PRI Discussion Paper Series (No.21A-3)

Population Mobility Structural Analysis and Population
Estimation Using a Quantitative Spatial Model

Economist, Policy Research Institute, Ministry of Finance

Wataru Takahashi

May 2021

The views expressed in this paper are those of theauthors and not those of the Ministry of Finance or
the Policy Research Institute.

Research Department Policy Research Institute, MOF
3-1-1 Kasumigaseki, Chiyoda-ku, Tokyo 100-8940, JapanTEL 03-3581-4111

Population Mobility Structural Analysis and Population Estimation Using a
Quantitative Spatial Model
Wataru Takahashi12
Policy Research Institute, Tokyo, Japan.
This article proposes a population forecasting using endogenous population model, which
incorporates a spatial model considering the spatially heterogeneous feature of agents andthe economy by employing quantitative spatial models. In this endogenous population
model, agents migrate to maximize their utility. This model is estimated using the twostage estimation approach, which is extensively used in quantitative spatial literature.
Estimated parameters are significant and almost consistent with the economic anddemographic stylized facts.
Using the parameters concerning migration and local utilities, we conduct projection
analyses for 2015-2125 across all prefectures in Japan, which is now experiencing
regionally asymmetric population decline. In the baseline projections, the population inless populated prefectures is mitigated slightly by introducing the young generation's
migration behavior. Counterfactual analyses are then conducted to break down the factorsof population decline in Japan. Among several factors, birth abandonments due to someconstraints and slow productivity growth after 1995 turned out to have severely impacted
demographics. The development of networks resulted in having negative impacts on
demographics though they had positive impacts on the welfare of economic agents inmany aspects.
Keywords: endogenous population model, spatial economics, quantitative spatial
economics, population mobility.
JEL classification: C21, J61, R10, R12, R23 P25.
1

Office of Econometric Analysis for Fiscal and Economic Policy, Research and Administration
Department, Policy Research Institute, Ministry of Finance, 3-1-1, Kasumigaseki, Chiyoda-Ku, Tokyo.
E-mail: wataru.takahashi@mof.go.jp.

2

The author is grateful to Prof. Motohiro Sato at Hitotsubashi University for his guidance and advice. The
author also would like to thank Prof. Shinji Yamashige for his advice on prior research. Many feedbacks in
Economic Theory and Policy Workshop(Aoyama Gakuin University),2019 Japanese Economic
Association Spring Meeting(Musashi University), Research in the Frontiers of Modern Economics
Workshops(Aoyama Gakuin University), are also helpful. We also received valuable comments fromprofessor Hirokuni Iiboshi (Tokyo Metropolitan University), professor Hisakazu Kato (Meiji University),
associate professor Masaki Nakahigashi (Niigata University) and professor Shinichi Nishiyama (Kyoto
University).The opinions in the article are solely of the author and do not represent the official views of theinstitution to which he belongs. This work was supported by KAKENHI 19160971.(Grant-in-Aid forYoung Scientists B).

11. Introduction
It has been a long time since Japan became an aging society with a declining birthrate. The populationof Japan began to decline in 2015. Economics tells us that a change in demographic structure will havea major impact on the economy and public finance. In particular, in finance, the multigenerational
overlapping generations model developed by Auerbach and Kotlikoff (1987) provides a detailed
analytical framework to examine the impact of changes in demographic structure, which has several
applications in Japan.3
However, the issue of sustainability has significant implications not only for the nation as a whole,
but also for regions. In 2007, the city of Yūbari in Hokkaido declared financial bankruptcy and was
designated by the government as a financial reconstruction organization, an event that is still fresh in
our memories. After the designation, Yūbari’s expenditure for public services was considerably
restrained and the welfare of its inhabitants worsened. Thus, the lack of sustainability of local
governments and economies contributes to regional inequality. In addition, lack of sustainability leads
to inequality between residents with different characteristics, as it is known that inhabitant traits affectinterregional migration and the attributes of local inhabitants (Moretti, 2011). Therefore, regional
populations and economies should be analyzed under a framework similar to that of macro-models.
Three factors should be considered in assessing sustainability in regions in this way. First, it isnecessary to consider the migration of economic agents. This is because interregional migration ismore intense than international migration. Second, as new economic geography4 points out, different
regions in the same country have different economic structures. The heterogeneity of these regional
economies as well as the of demographic structures should be taken into account. Third, theinterrelationship between regional populations and regional economies should be formulated. For
example, De la Croix and Gobbi (2017) argues that population density adversely affects fertility. In
addition, some studies regard population concentration as agglomeration and analyze its relationshipwith economic activity. Based on these studies, it is clear that there is a complex relationship between
current population, fertility rate, and economic activity in all regions.
Several studies have addressed regional economies in the study of sustainability using an
overlapping generations model. Börsch-Supan, Ludwig, and Winter (2006) analyzed the impact ofpension reforms in the three countries (Germany, France, and Italy) using the multiregional
overlapping generations model of INGENUE (2001). Fougerè et al. (2009) quantitatively assessed
how immigrant acceptance and educational spending affects the financial condition of Canadian states
and territories. However, these studies assume that the region is connected to larger regional units (the3 Braun and Joines (2015) is a typical example of a study in this area. The conclusion is that Japan's fiscal sustainability

cannot be achieved without fundamental structural reform, but it is also characterized by strong assumptions about thediscounted present value of public debt.
4
An early study is Krugman (1999). Comparatively recent theoretical developments include Fujita et al. (1999) and
Baldwin et al. (2003).

2

world and states) via capital and financial markets, and that equilibrium is achieved throughinterregional capital transfers and fund transfers induced by interest rates that vary regionally. In thesestudies, the immobility of economic agents is assumed, or the movement of people restored fromdemographic statistics is merely given exogenously. That is, although these previous studies consider
the heterogeneity of regional economies and demographic composition (the second factor), they do
not consider the endogenous migration of people (the first factor) or interactions between regional
populations and regional economies (the third factor).
The contribution of this paper is primarily related to the above-mentioned "first factor" and "third
factor." In this paper, we analyze a model combining the discrete selection model by McFadden (1974)
with a simple household utility maximization problem that is endogenous to the population. This
analytical method is in line with recent progress in quantitative spatial modeling (Redding and RossiHansberg, 2017). For this model, the indirect utility of inhabiting each prefecture in Japan at aparticular point in time is estimated by performing a guided analysis using panel data on two types ofinter-prefectural movement and traffic costs generated from a detailed traffic network. In addition,
parameters in the household model are estimated by structural estimation. These results are mostly
consistent with the intuition of the overlapping generations model.
This paper is structured as follows. In Chapter 2, two relevant research areas are surveyed. Chapter
3 formulates a theoretical model based on the above survey. Chapter 4 describes the estimation policiesof the formulated models and the estimation methods at each stage and describes the estimation results.
Chapter 5 conducts a counterfactual analysis of projections and populations based on the projected
total population and population by prefecture based on the various estimated parameters. Finally,
Chapter 6 describes conclusions and future policies.

2. Previous researchThis chapter describes previous studies of spatial equilibrium and endogenous population models. The
first part discusses the spatial equilibrium model often used in evaluating regional economic activity
and local government policies and the quantitative spatial model developed in recent years as anextension of it. The second part introduces examples of applying the endogenous population theorymodel, including an overlapping generations model, to multiple regions.
The purpose of the spatial equilibrium model is to derive the economic value of factors specific toa region (regional characteristics) through selection of economic agents capable of moving between
regions. Early studies by Rosen (1979) and Roback (1982) formulated economic agents to obtain
utility from consumer goods circulated between regions, (immobile) consumer goods fixed to regions,and factors specific to regions. In this spatial equilibrium, the utility of economic agents in all regions
is equalized by migration; furthermore, wages and non-migratory consumer goods prices are adjusted
to enable the derivation of the value of region-specific factors. Moretti (2011) and Diamond (2016)

3

discuss regional productivity using different types of regional characteristics, with the latterparticularly identifying differences in regional characteristics between cities in addition to wage
disparities. The spatial equilibrium model has also been applied in the field of public economics; asDiamond (2017) pointed out, the taxable rent of local governments depends on regional characteristics
as well as the elasticity of housing supply. In addition, Brueckner and Neumark (2014) showed that
economic agents’ preferences for desirable regional characteristics (proximity to the ocean, long
daylight hours) hampered "voting with their feet" and resulted in higher public sector wages.
The spatial equilibrium framework used in these studies consists of formulating relatively simple
equilibrium conditions and empirically evaluating policies using these formulated conditions. In
contrast, the quantitative spatial model used in this paper performs counterfactual hypothetical analysis
and simulation using structural estimation. Redding and Rossi-Hansberg (2017), who conducted asurvey of this study, show the applications of quantitative spatial modeling to various fields. The
commonality of these studies is that economic variables are stochastically formulated and follow an
extreme value distribution.5 Introducing such a probability distribution eliminated the complexity ofthe equilibrium condition of the new economic geography and facilitated multiregional andmultinational analyses.
Bryan and Morten (2015) and Morten and Oliveira (2016) can be cited as remarkable prior studies
in similar fields as this paper. Both studies analyzed the interrelationship between economic
development and labor migration by formulating the choice of employment area for economic agents.
These studies feature a detailed analysis of migration costs. Bryan and Morten (2015) estimate themigration costs of Indonesian regions by individual records and examine the impact of migration costs
on economic development through a comparison with the United States. Morten and Oliveira (2016)use Brazilian traffic network statistics to derive migration costs through the application of the fastmarching method (FMM) by Allen and Arkolakis (2014). Since endogeneity is assumed for economic
development and the development of traffic networks, natural experiments are being carried out using
data before and after the reorganization of traffic networks accompanying the construction of the newcapital. As will be explained later, detailed time-series data on traffic networks are available in Japan.
Therefore, a panel analysis applying Morten and Oliveira’s (2016) method using traffic network as an
operational variable will be conducted.
In order to analyze the population structure cited as a problem in the introduction, this paper
introduces a negative correlation between birthrate and population density, in addition to therelationship between population migration and economic development. This negative correlation has
been pointed out by De la Croix and Gobbi (2017) on a global scale, and Kato (2017) reported that a
5 For example, in the new economic geographic model, the formulation of international trade using the Fréchet

distribution by Eaton and Kortum (2002) is notable. On the other hand, in quantitative spatial analysis in recent years,
McFadden (1974) has carried out analysis using the standard Gumbel distribution (Weibull distribution), and the latter
framework is also followed in this paper, as will be described later.

4

strong correlation was observed in Japan. In addition, since Japan’s birthrate is declining and its
population is aging, a multiregional endogenous population model that formulates this correlation ispresented. Sato and Yamamoto (2005) conducted an early study in this area that formulates
demographic structures in economies with agglomeration effects on wages and congestion effects onchild-rearing costs using representative household models in two regions.
In the spatial equilibrium model introduced at the beginning of this chapter, equilibrium is achievedby concordance of utility in all regions. Therefore, it is possible to formulate a regional endogenous
population model such as the one proposed by Sato and Yamamoto (2005) that describes householdbehavior by maximizing utility under a spatial equilibrium model, and from which an endogenous
population-space equilibrium model is formulated. Sato (2007) formulated a birth-endogenous
multigenerational overlapping generations model based on a spatial equilibrium model. In the study,
consumption and child rearing of representative households in the region reflect additional congestioneffective additional costs, and productivity of representative enterprises in the region reflects
agglomeration effects. Therefore, the relationship between the regional population and birthrates
considered in this study depends on the relative strengths of the agglomerated economy and thedissipated diseconomy. If the productivity improvement by economies of scale is sufficiently large incomparison with the child-rearing cost of the region, the population is concentrated in the region withlarge scale, and the population density positively affects the scale of wages and population inflow,
while the birthrate is negatively affected.
Ishida, Oguro, and Yasuoka (2018) formulated an overlapping generations model under a spatial
equilibrium model. A two-region model is formulated here, but the formalized household behavior ismore detailed. The policy analysis of public childcare policy formulates congestion effects associatedwith land availability as well as housing costs, educational investment, and child-rearing costs. The
analysis shows that increasing population density negatively affects birthrate through congestion
effects in child-rearing costs. Here, the local government’s public childcare policy obtains the resultof reducing the negative effect on birth rate, while accelerating population inflow through the
improvement of local welfare. Therefore, this paper aims to formulate a multiregional endogenous
population model within the framework of quantitative spatial analysis, referring to Sato (2007) and
Ishida, Oguro, and Yasuoka (2018), and to gain an understanding of demographic structure andeconomic households in quantitative terms.

5

3. Theoretical model
In this chapter, we formulate a multiregional population endogenous model that deals with household
birth behavior and migration. In choosing a residential area, we adopt Morten and Oliveira’s (2016)
formula based on McFadden’s (1974) conditional logit model. This study formulates individual
preferences for each region by extreme value distributions and analyzes utility-based migration
quantitatively. In contrast, Sato (2007) and Ishida, Oguro, and Yasuoka (2018) theoretically derive anegative relationship between regional birthrates and populations by formulating an overlapping
generations model under a spatial equilibrium model. Based on these studies, this paper aims to gain
a quantitative understanding of the sustainability of a region by estimating the parameters of
population migration and population structure.

Formulation of population migration First, in accordance with Morten and Oliveira (2016), a
conditional logit model as a discrete selection as shown by McFadden (1974) is used to formulate theregional utility

at timein regionat that time. Here, individual utility

and the utilityof an individual

residing in the region

is described by:ε .
may be regarded as the fitness6 of individualfor region . Before choosing a residential area,

the fitness of an individual for the entire areaε,⋯,ε ,⋯,ε1∈

including their place of birth

is observed.

Individuals choose their areas of residence after birth and before engaging in economic activities in
specific areas. The residential area selection problem is formulated as follows:max

Equation (2) determines whether an individual
stays in region.

2

who resides in region

or migrates to another region. If individualfrom their place of birth, they must pay migration costsat the beginning of time

who was born in region

migrates

to initiate economic activity in region .

In solving the maximization problem, it is assumed that the fitnessof individualfor region

follows a standard type I extreme distribution (standard Gumbel distribution). That is, a randomvariablefollows a distribution defined by a probability density function

cumulative density function

and a

.

6

Based on interpretations of Moretti (2011), this term can also be interpreted as an additional preference by theindividual for a region.

6

exp

exp exp
exp exp.
.

This assumption can be regarded as equal to an idiosyncratic term in the utility of an individual in
McFadden’s (1974) formulation, and the corresponding indirect utility
idiosyncratic preferences (fitness)of individual

to the population expected utility

as addingof region . Therefore,

the model is consistent with the discrete selection problem from with the conditional logit model.
that an economic entity residing in region

If we follow McFadden (1974), the probability
chooses region

can be defined as follows:7

exp
∑
exp
Since ∑.

1 is fulfilled from the definition of3, the probability of migration,

satisfies the assumption of a random variable. In addition, due to the existence of migration costs
regional indirect utility

differs between places of birth, unlike conventional spatial equilibrium

8

formulations.

In addition, the regional labor force

at timeis determined depending on the individual's

regional distribution at the start of time , since the migration of population from region

to regionis directly governed by indirect utility. This regional distribution is equal to the young population at
the end of last year and can be defined asregion∈,,⋯,.9 Therefore, the entry of the labor force

using the young population,

into region

,

.

,

in eachis given by Equation (4) below.
4

1

Labor demandregionA single consumer good is produced in region . Technologies of typical firms inare expressed by the Cobb-Douglas production function.

7 See Appendix 1.2 for detailed derivation assumptions.
8 In the spatial equilibrium model, migration occurs until the indirect utility of all regions is consistent. On the other

hand, in the quantitative spatial model by Morten and Oliveira (2016), indirect utility among regions does notnecessarily coincide because of the constant movement of populations according to fitness ε and the existence oftravel costs.
9Since demographic dynamics are analyzed in Chapter 5, the various demographic indicators should be taken as given
here.

7

1

Here,

is the productivity at time.

5of the typical firms in region , and it is assumed that company

revenue belongs to an absentee company head and that the indirect utility of the individual is notaffected.
Capital markets are integrated across regions, and regional labor markets are perfectly competitive.
Therefore, regional wages

and capital rental costs

can be defined as follows. The subscript

is omitted because the rental costs are shared by the entire country.

log

log

log

log

log

log 1

1

log
log

,

6

.

7

Equations (6) and (7) can be combined to obtain Equation (8) corresponding to the labor demand curvein region .

is a function of capital rental costs

1

log

log

,

and can be regarded as a national trend.

log

1

1

log 1

log

In accordance with general assumptions made in urban economics, all real estate in region

8.

belongs

to absent landlords, and the land and other rent shall not affect the individual's utility function.

Housing market

According to studies of spatial equilibrium analyses of population migration, suchas Moretti (2011) and Diamond (2017), housing costs

at timeshall be defined by the trend term

and the reciprocal elasticity of housing supply to residential populations
the working populationof regionis entirely resident in the region. Here we assume that10

and that the resident

population in the region is consistent with the working population. Consequently, the housing market
in region

is given as follows:log
Formulation of utility

log

.

9

The endogenization of the population makes household utility

somewhat more complicated than in Morten and Oliveira (2016). Individuals who migrate to region
at the beginning of time

begin economic activity, earning income

in exchange for labor, and

10 As will be explained later, this process is somewhat valid because the "region" in this paper is a prefecture.

8

distributing income to consumption and child rearing. Let
consumer goods and let them raisechildren. In addition, it is assumed that each individual gains

utility through the regional characteristics
at timein region

be the individual’s consumption of. Thus, the expected utility of a representative individual

is defined as follows:

∙ exp

.

