Discussion Papers In Economics
And Business

A model of urban demography

Hiroshi Aiura

Yasuhiro Sato

Discussion Paper 09-18-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA model of urban demography

Hiroshi Aiura

Yasuhiro Sato

Discussion Paper 09-18-Rev.

November 2009

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A model of urban demography∗
Hiroshi Aiura†

Yasuhiro Sato‡

October 31, 2009

Abstract
This paper develops an overlapping generations model that involves the endogenous determination of fertility and an explicit city structure in order to analyze fully the social and naturalchanges in city populations. We provide conditions under which the model exhibits the spatial
features of demography observed in large Japanese cities. We also show by calibration thatthe low cost of obtaining human capital in Tokyo metropolitan area played a significant role inestablishing its urban primacy in Japan.

Keywords: urbanization, demography, migration, monocentric city
JEL Classification: J11, R11, R14, R23

1

Introduction

It has long been recognized that the structure of a city bears a relationship with the demography
within its areas. Statements on this issue are easily accessible: for example, the National Institute ofPopulation and Social Security Research of Japan [9] conduted a survey on the number of childrenthat couples are actually going to have and the number of children they would have wished to haveunder ideal conditions. Table 78 of this survey reports on the reasons why couples are going tohave fewer children than the ideal number. The table shows that whereas 16.8 percent of couples
who live in Densely Inhabited Districts (DIDs) chose the unaﬀordability of having a suﬃciently
spacious house as one of the reasons, only 5.4 percent of the couples who live in non-DIDs cited
this reason. These figures indicate that the city structure, through high land rents, may have asignificant impact on fertility within its areas.
It is also well known that the growth of cities can be traced to two sources − social and natural

population changes. The social changes represent the flows of populations into/out of the city∗

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.jp1whereas the natural changes describe the diﬀerence between the numbers of births and deaths
within the city. In order to understand what factors are important in making cities grow to bemegalopolises, it is necessary to obtain a tool that can deal with the city structure and its social
and natural population changes in a unified framework.
However, full-fledged analyses on the relationship between cities and their demographies fromthe viewpoint of economics have been scarce until recently. An noticeable exception is Shultz [13],which provides empirical results that support the interdependence of city structure and demography
by showing that advances in urbanization reduce the national total fertility rate.
Very recently, several studies uncovered possible nonnegligible interactions between city structure and demography. As for empirical evidences, Galor [5] presented stylized facts that imply(i) that urbanization and economic development started simultaneously, and (ii) that in the earlyphases of urbanization and economic development, the population growth rate had increased but
had then declined over succeeding years. Simon and Tamura [14] showed the existence of a negative
cross-sectional correlation between the price of living spaces as measured by rent per room andthe fertility rate for the Consolidated Metropolitan Statistical Areas in the United States over theperiod 1940—2000. The results described in these studies imply that city structure − including theland and housing markets − can play a major role in determining the demographic features of cities.
Theoretically, Zhang [16], Sato [10], Sato and Yamamoto [11], and Sato and Yamamoto [12] used

reduced form models to analyze the demographic impacts of urbanization .1 Zhang [16] and Sato andYamamoto [12] examined the impact of urbanization, caused by better opportunities for earnings
and education in the cities, on demography. Sato [10] and Sato and Yamamoto [11] investigated how
urbanization and demographic transition interrelate with each other via the merits of population
concentration (agglomeration economies) and its demerits (congestion diseconomies). All thesestudies showed that urbanization is accompanied by a decline in fertility, which is consistent withthe stylized facts described in Galor [5]. However, since the models developed in these studies haveno explicit spatial structure, their analysis does not shed any light on the spatial features of thedemography within cities.
In this paper, we contribute to this body of literature by developing an overlapping generations
model of endogenous fertility that involves explicit spatial structure. In particular, we focus on therelationship between the city size, the spatial patterns of fertility within the city, and the patternsof land consumption. For this purpose, we construct an overlapping generations model of endogenous fertility that involves the monocentric city structure à la Alonso [1]. In considering fertility
decisions, we follow the views of Becker [2], which regards having children as consumption, and not
as investment. Each household obtains utility from numéraire consumption, land consumption, and1

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

and examined the impact of the limitation of land availability on economic growth. However, their model does notdeal with land consumption or city structure.

2the number of its children. 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, andobtains utility from land consumption over the required level for child rearing. Population changes
arise not only from changes in the number of children (natural changes) but also from migration
into/out of the city (social changes). In such a model, the land rent is higher in the central partof the city, leading to lower land consumption and fewer numbers of children. Moreover, as thecity grows, the land rent gets higher, and land consumption and fertility diminish. These features
are consistent with the stylized facts that we shall describe in the next section and those shown in
Simon and Tamura [14].
By introducing human capital investment, we extend the basic model so that it can describe thegrowth of a city. We calibrate this extended model to replicate changes in population and demographic characteristics of three of Japan’s largest metropolitan areas (Tokyo, Osaka and Nagoya,
where Tokyo, Osaka and Nagoya are the primary, secondary and tertiary cities, respectively) during
the last half century. Based on these calibrated models, we explore the factors that have been significant in establishing the urban primacy of Tokyo. This is done by executing counterfactuals on
Osaka and Nagoya: we investigate what would happen if the value of one of the parameters of the
Osaka or Nagoya model took on that of the Tokyo model. Such counterfactuals reveal that Osakaand Nagoya could have grown as much as (or even more than) Tokyo had human capital investmentin these cities costed as little as it did in the case of Tokyo. Moreover, we find that the growth of
Osaka has been seriously limited by shortage of land supply, whereas Nagoya has the potential togrow more provided that its intra-city commuting infrastructure improves.
The remainder of this paper is organized as follows. Section 2 presents stylized facts on population changes and demographic characteristics of the three largest cities in Japan. Section 3 develops
the basic model and describes its properties. Section 4 extends the basic model to incorporate growth
factors and provide numerical analysis. Section 5 concludes.

2

Some stylized facts

This section provides an overview of basic stylized facts on the demography of metropolitan areasin Japan. We use data on the three largest Metropolitan Areas (MAs), which are Tokyo MA, Osaka
MA, and Nagoya MA.2 Tokyo, Osaka, and Nagoya are the first, second, and third largest cities in2Here, each MA consists of several prefectures:Tokyo MA=Tokyo+Kanagawa+Saitama+Chiba, Osaka

MA=Osaka+Kyoto+Hyogo+Nara, and Nagoya MA=Aichi+Gifu+Mie. These definitions are often used in analysis by the Ministry of Internal Aﬀairs and Communications, Japan. Although these are very coarse definitions, they
are superior to other definitions, taking the availability of various data into account.
Even if we were to use another definition of MAs, these three cities would always emerge as largest though we would
have diﬀerent sizes. For instance, by the definition of Urban Employment Area (UEA), the population size of Tokyo
UEA is 33.4 million UEA, that of Osaka UEA is12.1 million, and that of Nagoya UEA is 5.2 million (see Kanemoto3Japan. Figure 1 shows their sizes in terms of population from 1950 to 2007.

