Doshisha University Life Risk Research Center
Discussion Paper Series No. 2012-01

Sibling Rivalry and Fertility Dynamics
in the Gift Economy
Kazutoshi Miyazawa
Faculty of Economics
Doshisha University

Discussion Paper Series

Sibling rivalry and fertility dynamics in the gift
economyKazutoshi Miyazawa∗

Abstract
We present a theoretical model in which children transfer income totheir parent non-cooperatively among their siblings. In this gift economy,
we observe that fertility declines over time, capital accumulation can be
non-monotonic, and the steady-state equilibrium tends to be dynamicallyeﬃcient.

JEL Classification: J13, J14, O41
Keywords: Gift economy, Dynamic eﬃciency, Fertility, Sibling rivalry∗ Faculty of Economics,
kazu@mail.doshisha.ac.jp

Doshisha University,

Kamigyo,

Kyoto,

602-8580 Japan.

1Introduction

Intergenerational transfers within a family are one of the most controversial issues in public economics and population economics because they may neutralize
public policies related to income redistribution (Barro (1974)), and because they
may be a source of demographic transition (Galor and Weil (2000)). While descending transfers from parents to children, such as bequests, inter-vivos transfers, and education, have been analyzed extensively, research on ascending transfers from children to parents, such as gifts, attention, and informal care, have notbeen well developed, especially in the dynamic context. One reason is that thegift economy is dynamically ineﬃcient (Carmichael (1982)), which is a principle
rejected in most developed countries (Abel et al. (1989))1 .
Some of the exceptions include O’Connell and Zeldes (1993) and Wigger(2001). O’Connell and Zeldes (1993) show that the gift economy is dynamicallyeﬃcient if parents act as leaders, that is, if parents choose savings by using
a linear gift function that represents their children’s optimal response to thesavings choice. Perceiving their children as altruistic, parents have an incentive
to save less in order to receive more gifts. This induced undersaving makes the
economy dynamically eﬃcient. In the Romer (1986) endogenous growth model,
Wigger (2001) shows that the gift economy is dynamically eﬃcient in the sensethat the social return of capital is larger than the equilibrium growth rate.
The purpose of the paper is to give another rationale for the gift economy tobe dynamically eﬃcient. The focal points are endogenous fertility and sibling
rivalry2 . Parents choose the number of children they bear, taking gifts transferred from children as given. Each child chooses a gift to give to his parents,
taking the amount of gifts his siblings choose as given. In this scenario, aninteraction among the rates of fertility, gift-giving, and saving arises. Supposethat the fertility rate is low in equilibrium. Then the gift rate would be highbecause the decreased size of the family alleviates the free-rider problem within
the family. The saving rate would be low because parents rely on gifts from children in retirement. Consequently, the low saving rate could make the economy
dynamically eﬃcient. In a simple overlapping generations model, we show that
this scenario could be realized under fairly weak conditions. In addition to the
dynamic eﬃciency condition at a steady state, we also show that the fertility
rate declines over time and that capital accumulation could be non-monotonic in
the transition process. Intuitively, the fertility dynamics stems from a positive
relationship between the parents’ fertility decision and the children’s fertility decision. The non-monotonicity stems from a diﬀerence in the adjustment speedof capital and fertility.
The paper is organized as follows. In section 2, we introduce a basic model.
In section 3, we analyze the equilibrium dynamics of the rates of fertility, giftgiving, and saving. In section 4, we derive a dynamic eﬃciency condition whensiblings are non-cooperative. The final section concludes the paper.
1 The problem is also tackled in the two-sided altruism model (Abel (1987), Kimball (1987),
Laitner (1988), and Blackburn and Cipriani (2005) among others).
2 In a static model, Chang and Weisman (2005) show that sibling rivalry makes parental
transfers ineﬃcient.

1

2

The model

We use a two-period overlapping generations model with endogenous fertility. In
each period, identical individuals are newly born into the economy and live fortwo periods. In the first period, they supply one unit of labor and allocate theirincome among consumption, saving, child-rearing, and a gift to their parents. In
the second period, they retire from business and receive capital income and gifts
from their children to consume. In each period, identical firms produce a goodby employing capital and labor. Markets are competitive, and the economy isclosed.
We refer to the group of individuals that are born at period t as generationt. Denoting the population of generation t by Nt , and the number of children
each individual in generation t has by nt , we have
Nt+1
= nt
Nt

(1)