10

In addition, the budget constraint of this representative individual is defined as follows. In additiontoreferring to regional wages,

andrefer respectively to regional population density andhousing costs in region . Based on demographic assumptions and the results of De la Croix andGobbi (2017), child-rearing costs

increase as the population density of the region

increases.

.

11

Solving the optimization problem defined by utility (10) and budget constraints (11), we can derive
the conditions represented by Equations (12)–(14). These show the level of demand for consumer
goods, regional birthrate, and regional housing demandin regionat the time of utility

maximization of the representative individual.

12
1

13

1

The indirect utility function

14

can be derived by substituting Equations (12)–(14) into the utility

function (10):

1

∙ exp

15

The indirect utility of the representative individual refers to the expected utility level for choosing
regionat the beginning of time . In addition, the logarithm of Equation (15) gives the logarithmic

indirect utility functionof region

at time .

9

log

1

log
log

1

log

.

is the expected utility of moving to region

Based on the fact that16

at time , the following

Equation (17) can be obtained by summarizing and simplifying the equation.

,
The constant term

log

1

.

17

is a time-region idiosyncratic term.

is common among regions, andisgiven by Equation (18).

log

log

1

log

.

Formulation of individual behavior The time-region idiosyncratic term

18

resulting from theutility maximization problem discussed above corresponds to the regional utility of migration as
defined at the beginning of this chapter. In other words, it is possible to show the utility
residing in region

at timefor an individualof

as follows:

ε
log

log

1

is the fitness of the individual

As initially defined,log
to region.
at timeand follows thetype I extreme value distribution. The problem of region selection performed by the individual
the beginning of timefrom region

to region

1 exp

exp log∑atis defined in Equation (2), and the probability of migration of an individual

exp
∑

19

1 exp log(the migration probability) is derived through this maximization problem.

,
log

1

log

log
1

20

log

corresponding to the two-stage optimization problem

Equation (20) is the migration probabilityof the regional selection problem and the utility maximization problem by individuals. Here, theprobability that an individual residing in region
the economic variables of each region∈

will move to region

and costs of migrating10

at timeis described usingfrom region

to region .

Spatial equilibrium

In this chapter, quantitative spatial modeling is formulated inaccordance with Morten and Oliveira (2016) and others. It is possible to formulate the spatial
equilibrium by integrating the discussions so far. The equilibrium conditions consist of four equations.
∗

The regional equilibrium wage
regionprice∈, regional equilibrium migration probability∗

to region , regional equilibrium labor force∗∈of the period∈, and regional characteristics

∈

, ∈

from

, and regional equilibrium housing∈

are determined for a given regional productivity,∗∈, young population at the startby the equilibrium equations for them.

1. The regional labor market is in a spatial equilibrium state, and the regional equilibrium wage∗
for region is given by the following equation:
∈

1

∗

log

log

.

21

2. The regional housing market is in a spatial equilibrium state, and the regional equilibrium
housing price ∗ ∈ is given by the following equation:
∗

log

log ∗

3. The regional equilibrium migration probability

∗
, ∈

22
from region

to regionisgiven by the following equation:

exp log∗∑∗

exp log

log
∗

∗

log

1
∗

1

∗

23

is given by the following equation:

∈

∗

∗

log

∗

4. The regional equilibrium labor force entry

∗

log

,

.

24

4. Estimations
In this section, based on previous studies such as Ahlfeldt et al. (2015), Diamond (2016), and Morten
and Oliveira (2016), we conduct a two-stage estimation using the above spatial equilibrium conditions.
In the first stage, the expected indirect utility

in each region is derived using migration costs

derived from two types of domestic migration statistics and traffic network time series data. In thesecond stage, we derive parameters for analysis by applying spatial equilibrium conditions to thisindirect utility and by making structural estimates.

4.1 Stage 1: Induced estimation of indirect utility
In the first phase, we use migration statistics to estimate the expected indirect utility of Japan’s 47
prefectures.
11

Estimation equation

From the discussion so far, the probability

in region

as a destination is given by Equation (21):chooses region

exp
∑
exp

that an individual living.

25

from region

As can be seen from this formulation, costs of migration

to region

have asignificant impact on interregional migration. Therefore, it is necessary to formulate these migrationin the estimation.

costs

∈

26

is a fixed migration cost dummy, indicating that individuals need to pay when migrating
outside their home region.

∈11

prefectures in the same district
and region

at timeis the same district dummy and indicates that migration between
is comparatively easy.

is the migration time between region

and is derived from the time series data for the traffic network by a methodthat will be described later. Equation (22) is substituted into Equation (21) and the logarithm of bothsides is taken to obtain the following equation.

log

∈

log

exp

By rewriting this equation, we obtain Equation (23), which is the first-stage estimation equation.

log
Here, Π

∈

log ∑

exp

Π

is a fixed effect specific to the origin

in the fixed migration cost part in the estimation. The indirect effect.

and is absorbed, which is the object of thefirst-stage estimation, can be regarded as a fixed effect specific to the destination
Furthermore,

27

at time.

is the measurement error of the individual's probability of migration.

Intuitively, the sign of the coefficients is expected to be λ

0,

0,

0,

0. It is11 Here, we used a division that divided the 47 prefectures into eight districts. Except for the Hokkaido district, each

district includes multiple prefectures. The details are as follows: 1. Hokkaido district: Hokkaido; 2. Tohoku district:
Aomori, Iwate, Akita, Miyagi, Yamagata, Fukushima; 3. Kanto district: Ibaraki, Tochigi, Gunma, Saitama, Chiba,
Tokyo, Kanagawa; 4. Chubu district: Yamanashi, Nagano, Niigata, Toyama, Ishikawa, Fukui, Shizuoka, Aichi, Gifu,
Mie; 5. Kinki district: Shiga, Kyoto, Osaka, Hyogo, Nara, Wakayama; 6. Chugoku district: Tottori, Shimane, Okayama,
Hiroshima, Yamaguchi; 7. Shikoku district: Kagawa, Ehime, Tokushima, Kochi; 8. Kyushu district: Fukuoka, Saga,
Nagasaki, Kumamoto, Oita, Miyazaki, Kagoshima, Okinawa.

12difficult for an individual living in a particular prefecture to migrate to another prefecture. In addition,
the effort required for migration increases depending on the distance to the destination. Prefectures inthe same district have similar cultural backgrounds, and it is assumed that this effort will decrease
when the individual migrates within the same district.
Migration probability

Migration probability

is calculated from two types of data on inter-

prefectural migration: the "Basic Resident Register Migration Report" (Ministry of Internal Affairs
and Communications) and the "National Census Migration Tabulation" (Ministry of Internal Affairs
and Communications). The former has been reported every year since the 1950s but does not include
the statistics for Okinawa until the second half of the 1970s. The latter has been reported once every
decade since 199012 based on the census. The migration probability based on the Basic Resident
Register Migration Report

is defined as the proportion of the tabulated population 13 that

migrated to a given other prefecture, relative to the population of each prefecture at the start of theperiod of coverage every five years from 1975 to 2015. The probability of migration based on the
National Census Migration Tabulation

was calculated by tabulating the permanent residence five

years ago for each prefecture, and then calculating the ratio of the population migrating to eachprefecture to the population thus tabulated. This probability of migration is derived for 1990, 2000,
2010, and 2015, when demographic migration was tabulated. However, it should be noted that theapplicable period corresponding to the survey year differs due to differences in the tabulation methodin the Resident Basic Register Migration Report.
The Basic Resident Register Migration Report tabulates totals for all migrations from 1975 to 2015,
including those in Okinawa, and the time series for estimation is long. The National Census Migration
Tabulation has a short time series and does not capture all population movements until 2010,14 but isaccurate because it is based on the National Census, which is a full census.

Travel time

Here, we use the abundant time series data on the traffic network contained in the

National Land Numerical Information (Ministry of Land, Infrastructure, Transport and Tourism) andtwo pieces of legislation on the development of the traffic network to generate panel data on travel
time. The data produced by the Ministry of Land, Infrastructure, Transport and Tourism contains

time-series geographical data for various traffic networks including information on regular andirregular routes on Shinkansen (bullet train), trunk railways, highways, national roads, and maritime

12 While Japan conducts a census every five years, it conducts the migration tabulation only once every decade.
However, since 2010, population tabulation and migration tabulation have been implemented every five years, with themost recent being in 2015.
13 Since statistics for each fiscal year are totaled for five years, there is a possibility that the same individual migrates
a plurality of times.
14Population migration from 1990 to 1995 and from 2000 to 2005 is not included here.

13traffic. Following Allen and Arkolakis (2014),15 we applied the fast-marching method (FMM), a typeof image analysis method, to these geographical data and calculated travel times between prefectural
capitals every five years from 1975 to 2015.16 One of the contributions of this paper is that we obtained
annual data from a single source and used it to ensure consistency between the time series of the
explained variable and the time series of the travel time.17 Geographic information descriptions andmethodologies are given in the appendix, and here we present the geographic information used toderive travel time for 1975 and 2015 with the aim of demonstrating intuition about the procedure.

[Figure 1]

Figure 1 shows Japan's transport networks in 1975 and 2015. Basically, dark colors are allocated tolocations where there is a means of transportation with a high speed of travel. In order to guarantee
the existence of travel time as a solution, a constant speed is defined for land where there is no meansof transportation. However, we considered it impossible to pass through the sea. Comparing the traffic
network in 1975 (Figure 1a) with that in 2015 (Figure 1b), it is apparent that the traffic network is
being developed because the darkest lines, indicating the Shinkansen network, can be confirmed as
extending.

Endogenous problems

Utilizing the traffic network involves the problem of endogeneity.

Discussions of economic geography show that regional economies will develop as transportation costs
decline due to the development of traffic networks. However, it is also conceivable that an
economically developed interregional transport network will be further implemented. As a result, theunobservable elements of the migration probability

are assumed to correlate with indicators of

the traffic network. Morten and Oliveira (2016) avoided this problem by carrying out naturalexperiments18 on the construction of a new capital and the accompanying development of a newtransportation network and using data from before and after the development.
In this paper, we adopt an approach of constructing operational variables from two development

15 Allen and Arkolakis (2014) is an early study that used geographic data on traffic networks to calculate travel times

between two U.S. municipalities. This algorithm is developed with computational physics and calculates the transport
cost (transport time) using the least expensive route from all points to all points. ArcGIS is used to process geographicaldata.
16 For derivation, see Appendix 2.
17 For example, Allen and Arkolakis (2014) uses interregional logistics volume data from 2007 and economic anddemographic variables from the 2000 U.S. Census. In addition, the geographic information on transport networks issourced from different statistics aggregated at different times. (It uses data on river and canal networks from 1999, dataon railway networks from 2003, and data on highway networks from 2005.) Morten and Oliveira (2016) use panel dataon highway networks from 1940 to 2000, but does not consider other modes of transport.
18 Such endogenous treatment by natural experiments is often seen in previous studies. Ahlfeldt et al. (2015) uses datafrom the Berlin traffic network before and after the city was divided to control the causality of economic variables andtraffic networks.

14

plans established before the period covered by the first-stage estimation (1975 to 1985). The firstdevelopment plan is based on the National Shinkansen Railway Development Act (governed by the
Ministry of Land, Infrastructure, Transport and Tourism) enacted in 1970, and in 1973, thedevelopment plan for each of five planned Shinkansen lines, such as the Tohoku Shinkansen line, wasdecided. The purpose of this law is to construct a "Shinkansen railway network that organically andefficiently connects core cities throughout the nation, while being sufficient to form a national trunk
railway network.” In addition to covering all eight districts of Japan, the planned route network
connects almost all prefectural capitals.19 The second development plan was established under the
National Highway Act (governed by the Ministry of Land, Infrastructure, Transport and Tourism)
enacted in 1957, and the development plans were decided from 1962 to 1987, taking more time thanthe development plan of the planned Shinkansen lines. The objectives of these development plans are
similar to those of the Shinkansen network, aiming to construct highway networks that “constitute an
essential part of the national automotive transport network and that communicate areas of particularpolitical, economic and cultural importance and otherwise have particularly significant relationships
with national interests." These two maintenance plans were initiated before 1975 at the start of theperiod covered by the explanatory and explained variables.
As indicated in the purpose provisions cited above, each development plan is intended to connect
central cities in each district in Japan rather than connecting economically developed areas. Therefore,
these transport network development plans can be regarded as having been established independently
from the economic conditions of the time, and the transport network can be regarded as independentof the economic characteristics of the region. In addition, these are development plans, and it isassumed that they will not be affected by contrary causes and effects caused by economic activities
after their formulation. 20 That is to say, although these development plans affect subsequentinterregional travel time, it is assumed that they are unlikely to be affected by future economic
activities that are likely to correlate with individual migration probabilities.
The operational variablein the first stage is created by the following procedure. First, thedevelopment plan drawing is prepared by connecting the prefectural offices connected in eachdevelopment plan with a straight line.21 Then, the same procedures as in Allen and Arkolakis (2014)
are applied to the development plan diagram to find the shortest route between cities in the

19 The Shinkansen network connects 43 prefectural capitals, while Naha (Okinawa), Nara (Nara), Tsu (Mie) and Mito

(Ibaraki) are not connected.
20 From this point of view, the exogenous nature of the second development plan, the National Highway Act, is dubious

because it has been subject to revisions several times over a long period of time. The last revision was made in 1987
with the aim of dealing with the unipolar concentration in Tokyo and the depopulation of rural areas. Consequently, theresults of applying the development plan in the two-stage least squares method described below discuss only the Census
Migration Tabulation, which covers the period from 1990 to 2015.
21 The purpose of this procedure is to extract information on which cities are connected in development plans. Previous
studies such as Morten and Oliveira (2016) create links between all cities connected in the development plans(minimum spanning tree). Due to technical constraints, the above procedure is carried out in this paper.

15

development plan. This procedure is applied to each of the above two maintenance plans, and two
types of operating variables, "Shinkansen Development Plan" and "High-Standard Trunk Road
Network Plan," are prepared. The "simplified" development plan drawing prepared in the first stage isshown in Figure 2.
[Figure 2]

Again, land is allocated a constant speed. However, in order to derive solutions for Hokkaido andOkinawa, which are not connected in the development plans, routes with the same speed as on land22
were arranged.

First-stage estimation using the two-stage least squares methodA

two-stage

least-squareestimation (2SLS) is used to obtain the consequences responding to endogeneity for the transport
network. In the first stage, we estimate the following equation.

∈
Here,

is travel time and.

28

is an operational variable created by the procedure described

immediately above. Other variables are identical to the estimation formula for migration probability.
First, we describe the descriptive statistics for the variables used in the first stage of the estimation(Table 1). The number of inter-prefectural migrants also counts individuals who remain in the same
prefecture as having migrated. Thus, some values for migrant probabilities and number of migrants
are significantly larger relative to the mean and median, indicating that most prefectural residents
remain in the same prefecture. Since the locations of the prefectural capitals have not changed, thetravel distance remains unchanged with regard to time, whereas the travel time varies according to thedevelopment of the traffic network.23 The instrumental variables Shinkansen Development Plan andHigh-Standard Trunk Road Network Plan are invariant between times because they correspond to thedevelopment plan and have the same characteristics as travel time. In addition, the descriptive statistics
of migration data from the Basic Resident Register Migration Report and the National Census
Migration Tabulation differ somewhat depending on the source and survey date of primary information,and the method of calculating migration probability described above. Therefore, we must present theestimation results of both explained variables from the viewpoint of robustness.

22 These are: 1. the route between Sapporo (Hokkaido) and Aomori (Aomori) and 2. the route between Kagoshima(Kagoshima) and Naha (Okinawa).
23 Due to unit-time issues with the fast-marching method, the travel distance and travel time are equalized using theirrespective maximums, which are the distance from Sapporo (Hokkaido) to Naha (Okinawa) and the travel time from
Sapporo (Hokkaido) to Naha (Okinawa) in 1975.

16

[Table 1]

Table 2 below describes the estimation results based on the following estimation methods for bothmigration probabilities: 1. ordinary least squares (OLS), 2. instrumental variables (IV), 3. random
effect panel estimation (Panel), and 4. instrumental variables panel estimation (Panel-IV). Estimation
results of migration probabilities based on the Basic Resident Register Migration Report are shown in
the upper part, and estimation results of migration probabilities based on the National Census
Migration Tabulation are shown in the lower part. In implementing IV and Panel-IV, the Shinkansen
Development Plan and the High-Standard Trunk Road Network Plan are implemented in the first stage,
and the corresponding final estimation results are shown in (2) and (3), respectively. In addition, theresults of estimation using only "travel distance" in the first-stage estimation are shown in (1).
Regarding estimation results, all estimates are significantly negative for "fixed migration cost
dummy" and significantly positive for "same district dummy," yielding intuitive results. In addition,
while positively significant signs for "travel time" were derived in the estimations of OLS and Panel,
which do not use the instrumental variable method, negative significant coefficients for "travel time"
were obtained when using instrumental variable methods, i.e., in the cases of IV and Panel-IV, andresults suggesting the theory were obtained here as well.

[Table 2]

These estimates include, with some exceptions, three fixed effects: (1) origin-time fixed effects(origin-year FE), (2) destination-time fixed effects (destination-year FE), and (3) origin-destination
fixed effects (origin-destination FE). The regional time indirect utilityof the theoretical model

corresponds to (2) the destination-year FE. Regional time series data corresponding to this value areshown in Figure 3. Figure 3a shows the panel estimation result using the Shinkansen Development
Plan as an operational variable (Panel IV-2) as the indirect utility, and Figure 3b shows the panel
estimation result using the High-Standard Trunk Road Network Plan as an operational variable (PanelIV-3). The names appended to the time series data indicate the names of the prefectures.24[Figure 3]

24 As the estimation results for the Basic Resident Register Migration Report do not ensure the significance of using

the Shinkansen Development Plan as an operational variable, and the estimation results for the National Census
Migration Tabulation do not ensure the significance of Distance Only and the High-Standard Trunk Road Network Plan
as an operational variable, other estimation results were used. The reasons the significance of these variables is not
ensured are assumed to be: (1) the development plans were influenced by migration and economic development and(2) the development stage of the development plans was decided depending on the economic development of theconnected regions. In particular, the High-Standard Trunk Road Network Plan may be affected by (1) for the reasonsmentioned above.