[Insert Figure 1 around here]

The three cities have grown steadily, while keeping the order of their sizes unaltered. In 2007, thesize of Tokyo’s population was 34.8 million, Osaka’s was 18.4 million, while Nagoya had a populationof 11.3 million.
While the three cities have grown steadily during the past half century, their population sizes
have undergone some fluctuations, which is confirmed by the decomposition into their trend andcyclical components. We use the Hodrick-Prescott filter for this decomposition.3[Insert Figure 2 around here]

Figure 2 shows the trend (Figure 2-(a)) and cyclical (Figure 2-(b)) components of ln(population).
By eliminating the trend components, we can observe that each of three MAs has undegone a slightfluctuation in its size. Moreover, we know that Tokyo and Osaka have similar levels of cyclicalfluctuation whereas Nagoya has them to a lesser extent.
Such changes in population may originate from two kinds of sources: social and natural changes.
The former represents the to and fro migration of people and the latter represents the fertility
decisions of city residents. In this figure, we disregard the eﬀects of longevity, which we discuss indetail later. Figure 3 describes the social changes in the three MAs.

[Insert Figure 3 around here]

We know that these cities experienced enormous in-migration from other regions in the periodaround 1950-1970. After that, Tokyo has had a slight but steady rate of in-migration, Osaka has
experienced a slight net out-migration, and Nagoya has had almost zero net migration.
The total fertility rate is considered to be the primary source of the natural changes. Table 1
shows the total fertility rate for all prefectures that constitute our three MAs over the past half
century.

[Insert Table 1 around here]
and Tokuoka [7]).
3
We set the multiplier λ = 400, which is often used for annual data.

4

This table shows that the total fertility rate in all three MAs has steadily declined and that it islower for a larger city. Moreover, the three cities have a common spatial feature in that fertility islower in prefectures that are considered to be the central cities than for those that are considered
as suburbs. As is well known, the land rent/price is higher in a larger city, and land in the centerof a city has a higher rent/price than in the suburbs (see Table 2, for example).

[Insert Table 2 around here]

Put diﬀerently, the level of land rent is negatively associated with fertility both across cities andwithin each city. Since the level of land consumption is positively associated with the level of land
rent, it implies that land consumption and fertility have a positive correlation.

3

Basic model

3.1

Individuals

Consider a linear space on which there is one Central Business District (CBD), that is, we assume alinear monocentric city. We approximate the CBD by a point and assume that all workers commuteto the CBD. Without loss of generality, we assume that the residential area spreads only on theright hand side of the CBD.4 We index the location of the CBD as 0, and describe each location by
the distance x between it and the location of the CBD.
Time is discrete and each individual lives for two periods; a childhood and a parenthood. Eachindividual has a single parent. In the parenthood period, each individual is endowed with one
unit of time, which she spends on working and on child rearing. At the beginning of period t, shedecides on goods and land consumption (ct and dt ) and the number of her children (nt ). She exits
the economy at the end of period t. Nt individuals in the parenthood live in the city in period t.
This implies that nt Nt children are born in period t and grow to be parents in period t + 1. In thismodel, nt represents the total fertility rate.
We assume that individuals have an identical utility function of the Cobb-Douglas form and theutility of each individual depends on the level of goods and land consumption and on the numberof children:
Ut = α ln ct + β ln nt (dt − εnt ) ,(1)

where α, β and ε are positive constants and satisfy α + 2β = 1. There is only one kind of goods inthis economy, which we treat as a numéraire. In order to rear a child, one needs a certain amount ofland and we represent it as ε. An individual obtains utility from land consumption that is over the
4

Even if we suppose that the residential area spreads on both sides of the CBD, the results remain unaltered.

5level required for child rearing. This complementarity between land consumption and the numberof children brings forth the spatial characteristics of demography in a city.
In order to have nt children, each individual must spend bnt units of time, where b 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 on working. The
budget constraint for a working individual who resides at x distance from the CBD is given by(1 − bnt )I − τ x = ct + rt (x)dt ,(2)

where I denotes the wage income per unit of time and is a positive constant.5 τ represents thecommuting cost per unit distance and rt (x) is the market land rent at x distance from the CBD.
Although individuals are assumed to be price takers and take rt (x) as given, rt (x) is endogenously
determined later. We assume that land is owned by absentee landlords.6
The utility maximization gives the following demand functions:ct (x) = α (I − τ x) ,β
(I − τ x) ,nt (x) =bI + εrt (x)¶µε1dt (x) = β+(I − τ x) .
rt (x) bI + εrt (x)(3)

This leads to the indirect utility as follows:
Vt (x) = A + ln (I − τ x) − β ln rt (x) (bI + εrt (x)) .

(4)

where A is defined as A ≡ α ln α + 2β ln β.

We can see from (3) 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 the right 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. More importantly, a higher land rent leads toa smaller number of children and to smaller land consumption. This is because when rt (x) is high,the land requirement for child rearing forces individuals to have less children.

3.2

City structure and location equilibrium

We assume that children live with their parents, and that a working individual can move freely
within a city. The location equilibrium is attained in each period, and it requires that the indirect
utility be the same for all locations in a city:
Vt (x) = V t ,5∀x ∈ (0, xt ],(5)

In a later section, we explicitly endogenize the income by introducing a production structure and human capital

investment.
6
The assumption of absentee landlords is very standard in urban economics. See Fujita [4], for example.

6

where xt denotes the city fringe in period t and represents the spatial size of the city.
We follow the well-established tradition of urban economics in determining the market land rent
in a city by using the concept of "bid rent."7 The bid rent is the maximum land rent at location xthat each individual is willing to pay in order to reach her equilibrium utility level. We normalize theland rent outside of the city to one. This implies that the land rent at the city fringe rt (xt ) is equal to
1. From (4) and (5), we obtain the bid rent at x by solving A+ln (I − τ x)−β ln R (bI + εR) = Vt (xt )with respect to R, which yields
⎡
⎤
s
¶1/β
µ
I
−
τ
x
1 ⎣
⎦.
−bI + (bI)2 + 4ε (bI + ε)R=2εI − τ xt

The market land rent is then given by

rt (x) = max [R, 1]½
R if x ∈ (0, xt ].
=1 if x > xt

(6)

We readily obtain rt0 (x) < 0 and ∂rt /∂xt > 0 for x ∈ (0, xt ], which we summarize in the 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).
These are very standard results in the literature on monocentric models (Fujita [4]).
As shown in (3), diﬀerences in the land rent yield diﬀerences in the land demand and the fertility
rate. Substituting (6) into (3) and diﬀerentiating them with respect to x, we have that for x ∈ (0, xt ],
d0t (x) =n0t (x) =

´1/β
³
I−τ x
τ (bI)2 (1 − β) + 2αετ (bI + ε) I−τ
xt
> 0,
´1/β
³
I−τ x
(bI + ε) I−τ xt(bI + 2εrt (x))(7)ετ (1 − 2β) rt (x) − βτ bI.
(bI + εrt (x)) (bI + 2εrt (x))

The latter equation in (7) yields
n0t (x) > 0⇔

rt (x) >1>βτ bIετ (1 − 2β)βτ bI.
ετ (1 − 2β)

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

⇐

⇔

ε>

βbI
.
α

(8)

From (7) and (8), we obtain the following proposition.
7

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.

