Fertility, Regional Demographics, and Economic Integration
Hiroshi Goto1
Graduate School of Economics, Kobe University, 2-1, Rokkodai-cho, Nada-ku, Kobe, Hyogo, 657-8501,
Japan

Keiya Minamimura2
Graduate School of Economics, Kobe University, 2-1, Rokkodai-cho, Nada-ku, Kobe, Hyogo, 657-8501,
Japan

Abstract
To explain the links between population distribution and economic integration, we constructa spatial economics model with endogenous fertility. A higher population concentration
increases real wages and child-raising costs, thus lowering the fertility rate. However, people
migrate to more populated regions to obtain higher real wages. We show that mobility acrossregions results in more people ﬂowing into highly populated regions, but lowers fertility
rates there. The population growth path resembles a logistic curve in the early phase,but population decreases in the last phase. Additionally, economic integration leads topopulation concentration and decreases population size in the whole economy.
JEL classiﬁcation: F15, J13, R12, R23
Keywords: Population Change, Migration, Agglomeration, Trade freeness1
2

TEL: +81-78-803-7247, FAX: +81-78-803-6877, E-mail: hrs.9-12.gt.o.jp.ml@hotmail.co.jp.
TEL: +81-78-803-7247, FAX: +81-78-803-6877, E-mail: keiya.minami@gmail.com.

Preprint submitted to Kobe University Discussion Papers

February 14, 2014

1. Introduction
It is clear that regional population change is determined by the number of births, deaths,
and population migration. In fact, given an initial population size, we can describe completely its change over time as sequences of fertility, mortality, inﬂow, and outﬂow rates are
determined. Population geography as in Demoko et al. (1970), traditionally explains andpredicts regional population change simply by measuring these rates. Generally, such measurements can be credible, if fertility, mortality, inﬂow, and outﬂow rates are stable acrosstime and can be measured with high accuracy. However, economists might be dissatisﬁedwith simply measuring them as accurately as possible.
Economists usually suppose that fertility, mortality, and migration depend on economic
conditions (e.g., income, commodity prices, and levels of economic development,). For example, many theoretical macroeconomic studies focus on the relationship between economic
growth and fertility and most such studies ﬁnd that a negative relationship exists.3 New
Economic Geography (NEG) shows that real income tends to be high in regions with large
markets, and hence, the population tends to concentrate in these regions because of thispotential for higher real income.4 In short, economists may believe that the market has
the power to eﬀect population change and that it is, therefore, not suﬃcient to conduct ananalysis that ignores the market.
The purpose of this paper is to construct a simple benchmark model that can describe
regional population change in a market economy. Population change itself is a traditional
issue in economics. As early as 1798, Malthus already pointed out that population growthis curbed by the power of land to provide human subsistence.5 Since then, as mentioned
earlier, economists have conducted many studies on population change. However, in many

3

Galor and Weil (1996) argue that wages for women are increasing as economic growth progresses, whichraises the opportunity cost for child-rearing and decreases the fertility rate. Becker et al. (1990) proposethat people invest more in human capital and have fewer children with advancing economic growth.
4
For details on NEG, see Fujita et al. (1999).
5
Malthus was the ﬁrst to point out that the process of population growth takes the form of a logistic
curve. Even though our model does not consider land, it shows that the population growth path of the whole
economy takes the form of a logistic curve, which is induced by the market.

2

cases, they focused on only country-level population change (i.e., they ignored migration)or migration (i.e., they ignored fertility and mortality). In other words, thus far, natural
population change induced by fertility and mortality has been analyzed independently ofsocial population change induced by migration and, except for a few studies, these issues
have not been addressed together. In this paper, to ﬁll this gap, we propose a model thatconsiders not only natural population change but also social population change.6
We can easily image that in the era of Malthus, the movement of people and goods
between regions was diﬃcult. This would justify the exclusion of migration in favor of only
one closed region. However, this approach is clearly not appropriate in considering a modern
economy. Nowadays, people as well as goods can move among regions much more easily; thisis called economic integration. Yet, the majority of economic studies that address population
growth have neglected regional diﬀerences. Similarly, approaches that address economic
integration (e.g., NEG) have also largely ignored population growth, perhaps because of
the diﬃculty of addressing it. The absence of both migration and population growth fromthe literature has rendered research on the impact of economic integration on population
change. Therefore, we set out to explore the relationship between economic integration andpopulation change.
To construct a model to describe regional population change, we must address threeimportant facts. First, population change and economic conditions diﬀer radically acrossregions. In Figure 1(a), we plot total fertility rates of Japan’s 47 prefectures in 2010 according
to per-capita income. Regional diﬀerences are apparent in prefectures’ total fertility rates
and per-capita income per capita. Regions in which per-capita income is higher tend to havelower total fertility rates. On the other hand, net migration tends to be higher in regions thathave higher per-capita income (see Figure 1(b)). Thus, higher per-capita income may have
two opposite eﬀects on regional population changes: lower fertility and higher net migration.
The second important fact is that the existence of diﬀerences in regional population

6

Similar concerns are addressed by Sato and Yamamoto (2005) and Sato (2006). However, their model

assumes that fertility rate decreases by the externalities of urbanization. This is a shortcoming of theirmodel because we want to describe population change as adjusted by the market.

