Discussion Papers In Economics
And Business

Variety expansion and fertility rates
Akiko Maruyama Kazuhiro Yamamoto

Discussion Paper 07-29-Rev.3

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANVariety expansion and fertility rates
Akiko Maruyama Kazuhiro Yamamoto

Discussion Paper 07-29-Rev.3

February 2010
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANVariety expansion and fertility rates
Akiko Maruyama

Kazuhiro Yamamotoy

February 15, 2010

Abstract
To investigate how fertility rates interrelate with the modern economy,
we construct a simple model in which variety expansion of consumption
goods reduces fertility rates. In our model, variety expansion reduces therelative price of a composite of di¤erentiated goods compared to childrearing costs. Thus, parents raise the expenditure share for di¤erentiatedgoods and lower the number of children. We show that this model canbe applied to a growth model in which economic growth progresses withvariety expansion of consumption goods and fertility rates decrease witheconomic growth. Thus, we show a new mechanism for fertility decline,
and this mechanism can be applied to a growth model.

JEL classi…cation: J13; O10; F12
Kew words: Consumerism; Fertility rates; Variety expansion; Economic
growthPopulation Research Institute, Nihon University; Address: 1-3-2, Misaki-cho, Chiyoda-ku,
Tokyo 101-8360 Japan. E-mail address: maruyama.akiko@nihon-u.ac.jp.
y Corresponding author: Graduate School of Economics, Osaka University; Address: 1-7,
Machikaneyama, Toyonaka, Osaka, Japan; E-mail address: yamamoto@econ.osaka-u.ac.jp1

1

Introduction

In the modern world, fertility rates have decreased in developed countries. A
low fertility rate is an important feature that identi…es a modern developed
economy. For example, from 1960 to 2000, the total fertility rate respectively
decreased from 2.00 to 1.36 in Japan, from 2.72 to 1.64 in the U. K., from 2.37 to
1.36 in Germany, from 2.73 to 1.88 in France, from 2.20 to 1.54 in Sweden, andfrom 3.64 to 2.06 in the U. S. (Cabinet O¢ ce, Government of Japan (2004)).
Some researchers, e.g., Lutz (1996), have stated that, in modern industrialized countries, parents prefer to consume goods rather than to bear children.
He pointed out that "consumerism" is the basis of the decline in fertility rates
in modern developed countries. Lutz (1996) states:
Commentators often mention the increase in consumerism as a basic underlying cause for the recent fertility decline. The argument is that people would
rather invest in pleasures for themselves than in children; they would rather buy
a new car than have another child; they would rather spend their time watching
TV than changing diapers. (pp. 273)
Our purpose in this paper is to present a model in which "consumerism"
is a cause of fertility decline. 1 We present a model in which "consumerism"is induced by the "variety expansion" of consumption goods. In our model,
we assume that parents receive utility from the consumption of di¤erentiatedgoods and the number of their children.2 Parents allocate their …xed time toworking or rearing children. Thus, there is trade-o¤ between nominal income
and children. In our model, variety expansion lowers relative price of composite
of di¤erentiated goods. Then, variety expansion induces parents to extend theexpenditure share for di¤erentiated goods. With this mechanism, parents havefewer children when the variety of di¤erentiated goods that they can consume
increases. We show that this mechanism can be used to explain the decline infertility in the process of economic growth.
As we wrote above, "consumerism" is induced by the variety expansion of
consumption goods. In developed countries, there is a large variety of consumption goods, which enhances "consumerism". The variety of consumption goods
has increased with economic growth. In modern developed countries, the economy has grown continuously. The average GDP growth rate from 1981 to 2005
was 2.3% in Japan, 2.5% in the U. K., 1.9% in Germany, 2.0% in France, 2.2%
in Sweden, and 3.1% in the U. S. As Grossman and Helpman (1991) and Romer(1990) have shown, the variety of consumption goods increases with economic
growth in developed countries. Therefore, in those countries, the variety of
consumption goods increases with economic growth.
With economic growth, a large variety of consumption goods becomes an
1 In this paper, agents engaging in “consumerism” spend a larger share of their income(time) on consumption than on rearing children.
2 In much of the literature, e.g., Becker, Murphy, and Tamura (1991), Eckstein and Wolpin(1985), and Galor and Weil (1996) (2000), it is assumed that parents receive utility from thenumber of their children. We have followed their lead and assume that parents obtain utility
from the number of their o¤spring.

