GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.68

Technological Progress and Population Growth:
Do we have too few children?

Koichi Futagami and Takeo Hori

July 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Technological Progress and Population
Growth: Do we have too few children?∗
Koichi Futagami† and Takeo Hori‡

Abstract
Do we have too few children? We intend to address this question. In developed countries, the fertility rate has declined since WWII. This may causea slowdown in the growth of GDP in developed countries. However, important factors for the well-being of individuals are per capita variables, like percapita growth and per capita consumption. In turn, the rate of technologicalprogress determines the growth rates of per capita variables. If the population size is increasing, the labour inputs for R&D activity increase, andthus speed up technological progress. As individuals do not take accountof this positive effect when deciding the number of their own children, thenumber of children may become smaller than the socially optimal number ofchildren. However, an increase in the number of children reduces the assets
any one child owns: that is, there is a capital dilution effect. This works inthe opposite direction. We examine this issue using an endogenous growthmodel where the head of a dynastic family decides the number of children.
Keywords: Technological Progress, Fertility, R&D
JEL Classification Numbers: J1, O30∗

This paper is from an invited lecture at the Spring Meeting of the Japanese Economic Association held at Kyoto University on 7 June 2009. The authors are grateful to Reiko Aoki, Makoto
Saito, Etsuro Shioji and seminar participants at Hitotsubashi University for helpful comments onan earlier version.
†
Graduate School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 5600034, Japan. E-mail: futagami@econ.osaka-u.ac.jp‡
Department of Economics, Hitotsubashi University, 2-1, Naka, Kunitachi, Tokyo, 186-8601,
Japan. E-mail: t-hori@econ.hit-u.ac.jp

1

1 Introduction
Do we have too few children? We intend to address this question in this paper.
Developed countries experienced a decline in the fertility rate after WWII. Consequently, the population size of developed countries will decrease dramatically. As
is well known from standard growth theory, the natural rate of growth determines
the growth rate of GDP: that is, the sum of the rate of technological progress andthe rate of population growth. Therefore, it is often argued that the decline in population size due to the decrease in the fertility rate necessarily reduces the growthrate of GDP and potentially the level of welfare.1 Some economists insist that anincrease in the rate of technological progress—another factor that determines thegrowth rate of GDP—will counter the decline in the fertility rate, especially as
policies are required to promote technological growth.
This dual argument at first appears convincing. However, it does not consider
economic reasoning. First, the important variables for welfare are not GDP itself,
but rather per capita variables like GDP per capita or consumption per capita. In
fact, standard growth theory predicts that a decline in the population growth rate
leads to an increase in per capita variables like these, and therefore the growth rateof per capita variables is determined by the rate of technological progress.2
Second, we need to consider the endogenous mechanisms affecting the determination of economic growth. The dualism maintains that if a government could
increase the rate of technological progress, this would overcome the decline in thefertility rate, that is, the decrease in population growth. However, how should thegovernment promote technological progress? Standard growth theory (that is, theSolow model), cannot answer this question because the technological progress isbrought into the economy as manna from heaven. Therefore, we must resort toendogenous growth models to respond. In these models, the rate of technology
progress is endogenously determined. The essential problem then becomes, when
the population size is decreasing, can the government raise the rate of technological progress? Importantly, to boost technological progress there is a need for
1

The Japanese government recently appointed a minister of state for special missions to address
the declining fertility rate and undertake countermeasures.
2
See any standard textbook; for example, Barro and Sala-i-Martin (2004).

2

many researchers. A decrease in the population size may then reduce the numberof researchers working in research laboratories, and thus may lower the rate oftechnological progress.
To investigate this issue, we employ the model in Jones (1995). In this model,
population growth invokes technological progress. Therefore, we can show thatthe decline in population growth decreases the rate of technological progress andthus decreases the growth rates of GDP and consumption per capita. Consequently, it is questionable whether the government can increase the rate of technological progress when the rate of population growth is in decline. If this is thecase, should the government increase the rate of population growth to increase therate of technological progress?
However, the Jones (1995) model does not directly indicate that the government should raise the rate of population growth. First, the purpose of economic
policy is not to promote economic growth, but rather to increase the welfare levelof individuals. Therefore, we first consider how individuals derive their utility.
Individuals derive utility not only from the consumption of final goods, but also
from having children. An increase in the number of children naturally raises theirlevel of welfare. However, while children are a source of enjoyment for theirparents, raising children invokes pecuniary costs along with opportunity costs because their parents usually stop working or reduce their working time to rear them.
By taking account of these costs and the utility from having children, individuals
make a decision on how many children they have.
Although this is a completely private decision, should the government intervene in this largely private decision? The answer is that if market failure exists,
it is rational for a government to intervene in the decision-making process using
taxes or subsidies. The question is whether there is any market failure in the decisions made by individuals on the number of children they will have. Consider now
that researchers and engineers in private firms conduct research and development(R&D) activities. When parents decide upon the number of children, they do nottake into account the positive effects of population size on these R&D activities
in private firms. Consequently, because of this positive externality, the resource
allocation of the market equilibrium may differ from the socially optimal alloca3