360000Net migration
20000
40000

1.8

0

Total Fertility Rate
1.4
1.6

2500

3000
3500
Income per capita−1000 yen

4000−20000

1.2
2000

4500

Fitted values

2000

2500

3000
3500
Income per capita−1000 yen

4000

4500

Fitted values(a) Total Fertility Rate(b) Net migration

Figure 1: Total Fertility Rate and Net Migration in Japan in 2010change may change the population distribution across regions over time. Figure 2 describes
the Gini Concentration Ratio for Japan for 1947-2010, which shows an upward trend in thisperiod (we can see the same trend in other countries).7 This trend means that unequal population distribution across regions becomes larger over time and that people have gatheredin particular regions (e.g., Tokyo).
Finally, even though the total fertility rate diﬀers among regions, it has been declining
over time at the country level. Figure 2 shows the total fertility rate of Japan in 1947-2010.
It clearly indicates a negative trend of the total fertility rate in this period.8 Especially, we
should note that this decline in the total fertility rate seems to be associated with population
concentration. Actually, Galor (2011) points out that economic development increases thelevel of urbanization and Schultz (1985) shows that a progress of urbanization reduces fertility

7

The Gini Concentration Ratio is derived using the Lorenz curve, which plots the proportion of the total

population (on the vertical axis) that is cumulatively held in the total inhabitable area of regions (horizontalaxis). Note that the area share is measured by ordering regions according to population density. Here, we
use Japan’s 47 prefectures as regions, but Okinawa is excluded before 1972. Before 1975, inhabitable area
data are not available, so we use 1975 data for inhabitable area before 1975.
8
The total fertility rate declined sharply in 1966. This is because 1966 was a bingwu year according
to the Chinese calender; many East Asian people believe that children born in such years will have a bad
personality.

4

.5
.45

5

.35

.4
Gini

Total Fertility Rate
3
4

.3

2
1
1940

1950

1960

1970

1980

1990

2000

2010

Year
Total Fertility Rate

Gini

Figure 2: Gini Concentration Ratio and Total Fertility Rate in Japan from 1947 to 2010

rate.
Thus, the model that we construct in this paper should be able to explain these facts. To
this end, it must consists of multiple regions (at least two) and take into account migration,
fertility and trade. For this purpose, we construct a basic NEG model with endogenous
fertility.9
Today, NEG is the standard model to explain why the distribution of population andeconomic activities among regions is radically uneven. In the NEG model, real income tends
to be high in highly populated regions and people migrate to these regions to gain higher
real income. Figure 3 describes the relationship between population density and per-capita
income in Japan in 2010. This shows that highly populated regions tend to oﬀer higher
incomes. Moreover, as illustrated in Figure 1(b), net migration tends to be higher in regionswith higher real income. These facts justify use of NEG model to describe regional population
change.
To endogenize fertility, we employ a framework introduced by Becker (1965) that considers time allocation problem between working and child-rearing in which parents obtain
utility from the number of children.10 When substitutability between consumption goods
9
10

Mortality is excluded from our model for simplicity.
The endogenous fertility choice problem was studied by Becker and Lewis (1973), Eckstein and Wolpin(1985), Becker and Barro (1988), Barro and Becker (1989), Becker et al. (1990), Galor and Weil (1996),

5

4500
Income per capita−1000 yen
2500
3000
3500
4000
2000
0

2000
4000
Population density (per square kilometer)

6000

Fitted values

Figure 3: Population Density and Income per Capita (in Thousands of Yen) in Japan in2010

and children is strong, an increase in the real wage reduces the fertility rate by raising theopportunity cost of rearing children relative to the price of consumption goods. Note that the
NEG model employs monopolistic competition of the Dixit and Stiglitz (1977) type. Then,
population growth expands the variety of consumption goods which increases the real wage
and thus reduces the fertility rate.11 This mechanism is originally proposed by Maruyama
and Yamamoto (2010) which does not address multiple regions. We expand this model to
the NEG framework.
Then, using this model, we analyze the regional population change, focusing on the effects of economic integration (i.e., higher migration and trade freeness among regions). We
show that if people cannot migrate between regions, regional diﬀerences related to the initial
population disappear in the long run. This result is contrary to the aforementioned facts.
On the other hand, if migration is permitted, we obtain quite diﬀerent results. Even though
there are only subtle diﬀerences between regions, these diﬀerences become suﬃciently large
through migration with a snowball eﬀect as the population concentrates in a region withan initially lager larger population share. Moreover, the region in which the population is

Shoven (2008)), and so on.
11
For example, Docquier (2004) and Jones and Tertilt (2008) show negative relationship between income
and the fertility rate in the United States. Borg (1989) ﬁnds the same relationship in Korea.

6

concentrated has a higher real income, which results in a decreased fertility rate and increased net migration compared to less concentrated regions. Thus, in the long run, regions
exhibit diﬀerences in population change and economic conditions. Typically, higher trade
freeness brings about a more concentrated population, which leads to more regional diﬀerences. Additionally as population concentration lowers the fertility rate in large regions, thepopulation in the whole economy is suppressed. These results are consistent with the facts
and imply that economic integration has a huge impact on population change in regional
economies as well as the whole economy.
The remainder of this paper is organized as follows. In Section 2, we construct a basicmodel without time and generations. In Section 3, we present an extension of the modelthat introduces time and generations for demographic analysis. Numerical simulations are
conducted in Section 4 with several example. Finally, Section 5 concludes.
2. The Basic ModelIn this section, we construct a basic model without times and generations. Consider an
economy with a ﬁnite set of regions, R (the number of regions is represented by |R|). The
economy consists of one diﬀerentiated goods sector characterized by monopolistic competition following Dixit and Stiglitz (1977).
2.1. Preference and Demand
We assume that all individuals obtain utility from the consumption of a composite ofdiﬀerentiated goods, X, and their number of children, n. All individuals share the same
preference for the composite diﬀerentiated goods and number of children, which is represented by the following utility function:
[
] ρ1
ρ
ρ
U = αX + (1 − α)n ,0 < α < 1,(1)

where ρ is the substitution parameter, and in this speciﬁcation, σ (≡ 1/(1 − ρ)) represents
the elasticity of substitution between the composite diﬀerentiated goods and the number ofchildren. α represents the intensity of the preference for the consumption of diﬀerentiatedgoods. When ρ is close to zero (i.e., when the utility function is close to the Cobb-Douglasform), α becomes the expenditure share of diﬀerentiated goods.
7

We assume that the composite index X is, in turn, a constant elasticity of substitutionfunction deﬁned over a continuum of varieties of diﬀerentiated goods. Taking x(γ) and Γ as
the consumption of each available variety γ and the set of available varieties respectively, X
is given by

