Accounting for Economic Growth
in Japan and the United States:
A Bayesian Approach

Hideo Noda*
Koki Kyo†
Research Group of Economics and Management
No. 2009-E03
2009.6*†

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

Accounting for Economic Growth
in Japan and the United States:
A Bayesian Approach∗
Hideo Noda†
Faculty of Literature and Social Sciences, Yamagata University, Japan

Koki Kyo
Obihiro University of Agriculture and Veterinary Medicine, Japan

Abstract
We present a Bayesian model for statistical inference in the CES production
function, and its application to the empirical analysis of economic growth inJapan and the United States. Unlike the associated literature, we regard theeﬃciency parameter to be time-varying. By estimation of the time-varying
eﬃciency parameter using a Bayesian method, thus, we attempt to grasp thechanges in eﬃciency at macroeconomic level. It is found that Japan’s eﬃciency
performance has stagnated during the 1990s in particular, while in the UnitedStates it has risen. The empirical evidence suggests that a reform of nationalinnovation systems is a key issue in restoring Japan’s economic growth.
Keywords: CES production function, Box-Cox Transformation, Smoothness Priors, Bayesian Model Averaging, Economic Growth
JEL classification: C11, C22, C51, O40, O57∗

An earlier version of this paper was presented at the 66th Annual Meeting of the Japan Economic
Policy Association, May 2009. We are grateful to Terukazu Suruga for his useful comments. This
study was supported by Grant-in-Aid for Scientific Research (C) (21530193) from the Japan Society
for the Promotion of Science and by Cooperative Research Program (2016) from the Institute of
Statistical Mathematics.
†
Corresponding author. E-mail: noda@human.kj.yamagata-u.ac.jp1

1

Introduction

Most people, not only economists and policymakers, are interested in economic
growth in countries, because trends in economic growth have a direct impact on
people’s standard of living. In particular, the recent decline in the economic performance of many developed countries such as Japan and the United States necessitates
policy recommendations for the recovery of economic growth. In an attempt to respond such social demands, evidence from empirical studies becomes essential, andstatistical methods for quantitative analyses of economic ﬂuctuations take on an
increasingly important role in policy science. In the present paper, we propose an
extended production function approach for the quantitative analysis of economic
growth from a Bayesian perspective. Furthermore, we apply our proposed approachto an empirical analysis of the Japanese and U.S. economies.
Overall, earlier empirical studies on sources of growth in a macro economy can beclassiﬁed into two types. The ﬁrst is a growth accounting approach, which provides a
decomposition of the growth rate of output into components associated with changes
in factor inputs, and in a residual that reﬂects technological progress and otherelements (e.g., Young, 1995; Collins and Bosworth, 1996; Wang and Yao, 2003).
The second is often called a growth regression approach; it estimates regression
equations based on economic growth models (e.g., Barro, 1991; Mankiw, Romer, andWeil, 1992; Shioji, 2001). Nonetheless, these conventional approaches are consideredinadequate, as explained below.
As is well known, growth accounting analyses require data on the elasticity ofoutput with respect to conventional factors of production (like physical and humancapital), in addition to data on the output and factors of production. Given thediﬃculty of measuring the elasticity of output with respect to the factors of production directly, the elasticity is measured indirectly as each relevant factor’s shareof national income. In other words, growth accounting assumes that factor prices
are equal to social marginal products. However, the assumption that factor prices
coincide with social marginal products is unlikely to hold at all times in reality. If
the deviation between the factor prices and social marginal products is too large toneglect, then the results of growth accounting would naturally lack credibility.
Such assumptions are also made in many growth regression models. In addition,a problem with growth regression approaches is that they usually do not represent
the time-variations in variables reﬂecting technological change, such as total factor
productivity (TFP), appropriately. For example, Mankiw, Romer, and Weil (1992)
assume that the growth rate of TFP is constant at any point in time. This simpliﬁ-