The utility function of an individual in generation t is given by
Ut = ut + θut−1
where
ut = u(c1t , c2t+1 , nt ) = ln c1t + β ln c2t+1 + η ln nt
ut stands for an individual’s own lifetime utility, which consists of young-age
consumption c1t , old-age consumption c2t+1 , and the number of children nt .
β > 0 is a private discount factor, and η > 0 is a preference parameter attachedto the number of children. Besides his own lifetime utility, this individual cares
about his parent’s utility, ut−1 = u(c1t−1 , c2t , nt−1 ), which is weighted by an
altruistic parameter, θ > 0. This altruism causes children to transfer income totheir parent.
His budget constraints in the first and second period, respectively, are givenby(1 − gt − st − φnt )wt(1 + rt+1 )st wt + nt gt+1 wt+1

= c1t
= c2t+1(2)(3)

where gt and st stand for a gift rate and a saving rate, respectively. wt is a
wage rate in period t, and rt+1 is an interest rate in period t + 1. φwt is achild-rearing cost per child.
In addition, this individual is interested in his parent’s old-age consumption,
c2t . Because he or she has (nt−1 − 1) siblings, he would expect his parent’s
old-age consumption to be
c2t = (1 + rt )st−1 wt−1 + σnt−1 gt wt + (1 − σ)[gt wt + (nt−1 − 1)ḡt wt ](4)

where σ is a binary parameter. If σ = 1, then c2t = (1 + rt )st−1 wt−1 +nt−1 gt wt .
He expects his siblings to choose the same amount of gift as he chooses. We
refer to this case as siblings who are ‘cooperative’. If σ = 0, then c2t = (1 +rt )st−1 wt−1 + gt wt + (nt−1 − 1)ḡt wt . He chooses the gift gt , taking his siblings’
gift ḡt as given. We refer to this case as siblings who are ‘non-cooperative’.

2From equations (2) and (3), the lifetime budget constraint is given by(1 − gt − φnt )wt +nt gt+1 wt+1
c2t+1
= c1t +
1 + rt+1
1 + rt+1(5)

The optimization problem is to choose c1t , c2t+1 , nt , and gt to maximize
utility subject to equations (5) and (4), while taking nt−1 , st−1 , gt+1 , and ḡt asgiven.
The first-order conditions for c1t , c2t+1 , nt , and gt require
1
− μt
c1t
β
μt
−
c2t+1
1 + rt+1
μ gt+1 wt+1
η
− μt φwt + t
nt
1 + rt+1βθ[σnt−1 wt + (1 − σ)wt ]− μt wt
c2t

= 0
= 0
= 0
≤ 0 with = if gt > 0

where μt stands for a multiplier attached to equation (5).
Assuming that the gift is operative, we have, in a symmetric equilibrium,
st + φnt =∙¸
gt+1 wt+1=
nt φ −(1 + rt+1 )wt(1 + rt )st−1 wt−1
+ nt−1 gt =wtβ+η(1 − gt )1+β+ηη(1 − gt )1+β+ηβθ
(1 − σ + σnt−1 )(1 − gt )
1+β+η

(6)
(7)
(8)

Equation (6) implies that the propensity to consume based on the aftertransfer income (1 − gt )wt is constant and given by (1 + β + η)−1 . Equation (7)implies that the net marginal cost of having children is equal to the marginal
benefit at the optimum. Equation (8) implies that the marginal cost of gifttransfers is equal to the marginal benefit at the optimum.
The production technology is represented by a constant-returns-to-scale production function,
Yt = F (Kt , Lt )Assuming that factor markets are competitive and that capital fully depreciates in one period, we have
1 + rtwt= f 0 (kt )= f (kt ) − kt f 0 (kt )where kt = Kt /Lt is a capital-labor ratio, and f (kt ) = F (kt , 1) is per capita
output. For tractability, we assume:
Assumption 1: The income share of capital is constant over time:
kt f 0 (kt )=αf (kt )

3(9)Market clearing conditions for labor, capital, and good are given respectively
by
Nt
Kt+1
Yt

= Lt
= Nt st wt= Nt c1t + Nt−1 c2t + Nt+1 φwt + Kt+1

Because the model is closed, the goods market clearing condition can begiven by Walras’ law. From the capital market clearing condition, we havekt+1 =

st wt
nt

(10)

With equations (9) and (10), equations (7) and (8) become

µ

φnt

=

α
+ gt nt−1
1−α

=

¶

1−α
η
gt+1 st +(1 − gt )
α
1+β+η
βθ
(1 − σ + σnt−1 )(1 − gt )
1+β+η

(11)
(12)

Equations (6), (11), and (12) determine the law of motion of gt , st , and nt .