tion. That is, the number of children in the market equilibrium is smaller than in
the socially optimal allocation. Thus, some scope may exist for government intervention by granting households a subsidy for having children. Nevertheless, it
is not obvious whether the government should provide the subsidy to households
that want to have more children, for there exists yet another cost for households tohave a child: namely, if a household decides to have one more child, the amountof household capital that child owns becomes smaller. This reduces the capitalincome for each child. This is the capital dilution effect. Therefore, there is aneed for further analysis based on the Jones (1995) model.
Jones (2003) has already examined some of these issues by modeling thehousehold’s decision on the number of children in Jones (1995). In fact, Jones(2003) also argues that the number of children in the market equilibrium is smaller
than in the socially optimal allocation. However, Jones (2003) does not take into
account those R&D activities that target profit. We incorporate profit-maximizing
firms conducting R&D activities in our model. In particular, this modification caneasily overturn Jones’s (2003) result. Moreover, Jones’s (2003) analysis is limited
to steady state analysis. In contrast, the present analysis extends Jones (1995) byexamining the transition paths to the steady state.3 Because it takes a fairly longtime until the economy approaches the steady state, it is also important to examine the character of the transition paths. 4 We construct a dynamic system for themodel and conduct numerical simulations based on some plausible parameters.
We show that the number of children in the market equilibrium can become larger
than in the socially optimal allocation not only at the steady state, but also on thetransition paths.5 This implies that the government should not intervene in thedecisions of families by giving them subsidies to increase the number of children.
The rest of the paper is structured as follows: Section 2 sets up the model.
Section 3 constructs the dynamic system of the model. Section 4 derives thesocially optimal allocation. By conducting a numerical simulation, we compare
3

Jones (1995) also only examines the character of the steady state. Arnold (2006) examines
the dynamics of Jones’s (1995) growth model.
4See, for example, Steger (2003).
5
Jones (2003) also suggests this result for the steady state. However, he does not conduct aformal analysis.

4the number of children in the market equilibrium with that in the socially optimal
allocation. Section 5 provides some concluding remarks.

2 The ModelIn this section, we set up a model based on Jones (2003). A representative dynastic family populates the economy. Jones assumes that the government collects
lump sum taxes from households and uses them to pay wages for researchers. In
contrast to Jones (2003), we incorporate profit-maximizing private firms undertaking R&D activity. Consequently, there are three sectors: a final goods sector,
an intermediate goods sector, and an R&D sector. First, we consider the final
goods sector.

2.1 Final Goods Sector
The final good, Yt , is produced by the following production function:
Z At
1−αYt = LY,t
xαj,t d j, 0 < α < 1,(1)0

where LY,t and x j,t respectively, represent labour input and the input of the jthintermediate good at time t. At stands for the variety of intermediate goods attime t. If the R&D firms succeed in inventing a new variety, At increases. Perfectcompetition is supposed to prevail in the final goods market. Therefore, we obtainthe following profit-maximization conditions:Yt= wt ,
LY,t
1−α α
αLY,t
x j,t = p j,t ,(1 − α)(2)(3)

where wt and p j,t are the wage rate and the price of intermediate good j at time t,
respectively. We normalize the price of final goods to one. From (3), we obtainthe following demand function for intermediate good j:
1

x j,t = α 1−α

5

LY,t
.
1/(1−α)p j,t(4)

