GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.63

A model of urban demography

Hiroshi Aiura and Yasuhiro Sato

July 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanA model of urban demography∗
Hiroshi Aiura†

Yasuhiro Sato‡

June 27, 2009

Abstract
This paper develops an overlapping generations model that involves endogenous determination of fertility and explicit city structure. We provide conditions
under which there exists a unique steady state, which can replicate spatial features of demography observed in Japanese cities. We also provide comparative
steady state analysis by calibration.

JEL Classification: J11, R11, R23.
Keywords: city structure, land rent, fertility, demography∗

Yasuhiro Sato gratefully acknowledges financial support from JSPS and MEXT KAKENHI(18103002, 21243023, 21243021, 21330055, and 21730191). The usual disclaimer applies.
†

Faculty of Economics, Oita University (Japan); E-mail: h-aiura@cc.oita-u.ac.jp‡

Graduate School of Economics, Osaka University (Japan); E-mail: ysato@econ.osaka-u.ac.jp01

Introduction

It has long been recognized that city structure has relationship with demography. We
can easily find statements on this issue here and there: for example, the National
Institute of Population and Social Security Research, Japan [7] surveyed the number ofchildren that couples are going to have and the number of children that couples wish
to have under ideal conditions. Its table 78 reports the reasons why couples are goingto have fewer children than the ideal. The table shows that whereas 16.8 percent ofcouples who live in Densely Inhabited Districts (DIDs) choose the unaﬀordability of
having a suﬃciently spacious house as one of the reasons, this figure is 5.4 percent for
those who live in non-DIDs. These figures indicate that the city structure, via highland rent, may have significant impacts on fertility in cities.
However, full-fredged analyses from a viewpoint of economics have been scarce until
recently. An noticeable exception is Shultz [11], which provided empirical results that
support the interdependence of city structure and fertility by showing that advances inurbanization reduces national fertility.
Very recently, we observe that several studies uncovered possible nonnegligible interactions between city structure and demography by using a framework of economics.
As to empirical evidences, Galor [5] presented stylized facts that imply (i)that urbanization and economic development started simultaneously, and (ii)that in early phasesof urbanization and economic development, population growth rate increased, but it
declined in succeeding years. Simon and Tamura [12] showed the existence of a neg-1

ative cross-sectional correlation between the price of living space as measured by rent
per room and fertility for the Consolidated Metropolitan Statistical Areas in the UnitedStates over the period 1940—2000. The results described in these studies imply that city
structure that includes land and housing markets can play a major role in determining
demographic features in cities.
Theoretically, several studies used reduced form models to analyze the demographic
impacts of urbanization (Zhang [14]; Sato [8]; Sato and Yamamoto [9]; and Sato andYamamoto [10]).1 Zhang [14] and Sato and Yamamoto [10] examined impacts of urbanization caused by better opportunities for earnings and education in cities on demography. Sato [8] and Sato and Yamamoto [9] investigated how urbanization anddemographic transition interrelate with each other via merits of population concentration (agglomeration economies) and demerits of it (congestion diseconomies). All
these studies showed that urbanization is accompanied by declines in fertility, which isconsistent with the stylized facts described in Galor [5]. However, because the models
developed in these studies have no explicit spatial structure, their analysis does notshed any light on spatial features of demography within cities.
In this paper, we contribute this literature by developing an overlapping generations
model of endogenous fertility that involves explicit spatial structure. Especially, we
focus on the relationship between the city size, the spatial patterns of fertility within a1

Eckstein et al [3] developed an overlapping generations growth model that involves land as a

production input, and examined the impacts of the limitation of land availability on economic growth.
However, their model does not deal with land consumption nor city structure.

2

city, and land consumption.
For this purpose, we fist provide an overview of basic stylized facts on demography ofmetropolitan areas in Japan. We use data on three largest Metropolitan Areas (MAs),
which are Tokyo MA, Osaka MA, and Nagoya MA.2 Tokyo, Osaka, and Nagoya are thefirst, second, and third largest cities in Japan. Figure 1 shows the city size in terms of
population for the three largest MAs from 1950 to 2007.

[Insert Figure 1 around here]

These three cities have grown steadily, while keeping the order of size unaltered. Although they are diﬀerent in size, they have a common spatial feature of fertility. In
order to make this visible, Table 1 shows the total fertility rate for prefectures thatconsist of each MA during the past half century.

