An Empirical Study on the Japanese Prefectural
Economies Using Bayesian Statistical Models

Hideo Noda*
Koki Kyo†
Research Group of Economics and Management
No. 2009-E01
2009.1*†

Yamagata University (E-mail: noda@human.kj.yamagata-u.ac.jp)
Obihiro University of Agriculture and Veterinary Medicine (E-mail: kyo@obihiro.ac.jp)

An Empirical Study on the Japanese Prefectural
Economies Using Bayesian Statistical Models∗Hideo Noda†
Faculty of Literature and Social Sciences, Yamagata University, Japan

Koki Kyo
Obihiro University of Agriculture and Veterinary Medicine, Japan

Abstract
To examine trends in total factor productivity (TFP) and structural features in Japanese prefectures, this paper presents a new quantitative approachbased on a framework of Bayesian statistical models for the regional production
functions. When we apply our proposed approach to a regional analysis of the
postwar Japanese economy, the following results are obtained. As a regional
characteristic in terms of factor elasticities, through the periods of 1955-1973
and 1974-1995, the higher the public capital elasticities, the lower the private
capital elasticities in regions. The results also illustrate noteworthy changes in
TFP trends before and after the period of ﬁrst oil crisis. Speciﬁcally, the TFP
in each prefectural economy shows an upward tendency during the period fromthe mid-1950s to the early 1970s. Overall, however, it stagnated in the period
between the mid-1970s and the mid-1990s.
Keywords: Structural change, Regional growth in Japan, Total factor productivity, Regional production function, Smoothness priors
JEL classification: C11; O18; R11∗

The authors would like to thank Genshiro Kitagawa, Keisuke Osumi, Yoko Konishi, Yoshimitsu Yokoyama, and Minoru Hayashida for their useful comments on earlier drafts. This research
was supported by grant from the Institute of Statistical Mathematics (ISM) Cooperative Research
Program (2008-ISM·CRP-2013).
†
Corresponding author. E-mail: noda@human.kj.yamagata-u.ac.jp1

1

Introduction

Achieving sustained economic growth is an important issue that has become a policy
objective of many countries’ governments. Japan is no exception. Under the recent
economic conditions, there have been strong demands in Japan for policy proposals
that contribute to the promotion of economic growth.
When considering such policy issues, quantitative analysis of economic growth
based on statistical methodology is essential. Empirical studies of economic growth
are therefore becoming more important these days. Economic growth analyses done
after Solow (1956) often regard capital, labor and total factor productivity (TFP)
as the fundamental sources of economic growth. Of these sources, the chronologicalchanges in capital and labor are identiﬁable from the corresponding data. The
trends in TFP, however, cannot be observed directly from the data. In empiricalresearch of economic growth, therefore, the method used to measure TFP behavior
constitutes the crucial element to obtain and make useful implications. When we
review empirical studies of the Japanese economy, it would appear that the analyses
at the regional level are relatively-scarce compared with those at macro and industry
levels. Thus, the purpose of this paper is twofold: One is the development of a newquantitative approach for the analysis of regional economic growth. The other is the
elucidation of trends in TFP and of characteristic features in Japanese prefectural
economies, by the application of our proposed method.
For empirical studies of the Japanese economy using regional data, there aresome earlier report; for example, Akino and Hayami (1974) estimated the agricultural production function in Japan for the period 1880-1965 using cross-prefecturaldata. They also attempted to account for growth in agricultural output with estimates of factor elasticities. As a result, they conﬁrmed that about half of thelong-term rate of growth in agricultural output was accounted for by changes infour conventional inputs (i.e., labor, land, capital, and current inputs); one quarterby an increase in the level of education; and another quarter by an increase in public expenditure for agricultural research and extensions. Barro and Sala-i-Martin(1992) compared the process of regional economic growth and absolute convergenceacross prefectures in Japan and states in the United States, based on a frameworkof a neoclassical growth model. Absolute convergence means that poor prefecturesand states grow faster. Barro and Sala-i-Martin (1992) used two regional data sets,
speciﬁcally the 47 prefectures of Japan and the 48 states of the United States, to ex-2amine the convergence hypothesis. As a result, they found evidence of convergencein both countries for the periods between 1930 and 1987 for Japan and between 1880
and 1988 for the United States. Asada (1998) investigated the factors’ contributing
to output growth in prefectural economies based on a conventional growth accounting approach, using prefectural data for the period 1975-1990. It was conﬁrmedthat in many prefectures, the contribution of capital was relatively high; hence, themajority of prefectures corresponded to a capital-driven economy. Fukao and Yue(2000) estimated prefectural production functions using prefectural data for the period 1955-1995, and implemented a growth accounting method with estimates offactor elasticities. Based on their estimation results, they re-examined the convergence hypothesis of Barro and Sala-i-Martin (1992), and then concluded that theconvergence hypothesis was not be suitable for application to the Japanese economy.
Yamano and Ohkawara (2000) investigated the eﬀects of public investment on theregional economies of Japan using prefectural data for the period from 1970 to 1994.
Consequently they showed that the marginal productivity of public capital had recently declined in most depressed regions, whereas productivity in developed regions(e.g., Tokyo, Osaka) had increased slightly. Shioji (2001) constructed a conditional
convergence model and then estimated the long run eﬀect of public capital on output per capita using regional data of Japan and the United States; for the periodof 1955-1995 for Japan and 1963-1993 for the United States. As well, Shioji (2001)
performed a growth accounting exercise using the estimates obtained by econometricanalyses. Consequently, the estimation with disaggregated public capital indicatedthat the eﬀects of public capital on regional output in Japan and the United States
were not nearly as dissimilar as they appeared when one considered only aggregate
public capital. Utilizing growth accounting, it was conﬁrmed that public capital had
made modest contributions to the postwar growth of both Japan and the UnitedStates. Kataoka (2005), who focused on the period 1955-2000, examined the relationship between the regional allocation of public capital and regional and national
output in the postwar Japanese economy using the prefectural production function
approach. The results of his empirical analysis indicated that the Japan’s government allocated more public investment to higher productivity regions prior to theﬁrst oil crisis; whereas, after the onset of the ﬁrst oil crisis, more public investment was allocated to lower productivity regions. Miyara and Fukushige (2008)
estimated prefectural production functions with diﬀerently aggregated public capital using annual data from 1976 to 1997. Their estimation results indicated that3

productive public infrastructure diﬀers between prefectures. From the estimation
results, they note that policy-makers should consider prefectural diﬀerences in thetypes of productive public infrastructure.
From the viewpoint of the approaches of empirical study, earlier reports canbe roughly classiﬁed into two types. The ﬁrst type uses growth accounting and/or
econometric estimations of production functions (e.g., Akino and Hayami, 1974;
Asada, 1998; Fukao and Yue, 2000; Yamano and Ohkawara, 2000; Kataoka, 2005;
Miyara and Fukushige, 2008). The second type is mainly based on growth regression
models derived from economic growth theories (e.g., Barro and Sala-i-Martin, 1992;
Shioji, 2001).
However these approaches might be inadequate. First, growth accounting postulates that under the assumption of a perfectly competitive markets for production
factors, marginal productivity of each production factor is consistent with the corresponding factor price. Then, it indirectly measures the elasticity of output withrespect to production factors using the income shares of such factors. There is thesame assumption behind econometric estimations of growth regression models basedon neoclassical growth theory. In reality, however, such consistency might not always be achieved, and using indirect data that deviate from true values possibly
results in incorrect conclusions.
With regard the direct estimation of production function models, the condition
that the marginal productivity of each production factor coincides with the corresponding factor price need not be required. The traditional regression approach of
production functions is unable though to appropriately grasp trends in TFP. Because, for convenience, the approach generally assumes that log-TFP is a constant,a linear function of time or a quadratic function of time. Such ad hoc assumptions
regarding TFP trends imply that any understanding of the changes in TFP overtime is, in eﬀect, excluded from the analysis.
Considering such issues, Kyo and Noda (2006) conducted a statistical analysis of
TFP trends in Japan based on a Bayesian approach. Speciﬁcally, a Bayesian statistical model of Japanese regional production functions that regards TFP as a timevarying parameter was developed and a smoothness priors approach adopted whenestimating TFP trends; thereby, examining TFP changes in the regional economies
in Japan. Although Kyo and Noda (2006) took into account the structural heterogeneity among the regions, which means the case in which the elasticity of output
with respect to production factors varied across regions, the value of the elastic-4