2cation, which evidently lack ﬂexibility, implies that the behavior of TFP is restricted
to exponential growth. Furthermore, it should be noted that many growth regression models assume that an economy is in a steady-state. However, as stated inGalor and Zang (1997), it is hard to verify whether an economy is in a steady-state
or in a transition path.
As pointed out by Barro (1999), the advantage of the direct estimation of production function models is that it requires none of the assumptions mentioned above.
Nevertheless, the conventional econometric analyses of production functions are also
inadequate. Speciﬁcally, as in growth regression approaches, many earlier reports
assume that the growth rate of variables reﬂecting technological change is constantover time. Therefore, there is likely to be room for improvement in this frameworkin the context of the production function approach.
Considering these issues that have arisen from previous research, we present a
Bayesian model for statistical inference in a constant elasticity of substitution (CES)
production function, and its application to empirical analysis of economic growth
in Japan and the United States. Unlike the associated literature, in our framework
the eﬃciency parameter is regarded as a time-varying parameter from a Bayesian
perspective. By estimation of the time-varying eﬃciency parameter using a Bayesian
method, we attempt to grasp the changes in eﬃciency at macroeconomic level. Thisapplication of a Bayesian method allows a rigorous analysis of trends in a factor
reﬂecting technological change, compared with conventional approaches.
The rest of the paper is organized as follows. In Section 2, we construct ananalytical framework. Section 3 presents the basic scheme for parameter estimation.
In Section 4, we apply our proposed method to the empirical analysis of economic
growth in Japan and the United States. Section 5 concludes the paper.

2

Basic Setup

2.1

CES Production Function

In a given discrete time period t, the maximum amount of output Qt that can beproduced depends on both physical capital Kt and human capital Ht according toan aggregate CES production function− νρQt = ζ[ δKt−ρ + (1 − δ)Ht−ρ ],(1)where parameters ζ, δ, ρ and ν represent the eﬃciency parameter, the distribution
parameter, the substitution parameter and the returns to scale parameter respectively. In Eq. (1), we assume that γe > 0, 0 ≤ δ ≤ 1, ρ > −1, ν > 0.

3

A major concern for many people researching economic growth is ascertainingchanges in the eﬃciency parameter. Unlike the existing literature, we therefore
consider ζ to be a time-varying parameter ζt . To construct a statistical model for
the CES production function, we rewrite Eq. (1) as
Yt =
−ρ/ν

where Yt = (Qtνν− [ δKt−ρ + (1 − δ)Ht−ρ ]γt + εt ,
ρ ρ
−ρ/ν

− 1)/(−ρ/ν), γt = ζt(2)

, and εt denotes an error term. That

is, Yt is deﬁned by a Box-Cox transformation of Qt . The Box-Cox transformation
is used here to equalize the variance of εt over time, hence we can assume thatεt ∼ N(0, σ 2 ) with σ 2 being an unknown parameter. Zellner (1971) made use of
the Box-Cox transformation in analyzing production functions, while Chetty andSankar (1969) introduced a Bayesian method for parameter estimation of the CES
production function based on a similar transformation. However, neither paper
allowed the eﬃciency parameter to vary over time.

2.2

Bayesian Modeling via Smoothness Priors Approach

In this paper, we regard the γt to be a random variable from a Bayesian perspective,
and apply the smoothness priors approach, introduced by Kitagawa and Gersch(1996), in setting up a prior distribution for it. Speciﬁcally, a stochastic diﬀerence
equation of order 2 is used to deﬁne a prior distribution for γt as follows:
γt − 2γt−1 + γt−2 = ψt ,(3)

where ψt is a Gaussian white noise sequence with ψt ∼ N(0, σ 2 /d2 ), where d > 0
is an unknown parameter. It is also assumed that εt and ψt are independent ofeach other. In order to obtain a proper prior density of γt for t = 1, 2, . . . , n, it isrequired to introduce γ0 and γ−1 as unknown parameters. Thus, we have three newparameters γ0 , γ−1 and d.
We consider the parameters δ, ρ, ν, σ 2 , γ0 , γ−1 and d unknown constants. Whenthe values of these parameters are given, a system of Bayesian linear equations for
γt ’s can be constructed based on Eqs. (2) and (3). Then, we can obtain posterior distribution for γt ’s by using the Bayesian linear modeling method proposed
by Akaike (1980). To obtain robust estimates, we combine the Bayesian model
averaging method with the Akaike’s approach.