[∫
X≡

ρX

] ρ1

x(γ) dγ

X,

0 < ρX < 1,

Γwhere ρX is the substitution parameter for variety in diﬀerentiated goods and σX (≡ 1/(1 −ρX )) is the elasticity of substitution between any two varieties. A smaller ρX (i.e., a smaller
σX ) means that diﬀerentiated goods are more highly diﬀerentiated or that individuals havea stronger preference for variety.
We assume that individuals have one unit of time and that they allocate it to working andrearing children. Following Becker (1965) and others, in order to have a child, individuals
must spend time b to rear a child, where b is a positive constant. Then, given the wage rate
wi in region i and price pji (γ) for each variety that is produced in region j and sold in region
i, the budget constraint of individuals in region i becomes()∑ ∫pji (γ)x(γ)dγ ≦ wi (1 − bn),i ∈ R,

Γjj∈R

where Γj is the set of varieties produced in region j. The measure of Γj denoted by Nj isinterpreted as the number of varieties produced in region j.
Solving the utility maximization problem, individual demand for both the composite
index and for children in region i is given byµ(wi , Pi )wi, i ∈ R,Pi
1 − µ(wi , Pi )ni = n(wi , Pi ) ≡, i ∈ R;b

Xi = X(wi , Pi ) ≡

(2)
(3)

then, individual demand in region i for variety γ produced in region j can be written as()σXµ(wi , Pi )wiPixji (γ) = x(wi , pji (γ), Pi ) ≡, γ ∈ Γj , j, i ∈ R,
(4)Pi
pji (γ)where Pi is the price index for diﬀerentiated goods in region i, which is deﬁned by
Pi ≡

[
(
∑ ∫
j∈R

)] 1−σ1Xpji (γ)1−σX dγ

Γj

8

,

i ∈ R,(5)and µ(·) is the expenditure share of the composite diﬀerentiated goods, which is given byµ(wi , Pi ) ≡ασασ Pi1−σ=,ασ Pi1−σ + (1 − α)σ (bwi )1−σ
ασ + (1 − α)σ (bwi /Pi )1−σi ∈ R.

(6)

Let us denote the real wage in region i as ωi (≡ wi /Pi ). A rise in the real wage has
two opposite eﬀects on the number of children per individual. Clearly, because children aresuperior goods, the rise in the real wage increases the number of children (income eﬀect).
The rise in the real wage, however, increases the opportunity costs of having children, whichraises the relative price of children and reduces the number of children (substitution eﬀect).
If the elasticity of substitution between the composite diﬀerentiated goods and the numberof children is larger than one, this latter eﬀect is suﬃciently large to outweigh the former,
and thus, a rise in the real wage reduces the number of children. This can be checked easilybyωi ∂µ(ωi , 1)
= (σ − 1)(1 − µ(ωi , 1)),µ(ωi , 1) ∂ωii ∈ R,and (3). Thus, a rise in the real wage actually reduces the number of children per individual.
We propose this result.

Proposition 1. If σ > 1 holds, then a rise in the real wage reduces the fertility rate.

In the remainder of this paper, we restrict the range of parameters to 1 < σ < σX .
One reason for this restriction is, of course, to ensure that a rise in the real wage reduces
the fertility rate. Another reason is to ensure that demand for each variety decreases when
the price index falls. If the amount of composite diﬀerentiated goods is ﬁxed at a given
level, fall of the price index always reduces the demand for a single variety because it issubstitutable for other varieties. However, since fall of the price index also increases thedemand for composite diﬀerentiated goods, in general, it is unclear whether the demand fora single variety becomes smaller with a smaller price index. The restriction 1 < σ < σX
makes this point clear. Note that an increase in the range of varieties available reduces theprice index. We show in the following subsection that the equilibrium output of any active
ﬁrm, q ∗ , is constant and that the proﬁts of a single ﬁrm becomes positive or negative as its
9

output becomes larger or smaller than q ∗ . Thus, this limitation is important for the stabilityof the long-run equilibrium characterized by the zero-proﬁt condition.
Substituting the demand for composite diﬀerentiated goods and for children (see (2) and(3)) into the utility function (1), we can express the maximized utility as a function of thereal wage as follows:
Vi = V (ωi ) ≡ωi1[ασ + (1 − α)σ (bωi )1−σ ] 1−σ,i ∈ R.

We can also easily derive the following relationship:ωi ∂V (ωi )= µ(ωi , 1) > 0,Vi ∂ωii ∈ R.

(7)

In the Section 3, we assume that individuals move to the regions where they can gain higher
utility. Relationship (7) ensures that it is suﬃcient to compare the real wages among regions
to determine the movement of individuals.
2.2. Production
Next, we turn to the production side of the economy. Each variety γ of diﬀerentiatedgoods is produced by a ﬁrm indexed by γ under increasing-returns technology, using labor
only, and each ﬁrm behaves as monopolist. All ﬁrms in all regions use the same technology,
with a ﬁxed input of f and marginal input requirement a. Then, the labor input requirement
for the production of a quantity qj (γ) of any variety γ ∈ Γj at any given region j is given bylj (γ) = f + aqj (γ).
Diﬀerentiated goods can be shipped between regions, but they must incur transport costs
in shipment. For simplicity, we assume the iceberg form of transport costs: that is, if one
unit of any variety γ ∈ Γj of diﬀerentiated goods is shipped from region j to region i, 1/τji(τji ≧ 1, τjj = 1) units of it actually arrive.12 Thus, τji units of the variety must be sentfrom the origin for one unit to arrive at the destination. Given the demand for each varietyin (4) and transportation technology, the output of ﬁrm γ in region j is equal toqj (γ) =∑τji xji (γ)Li ,γ ∈ Γj , j ∈ R,i∈R
12

The iceberg form of transport costs is ﬁrst introduced by von Thünen (1826) and then formalized bySamuelson (1952).