2.2 Intermediate Goods Sector
A single firm produces each intermediate good. This firm is a monopoly and can
set the price of the intermediate good that it supplies. The monopoly is protectedby perfect patent protection. One unit of capital supplied by the family produces
one unit of the intermediate good. Therefore, the producer of the jth intermediate
good maximizes profit according to the following:
π j = p j x j − rx j ,subject to the demand function of the final good sector, (4), where r is the rental
rate of capital. This results in the following pricing rule:p ≡ pj =1r.
αHence, the price is the same for all intermediate goods j. Thus, the output levelsof all intermediate goods j are the same and given by:! 1
α2 1−αLY .
x ≡ xj =r
The profit of each intermediate good firm is given by:! 1
α1+α 1−απ x ≡ π j = (1 − α)
LY .
rα

(5)

(6)

2.3 R&D Sector
R&D activities are carried out using labour inputs according to the following technology:Ȧt = eδLA,t .

(7)

where LA,t is the labour input for R&D activities at time t. eδ represents the productivity level of R&D activities.6 Ȧt measures new intermediate goods. We assume
that the accumulated knowledge positively affects productivity in the following
manner:eδ = δAφt ,δ > 0,

60 < φ < 1,

(8)

If we incorporate the duplication effect into the innovation technology, the production function
becomes Ȧt = e
δ(LA,t )λ , 0 < λ ≤ 1. We neglect this effect for analytical simplicity.

6where φ represents a parameter that expresses the extent of knowledge spillover.
φ < 1 means there are decreasing returns in the production of new intermediate
goods. Perfect competition prevails in R&D races. Each R&D firm maximizes
its profit without considering this spillover effect. Therefore, the objective of thefirm becomes:
πA,t = PA,t Ȧt − wt LA,t ,where PA,t is the price of a blueprint of a newly invented intermediate good. Free
entry into the R&D race leads to the following zero-profit condition:PA,teδ = wt .
By using (8), we obtain:
PA,t δAφt = wt .

(9)

The discounted sum of profit of the intermediate good firm buying the blueprint
determines the price of the blueprint. That is, the following holds:
Z ∞Rτ
PA,t =
π x,τ e− u=t ru du dτ,

(10)

τ=t

where ru represents the return on assets at time u. By differentiating (10) withrespect to time t, we obtain the following no-arbitrage condition:
rt PA,t = π x,t + ṖA,t .

2.4 Dynastic Family
Individuals derive their utility not only from their own consumption but also fromthe utility of their children.7 Thus, parents care about the number of their children,
not just their own utility. We assume that the head of a representative dynastic
family maximizes the following:Z ∞Ut =u(cτ , Nτ )e−ρ(τ−t) dτ,

(11)

τ=t

where cτ is the consumption of a member of the dynasty at time τ, ρ(> 0) is therate of time preference, and Nτ is the number of members of the dynasty at time7

See Barro and Becker (1989) for this approach.

7

τ. We further assume that the instantaneous utility function of the head takes thefollowing form for analytical simplicity:8u(cτ , Nτ ) = log cτ + θ log Nτ ,(12)where θ(> 0) is the weight placed on the utility of the offspring.
We next formulate the budget constraint of the dynasty. Let nt denote the number of children at time t for a member of the family. We assume that each member
has one unit of time endowment and that rearing children requires time: that is,when rearing nt children, an individual member of the family must devote β(nt )units of time and he/she must give up the corresponding wage income. The rearing
cost function, β(nt ) satisfies the following conditions; β0 (nt ) > 0 and β00 (nt ) > 0.
Thus, the per capita stock of assets evolves according to the following equation:ȧt = (rt − nt )at + wt [1 − β(nt )] − ct .

(13)

Because there are Nt identical individuals and each individual has nt children,
the size of the family evolves according to the following equation:Ṅt = nt Nt .

(14)

The head of the family maximizes (11) subject to (13) and (14). The first-orderconditions are given by:1= λt ,ctµt Nt = λt [at + wt β0 (nt )],λ̇t = (ρ + nt − rt )λt ,θµ̇t = (ρ − nt )µt − ,
Nt

(15)
(16)
(17)
(18)

where λt and µt are costate variables associated with asset holding and the family
size, respectively. Furthermore, the following transversality conditions must be
satisfied: lim λt at e−ρt = 0 and lim µt Nt e−ρt = 0. The left-hand side of (16) rept→∞t→∞resents the shadow value of children in the family. The right-hand side of (16)represents the cost to have children. The first and second terms of (16) represent
the capital dilution effect and the opportunity cost, respectively.
8

See the appendix for the rationale underlying this functional form.