[Insert Table 1 around here]

We readily observe the following two facts: (i) that fertility is lower for a larger city,(ii) that fertility is lower for prefectures considered as the central cities than for thoseconsidered as suburbs. As is well known, the land rent/price is higher for a larger
city and the center of a city has higher land rent/price than suburbs (see Table 2, forexample).
2

Each MA consists of several prefectures: Tokyo MA=Tokyo+Kanagawa+Saitama+Chiba, Osaka

MA=Osaka+Kyoto+Hyogo+Nara, and Nagoya MA=Aichi+Gifu+Mie. This definition is often usedin analysis by Ministry of Internal Aﬀairs and Communications, Japan.

3

[Insert Table 2 around here]

Put diﬀerently, the level of land rent is negatively associated to fertility both across cities
and within each city. As is also well known, the level of land consumption is positively
associated to the level of land rent. Hence, this also implies that land consumption andfertility have positive correlation.
In order to analyze how city structure and demography can interact, we develop an
overlapping generations model of endogenous fertility that involves the monocentric city
structure à la Alonso [1]. In considering fertility decision, we adopt the view of Becker[2], which regards having children as consumption, not as investment. Each householdobtains utility from numeraire consumption, land consumption, and the number ofchildren. The key assumption here is the complementarity between land consumption
and the number of children: one needs a certain amount of land in order to rear a child,and obtains utility from land consumption over the required level for child rearing.
Population changes arise not only from changes in the number of children but also from
migration into/out of the city. In such a model, the land rent is higher in the central partof the city, leading to lower land consumption and fewer number of children. Moreover,as the city grows, the land rent gets higher and land consumption and fertility decreases.
These features are consistent with the stylized facts described above and in Simon andTamura [12]. Comparative steady state analysis provides impacts of changes in city
structure on demography in the city.
The remainder of the paper is organized as follows. Section 2 presents a basic model.
4Section 3 provides comparative steady state analysis. Section 5 concludes and suggests
future research directions.

2

Basic model

2.1

Individuals

Consider a linear space, on which there is one Central Business District (CBD), i.e., weassume a linear monocentric city. We approximate the CBD by a point and assume that
all workers commute to the CBD. Without loss of generality, we index the location of
the CBD as 0, and describe each location by the distance x between it and the locationof the CBD. Time is discrete and each individual lives for two periods; a childhood
and an parenthood. Each individual has a single parent. In the parenthood, eachindividual is endowed with one unit of time, which she spends on working and on childrearing. At the beginning of period t, she decides on goods and land consumption (ctand dt ) and her number of children (nt ). She exits the economy at the end of periodt. Nt individuals in the parenthood live in the city in period t. This implies that nt Ntchildren are born in period t and grow to be parents in period t + 1. In this model, nt
represents the total fertility rate.
We assume that Individuals have an identical utility function of the Cobb-Douglasform and the utility of each individual depends on the level of goods and land consump-5

tion and the number of children:
Ut = α ln ct + β ln nt (dt − εnt ) ,where α, β and ε are positive constants and satisfy α + 2β = 1. There is only one kindof goods in this economy, which we treat as a numeraire. In order to rear a child, one
needs a certain amount of land and we represent it as ε. An individual obtains utility
from land consumption over the required level for child rearing. This complementarity
between land consumption and the number of children brings forth the demographic
characteristics in a city.
In order to have nt children, each individual must spend bnt units of time, whereb is a positive constant. This assumption requires that nt must satisfy 0 ≤ nt ≤ 1/b.
Because each working individual is endowed with one unit of time, she spends 1 − bnt
units of time for working. The budget constraint for a working individual who resides
at x distant from the CBD is given by(1 − bnt )I − τ x = ct + rt (x)dt ,where I denotes the wage income per unit of time and is a positive constant. τ represents
the commuting cost per unit distance and rt (x) is the market land rent at x distantfrom the CBD. We assume that land is owned by absentee landlords.

6

The utility maximization gives the following demand functions:ct (x) = α (I − τ x) ,(1)β(I − τ x) ,bI + εrt (x)
¸
∙
ε
1
+
(I − τ x) .
dt (x) = βrt (x) bI + εrt (x)nt (x) =This leads to the indirect utility as follows:
Vt (x) = A + ln (I − τ x) − β ln rt (x) [bI + εrt (x)] .