4

3

Estimation Procedure

3.1

Basic Scheme

Our parameter estimation scheme is based on the procedure introduced in Jiang(1995), which is an application of Akaike’s approach (Akaike, 1980). For parameter
estimation, we need to rewrite Eq. (2) as follows:
Yt =

ν
ν
a − [ δKt−ρ + (1 − δ)Ht−ρ ]γt + εt ,
ρ
ρ

(4)

where a is a formal parameter. Eqs. (2) and (4) are equivalent under the constraint
a = 1.

(5)

We temporarily assume that a is a random variable with the prior distributiona = 1 + ξ,

(6)

where ξ ∼ N(0, σ 2 /λ2 ). To keep Eq. (5) it is required to put ξ = 0 into Eq. (6).
This can be realized by setting the variance σ 2 /λ2 = 0, or equivalently λ2 = ∞ for agiven value of σ 2 , since it is assumed that the mean of ξ is zero. However, the priordistribution for a is then not deﬁned well. This diﬃculty can be relieved by putting
λ = λ∗ with λ∗ being a suﬃciently large positive number.
From Eq. (4), the likelihood of γ = (γ1 , γ2 , . . . , γn )t and the other relatedparameters is given by(

f (q|γ, δ, ρ, ν, a, σ ) = J √
2

1

)n

2πσ 2

[

]

1
ν
exp − 2 ||y + Xγ − 1n a||2 ,2σρwhere J is the Jacobian of the transformation from Qt to Yt deﬁned by J =Πnt=1 (∂Yt /∂Qt ). The other symbols are deﬁned as follows: q = (Q1 , Q2 , . . . , Qn )t ,q = (Y1 , Y2 , . . . , Yn )t , X = diag(x1 , x2 , · · · , xn ) with xi = [δKi−ρ +(1−δ)Hi−ρ ]·(ν/ρ),

1n is a n-dimensional vector with all elements being one, and || ∗ || denotes the Euclidian norm. Furthermore, when the values of σ 2 , γ0 , γ−1 , d and λ are given, from
Eqs. (3) and (6) the prior densities for γ and a are given by(√

π1 (γ|σ 2 , b, d) =(2

π2 (a|σ ) =

√

d
2πσ 2
λ

)n
)

2πσ 2

5

[

]

d2
exp − 2 ||Dγ + Bb||2 ,
2σ
[

]

λ∗ 2
exp − 2 (a − 1)2 ,2σwhere··· ··· ··· 0..  −2 1 . . .
. 



.. 
 1 −2 1 . . .
. 


D=
,
.. 
.. .. .. ..

.
.
.
. . 
 0



 ..

.. .. .. ..
.
.
.
. 0 
 .
0 ··· 01 −2 1
1

0





1 −2
 0 1 




0
0


B=
.. 
 ..

 .
. 
0 0

and b = (γ−1 , γ0 )t . We then put θ t = (γ t , a) and obtain the marginal likelihood of
the related parameters as
L(σ 2 , δ, ρ, ν, b, d)
∫∫

=
(

∫

· · · f (q|γ, δ, ρ, ν, a, σ 2 )π1 (γ|σ 2 , b, d)π2 (a|σ 2 )dγda

=J √

)n

1

[

n ∗

2πσ 2

d λ det(W W )



where

t

−X

W =  −dD
0tn

ν
ρ 1n

− 12





]