ity was assumed to be constant throughout the period of their analysis. In otherwords, the analysis did not incorporate the possibility of structural change in regional
economies resulting from some kind of exogenous shock. From the perspective ofa modeling that is closer to reality, we extend the model of Kyo and Noda (2006)
to include structural change. As well, based on the extended model, we make an
attempt to estimate TFP trends and analyze the characteristic features of regional
economic structures. The approach using the Bayesian statistical model proposedin this study constitutes a new attempt that overcomes some obstacles that havehindered existing approaches as mentioned above. It also supports more detailed
analyses than existing approaches do. Therefore, ﬁndings that have been overlooked in earlier studies are expected to be uncovered. The ﬁndings of this study
are expected to contribute to the development of methods of quantitative analysis
applicable to regional economic research.
In this study, we examine the possibility of structural change between two subperiods, 1955-1973 and 1974-1995, using the Japanese Prefectural Database developed in the analysis of Fukao and Yue (2000) that covers annual data between 1955
and 1995. Estimating the elasticity of output with respect to each production factor, the results indicate that interregional variance of physical capital elasticity inthe private sector decreased and the variance of physical capital elasticity in the
public sector and human capital elasticity increased in both periods. The two subperiods also showed a strong negative correlation between physical capital elasticity
in the private sector and that in the public sector. We also ﬁnd a clear contrast
in TFP trends before and after the ﬁrst oil crisis. Speciﬁcally, in each prefectural
economy, TFP increased steadily during the period until the ﬁrst oil crisis, whereas
subsequently it stagnated.
The rest of this paper is organized as follows. In Section 2, we construct statisticalmodels for regional production functions. Section 3 describes the procedure used
for parameter estimations in the model. In Section 4, the statistical model and the
proposed approach for the estimation is applied by utilizing Japanese prefecturaldata. Then the estimation results are presented and discussed. Finally, in Section
5, we present a summary of our study and note future research directions.

52

Setup of the Model

2.1

The Regional Production Function Model

Suppose a situation in which a country’s economy is divided into m regions. In thei-th (i = 1, 2, . . . , m) region, given the technological state, output Qi (t) is produced
by inputting Ki (t) as the physical capital in the private sector (hereinafter called
private capital), Gi (t) as the physical capital in the public sector (hereinafter called
public capital), Hi (t) as human capital, and Li (t) as labor in the period t.
In this study, we apply Milbourne, Otto, and Voss’s (2003) speciﬁcation of macroproduction function to our regional analysis. Speciﬁcally, the regional production
function of the i-th region takes the following form.11−αi −βi −γi

Qi (t) = Ki (t)αi Gi (t)βi Hi (t)γi A∗i (t)Li (t)(i = 1, 2, . . . , m).

(1)

In Eq. (1), αi , βi and γi represent unknown parameters. Here it is assumed thatαi > 0, βi > 0, γi > 0 and αi + βi + γi < 1. In addition, A∗i (t) expresses alabor-augmenting measure of productivity.
It should be noted that the productivity indicator, A∗i (t), is aﬀected not only by
the technological level that reﬂects product and process innovations, but also by diverse factors such as national and regional institutions, central and local government
policies, economies of scale, and economies of agglomeration.2 Such a productivity
indicator might be divided broadly between the component reﬂecting the factors thatare unique to each regional economy such as the level of technological infrastructureor the eﬃciency of regional industries, the situation of industrial agglomeration andnatural conditions, and the part that reﬂects factors that are common among all
regions such as widely shared technical knowledge, the extent of the rule of law inthe country and the nationwide policies of the central government. From such a perspective, we assume that A∗i (t) includes two components: The component Ci that
does not change over time but which diﬀers among regions; the other componentA(t) that is common among the regions but which changes over time. That is, weconsider the following relation.
A∗i (t) = Ci A(t).
1
Milbourne, Otto, and Voss (2003), however, considered M types of public capital. Our mode
therefore corresponds to the case of M = 1 in their production function.
2
Hall and Jones (1999) collectively refer to various government policies and systems that aﬀect
the activities of economic agents as social infrastructure. Using cross-national analyses, they conﬁrmed that social infrastructure had an important role in long-term economic disparities acrosscountries.

6Here we normalize as A(0) = 1. Therefore, the relation A∗i (0) = Ci holds, and Ci
can be interpreted as the initial value of a labor-augmenting measure of productivityof the i-th region. When we deﬁne θi ≡ 1 − (αi + βi + γi ), the TFP of the i-th region
is expressed as {Ci A(t)}θi . Because A(t), a component of TFP, takes the samevalue in all regions and relies only on time, we refer to it as a time-varying factor of
productivity.
Logarithmic transformation of Eq. (1) is given byyi (t) = θi μi + αi x1i (t) + βi x2i (t) + γi x3i (t) + θi a(t),(2)

where yi (t) ≡ ln{Qi (t)/Li (t)}, x1i (t) ≡ ln{Ki (t)/Li (t)}, x2i (t) ≡ ln{Gi (t)/Li (t)},x3i (t) ≡ ln{Hi (t)/Li (t)}, a(t) ≡ ln A(t) and μi ≡ ln Ci . Furthermore, we augment
Eq. (2) to include a random disturbance:yi (t) = θi μi + αi x1i (t) + βi x2i (t) + γi x3i (t) + θi a(t) + εi (t),(3)

where εi (t) is a disturbance term. Here we assume that if i = j or t1 = t2 , thenεi (t1 ) and εj (t2 ) are independent of each other and that εi (t) ∼ N(0, σ 2 ) (i =1, 2, . . . , m). The symbol σ 2 denotes the variance of the disturbance, which is treatedas an unknown parameter.
We refer to the parameters that reﬂect the economic structure of the i-th region,
αi , βi , γi and θi , as structural parameters. In this paper, we setup a model inwhich the structural parameters take diﬀerent values before and after the period
at which changes to the economic structure occurred. Consider the annual data of
t = 1, 2, . . . , n for each variable, we assume that a structural change occurred during
period T + 1, and express the structural parameters from period 1 to period T as
(1)

(1)

(1)

αi , βi , γi

(1)

(2)

(2)

(2)

and θi ; from period T + 1 to period n as αi , βi , γi(2)and θi .

Then, Eq. (3) can be rewritten as follows:
(1)

(1)

(1)

(1)

(1)

yi (t) = θi μi + αi x1i (t) + βi x2i (t) + γi x3i (t) + θi a(t) + εi (t),(t = 1, 2, . . . , T )
(2)

(2)

(2)

(2)

(2)

yi (t) = θi μi + αi x1i (t) + βi x2i (t) + γi x3i (t) + θi a(t) + εi (t).
(t = T + 1, T + 2, . . . , n)(4)

In contrast to the model in Eq. (4), which takes into account structural change, themodel in Eq. (3) can be interpreted as a model without structural change.