17

If the indirect utility level described in Figure 3 is checked, utility is high in prefectures in the Kanto
and Kansai districts and in prefectures with central cities in regions with populations exceeding 1
million. In particular, the strong downtrend observed in the migration probabilities of the Basic
Resident Register Migration Report may be attributed to structural changes in migration due to thetime-series data extending over a long term.

4.2 Stage 2: Structural estimation of model parameters
estimated in the previous section to carry out

In the second stage, we use the indirect utility

structural estimates for each parameter of the theoretical model. The variables to be estimated are wage
elasticity to productivity in the region 1/ , elasticity of housing costs to regional labor supply,and preference parameters of utility functions,

Functions for estimation

,

,

,

.

The functions for structural estimation are derived from regional labormarket equilibrium (Equation (21)), regional housing market equilibrium (Equation (22)), and indirect
utility index

at timein region(Equation (18)). Following Ahlfeldt et al. (2015), the variables

in each region are defined as relative to the geometric mean,25 so the derived functions to be estimatedare defined in Equations (29)–(31) below.

Δ

Δ log

Δ log

Δ

Δ log

Δ
⁄

Δ

1

Δ log
̅

Δ log

1

⁄

Δlog

Δ

.

29

Δ

.

30

Δ

Δ

. 31

Additional discussions to interpret these equations are made prior to making structural estimates.
This is because, first, some parameters and economic variables are unknown. Second, additional25

We supplement for the regional indirect utility indicatorand regional characteristics. Since the utility levelis restored by an inverse
defined in Equation (18) has already been logarithmized, the regional indirect utility
logarithmic transformation, and then relativized by the geometric mean and logarithmized again. Moreover, we carriedout special leveling work because some regional characteristics
took negative values. Describing these tasks in
expressions produces the following:
̅

log

log

exp

.

exp
.

18

instrumental variables are required for structural estimation. Therefore, we will discuss two economic
variables separately, namely, (1) birth rate, (2) regional productivity and manipulation variables.

Subsidiary estimates (1): Discussion of child-rearing expenses

In this discussion, we make asimple estimate of the functional parameters of a functionthat defines the birthrate by

prefecture, incorporating the discussion of the negative causality between birthrate and population
density in De la Croix and Gobbi (2017).
De la Croix and Gobbi (2017) combined demographic statistics and various economic statistics of
44 developing countries26 to draw the conclusion that population density negatively impacts birthrates.
Based on the estimation equations used in this study, we formulate the estimation equation in Equation(32) below.

log 1

Here,

is the birthrate in region ,.

32

is the population density in region , andexplanatory variables.27 Each variable is logarithmized, and the estimate

representsshows the elasticity ofthe birthrate to population density. Following previous research, the "annual marriage rate by
prefecture (per 1,000 people)" ("Demographic Survey" (Ministry of Health, Labour and Welfare)) and"gross prefectural product per capita" ("Annual Report on Prefectural Accounts" (Cabinet Office)) areused as explanatory variables. The birthrate of the explained variable

uses the "number of births

by prefecture" for the working-generation population (25–71 years old) ("Demographic Survey"(MHLW)). The period covered by the estimation is 1989–2015. Fixed effects by year are introduced
to eliminate the effects of economic fluctuations and other elements.
Estimation results are shown in Table 3 below, which shows the estimation results of "basic
estimation" but does not introduce explanatory variables, and "extended estimation," which includes
explanatory variables.

[Table 3]

26 The same relation is highly plausible, as Sato (2007) and Kato (2017) have shown that it exists in Japan, boththeoretically and empirically.
27 De la Croix and Gobbi (2017) uses discrete data forbecause individual data from each country are used. For
this reason, it yields slightly different results to the analysis in this paper, such as making Poisson estimates andclustering by country and region.

19

In the basic estimation, positive coefficients were observed in the logarithmic term of population
density, and in the extended estimation, negative coefficients were observed, and results consistent
with the previous research were obtained in the extended estimation. For the extended estimation, anegative sign for population density is significant at the normal estimate, while the significancedisappears in the panel analysis of random effects. Therefore, the values of the normal estimate areused in the structural estimates that follow.28
From the estimation results, the relationship between the birthrate indicatordensity

can be defined as log

corresponding childcare cost function

3.59820.0136log 1

and population. In addition, thecan be defined as:

exp 3.5982 ∙.

33

We use this result in the structural estimates that follow.

Subsidiary estimates (2): Regional productivity estimates

In addition to estimating child-rearing costs by region, it is necessary to estimate regional productivity

when making structural

estimates. In this paper, we use the method of Hayashi and Prescott (2002), a prominent study ofJapanese productivity.
This method restores the total factor productivity (Solow residual) by growth accounting, and here,
the regional productivity

is restored by applying the procedure by prefecture. The data used hereare "production volume by prefecture (expenditure)" and "number of employees by prefecture" in the
Annual Report on Prefectural Accounts (Cabinet Office) from 1955 to 2016.29 Assuming that the
production function here is of the Cobb-Douglas type, we calculate the capital intensity

using the30

following equation:

1.

Table 4 contains information on the data used, the estimation method, and the results of theestimation. In particular, when panel estimation is carried out in the fixed effect model, capital intensity
is0.358, and regional productivity⁄

is restored using this.

28 Previous studies have yielded estimates of −0.110 in Sato (2007) and −0.141 or −0.157 in De la Croix (2017).

Although there were qualitative differences between regional and individual survey analyses, the estimates wereinterpreted as valid. In addition, the significance problem of panel analyses is a future problem.
29 There are several points to note in using these data. First, calculations in the Annual Report on Prefectural Accounts
have been revised in accordance with revisions to the definitions and calculation methods for national accounting.
Second, the aggregate value of "gross prefectural product" does not match the "gross domestic product," and there areconcern of errors in the "gross prefectural product."30
The production function corresponding to the gross prefectural productis. In addition, it isassumed that the capital intensity is the same in all prefectures.

20

[Table 4]

Operational variables

The three variables used are: (1) exogenous productivity shock on theregion (Bartik instrument), (2) labor market indicators, and (3) the ratio of available land to residential
land.
Bartik instruments are indicators presented in Bartik (1991) and are often used in spatial economics.
This indicator summarizes the impact on the regional economic structure of changes in the workforce
on a country-by-country basis in each industry. That is to say, where the workforce of the industry
in prefecture

at timeis, , , the Bartik instrumentof prefecture

at timeof the prefecture is defined as follows:

Δ log

,

,

, ,
,

.

34

In this paper, we use Bartik instruments as operational variables for regional labor markets andregional utility.
Morten and Oliveira (2016) applied the idea of Bartik instruments to compile integrated indicators
for the labor market by prefecture. This indicatorshows how strongly high wages in a particularprefecture attract workers from other prefectures.

.

35This indicator is useful as an operational variable because the responses of labor supplies to the same
productivity shock to different prefectures are assumed to be different. This paper applies thisoperational variable to the regional housing market because an individual who purchases housing has
already chosen the prefecture as the destination.
As a final operational variable, we use the ratio of the area available for residence to the area of theprefecturein accordance with Diamond (2017). The use of this operational variable issupported by previous studies on the relationship between the elasticity of housing prices and landavailability. This operational variable is used for individual indirect utility.
Taken together, the discussion of these identification restrictions allows the moment condition in
the second stage structural estimation to be expressed by Equation (36) for each corresponding21

operational variable. The variations of each error term are identical to those defined in Equations (29)–
(31).

0,

∈

,

0,

∈

,

0,

∈

,

36
.

In the following, we describe the estimation results for these moment conditions.

Results of estimates

For the purpose of consistency with the first phase of this section, here

we use indirect utility indicators restored from two statistics: the Basic Resident Register Migration
Report and the National Census Migration Tabulation. This means that Structural Estimate A includes
the time series from 1975 to 2015, and Structural Estimate B includes the time series from 1990 to2010.31
Before presenting the estimation results, we will explain the economic variables used in the second
stage. For the "wage rate", we use the "average monthly cash earnings" from the Basic Statistical

Survey of Wage Structure (MHLW). For regional productivity, we use the total factor productivity
restored in accordance with Hayashi and Prescott (2002) in subsidiary estimation (2). "Labor
population" is the population calculated by totaling the population of municipalities listed in the Basic
Tabulation Results for Population in the National Census from the ages of 25 to 71. As "monthly
rent/3.3 m²" is included in the Annual Report of the Retail Prices Survey (Ministry of Internal Affairs
and Communications), this is used as housing expenses
Regarding regional characteristics.

, we prepared three indicators: "amenity 1," "amenity 2," and"social capital." A discussion of these indicators is given in the appendix, and an overview of theindicators is given in this section. In deriving each "amenity," we used various statistical indices from1975 to 2015,32 which are published in Forms of the Prefectures as Seen in Statistics" (Ministry ofInternal Affairs and Communications). In addition, Social Capital Stock Estimation Data was used
when compiling "social capital." In this data, the net stock of social capital for the prefectures from1960 to 2014 is recorded in 16 categories. Since the traffic network is used as the first-stage operationalvariable, this paper examines 13 categories excluding "roads," "ports," and "air."Diamond (2016) is used as a reference for the process of obtaining regional characteristics’
31 Indirect utility indicators corresponding to the 2015 National Census Migration Tabulation have been omitted

because the calculation interval differs from other fiscal years and is not suitable for structural estimation.
32 In Forms of the Prefectures as Seen in Statistics (Ministry of Internal Affairs and Communications), the data series
differ from year to year. The data available for Structural Estimate A and Structural Estimate B differ because the period
covered differs. Twenty-one series were used for the Basic Resident Register Migration Report for 1975–2015, and 33
series were used for the National Census Migration Tabulation for 1990–2000. (See Appendix 3 for details.)

22

indicators from various statistical series. As the first stage, the series were divided by cases into sectors.
Concretely, statistical indices by prefecture were divided into the four sectors of: (1) environment, (2)education, (3) urban, and (4) welfare, and social capital was divided into the five sectors of (1)transportation, (2) living, (3) education, (4) environmental, and (5) industry. As a second stage,principal component analysis (PCA) was performed for each of these sectors to extract the firstprincipal component of each sector. PCA was then implemented for the principal components of these
sectors, and the regional characteristics thus derived are defined as "amenity 1" and "social capital."
In addition, "amenity 2" refers to an indicator extracted by performing PCA after extracting a statisticalseries that correlates significantly with the indirect utility function and compares the estimation resultwith the case when using "amenity 1" in order to confirm robustness.
Descriptive statistics for the second-stage estimates are presented in Table 5. Descriptive statistics
for Structural Estimate A and Structural Estimate B are described in the upper and lower rows,
respectively.

[Table 5]

Structural estimates of the second-stage estimates are presented in Table 6. Here, structural
estimates are made using the two indirect utility indicators derived in the first-stage estimation, theeconomic variables by prefecture described above, and the three regional characteristics. There areeight estimation results for each indirect utility because we also derive baseline estimates that do not
use regional characteristics.

[Table 6]

We consider the results of Table 6. The coefficient 1⁄
33

indicating "elasticity to wage productivity,"in Equation (29) can be interpreted as

and the coefficients are positively significant near 1 inall cases, indicating that wages rise due to increased regional productivity. The coefficientinEquation (30) can be interpreted as indicating "inverse elasticity of housing demand to housing costs."
This coefficientis also significantly positive in the vicinity of 1 in all cases of Structural

Estimation A and in reference estimates without the regional characteristics of Structural EstimationB, and significantly positive in the vicinity of 0.5 in other cases of Structural Estimation B). 34
Next, the results of Equation (31) will be explained. This result corresponds to each preference

33 Interpreting the regional production function used in this paper (Equation (5)), it becomes. In theconventional formulation, can be interpreted as capital intensity, but since the total factor productivity is restoredfrom the production function this time, it is probable that a value deviated from the normal value was derived.
34
This is consistent with formulations such as Moretti (2011) and Diamond (2017), in which housing prices increase
in response to population inflows.

23

parameter of the indirect utility of each consumer. "Preference for children" can be defined as an, and all coefficients

inverse function of the child-rearing cost function

for this parameterare significant, with coefficients around 0.8 being significantly positive in estimates of Structural
estimate A and 0.2–0.4 being significantly positive in estimates of Structural Estimate B. The
difference between the two is thought to be due to the difference in the period covered, and theestimates in the National Census Migration Tabulation for which the period covered is shorter reflect
the effects of the declining birthrate in recent years. The

1coefficient ofin Equation(31) shows the proportion of income spent on housing, indicating "preference for housing." Thecoefficients are positively significant, approximately 0.35 for Structural Estimation A, 0.4 forStructural Estimation B, and 0.25–0.3 for the rest.
We will describe estimation results on regional characteristics. In the case of both Structural
Estimation A and Structural Estimation B, there was no significance of regional characteristics.35 The
coefficients of explanatory variables for other economic variables are also robustly significant for the
introduction and selection of regional characteristics. Regarding Structural Estimate A and Structural
Estimate B, the coefficients of social capital are large for both. In contrast, in Structural Estimate A,
the coefficients of "amenity 1" following Diamond (2017) are relatively large, as are the coefficientsof "amenity 2,” which extracts significant ones in Structural Estimate B.
Finally, we discuss the validity of the parameters. The bottom of Table 6 shows the values of theparameters, ,restored from the respective estimation results. Regarding these values, it isdesirable that the following conditions are satisfied from the formulation of the individual utility
function in Equation (10):

0,

0,

1

0.

In Structural Estimate A using the Basic Resident Register Migration Report,, which corresponds

to "preference for consumption," is negative and does not meet the requirements of the utility function.
On the other hand, the restored values in Structural Estimate B using the Basic Tabulation Results forPopulation in the National Census satisfy all the above assumptions, indicating that the results arealmost robust with regard to the selection of regional characteristics. Therefore, the following
discussion uses the results for Structural Estimation B.

In this chapter, we conducted a two-stage analysis based on quantitative spatial economics. In the firststage estimation, indirect utility indicators by prefecture were derived for the two types of domestic
migration-related statistics by using the two-stage least-squares method with distance and travel time
35 Demographic migration arguments often do not specify the coefficients or significance of regional characteristics(Amenity) (Ahlfeldt et al. 2015, Diamond, 2017). However, this paper discusses it because it is formulated with acertain level of interest.

24

as operational variables. In the second-stage estimation, structural estimation was executed on theequilibrium condition formulated by the spatial equilibrium model to estimate the internal structure of
the utility of the individual who chooses residence between regions and to obtain various parameters.
In the next chapter, we will use these estimates to describe regional demographics.

5. Analysis of regional populations
In this chapter, we discuss the population dynamics of Japan's prefectures using the spatial equilibrium
model formulated in Chapter 3 and the model parameters derived in Chapter 4. In the first half of thischapter, we formulate population dynamics consistent with the spatial equilibrium model. In thesecond half, we derive the projected values of the population dynamics of the prefectures and confirm
the significance of introducing housing choices based on utility.

5.1 Spatial equilibrium and population dynamics
In Chapter 3, we formulated migration based on the utility function and utility of a simplified
household model that endogenizes population dynamics. If we describe our model based on theformulation of the spatial equilibrium in Equations (21)–(24), we see Figure 4 below.

[Figure 4]

In this figure, the squares represent the equations. In particular, those accompanied by numbers
correspond to the equilibrium Equations (21)–(24). The solid arrows point to the variables derived in
the spatial equilibrium model of this paper, while the dashed arrows indicate the variables that require
additional assumptions for derivation. Looking at Figure 4, we can see that there are three relationshipsthat determine the transition of regional economies: (1) population dynamics), (2) dynamics ofregional productivity (regional labor markets), and (3) dynamics of geographic factors. Of these, thelatter two are considered exogenously,36 and a detailed discussion of population dynamics is presented
here.

Discussion of demographic dynamics

As we have seen in Chapter 2, there are manystudies of multiregional demographic endogenous models in Japan. After describing how regional
populations are addressed in these previous studies, we describe the population dynamics required toderive regional populations in accordance with our formulation.
Sato and Yamada (2005) and Sato (2007) described population dynamics in a generic spatial
36 Geographic factors (regional characteristics

and migration costs) are of great significance for analysis
because they capture regional elements of population migration, but they are beyond the scope of the analysis in thispaper. In addition, following Hayashi and Prescott (2002), regional productivity is defined with the Solow residual.
Therefore, in formulating the dynamics of regional productivity, the formulation of the corporate sectors, including
additional data in the industrial sectors in particular, the relationship between household savings and investment, andthe capital migration equation, is required. This work is another task for later study.