10where Li is the number of individuals (workers) in region i. Then, with given prices in eachregion, ﬁrm γ’s proﬁt is given byπj (γ) =∑(pji (γ)xji (γ)Li − wj

f +a

i∈R

∑

)
τji xji (γ)Li.

i∈R

Each ﬁrm γ chooses its prices, pji (γ), to maximize proﬁt. It is well known that each ﬁrm
has a pricing rule given by the following form:pji (γ) = pji ≡aτji wj ,ρXγ ∈ Γj , i, j ∈ R,

(8)which represents the familiar result that each monopolistic ﬁrm will charge a markup overthe marginal cost.
We suppose that there is free entry and exit of ﬁrms. Then, as long as some ﬁrms areproducing, the proﬁts of any active ﬁrm must be driven to zero. Given pricing rule (8), theproﬁt of ﬁrm γ in region j becomes][aqj (γ)−f ,πj (γ) = wj
σX − 1γ ∈ Γj , i, j ∈ R.

Therefore, the zero-proﬁt condition ensures that the equilibrium output of active ﬁrms isconstant and common to every active ﬁrm in all regions as q ∗ = (σX − 1)f /a, which implies
that the associated equilibrium labor input becomes l∗ ≡ f + aq ∗ = σX f . Note that thelabor supply per individual in region j is µ(wj , Pj ). Therefore, in equilibrium, the numberof ﬁrms located in region j is given byNj∗ ≡µ(wj , Pj )λj L,
f σX

j ∈ R,(9)

where L is the total number of workers and λj (≡ Lj /L) is the share of workers in region j.
2.3. Price Index Eﬀect and Home Market Eﬀect
We can now explore the agglomeration force in the economy, which causes a concentrationof the population in particular regions. First, using the pricing rule (8), we write the priceindex (5) in the following form:
[
Pi =

∑(
j∈R

X
Nj p1−σji] 1) 1−σX[] 1−σ1Xa ∑1−σX=
ϕji Nj wj,
ρX j∈R11i ∈ R,(10)where ϕji (≡ τji1−σX ∈ (0, 1]) is a measure of the freeness of trade from region j to region i,which increases as τji falls and is equal to 1 when trade is free (i.e., τji = 1). Suppose now
that a fraction of ﬁrms in region j changes their location to region i, holding other things
constant. Then, the relationship between the change in the number of ﬁrms in region j andthat in region i is represented by dNj = −dNi . Using this relationship and diﬀerentiating
the above price index while keeping all other things constant, we obtain
1−σ
1−σ
wi
Ni ϕji wj X − wi X
Ni dPi=⋚ τji ,∑
1−σX ⋚ 0 ⇔Pi dNiσX − 1 k∈R ϕki Nk wk
wj

i, j ∈ R.

Thus, when transport costs from region j to region i are suﬃciently high, the price index
of region i declines because of the relocation of ﬁrms. In particular, if the wage rates inregion j and region i are the same, the price index in region i always falls. This is called
the Price Index Eﬀect; it means that the region with the larger number of ﬁrms has a lower
price index because a smaller proportion of this region’s consumption of diﬀerentiated goods
bears transport costs.
Next, because ﬁrms making no proﬁts is equivalent to the condition that they produce
the equilibrium output q ∗ , any active ﬁrm in region i must allocate q ∗ to consumers in eachregion, that is,∑τij xij (γ)Lj = q ∗ ,γ ∈ Γi , i ∈ R.

j∈R

Using the demand function for each variety (4) and the pricing rule (8), this equation yields
the following wage equation:(
wi = wiP ≡

a
ρX

[
) 1−σ
X
σ
X

1 ∑
ϕij µj Yj PjσX −1
σX f j∈R

] σ1

X

,

i ∈ R,(11)

where µj = µ(wj , Pj ) and Yj = wj Lj . Given the income levels, wage rates, price indexes,
and transportation costs, wiP gives the wage rate that ﬁrms can aﬀord to pay in region i.
In the short run, the actual wage rate in region i may diﬀer from wiP . If the actual wage
rate in region i is lower than wiP , ﬁrms in region i can gain rent due to being protected fromcompetition with other ﬁrms. In this case, however, through the entry of ﬁrms, the wage
rate in region i must be adjusted to wiP and the rent disappears in the long run. In thisregard, we call wiP the Wage Potential in region i; this is the wage rate that workers canpotentially gain in region i.
12Suppose that a fraction of the income in region j is transfered to region i and that allother things are constant. Then, the relationship between the change in the income level ofregion j and that of region i is represented by dYj = −dYi . Using this relation, diﬀerentiating
wiP given by (11), keeping all other things constant, we obtain the following relationship:
σ −1
σ −1
Yi dwiPYi µi Pi X − ϕij µj Pj X
µi=∑
σX −1 ⋛ 0 ⇔P
σXµj
wi dYik∈R ϕik µk Yk Pk

(

Pi
Pj

)σX −1⋛ ϕij ,

i, j ∈ R.

Thus, when the trade freeness from region i to region j is suﬃciently high, the wage potential
in region i rises because of the transfer of income. Speciﬁcally, if the price indexes and theexpenditure shares of diﬀerentiated goods are the same between regions j and i, the wage
potential always rises. This is called the Home Market Eﬀect; it means that, all other things
being equal, the region with the larger home market has the higher wage potential and thewage rate in this region tends to be high.13
Inflow of Consumers

Price Index EffectLower Cost of
LivingInflow of Workers

Larger Numberof Firms

Inflow of Firms

Larger Market

HigherPotential

Home Market EffectFigure 4: Circular Causality in Agglomeration

Figure 4 depicts the circular causality in the spatial agglomeration. Because the region
with the larger number of ﬁrms has the lower a cost of living, consumers migrate to that
region. This, in turn, induces an increase in the market size of the region with the larger
number of ﬁrms. On the other hand, because the region with the larger market has thehigher wage potential, workers and ﬁrms ﬂow into it. This makes the number of ﬁrms and

13Here, we consider the home market eﬀect keeping all other things constant. Since, in reality, other thingsneed not be constant, the home market eﬀect may not hold. Conditions under which the home market eﬀect
holds are studied by Davis (1998), Yu (2005) and Behrens et al. (2009).

