Doshisha University Center for the Study of the Creative Economy
Discussion Paper Series No. 2014-03

Grandparental child care, child allowances, andfertility
Kazutoshi Miyazawa (Doshisha University)

Discussion Paper Series

Grandparental child care, child allowances, andfertility
Kazutoshi Miyazawa∗

Abstract
Recent research on grandparenting reveals that it has a positive impact
not only on parental child care but also on the grandparent’s welfare. In
this study, we examine the eﬀect of child allowances on fertility by assuming that fertility is a joint product of both parental and grandparental childcare and that providing grandparental child care improves welfare. In doing so, we aim to establish a theoretical framework that more accurately
predicts the impacts of child care policies than that which is currently utilized in the literature (empirical evidence for which has been inconclusive
at best). We find that the fertility eﬀect of child allowances critically depends on individual preferences and household production technology. In
some cases, the fertility rate is monotonically decreasing or shaped like an
inverted U with respect to the size of child allowances. We therefore conclude that small child allowances can increase fertility in situations where
there is little initial parental child care. However, in situations where theinitial rate of parental child care is relatively large, or where grandparental
child care features as a key factor in household fertility production, childallowances can eﬀectively reduce the fertility rate.

JEL Classification: H21, J13, J14, O41
Keywords: Fertility, Child care, Grandparenting, Child allowance, Growth∗ Kazutoshi Miyazawa, Faculty of Economics, Doshisha University, Kamigyo, Kyoto 6028580 Japan. kazu@mail.doshisha.ac.jp

1

Introduction

Among the OECD countries, several, such as Japan, Korea, Germany, and Italy,
share a common concern regarding their total fertility rate as it is below the
replacement rate (2.07). Eﬀective and prompt public policies are needed toresolve this problem. One such potential policy, which standard endogenous
fertility models predict should be eﬀective, is a child allowance as it directly
lowers child care costs.1 Empirically, however, the eﬃcacy of this policy has
proven doubtful. Using the German Socio-Economic Panel (SOEP), Haan andWrohlich (2011) simulate women’s behavioral responses to changes in child carepolicies; they show that an increase in subsidized child care, conditional on
employment, has a positive impact on labor force participation, but its over-all
eﬀect on fertility is ambiguous.2 3
The purpose of this study is to bridge the gap between what theory would
predict and what the empirical evidence has shown. To do this, we introduce
an additional form of child care, which is provided by grandparents, into a standard endogenous fertility model.4 This seems to be a particularly appropriate
extension in that the importance of grandparenting has increased in developed
countries in line with the increase in healthy life expectancy (WHO, 2010), andthat the existence of informal child care providers has a significant impact on
not only labor force participation, but also on the demand for professional childcare (Blau and Robins, 1988; Hank and Kreyenfeld, 2003; and Kaptijn et al.,2010).
Recent research on grandparenting reveals its economic implications. Powdthavee(2011) shows a positive correlation between having grandchildren and self-rated
life satisfaction. Arpino and Bordone (2014) find that providing child care im1 Using an endogenous fertility model with pay-as-you-go public pensions, van Groezen etal. (2003) show that a generous child allowance has a positive impact on the fertility rate if thegovernment collects a lump-sum tax. Yasuoka and Goto (2011) arrive at the same conclusion,
though they recommend a consumption tax.
2 Positive fertility eﬀects are observed only in two subgroups, that is, highly educatedwomen and women having their first child.
3 Some papers, however, do find empirical evidence that child care policies increase fertility.
Milligan (2005) examines the fertility eﬀect of the Allowance for Newborn Children (ANC),
which was instituted in Quebec from 1988 to 1997; he shows that the fertility rate in Quebec
increased with the number of existing children and even rose by 12.0% in 1996.
Brewer et al. (2012) examine the fertility eﬀect of the Working Families’ Tax Credit(WFTC), which was introduced in the UK in 1999. They show: (1) among all women,
the fertility eﬀect is positive but insignificant. (2) Among women with a partner, the eﬀect issignificantly positive. (3) Controlling for the couples’ education levels, the reform increased
the birth rate among women with partners by 0.013, which implies that the fertility rate wasincreased by approximately 15%.
Cohen et al. (2013) examine how the probability of pregnancy among married women with
two or more children is associated with child subsidies and income. They show a NIS 150(approximately USD 34) monthly increase in the child subsidy leads to a 0.99% increase in
the probability of pregnancy.
4 Coall and Hertwig (2011) insist that grandmothers may have been the most productive,
experienced, and motivated helpers for reproducing mothers throughout human history. This
grandmother hypothesis is currently the most influential theory in explaining why human
female longevity extends beyond menopause.