7

Proposition 1 An individual residing more distant from the CBD consumes more land. If land
requirement ε for child rearing is suﬃciently large, she has more children.
The location of an individual has two eﬀects on her land consumption and on the number of herchildren. On the one hand, when an individual lives more distant from the CBD, she must bearhigher commuting costs, which reduces the income net of commuting costs. This has an eﬀect of
reducing land consumption and the number of children. On the other hand, the more distant fromthe CBD the location is, the lower the land rent is. A lower land rent enables an individual toconsume more land. This also induces her to have more children because a lower land rent implies
lower payments for the land needed for child rearing. With respect to land consumption, the lattereﬀect dominates the former, and one who lives farther away from the CBD always consumes more
land. This also holds true with regard to the fertility rate if the land requirement for child rearing is
suﬃciently large. These arguments provide one possible explanation for the stylized facts describedin Table 1.
From (6), we obtain the level of utility in the city as
V t = A + ln (I − τ xt ) − β ln (bI + ε) .

(9)Therefore, for a given number of individuals Nt in 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)(10)

D represents the land supply for each location that is exogenous in our model.
Letting xt (Nt ) denote xt that is determined by (10), we can examine x0t (Nt ), that is, how thecity population aﬀects the city fringe. Let Ω(xt ) denote the left hand side of (10). Ω(xt ) has thefollowing properties:Ω(0) = 0,lim Ω(xt ) = ∞,xt →I/τ0Ω (xt ) =D+
dt (xt )

(11)
Z xt
0

D
dt (x)2µ¶∂dt ∂rtdx > 0.
−∂rt ∂xt

As the city fringe xt increases, the market land rent rt at each location rises, which leads to a lower
land demand at each location and a larger Ω(xt ). Hence, Ω(xt ) is represented by an upward sloping
curve in the xt − Nt plane, as described in Figure 2.
[Figure 2 around here]8Since the right hand side of (10) is described by a horizontal line, we can see that xt is uniquelydetermined once Nt is given.
Now consider increases in the city population from Nt to Nt0 where Nt0 > Nt . Upward shifts of
the horizontal line lead to a higher xt . Moreover, we know from (11) the values of xt when Nt takes
extreme values (i.e., Nt is equal to 0 or converges to ∞).
x0t (Nt ) > 0,

(12)xt (0) = 0,I.
lim xt (Nt ) =Nt →∞τThe following lemma summarizes the results shown in (12).
Lemma 2 An increase in the number of individuals in the city enlarges the city area. There is nocity area if no one is in the city, and the city fringe can at most reach I/τ even if the city
population explodes.
Equation (3) shows that the city population size Nt aﬀects the land demand dt and the fertility
rate nt only through changes in the land rent rt , and that ∂dt /∂rt < 0 and ∂nt /∂rt < 0, which,combined with Lemma 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).
As seen in a standard monocentric city model, an increase in city population raises the land rent,
which depresses the land demand. Due to the land requirement for child rearing, this leads to alower fertility rate. Such relationships are consistent with the stylized facts presented in Section 2(tables 2 and 3) and in Simon and Tamura [14].

3.3

Population dynamics

We assume that there is migration into or out of the city depending on the diﬀerence in utility insideand outside the city. Such migration happens just before each period starts. More specifically,¢¡
M V t − v individuals who are ready to become adults flow 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.8 We assume that v and M are positive constants. This migration function
represents that the city attracts people if the people there enjoyed higher utility than the people
outside the city in the previous period; otherwise, the city loses people. We further assume that8

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

Zeng [15].

9¯V t ¯xt =0 > v. This implies that the land rent becomes suﬃciently low for people to flow into thecity if there are few people in the city.9 10

In period t, an individual residing at x has nt (x) children, who grow up to be parents in the

period t + 1. Therefore, the law of motion of population is given by
Z xt¡¢
Dnt (x)dx.
Nt+1 = M V t − v +dt (x)
0(13)

The first term represents the flow into/out of the city and hence the social changes in the city
population. The second term is the total number of children in the previous period, which represents
the natural changes in the city population. In this setting, the city has inflow of people if and only
if V t > v, which is equivalent to xt < [I − (bI + ε)β exp[v − A]]/τ . This requires that the distancefrom the CBD to the city fringe (the city area) is suﬃciently small compared to the real income

denominated by the commuting cost.
Both terms are functions of Nt and hence (13) can be described as

Nt+1 = Λ(Nt ),Z xt¡¢
Dnt (x)dx.
Λ(Nt ) ≡ M V t − v +dt (x)
0(14)

Since we can determine all the other variables in period t once we fix Nt and the law of motion of
population (14) determines Nt+1 for a given Nt , we have a steady state equilibrium if there exists
a steady state value of Nt . It is, if any, given by
N ∗ = Λ(N ∗ )(15)Let’s define Φ and Ψ as
Φ ≡βbI + ε

Ψ ≡ −

(

"

bI 2

τM
I + (bI + 2ε)
−
3/2
1/2
D
β (4ε) (bI + ε)

#)

,

(16)

βτ M (bI + 2ε).
D (bI + ε)

The following proposition establishes the existence of N ∗ and hence the steady state equilibrium,and provides the suﬃcient condition of its uniqueness and stability.
Proposition 3 There exists a steady state equilibrium of the model. It is unique and stable whenΨ > −1 and Φ < 1. These inequalities are satisfied when, for example, b and β are suﬃciently
small.

9
10

Remember that xt (0) = 0.
These assumptions are equivalent to A+ln (I)−β ln (bI + ε) > v, which is satisfied when the income I is suﬃciently

high.

10

Proof. See Appendix A.
Figure 2-(b) shows that the city population size fluctuates around the trends. We can obtain
suﬃcient conditions under which such fluctuations are possible in our model as a converging pathto the steady state. Let’s define Ξ as
µ
¶
DI
bDI 2
1
+
.
Ξ≡
τ bI + 2ε β(4ε)3/2 (bI + ε)1/2
Then we have the following proposition.
Proposition 4 Suppose that Ψ > −1 and Φ < 1. If the adjustment speed of migration is suﬃciently
high(i.e., M > Ξ), then the converging path to the steady state shows fluctuations around thesteady state.
Proof. See Appendix B.
This proposition implies that the population size fluctuates when the level of responses in terms of
the inflow/outflow of city population to changes in the utility diﬀerential is suﬃciently high.