(2)

where A is defined as A ≡ α ln α + 2β ln β.
We can see from (1) that a rise in wage income has two eﬀects on the total fertility
rate nt . One is the positive income eﬀect that is represented in the numerator of theright hand side. The other is the negative substitution eﬀect that raises the opportunity
cost of rearing children. This is described by the denominator of the right hand side.
These relationships are relevant in the later section. More importantly, a higher land
rent leads to a smaller number of children and to smaller land consumption. This isbecause when rt (x) is high, land requirement for child rearing enforces individuals tohave less children.

2.2

City structure and location equilibrium

We assume that children live with their parents and a working individual can movefreely within a city. The location equilibrium is attained in each period and it requires

7

that the indirect utility is the same for all locations in a city:∀x ∈ (0, xt ],Vt (x) = V t ,(3)where xt denotes the city fringe in period t. xt represents the spatial size of the city.
We follow the well-established tradition of urban economics in determining the land
rent in a city by using the concept of the "bid rent."3 The bid rent is the maximum land
rent at location x that each individual is willing to pay in order to reach her equilibrium
utility level. We normalize the land rent outside of the city to one. This implies thatthe land rent at the city fringe rt (xt ) is equal to 1. From (2) and (3), we obtain thebid rent at x by solving A + ln (I − τ x) − β ln R (bI + εR) = Vt (xt ) with respect to R,
which yields
R=

⎡

1 ⎣
−bI +
2ε

s

(bI)2 + 4ε (bI + ε)µI − τxI − τ xt

¶1/β

The market land rent is then given by
rt (x) = max [R, 1]½
R if x ∈ (0, xt ].
=1 if x > xt

⎤

⎦.
(4)

We readily obtain rt0 (x) < 0 and ∂rt /∂xt > 0 for x ∈ (0, xt ], which we summarize inthe following lemma.
Lemma 1 Within the city, the land rent is lower for a location more distant from the
CBD ( rt0 (x) < 0). As the city becomes larger, the land rent in the city rises( ∂rt /∂xt > 0).
3

The usage of the bid rent is very standard in urban economics. See Kanemoto [6] and Fujita [4]for a comprehensive discussion on the bid rent in monocentric city models.

8These are very standard results in the literature of monocentric models (Fujita [4]).
From (4), we obtain the level of utility in the city as
V t = A + ln (I − τ xt ) − β ln (bI + ε) .

(5)

Therefore, for a given number Nt of individuals in the parenthood, we can determine
all other endogenous variables once the city fringe xt is determined. The city fringe xt
is given by the land market clearing condition for a given Nt :
Z xt
0

D
dx = Nt .
dt (x)(6)

D represents the land supply for each location that is exogenous in our model. Letting
xt (Nt ) denote xt that is determined by (6), we can examine x0t (Nt ), that is, how thecity population aﬀects the city fringe:
Lemma 2 An increase in the number of individuals in the city enlarges the city area
( x0t (Nt ) > 0).
Proof. See Appendix A.
From Appendix A, we also know that xt (0) = 0 and limNt →∞ xt (Nt ) = I/τ : there isno city area if no one is in the city, and the city fringe can at most reach I/τ if city
population explodes.
Combined with Lemmas 1 and 2, (1) yields the following proposition:
Proposition 1 An individual residing more distant from the CBD consumes more land
( d0t (x) > 0). If land requirement for child rearing is suﬃciently large ( ε ≥ βbI/α),
she has more children ( n0t (x) > 0).
9

Proof. See Appendix B.
A location of an individual has two eﬀects on land consumption and on the number ofchildren. On the one hand, when an individual lives more distant from the CBD, shemust bear higher commuting costs, which reduces the income net of commuting costs.
This has an eﬀect of reducing land consumption and the number of children. On theother hand, the more distant from the CBD the location is, the lower the land rent
is. A lower land rent enables an individual to consume larger land. This also induces
her to have more children because a lower land rent implies lower payments for land
requirement for child rearing. With respect to the land consumption, the latter eﬀect
dominates the former, and one who lives farther away from the CBD always consumes
more land. This also holds true regarding the fertility rate if the land requirement forchild rearing is suﬃciently large.
Equation (1) shows that ∂dt /∂rt < 0 and ∂nt /∂rt < 0, which, combined with
Lemmas 1 and 2, prove the following proposition.