13

the size of the market in that region larger. Obviously, this circular causality leads to aconcentration of population in particular regions.
2.4. Equilibrium Wage Rate
The expenditure share of the composite of diﬀerentiated goods is the function of the wage
and the price index as in (6). From (9) and (10), region i’s price index can be expressed
as a function of L, ϕi = (ϕ1i , · · · , ϕ|R|i ), λ = (λ1 , · · · , λ|R| ), w = (w1 , · · · , w|R| ) and µ =(µ1 , · · · , µ|R| ): Pi = P (L, ϕi , λ, w, µ). Then, the equilibrium share of the composite ofdiﬀerentiated goods, µ∗i , is given byµ (wi , P (L, ϕi , λ, w, µ∗ )) = µ∗i ,i ∈ R.

Thus, we can express µ∗i as a function of L, ϕ1 , · · · , ϕ|R| , λ and w. It is easily veriﬁable
that µ∗i is linearly homogeneous for w.
In equilibrium, the wage rate in region i equals region i’s wage potential, that is, wi∗ = wiP .
Therefore, from (11), the equilibrium wage rate vector w∗ is determined by the following:[∑

ϕij λj µ∗j wj∗∑∗∗ 1−σXk∈R ϕkj λk µk (wk )
j∈R

] σ1

X

= wi∗ ,i ∈ R.

We can then express w∗ as a function of L, ϕ1 , · · · , ϕ|R| , and λ. Clearly, if w∗ is anequilibrium wage rate vector, for all c > 0, cw∗ is also an equilibrium wage rate vector.
∑
Hence, without loss of generality, we can normalize wage rates as i∈R wi = 1.
3. Extension of the Model for Demographic Analysis
In this section, we introduce times and generations to the model. Time is discrete, andeach individual lives for two periods: childhood and adulthood. We denote the numberof adults in region i in period t as Li,t and the total number of adults in time t as Lt∑(= i∈R Li,t ). In adulthood, individuals choose the regions they live in, supply labor inthose regions, and decide their amount of consumption and number of children.

143.1. Population Dynamics
At the beginning of period t, each adult in region i has ni,t children. In this model ni,t also
represents the fertility rate in region i at time t, and thus ni,t Li,t is the number of children
in region i at time t. The average fertility rate in the economy at time t, n̄t , is given by
n̄t ≡

∑

t ∈ N,

λi,t ni,t ,i∈Rwhere λi,t (≡ Li,t /Lt ) is region i’s share of adults (workers) at time t. Since n̄t Lt children
grow to adulthood in the economy in period t + 1, we have the following law of motion of
the total number of adults:
Lt+1 = n̄t Lt =

( t
∏

)
n̄r

L0 ,

t ∈ N.

(12)

r=0

3.2. Dynamics of Inter-regional Population Movement
Next, we consider inter-regional migration. We assume workers choose the regions inwhich they live and subsequently choose their number of children and amount of consumption
in that region. We ﬁrst deﬁne λprei,t+1 as region i’s share of pre-movement workers at timet + 1:
λi,t ni,t
ni,t Li,t=,n̄tj∈R nj,t Lj,tλprei,t+1 ≡ ∑i ∈ R, t ∈ N.

(13)

Similar to Fujita et al. (1999), we assume workers gradually move to regions where they cangain a higher real wage, that is, higher utility (see (7)). We capture this adjustment processby the following dynamics:(
λi,t+1 − λprei,t+1 = νwhere ω̄t (≡

∑

)
ωi,t
− 1 λi,t ,ω̄ti ∈ R, t ∈ N,i∈R λi,t ωi,t ) is the average real wage among regions at time t and ν (> 0) is

the adjustment parameter. Using (13), we can rewrite the above system as follows:14
(
))
(
ωi,t
ni,t
−1+ν
−1
λi,t , i ∈ R, t ∈ N.
λi,t+1 − λi,t =
n̄t
ω̄t
14

(14)

The dynamics given by (14) are essentially equivalent to the replicator dynamics that are often usedin evolutionary game theory (see Weibull (1995)). We introduce the natural change of the population into
these dynamics.

15

Then, we have the following law of motion of the number of workers in region i:
(
(
))
ωi,t
ni,t
+ν
−1
n̄t Li,t , i ∈ R t ∈ N.
Li,t+1 =
n̄t
ω̄t

(15)

3.3. Deﬁnition of Steady State
In a steady state, both the total number of workers in the economy, L∗ , and the share ofworkers in each region, λ∗i are stationary such that
Lt+1 = Lt = L∗ ,λi,t+1 = λi,t = λ∗i ,i ∈ R.

From (12) and (14), this steady state is given by n̄∗ = 1 and
( ∗
( ∗
))
ni
ωi
−1+ν
−1
λ∗i = 0, i ∈ R,
n̄∗
ω̄ ∗

(16)

where n∗i and ωi∗ are the fertility rate and the real wage, respectively, in region i corresponding
to L∗ and λ∗i .
Note that in the usual NEG model, spatial equilibrium is deﬁned as the state in which
no agents have an incentive to change location. In this regard, in spatial equilibrium, theshare of workers in each region, λSi , is deﬁned by)( S
ωi− 1 ≦ 0, i ∈ R,
ω̄ S
( S
)
ωi
− 1 λSi = 0, i ∈ R,
ω̄ S
where the superscript S represents the value in the spatial equilibrium. If the share ofworkers in each region is in spatial equilibrium, the real wage is equalized in all regions thathave a positive share, which leads to identical fertility rates in these regions. Therefore,(16) is satisﬁed in the spatial equilibrium. However, the steady state may not be the spatial
equilibrium. If some region has a higher real wage, there will be a population inﬂow, but
the fertility rate will be lower in that region. Therefore, it is possible for (16) to be satisﬁed
in a steady state that is not the spatial equilibrium. This result diﬀers completely from theusual NEG model.