4

Model with human capital

4.1

Introduction of human capital

In order to capture the continuous growth of metropolitan areas, we now introduce human capital
investment into our basic model. There are various ways of introducing human capital into ourmodel, and we adopt the most simple way. Each individual, in her parenthood, decides the level
ht of her human capital in addition to goods and land consumption (ct and dt ) and the numberof her children (nt ). We assume that she needs to spend Shσt in terms of numéraire in order toobtain human capital of level ht , where S > 0 and σ > 1, and that a worker whose level of humancapital is ht obtains the wage income wt ht . wt is the wage rate in the eﬃciency unit. In our
setting, the migration decision is taken before one determines the level of human capital. Of course,
there are some people who migrate after obtaining human capital, which implies the possibility ofheterogeneity of human capital among city residents. Further, the average income level within acity depends on whether people with high human capital flow into the city (See Mori and Turrini[8]). However, the model becomes highly intractable once we introduce both the human capital
accumulation within the city and a sorting possibility into the model. We assume the former only
because it is beyond the scope of our paper to develop a model that includes everything at a time.
Whereas the utility function is still given by (1), the budget constraint is now(1 − bnt )wt ht − τ x = ct + rt (x)dt + Shσt .
The resulting demand functions are determined by
11

(17)

ct (x) = α(wt ht (x) − τ x − Sht (x)σ ),(18)

β(bwt ht (x) + 2²rt (x))(wt ht (x) − τ x − Sht (x)σ ),rt (x)(bwt ht (x) + ²rt (x))β
nt (x) =(wt ht (x) − τ x − Sht (x)σ ),bwt ht (x) + ²rt (x)dt (x) =Sσht (x)σ−1 = (1 − bnt (x))wt .
We know from the first three equations that the other demands are determined once ht (x) is
determined. The last equation of (18) shows that an individual who invests more (i.e., has higherht ) has fewer children (i.e., smaller nt ), which implies that our model has the standard "quantityquality trade oﬀ" of children (see, e.g., Becker [2]).
Substituting the third equation of (18) into the last one, we have
wt − Sσht (x)σ−1
βbwt
.
=
σ
wt ht (x) − τ x − Sht (x)εrt (x) + bwt ht (x)This implicitly determines the level of human capital as ht (x) = h∗t (x; wt , rt (x)), where we explicitly
write that ht depends on wt and rt (x) for later reference. Substituting this into (18), we havedemand functions that lead to the following indirect utility:
V = A + ln(wt h∗t (x; wt , rt (x)) − τ x − Sh∗t (x; wt , rt (x))σ ) − β ln rt (x)(bwt h∗t (x; wt , rt (x)) + ²rt (x)).
Location equilibrium again requires (5), which we solve to obtain the equation that implicitly
determines the bid rent function:

R =

∙
½
1
∗
− bwt ht (x; wt , R)+ (bwt h∗t (x; wt , R))2 + 4²(bwt h∗t (x̄t ; wt , 1) + ²)2²⎫¶1/β #1/2 ⎬µ∗∗σwt ht (x; wt , R) − τ x − Sht (x; wt , R)×.
⎭
wt h∗t (x̄t ; wt , 1) − τ x̄t − Sh∗t (x̄t ; wt , 1)σ(19)

The market land rent function is given by (6).

In the basic model, we ignored the production side of the economy and assumed a fixed level ofincome in the city. Here, we endogenize the production decision, in which the production function
is given by
λ 1−λ
kt ,
yt = γ(1 − e−δHt−1 )lht

(20)

where γ, δ, and λ are positive constants. λ satisfies 0 < λ < 1. Ht−1 , lht and kt represent theaggregate level of human capital in the previous period, the human capital input and the physicalcapital input, respectively. In our framework, the total factor productivity depends on the result of
the past human capital accumulation. We assume that the physical capital market is global whereas
12the human capital market (i.e., the labor market in our model) is local. Without loss of generality,
we normalize the rental price of physical capital to one. The cost function is then wt lht + kt . The
first order conditions for profit maximization areµ¶kt 1−λ)= wt ,λγ(1 − e
lht
µ ¶λ
lht
−δHt−1(1 − λ)γ(1 − e
)
= 1.
kt
−δHt−1Eliminating the human capital to physical capital ratio from these equations, we obtain the wage
rate in the eﬃciency unit.
wt = λ(1 − λ)(1−λ)/λ [γ(1 − e−δHt−1 )]1/λ .

(21)

As the level of aggregate human capital in the city goes up, the wage rate also rises although it has
upper bound of λ(1 − λ)(1−λ)/λ γ 1/λ .

The city fringe xt is determined by the land market clearing condition (10) for a given Nt . The

law of motion of population is given by (13) and the law of motion of human capital is given byZ x¯t
Dht (x)dx,(22)
Ht+1 = ζHt +dt (x)
0which implies that human capital depreciates at the rate of 1 − ζ.

The level of population and aggregate level of human capital in the city is determined by (13)and (22), respectively. Once these values are determined, the other variables are well determined:(21) determines the wage rate, the market land rent is determined by (19) and (6), and the levelof human capital is given by ht (x) = h∗t (x; wt , rt (x)), which determines the demands (18). Finally,
the city fringe is determined by (10).

4.2

Numerical analysis

Using the model developed in the previous subsection, we explore numerically the factors thatmake a diﬀerence to the features of growth in the three largest metropolitan areas of Japan. We
proceed by employing two steps: first, we seek a parameter set for each MA under which thebasic model developed in Section 3 behaves in a consistent manner with the cyclical componentsof the population changes during the last five decades for that particular MA. We determine theparameters not related to the growth components in this step. Second, we extend this parameterset so that the urban growth model developed in Section 4 can replicate the total, social and naturalchanges of population for each MA.
The first step The expenditure share α of the numéraire consumption in the disposable income
is set to be the share of consumption expenditure net of transportation and education costs in the13disposable income for each MA (Annual Report on the Family Income and Expenditure Survey,
Ministry of Internal Aﬀairs and Communications, Japan). We take the average of this figure from1980 to 2008 for each MA.
We set the level of real income I as follows. First, we obtain the per household nominal income ofeach MA from 1963 to 2008 for every 5 years by using the figures for the Gross Prefectural Domestic
Income (Prefectural Accounts, Cabinet Oﬃce, Japan) and the number of households (Population
Census, Ministry of Internal Aﬀairs and Communications, Japan). The real income is derived by
dividing the nominal income by the GDP deflator. I is then calculated as the geometric mean of
the series of the per household real income.
The transport cost τ is determined so that the average share of commuting expenditure indisposable income at the steady state becomes the observed average share of transport related
expenditure in the disposable income from 1980 to 2008 for each MA (Annual Report on the Family
Income and Expenditure Survey, Ministry of Internal Aﬀairs and Communications, Japan).
We set the land supply D to the average areas of inhabitable land in 1970,80,90,2000 and 2005,
and normalize D for Tokyo MA to be one. This leads to D = 0.67 for Osaka MA and D = 0.784
for Nagoya MA.
The time cost b of child rearing is determined based on the total fertility rate in the central cityof each MA. We set it so that the steady state value of the number of children in the CBD (nt (0))
is equal to the average total fertility rate of the central city from 1950 to 2005 for every five years.
The land requirement ε for child rearing is set so that the model can satisfy ² ≥ βbI/α, underwhich nt 0 (x) > 0 for all x < xt . This ensures that the total fertility rate is lower in the central city
than in the suburbs in each MA as seen in Table 1.
Because we observe the cyclical fluctuations around the trend components for all MAs, we choose