1
b 2 ,
exp − 2 ||h − W θ||
2σ

(7)



y


h =  dBb  ,λ∗

0n  ,λ∗b = (γb t, ab)t = (W t W )−1 W t h denotes
0n is the n-dimensional zero vector, and θ

the mean of a posterior density for θ
f (θ|q; σ 2 , δ, ρ, ν, b, d, λ)f (q|γ, δ, ρ, ν, a, σ 2 )π1 (γ|σ 2 , b, d)π2 (a|σ 2 , λ),= ∫∫ ∫· · · f (q|γ, δ, ρ, ν, a, σ 2 )π1 (γ|σ 2 , b, d)π2 (a|σ 2 , λ)dγda(8)b ρb, νb, b
b and db for theb 2 , δ,given the related parameters. Therefore, the estimates, σrelated parameters can be obtained by maximizing the likelihood L(σ 2 , δ, ρ, ν, b, d)in Eq. (7).
Computationally, if the values of δ, ρ, ν and d are given, γ together with b and a
can be estimated simultaneously by using the least squares method (see for example,
Jiang, 1995). That is, when we set




y


z =  0n  ,
λ∗



−X O

V =  dD dB
O
0tn

ν
ρ 1n




0n  ,
λ∗





γ


β= b awith O denoting a zero matrix, the estimate,b ρ, ν, d) = (γb (δ, ρ, ν, d), ab t (δ, ρ, ν, d), bb(δ, ρ, ν, d))tβ(δ,t= (V t V )−1 V t z,

6

b 2 (δ, ρ, ν, d) = (1/n) ·for β is obtained, and then the estimate of σ 2 is given by σb ρ, ν, d)||2 . Note that the corresponding estimates γb ρ, ν, d)b (δ, ρ, ν, d), b(δ,||z − V β(δ,b(δ, ρ, ν, d) ≈ 1 of γ, b and a can be obtained from each corresponding partand ab ρ, ν, d). Thus, by substituting b = b(δ,b ρ, ν, d), and σ 2 = σb 2 (δ, ρ, ν, d) intoof β(δ,L(σ 2 , δ, ρ, ν, b, d), the partial likelihood of δ, ρ, ν and d can be obtained as[

1
L (δ, ρ, ν, d) = √2b (δ, ρ, ν, d)
2π σ

]n

∗

(

dn λ∗ det(W t W )− 2 exp −
1

n)
.
2

(9)

Correspondingly, a conditional poterior density, g(γ|q; δ, ρ, ν, d), for γ can be derived
form f (θ|q; σ 2 , δ, ρ, ν, b, d) in Eq. (8) byb ρ, ν, d), d)b 2 (δ, ρ, ν, d), δ, ρ, ν, b(δ,g(γ|q; δ, ρ, ν, d) = f (θ|q; σb(δ, ρ, ν, d).
under the condition that a = a

3.2

Using Bayesian Model Averaging Approach

Theoretically, the parameters δ, ρ, ν and d can be estimated using a numericalmethod, such as a quasi-Newton method, by maximizing the partial likelihood
L∗ (δ, ρ, ν, d) in Eq. (9). Depending on the function L∗ (δ, ρ, ν, d), it may sometimes
be hard to use such a numerical method directly to estimate each of the parametersδ, ρ, ν and d.
To avoid this diﬃculty, we use the Bayesian model averaging approach as follows.
Firstly, for a given integer I we take δi = i/(I +1) as a value of δ, so for i = 1, 2, . . . , I
we have I discrete values for δ that are uniformly distributed in the interval (0, 1).
On the other hand, by using the basic estimation scheme introduced in the preceding
subsection, for a value δi of δ together with the given values of ρ, ν and d, we havea set of Bayesian linear models which is denoted by Mi . Thus, we can obtain I setsof Bayesian linear models under i = 1, 2, . . . , I.
From the Bayesian model averaging perspective (see for example, Claeskens andHjort, 2008), we assume that all of the models {M1 , M2 , . . . , MI } are uncertain andall of them have equal uncertainties. Thus, we use a uniform prior for the models{M1 , M2 , . . . , MI } as follows:q(Mi ) =1,I(i = 1, 2, . . . , I)

where q(Mi ) is the prior probability for model Mi . As shown in the preceding
subsection a partial likelihood for model Mi is L∗ (δi , ρ, ν, d), as given in Eq. (9).