1

proves cognitive function, such as verbal fluency, both for grandmothers andgrandfathers. These findings suggest that grandparenting improves grandparents’ welfare; furthermore, similar results were found in studies conducted in
the US (Silverstein and Marenco, 2001) and Europe (Del Boca, 2002; Hank andBuber, 2009; Aassve et al., 2012).
In this study, we use an overlapping generations model in which grandparents’ help lowers the cost of child rearing. In our model, there are markets
for both child and grandchild care; parents buy one, and grandparents buy theother. The life of each individual features two periods; that in which he/she
is a parent and that in which he/she is a grandparent. In the first period, theindividual provides parental child care, taking as given the grandparental childcare provided by their own parents. In the second period, the individual provides grandparental child care and takes their children’s parental child care as
given. In this setting, an intergenerational, strategic interaction arises. For example, a young adult would decrease their parental child care if they expected
their parent to increase grandparental child care. This mechanism plays animportant role in examining a policy’s eﬀect on fertility. We assume that thechild care policy entails a subsidy for the purchase of parental child care, that
grandparental child care is not covered by the policy, and that the revenue forits enactment is collected by wage taxation.
In this model economy, we show that the equilibrium fertility rate may havean inverted U-shape with respect to the tax rate. The child care policy would
encourage parental child care, which would, in turn, have a positive impact
on fertility. However, assuming that grandparents take total expenditure on
child care into account, they would provide less child care per child in response
to increases in family size. This decrease in grandparental child care would
have a negative impact on fertility. Furthermore, the child care policy would
increase the price of child care. This price eﬀect would reduce the demand
for grandparental child care and thereby also negatively impact fertility. If
grandparenting is important in household production, then expanding the childcare policy could decrease fertility.
In terms of theory, this paper is related to the work of Hirazawa and Yakita(2009), Fanti and Gori (2009), and Cardia and Ng (2003). Assuming that children are a product of parents’ time and that there is a market for child care,
Hirazawa and Yakita (2009) analyze the fertility eﬀect of pay-as-you-go public
pensions in a small open economy. They show that an increase in the contribution rate increases the amount of time parents spend on child care and that thedemand for professional child care decreases if the elasticity of parental childcare time with respect to the contribution rate is small. They consequently
conclude that public pensions decrease the fertility rate when the negative impact on professional child care outweighs the positive impact on the time thatparents dedicate toward child rearing. Although these results are similar toour own, the mechanisms behind the models are diﬀerent in several respects.
First, our central concern is the introduction of an intergenerational interaction,
which Hirazawa and Yakita (2009) do not include. Our model demonstrates anegative correlation between parental child care and grandparental child care,2

a relationship that significantly impacts the eﬀectiveness of child care policies.
Second, as we focus on policy in particular, we are able to draw more direct implications. Third, because our model is constructed using a general equilibrium
framework, variations in pricing play an important role in our policy evaluation.
Specifically, we show that a child care allowance serves to increase the wage rate
and the price of child care. Hirazawa and Yakita (2009) do not consider any
such price eﬀects in their study.
Using a simple overlapping generations model with endogenous fertility,
Fanti and Gori (2009) show that a child tax may increase fertility. They argue
that, on one hand, the child tax decreases fertility because it directly increases
the cost of child rearing, but on the other hand, it has an income eﬀect which
serves to increase fertility. In the short run, disposable income increases because
the tax is paid back in a lump-sum transfer, while in the long run, the wage rateincreases because the tax-transfer policy stimulates private saving. Fanti andGori (2009) conclude that child taxes increase fertility when there is a strong
preference regarding the number of children, the share of capital income is large,
and when the rate of time preference is large. In our model, unlike in Fanti and
Gori’s (2009), the wage rate is increased by a subsidy for child care; this reduces
grandparental child care and thus generates a negative fertility eﬀect.
By introducing intergenerational time transfers into home good production
functions, Cardia and Ng (2003) examine what kind of child care policies eﬀectively increase long-run per capita income. In a calibrated economy, they showthat subsidizing grandparental time spent on child care is the most eﬀective policy when time transfers are operative; otherwise, subsidizing parents’ purchasesof professional child care is the most eﬀective. Our model is similar to thatof Cardia and Ng (2003), but the motivations are quite diﬀerent. Cardia andNg (2003) assume that the fertility rate is constant and that monetary transfers come in the form of a lump-sum bequest. In our model, the fertility rateis endogenous and grandparental monetary transfers are targeted toward childcare.
The rest of our study is organized as follows: In Section 2, we introduce
a basic model and derive equilibrium conditions. In Section 3, we present an
explicit solution by specifying household production technology. In Section 4,
we examine the eﬀects of child allowances on fertility, child care costs, growth,and welfare. Section 5 concludes the paper.