8

3 Market Equilibrium and Dynamics
Based on the model in the preceding section, we derive the market equilibrium
and construct a dynamic system. The goods market equilibrium is given by:
Yt = Ct + K̇t ,(19)

where Ct and Kt are aggregate consumption and capital stock, respectively. Denoting the per capita variables in lowercase, we can transform (19) into the following
per capita terms:k̇t = yt − ct − nt kt .

(20)

The equilibrium condition for capital is given by:
K t = A t xt .

(21)

We note here that there exist At varieties of intermediate goods at time t and thequantity x j,t is the same for all intermediate goods. The market equilibrium condition of the labour market is given by:
LY,t + LA,t = 1 − β(nt ) Nt .

(22)

To derive the dynamic system of the economy, let us define the followingevariables: χt ≡ ct /kt , zt ≡ At /kt , ζt ≡ µt Nt , νt ≡ A1−φt /Nt , PA,t ≡ PA,t /Nt , andgA,t ≡ Ȧt /At . In the appendix, we show that the following five equations constitutethe dynamic system of the economy:(!
)
1χ̇t = χt + 1 − 2 rt − ρ χt ,αrt żt = χt + nt + gA,t − 2 zt ,αζ̇t = ρζt − θ,ν̇t = (1 − φ)gA,t − nt νt ,




1
−
α
r
t
ėA,t = rt −eA,t .
P− nt  PeA,tα zt P

(23)
(24)
(25)
(26)
(27)

The appendix also shows that rt , nt , and gA,t are given by the solutions of thefollowing three equations:
!α
eA,t
P
α2 1−α,δ= (1 − α)
νt
rt
9

(28)

! 1
1 α2 1−α 1+ νt gA,t = 1 − β(nt ),
zt r
δ
1
1
ζt χt − 1 1−αβ0 (nt ) =
rt .
2α
zt
(1 − α)α 1−α

(29)
(30)

We now derive the balanced growth path (hereafter BGP). Section 5 examinese∗ ,the full dynamics. In this section, we characterize the BGP. The BGP, {χ∗ , z∗ , P
A∗∗r , ν , ζ , g∗A , n∗ } is determined by the following equations::!
1 ∗χ =ρ− 1− 2 r ,α∗rχ∗ = 2 − n∗ − g∗A ,αθζ∗ = ,
ρ
n∗
g∗A =
,
1−φ




1
1
−
α
∗
∗
 ,
n = r 1 −e∗ 
α z∗ P
∗

(31)
(32)
(33)
(34)
(35)

A

and the three equations, (28), (29), and (30). From (31), (32), and (34), the following holds:

2−φ ∗
n.
1−φ
Then, substituting (36) into (31), we obtain:
!
ρ
1
2−φ ∗
∗
χ = 2 + 2 −1
n.
α
α
1−φ
r∗ = ρ +

(36)

(37)

When θ = 1, we obtain a clear result for the decision of the dynastic family. By
using (29) and (30), we obtain the following equation:
β0 (n∗ )α2 ζ ∗ χ∗ − 1.
=
1 − β(n∗ ) − 1δ ν∗ g∗A 1 − α r∗
Due to (37) and (A8), we can rearrange this into the following:
1+αβ0 (n∗ ).
∗ =∗1 − β(n ) − lAρ(38)

If the allocation of labour input to the R&D sector is determined, (38) gives thesteady state number of children in the market equilibrium.
10

4 Socially Optimal Allocation
By solving the social planner’s problem, we can derive the socially optimal allocation, especially the optimal number of children. The social planner maximizes
the following welfare level of the dynastic family:Z ∞log cτ + θ log Nτ e−ρ(τ−t) dτ,
Ut =
τ=t

subject to the resource constraint and the production function of new varieties: k̇t = ktα At 1 − β(nt ) − lA,t 1−α − ct − nt kt ,(39)

Ȧt = δAφt LA,t = δAφt lA,t Nt ,(40)

and (14). The head of the dynastic family does not take account of (40) when
he/she optimizes. This is the source of the externality.
The first-order conditions of this maximization problem are given by:1= λt ,
ct"µ1,t Nt = λt (1 − α)(41)#yt β0 (nt )+ kt ,
1 − β(nt ) − lA,t
yt
µ2,t δAφt Nt = λt (1 − α),
1 − β(nt ) − lA,t!
yt
λ̇t = ρ + nt − α λt ,ktθµ̇1,t = (ρ − nt )µ1,t −− µ2,t δAφt lA,t ,Nt(1 − α)ytµ̇2,t = ρµ2,t − λt
− µ2,t φδAφ−1
t lA,t Nt ,
At