164. Numerical Examples
In this section, we analyze how economic integration aﬀects population demographics.
Since our model is highly non-linear, it is, unfortunately, diﬃcult to obtain analytical results.
Therefore, we employ a numerical simulation method and show some examples. For ouranalysis, it is useful to deﬁne the natural change in population, N Ci,t , and social change in
population, SCi,t , of each region at each time as
N Ci,t = (ni,t − 1)Li,t , i ∈ R, t ∈ N,()
ωi,t
− 1 n̄t Li,t i ∈ R, t ∈ N.
SCi,t = νω̄tFrom (15), the total change of workers in region i at time t, T Ci,t (≡ Li,t+1 − Li,t ), is given
by T Ci,t = N Ci,t + SCi,t .
4.1. The Case of Two-Region
Here, we show a two-region case, ﬁrst with no migration and then with migration. Figure 5 describes the demographics for no migration, in which workers cannot move between
regions. The economy initially has one unit of labor (workers) and region 1’s initial share ofworkers is set as 0.9. Then, we illustrate the dynamic paths of the share of workers (λi,t ),fertility rate (ni,t ), real wage (ωi,t ), number of workers (Li,t ), natural and social change (N Ci,tand SCi,t , respectively) in each region.
Because of the home market eﬀect and the price index eﬀect, the highly populated region
has a higher real wage and, thus, the real wage of region 1 is higher than that of region
2. From Proposition 1, this leads to a lower fertility rate in region 1 compared to region
2. Therefore, if workers cannot move between regions, the share of workers would equalize
gradually over time (see Figure 5(a)). On the other hand, the number of workers monotonically increases in regions 1 and 2, which raises real wages in both regions (see Figures5(c) and 5(d)). This rise of real wage results in fertility rates decreasing (see Figure 5(b)).
Natural changes in regions 1 and 2 become large at an early phase, but reach their peakat a certain point in time and thereafter decrease until zero (see Figure 5(e)). Because we
consider the case of no migration case, the social change is always zero (see Figure 5(f)).

17

1

7

1.3

0.9
1.25

6

0.8
1.2

0.7

5
1.15

0.6
λi

ni

0.5
0.4

ωi
1.1
1.05

4

3

0.3
1

0.2

2
0.95

0.1
0

0.9

t

(a) Share of workers

t

1

t

(b) Fertility rate30(c) Real wage

1

1

0.8

0.8

0.6

0.6

0.4

0.4

25

20

0.2

Li

NCi

15

0.2
SCi

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

10

5

−0.8

0

−1

t

(d) Number of workers

−0.8

t

−1

(e) Natural increaset(f) Social increase

Figure 5: Demographics with No Migration
σ = 1.2, σX = 4, α = 0.7, b = 0.25, ϕ12 = ϕ21 = 0.25, f = 1/3, a = 1/3, ν = 0

18

1

7

1.3

0.9

6
0.8
1.2
0.7

5

0.6
λi

ni

0.5

ωi

1.1

4

0.4

3
0.3
1
0.2

2

0.1
0

0.9

t

1

t

(a) Share of workerst(b) Fertility rate

40
35
30

(c) Real wage

1

1

0.8

0.8

0.6

0.6

0.4

0.4

25
0.2

Li

NCi

20
15

0.2
SCi

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

10
5
0

−0.8
−1

t

(d) Number of workers

−0.8

t

−1

(e) Natural increaset(f) Social increase

Figure 6: Demographics with Migration
σ = 1.2, σX = 4, α = 0.7, b = 0.25, ϕ12 = ϕ21 = 0.25, f = 1/3, a = 1/3, ν = 0.2

19

Next, we consider the case with migration, in which workers can move between regions.
Figure 6 describes the dynamic paths of the variables under the same parameters as the caseof no migration except for the migration adjustment parameter, ν.
In the case of no migration, if the home market eﬀect and the price index eﬀect always
make the real wage higher in the highly populated region, the fertility rate there will belower than in the less populated region since a higher real wage rate means a lower fertility
rate. Consequently, in this case, the home market eﬀect and the price index eﬀect make thepopulation distribution over the regions tend toward uniformity, as shown in Figure 5. We
call this dispersion force.15
However, if the migration of workers is permitted, the above argument may not hold.
In Figure 6, we set the initial share of workers to be almost the same in both regions, but
the share in region 1 is slightly larger.16 As the home market eﬀect and the price index
eﬀect make the real wage of region 1 higher than that of region 2, the social change, SCi,t ,is positive in region 1, but negative in region 2 (see Figure 6(f)). This agglomeration force
encourages the share of workers in region 1 to increase over time (see Figure 6(a)). On
the other hand, the natural change, N Ci,t , is small in region 1 and large in region 2 (seeFigure 6(e)) since a higher real wage reduces the fertility rate (dispersion force). When thediﬀerence in the share of workers between regions becomes large, the agglomeration force
and the dispersion force are balanced and the economy is in a steady state. Interestingly, inthis steady state, the social change of region 1 is positive, which means that this economy
is not in spatial equilibrium. In particular, in the steady state, the two regions diﬀer incharacteristics: one has a positive natural change but a negative social change, while theother has a positive social change but negative natural change.
The paths of the number of workers in the whole economy in the cases of no migrationand migration are described in Figure 7. In the case of no migration, the path has the formof a logistic curve (see Figure 7(a)). When there are few workers in the economy, since the15

In NEG, “dispersion force” is usually used to describe a power that provides a clear incentive to migratefrom large to small regions (e.g., congestion). In this paper, we use the term in a diﬀerent sense: here,“dispersion force” means a power that equalize population distribution.
16
We set λ1,0 = 1001/2000 and λ2,0 = 999/2000.