2

Model

We use a two-period overlapping generations model with two sectors of production. We denote the number of individuals who enter the economy in period t
as Nt (we say they belong to “generation t”). The law of motion for Nt is given
by
Nt+1
= nt
(1)
Nt
where nt stands for the fertility rate of generation t.
3

Two generations coexist in each period. In period t, the population of theyounger generation is represented by Nt , and the population of the older generation is Nt−1 .
We assume the following fertility household production function,nt = n(et , gt )(2)

where et and gt stand for parental and grandparental child care, respectively.
Here, et represents a choice variable for young adults in period t whereas gt
represents a similar variable for old adults. Equation (2) implies that births are
the outcome of intergenerational collaboration within families.

2.1

Household

The utility function of an individual in generation t is given by
ut = ln c1t + ρ ln nt + β[ln c2t+1 + θ ln(nt gt+1 )](3)

where c1t and c2t+1 stand for consumption in young and old adulthood, respectively. nt is the fertility rate given by equation (2), β ∈ (0, 1) is a private
discount factor, and ρ ∈ (0, 1) is a preference parameter attached to the numberof children. In addition, we introduce a preference for grandparenting in thesecond period. This can be interpreted as either giving in order to receive a“warm-glow” (Andreoni, 1989), or as the utility of grandparenting’s cognitive
benefits (Arpino and Bordone, 2014). A grandparent provides each child with
grandparental child care, gt+1 , the strength of which is measured by the parameter θ ∈ [0, 1). If θ = 0, our model is reduced to a standard endogenous
fertility model, such as those used by van Groezen et al. (2003) and Zhang andZhang (2005).
In the first period of life, each individual supplies one unit of labor to either the goods production sector or the child care production sector. They
divide their disposable income among consumption, savings, and the purchaseof professional child care. In the second period, they spend their capital income
only on either consumption or professional child care. The household budget
constraints are given by(1 − τ )wt
Rt+1 st= c1t + (1 − σ t )pet et + st
= c2t+1 + nt pgt+1 gt+1

(4)
(5)

where st is savings, wt is the wage rate in period t, and Rt+1 is the gross interest
rate in period t + 1. pet stands for the price of parental child care in period t,
and pgt+1 represents the price of grandparental child care in period t + 1. The
perceived price of grandparental child care is not pgt+1 but rather nt pgt+1 because
gt+1 represents the amount of grandparental child care per child. If gt+1 is a purepublic good (such that three generations inhabit a household) the price may bedenoted pgt+1 . In this model, however, we assume that grandparental child careis a pure rivalry good. Furthermore, in period t, the government subsidizes the4purchase of parental child care at a rate of σ t ∈ [0, 1) by imposing an income tax
with a constant rate of τ ∈ [0, 1). When we specify the household production
function, this tax-subsidy scheme can represent a child allowance. This point
shall be explained in greater detail in the next section.
The household maximization problem is formulated as
max ut = ln c1t + ρ ln n(et , gt ) + β[ln c2t+1 + θ ln(n(et , gt )gt+1 )]subject to the lifetime budget constraint,(1 − τ )wt = c1t + (1 − σ t )pet et +c2t+1 + n(et , gt )pgt+1 gt+1
Rt+1

(6)

taking grandparental child care (gt ) as given.
Assuming interior solutions, the first-order conditions require that
1
c1t
β
c2t+1
1 ∂nt
(ρ + βθ)nt ∂et
βθ
gt+1

= λt
λt
Rt+1
∙
¸
pg gt+1 ∂nt= λt (1 − σ t )pet + t+1
Rt+1 ∂etλt nt pgt+1
=
Rt+1
=

where λt represents a multiplier attached to equation (6).
Let us denote the elasticity of fertility with respect to parental child care as
εne,t =et ∂ntnt ∂et(7)Then, we can derive the demand for parental child care and grandparental
child care as follows5
et

=

gt+1

=

ρεne,t(1 − τ )wt
1 + β(1 + θ) + ρεne,t (1 − σ t )petβθ
(1 − τ )wt Rt+1
1 + β(1 + θ) + ρεne,tnt pgt+1(8)(9)

as well as the saving function,st =β(1 + θ)(1 − τ )wt
1 + β(1 + θ) + ρεne,t(10)5 In general, εne,t is a function of (et , gt ). Therefore, equations (8) and (9) implicitly solvefor et and gt+1 as response functions of gt .