(42)
(43)
(44)
(45)
(46)

where λt , µ1,t , and µ2,t are costate variables associated with capital, family size,
and the varieties of intermediate goods, respectively. Furthermore, the following
transversality conditions must be satisfied: limt→∞ λt at e−ρt = 0, limt→∞ µ1,t Nt e−ρt =0, and limt→∞ µ2,t At e−ρt = 0.
From (41) and (44), we obtain:hiċt = α(zt lY,t )1−α − ρ − nt ct .
11

(47)

From (39) and (47), we obtain:noχ̇t = χt − (1 − α) (zt lY,t )1−α − ρ χt .

(48)

Due to the definition of zt and (39), we obtain the following dynamics for zt :hiżt = χt + nt + gA,t − (zt lY,t )1−α zt .

(49)

We here define ζt ≡ µ1,t Nt and ψt ≡ µ2,t At . Then, from (45), we obtain:ζ̇t = ρζt − θ − ψt gA,t .

(50)(zt lY,t )1−αψ̇t = (1 − φ)gA,t + ρ ψt − (1 − α)
.
χt

(51)

From (46), we obtain:

From (43), we obtain the following relationship:(1 − α)(zt lY,t )1−αψt
 =δ .
1 − β(nt ) − lA,t χt
νt

(52)

Note that we have used the following definition of νt : νt ≡ A1−φt /Nt . As for the
dynamics of νt , we can obtain the same equation as (26). From (41) and (42), we
obtain:"#1yt β0 (nt )ζt =(1 − α)+ kt .
ct
1 − β(nt ) − lA,t

By using the definition of χt and zt , we can rearrange this relationship as follows:ζt χt − 1 = (1 − α)zt 1−α β0 (nt ).
[1 − β(nt ) − lA,t ]α(53)Noting that (A8) and lY,t = 1 − β(nt ) − lA,t , (48), (49), (50), (51), and (26), together with (52) and (53), constitute the dynamic system of the socially optimal
allocation.
We next derive the BGP of the socially optimal allocation. From (48), (49),(50), (51), and (26), the BGP of the social optimal, {χOP , zOP , ψOP , νOP , ζ OP , gOPA ,nOP }, is determined by the following: equations:χOP = (1 − α) (zOP lYOP )1−α + ρ,
12

(54)

χOP = (zOP lYOP )1−α − nOP − gOP
A ,

(55)

ρζ OP = θ + ψOP gOP
A ,
n
o
OP OP
(1 − φ)gOP= (1 − α)(zOP lYOP )1−α ,A +ρ χ ψ

(56)

gOP
A =

nOP
,
1−φ

(57)
(58)

and the relationship, lYOP = 1 − β(nOP ) − 1δ νOP gOP
A and the two equations; (52) and(53).
When θ = 1, we can obtain a clear result similar to (38) for the market equilibrium:

β0 (nOP )1=.
1 − β(nOP ) ρ(59)

In order to compare the socially optimal number of children at the steady statewith the number of children in the market equilibrium at the steady state, it isuseful to rewrite (38) using the share of researchers allocated to R&D activities.
This share is defined by lA ≡ sA [1 − β(n)]. Then, (38) is transformed into thefollowing form:(1 − s∗A )(1 + α)
β0 (n∗ )=.
1 − β(n∗ )ρBecause β0 (n)/[1 − β(n)] is an increasing function of n, if the following inequality
holds, (1− s∗A )(1+α) > 1, then the number of children in the market equilibrium atthe steady state is larger than the socially optimal number of children at the steadystate. That is, too many children exist in the market equilibrium at the steady state.
However, because the share of researchers is an endogenous variable, we conduct
a simulation approach to obtain a much clearer result. Moreover, we calibrate thetransition paths based on some plausible parameters in the following section.