3

Equilibrium and dynamics

3.1

Cooperative siblings

In this section, we analyze a benchmark case of σ = 1, that is, each child expects
his siblings choose the same amount of gift that he chooses.
The equilibrium is specified by equations (6), (11), andβθα+ gt =(1 − gt )
1−α
1+β+η

(13)

Equation (13) implies that the gift rate is constant over time. Equations (6)and (11) imply that the saving rate and the fertility rate are also constant overtime. Specifically, we have the following proposition.
Proposition 1 In the cooperative siblings model, the rate of fertility, gift-giving,and saving are constant over time and given respectively by
nt
gt
st

(β + η)θ − α(1 + β + η + βθ)φθ(1 − α)(1 + β + η + βθ)βθ − α(1 + β + η + βθ)= gC ≡(1 − α)(1 + β + η + βθ)α= sC ≡(1 − α)θ
= nC ≡The interior solutions requireθ>α(1 + β + η)
(1 − α)β

4

(14)
(15)
(16)

The process of capital accumulation is straightforward. Substituting equations (14) and (16) into equation (10), we havekt+1 =

φα(1 − α)(1 + β + η + βθ)f (kt )(β + η)θ − α(1 + β + η + βθ)(17)Given an initial condition, k0 , the capital-labor ratio converges monotonically to a unique steady state. Figure 1 illustrates the relationship between
fertility and capital accumulation in the cooperative siblings model.
[Figure 1 is here]

3.2

Non-cooperative siblings

In this section, we analyze a case of σ = 0, that is, each child chooses his gift
while taking his siblings’ gifts as given.
The equilibrium is specified by equations (6), (11), and
µ
¶
α
βθ
(18)
+ gt nt−1 =(1 − gt )1−α1+β+ηIn contrast to the cooperative siblings model, the gift rate depends on thenumber of siblings. This is due to the free-rider problem within the family. Eachchild has an incentive to decrease transfers to his parent if he has many siblings.
Equation (18) specifies a gift function gt = g(nt−1 ). Substituting this function into equations (6) and (11), we have nt = n(nt−1 ) and st = s(nt−1 ).
Specifically, we have the following proposition.
Proposition 2 In the non-cooperative siblings model, the law of motion of thefertility rate is given bynt =θ [η + β(1 − α)] nt−1β(1 − α)φθ + (1 + β + η)[α + (1 − α)φθ]nt−1
2(19)

If η + β(1 − α)(1 − φθ) > 0, then nt monotonically converges tonN =θ[η + β(1 − α)(1 − φθ)]
(1 + β + η)[α + (1 − α)φθ](20)

The gift rate and the saving rate converge monotonically to
gN

=

sN

=

βφθ − α(η + βφθ)
(1 − α)(β + η)
α[η + β(1 − α)(1 − φθ)](1 − α)(1 + β + η)[α + (1 − α)φθ]The interior solution requires
η + β(1 − α)αη<θ<β(1 − α)φβ(1 − α)φProof. See Appendix.

5

(21)
(22)

The process of capital accumulation is complicated by the fertility dynamics.
From equation (10), we havekt+1 =α Φ(nt )f (kt )θ Φ(nt−1 )

(23)

where

βθ
(24)
nt
Assume that the fertility rate in period t − 1 is higher than the steady state,nt−1 > nN . First, we know nt−1 > nt > nN from equation (19). Second, weknow Φ(nt ) > Φ(nt−1 ) from equation (24). Therefore the coeﬃcient of f (kt )in the right hand side of equation (23) is larger than α/θ in the process oftransition. Finally, using equation (19), we haveΦ(nt ) = 1 + β + η +β 2 (1 − α)φθ2 + (1 + β + η)[β + η + β(1 − α)φθ]nt−1
Φ(nt )=
Φ(nt−1 )[η + β(1 − α)][βθ + (1 + β + η)nt−1 ]which is increasing in nt−1 . Therefore, the coeﬃcient decreases monotonicallyand converges to α/θ in the process of fertility decline. The law of motion of
kt depends not only on the downward shift of the investment curve but also on
the initial condition.
[Figure 2 and 3 are here]
Figure 2 illustrates a case in which capital accumulation is non-monotonic.
Starting at a low initial condition, the capital-labor ratio increases steadily andmay go far beyond the steady state at any given time. Then, it decreases andconverges to the steady state. Figure 3 illustrates the relationship between
fertility and capital accumulation in the non-cooperative siblings model.