7

2.2

Parameters and Prior Distribution

Next, we describe the treatment of parameters and the construction of prior distribution. In this paper, the parameters of the model in Eq. (4) are classiﬁed
(1)

(1)

(1)

into two types; each is estimated using a diﬀerent method. First, αi , βi , γi ,
(1)

(2)

(2)

(2)

(2)

θi , αi , βi , γi , θi , μi (i = 1, 2, . . . , m) and σ 2 are treated as constant parameters and are estimated using the maximum likelihood method. Note that it(j)

is constrained by conditions as follows: αi
(j)
(j)
(j)
(j)
αi + βi + γi + θi = 1

(j)

(j)

> 0; βi> 0; γi(j)> 0; θi> 0 and(i = 1, 2, . . . , m; j = 1, 2). Then, a(t) (t = 1, 2, . . . , n) is

treated as a random variable parameter and is estimated using the Bayesian method.
Speciﬁcally, we introduce the smoothness priors approach to obtain a stable estimateof a(t) (t = 1, 2, . . . , n).3 Thus, we assume that a(t) changes smoothly over time, anduse the following ﬁrst order stochastic diﬀerence equation as a kind of smoothness
prior.
a(t) − a(t − 1) = ν(t),ν(t) ∼ N(0, σ 2 /η2 ).

(t = 1, 2, . . . , n)(5)

On the basis of the assumption that A(0) = 1, the relation a(0) = ln A(0) = 0 holds.
In Eq. (5), ν(t) represents a random disturbance with σ 2 /η2 being an unknown
variance. We also assume that ν(t1 ) and ν(t2 ) are independent of each other for
t1 = t2 and that εi (t) and ν(t) are independent of each other for any i and t.

3

Estimation Procedures of Parameters

3.1

Matrix-Vector Expression of the Model

The matrix-vector expression of the model in Eq. (4) is as follows:y i = μi bi + z i (S i ) + B i a + εi ,εi ∼ N(0n , σ 2 I n ). (i = 1, 2, . . . , m)

(6)

In Eq. (6), y i = (yi (1), yi (2), · · · , yi (n))t , a = (a(1), a(2), . . . , a(n))t , and εi =(εi (1), εi (2), . . . , εi (n))t . As well, bi and B i respectively denote an n-dimensional
vector and a diagonal matrix with the order n, which are deﬁned as


bi =

(1)

θi 1 T(2)
θi 1n−T





,

Bi =

(1)

θi I TOO(2)
θi I n−T,where 1T and 1n−T , respectively, denote T -dimensional and n − T -dimensional vec(1)

(1)

(1)

(1)

tors whose elements being 1, and O represent a zero matrix. S i = {αi , βi , γi , θi ,3

See Kitagawa and Gersch (1996) for details of smoothness priors.

8

(2)

(2)

(2)

(2)

αi , βi , γi , θi } is a set of structural parameters in the i-th region, and z i (S i )is deﬁned as follows:
⎡

(1)

(1)

⎤

(1)

αi x1i (1) + βi x2i (1) + γi x3i (1)
(1)
(1)
(1)
αi x1i (2) + βi x2i (2) + γi x3i (2)
..
.

⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
(1)
(1)
(1)
⎢
⎥
αi x1i (T ) + βi x2i (T ) + γi x3i (T )⎥.
z i (S i ) = ⎢
⎢ (2)
⎥
(2)
(2)
⎢ αi x1i (T + 1) + βi x2i (T + 1) + γi x3i (T + 1) ⎥⎢ (2)⎥⎢ α x (T + 2) + β (2) x (T + 2) + γ (2) x (T + 2) ⎥2i
3i
⎢ i 1i
⎥
i
i
⎢
⎥
..
⎢
⎥
.
⎣
⎦
(2)

(2)

(2)

αi x1i (n) + βi x2i (n) + γi x3i (n)In addition, I n represents the identity matrix of order n, and 0n denotes an ndimensional vector of zeros.
Here the prior of a in Eq. (5) is expressed as follows:
Da = ν,ν ∼ N(0n , (σ 2 /η2 )I n ).

(7)

In Eq. (7), ν = (ν(1), ν(2) . . . , ν(n))t and D denotes a square matrix of order n,which is deﬁned as

3.2

⎡

⎤

··· ··· 0⎢.. ⎥⎢ −1 1 . . .
. ⎥
⎢
⎥
⎢
.. ⎥
.. ..
D=⎢
.
. . ⎥
⎢ 0 −1⎥.
⎢ .
⎥.. .. ..
⎢ .
⎥
.
.
. 0 ⎦
⎣ .
0 · · · 0 −1 1
10

The Estimation Scheme I

We now introduce a procedure to estimate parameters other than structural ones,
based on the given values of the structural parameters. It is as an application of
the procedure proposed by Jiang (1995) that is based on the methodology of theBayesian modeling of Akaike (1980).
First, for i = 1, 2, . . . , m, the set of estimates of the structural parameters at the
(j)

(j)

(j)

estimation stage j is expressed as S i , and it is deﬁned as S (j) = {S 1 , S 2 , . . . , S (j)m }.
If S (j) is given, then a set of Bayesian linear models for a is constructed from Eqs.
(6) and (7). Here, μ = (μ1 , μ2 , . . . , μm )t , σ 2 and η are treated as hyperparameters. The posterior distribution of a is obtained immediately from this model; themode or average is expressed as a. Note that a and μ, which is the estimate of μ,depending on η and S (j) .

9

(j)

Now, given that S iand the value of constant η, a? and μ are simultaneously

derived by the following equation.4a(η, S (j) )μ(η, S (j) )= (W t W )−1 W t u,

where W denotes an (m+1)n×(n+m) matrix and u denotes an (m+1)n-dimensional
vector, which are deﬁned as
⎡

⎤

B1

b1

ηD

0n
..
.
..
.
0n

⎢
⎢ B
2
⎢
⎢ .
⎢
W = ⎢ ..
⎢
⎢
⎣ Bm

⎡

0n · · · 0n
. ⎥..
. .. ⎥
b2
⎥
⎥
.. ..
.
. 0n ⎥⎥,⎥..
⎥. bm ⎦· · · · · · 0n(j)y 1 − z 1 (S 1 )(j)y 2 − z 2 (S 2 )
..
.

⎤

⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
u=⎢
⎥.
⎢
⎥
⎢
⎥
⎣ y m − z m (S (j)
⎦
)
m

0n

Further, the estimate of σ 2 can be obtained by1
σ 2 (η, S (j) ) =
u−W
mn



a(η, S (j) )μ(η, S (j) )

 2

.

(j)

Given S i , the log-likelihood of η is deﬁned as

1 (η, S

(j)





 n
mn
1 
)=−
ln 2πσ 2 (η, S (j) ) + 1 − ln det(V t V ) + ln(η2 ).
2
2
2

(8)

In Eq. (8), V is represented as follows (m + 1)n × n matrix.
⎡

B1
⎢ B
2
⎢
⎢ .
.
V =⎢
⎢ .

⎤

⎥
⎥
⎥
⎥.
⎥
⎢
⎥
⎣ Bm ⎦

ηD,The estimate of η, that is η(S (j) ), is obtained numerically by maximizing 1 (η, S (j) ).
The estimates of the corresponding a, μ and σ 2 are given bya(S (j) ) = a(η(S (j) ), S (j) ),4

μ(S (j) ) = μ(η(S (j) ), S (j) ),(9)

σ 2 (S (j) ) = σ 2 (η(S (j) ), S (j) ).

(10)

See Jiang (1995) for details.

10

3.3

The Estimation Scheme II

Next, we present a scheme to obtain the estimates for the structural parameters
for given values of parameters other than structural ones. Because simultaneously
estimating the structural parameters for all regions is extremely diﬃcult, here we
consider estimating the structural parameters of each region one by one.
A set of all sample data excluding y i is expressed as Y (i) (i = 1, 2, . . . , m). As
shown in Eqs. (9) and (10), when S (j) is given, η(S (j) ), μ(S (j) ) and σ 2 (S (j) ) can beobtained. From Eqs. (6) and (7), in addition, the posterior distribution of a basedon Y (i) can be obtained as follows.


a|Y




−1
1
(i) t
(i)
∼N a ,(W)
W,
σ 2 (S (j) )
(i)

(i)

where



a(i) = (W (i) )t W (i)

−1

(11)

(W (i) )t u(i) .