2

important feature of a developed economy. Given these important featuresof a developed economy, in this paper, we construct a model in which variety
expansion a¤ects the fertility rates and variety expansion is induced by economic
growth.
Theoretically, our purpose is to present a tractable model in which the varietyof consumption goods increases with economic growth, thereby reducing thefertility rates. We show that important examples of variety expansion-type
growth models, such as those produced by Grossman and Helpman (1991) canbe applied to the endogenous fertility model. With this model, we present a newmechanism through which fertility rates are related with economic activities.
Our model of economic growth integrates the endogenous fertility model and
the Grossman-Helpman-Romer (GHR) type of the variety expansion growthmodel. The GHR growth model has a large in‡uence on the study of economic
growth. However, there are not any studies on endogenous fertility in the context
of the GHR model. Our model shows that the GHR model can be applied tothe model of an endogenous fertility rate.
Many studies have explained the mechanism of fertility decline in the processof economic development. Becker, Murphy, and Tamura (1991) wrote a seminal study that presents a model in which fertility is closely related with humancapital accumulation. In their model, parents obtain utility not only from consumption but also from the quantity and quality of their children. Each parent allocates his/her …xed time to working, parenting, and educating children.
Hence, there is a quantity-quality trade-o¤ of children. They show that, in theprocess of economic development, the value of education increases. Then, parents lower the number of children and allocate much of their time to educating
their children. Galor and Weil (2000) and Tamura (2002) showed that, with thisquantity-quality trade-o¤, fertility decline is induced by technological progress.
Kalemli-Ozcan et al. (2000) and Kalemli-Ozcan (2002, 2003) pointed out that,with a quantity-quality trade-o¤, fertility decline progresses by a decline in theinfant mortality rates.
In the literature cited above, it is commonly assumed that human capital accumulation plays an important role in the decline of the fertility rate.
Other explanations have also been given. Galor and Weil (1996) reported thata decline in the fertility rate is induced by a rise in relative wages earned by
women. They showed that the relative wage earned by women increases withcapital accumulation. Increasing the relative wages of women reduces the fertility rate by causing the cost of child-rearing to exceed the household income.
Sato and Yamamoto (2005) constructed a model in which urbanization induces
agglomeration economy and congestion diseconomies and the fertility rate decreases with urbanization. In their model, the agglomeration economy raises
parents’income, which raises the fertility rate by the income e¤ect and reduces
the fertility rate by the substitution e¤ect, and congestion diseconomies lower
parents’ income, which reduces the fertility rate. They showed that the substitution e¤ect and congestion diseconomies overcome the income e¤ect of an
agglomeration economy and urbanization and economic growth reduce the fertility rates. These studies examine important causes of the fertility decline in the3modern developed economy. However, as Lutz (1996) stated, "consumerism" isan important aspect of the modern developed economy which reduces the fertility rates. Thus, our paper presents another important mechanism of fertility
decline.
The organization of this paper is as follows. In Section 2, we present a simple
benchmark partial equilibrium model in which variety expansion lowers fertility
rates. In Section 3, we present a growth model that uses the mechanism of thebenchmark model. Section 4 is the conclusion.

2

A simple basic model

In this section, we present a basic model in which an agent reduces the numberof children with the variety expansion of di¤erentiated goods. In the following
section, we apply this basic model to the growth model.
We assume that agents obtain utility from the consumption of di¤erentiatedgoods and number of children:
U = u(C; m);

(1)

where
Z n
1
C=(
x(i) di) ;0

where n is the measure of variety of di¤erentiated goods, C is the composite of
consumption goods, x(i) is the consumption of di¤erentiated goods indexed i,and m is the number of children. We assume that 0 < < 1.
= 11 > 1
represents the elasticity of substitution among di¤erentiated goods.
We assume that agents have one amount of time that can be used for working
or rearing children. Following Becker (1965) and others, we assume that, if they
have a child, they have to use time to rear a child. Their budget constraint
becomesRn
w(1 R m) = 0 p(i)x(i)di;(2)nw = 0 p(i)x(i)di + w m:
We follow Dixit and Stiglitz (1977) and solve the utility maximization problem in two steps. First, regardless of the value of the composite of consumption
goods, C, each x(i) needs to be chosen so as to minimize the cost of attaining
C. This means solving the following minimization problem:Rn
min 0 p(i)x(i)di;1
Rn(3)
s:t: 0 x(i) di = C:
The …rst-order condition to this expenditure minimization problem gives equality of marginal rates of substitution to price ratios,