5

2.2

Production, government, and markets

The technology in the goods production sector is specified by a Cobb—Douglas
function,
Yt = F (Kt , Bt Lgt ) = AKtα (Bt Lgt )1−α(11)where Yt , Kt , and Lgt represent output, capital, and labor, respectively. Bt islabor augmenting technology, α ∈ (0, 1) is a share of capital income, and A > 0
is constant total factor productivity.
Competition in the factor markets require that
Yt
Kt

Rt

= α

wt

= (1 − α)

(12)
Yt
Lgt

(13)

In the child care production sector, one unit of child care is produced by one
unit of labor, that is,
Ytc = f (Lct ) = Lct(14)
where Ytc and Lct stand for output and labor. Without loss of generality, we
assume that the production technology for grandparental child care is the sameas that for parental child care. Consequently, competition in the child caresector makes the prices of parental and grandparental child care equal to thewage rate:
pet = pgt = wt
(15)
The government’s budget constraint is given byτ wt Nt = σ t pet et Nt(16)

The market clearing conditions for labor, capital, and child care are given
by
Nt
Kt+1
Ytc

= Lgt + Lct= Nt st= Nt (et + gt )

(17)
(18)
(19)

The above model is closed. The goods market clearing condition
Yt = Nt c1t + Nt−1 c2t + Kt+1
can be arrived at by applying Walras’ law.
Based on the ideas regarding labor augmenting technology put forward by
Arrow (1962), and specified by Romer (1986) and Grossman and Yanagawa(1992), we assume
Kt
Bt = g(20)
Lt
In equation (20), the gross interest rate is constant over time (Rt = αA),and per capita income, (Yt + pt Ytc )/Nt , grows at the same rate as wages.
6

3

Specified technology

In this section, we derive a balanced growth path equilibrium. In order to obtainan explicit solution, we specify equation (2) such thatnt = (et )ε H(gt )(21)

where H(0) > 0, H 0 ≥ 0, and ε > 0. This is a natural extension of thestandard endogenous fertility models used by van Groezen et al. (2003) andZhang and Zhang (2005). Without grandparental altruism (θ = 0), and whenε = 1, the cost of child care is proportional to the number of children in a family,(1 − σ t )pet nt /H(0). Therefore, a subsidy policy targeted at parental child carewould be the same as a child allowance.
First, we put forward a proposition regarding parental child care.
Proposition 1 We assume that household production technology is specified byequation (21). In a balanced growth path equilibrium, parental child care andthe subsidy rate are respectively given by
e∗t
σ ∗t

= e(τ ) = (1 − τ )eo + ττ= σ(τ ) =(1 − τ )eo + τ

(23)

eo =

ρε
∈ (0, 1)
1 + β(1 + θ) + ρε

(24)

where

(22)

represents the amount of parental child care that young adults would produce
without child care policies. Both e(.) and σ(.) increase as the tax rate increases.
Proof. From equation (21), we get εne,t = ε. From equations (8) and (15), weget
1−τ≡ eH (σt ; τ )(25)
et = eo1 − σtwhere eo is given by equation (24). The H in equation (25) stands for “household” in the sense that this function is derived from the household maximization
problem. eH (.) increases with σ t because the subsidy rate lowers the price ofparental child care. This positively sloped curve shifts downward when τ is
suﬃciently large because of the negative income eﬀect.
Next, from equations (15) and (16), we getet =τ≡ eG (σ t ; τ )
σt

(26)

The G in equation (26) stands for “government” in the sense that this function is derived from the government budget constraint. eG (.) decreases with σ tbecause total revenue is constant. When τ is large, this negatively sloped curve
shifts upward because the budget constraint is relaxed.
7