25equilibrium model. Sato and Yamada (2005) dealt with a dynamic population equilibrium that states,that the series of the populations in two regions,

utility series of the regions

the total population, and the indirect

satisfy three conditions.37 Taken together, these conditions lead toa balance between dynamic and spatial equilibrium of the population when the coincidence of the next
population and the number of children in the economy as a whole coincide with the equivalence theindirect utility of each region.
In Sato (2007), demographic and economic steady-state conditions were analyzed, and the variables
are time-independent. Since there are households and enterprises in the economy, wages and interest
rates are determined from the conditions for maximizing enterprise profits, and consumption in eachperiod and the birthrate are determined from the conditions for maximizing household utility. In Sato(2007), the number of young people,

in region

is determined based on (1) the regional utility-

equalization condition and (2) the migration-stable condition.38 Wages and birthrates are given by the
number of young people,

in each region. Since the elderly population is consistent with the young

population in the previous period, steady-state definitions can be made by these two conditions.
In the above spatial equilibrium model, indirect utility coincides by households choosing a region
through utility maximization. On the other hand, the new economic geography (quantitative spatial)
model corresponding to the present study has wider variables to be analyzed in equilibrium and derives
an equilibrium in which, as seen in Morita and Yamamoto (2018), corporate profits are consistent ineach region and an equilibrium in which, as shown in Goto and Minamimura (2018), real wages are
consistent in each region.39 New economic geography also uses replicator dynamics as a convergenceprocess to equilibrium, whereas Goto and Minamimura (2018) formulated so that the regional
population converges to a population level that corresponds to an equilibrium level in real wages.40,
and the total population , and the indirect utility of each region, , (1) the congruence of the populations in the two regions and the total population, (2) the law of motion for
population, and (3) the no-migration condition is defined.
,
1
,
2
.
3
,
are the birthrates in each region.
38 (1) The regional utility-equalization condition and (2) the migration-stable condition is defined as follows.
isthe young population and, is the regional utility. The migration-stable condition was introduced by Sato (2007),
which dealt with economies in three or more regions.
̅,1
37 For the populations in the two regions

0.

,

2

39 For example, the real wage equilibrium condition in Goto and Minamimura (2018) is described in (1) below.

0,

1

0.

1

is the real wages in region andis the average. Furthermore,is the population share of the region. The
second equation is introduced, and its equilibrium condition also encompasses the case of full agglomeration.
40 Goto and Minamimura’s (2018) steady state of the population is formulated as follows:
∗
1,
1
∗

∗

∗

1

∗

26

1

∗

0.

2

What is particularly discussed in these formulations is the steady state of spatial equilibrium andpopulation. Under spatial equilibrium, the real wages∗∗

in each region coincide, and the birthrate

also coincides because opportunity costs coincide. Therefore, the steady-state condition ofdemographic dynamics in each region (Note 36, Equation (2)) is an identity, and under spatial
equilibrium, the steady state of the population is always realized only by the fact that the birthrate
satisfies the conditions. On the other hand, the steady-state conditions of the population are said to be
met even when real wages∗

do not coincide among regions.41 Therefore, no spatial equilibrium

can be achieved under a population steady state. In subsequent formulations of population dynamics,
Japan's prefectural population and regional economy will be predicted using projections from thecurrent economic conditions (2015) rather than balanced welfare assessments or population steadystate derivations,42 taking into account this spatial equilibrium and the inconsistency with population
steady states.

Population dynamics in this paper

In the following, we discuss the population dynamics

corresponding to the quantitative spatial model of this paper. Before discussing, we list the variables
pertaining to population that we dealt with in Chapters 3 and 4 and their definitions.

1. Working population:

Population aged 25–75 years who commence work in region
2. Entry into the labor force:

Population aged 25–29 years who select region
3. Younger population,

after region selection at time .

based on region selection at time .

:Population aged 25–29 years who participate in region selection at time
4. Birthrate

in region.

:

The number of children an individual chooses after region selection at time
5. Population density

in region .

:Population density after region selection at timein region .

Various variables are added to these definitions and some assumptions are made to describe population
dynamics. Since the time series of structural estimates up to Chapter 4 was in five-year units, we will
also discuss in five-year units.

Here, the birthrate at time in each region is, and the average birthrate among regions is . corresponds tothe degree of freedom of migration and is a general formulation in new economic geography. Under these conditions,the number of adults and population share in each region remain constant.
∗
41 In this case, the higher the real wage
in a region, the lower the birthrate ∗ . This corresponds to the current
situation in Japan.
42 These theoretical considerations are challenges for the future.

27

First, we define the population by generation. In this paper, we define three generations: the juvenile
generation population
population,, the working generation population,,, and the elderly generation. The juvenile generation population includes the population from 0 to 24 years old,

the working generation population includes the population from 25 to 74 years old, and the elderly
generation population includes the population 75 years and older.43
The juvenile generation population,

is defined by the following Equation (37) with respect tothe juvenile generation population in the previous period

,

,

:

,

.

,

37

corresponds to a population (cohort) of individuals born in the previous periodHere,
1 in region. In addition, the young population

are 25 to 29 years old at timein regionis the individuals who,

and who participate in region selection during this period.

Next, we define the working generation population. The population of this generation can be,

described by Equation (38) below using the entry of the working population, and the working generation population of the

population of the retiring cohort
previous period

,

.

,

While region

at time , the

.

,

38

from all prefectures in Japan at time , workers

welcomes the population ofaged 75–79 will retire and transition to the elderly generation. We assume that the working generation
population,in this region

is consistent with the working population.

,in region.44

39

That is, housing demand in the regional housing market at timeis defined by this generation

population.
Finally, we define the elderly population,. This generation includes the population 75 years

or older, and its dynamics are formulated as shown in Equation (40).

1

,

,

.

40

43 Structural estimates assume that the working population is 25–71 years old, which is based on the age groups inthe census. On the other hand, we made the above assumption because we needed to describe the population structure
every five years.
44
The consumption, child-rearing, and housing purchases described in the utility function of Equation (10) areconsidered to be reasonable assumptions because the majority are purchased during this generation (25 to 75 years old).

28

is the population 75–79 years old at time , which is consistent with the retirement
population above. In addition, the population of 1
population in the previous period

from the elderly generation,

shall remain in the economy at time,

. Here,corresponds to the mortality rate of the elderly generation population.45
Finally, we formulate the migration of the young generation in a quantitative spatial economic
model and connect the quantitative spatial model with population dynamics. The population entering
the economy in region

at the beginning of timeis the young population,. The definition isrestated in Equation (41) below.

41,

That is, individuals aged 25 to 29 who enter the economy at the beginning of time

make region

selections through the utility maximization described in Chapter 3. As a result of the region selection,
the probability

that an individual in region

result, the labor population entry in region

will migrate to region

is determined, and as a

as shown in Equation (42).

is defined as

,

.

42

The above formulations fully describe the population dynamics by generation in the economy. By
aggregating these, the total populationof region,

As the total population of the region
population variables: 4. birthrate (at time,,is defined.

.

43

has been derived, it is possible to define the remaining) and 5. population density (). The area of the prefecture

remains unchanged, and the child-rearing cost function is considered not to change. Under thisassumption, the population densityand birthrate

are defined by the total populationforthe period.

.

44

45 Needless to say, the mortality rate should also be defined for the juvenile generation population and the working

generation population, but these are omitted here because they complicate the relationship between the regional
economy and the regional population.

29

exp
Here,

is the area of prefecture.

3.5982.

45

and remains unchanged. In addition, since the wage rate

assumed to be determined by regional productivity, the birthrate
determined when the total regional populationisin population dynamics is

is given.

As noted above, this section derives population dynamics consistent with the spatial models
formulated and estimated through Chapters 3 and 4. Finally, we present a conceptual diagram of theformulation of population dynamics.

[Figure 5]

The formulation given here makes it possible to describe the population structure in all prefecturesconsistently with spatial equilibrium.

Assumptions about population dynamics and economic models

As

a

final

discussionofpopulation dynamics, we examine the assumptions made in Chapter 3 to accurately correspond theformulation of the individual utility function and indirect utility function46 with population dynamics.
According to the formulation of population dynamics, those who select regions at timearelimited to those aged 25 to 29 in the current period. On the other hand, the individual utility function
and the indirect utility function
regionshow that the utility obtained by an individual's residence inis defined by various economic variables at time . Given that individuals make a region

selection only on this one occasion due to the settings of region selection, the young population at timein regionof residing in the region in a myopic

derives the (logarithmic) indirect utility

fashion. In other words, in this paper, we assume that individuals only infer the lifetime utility obtainedby residing in region

using the present economic conditions in region
47

lifetime utility based on perfect foresight.

and do not maximize theThis can be formulated as in Equation (46) below. In

addition, since individuals observe the attributes of each region prior to migration, the observed
economic characteristics of region

are the already-realized economic characteristics of the previous

period. Therefore, we assume that the indirect utility

corresponding to Equation (46) is as given

in Equation (47) below.
46 For the purposes of discussion, (1) the individual utility function and (2) the indirect utility function are restatedbelow.
∙ exp

1
∙ exp

1

2

47 This idea is also observed in Kennan and Walker (2011), a more detailed formulation of migration. They are

considering the lifetime income maximization behavior of individuals dynamically, but they make some
simplifications on the assumption of verification and replacing this dynamic optimization problem with an expected
lifetime income maximization problem.

30

exp
∑

.

exp

46
∙ exp

1

.

47

Additional assumptions about individual birth behavior are also needed. Based on formulation of
the utility function, it is interpreted that an individual who selects region
birthsdetermines the lifetime

between the ages of 25 and 29 based on economic variables at this time (time ). On theother hand, the subsidiary estimation (1) in Section 4.2 defines "births by prefecture" for the working. This is in accordance

generation population and estimates the function of child-rearing costswith the general assumption that households raise children while working. Demographically strong
assumptions are made to ensure consistency between these settings. That is, we assume that myopic
individuals determine their lifetime births

between the ages of 25 and 2948 and actually raisechildren during this period. Under this assumption, the equation for fertility(Equation (45)) in the

projection is modified as follows:

1

exp

3.5982.

.

48

here shows the ratio of the number of births by parents aged 25 to 29 to the total number ofbirths in region

at time , and the adjusted birthrateshows the birthrate when all births in theregion are given by the young population.
The discussion in this section suggests that the following assumptions need to be made to ensure
consistency between the spatial equilibrium model in this paper and population dynamics.

1. Individuals in this paper make migration choices based on regional utility only once in their
lifetimes. Individuals are myopic, and individuals who choose to migrate at time

derivetheir utility and infer their lifetime utility using only the economic variables for region
time

at

1.

2. Individuals in this paper give birth and rear children only in the young period (25–29 years).

48 As the lifetime births are determined based on the economic variables (population density and wage rate) of eachregion at the time point, the formulation of the estimation continues to be valid. In addition, since the definition rangeof the fertility rate of women, which is summarized in the "Demographic Health Centers and Municipalities" (describedlater), is 15 to 49 years old, it is necessary to further divide the ratio of the population aged 25 to 75 years old and the
population aged 15 to 49 years old in the actual calculation.

31

These are relatively strong assumptions that have been made to ensure consistency between the spatial
equilibrium model and population dynamics. As such, these need to be taken into account in making
estimates and interpreting the results of estimates.

5.2 Population Forecast by Prefecture Based on the Quantitative Spatial Model
The definitions of population dynamics in the previous section make it possible to describe economic
and population dynamics by prefecture in Japan. In this section, we use projections to forecast theeconomic and demographic trends in each prefecture, starting in 2015.

Various statistical and additional assumptions used in forecasting First,

weexplainvariousstatistical indices used in the forecasting. In making forecasts, it is necessary to first describe thecurrent state of the economy and population in the prefecture at the start point (2015). For the sourcesof data used to derive regional economies, we continue to use the statistics used in Structural Estimate
B in Section 4.2. Migration can be restored and predicted from the interregional migration costs
estimated in Section 4.1 and the indirect utility

derived from the spatial economic model.

Therefore, what is additionally required for forecasting is a detailed population structure andpopulation dynamics statistics. For the population of each prefecture by five-year age group in 2015,
we used "population by sex, prefecture, municipality, and age (five-year age groups)" listed in"Population Dynamics by Health Center and Municipality" (MHLW). The "total fertility rate andbirthrate by maternal age group" in the same statistics are used to calculate the fertility rate. Thisbirthrate by age group is multiplied by the population by age group to obtain the share of the numberof births by age group and derive the modified birth rateused to derive the mortality rate

. The 2015 Life Table by Prefecture wasof the elderly generation.

Several further assumptions are made about the additional matters required when using thequantitative spatial model for forecasting. Looking at the quantitative spatial model described in
Figure 4 and its equilibrium conditions described in Chapter 3, regional productivity
young generation population∈, and migration costs

,

∈
, ∈

∈

, the

at the beginning of the period, regional characteristics
should be given at the start of time . Of these, since theyoung generation population is given by population dynamics, other economic variables are assumed
as follows.
1. Regional characteristics
2. Regional productivity∈∈

and migration costs, ∈are constant across time.

will grow at the average growth rate for each region from1975 to 2015.

32

Prefectural population forecast by projection In the forecast, we use the spatial model shown in
Figure 4 and the model with various assumptions in this chapter added to the population dynamics
shown in Figure 5. For the parameters, we used a combination consistent with assumptions from theestimation results of structural estimates in Chapter 4.49 As a result, the following prediction results
were obtained. Figure 6 uses the parameters in the results for "amenity 1" and Figure 7 uses theparameters in the results for "social capital" with social capital as the regional characteristics. The lefthand side of each chart shows the "total Japan population forecast" aggregating the populations ofeach prefecture, while the right-hand side of the charts shows the population growth rates for eachprefecture in the estimation period (2015–2125). For each parameter combination, population
dynamics in the absence of demographic migration 50 are derived as a reference estimate and areappended to the chart. These values are expressed as the total using dashed lines in the total population
forecasts and as the growth rate using horizontal axis values in population growth rate comparisons.
It should be noted that the estimates are higher than the level in the Future Estimated Population(Cabinet Office), reflecting the fact that deaths up to the age of 75 are not considered and that theeffects of the age structure on birth are simplified.

[Figure 6][Figure 7]

In common with the results of Figure 6 and Figure 7, it can be pointed out that the total populationin the presence of migration is smaller than in the absence of migration, and that when migration isformulated, the population decline in urban areas with a large initial population accelerates, whereas
the population decline rate in regions with a small population is greatly eased. These conclusions canbe regarded as robust results for the formulation of regional characteristics, as they are derived fromentirely different regional characteristics.
The latter result regarding population growth is consistent with intuition. In the endogenous
population model, individuals yield more utility than the number of children, so in the model in thischapter, where child-rearing costs correlate with population density, urban utility is relatively low. In
addition, an increase in housing costs due to an increase in the size of the working population also
causes a decline in the utility of urban areas.
It is difficult to interpret the former result, in which the total population becomes low as a result of
migration to regions where it is easy to raise children, against the result of the population growth rate,
49 See Chapter 4 for details. "Amenity 1" and "social capital" in the results based on the National Census Migration

Tabulation (Structural Estimation B) are applicable.
50 This refers to the situation in which region selection based on utility at ages 25–29 is not made and the population
born in each prefecture resides in the prefecture until death.

33

for which the utility function is easily formulated and interpreted. To deepen understanding of theresults, we forecasted the population using the estimation results for "amenity 1" under the assumptionthat regional productivity∈

would grow at the average growth rate for all regions over theentire period from 1975 to 2015. This assumption eliminates differences in productivity growth rates
among regions and states that productivity in all regions grows uniformly. The results under thisassumption are shown in Figure 8.

[Figure 8]

Figure 8 shows that while the relationship of population growth rates to base estimates in Figures 6
and 7 was maintained, the decline in the total population was relatively restrained, and the differencein personal income due to differences in productivity accelerated population decline. In other words,
in urban areas, the cost of raising children is high, but the expected income is high, whereas in rural
areas, the cost of raising children is low, but the income, which is the source of funding for raisingchildren, is also low. As a result, the population growth rate in the country as a whole is lower
compared to intuition.
Finally, we describe the differences in the results due to the selection of regional characteristics.
Appendix 3 discusses the levels of "amenity" and "social capital" used in estimates and forecasts. The"amenity" level is low in urban areas and high in rural areas, consistent with intuition, but there is asimilar trend in capital. This is reflected in the "comparison of population growth rates" in Figures 6
and 7. Excluding prefectures with relatively little change in growth rates compared to the baseestimates, the signs and degree of change in growth rates do not differ significantly between Figures
6 and 7. This may be due to the application of a method to select "social capital" related to migration.
Since it is assumed that there is a large correlation between social capital stock by prefecture and totalfactor productivity by prefecture, it is possible to derive estimation results completely different toFigure 7 through the refinement of the formula for the regional labor market (corporate sector) and thestructural estimation using social capital indicators. However, the fact that the results did not differ
significantly between the assumption of "amenity" and of "social capital" for regional characteristics
also shows that the results are robust regarding the choice of regional characteristics and can beinterpreted positively.
Through the forecast by projection, the dynamics of the population by five-year age groups in eachprefecture were obtained. However, since it is difficult to present all the results, this paper shows in
Figure 9 the juvenile generation population (0–24 years old: dark color), the working generation
population (25–74 years old: light color), and the elderly generation population (75 years and older:
middle color) in the prefectures of Tokyo and Tottori, where the largest divergence in population
growth rate was observed in the standard estimation and each simulation of Figures 6–8.