(12)

In Eqs. (11) and (12), W (i) and u(i) , respectively, stand for an mn × n matrix andan mn-dimensional vector. Speciﬁcally, W (1) , W (2) , . . . , W (m) , u(1) , u(2) , . . . , u(m)are deﬁned as⎡⎢⎢⎢
W (1) = ⎢
⎢
⎢
⎣

⎡

⎤

B2
B3
..
.
Bm
η(S (j) )D

⎡

⎥
⎢
⎥
⎢
⎥
⎢
⎥ , W (2) = ⎢
⎥
⎢
⎥
⎢
⎦
⎣

(j)

y 2 − μ2 (S (j) )b2 − z 2 (S 2 )(j)y 3 − μ3 (S (j) )b3 − z 3 (S 3 )
..
.

⎤

B1
B3
..
.
Bm
η(S (j) )D
⎤

⎡

⎥
⎢
⎥
⎢
⎥
⎢
⎥ , · · · , W (m) = ⎢
⎥
⎢
⎥
⎢
⎦
⎣

⎡

B1
B2
..
.
B m−1
η(S (j) )D

⎤
⎥
⎥
⎥
⎥;
⎥
⎥
⎦

(j)

y 1 − μ1 (S (j) )b1 − z 1 (S 1 )(j)y 3 − μ3 (S (j) )b3 − z 3 (S 3 )
..
.

⎤

⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
(1)
(2)
u =⎢
⎥, u = ⎢
⎥,
⎢
⎥
⎢
⎥
⎢
⎥
⎢
(j)
(j) ⎥
⎣ y m − μm (S (j) )bm − z m (S (j)
⎦
⎣
⎦
)
−
μ
(S
)b
−
z
(S
)
y
m
m
m
m
m
m

0n

⎡

0n

(j)

y 1 − μ1 (S (j) )b1 − z 1 (S 1 )(j)y 2 − μ2 (S (j) )b2 − z 2 (S 2 )
..
.

⎤

⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
(m)
=⎢
···,u
⎥.
⎢
⎥
⎢
⎥
(j)
⎣ y m−1 − μm−1 (S (j) )bm−1 − z m−1 (S m−1 ) ⎦0nHere μi (S (j) ) (i = 1, 2, . . . , m) denotes the i-th element of μ(S (j) ).

11

We now undertake Householder transformation of the mn × (n + 1) matrix[ W (i) u(i) ] by using an appropriate mn × mn orthogonal matrix H as follows:Hwhere W

(i)



W





u(i)

(i)

=


W
O

(i)

 (i)
u
v (i)



,

 (i) denotes an nrepresents an upper triangular matrix of order n, u

dimensional vector, and v (i) is an appropriate vector. Based on this transformation,
Eqs. (11) and (12), respectively, can be rewritten as follows:





−1
1
 (i) )t W
 (i)a|Y (i) ∼ N a ,(W,
σ 2 (S (j) )(i)a(i) = (W(13)(i) −1 (i)


)

u .

Let f (i) (a|Y (i) , S (j) ) be the posterior distribution density function of a, which
is deﬁned by Eq. (13), and let fi (y i |S i , a) be the likelihood function of S i and aderived from the model in Eq. (6). Then, the marginal likelihood of S i is given byf (y i |S i , Y (i) , S (j) ) =fi (y i |S i , a)f (i) (a|Y (i) , S (j) )da.

Consequently, the log-likelihood of S i is given by

2 (S i )= ln f (y i |S i , Y (i) , S (j) ) ||u − W a ||2
n 
i
i i
= − ln 2πσ 2 (S (j) ) −2
2σ 2 (S (j) ) 1 
2 
1 
 (i) ),− ln det(W ti W i ) + ln det(W22where ai is deﬁned asai = (W ti W i )−1 W ti ui .
In addition, W i and ui represent 2n × n matrix and 2n-dimensional vector, whichare, respectively, deﬁned as


Wi =

Bi

W

(i)





,

ui =

(j)

y i − μi (S (j) )bi − z i (S i )
 (i)
u



.

Then, at the (j + 1)-th estimation stage, the structural parameters in the set
S i can be estimated numerically by maximizing

12

2 (S i ) for i = 1, 2, . . . , m.

At thisstage, corresponding to the deﬁnition of S i , the set of the estimates for the structuralparameters is described as
(j+1)

Si



(1)

(1)

(1)

(1)

(αi )(j+1) , (βi )(j+1) , (γi )(j+1) , (θi )(j+1) ,

=

(2)

(2)

(2)

(2)



(αi )(j+1) , (βi )(j+1) , (γi )(j+1) , (θi )(j+1) .
Note that the above maximization of the log-likelihood is realized under conditionsof constraint on the structural parameters mentioned in Subsection 2.1. It should
also be noted that the above process of parameter estimation is considered for themodel in Eq. (4), thus the process of parameter estimation for the model in Eq. (3)can be considered similarly but in a more simple way.

3.4

Algorithm for Parameter Estimation

Based on the parameter estimation schemes I and II mentioned above, an algorithm
for parameter estimation can be summarized as follows:
Step 1Give a set of appropriate initial values for the structural parameters. Set up,
(1)

(1)

(1)

(1)

for instance, (αi )(0) = 0.25, (βi )(0) = 0.25, (γi )(0) = 0.25, (θi )(0) = 0.25,
(2)

(2)

(2)

(2)

(αi )(0) = 0.25, (βi )(0) = 0.25, (γi )(0) = 0.25, (θi )(0) = 0.25 (i =1, 2, . . . , m). Then, calculate μ(S (0) ), σ 2 (S (0) ) and η(S (0) ), for given S (0)based on estimation scheme I. Further, calculate 1 (η(S (0) )) = 1 (η(S (0) ), S (0) )according to Eq. (8).
Step 2

Perform the following operations for j = 1, 2, · · ·.
(j)(i) For given S (j−1) , μ(S (j−1) ), σ 2 (S (j−1) ) and η(S (j−1) ), obtain S i , atentative estimate of S i , for i = 1, 2, . . . , m using estimation scheme II.
(ii) Using estimation scheme I, calculate μ(S (j) ), σ 2 (S (j) ) and η(S (j) ) for thevalue of S (j) that has been obtained.
(iii) Calculate the value of

1 (η(S(j))) =1 (η(S(j)), S (j) ) according to Eq.

(8).
(iv) If

1 (η(S(j))) >1 (η(S(j−1))) holds, return to (i) and continue the oper-

ations; otherwise, proceed to the next step.
Step 3Use S = S (j−1) as the set of estimates of the structural parameters in S
and use a = a(S), μ = μ(S), σ 2 = σ 2 (S) and η = η(S), respectively, as theestimates of a, μ, σ 2 and η. In addition, calculate the estimate of Ci according
to Ci = exp(μi ) (i = 1, 2, . . . , m).