The equilibrium subsidy rate is given by eH (σ t ; τ ) = eG (σ t ; τ ). Solving this,
we obtain equations (22) and (23). An analysis of τ with comparative statistics
shows that the equilibrium point (σ ∗t , e∗t ) moves upward and to the right, which
implies that both parental child care and the subsidy rate are positively related
to the tax rate.
Second, we put forth a proposition regarding grandparental child care.
Proposition 2 We assume that household production technology is specified byequation (21). In a balanced growth path equilibrium, grandparental child careis negatively related to parental child care, that is,gt∗ = g(et ) = d(1 − et )

(27)

αθ
∈ (0, 1)
1−α+θ

(28)

where
d=

Proof. From equation (21), we get εne,t = ε. We obtain the demand for
grandparental child care with equations (9), (10), and (15) as follows
gt+1 =

θ Rt+1 st
1 + θ nt wt+1

which implies that expenditures on grandparental child care are a constant shareof capital income.
From equation (18), we obtain kt+1 = st /nt where kt = Kt /Nt stands forcapital per capita. Using this, we getgt =αθxt ≡ g H (xt )
1+θ

(29)

where xt = Akt /wt denotes the ratio of per capita output in the goods production sector to the wage rate in period t. g H (.) increases as xt increases because
xt is positively related to capital income, and negatively related to the price ofgrandparental child care.
On the other hand, based on equations (13), (14), (17), and (19), the wage
rate in period t can be given bywt =(1 − α)Akt
1 − et − gt

(30)

Therefore, we get
gt = 1 − et − (1 − α)xt ≡ g L (xt ; et )(31)

The L signifies “labor” in the sense that this function is derived from thelabor market condition. g L (.) decreases as xt decreases because the demand
for labor in the goods production sector, 1 − et − gt , increases with xt . When8

et is suﬃciently large, this negatively sloped line shifts downward because thedemand for labor is constant for a given xt .
The equilibrium is given by g H (xt ) = g L (xt ; et ). By solving this, we obtainequation (27). An analysis of et with comparative statistics shows that theequilibrium point (x∗t , gt∗ ) moves downward and to the left, which implies bothgrandparental child care and the output-wage ratio are negatively related toparental child care.
Equation (27) shows that grandparental child care is independent of thefunctional form H(.) in equation (21). This is because, in general, the taxsubsidy scheme aﬀects the demand for grandparental child care in three ways
(See equation (9)): First, the tax burden decreases grandparental child carethrough the negative income eﬀect. Second, the tax-subsidy scheme increases
both the wage rate and the price of grandparental child care; this price eﬀect
arises from the reallocation of labor. Because the policy increases the demandfor child care, labor moves from the goods sector to the child care sector. The
subsequent scarcity of labor in the goods sector serves to increase the wage
rate. While this increase has a positive income eﬀect for young adults, it has
a negative price eﬀect for older adults because the price they pay to care for
their grandchildren is equal to the wage rate. Third, the fertility rate increases.
Because the fertility rate is included in the perceived price of grandparental
child care, it has an additional negative price eﬀect on demand.
In our specified economy, some eﬀects cancel each other out. Based onequations (22), (27), (28), and (30), the after-tax wage rate in period t is givenby(1 − α)Akt(1 − τ )wt =(1 − d)(1 − eo )Because the right-hand side does not contain policy variables, we know that
there is no net income eﬀect on the tax-subsidy scheme. On the other hand,using equations (15), (27), (28), and (30), we find that the perceived price ofgrandparental child care can be written asnt pgt+1 =(1 − α)Ant kt+1(1 − d)(1 − et+1 )As nt kt+1 (= st ) is also independent of the tax-subsidy scheme, we find thatthe price of grandparental child care is inversely proportional to (1 − et+1 ).
This implies that the demand for grandparental child care, gt+1 , is proportional
to (1 − et+1 ) because, as we assume a logarithmic utility function, the priceelasticity of demand is equal to one.
Proposition 2 states that there is, by means of a market mechanism, astrategic interaction between the diﬀerent generations. That is, the tax-subsidy
scheme increases parental child care (et+1 ) which, in turn, increases the perceived price of grandparental child care and thereby decreases demand.