7

Thus, a conditional posterior probability for model Mi is given byp(Mi |q; ρ, ν, d) ==L∗ (δi , ρ, ν, d)q(Mi )∗l=1 L (δl , ρ, ν, d)q(Ml )L∗ (δi , ρ, ν, d).
∑I∗
l=1 L (δl , ρ, ν, d)∑I

Moreover, from the basic estimation scheme mentioned above, we can see thatb (δi , ρ, ν, d) is as the posterior
for given δi , ρ, ν and d, the posterior mean of γ is γmean of γ, deﬁned by∫∫b (δi , ρ, ν, d) =γ∫

· · · γg(γ|q; δi , ρ, ν, d)dγ.

Therefore, based on the Bayesian model averaging approach a synthetic estimate forγ is (see, Draper, 1995):bb (ρ, ν, d) =γI∑b (δl , ρ, ν, d)p(Ml |q; ρ, ν, d).
γl=1Now, we consider the quantityL̄(ρ, ν, d) =I∑L∗ (δl , ρ, ν, d)q(Ml )

l=1

=

I
1∑
L∗ (δl , ρ, ν, d)
I l=1

(10)

to be a measure of goodness of ﬁt for the model with respect to the parameters ρ, νand d. Thus, the estimates ρb, νb and db of ρ, ν and d can be obtained by maximizing
the mean likelihood L̄(ρ, ν, d) in Eq. (10) with respect to these parameters. Hence,
the ﬁnal estimate for γ is obtained as
b
b
b
b
b
b
b = (γb (ρb, νb, d),b1 , γb2 , . . . , γb n )t = γγ

and the estimate for δ is given by
δb =

I
∑

b
δl p(Ml |q; ρb, νb, d).

l=1
−ρ/ν

As well, from γt = ζt

−ν/ρ
bt
.
is given by ζbt = γb

, the estimate for the time-varying eﬃciency parameter ζt8

4

Empirical Analysis

4.1

Data

Our samples for Japan and the United States cover a 42-year period (1960-2001).
We use real GDP as a measure of output. In our statistical analysis, the real GDP(market prices, billions of national currency at 1995 prices), physical capital (billionsof national currency, beginning-of-year stock) and the number of employed persons
are data presented in Kamps (2006). For the data of physical capital, we use theaggregate of Kamps’ (2006) private and public capital to represent physical capital.
The data of human capital resulting from schooling are constructed as follows.
As in Hall and Jones (1999) and Caselli (2005), we assume that human capital pere is measured by the equation
worker, h,e = exp[ϕ(s)],hwhere s is the average number of years of schooling. As stated in Hall and Jones(1999), in this speciﬁcation ϕ(s) denotes the eﬃciency of a unit of labor with s
years of schooling relative to one with no schooling (ϕ(0) = 0). The derivative ϕ′ (s)is the return to schooling estimated in a Mincerian wage regression: an additional
year of schooling raises a worker’s eﬃciency proportionally by ϕ′ (s). For the ﬁrst
four years of schooling, we assume that the rate of return of schooling is 11.7 percent, which corresponds to the average of sub-Saharan African countries reported by
Psacharopoulos and Patrinos (2004). We also assume a rate of return of schoolingof 9.7 percent for the next four years of schooling, and a rate of return of schoolingof 7.5 percent for schooling beyond the eighth year. The values of 9.7 percent and7.5 percent are, respectively, the average for the world as a whole and the average ofOECD countries from Psacharopoulos and Patrinos (2004). Thus, ϕ(s) is speciﬁed
as follows:
 0.117 × s

if s ≤ 4,
0.117 × 4 + 0.097 × (s − 4)
if 4 < s ≤ 8,
ϕ(s) =
 0.117 × 4 + 0.097 × 4 + 0.075 × (s − 8)
if s > 8.
Therefore, the data for aggregate human capital (skill-adjusted labor) are produced according to P · exp[ϕ(s)], where P denotes the number of employed persons.
In the present paper, we use data for s constructed by Klenow and Rodriguez-Clare(2005). Because of source data constraints, however, we could not obtain the dataof human capital per worker in 2001 for both Japan and the United States. Consequently, we generate the data for Japan by using a linear extrapolation based on the9fact that in both Japan and the United States, human capital per worker increased
linearly in the period 1985-2000.
Figure 1 shows the changes in growth rate of GDP, physical capital and humancapital from 1961 to 2001 in Japan.
.14
GDP
Human Capital
Physical Capital