34

[Figure 9]

In the case of basic estimation, the population of Tokyo declined moderately after reaching its
maximum in around 2040, still remaining above 10 million at the end of the estimate.51 In the case of
Tottori Prefecture, on the other hand, the population is predicted to decline sharply from 570,000, andat the end of the estimate, it is almost halved to less than 300,000. By contrast, when region selection
is introduced, the young population (aged 25–29) leaves Tokyo, where child-rearing costs are high,and flows into Tottori, where these costs are low. As a result, while the working-generation populationin Tokyo decreases, the rapid decline in this population in Tottori that was observed in the standard
estimate is restrained. As a result, the Tokyo population declines sharply from around 2040, reaching
8 million to 9 million at the end of the estimate. In Tottori, the population increased until around 2060
through the inflow of the working generation population and the addition of their children to thejuvenile generation population, and as of the end, the population remained at 500,000. Comparing theresults between the simulations, it can be seen that the effect on the population structure of Tokyo is
greater when "social capital" is defined as a regional characteristic (the case of Figure 7) than when
"amenity" is defined as a regional characteristic (the case of Figure 6). In addition, when the
productivity growth rate is assumed uniformly (the case in Figure 8), the population growth rate inTokyo is further restrained compared to the case when "amenity" is taken as a regional characteristic.
This fact reaffirmed that in the formulation of the quantitative spatial model in this paper, thepopulation growth rate greatly depends on the productivity growth rate.
In this section, we predicted the population envisaged for a century in the future by implementing
a projection combining a quantitative spatial model and population dynamics. Counterintuitively, it
has been found that the introduction of region selection according to the quantitative space model
results in a tendency to select areas where child rearing is easy, while conversely accelerating thedecline in the total population of Japan as a whole. The ideal way for policies to curb the decline inJapan's total population based on the above is discussed below. We would also like to analyze which
geographic and economic factors caused a decline in Japan's total population and a decline in regional
populations. With regard to these matters for examination, this paper uses counterfactual analysis,
which is a characteristic of quantitative space models, to analyze various factors that characterize theJapanese economy in the second half of the 20th century, such as (1) a decrease in birthrate, (2)regional disparities in birthrate, (3) improvement of traffic networks, and (4) sluggish productivity.

51 As

mentioned above, the total population is higher than the "Future Estimated Population" (Cabinet Office),reflecting the fact that the mortality rate before age 75 and the detailed structure of the birthrate were abandoned. This
trend can also be seen in estimates in Tottori.

35

5.3 Analysis of population change factors and population policies by counterfactual analysis
As noted above, this section discusses Japan's demographic factors and desirable demographic policies
by conducting counterfactual analysis. One feature of quantitative spatial models is counterfactual
analysis using structural estimates for equilibrium conditions in multiple regions.
For example, Ahlfeldt et al. (2015) used city block data from before and after the division andreunification of Berlin and found that changes in urban structures observed in land prices and otherelements occurred endogenously through the effects of physical division and the restructuring oftransport networks and the accumulation and diversification of productivity and regional
characteristics. This study shows endogeneity through two counterfactual analyses assuming
exogenous regional characteristics and productivity. In the first counterfactual analysis, formal
simulations are conducted to replace productivity, regional characteristics, population density, andmore in the data after division and reunification with the data before to show that the restructuring ofurban structures as a result of segmentation and integration was not brought about only by physicaldivision. The second counterfactual analysis shows that the effects of division and reunification on
urban structures were caused by the agglomeration of productivity and regional characteristics rather
than by the basic economic conditions of the region, and further clarifies the details of changes inregional structures, such as the relative magnitude of the effects of externalities on productivity andhousing and the effects of transport network restructuring.
Bryan and Morten (2015) and Morten and Oliveira (2016) also assessed regional economic
structures based on structural estimates. First, Bryan and Morten (2015) pointed out that if Indonesia
had had a transport network comparable to that of the United States in the past, its GDP would be 45%
to 60% higher than at present, emphasizing the importance of developing transport infrastructure.
Morten and Oliveira (2016) also quantitatively assessed the impact of the traffic network associatedwith the construction of the new capital by deriving and evaluating the migration costs if the traffic
network were absent and if the traffic network centered on the former capital were maintained. Further,
they explained the effectiveness of regional development policies as due to the improvement oftransportation infrastructure, which affects the wage elasticity and trade of the local population.
Based on the analytical methods of these previous studies, we clarify how changes in various
parameters in spatial economic models affect regional population dynamics and the total population.
The scenarios considered address the following four matters discussed in the context of economic anddemographic issues.
The first is a decrease in the birthrate. According to the 2019 Annual Totals of Monthly Reports on
Demographic Statistics (approximate figures) (Ministry of Health, Labour and Welfare), the total
fertility rate, which was 1.76 as of 1985, was 1.36 as of 2019. It has generally been pointed out thattrends towards remaining single, marrying later, and bearing children later are factors in the decreasein the birthrate, but this discussion assumes that the difference between the "ideal number of children"36and the "intended number of children" in the 15th Basic Survey on Birth Trends (National Institute ofPopulation and Social Security Research) has improved. In other words, we analyze the case in which
the adjusted birth rate

improved due to a 13% decrease in child-rearing costs in Equation (48) and

derive the population dynamics if the birthrate improved. 52 Let this scenario be "(1) increased
birthrate." The second analysis deals with regional birthrate disparities. As pointed out in the 15th
Basic Survey on Birth Trends, the total fertility rate is highest in Okinawa at 1.82 and lowest in Tokyo
at 1.15. One often-mentioned hypothesis in this regard is that low birthrates in heavily populated urban
areas and high birthrates in less-populated rural areas result in low birthrates in Japan as a whole. This
work deals with scenarios in which only regional birthrate disparities are eliminated by treating
variances in regional birthrates

with mean-preserving contractions at each time point and deriving

regional birthrates with halved standard deviations. This scenario is called "(2) improvement ofregional disparities in birthrates." The next element we deal with is development of the traffic network.
The above-mentioned studies of quantitative spatial models have formulated the development oftraffic networks in detail.53 Some also believe that the development of traffic networks has resulted ingreater concentration as a result of reducing travel costs in urban and rural areas. Therefore, in thiswork, we examine how migration costs affect the population dynamics of prefectures and the
population dynamics of regions through use of migration costs in the traffic network as of 1975. This
scenario is called "(3) verification of traffic network effects." Finally, we examine the slump in
productivity growth. As Hayashi and Prescott (2001) pointed out, one of the factors behind the slump
in the Japanese economy since the 1990s is the decline in the growth rate of total factor productivity.
Total factor productivity not only determines the calculation of the economy as a whole but also affects
the number of births by determining individuals' incomes. If the decline in the income of the younger
generation leads to a declining birthrate, an increase in productivity may raise the birthrate. As a final
counterfactual analysis, therefore, we analyze the scenario in which the total factor productivity
growth rate in each prefecture is the growth rate from 1975 to 1995, not the growth rate from 1975 to2015. This scenario is called "(4) productivity improvement."
Figures 10–13 show projections up to 2125 under these scenarios: (1) increased birthrate; (2)improvement of regional disparities in birthrates; (3) verification of traffic network effects; and (4)productivity improvement. As in the previous section, the left-hand side of each chart shows the Japan
total population forecast, which is an aggregate of the population of each prefecture, and the righthand side shows each prefecture’s population growth rate over the estimation period (2015–2125).

52 In the 15th Basic Survey on Birth Trends, the "ideal number of children" was 2.32, the "intended number of children"was 2.01, and the scope of the decline in child-rearing expenses corresponds to this. If the divergence between the two
indicates an obstacle to child rearing, this scenario can be regarded as indicating the greatest theoretically expected
effect of child-rearing support alone.
53 Trading costs are the most important factor in this field because of the history of the establishment of new trade
theory and new economic geography, which form the basis of the quantitative space model. See Krugman (1980), etc.

37

[Figure 10][Figure 11][Figure 12][Figure 13]

Figure 10 analyzes scenario (1) increased birthrate. When the obstacles to child rearing are
eliminated, the effect is large; besides the total population increasing compared to the standard estimate,
population decline is suppressed in all prefectures except Tokyo compared to the standard estimate
without migration. This is because the "number of children" is reflected in the region selection by theyoung population; thus, the population is flowing into districts where the density is low. Although the
population decline in Tokyo is larger than in the standard estimate, the extent of the decline has been
reduced compared with the case where the regional characteristics shown in Figure 6 are defined as"Amenity 1," reflecting the rise in the birthrate. Next, we analyze scenario (2) improvement of regional
disparities in birthrates, described in Figure 11. In this case, where the standard deviation of birthrates
is assumed to be half, there is little change in the total population dynamics and population growthrate, except that the total population decline in 2125 is slightly suppressed (about 500,000) comparedwith the amenity shown in Figure 6. Even if regional differences in birthrates improve exogenously,
as shown, there is little effect on population dynamics. Therefore, it can be confirmed that the impactof regional birthrate disparities on the total population is not very large.
Figure 12 is an analysis of the effects of traffic network development. Although the decline in the
total population is approximately 1 million people smaller than in the case in Figure 6, there is littlechange. The major change here is the regional population growth rate; in addition to the population
decline turning to growth in Yamanashi and Tottori, the population growth rate has improved in all
prefectures except Tokyo compared to the base estimate. The development of traffic networks will
increase the young population migrating from areas with high (expected) indirect utility to areas where
it is lower by reducing travel costs. From these facts, we can confirm simultaneously that this effecthas caused population concentration in urban areas and population outflow from rural areas, and that
it will continue to be a cause of population decline in rural areas in the future. It should be noted,
however, that one of the conclusions of new economic geography is that the development of traffic
networks will promote interregional trade and improve the productivity and welfare of the economy
as a whole.54 It is also known that the development of traffic networks has little impact on the future
decline in total population, and therefore caution must be exercised in assessing the negative effects

54 The model in this paper assumes that social welfare in the absence of traffic network development will be reducedby the higher travel cost, since the net utility of residing in the region is obtained from the difference between theindirect utility and the travel cost. Social welfare is lower than in the standard estimate and the case of Figure 6
because the above-mentioned trends in the corporate sector are also conceivable.

38

of traffic networks development. Finally, we examine the effects of productivity growth improvementin Figure 13. Compared with Figure 6, the decline in the total population has been substantiallyrestrained, almost to the same level as the standard estimate without an assumption of population
migration. In addition, although not as much as in the case of the transport network development effect,
the population growth rate in rural areas is improving, and the population growth rate is high in most
prefectures, mainly in rural areas, compared to the standard estimate. This is because the number ofchildren increased in each prefecture due to an increase in disposable income resulting from increased
productivity in rural areas where the population is low and child rearing is comparatively easy. In
Figure 8, the productivity growth rate was assumed to be the national average, but recalling that evenin this case, the total population and the rural population growth rate showed improvement, it can besaid that both the large variation in the growth rate of total factor productivity across prefectures andthe sluggishness in the growth rate itself contributed to the decline in Japan's total population.
As in the previous section, we illustrate the time series of population by age group in Tokyo andTottori, which fluctuate more than the standard estimate. Figure 14 shows the population dynamics ofTokyo and Tottori in response to the following scenarios: (1) increased birthrate; (2) improvement ofregional disparities in birthrate; (3) verification of traffic network effects; and (4) productivity
improvement.

[Figure 14]

In the case of the increased birthrate, the improvement in birthrate restrains the decline in the juvenile
generation population, and in particular, the movement of the young population to Tokyo increases.
As a result, the population growth rate improves mainly in the juvenile generation population and theworking generation population, compared with Figure 6, in which the other conditions are identical.
In the case of improvement of regional disparities in birthrates, the decline in Tokyo’s population has
somewhat increased, while Tottori’s population dynamics remain mostly unchanged. This may be
attributed to the fact that the indirect utility of individuals in regions with high child-rearing costs has
decreased by compulsorily compensating for the divergence between "ideal number of children" and"intended number of children." This has a negative impact on Tokyo’s total population by preventing
the inflow of the young generation population.
Looking at the case of verification of traffic network effects, it can be seen that the total population
growth rates of Tokyo and Tottori have improved compared with the case of Figure 6. With regard to
Tottori, the fact that the outflow of the young population to urban areas was restrained by the lack ofimprovement of the traffic network seems to be a factor in the improvement. On the contrary, thepopulation decline in Tokyo can be seen to have been halted because the population flowing out isrestrained by avoiding the high cost of raising children. Finally, we observe the effect of productivity

39

improvement on regional population. In this scenario, disposable income increases in each region as
the region is not affected by the decline in total factor productivity growth rate in the second half ofthe estimation period (1996–2014). This increases the number of children, particularly in rural areas
where child-rearing costs were low, but income was also low. This effect is still exerted in Tokyo as
well; therefore, the juvenile generation population is seen to be larger in both regions than in thestandard estimate.
In this section, we examined the factors behind Japan's population decline and countermeasuresagainst population decline by describing the future dynamics of Japan's total population and regional
population, especially when the obstructors to economic and demographic growth seen in Japanesesociety since the 1990s did not exist or were ameliorated. As a result of the examination, in the rangeof the formulations in this paper, abandonment and delay of childbirth due to various obstructors tobirth selection exerted a greater effect than regional differences in birthrate, and it was shown that
improvement of the traffic network affected the population dynamics, such as the number of births,
while improving economic welfare. Regarding the total factor productivity growth rate, it was also
found that both the sluggish growth of the growth rate itself and regional disparities in the growth rate
worsened population dynamics.

6. Conclusion
In this paper, we first formulated and estimated a quantitative spatial model and predicted prefectural
populations based on the estimates, taking into account the sustainability of regional economies inrecent years. In preparing the quantitative space model, we formulated migration that could affect thesustainability of regional economies and regional populations, heterogeneity of regional economies,
and the relationship between regional economies and population structures. Two-stage estimation was
carried out within the framework of the quantitative space model. First, migration between prefectures
was estimated in reduced form using two types of statistics, and the fixed effect on each point in timeof each destination prefecture was used as the expected utility of migrating to the region. Furthermore,
under the assumption that each individual makes a region selection once in his or her life andmaximizes his or her utility, we formulated and made structural estimates of utility using a discrete
selection model. Second, in the population forecast by prefecture, the parameters on migration andutility function obtained from the first-stage estimation were applied to simple population dynamics,
and the population dynamics from 2015 to 2125 were derived by projection.
In the first quantitative spatial model estimation, we were able to obtain highly significant
estimation results for analysis of long-term migration in Japan and for analysis of increases in housing
prices and child-rearing costs due to population inflow. This result enabled us to formulate and analyze
the relationships between migration, heterogeneity of regional economies, and regional economies anddemographic structures. In the second population dynamics forecast, regional population projections

40

were made under certain assumptions concerning regional characteristics and regional productivity,and counterfactual analyses were conducted on various factors of the declining birthrate and aging
population. As the result, it was shown that various obstructors of birth selection affected the
population structure of regions rather than regional differences in birthrate, and that development of
the traffic network affected the population dynamics such as the number of births while also improving
economic welfare. Regarding productivity, it was found that both regional differences in productivity
and slowing growth rates restrained the population.
The contribution of this paper is described by field. While conventional studies have formulated
regional differences in population structures, such as birthrates, under spatial equilibrium models, thispaper uses a quantitative spatial model to directly analyze the relationship between migration and birth.
In the spatial equilibrium model, the economy is said to be moving towards equilibrium and population
migration towards disappearance since the expected utilities of individuals in every region are
equalized in the equilibrium, whereas in the formulation in this paper, the disparity in expected utilityof individuals derived from regional economic conditions does not necessarily disappear, enabling
analysis of population migration. In regional economic analyses as well, the author is not aware of any
existing studies that apply to Japanese cases Allen and Arkolakis’s (2014) quantification of geographic
travel costs or the formulation of quantitative space models following Ahlfeldt et al. (2015) andDiamond (2016), and that significant results on the relation between transportation networks anddemographics were obtained is also a contribution of this paper. In addition, in the population
estimation, it is considered to be an achievement that an economic foundation was given to the social
and natural changes in population.
After pointing out the elements that could not be dealt with completely in this paper, we describe
research tasks for the future. The first is the dynamization of the problem and the introduction of an
overlapping generation structure. In this paper, we assume that individuals residing in different regions
have made region selections by migration at the same age, once in their lifetime. In the real world,
however, individuals make multiple migration choices at different times. Because Kennan and Walker(2011) formulated migration by young groups based on dynamic optimization issues, while making
assumptions about region selection, it may be helpful to extend the model on the basis of these studies.
Many of the long-term financial analyses, including Auerbach and Kotlikoff (1987), have an
overlapping generation structure. In analyzing the fiscal sustainability of regional economies and theinterrelationship between fiscal and population structure, although it is desirable to introduce anoverlapping generation structure, there are many issues such as how to set regional capital levels,
savings levels, interest rates, and other requirements.
Second, a detailed formulation of the industrial sector is also an issue to be considered. In an
advanced study of the quantitative spatial model of Ahlfeldt et al. (2015), industrial agglomerations
were analyzed following new economic geography. Instead of considering these agglomerations, this41paper derived the productivity growth rate of a region. On the other hand, it can be said that moreelaborate general equilibrium analyses open up the possibility of deeper discussion of the relationshipbetween productivity and population, as well as analyses of economic growth and welfare, such as thebalance between accumulation and dispersion effects. Finally, a refinement of population dynamics
estimates can be raised. In the formulation of this paper, we discard the mortality rate up to the working
generation and the age structure of the birthrate for population dynamics as well as for population
migration. In addition to refining the optimization problem, it is also envisioned that these derivations
will allow for more elaborate demographic projections.

42

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44Charts
Figure.1: Japan's transportation network in 1975 and 2015(a) In 1975(b) In 2015

NOTE) Created by the author using ArcGIS10 and QGIS3 using "Numerical Information on Land" (Ministry of Land,
Infrastructure, Transport and Tourism).

Figure.2: Outline of the transportation network improvement plan in Japan(a) Shinkansen Development Project(b) High Standards Trunk Road Network ProjectNOTE) Prepared by the author using ArcGIS10 and QGIS3 using data provided by the Ministry of Land, Infrastructure,
Transport and Tourism. "Shinkansen improvement plan" is a "high standard trunk road network plan" which the authordraws based on each improvement plan drawing.

45

Figure.3: Regional Indirect Utility Indicators (First-Stage Estimation Results)
1.

Indirect Utility Indicators Based on the Basic Resident Register Population (1975-2015)2.