Proposition 2 The land rent is higher in a larger city, where an individual consumes
less land and has less children ( ∂rt /∂Nt > 0, ∂dt /∂Nt < 0 and ∂nt /∂Nt < 0).
These relationships are consistent with the stylized facts presented in Introduction andin Simon and Tamura [12].

10

2.3

Population dynamics

We assume that there is migration into or out of the city depending on the utility
diﬀerence between inside and outside of the city. This migration happens just before¡¢
each period starts, and M V t − v individuals who are ready to become adults flows
into the city just before period t + 1 starts, where v (> 0) is the utility level of people

outside of the city and M represents the adjustment speed of migration.4 We assume
that v and M are positive constants. This migration function represents that the city
attracts people if people there enjoyed higher utility than people outside of the cityin the previous period, and the city loses people otherwise. We further assume that¯V t ¯xt =0 > v. This implies that the land rent becomes suﬃciently low for people to flow

into the city if there are few people in the city.56

In period t, an individual residing at x has nt (x) children, who are grown up to beparents in period t + 1. Therefore, the law of motion of population is given by¡¢
Nt+1 = M V t − v +Z xt0

Dnt (x)dx.
dt (x)(7)

The first term represents the flow into/out of the city and the second term is the total
number of children in the previous period.
The following proposition establishes the existence of the steady state equilibrium
4

For the discussion on the stability of spatial equilibrium under this type of migration function, see

Tabuchi and Zeng [13].
5Remind that xt (0) = 0.

6These assumptions are equivalent to A + ln (I) − β ln (bI + ε) > v, which is satisfied when theincome I is suﬃciently high.

11and the suﬃcient condition of its uniqueness and stability.

Proposition 3 There exists a steady state equilibrium of the model. It is unique andstable when b and β are suﬃciently small.

Proof. See Appendix C.

3

Numerical analysis

In this section, we provide comparative steady state analysis by calibration. In so doing,
we fix parameter values so that the model can replicate demographic characteristics ofTokyo Metropolitan Area in Japan for the past half century. As shown in Introduction,
population of Tokyo MA have grown steadily during the past half century. Because ourmodel does not have growth factor, it would be natural to eliminate trend components
from the population data when fitting our model to the data.
Although Tokyo MA have grown steadily during the past half century, its population size has shown slight fluctuations, which is confirmed by decomposition into trend
components and cyclical components. We use Hodrick-Prescott filter for the decomposition.7[Insert Figure 2 around here]

Figure 2 shows the trend and cyclical components of ln(population). By eliminating
7

We set the multiplier λ = 400, which is often used for annual data.

12

the trend components, we can observe that city size slightly fluctuates for Tokyo MA.
We use the cyclical components of ln(population) of Tokyo MA.
α is set to be 0.62, which comes from the share of consumption expenditure net oftransportation and education costs in the disposable income for the year 2007 (Annual
Report on the Family Income and Expenditure Survey, Ministry of Internal Aﬀairs andCommunications, Japan). This implies that β = 0.19. We set the level of nominal income I to be 3693.1, which is made as follows. First, we obtain the per household incomeof Tokyo MA from 1955 to 2005 for every 5 years by using Gross Prefectural Domestic
Income (Prefectural Accounts, Cabinet Oﬃce, Japan) and the number of households(Population Census, Ministry of Internal Aﬀairs and Communications, Japan). I is thencalculated as the geometric mean of the series of the per household income.8 Other parameters are set so that they can satisfy the following conditions: (i) ² ≥ βbI/α, underwhich nt 0 (x) > 0 for all x < xt , (ii) −1 < Ψ < 0 < Φ < 1 (see Appendix C for thedefinitions of Ψ and Φ), which ensure the existence and stability of the steady state, (iii)
the average share of commuting expenditure in income (= τ x/I ) does not exceed 0.1,
which comes from the fact that the share of transport related expenditure in the disposable income is around 0.06 (Annual Report on the Family Income and Expenditure
Survey, Ministry of Internal Aﬀairs and Communications, Japan), (iv) we assume thateach generation takes ten years and the series of simulated population for six generations can replicate the observed variance of the cyclical components of ln(population)
8

justification to be written

13

of Tokyo Metropolitan Area from 1950 to 2007.9
We normalize the land supply D of each location to be one. The time and land
requirements b and ε for child rearing are set to equal to 0.19 and 750, respectively.
These values ensure (i) and (ii) to hold true. (iii) requires the per distant commuting
cost τ to be 0.05. In order for (iv) to hold true, we set the utility level v of the outsideof the city and the adjustment speed M of migration to 5.75 and 69.405, respectively.
The following figure shows the simulated fluctuations in ln(population) around thesteady state value of it for six generations. The initial city population N0 is 7.4175,
which is ln(population) of Tokyo MA for the year 1950.10[Insert Figure 3 around here]