5 Simulation of Transition Paths
To analyse the model numerically, we specify the function β(·) as:β(n) = βn2 ,where β is a positive constant.
13

(60)

As a benchmark, we choose the parameter values as follows. The time preference rate, ρ, is set to 0.05, which is a conventional value in the growth literature.
The weight on the utility of offspring, θ, is equal to 1, because we analyse thecase where θ = 1 holds in Section 4. We assume that α is equal to 1/1.25, which
implies that the mark-up of the intermediate goods sector is 1.25. The values of βand δ are chosen so that the population growth rate of the market equilibrium at
BGP becomes sufficiently close to 0.01. Then, we obtain β = 1300 and δ = 0.3.
Finally, φ is set to 0.5 so that the annual growth rate of the economy, g∗A , is around
0.02.
Under this parameter set, the population growth rate of the market equilibrium
is equal to 0.0099, which is larger than that of the socially optimal allocation,
nop = 0.0077. In this benchmark case, the number of children in the market equilibrium at the steady state is larger than the socially optimal number of children
at the BGP. That is, there are too many children in the market equilibrium at thesteady state. Because the growth rates are given by (34) and (58), it is expected∗that in our numerical example, g∗A is larger than gop
A . As expected, gA is 0.197

and larger than gop
A = 0.0147. The market equilibrium attains a higher growthrate than the socially optimal growth rate. The share of labour allocated to R&D
activities in market equilibrium s∗A = 0.185 is smaller than the socially optimal∗
share sop
A = 0.201 and satisfies the condition (1 − sA )(1 + α) > 1.

The next question is whether the number of children in the market equilibriumis larger than the socially optimal number of children along all the transition paths
to the BGP. For this purpose, we analyse the transition paths by using the relaxation algorithm.9 In the following numerical examples, the initial values of thestate variables are chosen as z(≡ At /kt ) = 0.015 and ν(≡ At 1−φ /Nt ) = 2.9 so thatin the benchmark market equilibrium, the population growth rate decreases overtime and the share of labour allocated to R&D activities increases over time.
Figure 1 presents the results for the benchmark case. The upper panel shows
the transitional paths of the number of children (the population growth rate). The
9Trimborn et al.
(2008) detail the relaxation algorithm.
They also provide MatLab programs for the relaxation algorithm, freely downloadable at http://www.rrz.unihamburg.de/IWK/trimborn/relaxation.htm.

14

solid line stands for the number of children in the market equilibrium while thedotted line stands for the socially optimal number of children. The panel showsthat along all transition paths to the BGP, the number of children in the market
equilibrium is larger than the socially optimal number of children in this benchmark economy.
The middle and lower panels of Figure 1 depict the transitional paths of gA
and sA , respectively. From the lower panel, we know that the labour share allocated to R&D activities in the market equilibrium increases over time along thetransition path whereas it decreases over time in the socially optimal allocation.
The share of labour allocated to R$D activities is also smaller than the socially
optimal share along the transition path. The growth rate of the market equilibriumis smaller than the socially optimal growth rate during the early stage of transition
due to the smaller share of labour allocated to R&D activities in the market equilibrium. During the later stages of transition and at the BGP, however, the growthrate of the market equilibrium becomes higher than the socially optimal rate. Thisis because the differences between the shares of labour allocated to R&D activities of the market equilibrium and that of the social optimum decrease over timealong the transition path, and because the higher population growth of the market
equilibrium positively affects the growth rate.
In Figures 2–4, we present numerical examples other than the benchmark case.
Figure 2 shows where the value of ρ is increased and decreased from the benchmark level. Figures 3 and 4 show where the values of φ and θ are increased anddecreased from the benchmark levels. All figures show that not only at the BGP
but also along the transition path to the BGP, the number of children in the market
equilibrium is larger than the socially optimal number of children. In all of the
figures, the growth rate and the share of labour allocated to R&D activities exhibit
similar transitional paths. The only exception is when the value of ρ is increased
from the benchmark. When ρ is 0.07, the share of labour allocated to R&D activities in the market equilibrium becomes higher than that of the social optimum inthe later stage of transition (the lower left panel of Figure 2). However, even inthis exceptional case, the number of children in the market equilibrium is larger
than the socially optimal number of children along the transition path.
15

6 Concluding Remarks
In this paper, we incorporated profit-maximizing innovating firms and thus extended the model in Jones (2003). Importantly, the present model is much more realistic than Jones (2003), where the labour allocation is fixed. Moreover, we haveexamined transition paths by conducting simulation analysis. We have shown thatthe number of children in the market equilibrium is larger than in the socially optimal allocation. Based on plausible parameters, this is not only at the steady state
but also on the transition paths.
The usual argument is that because the number of children that parents actually have is smaller than the number of children they want to have, the government
must subsidize parents so they can have more children. However, our result implies that the government should be cautious with policy when intervening in theprivate decisions of parents with respect to the number of children.
Finally, we provide some directions for future research. First, in the present
model, we assume that the number of children take real numbers. However, in
reality the number of children must be nonnegative integers. Therefore, there may
be a substantial difference between having two children and having three children.
If so, we must investigate this issue by taking into account the integer problem.
Second, a continuous decrease in the population leads to zero population size.
Given at least some population size must exist for economic activities and human
life, we must consider the possibility of an optimal population size. These areimportant issues to be considered in future research.