3.3

Comparison

In this section, we compare the steady-state equilibrium when siblings are cooperative and when they are not. The following propositions summarize theresults.
Proposition 3 For a small child-rearing cost φ, there exists a pair of (θ, θ)such that nC > 1 and nN > 1 for any θ ∈ (θ, θ).
Proof. See Appendix.
Proposition 4 For any θ ∈ (θ, θ), we have(i) g N < g C(ii) sN > sC(iii) kN > kC
Proof. See Appendix.

6

The reason is simple. First, other things be equal, non-cooperative siblings
evaluate their own gifts less than cooperative ones if the number of siblings
is strictly larger than one. Therefore, the steady-state gift rate in the noncooperative siblings model is lower than that in the cooperative siblings model.
Second, the lower the gift rate is, the higher the saving rate. A lower gift
rate implies that disposable income in the working period is larger, and that
disposable income in the retirement period is smaller. Therefore, individuals
have incentives to save in order to smooth their consumption. Third, the higher
is the saving rate, the higher the steady state capital-labor ratio. Comparing
the steady states, the wage rate is higher and the interest rate is lower in the
non-cooperative siblings model.
One might imagine that the steady state fertility rate in the non-cooperative
siblings model is always lower than the fertility rate in the cooperative siblings
model because individuals have to increase private savings. The answer is no.
From equations (A11) and (A12) in Appendix, we have∙¸nC − 1
1+β+η
α
=
1+
(25)
nN − 1
1 + β + η + βθφθ(1 − α)Because equation (25) is decreasing in θ, it can be true that nN > nC > 1
for a large θ. Specifically, we have the following proposition.
Proposition 5 Assume that θ ∈ (θ, θ). Then, nN > nC > 1 for any θ > θ̃ andnC > nN > 1 for any θ < θ̃, wheresα(1 + β + η)
θ̃ =
(26)
(1 − α)βφ4Dynamic eﬃciency

One of the most controversial issues in the gift economy is that the steadystate equilibrium tends to be dynamically ineﬃcient (Abel (1987), O’Connell
and Zeldes (1993)). Because our model includes a choice of fertility, one might
imagine that the assumption of endogenous fertility is critical to the dynamic efficiency condition. We show, however, that it is not fertility, but rather strategic
behavior among siblings, that aﬀects the dynamic eﬃciency condition.
From equations (9) and (10), we have, at a steady state,
1+r
α 1
=
n
1−αs

(27)

Equation (27) provides a simple rule for the dynamic eﬃciency condition:
the economy is dynamically eﬃcient if and only if the equilibrium saving rate,s, is smaller than the ratio of the share of capital income to the share of labor
income, α/(1 − α). Specifically, we have the following proposition.
Proposition 6 In the cooperative siblings model, the steady-state equilibrium
is dynamically eﬃcient if and only if θ > 1.
In the non-cooperative siblings model, the steady-state equilibrium is dynamically eﬃcient if and only ifθ>

β + η − α(1 + 2β + η)φ(1 − α)(1 + 2β + η)
7(28)

Proof. Substituting sC in equation (16) into equation (27), we have (1 +r)/n = θ. Therefore θ > 1 is a necessary and suﬃcient condition for dynamic
eﬃciency.
Substituting sN in equation (22) into equation (27), we have
1+r
(1 + β + η)[α + (1 − α)φθ]=nη + β(1 − α)(1 − φθ)
which is larger than one if and only if equation (28) is satisfied.
The condition θ > 1 is well-known and often criticized because altruism toparents is too strong (Michel et al. (2006)). The condition in equation (28)
is one of the contributions of this paper. Suppose that the preference for thenumber of children is weak enough to make the right hand side of equation (28)
negative, that is,αη<(1 + β) − β(29)
1−αThen, the steady-state equilibrium in the non-cooperative siblings model
is dynamically eﬃcient for all θ > 0, given that gifts are operative3 . The
reason is simple. The smaller η is, the smaller the equilibrium fertility rate.
Because the free-rider problem is alleviated, children increase giving gifts totheir parent, which in turn decreases the parent’s private saving because they
expect to receive gifts from their children in the future. Therefore, the interest
rate tends to increase beyond the depressed fertility rate in the capital market.