the adjustment speed M of migration such that the following two criteria are satisfied; (i) Ψ > −1and Φ < 1, which ensure the existence and stability of the steady state, and (ii) M > Ξ, whichleads to the fluctuations in the converging path to the steady state.
The utility level outside the city, v, determines the overall degree of social change in the city
population. We assume that each generation spans ten years, and determine v so that the seriesof simulated populations for six generations can replicate the observed variance in the cyclicalcomponents of ln(population) of each MA from 1950 to 2008.
Finally, we set the initial number of adult population N0 as the number of people over the ageof 14 in 1950 for each MA divided by 2. Appendix C reports the parameter values for the basicmodel.
The second step When setting the parameters of the urban growth model, we use the values
determined in the first step for all parameters other than v. Note here that the first step ignores thesocial changes arising from income growth in the city, which is relevant in the observed population
14

changes and in the urban growth model. Since v controls the overall degree of social population
change, the value of it determined in the first step must be modified in the urban growth model.
The additional parameters are determined as follows. The share of labor λ in the production
function comes from the average share of labor income for each MA from 1955 to 2006 (Prefectural
Accounts, Cabinet Oﬃce, Japan).
The parameters related to the human capital accumulation S, σ, and ζ are determined so thatRxt (x)2Nt + 0 t Dn
dt (x) dx can replicate the total population figures from 1950 to 2000. One parameter is
Rx
0 (x)used to adjust the initial total population 2N0 + 0 0 Dn
d0 (x) dx, one is used to determine the overall
movement of the total population, and the last parameter determines the pace of changes in the
total population.
The utility level v outside the city, the production parameters γ and δ, and the initial level ofaggregate human capital H0 are set as follows. We first set the same value of γ for all MAs at
λ k 1−λ , the term (1 − e−δHt−1 )a moderate level. In the production function yt = γ(1 − e−δHt−1 )lht
t

represents the productivity related to the human capital and γ describes the productivity withorigins other than human capital. We assume that the productivity diﬀerence comes only from theproductivity related to the human capital, which reflects the diﬀerence in access to informationand education. We then determine v, δ, and H0 to produce the observed net in-migration (social

population change) from 1950 to 2000. Here, because we assume that each generation spans ten
years, we calibrate the net in-migration for each ten-year period. Again, one parameter is used toadjust the initial level of net in-migration, one is used to determine its overall movement, and thelast parameter determines the pace of changes.
Table 3 reports the resulting parameter values for the urban growth model.

[Insert Table 3 around here]

A few comments are in order. First, the commuting cost (τ ) takes a similar value in Tokyo andOsaka, and a smaller value in Nagoya, which is considered to reflect the diﬀerences in the development of public transportation such as the subway system. Second, Tokyo has the largest land
supply (D) and Osaka has the most severe land shortage. Third, the time and land requirements
for child rearing (b and ε) are the highest in Tokyo and the lowest in Nagoya, whereas the costs ofhuman capital investment (S and σ) are the highest in Nagoya and the lowest in Tokyo. Hence,
our calibrated models imply that one is most likely to have less children and invest more in humancapital in Tokyo and is least likely to do the same in Nagoya. Finally, the adjustment speed of
migration (M ) is the highest in Tokyo and the lowest in Nagoya, which is considered to reflect thefact that Tokyo is most well connected to all regions in Japan.
The results of a numerical realization of the model are presented in Figures 5 to 7.

15

[Insert Figures 5 to 7 around here]

These figures prove that our model can replicate the observed total, social and natural changes
in the populations of the three MAs qualitatively and, often, quantitatively. (a) and (b) of eachfigure show that our model can replicate well the rapid increase in city populations between theyears 1950 and 1970, followed by the slowdown in population growth, and the corresponding net inmigration. In (c) of each figure, the calibrated series of total fertility rates in the CBD decline overtime but less so than the observed values. This is because our model does not have a component
representing gains of longevity during the past five decades. Without longevity eﬀects, in orderto support the observed population increases, we must have smaller declines in the total fertility
rate. (d) of each figure shows that the total fertility rate is lower for locations closer to the CBD.
Moreover, as a city grows (xt increases), the overall level of the total fertility rate declines.
Counterfactual analysis By using the calibrated models, we examine the factors that are significant in making a diﬀerence to the growth patterns of the three MAs. We do this by employing
a counterfactual analysis, as follows. We explore how Osaka MA or Nagoya MA would behave if
one of the parameters took on the same value as that of Tokyo MA. For example, we simulate
how Osaka MA would look like if it had the same inhabitable area as Tokyo MA. In such a case,
we use the parameter set of Osaka MA and change only D from 0.67 to 1, and simulate it for six
generations. Figures 8 and 9 describe the results of our counterfactual analysis.

[Insert Figures 8 and 9 around here]

Figures 8 and 9 show the counterfactual analysis on Osaka MA and Nagoya MA, respectively.
Osaka MA would have been as large as Tokyo MA if it were possible to obtain human capital in
Osaka for as little cost as in Tokyo (if Osaka had had the same value of S or σ). Furthermore, theresult with respect to the land supply D proves that the growth of Osaka MA might have been
limited because of the shortage of land endowment. As for Nagoya MA, the cost of obtaining humancapital (S or σ) again has a great eﬀect on the growth of the city. Here, whereas D does not have asignificant impact, the commuting cost τ does. Given the cost structure of obtaining human capital,
the production structure of Tokyo MA may be harmful to the growth of Osaka and Nagoya. Underδ, λ, and ξ of Tokyo, Osaka and Nagoya grow less than they have in reality. Other parameters
including initial population (N0 ) and human capital (H0 ) have little of no impact on city growth.
Put diﬀerently, Osaka and Nagoya would have grown as large as Tokyo if human capital could havebeen obtained as easily as in Tokyo. Smaller land endowments and larger commuting costs havealso limited the growth of Osaka and Nagoya, respectively.
16

5Concluding remarks

This paper provided a model with which we could analyze the interaction between city structuresand demographic factors. This enabled us to treat the social and natural changes in city populations
within a unified framework. In the developed model, we supposed the monocentric city structure
and a complementarity between land consumption and having children. We showed conditions
under which our model could replicate the observed demographic characteristics in the three largest
metropolitan areas of Japan. A counterfactual analysis revealed that the low cost of obtaining
human capital in Tokyo has been the major factor in establishing its urban primacy, and that otherfactors such as initial levels of population and human capital have little eﬀect on urban growth.