4

1

x(i)
x(j)
We substitute (4) to

Rn
0

p(i)
:
p(j)

=

1

(4)

1

= C; and get

x(i) dix(j) = R np(j)1=(

p(i) =(
01)
1) di 1=C:

(5)This simply represents the compensated demand function for the j th variety
of a consumption product. We can also derive an expression for the minimum
cost of attaining C. From (5), we can derivep(j)x(j) = R np(j) =(

1)C:1) di 1=p(i) =(
0

We integrate the above equation over j and obtain
Z np(j)x(j)dj =

0

Z n

(

p(i) =(

1)

1)=

di

C:

0

We de…ne the term multiplying C on the right-hand side (RHS) of the above
equation as a price index:
P

Z n

(

p(i) =(

1)

Z n

1)=

di

=

0

1=(1

p(i)1

di

)

:

(6)

0

With (6), the price index times the quantity composite is equal to the expenditure:
Z np(i)x(i)di = P C:
(7)0

In our setting, the consumer has a preference for the variety of di¤erentiatedgoods. We can derive (use Leibniz rule)3
Z n
1
1
@P
=
(
p(i) 1 di) p(n) 1 < 0:
(8)
@n
0

Thus, if the variety of di¤erentiated goods increases, the expenditure whichenables a consumer to obtain a given amount of C decreases.
Demand for x(j) is derived as follows:
x(j) =

p(j)
P

1
1

C:

(9)

3 In this section, we ignore the production side of the economy and assume that each price ofdi¤erentiated goods is constant. In Section 3, we consider the production side of the economy.

5The next step of the consumer’s problem is to determine the number ofchildren and the expenditure for C. We substitute (7) into the budget constraint,
and the second step problem is de…ned as
max U = u(C; m);s:t:w = P C + w m:
We de…ne the solutions of the above problem as follows:
C = C(P; w ; w);(10)m = m(P; w ; w):

(11)

and

These two are demand functions for a composite of consumption goods and children. We substitute (10) and (11) into the budget constraint and di¤erentiate
it with P :@m(P; w ; w)
@C(P; w ; w)
+w
= 0:
@P
@P
Equation (12) can be rewritten as:
C +P

w @m(P; w ; w)
=
C
@P

1

@C(P; w ; w) P
=
@P
C

1+ :

(12)

(13)

;w)
@C(P;w ;w) P
Equation (13) shows that, if > 1, @m(P;w
> 0.
@P
@P
C is theprice elasticity of the demand for the composite of consumption goods.
From (8), P is a decreasing function of consumption variety n. Therefore,;w) @P;w)> 0, @m(P;w
if @m(P;w@P@P@n < 0: the number of children decreases withvariety expansion. The discussion above leads us to the next proposition:

Proposition 1 If
expansion.

> 1, the number of children decreases with the variety

We show that the variety expansion reduces the fertility rate, if > 1. In
our setting, the consumer has a preference for a variety of di¤erentiated goods.
When the variety of di¤erentiated goods increases, the price index of di¤erentiated goods decreases (see (8)). Thus, the variety expansion lowers the relative
price of the composite of di¤erentiated goods and raises the expenditure sharefor di¤erentiated goods. Therefore, the variety expansion raises the relative
price of children and reduces the expenditure share for children. With thismechanism, the variety expansion reduces the fertility rate.
Here, we specify the utility function as a CES function:
1

U = u(C; m) = C + m

:

(14)

We assume that 0 < < 1. = 1 1 > 1 represents the elasticity of substitution
between children and the composite of di¤erentiated goods.
6With this speci…cation, the demand function for the composite of di¤erentiated goods and children becomew

C=

P + (w ) 1
w

m=

P1

;

1

(15)

:
(16)
1
P 1 (w ) 1 + w
The price elasticity of the demand for a composite of di¤erentiated goods under
this demand function is derived as follows:
CES =P + 1 1 (w ) 1
P + (w ) 1

P1
P1

1

1

:

(17)

Equation (17) shows that CES > 1, if 0 < < 1. Thus, under the CES utility
function, variety expansion reduces the number of children.
The simple partial equilibrium model in this section shows that variety expansion reduces the number of children of a consumer. In Sections 3, we showthat this simple mechanism can be used in the growth model. For the analytical
simplicity, we assume that the utility function is a CES form in next section. In
that section, we show that the fertility rate decreases with economic growth.