20

L

60

60

60

50

50

50

40

40

40

L

30

L

30

30

20

20

20

10

10

10

0

0

t

(a) No-migration caset(b) Migration case0t(c) Shadow area of (b)Figure 7: Number of Workers in the Whole Economy
σ = 1.2, σX = 4, α = 0.7, b = 0.25, ϕ12 = ϕ21 = 0.25, f = 1/3, a = 1/3 ν = 0.2

real wage is low the fertility rate is high and the population grows rapidly. As the population
grows, the real wage becomes higher and this suppresses the fertility rate (Proposition 1).
Therefore, population growth gradually decreases and the economy reaches a steady state.
However, in the case of migration, the shape of the path diﬀers, as in Figure 7(b). The
population becomes largest along the path before the steady state and then decreases. In theearly phase, because the diﬀerence in the shares of workers between regions is not sizable,
the real wages in both regions are nearly the same. Hence, the migration rate of workers
is small and the population growth path in the economy is similar to that of the case of
no-migration. Figure 7(c) illustrates this phase, which corresponds to the shaded area of
Figure 7(b). In this phase, the path of the population growth is also the form of a logistic
curve.
After the early phase, however, workers migrate to the region with the higher real wage
and gradually congregate there, which increases the real wage in the highly populated region(region 1) and decreases the real wage in the less-populated region (region 2). This brings
about further migration and reduces the fertility rate in region 1. When the diﬀerence
between the shares of workers becomes large enough, the fertility rate is less than 1 in region
1. Thus, total population growth becomes negative, even though region 2 has a fertility rateover 1, and this overall negative growth continues until the negative natural change in region
1 is canceled out by positive natural change in region 2.

21

4.2. How Integration of the Economy Aﬀects Population size and Spatial Structure
Next, we show how integration of the economy aﬀects population size and spatial structure. We examine how the share of workers in each region and the number of workers in
the whole economy are aﬀected by changing transportation costs (trade freeness) and themigration adjustment. We set the initial distribution of workers and all parameters to be
the same as in the migration case in Section 4.1, except for ϕ and ν.
1

50

0.9

48

0.8

46

0.7

44

70

65

60

i

55

42

0.6
λ

L

0.5

LMax
50

40

0.4

38

0.3

36

0.2

34

0.1

32

0

30

45

40

0

0.1

0.2

0.3

0.4

0.5
φ

0.6

0.7

0.8

(a) Share of workers

0.9

1

35

0

0.1

0.2

0.3

0.4

0.5
φ

0.6

0.7

0.8

(b) Number of workers

0.9

1

30

0

0.1

0.2

0.3

0.4

0.5
φ

0.6

0.7

0.8

0.9

1

(c) Max number of workers

Figure 8: Eﬀects of the Change of Trade Freenessσ = 1.2, σX = 4, α = 0.7, b = 0.25, f = 1/3, a = 1/3, ν = 0.2

First, we show the eﬀects of changing trade freeness.17 Figures 8(a) and (b) describe theshare of workers in each region and the number of workers in the whole economy, respectively,in the steady state corresponding to ϕ. Increased trade freeness increases the steady-stateshare of workers in region 1. Above ϕ = 0.4, the economy completely agglomerates. Moreover, the number of workers in the economy decreases as trade freeness increases. Note thatan increase in trade freeness raises the real wage not only directly, but also indirectly by
inducing population concentration. This lowers the fertility rate. After complete agglomeration, trade freeness does not aﬀect the number of workers in the steady state since, in thiscase, there is only one region in the economy. On the other hand, the maximum number ofworkers on the transition path decreases monotonically with increased trade freeness. Thisis shown in Figure 8(c), which illustrates the maximum number of workers on each transition
path. Since the number of workers reaches the maximum level on the path before workers
17

We assume symmetry trade freeness: ϕ12 = ϕ21 = ϕ

22

agglomerate in region 1, the maximum number of workers in the economy is inﬂuenced bytrade freeness.
55

1

55

0.9

50

0.8

50

0.7

45

0.6
λ

i

45
LMax

L

0.5

0.4

40

40

35

35

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5
ν

0.6

0.7

0.8

(a) Share of workers

0.9

1

30

0

0.1

0.2

0.3

0.4

0.5
ν

0.6

0.7

0.8

(b) Number of workers

0.9

1

30

0

0.1

0.2

0.3

0.4

0.5
ν

0.6

0.7

0.8

0.9

1

(c) Max number of workers

Figure 9: Eﬀects of Change of the Migration Adjustment Parameterσ = 1.2, σX = 4, α = 0.7, b = 0.25, ϕ12 = ϕ21 = ϕ = 0.25, f = 2/3, a = 1/3

Next, we show the eﬀects of changing the migration adjustment parameter, ν. Figures
9(a) and (b) describe the share of workers in each region and the number of workers in thewhole economy, respectively, in the steady state corresponding to ν. It is obvious that theshare of workers in the steady state is aﬀected by the migration adjustment parameter. If
ν is small, it is diﬃcult for workers to move even if there is a diﬀerence in the real wages
between regions. However, the fertility rate is lower in the highly populated regions; thisoccurs irrespective of the level of ν. Hence, the distribution of workers between regions
tends to be uniform, as in the case of no migration in the steady state. Conversely, if ν is
suﬃciently large, that is, if it is easy to move, a positive social change always overcomes anegative natural change in the large region. Thus, the steady state of the spatial structureof the economy is full agglomeration. Given a moderate ν, the economy does not converge to
be both symmetric and in full agglomeration. In the steady state, the social change balanceswith the natural change, and the economy is in partial agglomeration.
When ν is small, an increase of ν does not aﬀect the steady state number of workers
because the distribution of workers is always uniform. Given intermediate ν, which is between
symmetry and complete agglomeration, an increase of ν decreases the steady state numberof workers because it increases the share of workers in region 1. When ν is suﬃciently large,
as in the case of a small ν, an increase of ν does not aﬀect the steady state number of workers
23because workers fully agglomerate to region 1, while the number of workers is smaller thanin the symmetry case.
If the steady state share of workers is the same between regions, the maximum numberof workers on the transition path equals the steady state number of workers in the whole
economy. However, it decreases as the migration adjustment parameter increases if ν is larger
than in the symmetric case. When this occurs, the steady state distribution of workers is not
uniform and the number of workers reaches its maximum level before reaching the steadystate. This is shown described in Figure 9(c).
From the above discussion, we see that if the economy is more integrated, the number ofworkers decreases. This is why economic integration not only raises the real wage directly(consider the eﬀects of rise of ϕ) but also induces the spatial agglomeration which in turnraises the real wage. Hence, the fertility rate decreases, which results in a decline in thenumber of workers. This result cannot be obtained in models which consider only natural
population change or only social population change.
4.3. The Case of Multi-Region
Finally, we examine the multi-region case. If there are many regions, we can consider
various geometry of the economy. Here, we focus on the special artiﬁcial case known as the
Racetrack Economy in which each regions are laid out symmetrically around the circumference of a circle.18 Transportation is possible only along the circumference within the shortest
distance. Since we employ iceberg-type transportation costs and regions are located along
the circumference at even intervals, the trade freeness between regions i and j becomes
ϕ|i−j|if |i − j| ≦ |R|/2
ϕij =
,