Indirect Utility Indicators Based on Census Population Movements (1990-2015)

NOTE) These tables describe the results of the first-stage guidance system estimation. Regional indirect utility
indicators correspond to destination fixed effects in Equation (23). Panel data estimation using the two-step leastsquares method using the development plan formulated by the Japanese government as an operational variable is being
implemented.

46

Figure.4: Structure of the spatial equilibrium modelNOTE) This figure was made from the theoretical model of this paper.

Figure.5: Schematic diagram of demographic dynamics

NOTE) This figure was made from the theoretical model of this paper.

47

Figure.6: Simulation Results (1) Population Dynamics (Amenity)(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

Figure.7: Simulation Results (2) Population Dynamics (Social Infrastructure)(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

Figure.8: Simulation Results (3) When Productivity Growth Rates Are Uniform (Amenity)(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

48

Figure.9: Simulation Results (1-3) Comparison of Regional Populations(Tokyo and Tottori Prefectures)
1. Baseline Estimate-Without Population Movements(a) Tokyo(b) Tottori Prefecture

2. Simulation Results (1) Population Dynamics (Amenity)(b) Tokyo(b) Tottori Prefecture

3. Simulation Results (2) Population Dynamics (Social Infrastructure)(a) Tokyo(b) Tottori Prefecture

4. Simulation Results (3) When Productivity Growth Rates Are Uniform (Amenity)(a) Tokyo(b) Tottori PrefectureNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

49

Figure.10: Simulation Results (4) Counterfactual Analysis (1): Increased Fertility Rate(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

Figure.11: Simulation Results (5) Counterfactual Analysis (2):Improvement of regional disparities in fertility rates(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

Figure.12: Simulation Results (6) Counterfactual Analysis (3):Traffic Network Effectiveness Verification(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

50

Figure. 13: Simulation Results (7) Counterfactual Analysis (4): Productivity Improvement(a) Total Japan Population Forecast (2015-2125)(b) Comparison of population growth rateNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

Figure. 14-1: Simulation Results (4-7) Comparison of Regional Population (1):(Tokyo and Tottori Prefectures)
1. Simulation Results (4) Counterfactual Analysis (1): Increased Fertility Rate(a) Tokyo(b) Tottori Prefecture

2. Simulation Results (5) Counterfactual Analysis (2): Improvement of regional disparities in
fertility rates(a) Tokyo(b) Tottori PrefectureNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

51

Figure. 14-2: Simulation Results (4-7) Comparison of Regional Population (2)(Tokyo and Tottori Prefectures)
3. Simulation Results (6) Counterfactual Analysis (3): Traffic Network Effectiveness Verification(a) Tokyo(b) Tottori Prefecture

4. Simulation Results (7) Counterfactual Analysis (4): Productivity Improvement(a) Tokyo(b) Tottori PrefectureNOTE) The author's work is based on the spatial economic dynamics and demographic dynamics of this paper.

52

Table. 1: Descriptive statistic for each variable in the first-stage estimation
Estimation of "Basic Resident Register Population"
Number of
movements
Max
Min
Mean
MedianS. Deviation

11264664
19
55469
1576
442383

Transfer
probability
0.95520.00000.02130.00080.1282

Distance
1.00000.00000.24000.20600.1669Travel time

Shinkansen
Project

Highway plan

1.00000.00000.13450.10200.13011.23180.00000.17140.12760.17330.74790.00000.15080.12570.1153

Estimation of "Census Population Movement Aggregation"
Number of
movements
Max
Min
Mean
MedianS. Deviation

12607447
14
57231
828
502962

Transfer
probability
0.97570.0001
0.02130.00040.1362

Traveldistance
1.00000.00000.24000.20600.1669Travel time
0.89960.00000.12900.12760.1257

NOTE) The "number of movements" indicates the number of inhabitants transferred from one prefecture to another. The"movement probability" indicates the movement probability of the inhabitant to each prefecture when the prefecture of the transfer
source is given. In the "Basic Resident Register Population Estimate", these values are derived using the 1975-2015 "Basic
Resident Register Population Movement Report Annual Report" (Ministry of Internal Affairs and Communications). In the"Census Population Migration Aggregation", these values are derived using the "Census: Population Migration Aggregation" for1990, 2000, 2010 and 2015 (MIC). "Distance" is the Euclidean distance between prefectural capitals, and "travel time" is theestimated travel time calculated every five years by Fast Marching Method (FMMs) using various data from 1975-2015 in the
National Land Numerical Information (MLIT). "Shinkansen Project" and "Expressway Project" are calculated by FMM in thesame way as "Travel Time" using the geographic data of the traffic network planned by "National Shinkansen Railway
Improvement Act" (1970-1973) and "National Land Development Trunk Expressway Construction Act" (1962-1987).

53

Table. 2: Results of the first-stage estimation
Estimation results based on "Basic Resident Register Population" (1975-2015)
OLS
Movable fixed cost dummySame local dummy
Trend
Travel time

-5.8824(0.1087)
2.4735(0.1341)0.0322(0.0072)
0.9344(0.1998)

IV-1
***
***
***
***

-5.5013(0.0316)
1.0943(0.0160)
1.1888(0.0164)-11.4353(0.0908)

IV-2
***
***

***

-5.5932(0.0312)1.1496(0.0157)
1.1260(0.0161)-10.8094(0.0870)

IV-3
***
***
***
***

-5.7640(0.0309)
1.2523(0.0156)
1.0092(0.0159)-9.6459(0.0845)

Panel
***
***
***
***

Constant term
Origin-Year FE
Destination-Year FE
Origin-Destination FE

Observation
F-value
R-Squared-5.8824(0.1087)
2.4735(0.1341)0.0149(0.0487)
0.9344(0.1998)
0.1555(0.4173)

***
***

***

PanelIV-1

PanelIV-2-3.8356(0.0821)0.4258(0.1020)-0.1210(0.0491)-3.4402(0.1012)
1.1260(0.4221)-4.4133(79.787)
1.0037(79.824)-0.0826(5.2985)-2.2055(170.56)
0.8519(37.828)

***
***
**
***
***

PanelIV-3-3.8356(0.0821)0.4258(0.1020)-0.1210(0.0491)-3.4402(0.1012)1.1259(0.4221)

Yes
Yes
Yes

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
Yes
Yes

Yes
Yes
Yes

Yes
Yes
Yes

Yes
Yes
Yes

19881
0.00000.999519881
0.00000.884619881
0.00000.886019881
0.00000.886519881
0.00000.991919881
0.00000.991719881
0.00000.991819881
0.00000.9917

PanelIV-1

PanelIV-2

***
***
**
***
***

Estimation Results by "Population Migration Aggregation in National Census" (1990-2015)
OLS
Movable fixed cost dummySame local dummy
Trend
Travel time

-6.8194(0.3167)2.3295(0.3369)
0.0524(0.0316)0.9335(0.6503)

IV-1
***
***
*

-6.2359(0.0447)
1.1098(0.0226)
2.7219(0.0426)-11.8149(0.1333)

IV-2
***
***
***
***

-6.3291(0.0442)
1.1663(0.0223)
2.5716(0.0415)-11.1537(0.1276)

IV-3
***
***
***
***

-6.4959(0.0438)
1.2674(0.0221)2.3025(0.0409)-9.9700(0.1239)

Constant term
Origin-Year FE
Destination-Year FE
Origin-Destination FE

Observation
F-value
R-Squared

Panel
***
***
***
***

-6.8194(0.3017)2.3295(0.3210)0.1325(0.0490)0.9335(0.6292)-0.3204(0.1838)

***
***
***

*

-7.5394(399.7242)
3.0497(399.8807)
0.1808(26.7953)
2.4867(862.3998)-0.4521(73.1103)-4.8156(0.1153)
0.3249(0.1417)-0.0018(0.0453)-3.3897(0.1468)
0.0461(0.1783)

***
**

***

PanelIV-3
14.3279(423.74)-13.0277(196.41)
8.5402(196.48)0.5487(13.166)-1.4559(35.924)

Yes
Yes
Yes

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
Yes
Yes

Yes
Yes
Yes

Yes
Yes
Yes

Yes
Yes
Yes

19881
0.00000.999719881
0.00000.901919881
0.00000.903119881
0.00000.903819881
0.00000.994419881
0.00000.994419881
0.00000.994419881
0.00000.9940NOTE) Standard errors are shown in parentheses. Dependent variables are migration probabilities (upper) calculated from the 1975-2015 Basic Resident Register Population Movement Report, andmigration probabilities (lower) calculated from the 1990-2010 Census: Population Movement Aggregate. See Section 4.1 for an explanation of "fixed moving costs" and "the same local dummy." Allestimation equations include year-destination dummies as an indicator of year-by-year regional fixed effects, but the values are omitted. In IV and PanelIV, a two-step least-squares method is used. In thefirst step, the "travel time" is regressed to the "distance" and other instrumental variables. In IV-1 and PanelIV-1, only "distance" is regressed. "Distance" and "Shinkansen Project" are used in IV-2 andPanelIV-2, while "Distance" and "Expressway Project" are used in IV-3 and PanelIV-3. Panel and PanelIV also use random-effects modeling to introduce fixed-effect dummies.

54

Table. 3: Estimation Results of Auxiliary Estimation Equations (1: Birth Rate Model)
OLS

Log(density)

XT

Basic
estimation

ExtendedestimationBasic
estimation

0.0050(0.0017)-0.0136(0.0030)
0.0457(0.0054)-0.0478(0.0004)-3.5982(0.0181)0.0146(0.0052)***

Marriage
Log(PDP)
Constant-3.5118(0.0132)

Observation
F-value
R-Squared1269
0.00000.5618

***

***

***

***
***

-3.5671(0.0277)***1269
0.00000.6050***1269
0.00000.7213Extendedestimation-0.0045(0.0096)0.0861(0.0129-0.3086(0.0515)-3.5856(0.0823)

***
***
***

1269
0.00000.8125NOTE) Standard errors are shown in parentheses. The dependent variables are the birth rate indicators
derived from "births by prefecture" in the "Vital Survey" (Ministry of Health, Labour and Welfare) and"birthrate indicators derived from "population, demographics and households based on the basic resident
register" (Ministry of Internal Affairs and Communications) for "population aged 25-71". The explanatory
variables are "population density" and "annual marriage rate by prefecture (per 1,000 population)" in the"Vital Survey" and "gross prefectural production per capita" in the "Prefectural Economic Accounts"(Cabinet Office).

Table. 4: Estimation Results of Auxiliary Estimation Equations (2: Growth accounting)
Level
OLS
Labor
CapitalObservation
F-value
R-Squared

0.4951(0.0057)
0.5321(0.0048)1786
0.00001.0000Differential
XT

***
***

0.3744(0.0902)0.5396(0.0150)

OLS
0.2953(0.1191)
0.3616(0.0269)

***
***

1786
0.00000.93931739
0.00000.2209XT*****

0.2997(0.0715)0.3577(0.0313)

***
***

1739
0.00000.0725NOTE) Standard errors are shown in parentheses. "Prefectural GDP" from "Prefectural accounts" (Cabinet
Office) is used as the dependent variable. Statistics on workforce and capital are based on the Prefectural
Resident Economic Calculations (cabinet) and Tokui (2013). Data from 1975-2012 were used in thecalculation of capital intensity, and data from 1975-2014 were used in the derivation of TFP by region.

55

Table. 5 (a): Descriptive Statistics for Variables in the Second-Stage Estimation(Basic Resident Register)UtilityMax
Min
Average
S. Deviation

2.913-3.672-0.825
1.272
Amenity 1

Max
Min
Average
S. Deviation

3.884-3.5830.000
1.414

Regional
productivity

Wage rate
513.369
139.674303.81771.8624.81792.00682.68050.3622

Amenity 2

Social capital

4.580-4.2350.000
1.718

11.298-2.4990.000
2.208

Working
population
884704834764316378211579977
Bartikmanipulated
variables
0.091-0.134-0.0290.046

Population
density

Residential
expenses (rent)

9528.533
67.988862.619
1360.064

9230.000
987.000
3476.000
1346.000

Labour Market
Indicators

Percentage ofresidential area

5.544
1.442
5.119
1.022

0.698
0.163
0.368
0.145

NOTE) All variables are logarithmically transformed into prefectural data, with differences between 1975 and 2015. "Utility" is the collective
indirect utility of each prefecture derived as a fiscal year regional fixed effect by prefecture. "Wage rate" is based on "Average Monthly Cash Salary"(Ministry of Health, Labour and Welfare). "Regional productivity" is derived by applying growth accounting to "gross prefectural product" in"prefectural accounts" (Cabinet Office). The "number of workers" refers to the labor force population of each prefecture (25-71 years old) derived
based on the Census. Population density was derived from "Prefectural area" (MLIT). "House rent" refers to "rent per 3m per month and 3. 2" in the
Annual Report of the Retail Price Statistics Survey (Ministry of Internal Affairs and Communications). "Amenity" is a figure obtained by dealing
with various indices listed in the "Statistics of Prefectural Sugata" (Ministry of Internal Affairs and Communications) by principal component
analysis (Principal Component Analysis) based on the method of Diamond (2016). "Social capital" is the figure obtained by applying the same
method as "amenity" in the "Net Capital Stock" in the "Social Capital Stock Estimation Data" (Cabinet Office). "Bartik instruments" refer to
instrumental variables derived using the "number of employees and employees by major industry classification" (MHLW) based on the method of
Bartik (1999). "Labour market index" is an integrated indicator of the labor market introduced by Morten and Oliveira (2016), "habitable area ratio"
is an index calculated by "prefectural area" and "habitable area ratio" of "prefectural area" (Ministry of Land, Infrastructure, Transport and Tourism),"and is used in various spatial economic analyses, including Diamond (2017).

Table. 5 (b): Descriptive Statistics for Variables in the Second-Stage Estimation (Census)
UtilityMax
Min
Average
S. Deviation

0.3717-1.9303-0.8409
0.5681
Amenity 1

Max
Min
Average
S. Deviation

3.1313-3.43160.00001.3595

Regional
productivity

Wage rate
387.927139.674226.3039
56.9582

4.01892.00682.53340.3026

Amenity 2

Social capital

3.7595-4.92900.00001.8097

1.8386-10.9600
0.00002.0115Working
population
73026543502681481995
1351430
Bartikmanipulated
variables-0.0063-0.1343-0.06290.0395Population
density

Residential
expenses (rent)

5470.6367.99
588.971042.885397.0000
987.0000
2195.4184
782.2333

Labour market
indicators.

Percentage ofresidential area

5.54414.27134.79840.2182

0.69840.16330.37160.1476

NOTE) All variables are logarithmically transformed into prefectural data, with differences between 1990-2000 and 2000-2010. "Utility" is thecollective indirect utility of each prefecture derived as a fiscal year regional fixed effect by prefecture. "Wage rate" is based on "Average Monthly
Cash Salary" (Ministry of Health, Labour and Welfare). "Regional productivity" is derived by applying growth accounting to "gross prefectural
product" in "prefectural accounts" (Cabinet Office). "Labor population" refers to the labor force population of each prefecture (25-71 years old)derived based on the Census. Population density was derived from "Prefectural area" (MLIT). "Rent" is the "Rent per month/3. 3m2" published inthe "Annual Report of the Retail Price Survey" (Ministry of Internal Affairs and Communications). "Amenity" is a figure obtained by dealing withvarious indices listed in the "Statistics of Prefectural Sugata" (Ministry of Internal Affairs and Communications) by principal component analysis(Principal Component Analysis) based on the method of Diamond (2016). "Social capital" is the figure obtained by applying the same method as"amenity" in the "Net Capital Stock" in the "Social Capital Stock Estimation Data" (Cabinet Office). "Bartik maneuvering variables" refers to the
maneuvering variables derived using the "number of employers by industry class" (MHLW) based on Bartik (1999) method. "Labour market index"
is an integrated labor market index introduced by Morten and Oliveira (2016), "habitable area ratio" is an index calculated by "prefectural area" and"habitable area ratio" of "prefectural area" (Ministry of Land, Infrastructure, Transport and Tourism)," and is used in various spatial economic
analyses, including Diamond (2017).

56

Table. 6: Estimation Results of Phase II Estimation (Basic Resident Register and Population Census)(Structural Estimate A) Basic Resident Register Population Movement Report (19752015)
Standard
Amenity 1
Amenity 2
Social capital
estimation
Wage elasticity
Residential Demand Reverse
Elasticity
Child preferences
Dwelling preference

0.9983(0.0035)
0.9482(0.0268)
0.8216(0.1123)
0.3530(0.0543)

Regional characteristics

***
***
***
***

0.9983(0.0035)
0.9486(0.0268)
0.7921(0.1456)
0.3684(0.0734)
0.1068

***
***
***
***

0.9983(0.0035)
0.9486(0.0268)
0.7907(0.1484)0.3690(0.0751)
0.0760

***
***
***
***

0.9983(0.0035)
0.9486(0.0269)
0.8084(0.1210)0.3606(0.0596)0.4766

***
***
***
***

(Structural Estimate B) Census Population Movements (1990-2010)
Standard
estimation
1.0005(0.0064)0.9592(0.0370)
0.6086(0.1892)
0.4270(0.0944)

***
***
***
***

Amenity 1
0.9992(0.0065)0.5248(0.1143)0.6186(0.2019)0.2532(0.0961)
0.3229

***
***
***
***

Amenity 2
0.9992(0.0065)0.5291(0.1135)
0.6151(0.2046)
0.2667(0.1002)
1.0584Social capital
***
***
***
***

0.9992(0.0065)
0.5363(0.1121)0.6009(0.1999)0.2903(0.0979)
0.6445

Annual fixed effect

No

No

No

No

Yes

Yes

Yes

Yes

Number of observations
Over-discrimination test value423
0.6977423
0.6075423
0.6075423
0.60751414.2330141
0.0634141
0.0625141
0.0609-0.1746
0.8216--0.1605
0.79210.1068-0.1597
0.79070.0760-0.16900.80840.4766-0.0356
0.6086-0.1282
0.6186
0.32290.11820.61511.0584

0.10880.60090.6445

***
***
***
***

Model parameter restoration value

NOTE) Standard errors are shown in parentheses. Equations (29)-(31) in Section 4.2 are used in all estimates. The "Basic Resident Register Population Movement Report" and the "Standard Estimationof Indirect Utility Indicators for the Census Population Movement Aggregate" that do not take into account regional characteristics, and the results of descriptions of "Amenity 1", "Amenity 2", and "Social
Infrastructure" that take into account various regional characteristics (GMM Estimation Results) are presented. In addition, the parameters in Equation (31) are restored according to the estimated values.