3

A growth model

In this section, we apply a basic model in the above section to a growth model.
Our model in this section is an overlapping generation model in which agentslive two periods, childhood and adulthood. In childhood, agents do not perform
any economic activities; they are reared by their parents. In adulthood, theagents consume di¤erentiated goods and determine the number of children they
will have. Thus, the consumption behavior of agents is the same as that in themodel in Section 1. We assume that labor is the numeraire, w = 1. There is acontinuum of population, Lt , at period t. The utility function of adult agentsat period t ishi1
Ut = Ct + mt
;

where

Ct = (

Z nt

(18)

1

xt (i) di) ;0

4

and their budget constraints are
1

mt =

Z nt

pt (i)xt (i)di:0

4 In our model, we assume that there is no capital. As we show later, the …rms obtain zeropro…ts. Thus, the income of agents is composed only with their wages.

7

The same procedure that derives (15) and (16) leads us to the demand
functions for the composite of di¤erentiated goods and children:
1

Ct =

;

1

(Pt + Pt1
1

mt =

:

1

Pt1

(19)

)

1

(20)

+

1

Substituting (19) into (9), we can derive a consumer’s demand function for eachdi¤erentiated good:1

Pt1

xt (j) =pt (j)

1
1:

1(Pt + Pt1

(21)

)

1

In this growth model, we follow Grossman and Helpman (1991) and assume
that the variety of di¤erentiated goods increases with innovation activities. In
our model, we assume that the innovation activity produces a patent. With thispatent, a …rm produces a variety of di¤erentiated goods with a constant returnto scale technology: x(i) units of labor can produce x(i) units of di¤erentiatedgoods. To produce a patent in period t requires It units of labor. Here, we
assume that It = n 1 ,1. It = n 1 means that there is knowledge
t 1
t 1
accumulation in this economy. As in standard endogenous growth models, suchas those developed by Grossman and Helpman (1991) and Romer (1990), foreach innovation …rm, this knowledge accumulation is externality. The term of
the validity of a patent is one period. After the term of the validity of a patent,di¤erentiated goods are produced by perfectly competitive …rms. The price of
those goods is p = 1. At the validity term of the patent, …rms set the standard
1mark-up price. Each …rm sets the price pt (i) = pt (j) =1 = , which is nota¤ected by n.
Rn
11
Here, Pt = ( 0 t p(i) 1 di)= (nt 1 + kt p 1 ), where nt = nt 1 + kt
and kt represents the number of varieties that are created at period t. Using
this and (21), the demand qt faced by a …rm that is created at period t is
(nt 1 + kt p

qt =(nt 1 + kt p

1

)

1

1

)

1

p

1
1

+ (nt 1 + kt p

1

) (1

Lt :

1
)

(22)

1

Then, the instantaneous pro…ts of a …rm are
t =(

1

p

)

1 (n

t 1 + kt p

11
1

) + (nt 1 + kt p

Lt :
1

) (1

)

The free-entry condition in the innovation market means that It =(23), the free-entry condition is written as

8

(23)

1

t.

From

(

1
nt 1

1

) =(

1

p

)

1 (n

t 1 + kt p

1

1

Lt :

) + (nt 1 + kt p

1

) (1

)

(24)

1

The right-hand side (RHS) of (24) is the pro…t of a …rm, and the left-hand side(LHS) is the cost for an innovation to produce a patent. If, the RHS isa decreasing function of nt . This shows that intensive competition reduces theoperating pro…ts of a …rm. In addition, the RHS is an increasing function of
the population, Lt . This is because a large population results in a large market
that brings a …rm large pro…ts. If
(

1
nt 1

) <(

1

1

p

)

1 n

1

Lt ;

t 1 + nt 1

(1

)

(25)