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.
[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] Kanemoto, Y. and K. Tokuoka, Proposals for the standards of metropolitan areas of Japan,
Journal of Applied Regional Science 7, 1-15 (in Japanese).
[8] Mori, T. and A. Turrini, 2005, Skills,agglomeration and segmentation, European Economic
Review 49, 201-225.
[9] The National Institute of Population and Social Security Research, Japan, 11th National Fertility Survey (Dai 11-kai Syussei Doukou Kihon Chosa), 2003.
[10] Sato, Y., 2007, Economic geography, fertility, and migration, Journal of Urban Economics 61,
372-387.
[11] Sato, Y. and K. Yamamoto, 2005, Population concentration, urbanization, and demographictransition, Journal of Urban Economics 58, 45-61.
17

[12] Sato, Y. and K. Yamamoto, 2009, Urbanization and poverty trap, In: L. M. De Smet (eds.)
Focus on Urbanization Trends (NOVA Science publisher).
[13] Schultz, T. P., 1985, Changing world prices, women’s wages, and the fertility transition: Sweden, 1860-1910, Journal of Political Economy 93, 1126-1154.
[14] 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.
[15] Tabuchi, T. and D.-Z. Zeng, 2004, Stability of spatial equilibrium, Journal of Regional Science44, 641—660.
[16] Zhang, J., 2002, Urbanization, population transition, and growth, Oxford Economic Papers 54,
91-117.

Appendix A: Proof of Proposition 3.
Remind here that we can determine all the other variables in period t once we fix Nt and that thelaw of motion of population (13) determines Nt+1 for a given Nt . Therefore, we have a steady state
equilibrium if there exists N ∗ that satisfies (15).
Λ(Nt ) is rewritten as¡¢Λ(Nt ) = M V t − v +Z xt

Dnt (x)dx
dt (x)0Z xt¡¢
Drt (x)dx.
= M Vt−v +bI+ 2εrt (x)0

From (6), (9) 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 N ∗ that satisfies (15).
Moreover, the uniqueness and stability is ensured if −1 < Λ0 (Nt ) < 1. We readily obtainΛ0 (Nt ) = ∆x0t (Nt ),
Z xt
D
τM
∆ ≡
−
+
Ωdx,
bI + 2ε I − τ xt
0
∂rt (x)bDIΩ ≡2 ∂x .
(bI + 2εrt (x))
t
18

(A1)

Note that
0 ≤ Ω

(A2)
τ bDI (bI + ε)=1/β(I−τ x)(I−τ xt )1+1/β∙´1/β ¸3/2³
2I−τ xβ (bI) + 4ε (bI + ε) I−τ xt
τ bDI (I − τ xt )1/(2β)−1<β (4ε)3/2 (bI + ε)1/2 (I − τ x)1/(2β)
τ bDI
.
3/2
β (4ε) (bI + ε)1/2 (I − τ xt )≤Note next that (10) yields
0 < x0t (Nt )(=<D (bI + ε)+β (I − τ xt ) (bI + 2ε)Z xt0

β (I − τ xt ) (bI + 2ε).
D (bI + ε))−1
2ε2 D (rt (x))2 + bDI (bI + 2εrt (x)) ∂rt (x)dx∂xtβ (I − τ x) (bI + 2εrt (x))2

Consider the case in which ∆ ≥ 0. From Lemma 2, we have0 ≤ Λ0 (Nt )
"
#
Z xt
τ bDI
τM
D
< −
+
+
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 bDIxt
D
τM
+
+
< −
3/2
1/2
I − τ xt bI + 2ε β (4ε) (bI + ε) (I − τ xt )D (bI + ε)
(
"
#)
β
τM
bI 2
=
−
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 )µ¶D
β (I − τ xt ) (bI + 2ε)τM>−
bI + 2ε I − τ xt
D (bI + ε)∙¸βτ M (bI + 2ε)=
I − τ xt −bI + εDβτ M (bI + 2ε).
≥ −D (bI + ε)Let Φ and Ψ denoteΦ ≡βbI + ε

Ψ ≡ −

(

"

bI 2

τM
−
I + (bI + 2ε)D
β (4ε)3/2 (bI + ε)1/2

βτ M (bI + 2ε).
D (bI + ε)
19

#)

,

We have −1 < Λ0 (Nt ) < 1 if Ψ > −1 and Φ < 1.

From the facts that lim(b,β)→(0,0) Φ =lim(b,β)→(0,0) Ψ = 0, we have Proposition 3.

Appendix B: Proof of Proposition 4.
We know from (A1) that Λ0 (Nt ) = ∆x0t (Nt ). Because x0t (Nt ) > 0, we observe fluctuations in theconverging path if ∆ < 0. Equations (A1) and (A1) show that
∆ <
=
<
=

Z xt
τM
τ bDI
D
−
+
dx
3/2
bI + 2ε I − τ xt
β (4ε) (bI + ε)1/2 (I − τ xt )0Ã!
D(I − τ xt )τ bDIxt1− τM +I − τ xt
bI + 2εβ (4ε)3/2 (bI + ε)1/2
" Ã
!
#
1
DI
bDI 2
τ
+
−M
I − τ xt τ bI + 2ε β (4ε)3/2 (bI + ε)1/2τ(Ξ − τ M ) .
I − τ xt

Therefore, we have ∆ < 0 if
M > Ξ.

Appendix C: Parameter values for the basic model.
[Insert Table C around here]20

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.

Tokyo MA

Osaka MA

Nagoya MA

α

0.64

0.66

0.65

τ

0.000110.000095

0.00021

D

1

0.67

0.78

b

0.01

0.001

0.0001

740

670

640

v

6.62

6.56

6.46

M

26647

20935

10731

2N

4466.43388.72145.4

λ

0.68

0.66

0.61

S

44

50

68

σ

1.78

1.92

2.25

ς

0.75

0.9

0.94

γ

185

185

185

δ

0.000020.000029

0.000029

H

175000140000120000

Table 3. Parameter values for the urban growth model.

Tokyo MA

Osaka MA

Nagoya MA

α

0.64

0.66

0.65

I

4876.54613.04528.9τ

0.000110.000095

0.00021

D

1

0.67

0.78

b

0.01

0.001

0.0001

740

670

640

v

6.2592

6.2885

6.249

M

26647

20935

10731

N

4466.43388.72145.4

Table C. Parameter values for the basic model.