.12
.10
.08
.06
.04
.02
.00
-.02
1965

1970

1975

1980

1985

1990

1995

2000

Figure 1: Growth rates of GDP, physical capital and human capital in Japan

In Japan, the average annual growth rates of physical capital during the periods
1960-1969, 1970-1979, 1980-1989 and 1990-2001 are 6.84, 8.57, 4.57 and 3.31 percent,respectively. The average annual growth rates of human capital during these periods
are 1.13, 1.73, 1.45 and 0.62 percent, respectively.
Figure 2 displays the changes in growth rates of GDP, physical capital and humancapital, respectively, from 1961 to 2001 in United States.
.10
GDP
Human Capital
Physical Capital

.08
.06
.04
.02
.00
-.02
-.04
1965

1970

1975

1980

1985

1990

1995

2000

Figure 2: Growth rates of GDP, physical capital and human capital in U.S.

10

In the United States, the average annual growth rates of physical capital during
the periods 1960-1969, 1970-1979, 1980-1989 and 1990-2001 are 4.05, 3.33, 2.90 and
2.98 percent, respectively. The average annual growth rates of human capital during
these periods are 2.70, 4.19, 1.73 and 1.41 percent, respectively.

4.2

Estimation Results

Table 1 presents the estimation results of the returns to scale parameter (ν) the
distribution parameter (δ) and the substitution parameter (ρ) for Japan and theUnited States.
Table 1: Estimation results of ν, δ and ρ
Japan

U.S.

ν
b

0.950

0.950

b
δ

0.089

0.248

ρb

−0.360−0.810For both Japan and the United States, the estimate of the returns to scale parameter is 0.950. Hence, the CES production function in both countries is regarded
as decreasing returns to scale during the period 1960-2001. As well, from the estimates of substitution parameters, the elasticities of substitution in Japan and theUnited Stated are 1.563 and 5.263, respectively. That is, the elasticity of substitution between physical and human capital in the United States is more than threetimes that of Japan’s. Furthermore, Japan’s estimate of distribution parameter isvery small compared with that of the United States’, that is, 0.089 versus 0.248.
From Eq. (1), we ﬁnd that if δ is large, then elasticity of output with respect tophysical capital is large and elasticity of output with respect to human capital issmall. Therefore, all other things being equal, Japan’s small estimate of distribution
parameter suggests that elasticity of output with respect to physical capital is smallin comparison with that of the United States.
Table 2 shows the estimates of the eﬃciency parameter (ζt ) for Japan and theUnited States.
We set ζt∗ = log(ζbt /ζb1 ) to be the eﬃciency index at time t, so that ζ1∗ corresponds
to the value of the eﬃciency index in 1960. Figures 4 and 5 illustrate the changes
in eﬃciency indices from 1960 to 2001 in Japan and the United States, respectively.
As shown in Figure 4, the eﬃciency index for Japan rose rapidly during the 1960s.
When we consider the factors underlying this trend, the steady increase might be11

Table 2: Estimation results of ζt
Japan

U.S.

Japan

U.S.