We see from Figure 5 that the economy almost converges to the steady state at t = 6.
Moreover, from generation 0 to 6, the city population fluctuates around the steadystate, and the degree of fluctuation is decreasing. These features are consistent withthe cyclical components of ln(population) of Tokyo MA.
Some comparative steady states are provided below. Table 3 summarizes the results.

[Insert Table 3 around here]

An increase in the share α of numeraire consumption expenditure decreases thefertility rate nt . Still, the city grows both in terms of population Nt and space xt
910

The variance of the cyclical components is 1.6 × 10−5 .
The simulated steady state value of ln(Nt ) is 1.993.

14

because of increases in the migration into the city. Accordingly, the land rent rt rises
and land consumption dt decreases.
An improvement in the city transportation system (a decline in τ ) increases the
migration into the city and enlarges the city size (Nt and xt ). In the neighborhood of
the CBD, rt declines, and nt and dt increase although the opposite holds true in thesuburbs.
A reduction in the land requirement ε for child rearing increases dt but decreases nt
and the migration into the city, leading to a smaller city size and lower rt .
An improvement in the environment of child rearing (a decline in b) increases nt
and migration into the city, which enlarges the city size and higher rt .
Although a higher income I reduces nt , a better job opportunity induces more people
to move into the city. The latter eﬀect dominates the former, leading to a larger city
size. This raises rt to reduce dt .
An increase in the land supply D for each location increases both nt and migration
into the city, which leads to a larger city population. However, a larger land supply
enable more people to reside at each location, and the city fringe xt declines, implying
decreases in the city size in terms of space. In accordance with this, rt decreases anddt increases.
A decrease in the utility level v outside of the city and an increase in the adjustment
speed M of migration have qualitatively the same eﬀects. They decrease nt but increase
the migration into the city. The latter dominates the former, leading to a larger city
size, a higher rt , and lower dt .
15

4Concluding remarks

This paper provided a model with which we can analyze the interaction between city
structure and demographic factors. In the developed model, we suppose the monocentric city structure and the complementarity between land consumption and having
children. We showed conditions that ensure the existence and uniqueness of a steadystate, and characterized the steady state. We further provided comparative steady state
analysis by calibration.
It’s worth pointing out an important possible direction of future research. Since
our model does not have any growth factor, it cannot replicate the steady growth of
metropolitan areas in Japan. Including growth factors such as human capital accumulation or technological progress would enables us to uncover possible impacts of city
structure on demography in such a growth path.

References[1] Alonso, W., 1964, Location and Land Use, Cambridge, MA, Harvard University
Press.
[2] Becker, G. S., 1965, A Theory of the allocation of time, Economic Journal 75,
493-517.
[3] Eckstein, Z., S. Stern and K. I. Wolpin, 1988, Fertility choice, land, and the Malthusian hypothesis, International Economic Review 29, 353-361.
16

[4] Fujita, M., 1989, Urban economic theory: Land use and city size, Cambridge University Press, Cambridge.
[5] Galor, O., 2005, From Stagnation to Growth: Unified Growth Theory, in the
Handbook of Economic Growth (P. Aghion and S. Durlauf eds.).
[6] Kanemoto, Y., 1980, Theories of urban externalities, North-Holland, Amsterdam.
[7] The National Institute of Population and Social Security Research, Japan, 11th
National Fertility Survey (Dai 11-kai Syussei Doukou Kihon Chosa), 2003.
[8] Sato, Y., 2007, Economic geography, fertility, and migration, Journal of Urban
Economics 61, 372-387.
[9] Sato, Y. and K. Yamamoto, 2005, Population concentration, urbanization, anddemographic transition, Journal of Urban Economics 58, 45-61.
[10] Sato, Y. and K. Yamamoto, 2005, Urbanization and poverty trap, DEE Discussion
Paper No.05-2, Graduate School of Environmental Studies, Nagoya University.
[11] Schultz, T. P., 1985, Changing world prices, women’s wages, and the fertility
transition: Sweden, 1860-1910, Journal of Political Economy 93, 1126-1154.
[12] Simon, C. J. and R. Tamura, 2009, Do higher rents discourage fertility? Evidence
from U.S. cities, 1940-2000, Regional Science and Urban Economics 39, 33-42.
[13] Tabuchi, T. and D.-Z. Zeng, 2004, Stability of spatial equilibrium, Journal of
Regional Science 44, 641—660.
17