Population (thousands)

40000
35000
30000
25000
20000
15000
10000
5000
0
1950

1960

1970

1980

1990

2000

Year

Figure 1. Population of three largest metropolitan areas in Japan. Bold line: Tokyo, solid line:
Osaka, dotted line: Nagoya.

trend components of ln(population)

17.5
17

16.5
16

15.5
15

14.5
1950

1960

1970

1980
year

1990

2000

1990

2000

(a) Trend components of ln(population).

cyclical components of ln(population)

0.015
0.01
0.005
0

-0.005-0.01-0.015
1950

1960

1970

1980
year

(b) Cyclical components of ln(population).
Figure 2. Trend and cyclical components of ln(population) of three largest metropolitan areas inJapan. Bold line: Tokyo, solid line: Osaka, dotted line: Nagoya.

Net in-migration (thousands)

450
400
350
300
250
200
150
100
50
0
-50
-100
1950

1960

1970

1980

1990

2000

Year

Figure 2. Net in-migration of three largest metropolitan areas in Japan. Bold line: Tokyo, solidline: Osaka, dotted line: Nagoya.

Nt’

Nt

0

-x (N ) tt xt(Nt’)-xt

Figure 4. Determination of the city fringe xt for a given number of individuals N t .

35000population (thousands)

30000
25000
20000
15000
10000
5000
0
1950

1960

1970

1980

1990

2000

year

(a) Gray line: observed population of Tokyo MA. Dots: simulated population of Tokyo MA.

net in-migration (thousands)

4000
3500
3000
2500
2000
1500
1000
500
0
1950

1960

1970

1980

1990

2000

year

(b) Triangles: observed net in-migration of Tokyo MA for each ten years (the first triangle
represents the figure for ten years from1950 to 1960). Dots: simulated net in-migration ofTokyo MA.

total fertility rate in the CBD

3
2.5
2
1.5
1
0.5
0
1950

1960

1970

1980

1990

2000

year

(c) Triangles: observed total fertility rate of the CBD of Tokyo MA (Tokyo prefecture). Dots:
simulated total fertility rate of the CBD of Tokyo MA (x=0).

n
1.4
1.2
1.0
0.8
0.6
0.4
0.2
5. μ 106

1. μ 107

1.5 μ 107x(d) Simulated total fertility rate within Tokyo MA. A line located higher in the graph describes
the fertility rate of an earlier generation.
Figure 5. Calibration results of Tokyo MA.

25000population (thousands)

20000

15000

10000

5000

0
1950

1960

1970

1980

1990

2000

year

(a) Gray line: observed population of Osaka MA. Dots: simulated population of

net in-migration (thousands)

Osaka MA.

1600
1100
600
100
-400
1950

1960

1970

1980

1990

2000

year

(b) Triangles: observed net in-migration of Osaka MA for each ten years (the first
triangle represents the figure for ten years from1950 to 1960). Dots: simulated net
in-migration of Osaka MA.

total fertility rate in the CBD

3
2.5
2
1.5
1
0.5
0
1950

1960

1970

1980

1990

2000

year

(c) Triangles: observed total fertility rate of the CBD of Osaka MA (Osaka
prefecture). Dots: simulated total fertility rate of the CBD of Osaka MA (x=0).

n
1.4
1.2
1.0
0.8
0.6
0.4
0.2
2.0 μ 106

4.0 μ 106

6.0 μ 106

8.0 μ 106

1.0 μ 107

1.2 μ 107x(d) Simulated total fertility rate within Osaka MA. A line located higher in thegraph describes the fertility rate of an earlier generation.
Figure 6. Calibration results of Osaka MA.

12000population (thousands)

10000
8000
6000
4000
2000
0
1950

1960

1970

1980

1990

2000

year

(a) Gray line: observed population of Nagoya MA. Dots: simulated population of
Nagoya MA.
500
net in-migration (thousands)

450
400
350
300
250
200
150
100
50
0
1950

1960

1970

1980

1990

2000

year

(b) Triangles: observed net in-migration of Nagoya MA for each ten years (the first
triangle represents the figure for ten years from1950 to 1960). Dots: simulated net
in-migration of Nagoya MA.

total fertility rate in the CBD

3.5
3
2.5
2
1.5
1
0.5
0
1950

1960

1970

1980

1990

2000

year

(c) Triangles: observed total fertility rate of the CBD of Nagoya MA (Aichi
prefecture). Dots: simulated total fertility rate of the CBD of Nagoya MA (x=0).

n
1.4
1.2
1.0
0.8
0.6
0.4
0.2
1 μ 106

2 μ 106

3 μ 106

4 μ 106

5 μ 106

6 μ 106x(d) Simulated total fertility rate within Nagoya MA. A line located higher in thegraph describes the fertility rate of an earlier generation.
Figure 7. Calibration results of Nagoya MA.

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000
year

year

10000
5000
0
1950 1960 1970 1980 1990 2000

1
0.5
01950 1960 1970 1980 1990 2000

20000
15000
10000
5000
0
195019601970198019902000

total fertility rate in the CBD

35000
25000

15000
10000
5000
0

year

total fertility rate in the CBD

20000

35000
30000
25000
20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000
year

total fertility rate in the CBD

population (thousands)

25000

1950 1960 1970 1980 1990 2000

population (thousands)

cost of child rearing: b

35000
30000

4000
3000
2000
1000
0
-1000
1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
-500 1950 1960 1970 1980 1990 2000
year

year

year

30000

net in-migration (thousands)

15000

1.5

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

net in-migration (thousands)

20000

2

4000
3000
2000
1000
0
-1000
-2000
-3000
1950 1960 1970 1980 1990 2000

year

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000year

net in-migration (thousands)

25000

2.5

4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000

year

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

Figure 8. Counterfactuals on Osaka MA.
□: Tokyo, △: Osaka, ◆: Counterfactual Osaka

year

year

net in-migration (thousands)

30000

total fertility rate in the CBD

population (thousands)

35000

year

land requirement:

net in-migration

net in-migration (thousands)

total fertility rate in the
CBDpopulation (thousands)

35000
30000
25000
20000
15000
10000
5000
0

total fertility rate inCBD

195019601970198019902000

population (thousands)

consumption share: α

initial human capital: H

initial population: N

population

4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000
year

5000
0

1
0.5
0
1950 1960 1970 1980 1990 2000

total fertility rate in the CBD

year

35000
30000
25000
20000
15000
10000
5000
0
195019601970198019902000

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

population (thousands)

cost of obtaining h : S

total fertility rate in the CBD

year

35000
30000
25000
20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000
year

2000
1000
0
-1000
-2000
-3000
1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
-500

1950 1960 1970 1980 1990 2000
year

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000

year

total fertility rate in the CBD

population (thousands)

commuting cost: τ1950 1960 1970 1980 1990 2000

4000
3000

year

year

35000
30000
25000
20000
15000
10000
5000
0

year

year

total fertility rate in the CBD

population (thousands)

195019601970198019902000

4000
3500
3000
2500
2000
1500
1000
500
0
-500 1950 1960 1970 1980 1990 2000

year

year

35000
30000
25000
20000
15000
10000
5000
0

net in-migration (thousands)

10000

1.5

net in-migration (thousands)

15000

net in-migration (thousands)

20000

2

net in-migration (thousands)

