz G <

16(27)

As demonstrated by Yakita (2023), when the time productivity related to formal
support and informal support differ, increases in old-age support might not lower thefertility rate, depending on the magnitude of management costs.
14

In those equations, H > 0 . 17 If the management cost factor M in formal old-age
support provision is sufficiently small, i.e., if M < H , then increases in formal old-age
support decrease labor employment in the consumption goods production sector, i.e.,dl / dz G < 0 . In this case, the market per-worker labor time employment, which is thesum of employments of the government and the consumption goods production sector,l G + l , does not necessarily increase, although government employment increases. Also,
if M = H , then changes in formal old-age support do not alter the labor employment in
the consumption goods production sector, i.e., dl / dz G = 0 . If the management cost
factor is great, i.e., if M > H , then the increased formal old-age support expands
employment in the consumption goods production sector, i.e., dl / dz G > 0 . In this case,the market labor supply of the working generation increases. 18
Next, we consider the effect on the balanced growth path. First, from (22), the physicalcapital–human capital ratio decreases with labor employment in the consumption goodsdυ−α (1 − δ ) υproduction sector, =i.e.,[] < 0 . Second, we obtaindl
1 − α (1 − δ ) ld (ht +1 / ht )h /h−1d (υ / l )and=[](υ / l 2 ) < 0 from= αδ t +1 t > 0 from (23)dl
1 − α (1 − δ )υ /ld (υ / l )(24). Therefore, the balanced growth rate decreases along with the labor employed in the
consumption goods production sector, i.e., d γ / dl < 0 . Together with (26), if the
management cost factor M in formal old-age support provision is sufficiently small(great), then increases in formal old-age support raise (lower) the balanced growth rate.
Therefore, if the management cost factor M in formal old-age support provision is
sufficiently small (great), then increases in formal old-age support raise (lower) thephysical capital–human capital ratio. Summing up, we can obtain the following result:
Proposition 1 When the management costs in formal old-age support provision are
sufficiently small (great), an increase in formal old-age support provision increases(decreases) the balanced growth rate and also raises (lowers) the physical capital–human17

It is noteworthy that the right-hand side of the condition in (27) includes the
management cost factor, i.e., H ( M ) . For analytical purposes, we assume the existenceof a unique M ∈ (0, M max ) satisfying (27). The possibility that dl / dz G < 0 for allM ∈ (0, M max ) cannot be ruled out a priori. An explanation for this assumption is given
in Appendix A.
18
Based on the analysis reported by Aya (2014) and assuming different time efficiency
between formal and informal long-term care provision, Yakita (2023) obtains M = 0.55
for Japan.
15

capital ratio.
The explanation underlying the results described above can be presented as described
below. Increases in formal old-age support z G require tax increases for financing them,
T . The increased tax burden decreases the number of children n and lifecycle savingss (see (7) and (9)). The smaller number of children increases old-age support time inputs,
formal and family, l G and l . Because total support z remains constant, the increaseof formal support works to decrease family old-age support time.
When the management cost factor of formal old-age support provision M is great,
the decreases in the fertility rate and lifecycle savings become large through the large
income effects (see (7) and (9)). Therefore, although the time inputs for old-age supportl G + l =z / ( µ n) increase, child rearing time is affected by the increased tax burden, i.e.,the management cost factor M . When M is great, the increased formal old-age
support, together with the lowered fertility rate, largely lowers child rearing time=φnε − λδ
z + Mz G(1 −) . The great decrease in the fertility rate might engender aµn1+ ε + ρ

higher labor time supply in the consumption goods production l =1 − (l G + l ) − φ n (seetime constraint (1)), possibly overwhelming the effect of increases of total old-age support
time l G + l . This result might be interpreted as that individuals might increase themarket labor supply largely to compensate the largely decreased disposable income. The
comparative statics result presented in (27) reflects these effects. Because the effects of
the increased formal old-age support on lifecycle savings and the fertility rate are both
great, the effect on per-worker physical capital k is indeterminate (see (15)). If the perworker physical capital changes slightly, then the physical capital–effective labor ratio in
the consumption goods production sector υ / l = k / (hl ) decreases with an increase of
the formal old-age support (see (24)). This effect lowers the wage rate (see (12)), thereby
decreasing parental education expenditures on children (see (8)). Therefore, when the
management cost factor is great, the increased formal old-age support deters the balanced
growth rate (see (23)).
By contrast, when the management cost factor M is small, the decrease of thefertility rate caused by increases in formal old-age support is small because of the small
increase of the tax burden. The individual’s time freed by the increased formal old-age
support works to increase the disposable income, thereby working to increase savings and16the number of children (see (7) and (9)). Therefore, in this case, the labor time supply to
consumption goods production decreases because of the time constraint (1). The
comparative statics results in (27) also reflect these effects. The decrease of lifecycle
savings is also small because of small tax increase. The decreased labor supply might
increase the physical capital–effective labor ratio (see (24)). Thereby, the wage rateincreases and therefore parental education expenditure also increases (see (8)).
Consequently, when the management cost factor is small, increases in formal old-age
support raise the balanced growth rate (see (23)). It is noteworthy that, in this case of asmall management cost factor, because of a small burden the management costs, thefamily old-age support time freed by increased formal old-age support works to increase
the disposable income of individuals greatly.
The result presented above contrasts to the result obtained by Miyazawa (2010), who
assumes strategic bequest motives. He shows, assuming exchange motives for bequests
and a constant population scale, that even if formal old-age support provision is less costefficient than family support, the increased formal old-age support boosts economic
growth: The positive valuation effect of formal old-age support dominates thedisincentive effect in the strategic behavior. Canta et al. (2016) also demonstrate that if
family long-term care (LTC) insurance is sizably cost-efficient, then a payroll tax increase
for formal LTC financing might increase the capital stock because individuals compensate
the crowded-out family help by increasing savings, in a model of constant population
with a family norm. By contrast, we assume endogenous fertility decisions of individuals
in addition to parental educational expenditure, engendering endogenous economic
growth. By contrast, the present analyses illustrate that even cost-inefficient public oldage support might accelerate economic growth because of endogenous fertility decisionsof individuals.