1

innovation occurs at period t, since the pro…t of a …rm when kt = 0 is larger thanthe cost of an innovation. If, the right-hand side of (25) is a decreasing
function of nt 1 . From (20), if innovation occurs at period t, the fertility rate
in period t becomes
1

mt =

(

(nt 1 + kt p

1

1)
1)

) (

:

1

(26)

+

1

5
t
Since < 1 and < 1, @m
Therefore, (26) shows that the fertility rate@kt < 0.
decreases with innovation. Innovation activities raise the variety of di¤erentiatedgoods, which lowers (raises) the relative price of the composite of di¤erentiatedgoods (children). Therefore, when innovations progress, fertility rates decrease.

Proposition 2 Fertility rates decrease with innovation.
On the other hand, if
(

1
nt 1

)

(

1

1

p

)

1 n

1

Lt ;

t 1 + nt 1

(1

)

(27)

1

innovation does not occur in period t. If (27) is satis…ed and
mt =

1
(

nt 1 (

1)
1)

1;

1

(28)

+

1

innovation never occurs in period t or the following periods. In this case, the1
fertility rate is constant at mt =after period t.
(
1)
nt

1

(

1)

1

1

+

The above discussion demonstrates that there are two cases in the model:
innovation and decline in fertility rates and no innovation and constant fertility
5 < 1 means that the elasticity of substitution among di¤erentiated goods is larger than1.
< 1 represents that the elasticity of substitution between composite of di¤erentiatedgoods and children is larger than 1.

9

rates. To see them explicitly, we study the dynamic property of the economy.
There are two state variables, Lt and nt , which follow dynamic equations.
Population, Lt , follows Lt+1 = mt Lt . We substitute (26) and kt = nt nt 1
into Lt+1 = mt Lt and derive1

Lt+1 =((nt 1 + (ntnt 1 )p

1

) (

1)
1)

1
1

Lt :

(29)

+

Next, we derive the dynamic equation of variety nt . nt increases with innovation activities. Thus, the dynamic equation of nt is a free-entry condition of
the innovation market. We substitute kt = nt nt 1 into (24) and obtain

(

1
nt 1

) =(

1

p

)

1 (n

t 1 + (nt

nt 1 )p1

1
1) + (nt 1 + (ntLt :nt 1 )p

1

) (1

)

(30)
(29) and (30) express the dynamic property of the model. To analyze the dynamic property, we create a phase diagram.
The phase diagram is depicted in Figure 1. In Appendix 1, we show that the(1

locus of the constant population, LLt+1= 1, requires nt = ( 1 1 ) (1
t

)
)

when

1

innovation is not carried out and the locus of constant population becomes adownward-sloping curve when innovation activities are carried out. In Appendix
2, we show that, under the assumption of, we can depict a downwardsloping curve of nnt t 1 = 1 as in Figure 1.
In Figure 1, we can observe four regions. In Region 1, population increases,
and innovation does not occur. In this region, the population is small. Since thepopulation is small, the market for di¤erentiated consumption goods is small.
The small markets result in low pro…ts. Thus, innovation does not occur inthis region. In Region 1, the variety of di¤erentiated goods is also small. The
relative price of the composite of di¤erentiated goods is therefore high. Thus,
the fertility rate is high, mt > 1. Since the fertility rate is high, the population
increases with time.
In Region 2, the population increases, while innovation activities are carriedout. In Region 2, the population becomes large. Therefore, the markets fordi¤erentiated consumption goods are large enough to induce innovation activities. Thus, the variety of di¤erentiated goods increases with time. In addition,in this region, the variety of di¤erentiated goods is relatively small. Thus, fertility rates are high, mt > 1. Then, in Region 2, both the population andthe variety of di¤erentiated goods increase with time. However, fertility rates
decrease with innovation. Therefore, in Region 2, fertility rates decrease withtime. Galor (2005) reported that fertility rates start to decrease when economic
development takes o¤. Region 2 of our model is consistent with this phenomenon. Our model shows that a large population enables economic development,
and economic development increases as a result of variety expansion. Variety
expansion reduces fertility rates.