25000

2.5

net in-migration

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

Figure 8. Counterfactuals on Osaka MA (continued).
□: Tokyo, △: Osaka, ◆: Counterfactual Osaka.

year

year

net in-migration (thousands)

30000

total fertility rate in the CBD

population (thousands)

total fertility rate in CBD

35000

1950 1960 1970 1980 1990 2000

population (thousands)

land supply: Dutility out of the city: v

speed of adjustment: M

population

7000
6000
5000
4000
3000
2000
1000
0
-1000
1950 1960 1970 1980 1990 2000
year

1
0.5
0
1950 1960 1970 1980 1990 2000
year

5000
0
1950 1960 1970 1980 1990 2000

population (thousands)

40000
30000
20000
10000
0
-10000-20000-30000195019601970198019902000
year

3

net in-migration (thousands)

10000

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000
year

7
6
5
4
3
2
1
0
1950 1960 1970 1980 1990 2000

25000
20000
15000
10000
5000
0
-5000
1950 1960 1970 1980 1990 2000year

net in-migration (thousands)

15000

total fertility rate in the CBD

20000

total fertility rate in the CBD

25000

net in-migration (thousands)

2.5
2
1.5

35000
30000

net in-migration

3.5
3

year

year

labor share: λtotal fertility rate in CBD
total fertility rate in the CBD

population (thousands)

80000
70000
60000
50000
40000
30000
20000
10000
0
1950 1960 1970 1980 1990 2000

population (thousands)

productivity of H : δelasticity of cost of h : σpopulation

6000
4000
2000
0
-2000
-4000
-6000
1950 1960 1970 1980 1990 2000
year

10000
5000
0
-5000
-10000-150001950 1960 1970 1980 1990 2000year

1950 1960 1970 1980 1990 2000
year

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000
year

Figure 8. Counterfactuals on Osaka MA (continued).
□: Tokyo, △: Osaka, ◆: Counterfactual Osaka.

net in-migration (thousands)

35000
30000
25000
20000
15000
10000
5000
0

total fertility rate in the CBD

population (thousands)

depreciation of H : ζ

year
4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000year100005000
01950 1960 1970 1980 1990 2000

1
0.5
01950 1960 1970 1980 1990 2000

30000
25000
20000
15000
10000
5000
0

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

30000
25000
20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000

15000
10000
5000
0
1950 1960 1970 1980 1990 2000

total fertility rate in the CBD

population (thousands)

20000

2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

year

-2000
1950 1960 1970 1980 1990 2000

4000
3500
3000
2500
2000
1500
1000
500
0
1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000
year

2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
-500
1950 1960 1970 1980 1990 2000
year

total fertility rate in the CBD1950 1960 1970 1980 1990 2000

0
-1000

2.5

year

35000
30000
25000
20000
15000
10000
5000
0

1000

year

35000
25000

2000

year

2.5

year

30000

3000

year

total fertility rate in the CBD

population (thousands)

year

35000

4000

year

total fertility rate in the CBD

population (thousands)

35000

195019601970198019902000

population (thousands)

land requirement:

cost of child rearing: b

consumption share: α

initial human capital: H

year

net in-migration (thousands)

15000

1.5

net in-migration (thousands)

20000

2

net in-migration (thousands)

25000

2.5

net in-migration (thousands)

30000

net in-migration

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

Figure 9. Counterfactuals on Nagoya MA.
□: Tokyo, △: Nagoya, ◆: Counterfactual Nagoya

year

net in-migration (thousands)

35000

total fertility rate in CBD
total fertility rate in the CBD

population (thousands)

initial population: N

population

4000
3500
3000
2500
2000
1500
1000
500
0
-500
-1000
1950 1960 1970 1980 1990 2000year

population (thousands)

year

35000
30000
25000
20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000

01950 1960 1970 1980 1990 2000
year

2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

2
1.5
1
0.5

25000
20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000

01950 1960 1970 1980 1990 2000

3
2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000

total fertility rate in the CBD

population (thousands)

cost of obtaining h : S

35000
25000

1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
1950 1960 1970 1980 1990 2000
year

4000
3500
3000
2500
2000
1500
1000
500
0
1950 1960 1970 1980 1990 2000

year

year

30000

4000
3000
2000
1000
0
-1000
-2000
-3000
-4000

year

total fertility rate in the CBD

population (thousands)

2.5

35000
30000

1950 1960 1970 1980 1990 2000

year

total fertility rate in the CBD1950 1960 1970 1980 1990 2000

4000
3500
3000
2500
2000
1500
1000
500
0
-500

year

3
2.5

year

35000
30000
25000
20000
15000
10000
5000
0

net in-migration (thousands)0

net in-migration (thousands)

5000

1
0.5

net in-migration (thousands)

10000

1.5

net in-migration (thousands)

15000

2

net in-migration

3
2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

year

year

Figure 9. Counterfactuals on Nagoya MA (continued).
□: Tokyo, △: Nagoya, ◆: Counterfactual Nagoya

year

net in-migration (thousands)

20000

total fertility rate in the CBD

25000

2.5

total fertility rate in the CBD

population (thousands)

30000

year

commuting cost: τ

total fertility rate in CBD

35000

1950 1960 1970 1980 1990 2000

population (thousands)

land supply: Dutility out of the city: v

speed of adjustment: M

population

9000
8000
7000
6000
5000
4000
3000
2000
1000
0
1950 1960 1970 1980 1990 2000year

4000020000
01950 1960 1970 1980 1990 2000

total fertility rate in the CBD1950 1960 1970 1980 1990 2000

3
2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

1950 1960 1970 1980 1990 2000

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

1950 1960 1970 1980 1990 2000

20000
15000
10000
5000
0
1950 1960 1970 1980 1990 2000
year

4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-5000
1950 1960 1970 1980 1990 2000
year
10000
5000
0
-5000
-10000-15000-20000-25000-30000195019601970198019902000
year

year

total fertility rate in the CBD

population (thousands)

depreciation of H : ζ

year

35000
30000
25000
20000
15000
10000
5000
0

25000

year

total fertility rate in the CBD

population (thousands)

labor share: λ

year

35000
30000
25000
20000
15000
10000
5000
0

30000

year

year

35000
30000
25000
20000
15000
10000
5000
0

net in-migration (thousands)

60000

net in-migration (thousands)

80000

net in-migration (thousands)

100000

7
6
5
4
3
2
1
0

net in-migration

2.5
2
1.5
1
0.5
0
1950 1960 1970 1980 1990 2000

year

year

Figure 9. Counterfactuals on Nagoya MA (continued).
□: Tokyo, △: Nagoya, ◆: Counterfactual Nagoya

net in-migration (thousands)

120000

total fertility rate in the CBD

population (thousands)

total fertility rate in CBD

1400001950 1960 1970 1980 1990 2000

population (thousands)

productivity of H : δelasticity of cost of h : σpopulation

4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-5000
195019601970198019902000
year