ϕ|R|−|i−j| if |i − j| > |R|/2
where ϕ ∈ (0, 1] is constant.
In this economy, we can easily conﬁrm that the uniform distribution of workers, called
the Flat Earth in Fujita et al. (1999), is always a spatial equilibrium. However, the ﬂat
earth is not always sustainable; that is, the symmetric spatial equilibrium may turn out to18

The racetrack economy is ﬁrst introduced in NEG by Krugman (1993).

24

1

0.2

0.6

0.1

0.5

0

0.4

0.6

−0.1

0.3

0.5

−0.2

0.2

0.9
0.8
0.7

0.4

−0.3

0.1

0.3
−0.4

0

0.1

−0.5

−0.1

0
0

−0.6
0

−0.2
0

0.2

10

20

30

40

50

60

70

80

90

10

20

30

40

50

60

70

80

100

(a) Share of workers

90

100

(b) Natural increase

10

20

30

40

50

60

70

80

90

100

(c) Social increase

Figure 10: Multi-Region Case
σ = 1.2, σX = 4, α = 0.7, b = 0.25, ϕ = 0.95, f = 1/3, a = 1/3, ν = 0.2, |R| = 100

be unstable. As Figure 10 shows, we start the simulation from an almost ﬂat but randomly
deviated distribution of workers.19 Even though the deviation is very small, the circularcausality of agglomeration can break the ﬂat earth: an almost even distribution of workers
eventually develops local concentrations of workers. Figure 10(a) shows this process. One
hundred regions are arranged along the front axis in numerical order and the share of workers
in each region is indicated by the vertical axis. The almost ﬂat earth evolves over time intoa very uneven spatial structure in which workers become concentrated in two regions thatare positioned opposite to each other on the circumference.
While this result may appear to be the same as that obtained by Fujita et al. (1999),
there are some major diﬀerences. Figure 10(c) illustrates the social change of each region overtime. It shows that workers migrate from small regions to large regions and that the socialchange of the largest region is extremely positive even after a suﬃciently long time. As aresult, the region that initially has the largest number of workers tends to be extremely large
in the steady state. It has been noted that workers ﬂow from the smaller of two regions inwhich workers are concentrated. The smaller region has a large share of workers in the steadystate because its natural change is highly positive. Figure 10(b) shows the natural changeof each region over time. When workers concentrate in one region, the real wage becomes

19

First we draw Li randomly from the interval [0.95, 1.05]. Then we normalize the total number of workers

to one.

25

lower in the opposite region since a larger fraction of consumption must bear transportation
costs. This leads to a higher fertility rate and increases the natural change in the region
that is positioned opposite to the region with the largest number of workers. Therefore, thetwo regions in which workers are concentrated have markedly diﬀerent characteristics andsizes. These results can be obtained only by considering the endogenous fertility rate, which
is ignored in Fujita et al. (1999).
5. Conclusion
In this paper, we constructed a model to describe regional population changes in s market
economy. Using this model, the eﬀects of economic integration on population change wereanalyzed.
If workers can migrate among regions, then regional diﬀerences in the real wage, the natural population change, and the social population change are become larger with a snowballeﬀect, even though, there are only subtle diﬀerences initially. The population concentrates
in the region that initially has a larger population share. The region in which the population
concentrates has a higher real wage, which results in a lower fertility rate and higher net
migration compared to other less-concentrated regions. Thus, in the long run, regions diﬀer
in population change and real income. In particular, the diﬀerence in the real wage in thelong run means that the steady state is not spatial equilibrium, that is, workers have an
incentive to change their location. This result diﬀers widely from the usual NEG model andis consistent with the facts we presented in the introduction, which enhances the legitimacyof our model. In addition, we derived a prediction for the population growth path of thewhole economy: it resemble a logistic curve in the early phase, but the population decreases
in the last phase.
We also showed the following: (i) highly freeness to migrate and trade of goods lead topopulation concentration, (ii) population concentration leads to a lower fertility rate in highly
populated regions, and, as a result, (iii) economic integration decreases the total population.
If inter-regional migration is permitted, workers will move to regions where they can earnhigher real wages. This increase the populations of regions with higher real wages. Moreover,the existence of transportation costs leads to higher real wages in highly populated regions
26

compared to less-populated regions since a larger fraction of goods must bear transportation
costs in the latter case. This circular causality induces a population concentration in aparticular region. Speciﬁcally, we showed that the more the transportation cost decreases,
the more the population is concentrated. On the other hand, higher real wages in highlypopulated regions lead to an increase in child-rearing costs and a decrease in the fertility
rate. Therefore, increased economic integration leads to a smaller population in the whole
economy.
However, in this paper, we dealt only with two overlapping generations, that is, childhood and adulthood. We paid little attention to the composition of the population. In
demographic studies, the composition of the population is typically a matter of primary
importance, as its structure changes over time just as its size and distribution do. However,for simplicity, we omitted population structure from our analysis. In spite of this limitation,
we believe that our analysis identiﬁes new aspects of the relationship between economic
integration and population change. We leave the consideration of population structure for
future research.
Acknowledgments
The authors thank to Nobuaki Hamaguchi, Tamotsu Nakamura, Noritsugu Nakanishi,
Yamamoto Kazuhiro, Dao-Zhi ZENG, and participants of the 27th Annual Conference of
ARSC at Kyoto University. Needless to say, any errors remaining in this paper are theresponsibility of the authors.
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