5

Conclusions

While intergenerational transfers from parents to children have been extensively
analyzed in public economics and population economics, intergenerational transfers from children to parents have not been well developed, especially in the
dynamic context. One of the reasons could be a theoretical limitation insofar
as gift economies tend to be dynamically ineﬃcient. Alternatively, unrealistic
altruism towards the parent must be assumed to make gifts operative. By incorporating non-cooperative behavior among siblings into a simple overlapping
generations model, we tried to overcome this theoretical limitation. In a noncooperative siblings model, we show that the fertility rate declines over time,
capital accumulation can be non-monotonic, and the steady-state equilibrium
tends to be dynamically eﬃcient. Further investigation into strategic behaviors
within families could be worth-while to make the gift economy model applicable
to issues in public economics and population economics.

3 The condition (29) is not so restrictive.

Assume that the capital share is α = 1/3 andthat the private discount factor is β = 2/3 (which implies an annual discount rate is 1.3 percent when one period is 30 years). Then, equation (29) implies η < 1/6.

8

Acknowledgments
I would like to thank Hikaru Ogawa, Tatsuya Omori, participants at the 26th
annual congress of ARSC at Aomori Public College, and seminar participantsat Nagoya University for useful comments. The financial support of the Japan
Society for the Promotion of Science (No.22530190) and the Kampo Foundation
are gratefully acknowledged. The remaining errors are of course mine.

References[1] Abel, A.B. (1987) Operative gift and bequest motives, American Economic
Review 77, 1037-1047.
[2] Abel, A.B., Mankiw N.G., Summers L.H., Zeckhauser R.J. (1989) Assessing
dynamic eﬃciency: theory and evidence, Review of Economic Studies 56,
1-20.
[3] Barro, R.J. (1974) Are government bonds net wealth?, Journal of Political
Economy 82, 1095-1117.
[4] Blackburn, K., Cipriani, G.P. (2005) Intergenerational transfers and demographic transition, Journal of Development Economics 78, 191-214.
[5] Carmichael, J. (1982) On Barro’s theorem of debt neutrality: the irrelevance of net wealth, American Economic Review 72, 202-213.
[6] Chang, Y-M., Weisman, D.L. (2005) Sibling rivalry and strategic parental
transfers, Southern Economic Journal 71, 821-836.
[7] Galor, O., Weil, D.N. (2000) Population, technology, and growth: From
Malthusian stagnation to the demographic transition and beyond, American Economic Review 90, 806-828.
[8] Kimball, M.S. (1987) Making sense of two-sided altruism, Journal of Monetary Economics 20, 301-326.
[9] Laitner, J. (1988) Bequests, gifts, and social security, Review of Economic
Studies 55, 275-299.
[10] Michel, P., Thibault, E., Vidal, J-P. (2006) Intergenerational altruism andneoclassical growth models, Kolm, S-C., Ythier, J.M. eds., Handbook of
the Economics of Giving, Altruism, and Reciprocity, Volume 2, Elsevier,1055-1106.
[11] O’Connell, S.A., Zeldes, S.P. (1993) Dynamic eﬃciency in the gifts economy, Journal of Monetary Economics 31, 363-379.
[12] Romer, P.M. (1986) Increasing returns and long-run growth, Journal of
Political economy 94, 1002-1037.
[13] Wigger, B.U. (2001) Gifts, bequests, and growth, Journal of Macroeconomics 23, 121-129.

9Appendix[Proof of Proposition 2]
From equation (18),gt =αβθ − 1−α(1 + β + η)nt−1
βθ + (1 + β + η)nt−1(A1)

From equations (6) and (11),
st

=

φnt

=

β(1 − gt )¡¢(1 + β + η) 1 + 1−α
α gt+1¶µ
1 − gtββ+η−1+β+η
1 + 1−αα gt+1(A2)(A3)

First, substituting equation (A1) into equation (A3), we havent =θ[η + β(1 − α)]nt−1β(1 − α)φθ + (1 + β + η)[α + (1 − α)φθ]nt−1
2(A4)

which is equation (19) in the main body.
Let us define a function of n by
f (n) =An
B + Cnwhere A, B, and C are all positive constants. The function is increasing andconcave in n ≥ 0 and has a upper bound of A/C.
Equation n = f (n) has a unique positive solution if and only if
f 0 (0) > 1 ⇔ A > B
From equation (A4), this condition is equivalent toη + β(1 − α)(1 − φθ) > 0(A5)