13

4

Regional Analysis of the Japanese Economy

4.1

Data

The empirical analysis in this paper is performed using data available from theJapanese Prefectural Database developed in the analysis of Fukao and Yue (2000).5
Although most annual data in the Japanese Prefectural Database cover the period
between 1955 and 1995, the data for some variables for Okinawa are available only
for the years after 1972 or 1973. Therefore, this paper analyzes 46 prefectures, andexcludes Okinawa.
The corresponding relations among the variables in our model and the data fromthe Japanese Prefectural Database are as follows: We use the gross prefectural domestic expenditure as Qi (t) data, private capital stock by prefecture as Ki (t) data,
public capital stock (industrial infrastructure) by prefecture as Gi (t) data, humancapital by prefecture as Hi (t) data, and the number of domestic employed persons
by prefecture as Li (t) data. The values of the gross prefectural domestic expenditure, private capital stock by prefecture and public capital stock by prefecture aremeasured in real terms using 1980 as the base year.
The method of developing the data described above is explained in the appendix
of Fukao and Yue (2000); however, we here provide supplementary remarks on thehuman capital data, particularly that in the Japanese Prefectural Database. In
conventional economic growth literature, human capital is deﬁned as the stock ofknowledge and skills acquired by individuals through school education, on-the-job
training (OJT) and oﬀ-the-job training (Oﬀ-JT), and other means.6 As might beeasily imagined, therefore, the actual measurement of human capital is extremely
diﬃcult. A data developer consequently faces issues of the perspective from which
the human capital should be expressed numerically, and must establish an indicator
5

As evident from the formulation of the model in this study, data of variables related to factors of
production such as private capital, public capital, human capital, and labor at a prefectural level arenecessary for the estimation of model parameters. These factors of production, private capital, public capital and labor are readily available; however, hardly any prefectural level human capital data,
which are time-series data covering several decades, exist. Therefore, considering the availability andaccuracy of data, this study used sample data from the Japanese Prefectural Database. The data
are downloadable from http://www.ier.hit-u.ac.jp/Japanese/publication/database.html
6
In general, OJT refers to education and training given while engaging in daily work; Oﬀ-JT
refers to education and training provided while temporarily away from regular work. According
to the National Institute of Science and Technology Policy (2004), the percentage of implementing
Oﬀ-JT among Japanese ﬁrms exceeds the rate of implementing OJT, suggesting a characteristicthat workers in Japan tend to acquire a large part of their new knowledge in the environment
outside their workplace. Such a tendency is expected to increase in the future as the number of
specialized graduate schools increases.

14

of some kind to understand workers’ skills and ability levels; thus needing to becreative in data development.
Gernerally, Fukao and Yue (2000) considered that individual skills and ability
levels were reﬂected in workers’ wages and developed their human capital data basedon a wage index that takes into account workers’ educational background and sex.
More speciﬁcally, hi,t , which is the human capital per worker in prefecture i in year
t , is measured as4hi,t =k=1

2

s=1 Ωt,s,k · Ei,t,s,k

Li,t

,

where suﬃx s denotes sex that s = 1 is male and s = 2 is female; suﬃx k represents
educational background and k = 1, 2, 3, 4, respectively, denote completion of juniorhigh school or below, high school, junior college or technical college, and undergraduate or higher courses. With such deﬁnitions of s and k, Ωt,s,k represents the wage
index in year t of workers whose sex is s and educational background is k, Ei,t,s,k
represents the number of occupied persons in prefecture i in year t, whose sex iss and educational background is k, and Li,t is the number of employed persons in
prefecture i in year t . For instance, Ωt,1,2 indicates the wage index in prefecture
i in year t of male workers who had completed high school education and Ei,t,2,3means the number of occupied women in prefecture i in year t who had completeda junior college or technical college curriculum. Here the wage index, Ωt,s,k , of eachyear is deﬁned as a result of dividing the wage, wt,s,k , of workers whose sex is s andeducational background is k by the wage, wt,1,1 , of male workers who had completed
junior high school or below, that is Ωt,s,k = wt,s,k /wt,1,1 .

4.2

Results and Discussion

As a preliminary discussion, we examine the model performance of Eqs. (3) and (4).
With regard to the model of Eq. (4), we focus on the ﬁrst oil crisis in 1973, andconsider the case that structural parameters between two sub-periods, called Period
I (1955-1973) and Period II (1974-1995), respectively, may have diﬀerent values.
For the values of Akaike’s Information Criterion (AIC), we obtain −6991.8 for themodel of Eq. (3) and that −7298.6 for the model of Eq. (4).7 According to the7

In our model, the AIC value is calculated according to the following equation.
AIC = −2 × 1 (η(Ë (j−1) )) + 2 × the number of free parameters,

where the number of free parameters of a model without the structural change is 3m+2; the numberof free parameters of a model that takes into account the structural change is 6m + 2.

15

minimum AIC method proposed by Akaike (1974), the model with a smaller AIC
value is considered better. The model of Eq. (4) that incorporates structural change
is therefore regarded as a better model. Hence, we discuss as follows the estimation
results from the model of Eq. (4).
Tables 1 and 2 present the estimated values of each production factor elasticity
based on the model of Eq. (4) in Japans 46 prefectures, excluding Okinawa, for
Period I and Period II.

Table 1: Estimates of production factor elasticities during Period I (1955-1973)
α

β

γ

α

Hokkaido

0.50809

0.067390.27501

Mie

Aomori0.576970.14058

0.12177

Shiga

β

γ

0.15959

0.32946

0.209310.74638

9.50E-110.20931

Iwate

0.00259

0.28786

0.37815

Kyoto

0.587170.05682

0.21681

Miyagi0.25006

0.229850.27733

Osaka

0.09673

0.334450.25711

Akita

0.10256

0.22772

0.38527

Hyogo

0.61062

9.57E-110.29017

Yamagata
Fukushima

0.023800.802910.307430.00297

0.35569
0.19411

Nara
Wakayama

0.256320.22084

0.20958
0.350260.29001
0.11358

Ibaraki

0.632160.04478

0.23672

Tottori

0.810760.089250.00036

Tochigi

0.18528

0.32399

0.21850

Shimane0.57795

0.072420.25807Gunma

0.464280.16933

0.20049

Okayama

0.28701

0.21974

0.28001
0.13240

Saitama

9.64E-11

0.511220.09436

Hiroshima

0.709280.07761

Chiba

8.43E-11

0.58980

8.43E-11

Yamaguchi

0.027800.30960

0.30494Tokyo

0.43865

9.36E-11

0.41765

Tokushima

0.33294

0.28760

0.13109

Kanagawa

0.02879

0.343440.28469

Kagawa0.57659

0.12526

0.14644

Niigata

0.429300.10146

0.29703Ehime

0.042070.29846

0.31604

Toyama0.446360.166720.23127

Kochi

0.34988

0.149590.30348

Ishikawa
Fukui

0.146110.419390.29489
0.11957

0.265570.30715

FukuokaSaga

0.717450.01148

9.34E-11
0.34123

0.247910.31454

Yamanashi

0.431080.209970.16006

Nagasaki

0.140820.25287

0.33445

Nagano1.01E-100.365030.28461

Kumamoto

0.38156

0.195470.21742Gifu

0.56816

0.09797

0.20588

Oita

0.23326

0.284840.22454

Shizuoka

0.224970.169640.38876

Miyazaki

0.290650.23761

0.22993Aichi

0.77068

1.11E-100.17333

Kagoshima

0.419710.18044

0.21547

From Tables 1 and 2, we ﬁnd that production factor elasticity takes various
values among the prefectures. As well, it is conﬁrmed that such elasticity takes
diﬀerent values through Periods I and II, suggesting the occurrence of structuralchange. Regarding the variance of the elasticity of each production factor among
prefectures, the variance of private capital elasticity was 0.064 in Period I and 0.020

16

Table 2: Estimates of production factor elasticities during Period II (1974-1995)
α