57Appendix
1. Derivation of equations
1.1. Derivation of the indirect utility function (15)
In deriving the indirect utility (expected utility) obtained by the agent living in the region , the agent
solves the following utility maximization problem with the utility function (10) constrained by thebudget constraint (11).

max
,

∙ exp

,

The Lagrangian

,

. .

.

corresponding to this maximization problem can be defined as follows.
∙ exp

∙

.

The first-order conditions for the problem are then given by

∙ exp

0.

A1

∙ exp
1

0.
∙ exp

A2
0.

and the housing

By combining (A1) and (A2), and (A3) and (A1), the number of children
demandin terms of the consumption

1

∙

1

are derived.

1

⟺
1

∙

A3

1

⟺

.
1

A4
.

A5

By substituting (A4) and (A5) into the budget constraint (11), the consumption level forincomeforcan be defined.

1

.

0.

58

⟺

.

A6

Substituting this equation (A6) back to (A4) and (A5), we yield equations (12)-(14). The indirect
utility function (15) can be derived by substituting (12)-(14) into the utility function (10).

1.2. Derivation of the migration function (3)Following the discrete choice model formulation by McFadden (1974)55, the migration equation of anagent born in a specific region is derived from its regional fitness and indirect utility. In addition tothis approach, we discuss the derivation method in the formalized quantitative spatial model of Eaton
and Kortum (2002), which can be granted as a technical supplement to this paper.

Discrete choice model by McFadden (1974)∈

and chooses

Suppose an agent in a group has the characteristic,⋯,from the option. Also, the individual utility function can be

described in the following format.

,

,

.

A7

describes the non-stochastic, representative preference of the population, andprobabilistic, idiosyncrasies of individual preference for . The probability

that an individual, who

is randomly selected from the population corresponding to the characteristicchooses the option

represents theand the option,

can be defined as follows.

| ,
,

,

,

,

for ∀

We define further the simultaneous cumulative distribution functionof,1, ⋯ , , and its partial derivativefordensity function). For,with respect to,

The members of the group have the utility,⋯,

A8
for the values56(simultaneous probability, the equation (8) can be written as follows.

,⋯,

option,⋯,.

. It is assumed that

,

.

,

,

A9

for the possible choices incorresponding to individual preference's peculiarity

55 The definition of the utility function in McFadden (1974) is different from this paper, but the derived migration
equations are the same ones.
56 In deriving equation (A12) from the definitions of equations (A7) and (A9), McFadden (1974) defines threeprinciples for discrete choice models. They constst of (1) Independence of Irrelevant Alternatives, (2) Posititvity, (3)Irrelevance of Alternative Set Effect. Here, all are satisfied by the formulation of the utility function, the specific term
and the assumptions about the nature of option .

59

is an independent and identical distribution (i.i.d) in the standard Gumbel distribution 57 . That is,,satisfies the following equation.

,,

If we set

exp

exp

.

A10

here, we can define the simultaneous probability density function

,⋯,

exp

exp

exp

exp

for .

exp

exp

exp

Replacing this with equation (A9) and rearranging58 it, the option. A11's selection probability

isgiven as follows.

exp

exp

exp

exp

exp

exp

exp

exp

exp

∙

exp

exp

exp

exp

∙ exp

exp

exp

exp

exp

exp
∑

exp

.

∙ exp

exp

exp

A12

The discussion in this paper can be described easily by applying the equation (A12). The group

57 In the original paper, it is called the Weibul distribution, but here it is calles the standard Gumbel distibution according

to the convention of the quantitative spatial model. The Type I Extreme Value distribution is the Gumbel distribution,and its density function and cumulative density function can be described as
1
exp
exp exp,
exp exp.
The standard Gumbel distribution is defined as the Gumbel distribution with
1,
0.
58 In the 1st and 2nd columns, the integration by substitution is performed withexp
, and the first exp
isis derived based on the integration formula.
removed from ⁄1⁄ . In the 2nd - 4th columns, the value ofSince the internal value of |∙| is 0-1=-1, the result in the 5th column is derived.

60

residing in the region
regions

at the start of the time

selects the desired region for residence from the,⋯,1, ⋯ , . The indirect utilities of the region at that time are,⋯,

the travel costs

must be borne in selecting the region, the options for the,⋯,individuals in this group are

selects the region. The options are the expected utilities ofadditionally observes its preferences ε

the regions, and the individualthat these preferences εbut since,⋯,ε. It is assumed

follows the standard Gumbel distribution, and the utility when individualis given by. The individual utility maximization problem, inthis case, is the discrete choice problem of McFadden (1974). According to the equation (A12), thefollowing probability is derived.

exp
∑
exp
is precisely the same as.

in the equation (13).

Derivation method in quantitative spatial models

In the formulation of McFadden (1974),,⋯,

the simultaneous cumulative distribution function

for all options was formulated but inquantitative spaces such as Eaton and Kortum (2002) and Ahlfeldt et al. (2015). The model takes
different approaches based on ordered statistics and extreme value theorem. Since this paper is aquantitative spatial model paper, we describe this approach and conclude the supplementary discussion
on the derivation process.
The probability

that an individual with the characteristic , who is randomly selected from thegroup, selects the option

from the setof the options can also be defined by the following

equation (A13).

Pr

We continue to assume that the error term

max
∈ ⁄

,

.

A13

the standard Gumbel distribution is anextreme value distribution (the Standard Type I extreme value distribution); it has the maximum
stability. Maximum stability means that the maximum value of a random variable sequence that
follows an extreme value distribution also follows the same extreme value distribution. Therefore, thefollowing relationship is established for the cumulative distribution function of the standard Gumbel
distribution

and the cumulative distribution function of its maximum value

Pr

exp
61

exp

.

.

A14

Pr max

exp

∈

exp

.

A15

The same result as the first equation of (A12) can be obtained by performing the equation
conversion 59 for the above equation (A13) of the selection probability. Then, the migration

probability (3) can be derived by following the same transformation as the equation (A12).

Pr

max
∈ ⁄

Pr

max
∈ ⁄

∈ ⁄

exp

exp

exp

exp

exp

exp

exp

A16

exp

exp

.

This method of transforming optimization problems for economic variables (utility maximization
problems and cost minimization problems) into a conditional logit model using the extreme value
distribution properties has been widely applied. For example60, with this method, Eaton and Kortum(2002) formulate the selling price under free trade with multiple countries, whereas Ahlfeldt et al.
(2015) analyze employment and residence choice among multiple districts in the city.

2. Fast Marching Method (FMM)
This section outlines the Fast Marching Method (FMM), a method for deriving the travel cost index
used as the travel time and instrumental variables in estimating interregional population migration,
59 Lines 1-2 are operations within the inequality sign. In the 2-3 lines, since

is a random variable, it is defined by. Furthermore, in lines 4-5,
0 holds, and the terms related to thethe probability density function
random variable
outside the brankets andinside the brackets can be summarized. Also, since the randomvariablesandare randomly selected from the same independent and identical distribution, the subscripts are
eliminated.
60 The selling price of Eaton and Kortum (2002) is the problem of minimizing the cost of purchasesers in the target
country, and the choice of employment/residence of Alhfeldt et al. (2015) is the problem of maximizing the utility ofresidents. In addition, the derivation of equilibrium conditions without using extreme value theory was limited to three
countries at most, and the description was complicated.

62

and details the derivation process.

2.1. Explanation of Fast Marching Method
Originally, FMM is an analysis method for image analysis for analyzing the shortest distance problem
on a solid surface. Allen and Arkolakis (2014) applied this to a two-dimensional transportation
network and applied the time required for the shortest path to the derivation. Since then, it has been
widely used as the primary analysis method in quantitative spatial analysis.
Let∈

be a coordinate system on the plane. Suppose we define the time61 required for thefunction : →∈to pass a given point

cost minimization problem from point
∈

|

0

,

1

on the plane. Define the solution

to point . Here,is a set of paths from point: 0,1 ⟶,of the travelis the path and Γ ,≡to point , which is continuous and can bedifferentiated once.

,

inf

∈

.

,

,

For those paths, a geographic travel cost,

A17

is defined 62 for all regional

combinations. This geographic travel cost is known to have two properties. One is symmetry, where,,

holds for all pairs , of points. The other is the continuity63 of travel costs due tothe continuity of the terrain.
,

The solution to this geographical movement cost
The gradient,is defined for the point .

‖

, ‖
,

Utilizing this property, the transfer costsame cost curve

can be defined by the following formula.

|

,

.

A18

can be defined by the following process. Here, thethat can be moved from the pointat the same cost (time)

is usedas the derivation standard.

1. Define the equal-cost curve|,

corresponding the costbased on the travel

2. Expand the equal-cost curve in the form corresponding to the costand define the

cost

.

equal-cost curve′|. The expansion width is given by ⁄, ′61 It is interpreted as transpottation costs in the new trade theory.

:⟶ 1, ∞ must be monotonically increasing ans satisfy
The formulation is as follows.
lim,,

62 The function
63

→

63

0

1.

accordingto the expansion destination .
∈3. Repeat steps 1 and 2 until the equal-cost curve reaches all points inIn the next section, the travel cost index.

, , which is derived by applying the above process tothe travel costs defined for each point.

2.2. Derivation method of travel time index
In this paper, following Allen and Arkolakis (2014), the travel cost index,

five years from 1975 to 2015. In defining the travel costat the time , the time-at the point

was derived every

series geographical data on the transportation means described in “Numerical Land Information”(Ministry of Land, Infrastructure, Transport, and Tourism) was used. As means of transportation,“Shinkansen (bullet train),” “Railway,” “Expressway,” “National Road,” “Regular Sea Route,”“Irregular Sea Route,” “Land,” and ”Sea” in order of expected movement speed are considered64.
For each means of transportation, moving speed is assumed, and then the reciprocal of the relative
moving speed is used as the moving cost of the means of transportation. Let the travel cost of thefastest means of transportation at point

be the travel cost

. Here, each pixel on thegeographically referenced image of the Japanese map is designed as point , and pixel

is coloredwith a shade color corresponding to the fastest means of transportation existing at point

recorded on“National Land Numerical Information” at time . Note that the coloring is performed ib a grayscalecorresponding to the value∈ 0,255 , and the darker the color, the faster transportation means exist.

The following table (Table A1) shows the code/series name, estimated travel speed, transportation cost
, and coloring valueon the “National Land Numerical Information” of each means oftransportation.

[Table A1]

Of the series of data, “Shinkansen,” “Railway,” and “Expressway” are one-year time series data,“National Roads,” “Regular Sea Routes,” “Irregular Sea Routes,” “Land,” and “Sea” are the data at
the time of creation. The range defines the coloring value for the convenience of output by ArcGis and
MATLAB. The ArcGIS output images for 1975 and 2015 are shown in Figure 1 and Figure 2.
Next, apply FMM to the nine images corresponding to the transportation network from 1975 to 2015,
and derive the travel cost,

between the prefectural capitals of each prefecture. In deriving, theMATLAB code of Kroon (2009) used by Allen and Arkolakis (2014) was modified as necessary. The
structure of the code is described below.
64 Each sea route is required when moving from each prefecture other than Honshu to the prefectures of Honshu. In

addition, the cost of movinf land is ser to guarantee the existence of a solution when there is no means of transportation
at a specific point. Further, as will be described later, it is impossible to pass on the sea without a route.

64Step1. Image capture into MATLAB and identification of primary elements.
(a) Capture prefectural office location coordinates (latitude/longitude) and image.
(b) Define the colored range of transportation.
(c) Specify the coordinates of the four corners of the image.
(d) Convert between departure point coordinates and destination coordinates.
Step.2 Allocation of travel speed and estimation of travel time from all departure points
to destinations.
(a) Create a combination of departure point and destination (excluding repetition).
(b) Create transportation costin all image coordinates.
, ,(c) Create movement cost index

from each departure point

to allfrom each departure point

to allimage coordinates.
, ,(d) Extract travel cost index
destinations.

By definition, there are only one means of transportation in each image coordinate, so thetransportation costis the sum of the defining functions of all means of transportation multiplied by

the cost. In this paper, the travel cost index for all combinations derived in this process is used in thefirst-stage estimation.

3. Regional Characteristics
Here, we will explain the method of deriving the regional characteristics used in the previous studies
survey that played an essential role in the first-stage estimation of this paper and the actual analysis
and their properties.

3.1. Discussion
While local amenities play an important role in spatial equilibrium, their concepts differ from study to
study. Here, we will supplement the discussion on regional characteristics in this paper by organizing
previous studies that mention regional characteristics.
According to Rosen (1976) and Roback (1982), which are early studies of the spatial equilibrium
model, workers benefit from nominal wages, housing costs, and regional characteristics. The labormarket is region-specific, with shocks to labor demand reflected in nominal wages, while shocks tolabor supply are reflected in housing costs through population fluctuations. While workers move freely,
the land is a fixed element, and regional characteristics are formulated as an evaluation index forworkers for this fixed element.
Redding and Sturm (2008) also mention regional characteristics. This study is an extension of65

Helpman (1998) to multiple regions, and regional characteristics are defined as an extension of the
non-tradable goods (housing) in his economic geography model. In the new economic geography ofKrugman (1991), these non-tradable fixed elements (or endowed resources) are thought to act as afactor in the dispersion force against the agglomeration forces. The effects of a diversified economy
include intensifying price competition with neighboring companies due to the choice of businesslocation (market congestion effect) and rising living costs due to rising fixed factor prices due topopulation growth (overcrowding effect). However, unlike the spatial equilibrium model of Roback(1982), in new economic geography, lands that do not have the same characteristics in advance areformulated to produce characteristics as a result of economic activity.
Redding and Rossi-Hansberg (2017) discuss this difference. They set the regional characteristics as“any characteristic that makes a location more or less desirable place of residence” and gives a generaldefinition. In particular, they classify these characteristics into exogenous agglomerations thatconsider climate change, water resources, and physical geographic factors, and endogenous
agglomerations that respond to endogenous shocks to the local economy, such as crime rates.
Based on the above theoretical treatment of regional characteristics, what variables are used asregional characteristics in previous studies is summarized (Table A2).

[Table A2]

In Roback (1982) and early study of the spatial equilibrium model, both exogenous habitability suchas weather and endogenous habitability such as crime rate and unemployment rate are used without
distinction. This tendency is also seen in Albouy (2013,2016), who has developed the spatial
equilibrium model of Roback (1982) and analyzed the quality of life (QOL).
The variable selection of regional characteristics in other studies depends on the study's interests andthe formulation of the model. Brueckner and Neumark (2014) and Diamond (2017) formulate natural
factors to be reflected in government taxation behavior, considering only exogenous regional
characteristics. Since the goal of Ahlfeldt et al. (2015) is to estimate residential externalities and labor
externalities using the natural experiment of Berlin’s division and variables related to exogenous
regional characteristics and transportation, variables related to the post-unification government(endogenous regional characteristics) are adopted. Diamond (2016) also argues that widening regional
disparities in worker quality in the United States is due to the accumulation of regional characteristics
associated with highly skilled workers, with variables relating exclusively to endogenous regional
characteristics being selected in her analysis.
What is noteworthy as a tendency seen in all previous studies is that there is not much detail about
how individual regional characteristics affect utility. Within each study, indicators that positively affectutility (green areas, sunny days) and indicators that have a negative effect (noise, crime rate) are often

66

evaluated in the same framework. This tendency may be because regional characteristics are
instrumental variables 65 for other variables formulated in the estimation. To address this issue,
Diamond (2016) divided 13 regional characteristics into six groups and performed principal
component analysis (PCA) in each group for all groups. One virtual regional characteristic is createdby performing the principal component analysis again. This procedure allows Diamond (2016) toanalyze the nature of the regional characteristics themselves.

This section has developed a survey on the economic definition and concrete indicators regional
characteristics (Amenity) in the spatial equilibrium/quantitative spatial model. In particular, themethod that uses the principal component analysis of Diamond (2016) step by step is a technique thatshould be noted when dealing with regional characteristics.

3.2. Derivation process of regional characteristics/ social capital levels
This section describes the process of deriving the regional characteristics “Amenity1,” “Amenity2”,
and “Social Capital” used in Section.4 of this paper. As described in the previous chapter, the method
using principal component analysis by Diamond (2016) was applied in the derivation.
Candidates for regional variables used for each “Amenity” are extracted and processed according todata development status based on ”Social Demographic System; Statistics of Prefectures andMunicipalities.”(Ministry of Internal Affairs and Communications). The variable candidates used for“Social Capital” were “Net Capital Stock by 16 sectors.”(Cabinet Office). The following table (Table
A3) summarizes the details of each of these variables.