[14] Zhang, J., 2002, Urbanization, population transition, and growth, Oxford Economic Papers 54, 91-117.

Appendix A: Proof of Lemma 2.
Let Ω(xt ) denote the left hand side of (6). Ω(xt ) has the following properties:Ω(0) = 0,lim Ω(xt ) = ∞,xt →I/τDΩ (xt ) =+dt (xt )
0

Z xt
0

D
dt (x)2µ¶∂dt ∂rt−dx > 0.
∂rt ∂xt

Hence, (6) determines xt once Nt is given as described in the following figure.

[Figure A1 around here]

From this figure, we readily know thatx0t (Nt ) > 0,xt (0) = 0,
lim xt (Nt ) =

Nt →∞

18

I
.
τ

Appendix B: Proof of Proposition 1.
Substituting (4) into (1) and diﬀerentiating them with respect to x, we have that forx ∈ (0, xt ],

³
´1/β
I−τ x
τ (bI)2 (1 − β) + 2αετ (bI + ε) I−τ
xt> 0,d0t (x) =
´1/β
³
I−τ x
(bI + ε) I−τ xt[bI + 2εrt (x)]n0t (x) =ετ (1 − 2β) rt (x) − βτ bI.
[bI + εrt (x)] [bI + 2εrt (x)]The latter equation yields
n0t (x) > 0⇔

rt (x) >βτ bI.
ετ (1 − 2β)

From the fact that rt (x) ≥ 1, this leads to
n0t (x) > 0

⇐

1>

βτ bI
ετ (1 − 2β)

⇔

ε>

βbI
.
α

Appendix C: Proof of Proposition 3.
Note here that we can determine all the other variables in period t once we fix Nt andthat the law of motion of population (7) determines Nt for a given Nt−1 . Therefore, wehave a steady state equilibrium if there exists a steady state value of Nt . It is, if any,
given by the intersection of Nt+1 = Nt and (7).
(7) is rewritten as
Nt+1 = Λ(Nt ),¡¢Λ(Nt ) ≡ M V t − v +
19

Z xt
0

Drt (x)dx.
bI + 2εrt (x)From (5), (4) and the results that xt (0) = 0 and limNt →∞ xt (Nt ) = I/τ , we have´
³ ¯
Λ(0) = M V t ¯xt =0 − v > 0,lim Λ(Nt ) = −∞.

Nt →∞These establish the existence of at least one intersection of Nt+1 = Nt and (7).
Moreover, the uniqueness and stability is ensured if −1 < Λ0 (Nt ) < 1. We readily
obtainΛ0 (Nt ) = ∆x0t (Nt ),
Z xt
τM
D
−
+
Ωdx,
∆ ≡
bI + 2ε I − τ xt
0
∂rt (x)DbI.
Ω ≡2[bI + 2εrt (x)] ∂xt
Note that
0 ≤ Ω

(A1)
1/β

(I−τ x)τ bID (bI + ε) (I−τxt )1+1/β=∙´1/β ¸3/2³
2I−τ xβ (bI) + 4ε (bI + ε) I−τ xt<τ bID (I − τ xt )1/(2β)−1β (4ε)3/2 (bI + ε)1/2 (I − τ x)1/(2β)
τ bID
.
≤
3/2
β (4ε) (bI + ε)1/2 (I − τ xt )Note next that (6) yields
0 < x0t (Nt )(=<D (bI + ε)+β (I − τ xt ) (bI + 2ε)β (I − τ xt ) (bI + 2ε).
D (bI + ε)