101

In Region 3, the population decreases, and innovation activities are carriedout. In Region 3, the population is large, and the market for di¤erentiatedgoods is large. Since the market for di¤erentiated goods is large, innovation
progresses. Thus, the variety of di¤erentiated goods increases with time in thisregion. However, since the variety of di¤erentiated goods is large, fertility rates
decrease, mt < 1. The population then decreases with time in this region.
In addition, the fertility rate decreases with innovation activities. Thus, thefertility rate decreases with time in Region 3. As noted in the Introduction, thetotal fertility rates in developed countries are low, and population decreases withtime. In addition, economic growth continues in developed countries. Region 3of our model is consistent with these phenomena.
In Region 4, the population decreases, and innovation stops. In Region 4,
the population decreases. Since the market for di¤erentiated goods decreases, it
is impossible to obtain pro…ts with innovation. Thus, innovation activities are
not conducted, and the variety of di¤erentiated goods is constant in this region.
In Region 4, there is a large variety of di¤erentiated goods, which lowers fertility
rates, mt < 1. Thus, the population decreases with time in this region. Since
innovation does not occur, the fertility rate is constant in Region 4. 6
In Figure 1, we depict an example case of a dynamic path that starts from
Region 1. In this path, in the early periods, the population increases as long as
there are not any innovations (Region 1). An increase in population enlarges themarket for di¤erentiated goods. When the economy switches from Region 1 to
2, innovation activities start. In this region, the population increases (mt > 0),
while the fertility rates decrease with innovation (see Proposition 1). When theeconomy reaches Region 3, the population starts to decrease, and fertility rates
continuously decrease with innovation (mt < 1). The important features ofdeveloped countries in the modern world are declines in fertility rates and continuous economic growth. Therefore, developed countries in the modern world
are in Region 3. If the economy reaches Region 4, innovation activities stop, andpopulation decreases at a constant rate. Region 4 may not be realized in themodern world. Our model shows the only one possibility for the future economy: continuous decline in fertility rates and a curtailment of economic growth.
In this dynamic path, fertility rates decrease with economic growth (Regions 2
and 3). This result is consistent with experiences of economic development (SeeGalor (2004)). 7
6 Region 4 of our model may not be realized in the real world. Our model shows thepossibility of decline in population and the curtailment of economic growth. In much of theliterature devoted to the analysis of endogenous fertility rates, such as that by Becker, Murphy,
and Tamura (1991) and Galor and Weil (2000), the fertility rates and population continued
to decrease. Lutz (1996) pointed out the possibility of a continuous decline in population in
the future.
7 In our model, there are many possibilities of dynamic paths. For example, if the economy
starts from Region 4, the variety is constant, and population decreases with time.

11

4Conclusion

We presented a simple model in which variety expansion reduces fertility rates.
We showed that this simple model can be applied to the model of economic
growth. In the model of economic growth, fertility rates decrease with innovation activities, which are the engine of economic growth. Thus, our model
presented a new mechanism of fertility decline.
In the modern world, fertility rates in developed countries are low and decrease with time. Our model clari…ed that low fertility rates in developed countries are to be attributed to a large variety of consumption goods. Developed
countries have experienced variety expansion by innovation activities, which
reduce fertility rates.
Our model is very simple. Thus, we can extend the model in some directions.
In our growth model, parents solve a static problem: when they decide number of children, they do not consider their future consumption. If we assume
that agents live three period, they must consider their old time consumption
when they decide number of children. Under this extended model, suppose that
individuals expect that price index falls in the near future. In this case, they
may decrease the number of children, since they optimally substitute current
consumption for future consumption. The timing matter. Further, the scenariothat the expectations for future innovation decreases current fertility is itself
interesting. This extension is one important problem. We can study the effects of public policy, such as subsidies for rearing children. We can extend thegrowth model and study human capital accumulation, as in Becker, Murphy,
and Tamura (1991). This extension analyzes the quantity-quality trade-o¤ thatis an important mechanism of the fertility decline in developed countries. Finally, we consider the relationship between the fertility rate and urbanization
with the mechanism used in this model. Urbanization is an important aspect of
economic development. Sato and Yamamoto (2005) showed that demographictransition and urbanization are closely related. These extensions are importantfor future studies.