β

γ

α

β

γ

Hokkaido

0.28169

0.108850.37806Mie

0.40189

0.172380.25486

Aomori0.38930

9.32E-11

0.41164

Shiga

0.58756

0.090720.22059

Iwate

0.53754

1.01E-100.36713

Kyoto

0.48519

9.34E-11

0.38161

Miyagi0.420480.02425

0.38549

Osaka

0.43550

0.076470.32586

Akita

0.43911

1.03E-100.41538

Hyogo

0.48519

9.34E-11

0.38161

Yamagata

0.41000

9.94E-11

0.42565

Nara

1.11E-100.39487

0.26182

Fukushima

0.419260.238120.18302

Wakayama

0.30319

1.08E-100.45440

Ibaraki0.25589

0.33260

0.14739

Tottori0.40065

0.06134

0.34003

Tochigi
Gunma

0.09061
0.21943

0.56801
0.34298

9.54E-11
0.15703

Shimane
Okayama

0.48783
0.144140.01799
0.39646

0.368120.14630

Saitama

0.196970.27003

0.25625

Hirishima

0.214880.23732

0.27655

Chiba

0.447800.126890.24245

Yamaguchi

0.386440.17550

0.28630Tokyo

0.47050

1.02E-10

0.39138

Tokushima

0.374100.113960.31149

Kanagawa

0.05259

0.390300.22143

Kagawa0.47368

0.003810.38153

Niigata

0.49189

1.05E-100.36736

Ehime

0.50209

1.13E-100.40017

Toyama0.286460.20432

0.27723Kochi

0.43534

0.00294

0.40624

Ishikawa

0.49326

9.79E-11

0.36707

Fukuoka

0.38395

0.103380.33063Fukui

0.36398

0.14515

0.31111

Saga

0.46655

1.01E-100.39598

Yamanashi

0.35904

0.18469

0.25202

Nagasaki

0.45787

1.11E-100.40882

Nagano0.235140.285430.21901

Kumamoto

0.264250.23233

0.25558

Gifu
Shizuoka

0.231240.227610.26000
0.499640.25461
0.01576

Oita
Miyazaki

0.24378
0.392880.32557
0.074360.17753
0.35808

Aichi

0.16406

0.455910.06971

Kagoshima

0.384310.103510.32048

in Period II. While that of public capital elasticity was 0.019 in Period I and 0.026
in Period II, and that of human capital elasticity was 0.008 in Period I and 0.019 in
Period II. A comparison of Period I and Period II therefore indicates a decrease in
the variance of private capital elasticity and an increase in the variance of humancapital and public capital elasticity.
The characteristics of a structural changes in prefectures in terms of production
factor elasticity are shown in Figure 1 based on the estimation results presented inTables 1 and 2.

17

18
(a)

Figure 1: Characteristics of structural changes in 46 prefectures, excluding Okinawa

(b)

(c)

In Figure 1, (a), (b) and (c) correspond to private capital elasticity, public capitalelasticity, and human capital elasticity, respectively. The positions of the prefecturesin which the elasticity of each factor increased from Period I to Period II are shown inyellow and the positions of prefectures in which the elasticity declined are shown ingray. In the dynamics of the structural changes in the prefectural economies viewed
by factor elasticity, the structural movement inherent in each prefecture is visible
from the graph in Figure 1. The directions of the changes in the elasticity of eachfactor sorted by color reveal the characteristic that some neighboring prefectures
made changes in the same direction, which suggests close economic correlations
among these neighboring prefectures.
In an attempt to understand the structural characteristics of regional economies
in Japan, we now examine what kind of relevance exists among the elasticity values
estimated for each factor. Tables 3 and 4 present the correlation coeﬃcients of thefactor elasticity during Period I and Period II, respectively.

Table 3: Correlations among production factor elasticities during Period I (1955-1973)
Private capital (α)

Public capital (β)

Private capital (α)

1.000Public capital (β)−0.881

1.000

Human capital (γ)−0.338−0.139Human capital (γ)

1.000Table 4: Correlations among production factor elasticities during Period II (1974-1995)
Private capital (α)

Public capital (β)

Private capital (α)

1.000Public capital (β)−0.8691.000

Human capital (γ)

0.638−0.931

Human capital (γ)

1.000From Tables 3 and 4, a high negative correlation between private capital elasticity
and public capital elasticity is shown through both Periods I and II. Regarding therelation between private capital elasticity and human capital elasticity, a negative
correlation is found during Period I, although a positive correlation appears during
Period II. However, the absolute values of the correlation coeﬃcients are not verylarge. Regarding the relation between human capital elasticity and public capitalelasticity, although Period I shows a low negative correlation, Period II presents ahigh negative correlation. The existence of the high negative correlation between

19

private capital elasticity and public capital elasticity throughout the two periods
is particularly noteworthy. This, in general, suggests the characteristic of regional
economies in Japan that the lower the private capital elasticity in the region, thehigher the public capital elasticity.
i , of the labor-augmenting meaNext, we compare and study the initial value, C

sure of productivity among prefectures. Figure 2 represents a graph drawn basedon the estimation results of the initial values of productivity indicators for 46 prefectures, excluding Okinawa.

Mie
Shiga
Kyoto
Osaka
Hyogo
Nara
Wakayama
Tottori
Shimane
Okayama
Hirosima
Yamaguchi
Tokushima
KagawaEhime
Kochi
Fukuoka
Saga
Nagasaki
Kumamoto
Oita
MiyazakiKagoshima

Hokkaido
Aomori
Iwate
Miyagi
Akita
Yamagata
Fukushima
IbarakiTochigi
Gunma
Saitama
Chiba
Tokyo
Kanagawa
NiigataToyamaIshikawa
Fukiui
Yamanashi
NaganoGifu
ShizuokaAichi
0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70 80

Figure 2: Estimated initial values of the labor-augmenting measure of productivity

Figure 2 shows a prominently high estimated level for Tokyo. The second highest
value is the estimate for Osaka, which, however, is less than half the estimated value
for Tokyo. The productivity indicator of Tokyo in the mid-1950s, therefore, appears
to have already been much higher than those of other prefectures. After Tokyo andOsaka, the estimates for such prefectures as Kanagawa, Kyoto, and Aichi are high.
With a few exceptions, prefectures having an ordinance-designated city are generally
ranked highly. For a comparison of the estimates for prefectures other than the socalled three major metropolitan areas that have Tokyo, Osaka and Nagoya as their20

cores, we can not ﬁnd any salient diﬀerences.
How can such estimation results of the initial values of the productivity indicators
be interpreted? For this issue, the detailed analysis made by Fujita and Tabuchi(1997) of the structural background of regional economic growth in Japan afterWorld War II (hereafter WWII) presents some useful suggestions. Following Fujita
and Tabuchi (1997), we give a brief historical overview of regional growth in postwar
Japan.8 The regional economies in Japan after WWII generally developed throughthe following process. In Japan in 1946, immediately after the war, GNP declined toapproximately half of its peak before the war; there was an economic boom during
the Korean War (1950-1951) and by the mid-1950s GNP had recovered to the prewar peak level. The condition of the regional economies in Japan around the ﬁrst
half of the 1950s is characterized by the term Tokyo-Osaka bipolar regional system.
The subsequent period from the mid-1950s to the ﬁrst half of the 1970s was a timeof rapid growth in the Japanese economy. During this period, the major sector
in Japan’s manufacturing industry shifted from the light industries to the heavy
and chemical industries, with an accompanying conversion from the Tokyo-Osaka
bipolar regional system to the Pacific industrial belt system (extending from Tokyo
to Kitakyushu). Because of the eﬀect of the ﬁrst oil crisis, leading industries inJapan shifted from the heavy and chemical industries to the high technology andservice industries that were more energy-eﬃcient and internationally competitive
after the mid-1970s. In other words, the importance of more knowledge-intensive
production activities increased and therefore a knowledge-based economy developed.
During this transition process, Japan’s regional economic system shifted from the
Paciﬁc industrial belt system to the Tokyo monopolar system. Since then, a regional
system of the sole concentration being in the Tokyo metropolitan area with Tokyo
as its core has continued to date.
Year 1955, which is the initial year of the period (1955-1995) that is analyzed inthis study, is the period of transition from the Tokyo-Osaka bipolar regional system
to the Paciﬁc industrial belt system described earlier. Fujita and Tabuchi (1997)
state that the year 1955 marked the point when the rate of net migration to the threemajor metropolitan areas of Tokyo, Osaka, and Nagoya began to increase sharply;
the rate of net migration to the Tokyo area was the highest among the three major
cities. When particularly addressing the share of output of the manufacturing in8
See Falth (2005), for instance, for the history of Japanese economic growth from the Edo period
until the postwar recovery period.