Z xt
0

(A2)
)−1

2ε2 D [rt (x)]2 + bID [bI + 2εrt (x)] ∂rt (x)dx∂xtβ (I − τ x) [bI + 2εrt (x)]220

Consider the case in which ∆ ≥ 0. From Lemma 2, we have0 ≤ Λ0 (Nt )
#
"
Z xt
τ bID
D
τM
dx x0t (Nt )
+
+
< −
3/2
1/2
I − τ xt bI + 2ε
β (4ε) (bI + ε) (I − τ xt )0#"
β (I − τ xt ) (bI + 2ε)τ bIDxt
D
τM
+
+
< −
3/2
1/2
I − τ xt bI + 2ε β (4ε) (bI + ε) (I − τ xt )D (bI + ε)
(
"
#)
bI 2
τM
β
I + (bI + 2ε)
.
−
=
3/2
1/2
bI + εDβ (4ε) (bI + ε)Consider next the case in which ∆ < 0. In this case, we obtain0 > Λ0 (Nt )¶µβ (I − τ xt ) (bI + 2ε)τMD−>bI + 2ε I − τ xt
D (bI + ε)∙¸
τ M (bI + 2ε)βI − τ xt −=bI + εDβτ M (bI + 2ε).
≥ −D (bI + ε)Let Φ and Ψ denoteΦ ≡βbI + εΨ ≡ −(I + (bI + 2ε)

"

bI
3/2

β (4ε)

2
1/2(bI + ε)

−

τM
D

#)

,

βτ M (bI + 2ε).
D (bI + ε)

We have −1 < Λ0 (Nt ) < 1 if −1 < Ψ < 0 < Φ < 1. From the facts that lim(b,β)→(0,0) Φ =lim(b,β)→(0,0) Ψ = 0, we have Proposition 3.

21

Tokyo MA
year

Central city

Suburbs

Tokyo

Saitama

Chiba

Kanagawa

1930

3.51

5.33

5.05

4.34

1950

2.73

3.92

3.59

3.25

1960

1.7

2.16

2.13

1.89

1970

1.96

2.35

2.28

2.23

1980

1.44

1.73

1.74

1.7

1990

1.23

1.5

1.47

1.45

2000

1.07

1.3

1.3

1.28

Osaka MA
year

Central city

Suburbs

Osaka

Kyoto

Hyogo

Nara

1930

3.21

3.59

3.94

4.39

1950

2.87

2.8

3.08

3.08

1960

1.81

1.72

1.9

1.87

1970

2.17

2.02

2.12

2.08

1980

1.67

1.67

1.76

1.7

1990

1.46

1.48

1.53

1.49

2000

1.31

1.28

1.38

1.3

Nagoya MA
year

Central city

Suburbs

Aichi

Mie

Gifu

1930

4.6

5.01

5.47

1950

3.27

3.33

3.55

1960

1.9

1.95

2.04

1970

2.19

2.04

2.12

1980

1.81

1.82

1.8

1990

1.57

1.61

1.57

2000

1.44

1.48

1.47

Table 1: Total fertility rate in three largest metropolitan areas in Japan.

Centralcity
Tokyo MA
Osaka MA
NagoyaMATokyo
1081.7
Osaka
749.4
Aichi
427.4

Suburbs
Saitama Chiba
525.3
Kyoto
538.2
Mie

Kanagawa

417.3

796.9

Hyogo Nara
487.5

422.4

Gifu
227

246.6

Table 2: Value of land for housing per 3.3 ㎡ (in thousand yen) in three largest
metropolitan areas in Japan for the year 2000.

changes in endogenous variables
parametersNNet
migration

x

c

d

n

r

α
ε

+

+

+

+

－

－

+

－

－

－

0

+

－

－

τ

－

－

－

－

C: －

C: －

C:

S:

S:

S: －

b

－

－

－

0

D

+

+

－

0

I

+

+

+

M

+

+

v

－

－

+

C: －

+

－

－

+

+

－

+

－

－

+

+

0

－

－

+

－

0

+

+

－

S:

+

Table 3: Comparative steady states.
+ and－describe that the variable increases and that it decreases, respectively.
C and S denotes the central city and the suburbs, respectively.

+

Figure 1: Population of three largest metropolitan areas in Japan.

Nt’

Nt

0

-x (N ) tt xt(Nt’)-xt

Figure 2: Determination of the city fringe xt for a given number of individuals N t .

Figure 3: Trend and cyclical components of ln(population ) in Tokyo MA in Japan.

Figure 4: Simulated population.