Acknowledgements
We are grateful to Naohito Abe, Reiko Aoki, Takeo Hori, Ryo Horii, Junichiro
Ishida, Shingo Ishiguro, Shigeki Kano, Daiji Kawaguchi, Akiomi Kitagawa,
Hajime Kobayashi, Yoko Konishi, Yasutomo Murasawa, Masaki Nakabayashi,
Tomohiko Noda, Daisuke Oyama, Yoshinao Sahashi, Tatsuhiro Shichijo, Ken
Tabata, Minoru Tachibana, Kan Takeuchi, Katsuya Takii, Akira Yakita, Midori
Wakabayashi, and anonymous referees for their helpful comments and suggestions. The …rst author acknowledges the support by a grant obtained by the
Nihon University Population Research Institute from the "Academic Frontier"
Project for Private Universities: matching fund subsidy from MEXT (Ministryof Education, Culture, Sports, Science and Technology), 2006-2010. The second
author acknowledges the …nancial support from a JSPS Grant-in-Aid for Scien-

12ti…c Research (A) and a Grant-in-Aid for Young Scientists (B). Any remaining
errors are the responsibility of the authors.

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Appendix 1
Here, we de…ne zt = nt 1 + (nt(29) and (30) as follows:

nt 1 )p

Lt+1
=
Lt
(
Assuming that

1
nt 1

) =(

1

. With this de…nition, we can write

1
zt

(1
(1

1
1

)
)

;
1

p

)

(31)

+

1

1
1

Lt :

zt + zt (1

)

(32)

1

, Equation (32) has a unique solution,zt = z(nt 1 ; Lt );t 1 ;Lt )t 1 ;Lt )> 0 and @z(n@L> 0. From (31), the locus of constantwhere @z(n@nt 1tpopulation is given byz(nt 1 ; Lt ) = (

1

(1
1

) (1

)
)

:

(33)

1

Obviously, the locus of (33) is a downward sloped curve in the space of (nt 1 ; Lt ).
=1
On the other hand, when innovation activities are not carried out, LLt+1trequires
n =(

1

(1
1

) (1

)
)

:

(34)

1
(1

(33) and (34) are connected with each other when z(nt 1 ; Lt ) = n = ( 1 1 ) (1
1

14

)
)

.

Appendix 2
From (30), nnt t 1 = 1 requires that1( ) =(
n

1

1

)(n1

1

p

)

1

Lt :

1 n + n (1

)

(35)

1

(35) is written as
Lt = (
@(n1

(1
)

(1

+n

p

1

+n

1

)

)
1

):

(36)

(

1) n1

1

)

@n

(1
)

(1

=(

n
(

(1
(1

)
)

1

1)

) n1

1

(

1

p

n

)

(1
(1
)

)

(1

0.

). Since 1 >1)) n +1 <nt
nt 1 = 1 is a1
10, since we assume thatand
1. Thus, the line
downward-sloping curve as in Figure 1. We de…ne R(n) = (
(1
)

n (1

1

and

1,

(1

)

)(n

+

1

< 1

. Thus,< 0 is satis…ed. Therefore, limn!0 R(n) = 1 and limn!1 R(n) =We substitute (29) into (30) and get
Lt =( nt1 1 ) (
p

1)

(

1

1

(1
1

1

) (1

)
)

+(

1

1

1

) (1

)

:

1

(37)

1

From (36), we can derive that
(1
1)
(1) (
)(nt + nt (1Lt = ( nt
1
p 1

)
)

( 1 1 ) (1)

nt + nt

1

1

)
)

1

( 1 1 ) (1)), while equa-1(1tion (38) is the locus of nnt+1= 1. Since nt
t

(38)
(1

= 1 when nt
Equation (37) is the locus of LLt+1
t
(1
(1

):

1

( 1 1 ) (11

. Since nt 1

)
)

(1

, ( 1 1 ) (1

)
)

+

1

nt , ( nt1 1 )( n1t ) .

Therefore,( nt1 1 ) (
p

1)

1
1

(

1

(1
1
1

) (1

)
)

+(

1
1

) (1

1

)

1

(1
(1) (
1)
( nt
)(nt +nt (11p 1Thus, the RHS of (37) is larger than the RHS of (38), which means thatthe locus of LLt+1= 1 always locates above the locus of nnt+1= 1 when nt
tt(1( 1 1 ) (1)). Thus, the phase diagram can be depicted as in Figure 1.

1

15

)
)

1

):