21

dustry in 1955, that of the Tokyo metropolitan area was also the highest, followedby the Osaka and Nagoya areas. While Fujita and Tabuchi (1997) emphasize theagglomeration economies in the Tokyo area after the mid-1970s, comprehensive consideration of the results of their analysis suggests that the agglomeration economies
in the Tokyo area in the mid-1950s were already far greater than in other regions.
The agglomeration economies in the Osaka and Nagoya metropolitan areas should
also have been relatively large, following Tokyo. Therefore, the results of the estimation of the initial values of productivity indicators are most likely to reﬂect theextent of agglomeration economies. In other words, the estimation results related
to the initial values of productivity indicators are considered to be mostly consistent with the extent of the agglomeration economies in each prefecture during themid-1950s.
Next, we investigate the trends in TFP. In our model, the TFP in the prefectures in each period is determined by Ci ，θi and A(t). Recall that, based on theassumption, Ci remains constant through all periods and θi takes a constant value
in each Period (namely Periods I or II). For this reason, the direction of the changes
in prefectural TFP in each Period is speciﬁed by A(t). Then, depending on the differences in the size of Ci and θi of each prefecture, the TFP of each prefecture shifts
in parallel. Thus, we can calculate the direction of changes in prefectural TFP byinvestigating the movement of the time-varying factor of productivity, that is A(t).
In Figures 3 and 4, the vertical axis represents the natural logarithm of the estimated time-varying factor of productivity (i.e., a(t) = ln A(t)); the horizontal axis
represents the years. The estimated values of the time-varying factor of productivity
during Periods I and II are shown.

2.40
1.90
1.40
0.90
0.40
-0.10
1955 1957 1959 1961 1963 1965 1967 1969 1971 1973

Figure 3: Changes in the time-varying factor of productivity during Period I (1955-1973)

22

2.40
1.90
1.40
0.90
0.40
-0.10
1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994

Figure 4: Changes in the time-varying factor of productivity during Period II (1974-1995)

The changes in the time-varying factors of productivity during Periods I and II
indicate the following characteristics. First, Figure 3 shows that the time-varying
factor of productivity shifts upward through Period I. The movement of the timevarying factor of productivity during Period II observed in Figure 4 implies a generally sluggish trend during this period, albeit with slight increases at the end of
the 1970s and at the end of the 1980s. During the 1990s, in particular, a sign of aslow decline in the trends appears. Hence, the changes in the TFP trends of eachprefecture can be inferred as follows based on information related to changes in suchtime-varying factors of productivity. In other words, the TFP of each prefecture
showed a trend to increase from the mid-1950s to the ﬁrst half of the 1970s. Afterthe mid-1970s, however, the trend changed toward a state of stagnation.
Based on the estimation results, we consider the factors underlying such TFP
trends. Because TFP is unquestionably simultaneously aﬀected by various factors
and because our estimation results alone cannot provide deﬁnitive conclusions, theinterpretation requires careful consideration. Considering this, we might be able
to make a general interpretation as described below. First, the steady increase in
TFP during Period I (1955-1973), suggested by Figure 3, might be reﬂected bytechnological progress based primarily on technology imported from North America
and Europe. As pointed out by Ohkawa and Rosovsky (1973), Peck and Tamura(1976), Wakasugi (1986), Goto (1993) and others, Japanese industries aggressively
adopted advanced foreign technologies in the post-WWII era and actively invested inresearch and development (R&D) in an eﬀort to create improved technology basedon that adopted. The R&D activities based on proﬁt motives of ﬁrms not only
raise the technology levels of these ﬁrms, but promoted technological progress of the23macro economy as the technical knowledge with extensive applicability among othernew technologies spread to other industries beyond immediate boundaries.
Here we brieﬂy give some background to the technological adoption that occurred in Japan in the post WWII era, according to Wakasugi (1986). During thechaotic period during and after WWII, Japan’s institutions allowed no introduction of technology from abroad. It was not until 1950 that the import of overseas
technology began again. Figure 5, depicting the number of approved contracts forimportation of foreign technology, represents the combined total of such contracts
in the chemical industry, and those dealing with petroleum products, steel and iron,and general, electrical and transport machinery of Japanese ﬁrms during the period
from 1950 to 1969.
1400
1200
1000
800
600
400
200
0
1950 1952 1954 1956 1958 1960 1962 1964 1966 1968

Figure 5: Numbers of contracts approved for the introduction of foreign technology
Note: The number of contracts includes only ‘class A’ technical agreements having
a contractual or payment term of one year or longer.
Source: Data from Wakasugi (1986)．

As might be evident from Figure 5, there was no increasing trend in the number
of approved contracts for the importation of foreign technology indentiﬁed during the1950s. This is attributable to the considerable restrictions imposed on technology
import by the then Japanese government. In 1959, approval for the adoption ofoverseas technology was relaxed. Subsequently in 1968, adoption of technology
was completely deregulated. The number of technological adoption cases increased
substantially because of this series of policy changes.
The remarkable eﬀect of the government’s approval system for the introductionof overseas technology on the formation of the industrial structure of the Japanese
economy is worthy of mention. Wakasugi (1986) states that in 1950, the Ministry24

of International Trade and Industry announced 33 technologies for which the adoption was to be encouraged. Such technologies were related mostly to the heavy
and chemical industries such as chemicals, steel and machinery. The technology list
added to 1950 was also concerned the rationalization and modernization of production processes in the heavy and chemical industries. Although it did not mean thatthe technologies of sectors not named in the technology list were prohibited frombeing adopted, the technologies that could be adopted in the heavy and chemicalindustries are thought to have been biased towards at the time when technologicaladoption required strict permission from the government. In other words, because
the promotion of the heavy and chemical industries was one economic goal of thegovernment, the development of Japan’s heavy and chemical industries was formed
according to the government’s intentions for its policies.
The migration of numerous workers from the agricultural sector to the manufacturing sector during this time also constitutes a signiﬁcant factor. The increasein the labor movement from the agricultural sector, with relatively low TFP, to the
manufacturing sector, with higher TFP, increased the eﬃciency of resource allocation and promoted TFP growth at the macro level.
We now turn to discuss the causes aﬀecting the slowdown of TFP during Period
II (1974-1995). As apparent from a comparison of Figures 3 and 4, the movementof the time-varying factor of productivity stagnated at the time of the ﬁrst oil crisis; negative growth was then experienced during some periods. Thus, after 1974,
prefectural TFP also is considered to have entered a phase of stagnation.
How can we interpret the slowdown in TFP growth during Period II (19741995)? As argued in Moriguchi (1988), for example, the ﬁrst oil crisis gave rise to a
substantial decline in capacity utilization in energy-intensive industries such as the
petrochemical and the, petroleum reﬁning rerated industries. It took a fair amountof time to restore capacity utilization to its functional level. So, the deterioration inenergy eﬃciency led to the obsolescence of some existing production equipment.
In the early 1970s, the transition of the foreign currency exchange system fromﬁxed to ﬂoating rates was also a remarkable occurrence. This led a swift appreciation of Japanese yen in terms of US dollars, and exaggerated the response ofgovernment and the Central Bank of Japan at that time, such as in the excessive
public investment by the government and the excessive increase of the money supply
by the Central Bank of Japan. As a result, such policy failures gave rise to highinﬂation, which the ﬁrst oil shock promoted.