9

4

The eﬀects of child allowances on fertility, childcare costs, growth, and welfare

4.1

Fertility

In this section, we examine how the child allowance policy aﬀects the fertility
rate. From equation (21), we can see that the fertility rate may be given asnt = e(τ )ε H(g(e(τ ))) ≡ n(τ )(32)where e(τ ) is given in Proposition 1, and g(e) is given in Proposition 2. Becausee0 > 0 and g 0 < 0, it is clear that the relationship between fertility and the tax
rate is not monotonic and depends on the curvature of H. For example, assume
that H is convex; at a low tax rate, a marginal increase would raise the fertility
rate because the positive eﬀect on parental child care would dominate the negative eﬀect on grandparental child care. Conversely, the marginal eﬀect would
be reversed at a higher tax rate. As such, we arrive at following proposition:
Proposition 3 The fertility rate in equation (32) increases (decreases) with the
tax rate if and only if
e(τ )
ε
< (>)
(33)
1 − e(τ )εng (τ )where εng = (g/n)(∂n/∂g) represents the elasticity of fertility with respect tograndparental child care.
Proof. We control e instead of τ because e = e(τ ) is a linear, increasing
function. By diﬀerentiating n = eε H(g(e)) with respect to e, and using the factthat g = d(1 − e), we getµ¶e
n0 (e) = eε−1 H ε − εng
1−ewhere εng = (g/n)(∂n/∂g) = gH 0 /H. Therefore, n0 (e) R 0 ⇔ e/(1−e) Q ε/εng .
In order to intuitively understand Proposition 3, we further assume that
H(g) = D (μ + g)

φ

(34)

where φ ≥ 0, μ ≥ 0, and D > 0 are constant parameters. This functional form
is similar to that used by Palivos and Varvarigos (2010, 2013). The eﬀect of achild allowance on fertility is summarized in the following proposition:
Proposition 4 We assume that household production technology is specified byequations (21) and (34). As such, the fertility function n(τ ) in equation (32) is(i) monotonically decreasing if e∗ ≤ eo , (ii) inverted U-shaped if eo < e∗ < 1,10and (iii) monotonically increasing if e∗ ≥ 1, where eo is given by equation (24)and e∗ is given by
1 + μd(35)e∗ =1 + φεIn case (ii), the fertility rate is maximized atτ=e∗ − eo
1 − eo(36)

Proof. From equation (34), we know that the elasticity is given by εng =φg/(μ + g). When we combine this with equation (27), we find that equation(33) is equivalent to
1 + μd≡ e∗e(τ ) < (>)
1 + φε(i) If we assume that e∗ ≤ eo then n(τ ) is monotonically decreasing for anyτ ≥ 0 because eo < e(τ ) for any τ > 0.
(ii) If we assume that eo < e∗ < 1, then n0 (τ ) > 0 if e(τ ) < e∗ , and n0 (τ ) < 0∗
if e < e(τ ). Consequently, the fertility rate is maximized at e(τ ) = e∗ , whichyields equation (36).
(iii) If we assume that 1 ≤ e∗ , then e(τ ) < e∗ for any 0 ≤ τ < 1. Therefore,n(τ ) is monotonically increasing.
The endogenous fertility models’ theoretical prediction that child allowances
would increase fertility would be true if and only if e∗ > 1. Otherwise, a childallowance may have a negative impact on fertility. One of the critical parametersis the strength of grandparents’ preference for child care, as outlined in thefollowing proposition.
Proposition 5 We assume that household production technology is specified byequations (21) and (34). As such, child allowances increase fertility if and only
if the preference for grandparenting is relatively small,θ < θ̂ =(1 − α)με
αφ − με(37)

where it is assumed that α > με/φ. If α ≤ με/φ, then child allowances would
increase fertility for any θ ≥ 0.
Proof. From equations (28) and (35), we can see that e∗ > 1 is equivalent to
με
αθ
<
1−α+θ
φ
The left-hand side increases as θ increase, and it is strictly smaller than α.
Therefore, if α ≤ με/φ, then we get e∗ > 1. If we assume that α > με/φ, thenwe obtain equation (37).

11

[Figure 1 and 2 are here]
Figure 1 illustrates the relationship between the tax rate and the equilibrium
fertility rate under certain conditions.6 In this example, we obtain eo = 0.14
and e∗ = 0.31, which implies that fertility has an inverted U-shape. Basedequation (36), we know that the fertility-maximizing tax rate is τ ∗ = 0.20 (anygreater rate would decrease fertility).
Figure 2 illustrates the relationship between the tax rate and the allocation of labor. It shows employment in the parental child care sector, e(τ ), thegrandparental child care sector, g(τ ), and the goods sector, 1 − e(τ ) − g(τ ) (indescending order). When τ = 0, 14.3% of the labor force is employed in theparental child care sector, 17.1% is employed in the grandparental child caresector, and 68.6% is employed in the goods sector. As the tax rate increases,e(τ ) increases while g(τ ) and 1 − e(τ ) − g(τ ) decrease. When τ ∗ = 0.20, thelabor allocation is given by 31.4%, 13.7%, and 54.9%, respectively. This example demonstrates that the child allowance policy would have a non-negligible
impact on the allocation of labor.
These observations raise one concern: If, for any reason, the mobility oflabor were to be limited, a child allowance would impose a heavy burden on theeconomy. This, in turn, implies that the child allowance should be smaller thanthat which we used in our example.