Appendix
A1
In this appendix, we show how the objective function (11) and the instantaneous
utility function (12) are derived. Consumption at time t by a member of the dynastic family is given by ct . Because a member has nt children, the utility of thismember, Ut is defined as:
Ut = u(ct ) · dt + (1 − ρdt)Υ(nt dt) · (nt dt)Ut+dt ,
16

(A1)

where Υ(nt dt) is the degree of altruism of parents toward their children. For sim1−σtplicity, we assume the following functional forms: u(ct ) = c1−σand Υ(nt dt) =(nt dt)−ε .
We assume that the size of the dynastic family at time t is one, that is, Nt = 1.
Then, we have Nt+dt = nt dt, Nt+2dt = nt dt × nt+dt dt, and so on. Therefore, by using(A1), we can obtain the following discounted sum of Ut :Ut = u(ct )·dt +(1−ρdt)u(ct+dt )(Nt+dt )1−ε ·dt +(1−ρdt)2 u(ct+2dt )(Nt+2dt )1−ε ·dt +····.
(A2)
Because of the assumption of the functional form, we have:u(c)N 1−ε =1−εcN 1−σ

1−σ

1−σ

−1

.

As in Barro and Sala-i-Martin (2004, p410), we add the term -1 in the numerator
so that we obatin the log-utility form as σ approaches 1. When σ approaches 1
by keeping the ratio (1 − ε)/(1 − σ) constant, then its limit becomes:1−ε

cN 1−σ
lim

1−σ

−1

1−σ

σ→1

= ln c + θ ln N,(A3)

where θ ≡ (1 − ε)/(1 − σ). Consequently, (A2) can be rewritten as follows:
Ut = [ln ct +θ ln Nt ]dt+(1−ρdt){[ln ct+dt +θ ln Nt+dt ]dt+(1−ρdt)[ln ct+2dt +θ ln Nt+2dt ]dt+···}.
Neglecting the higher-order terms of (dt)n (n > 1), (A1) finally becomes:
Ut = [ln c + θ ln N] · dt + (1 − ρdt)Ut+dt .
Dividing both sides of this equation and taking the limit of dt → 0, we obtain thefollowing differential equation:·

Ut = ρUt − (ln ct + θ ln Nt ).
By integrating this differential equation, we can obtain the desired result.

17

A2
In this appendix, we derive the dynamic system of the economy. Noting that thesame amount of the intermediate goods is produced, we can transform (1) into thefollowing production function in intensive form:
yt = ktα (At lY )1−α ,(A4)where lY ≡ LY /N. Thus, (20) becomes:k̇t = ktα (At lY )1−α − ct − nt kt .

(A5)

From (15) and (17), we obtain:ċt = (rt − ρ − nt )ct .

(A6)

By using (5) and (21), and defining zt ≡ At /kt , we obtain: 1
zt lY,t = α2 /r 1−α .

(A7)

By defining χt ≡ ct /kt and using (A7), we obtain (23). We next derive thedynamics of zt , (24). By defining ζt ≡ µt Nt , we obtain (25). By similarly definingeνt ≡ A1−φt /Nt , we obtain (26). Again, by defining PA,t ≡ PA,t /Nt and using (6) and(A7), we obtain (27).
By using (2), (9), and (A7), we obtain (28).
From the definition of νt , (7), and (8), we obtain:
1
LA,t /Nt ≡ lA,t = νt gA,t .
δ(A8)

Substituting (A7) and (A8) into (22), we obtain (29).
From (15) and (16), we obtain:
kt + wt β0 (nt )
.
(A9)
ct

α
By using (2), and (A4), we obtain (1 − α) AtkltY,t = wAtt . Substituting this into (A9),ζt =using (A7), and rearranging, we obtain (30).