5. Concluding Remarks
We have demonstrated that the effects of formal old-age support on economic growthdepend crucially on the magnitude of management costs in formal old-age support
provision. If formal old-age support is provided cost-inefficiently with lower management
costs, then increases in formal old-age support raise the balanced economic growth rate
and the physical capital–human capital ratio, but that increased formal old-age support
lowers the long-term fertility rate. By contrast, if formal old-age support provision is
sufficiently cost-inefficient, then the increased formal old-age support deters economic
growth and lowers the physical capital–human capital ratio in the long term. The present
17analyses have policy implications for old-age support program design in population-aging
countries. For population-aging economies, making formal old-age support production
cost-efficient engenders high economic growth along with lowering of the fertility rate. 19
The main message from our analyses is that the negative effect of management costson the economy might be offset by labor time efficiency of formal old-age support
provision. 20 With more labor-time efficient formal old-age support, an increase in formal
old-age support needs a smaller increase of labor time input of the government sector,
freeing larger time of individuals from family old-age support and thereby suppressing
decline in fertility. The freed time might increase the wage income of individuals andtherefore human capital accumulation, consequently accelerating economic growth, if thetax burden increase caused is depressed by higher labor time efficiency of formal old-age
support provision.
Three remarks can also be made. First, we have not considered child policies because
this study emphasizes investigation of the effects of formal old-age support. Government
pursues a better tradeoff between elderly care and child care in policy making in the realworld (e.g., Yakita, 2018; Connelly, 1992). Second, we have not considered theprobability of becoming dependent and the uncertain lifetime of elderly people. Such aconsideration is presented rather commonly in the literature (e.g., Cremer and Roeder,
2017; Yakita, 2023). This paper especially emphasizes the effects of formal old-age
support on economic growth. Third, because of extended longevity, grandparents mightbear the burden of child rearing care, at least partly. Not only might grandparent’s childcare enable parents to supply more labor time (Miyazawa, 2016), but also grandchildren’scare for grandparents might reduce the necessity for formal elderly care by stimulatingthem cognitively (Arpino et al., 2014; Rapp et al., 2023).

Appendix A: Management cost factor M satisfying condition M = H of (27)
We have the following from (25):z G dnzG M[≡ Λ] .
=−n dz G µ n 2 φ (1 + ε + ρ ) − ( z + Mz G )ε − λδ

(A1)

The right-hand side of the condition of (27) can be rewritten as19

Effects of time efficiency of old-age support are briefly discussed in Appendix B.
When time efficiency in formal old-age provision is higher, the tax burden becomes
smaller (see (10)). Yakita (2023) considers a case in which time efficiency differs between
public and family old-age support sectors.
20

18

H=

(1 + ρ + λδ )Λ
,(ε − λδ )(1 + Λ )(A2)from which we obtaindH 1 + ρ + λδ
1
dΛ
=
.
2
dM
ε − λδ (1 + Λ ) dM

(A3)

From (A1) we obtainφ (1 + ε + ρ ) dn
Λ[2nµ− zG ]
dΛ Λ
dM
ε
λδ
−
{1 −
}> 0.
=
dM M
zG

(A4)

where we have from (16’):dn
z G φ (1 + ε + ρ ) z + Mz G=−(− ) / [
]< 0.
dM
µn
ε − λδµ n2(A5)

The right-hand side of the condition H increases with management cost factor M , i.e.,dH / dM > 0 . Because we have H = 0 when M = 0 , the condition (27) is satisfied
with equality.
As M increases, both sides increase. The right-hand side is apparently nonlinear.
From (A1) and (A5), we obtain Λ → 0 as M → 0 . From (A4), we also haved Λ / dM → ∞ as M → 0 . Therefore, one might have dH / dM → ∞ as M → 0
from (A3).

45°
1+ ε + ρε − λδ

H (M )



0

M max

M M =H

Figure A1 M satisfying condition M = H in (27)

Appendix B: Effects of Increased Support Time Productivity
19

M

This appendix presents an analysis of the effects of increases in the support-time
productivity of informal old-age support. It is noteworthy that, for these analyses, thesupport-time productivity is equal in both informal and formal old-age support provision.
From (16), we have

dn
=
dµ

z + Mz G
2 φ (1 + ε + ρ )nµ [ε − λδ

z + Mz G
−
]
µ n2

> 0.

(A6)

Because increases in the support-time productivity of old-age support provision reduce
old-age support time, the freed time might be allocated to child rearing. From (20) we
also obtaindl (1 + ε + ρ ) z − (ε − λδ ) Mz G
=
> 0.
dµ
µ 2 n(1 + ε + ρ )(A7)The increased productivity increases labor employed in the consumption goods
production sector. Therefore, together with (22), the physical capital–human capital ratio
decreases. From (23) and (24), the balanced growth rate becomes lower. A brief
explanation of these results is the following. The increased labor in the consumption
goods production sector lowers the wage rate. Therefore, because of the greater numberof children, parents reduce their per-child education expenses and also savings. However,
the results are not comparable to those reported by Yakita (2023), who assumes different
labor-time productivity between informal and formal old-age support.

Acknowledgments
Financial support from the Japan Society for the Promotion of Science (KAKENHI Grant
No. 22K001544) is gratefully acknowledged.

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