25

Further, as pointed in Valdés (2003), from the mid-1970s some Japanese industries became the owners of worldwide leading-edge technology. This means that thepace of imitation-based technological improvement declined. Hence, such situation
led to a high requirement for the development of original state-of-art technology inJapan. As is well known, however, creating new technology oneself compared withintroduction of imitation technology requires large R&D budgets. As well, it also
has a high possibility of failure compared to the introduction of imitation technology.
For that reason, it follows that the speed of technological progress would slow.
Thus, there were policy failures and confusions associated with the exogenous
shocks of the ﬁrst oil crisis and the transition to a ﬂoating exchange system. In
addition, by reaching positions at the technological frontier, the advantages of backwardness were mostly lost. As well, it should be noted that since the 1970s Japan
has had a rapidly aging population. For example, Noda (2007) analyzed the relationship between population aging and technological progress using an endogenous
innovation model, and found that population aging may have a negative impact on
the rate of technological progress at a macroeconomic level. From the viewpointof Noda’s (2007) model, hence, rapid population aging since the 1970s might havecontributed to the decline in the rate of technological progress in Japan. For the
early part of Period II (1974-1995), we infer that the slowdown in the TFP growth
can be attributed to a mixture of causes, as mentioned above.
Next, we refer the slowdown of TFP growth after the 1990s. In addition to
the continuing aging problem, we considered the following causes. In Hayashi and
Prescott (2002), for instance, the decline in the TFP trend in the Japanese economy
in the 1990s was interpreted as being caused by loan policies for ineﬃcient ﬁrms
and declining industries. Sakuragawa (2003) more speciﬁcally developed the interpretation of Hayashi and Prescott (2002) and convincingly demonstrated, based on
various data, that the major cause of the decline in the TFP trends in Japan during
the 1990s was the ineﬃciency of resource allocation. The argument of Sakuragawa(2003) can be summarized as follows: Bad-loan problems surfaced in Japan at thetime of the collapse of the bubble economy at the beginning of the 1990s; the facts
that no short-term solution was introduced and that the problem persisted for a longtime had negative eﬀects on the Japanese economy. For instance, additional loans
were provided to bad borrowers and the over-banking phenomenon with excessive
deposits promoted the soft budget problem, which preserved a considerable numberof ineﬃcient ﬁrms, thereby hindering the movement of normal production factors

26

from low-productivity industries to high-productivity ones. The inevitable results
were to prolong the problem of bad loans and to further increase ineﬃcient resource
allocation. The series of such negative eﬀects is a conceivable cause of the sluggish
TFP growth during the 1990s. Sakuragawa (2003), therefore, asserts that promptly
solving bad loan problems using an appropriate policy mix is important.
What other policies are feasible for an improvement of TFP and therefore promotion of economic growth? One might be industrial policies related to R&D subsidies. As emphasized by endogenous growth theory (see, for example, Grossman
and Helpman, 1991; Aghion and Howitt, 1998; Barro and Sala-i-Martin, 2004),
technical knowledge and information accumulated in R&D activities contribute to
the technological advancement of the macro economy through positive externality;
namely, spillover eﬀects.9 Nakamura (2003) analyzed the external eﬀects of technical knowledge in Japan’s manufacturing industry between 1968 and 1996 using
econometric analysis and discovered that the spillover of technical knowledge had
been a major determinant of TFP growth, even after the 1990s. According to ouranalysis, however, TFP growth during the 1990s was stagnant. It seems very apparent that TFP reﬂects the eﬀects of multiple factors and that the slow TFP growth inthe1990s must have been strongly aﬀected by problems related to the bad loan issue
described earlier. Nevertheless, the possibility that the conventional policy of R&D
subsidies was ineﬃcient is also suggested. As pointed out also by Nakamura (2003),
that depending on the ﬁeld, spillover of technical knowledge alone creates diverse
degrees of impact on other industries. Therefore, prioritizing those technical ﬁelds
that give greater impact in the allocation of subsidies is considered eﬃcient. Meanwhile, the sluggish TFP might have been caused in part by inadequate allocation ofeven eﬃcient subsidies. In the future, therefore, understanding the relative extent
of technical knowledge spillover in each industry before implementing any policy ofappropriate R&D subsidies is necessary.

9Part of the technical knowledge is protected for a certain period as intellectual property suchas patents and utility models and is usually not available for use unless payment for a license ismade. However, we can see cases where technical knowledge spreads easily in such a way that does
not require payment. Suppose, for example, that a ﬁrm has developed a next-generation technology
and has succeeded in developing a product from this technology. When this new product becomes
available in the market, rival ﬁrms are able to learn the technology used for the new product throughreverse engineering. In this way, because the non-excludability of technical knowledge is limited,
the use of the knowledge without paying a fee to the innovator, that is the spillover of technicalknowledge, occurs.

27

5Concluding Remarks

In this study, we have developed a new quantitative approach based on a frameworkof Bayesian statistical models for regional production functions. As well, by estimating the prefectural production functions using annual data from the Japanese
Prefectural Database covering the period 1955-1995, we have in the main examined
trends in TFP and the structural features in the Japanese prefectures. Our approach does not estimate the growth rate of TFP as a residual, as might be done instandard growth accounting analysis, nor does it assume strong constrains on thegrowth rate of TFP, as might be done in the traditional econometric analysis ofproduction functions. The approach that we propose has the following speciﬁc characteristics: (i) smoothness priors are used and trends in TFP are estimated basedon a Bayesian method; (ii) the eﬀects of a structural change caused by the ﬁrst oilcrisis are examined; and (iii) the values of the elasticity of output with respect toproduction factors vary among prefectures.
The main results can be summarized as follows: We considered the inﬂuencesin the ﬁrst oil crisis and then analyzed structural changes between two sub-periods:
Period I (1955-1973) and Period II (1974-1995). Estimating the elasticity of output
with respect to production factors, clear diﬀerences in the factor elasticity were identiﬁed among the regions. Thus our estimation result questions some earlier reports’assumptions that factor elasticity values among regions are identical. Regarding
the variance of factor elasticity among prefectures, the variance of private capitalelasticity decreased and the variance of human capital elasticity and public capitalelasticity increased during both periods. As well, a high negative correlation was
found between private capital elasticity and public capital elasticity in both periods,
which suggested a regional characteristic of Japan that, in general, the lower theprivate capital elasticity in the region, the higher the public capital elasticity.
Further, we applied the smoothness priors approach to examine the TFP trend
in each prefecture during the period 1955-1995. As a result, we found that the
TFP in each prefecture increased from the mid-1950s to the early 1970s. During
the period between the mid-1970s and the mid-1990s, however, a slowdown in the
TFP growth was identiﬁed. In particular, TFP trends declined after the 1990s. We
can therefore interpret that, in Japan, although TFP promoted economic growthfrom the mid-1950s to the early 1970, its contribution diminished during the period
from mid-1970s to the mid-1990s. A slow decline in the TFP trends was apparent

28

particularly after the 1990s, and the eﬀects of ineﬃciency in resource allocation in
the production factor and ﬁnancial markets associated with the bad-loan problems
during this period might have been background factors for this situation.
Finally, we refer to some future issues. Although we used the Japanese Prefectural Database, we must bear in mind that these data do not include the capitalutilization ratio or working hours. Therefore, we will attempt to reconsider the results here in future work after taking into account the capital utilization ratio andworking hours and by utilizing data extended to the most recent possible year. In
addition, an important issue is the speciﬁcation of production function, and we needto examine more general cases such as the CES production function.

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