If equation (A5) is satisfied, then nt monotonically converges tonN =θ [η + β(1 − α)(1 − φθ)]
(1 + β + η)[α + (1 − α)φθ](A6)which is equation (20) in the main body.
Second, equation (A1) implies the gift rate monotonically converges to
gN

=
=

α
βθ − 1−α(1 + β + η)nN
βθ + (1 + β + η)nN
βφθ − α(η + βφθ)
(1 − α)(β + η)

which is equation (21) in the main body. The interior condition requiresθ>αηβφ(1 − α)10(A7)Finally, we examine the law of motion of the saving rate. Substituting
equation (A1) into equation (A2), we have
st =

αnt−1
βθ + (1 + β + η)nt·
(1 − α)θ βθ + (1 + β + η)nt−1(A8)

Therefore the saving rate converges tosN

αnN(1 − α)θ
α[η + β(1 − α)(1 − φθ)](1 − α)(1 + β + η)[α + (1 − α)φθ]

=
=

(A9)

which is equation (22) in the main body.
To examine the dynamics, substituting equation (A4) into equation (A8),
we have©
ªαnt−1 β 2 (1 − α)φθ2 + (1 + β + η)[β + η + β(1 − α)φθ]nt−1©
ªst =(1 − α)[βθ + (1 + β + η)nt−1 ] β(1 − α)φθ2 + (1 + β + η)[α + (1 − α)φθ]nt−1(A10)
Log-diﬀerentiating equation (A10) with respect to nt−1 , we have
∂ ln st
∂nt−1

=

1

1+β+η
nt−1
βθ + (1 + β + η)nt−1(1 + β + η)[β + η + β(1 − α)φθ]+ 2
β (1 − α)φθ2 + (1 + β + η)[β + η + β(1 − α)φθ]nt−1(1 + β + η)[α + (1 − α)φθ]−β(1 − α)φθ2 + (1 + β + η)[α + (1 − α)φθ]nt−1−

It can be shown that the sum of the first and second terms is positive, andthat the sum of the third and fourth term is also positive. Thus, we know
∂st /∂nt−1 > 0 for any nt−1 . Because the path of nt is monotonic, the path of
st is also monotonic.
[Proof of Proposition 3]
From equations (14) and (20), we havenC − 1 =nN − 1 =(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ]φθ(1 − α)(1 + β + η + βθ)(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ](1 + β + η)[α + (1 − α)φθ]

(A11)
(A12)

Therefore the steady-state fertility rates are larger than one if and only if(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ] > 0(A13)

Rearranging terms, equation (A13) is a quadratic inequality of θ such asβ(1 − α)φθ2 − [η + β(1 − α) − (1 + β + η)(1 − α)φ]θ + α(1 + β + η) < 0 (A14)
Denote the discriminant by D,
D = [η + β(1 − α) − (1 + β + η)(1 − α)φ]2 − 4βα(1 − α)(1 + β + η)φwhich is positive for a small φ.
11

Then, condition (A14) is satisfied for any θ ∈ (θ, θ), where√η + β(1 − α) − (1 + β + η)(1 − α)φ − Dθ =2β(1 − α)φ√η + β(1 − α) − (1 + β + η)(1 − α)φ + Dθ =2β(1 − α)φ[Proof of Proposition 4]
From equations (15) and (21), we havegC − gN =β {(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ]}(1 − α)(β + η)(1 + β + η + βθ)

which is positive if equation (A13) is satisfied. Therefore, g C > g N for anyθ ∈ (θ, θ).
From equations (16) and (22), we havesC − sN = −α {(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ]}(1 − α)θ(1 + β + η)[α + (1 − α)φθ]which is negative if equation (A13) is satisfied. Therefore, sN > sC for anyθ ∈ (θ, θ).
Comparing the coeﬃcients of f (kt ) in equations (17) and (23), we have=αφα(1 − α)(1 + β + η + βθ)−θ(β + η)θ − α(1 + β + η + βθ)
α {(β + η)θ − (1 + β + η + βθ)[α + (1 − α)φθ]}θ[(β + η)θ − α(1 + β + η + βθ)]

which is positive if equation (A13) is satisfied. Therefore, kN > kC for anyθ ∈ (θ, θ).

12

Figure 1. Fertility dynamics and capital accumulation in the cooperate siblings model

n

nC

O

kC

k

Figure 2. Capital accumulation in the non-cooperative siblings model

k

k

k

O

k

kN

k

k

f k

f k

k

Figure 3. Fertility dynamics and capital accumulation in the non-cooperate siblings model

n

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