4.2

Child care costs

In this section, we examine how the child allowance policy aﬀects child carecosts. Using equations (22) and (23), we find that the price that young adults
must pay for the care of each child is given by(1 − σt )pet et(1 − τ )wt eo=ntnt
Based equation (29), for older adults the present value of the cost of eachchild’s care is given by
pgt+1 gt+1
θ st
=
Rt+1
1 + θ nt
Because (1 − τ )wt and st are independent of the child allowance policy, bothparental and grandparental costs are negatively related to the fertility rate. In
Figure 1, for example, the cost of child care decreases if τ < 0.20, and increases
if τ > 0.20.
6 The assumed parameters are β = 0.5, ρ = 0.6, θ = 0.6, α = 0.4, μ = 0.05, ε = 0.5,
φ = 1.5, and D = 25.

12

4.3

Per capita income growth

The growth rate of per capita income is given byβ(1 − α + θ)A
kt+1
=
kt
(1 + β + βθ)n(τ )(38)Equation (38) shows that the growth rate is negatively related to the fertility
rate, which is the same conclusion reached by standard endogenous fertility
models. However, our model provides diﬀerent policy implications because thefertility function is not monotonic. In Figure 1, for example, the child careallowance lowers the growth rate if 0 < τ < 0.20, and raises it if τ > 0.20. Witha high tax rate (such that τ > 0.46), the growth rate of per capita income would
be greater with the child policy than without it.

4.4

WelfareWe may examine the eﬀect of a child allowance on a country’s overall welfare
by first examining its eﬀect on fertility, child care costs, and the growth rate.
Suppose that the government introduces a child allowance in period T , while
keeping the tax rate at τ < τ ∗ from that point onward. The welfare eﬀect
that this policy would have on each generation can be inferred as follows: First,
future generations would suﬀer due to decreases in the growth rate. Second,
generation T − 1 would be worse oﬀ because the wage rate in period T (whichis the price of grandparental child care) would increase. Third, generation T
would be better oﬀ because the increased fertility rate would lower the cost thatthey would have to pay for child care (although the reduction in generationT − 1’s grandparental child care would have a negative welfare eﬀect). In sum,
introducing a child allowance would not produce a Pareto improvement. Additional policy instruments are required to compensate for the losses suﬀered bygeneration T −1 (such as a price subsidy, a public pension, or a public debt) andto oﬀset the negative growth eﬀect that a child allowance would create (such asa subsidy for savings).

5

Concluding remarks

In this study, we try to bridge the gap between the theoretical prediction thatchild allowances should improve the fertility rate and the inconclusive empiricalevidence to that eﬀect. Focusing on grandparenting, we show that the eﬀect thatchild allowances have on the fertility rate depends on individual preferences
and household production technology. If there is little parental child care to
begin with, then introducing a small child allowance would encourage fertility.
However, child allowances serve to depress fertility if the initial rate of parental
child care is relatively large, or if grandparental child care is a key factor inhousehold fertility production. Our results suggest governments should accountfor not only the parental child care environment, but also for the grandparental
environment in their endeavors to raise their fertility rate.
13

Acknowledgments
I would like to thank Ken Tabata, Hikaru Ogawa, Akira Nishimori, Makoto
Hirazawa, Tadashi Yagi, and seminar participants at Kwansei Gakuin University, Chukyo University, and Nagoya University for their useful comments on an
earlier version of this paper. This research is financially supported by the Japan
Society for the Promotion of Science (No.26380381). Any remaining errors are,of course, my own.

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16

Figure 1. Fertility rate

n
1.2

1.1

1.0

0.9

0.8
0.0

0.1

0.2

0.3

0.4

0.5

tau

Figure 2. Labor allocatione,g,1-e-g

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0

0.1

0.2

0.3

0.4

0.5

tau