18

A3
In this appendix, we derive the equation that determines the socially optimal number of children at the steady state, (59).
From (54), (55), we obtain:(1 − α) (zOP lYOP )1−α + ρ = (zOP lYOP )1−α − nOP − gOP
A .
By using (58), this relationship can be rearranged as follows:α(zOP lYOP )1−α = ρ +2 − φ OPn .
1−φNext, substituting (A10) into (55) and using (58), we obtain:"#
1
2 − φ OPOPχ =
ρ + (1 − α)
n .
α
1−φ

(A10)

(A11)

From (57), (A10), and (A11), we obtain:OP(1 − α) ρ + 2−φ
n
1−φ
h
i.
ψOP =
OP
(ρ + nOP ) ρ + (1 − α) 2−φ
n
1−φ

(A12)

Substituting (A12) into (52) and noting that lY = 1 − β(n) − lA and (A11), we
obtain:2−φ OP(zOP )1−α δ ρ + 1−φ n=.
α (ρ + nOP )νOP
(lYOP )α(A13)

By multiplying both sides of (A13) by lYOP and substituting(A10) into them, we
obtain:νOP =δ[1 − β(nOP )]nOPρ + 2−φ
1−φ

.

(A14)

From (53) and (56), we obtain:
gOP(zOP )1−α 0 OP
θ OP
OP OP
(1 − α) OP α β (n ) = ζ χ − 1 = χ − 1 + A ψOP χOP .
ρ
ρ
(lY )
Substituting (58), (A11), and (A12) into this relationship, we obtain:"#(zOP )1−α 0 OPθ θ−α
2 − φ OP
(1 − α) OP α β (n ) =ρ + (1 − α)
n
ρα
θ
1−φ
(lY )
!
2 − φ OP
1−α
nOP
ρ+
n .
+
αρ(1 − φ) ρ + nOP
1−φ
19

(A15)

When θ = 1, we can rearrange (A15) as follows:!
!
(zOP )1−α 0 OP
2 − φ OP1nOP
1
(1 − α) OP α β (n ) =
ρ+
n
1+
.
ρ
1−φ
1 − φ ρ + nOP(lY )Substituting (A13) into this and using (A14), we obtain (59).

20References
Arnold, Lutz G. (2006) “The dynamics of the Jones R&D growth model,” Review
of Economic Dynamics, 9, 143–152.
Barro, Robert J. and Xavier Sala-i-Martin. (2004) “Economic Growth,” The MIT
Press, Cambridge.
Barro, Robert J. and Gary S. Becker (1989) “Fertility Choice in a Model of Economic Growth,” Econometrica, 57, 481–501.
Jones, Charles I. (1995) “R&D-based Models of Economic Growth,” Journal of
Political Economy, 103, 759–784.
Jones, Charles I. (2003) “Population and Ideas: A Theory of Endogenous
Growth,” in Aghion, Frydman, Stiglitz, and Woodford (eds.) Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S.
Phelps (Princeton University Press).
Steger, Thomas M. (2003) “The Segerstrom Model: Stability, Speed of Convergence and Policy Implications,” Economic Bulletin, 15, 1–8.
Trimborn, Timo, Karl-Josef Koch and Thomas M. Steger. (2008) “Multidimensional Transitional Dynamics: A Simple Numerical Procedure,” Macroeconomic Dynamics, 12, 301–319.

21

xb10

-3

11
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n 9
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0.04

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t

0.5
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sA 0.3
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t

Figure 1. Benchmark Case

22

xb10

-3

10

0.014

9

0.013

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n

n 0.012
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0.011

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g 0.03

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t
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sA
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0
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20

40

60

80

100

0.1
0

20

t

40
t

(a) ρ = 0.07(b) ρ = 0.03
Figure 2. Effects of Changes in ρ

23

-3

-3

xb10

xb10
11

11

10

10

9

n

n 9

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8

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g
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sA

0.2
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0.1
0
0

20

40

60

80

100

0.1
0

20

40

t

t

(a) φ = 0.7(b) φ = 0.3
Figure 3. Effects of Changes in φ

24

-3

xb10
10

0.015
0.014

8
0.013

n

n
0.012
6
0.011
0.01

4
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sA

sA 0.3

0.1

0.2

0
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20

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60

80

100

0.1
0

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40

t

t

(a) θ = 1.2(b) θ = 0.9
Figure 4. Effects of Changes in θ

25