[Table A3]

Estimates using the “Basic Resident Register Population Migration Report” (Ministry of Internal
Affairs and Communications) are from 1975 to 2015, and estimates using the “Census Population
Movement Aggregate” (Ministry of Internal Affairs and Communications) are from 1990 to 2015.
Therefore, the economic variables used in this derivation, recorded in ”Social Demographic System;
Statistics of Prefectures and Municipalities,” that cover the target periods are different among eachestimation. As a result, the economic variables used to derive “Amenity” are different. Also, various
economic variables are processed into ratios in order to align the units. Social capital uses theaggregated values66 of 16 departments of “Net Capital Stock” in “Social Capital Stock Estimate data”(Cabinet Office) and is defined by department name.
We created variables "Amenity 1", "Amenity 2", and "Social Capital" by performing the following
65 Albouy (2013, 2016) explicitly mentions regional characteristics as “instrumental variables”.
66 Here, we use the one adjusted to the 2010 level using the time-series of the “Regional Consumer Price Index”(Ministry of Internal Affairs and Communications) of the “Consumer Pice index.”67

procedure on each of these variable sets using Principal Component Analysis (PCA) in Diamond(2016). As the first stage of the procedure, the variables are divided into groups. Regarding theclassification of Diamond (2016) and Table A1, we classified those statistical indicators by prefectures
into four divisions, (1) “Environment,” (2) “Education,” (3) “Urban,” (4) “Welfare.” For social capitals,
five departments were created ((1) “Transportation,” (2) “Living,” (3) “Education,” (4) “Environment,”(5) “Industry”), and variables are classified into these departments. Furthermore, as the second step,principal component analysis (PCA) was performed for each department, and only the first principal
component of each department was extracted. In the third stage, the first principal component of thecharacteristics was extracted by performing PCA on each department's principal components. This
component was designed as the regional characteristic. Regarding “Amenity2”, the sameprocessing was performed after extracting only those having a significant effect on the probability of
population migration between prefectures. Table A4 shows the case classification for each sector ineach estimation and shows the correspondence between economic variables and sector names and thepresence or absence of explanatory power (10% significant) for the probability of population migration.
Although it is not used in the estimation of this paper, it is also added that there is an explanatory
power for the population migration probability of each sector of “Social Capital Net Stock.”[Table A4]

Finally, the nature of these regional characteristics is described. In Table A5, the ranking of the timeseries averages of “Amenitt1” and “Social Capital” by prefecture is stated. As can be inferred from
this, regional characteristics tend to be low in urban areas and high in rural areas, and it can be said
that the regional characteristics in this paper show a diversification effect. Regarding the point that adiversification effect can be seen in social capital, the industrial sector's formula, which is expected tocontribute to the social capital stock, is limited to a simple one using our productivity. It seems thatthe background is the accumulation effect could not be dealt with here.

[Table A5]

68

Charts of Appendix
Table. A1: Geographical reference data of transportation network in"National Land Numerical Information."
Estimated
Movement Speed

Means of Transportation

Code

Series Name

Shinkansen (Bullet Trains)
Railway
Expressway
National Road
Regular Sea Routes
Irregular Sea Routes
Land
Sea

N02
N02
N06
N01
N09
C20
-

Train(Shinkansen)
Railway (Ordinary railway)
High-speed road time series
Road data
Regular Passenger route data
Route data--

240km/h
120km/h
80km/h
60km/h
40km/h
30km/h
15km/hImpassible

Movement
Cost
1
2
3
4
6
8
16
1000

Coloring Value
0~22
23~50
51~72
73~97
98~107108~155
156~189
190~

NOTE) Created by the author using "Numerical Information on Land" (Ministry of Land, Infrastructure, Transport, andTourism).

Table. A2: Regional characteristics in previous studies
StudiesTargetCategory

Regional Characteristics

Roback (1982)

Impact of regional characteristics on wages.

Spatial equilibrium model

Total Crime rate, Particulate Level, Local Unemployment Rate,
Population Density, Heating Degree Days, Total Snowfall, Numberof Cloudy Days, Number of Clear Days

Albouy (2013)

Impact of regional characteristics on QOL.

Spatial equilibrium model

Heating and Cooling Degree Days, Sunshine, Average Slope,
Coastal Proximity, Violent Crimes, Property Crimes, Air Quality
Index, Bars and Restaurants, Arts and Culture Index

Brueckner and Neumark (2014)(Diamond, 2017)

Impact of regional characteristics on public sector
wages.

Spatial equilibrium model

Warm index:
Deviation from the average temperature and 20 degrees Celsius.
Dryness index:
Negative value of average precipitation.
Proximity index:
Average distance from nearby coastlines and major rivers.
Population density:
Population density in the census tract.

Ahlfeldt et al. (2015)

The estimate of accumulation/dispersion effectfor each block.

Spatial equilibrium model

Block and Land area and Centroids, Distance to the Nearest U-Bahn
and S-Bahn Stations, Green areas; Public Parks, Forests and otherGreen Public Areas, Water Areas; Canal, River, and Lade, Schools,
Noise, Land Use; Commercial, Residential, Industrial and Mixed,
Second War Destruction, Listed Buildings, Urban Regeneration
Policies Post- Reunification, Government Buildings PostReunification, Wages

Diamond (2016)

Factors and impacts of regional disparities inlabor skills.

Quantitative Spatial model

Apparel Stores, Eating and Drinking Places and Movie Theaters per1000 Residents, Busses per Capita, Public Transit Index, Average
Annual Daily Traffic-Interstates, Average Annual Daily Traffic-Major
Urban Roads, Poverty and Violent Crimes per 1000 Residents, EPA
Air Quality Index, Government Spending on Parks and Recreation per
Capita, Government Spending on K-12 Education, Pupils Enrolled in
Public K-12 Education, Number of Enrolled in Public K-12
Education, Patents Per Capita, Employment Rate

Albouy (2016)

Impact of regional characteristics on quality oflife, wages, and corporate productivity.

Quantitative Spatial model

Heating and Cooling Degree Days, Sunshine, Average Slope, Coastal
Proximity, Latitude, Wheaton Residential Land Use Regulatory
Index; from Gyokuro, Saiz, and Summers (2008)

NOTE) This table was made from the theoretical model of this paper.

69

Table. A3: Source and Processing Procedure of Regional Characteristics
Name of Variables

Processing Process

Statistical Series Name

Source“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Labor Force Survey"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Labor Force Survey"(Ministry of Internal Affairs and Communications)"White Paper on Police" (National Police Agency)"Census Report"(Ministry of Internal Affairs and Communications)"Community Health Business Report(Health Center Management Report)""Census Report"(Ministry of Internal Affairs and Communications)"Community Health Business Report(Health Center Management Report)""Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)

Amenities (Basic Resident Register Population)
Average Temperature-“Average Temperature”

Highest Temperature-“Highest Temperature”

Lowest Temperature-“Lowest Temperature”

Percentage of Daylight Hours“Daylight Hours(yearly)”/”Hours(yearly)”“Daylight Hours(Yearly)”

Block Park Area Ratio"Block park area" / "total area"“Block park area.”"Total area."

Neighborhood Park Area Ratio"Neighborhood park area" / "total area."“Neighborhood park area.”"Total area."Sports Park Area Ratio"Athletic park area" / "total area"“Athletic park area.”"Total area."

Green Space Area Ratio"Green area" / "Total area"“Green area.”"Total area."

Green Road Area Ratio"Green road area" / "total area"“Green road area.”"Total area."

Highschool Graduation Rate

“Final education population (high school / old middle
school)” / “total population”“Final Education Population(High School / Old Middle School)”"Total population."

Vocational School Graduation Rate

"Final education population (junior college / technicalcollege)" / "total population"“Final Education Population(Junior College / Technical College)”"Total population."

University/Graduate School
Graduation Rate

“Final education population (university / graduate
school)” / “total population”“Final Education Population(University / Graduate School)”"Total population."Labor Force Population Ratio“Labor population” / “total population”“Labor population.”"Total population."

Unemployment Rate---

“Unemployment rate.”

Number of crimes per 1000 people

1000 x "Number of criminal offenses recognized" /"Total population""Number of criminal offenses recognized.""Total population."

Number of pregnant women
instructors per 1000

1000 x "Pregnant women's health guidance staff.""Pregnant women's health guidance staff.""Total population."

Number of infant health instructors
per 1000/"Total population""Infant health guidance instructed personnel.""Total population."

Number of hospitals per 1000
people1000 x "Infant Health Guidance Instructed Personnel.""Number of hospitals.""Total population."

Number of clinics per 1000/"Total population""Number of general clinics.""Total population."

Number of hospital beds per 1000
people1000 x "number of hospitals" / "total population""Number of hospital beds.""Total population."

Number of clinic beds per 1000
people1000 x "Number of general clinics" / "Total
population""Number of beds in general clinics.""Total population.""Main lake area" / "Total area."“Main lake area.”"Total area."

Amenities (Census)
Lake Area Ratio

70"Area adjustment by prefecture, city, ward, town, and village."(Geographical Survey Institute)"Area adjustment by prefecture, city, ward, town, and village."(Geographical Survey Institute)Forest Area Ratio“Forest area.”"Total area.""Forest area" / "Total area"

Average Temperature-“Average Temperature”

Highest Temperature-“Highest Temperature”

Lowest Temperature-“Lowest Temperature”

Average Temperature-“Average Temperature”

Number of Retail Stores per1000 People

1000 x "Number of retail stores" / "Total population""Number of retail stores.""Total population."

Sales Floor Area per Retail Store"Retail sales floor area" / "Number of retail stores.""Retail sales floor area.""Number of retail stores."

Road Density"Actual road extension" / "total area"“Actual road extension.”"Total area."

Block Park Area Ratio"Block park area" / "total area"“Block park area.”"Total area."

Neighborhood Park Area Ratio"Neighborhood park area" / "total area."“Neighborhood park area.”"Total area."Sports Park Area Ratio"Athletic park area" / "total area"“Athletic park area.”"Total area."

Green Space Area Ratio"Green area" / "Total area"“Green area.”"Total area."

Green Road Area Ratio"Green road area" / "total area"“Green road area.”"Total area."

Highschool Graduation Rate

“Final education population (high school / old middle
school)” / “total population”“Final Education Population(High School / Old Middle School)”"Total population."

Vocational School Graduation Rate

"Final education population (junior college / technicalcollege)" / "total population"“Final Education Population(Junior College / Technical College)”"Total population."

University/Graduate
Graduation Rate

“ Final education population (university / graduate
school)” / “total population”“Final Education Population(University / Graduate School)”"Total population."

SchoolJunior High School Enrollment Rate
High School Enrollment Rate-

“Junior high school enrollment rate”-“High school student enrollment rate”

Learning Time Ratio“Average study time” / 24

“Average time of study”

Commuting / School Time Ratio“Average time for commuting to work/school” / 24

Labor Force Population Ratio“Labor population” / “total population”“Average time for commuting to work /school”“Labor population.”"Total population."

Unemployment Rate---

“Unemployment rate.”

Number of crimes per 1000 people

1000 x "Number of criminal offenses recognized" /"Total population""Number of criminal offenses recognized.""Total population."

Birth to Population Ratio“Number of births” / “total population.”“Number of births.”"Total population."

1000 x "Pregnant women's health guidance staff.""Pregnant women's health guidance staff.""Total population."

Number of infant health instructors
per 1000/"Total population""Infant health guidance instructed personnel.""Total population."

Number of hospitals per 1000 people

1000 x "Infant Health Guidance Instructed Personnel.""Number of hospitals.""Total population."

Number of pregnant
instructors per 1000

women

71

"Area adjustment by prefecture, city, ward, town, and village."(Geographical Survey Institute)"Area adjustment by prefecture, city, ward, town, and village."(Geographical Survey Institute)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)“Meteorological Observatory Observation Data”(Japan Meteorological Agency)"Economic Census" Business / Corporate Statistics Survey "(Ministry of Internal Affairs and Communications)"Census Report" (Ministry of Internal Affairs andCommunications)"Commercial Statistics" (Ministry of Economy, Trade, andIndustry)"Economic Census" Business / Corporate Statistics Survey "(Ministry of Internal Affairs and Communications)"Annual Road Statistics" (Ministry of Land, Infrastructure,
Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"City Planning Status Survey" City Planning Annual Report "(Ministry of Land, Infrastructure, Transport, and Tourism)"Area adjustment by prefecture, city, ward, town, and village"(Geographical Survey Institute)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Census Report (Large-scale Survey)"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"School Basic Survey" (Ministry of Education, Culture,
Sports, Science, and Technology)"School Basic Survey" (Ministry of Education, Culture,
Sports, Science, and Technology)"Social Life Basic Survey" (Ministry of Internal Affairs andCommunications)"Social Life Basic Survey" (Ministry of Internal Affairs andCommunications)"Labor Force Survey"(Ministry of Internal Affairs and Communications)"Census Report"(Ministry of Internal Affairs and Communications)"Labor Force Survey"(Ministry of Internal Affairs and Communications)"White Paper on Police" (National Police Agency)"Census Report"(Ministry of Internal Affairs and Communications)"Vital Statistics"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Community Health Business Report(Health Center Management Report)""Census Report"(Ministry of Internal Affairs and Communications)"Community Health Business Report(Health Center Management Report)""Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)Number of clinics per 1000/"Total population""Number of general clinics.""Total population."

Number of hospital beds per 1000
people1000 x "number of hospitals" / "total population""Number of hospital beds.""Total population."

Number of clinic beds per 1000
people1000 x "Number of general clinics" / "Total population""Number of beds in general clinics.""Total population."

Number of Doctors per 1000

1000 x "number of doctors" / "total population""Number of doctors.""Total population."

Medical Expenses (Logarithmic)

Logarithmic conversion of "medical expenses(national medical expenses)"“Medical expenses(national medical expenses)”"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Medical Facility Dynamics Survey"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"Physician / Dentist / Pharmacist Statistics"(Ministry of Health, Labor and Welfare)"Census Report"(Ministry of Internal Affairs and Communications)"National Medical Expenses"(Ministry of Health, Labor and Welfare)

9. Schooling Facilities
10. Social Education
11. Hydraulic Engineering/Restoration
12. Coast

13. Agriculture/Forestry/Fisheries
14. National Forest
15. Industrial Water
16. Government Buildings

Social Capital ( ”Net Social Capital Stock by 16 sectors”)
1. Road
1. Port
2. Aviation
3. Pulbic Housing

5. Sewerage
6. Waste
7. Water Supply
8. City Park

NOTE) This table was made using ”Social Demographic System; Statistics of Prefectures and Municipalities.” (Ministry of Internal
Affairs and Communications) and “Net Capital Stock by 16 sectors”(Cabinet Office).

Table. A4: Classification of regional characteristics
Name of Variables

Divisions

Significance(10%)

Amenities (Basic Resident Register Population)
Average Temperature
Highest Temperature
Lowest Temperature
Percentage of Daylight Hours
Block Park Area Ratio
Neighborhood Park Area Ratio
Sports Park Area Ratio
Green Space Area Ratio
Green Road Area Ratio
Highschool Graduation Rate
Vocational School Graduation Rate
University/Graduate School Graduation Rate
Labor Force Population Ratio
Unemployment Rate
Number of crimes per 1000 people
Number of pregnant women instructors per
1000
Number of infant health instructors per 1000
Number of hospitals per 1000 people
Number of clinics per 1000
Number of hospital beds per 1000 people
Number of clinic beds per 1000 people

○
○

(1)

Environment(2)

Education

(3)

Urban

(4)

Welfare

○
○
○
○
○
○
○
○

Amenities (Census)
Lake Area Ratio
Forest Area Ratio
Average Temperature
Highest Temperature
Lowest Temperature
Average Temperature
Number of Retail Stores per1000 People
Sales Floor Area per Retail Store
Road Density
Block Park Area Ratio
Neighborhood Park Area Ratio
Sports Park Area Ratio
Green Space Area Ratio
Green Road Area Ratio
Highschool Graduation Rate
Vocational School Graduation Rate
University/Graduate School Graduation Rate
Junior High School Enrollment Rate
High School Enrollment Rate○
(1)

Environment

(3)

Urban

(1)

Environment

○
○

○

○

(2)

Education○72

Learning Time Ratio
Commuting / School Time Ratio
Labor Force Population Ratio
Unemployment Rate
Number of crimes per 1000 people
Birth to Population Ratio
Number of pregnant women instructors per
1000
Number of infant health instructors per 1000
Number of hospitals per 1000 people
Number of clinics per 1000
Number of hospital beds per 1000 people
Number of clinic beds per 1000 people
Number of Doctors per 1000
Medical Expenses (Logarithmic)

(3)

○
○

Urban

○
○
○
(4)

Welfare○○Social Capital
Road
Port
Aviation
Public Housing
Sewerage
Waste
Water Supply
City Park
Schooling Facilities
Social Education
Hydraulic Engineering/Restoration
Coast
Agriculture/Foresty/Fishcers
National Forest
Industrial Water
Government Buildings(1)

Transportation○(2)

Living(3)

Education(4)

Environment

(5)
(6)

Industry
Living

○
○
○
○
○
○
○

NOTE) Created by the author based on Diamond (2016).

Table. A5: Local Characteristics
Amenities
Top 5 Prefectures
Bottom 5 Prefectures

Social Capitals
Top 5 Prefectures
Bottom 5 Prefectures

1. Nagano1. Iwate
2. Yamagata
3. Shizuoka
4. Aomori

1. Tottori
2. Yamanashi
3. Shimane
4. Nara5. Miyazaki

43. Fukuoka
44. Nara
45. Osaka
46. Kyoto
47. Tokyo

43. Kanagawa
44. Hokkaido
45. Aichi
46. Osaka
47. Tokyo

NOTE) The author created this table.

73