1960

1.2546

0.0376

1981

3.38920.04171961

1.37130.03851982

3.41970.04201962

1.48440.03951983

3.45120.04311963

1.60550.0405

1984

3.51670.04441964

1.71980.0416

1985

3.59880.04551965

1.82850.04271986

3.67890.04611966

1.9626

0.0437

1987

3.78500.04661967

2.13280.04431988

3.91700.04711968

2.33690.0447

1989

4.03400.04761969

2.55540.04481990

4.11940.04791970

2.74490.04491991

4.14220.04821971

2.88030.04541992

4.11780.04881972

3.00870.04601993

4.09160.04951973

3.08690.04641994

4.09350.05011974

3.09170.04621995

4.13050.05081975

3.10770.04591996

4.1871

0.05171976

3.14000.04551997

4.20230.05271977

3.19060.0447

1998

4.18820.05401978

3.25880.04381999

4.20570.05531979

3.32280.04272000

4.24480.05651980

3.35850.04182001

4.25310.0573

mainly reﬂected in the technological changes arising out of the technology imported
from North America and Europe. Since the 1970s, however this growth rate has
declined sharply, particularly from the 1990s onward. As is well documented, fromthe mid-1970s, a number of Japanese industries became the owners of leading-edge
global technology. This meant that the pace of imitation-based technological improvement declined, and the development of original state-of-art technology in Japan
increased. However, independently creating new technology compared with introducing imitation technology requires large research and development budgets, andalso has a higher probability of failure, leading to a decreased rate of technologicalprogress. The average annual growth rates of the eﬃciency index for Japan during
1960-1969, 1970-1979, 1980-1989 and 1990-2001 are 8.23, 2.15, 2.06 and 0.29 percent, respectively. On the other hand, as shown in Figure 5, the eﬃciency index for
the United States rose moderately during the 1960s, then gradually declined during
the 1970s. Overall, however, there is an upward tendency from the 1980s onward.
For the United States, the average annual growth rates of the eﬃciency index in the

12

1.4
Efficiency index

1.2
1
0.8
0.6
0.4
0.2
0
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000

Figure 3: Eﬃciency index in Japan

Efficiency index

0.5
0.4
0.3
0.2
0.1
0
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000

Figure 4: Eﬃciency index in U.S.

four relevant sub-periods are as follows: 1.97 percent for 1960-1969; -0.56 percent
for 1970-1979; 1.46 percent for 1980-1989; 1.65 percent for 1990-2001.
To examine the eﬃciency index further, we focus attention on the data of Japan’s
technological knowledge stock obtained from the Cabinet Oﬃce (2002). The technological knowledge stock is deﬁned as the accumulated technological knowledge of theenterprises which have been pursuing research and development (R&D). We havethe data of Japan’s technological knowledge stock for 1973-2000. Let ωt be technological knowledge stock. We deﬁne ωt∗ = log(ωt /ω1 ) at time t. It follows that ω1∗ = 0,
which corresponds to the value in 1973. In addition, we compute ζt∗ = log(ζt /ζ1 )using the estimates of ζt for 1973-2000. Figure 5 depicts the relationship between
the eﬃciency index and the technological knowledge stock.
As shown by the scatter plot in Figure 5, there is a high positive correlation
between the eﬃciency index and the technological knowledge stock; the coeﬃcientof correlation being 0.97. Therefore, we infer that Japan’s eﬃciency index may reﬂect
the eﬀectiveness of investment in R&D and the government’s innovation policies.

13

0.80
0.70
0.60
0.50

ω* 0.40
0.30
0.20
0.10
0.00
0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

ζ*

Figure 5: The eﬃciency index and the technological knowledge stock in Japan

5

Concluding Remarks

In the preceding sections, we have presented an extended production function approach based on Bayesian modeling and its applications to empirical analysis ofJapan and the United States. Unlike earlier studies, in our framework the eﬃciency
parameter is regarded as being time-varying rather than constant. By estimating thetime-varying eﬃciency parameter using a Bayesian method, we attempt to understand the changes in eﬃciency performance at macroeconomic level. Our approachenables a rigorous investigation of the dynamic behavior of a technical factor that
contributes to economic growth.
The main results can be summarized as follows. The average annual growth
rates of the eﬃciency index for Japan during 1960-1969, 1970-1979, 1980-1989 and1990-2001 are 8.23, 2.15, 2.06 and 0.29 percent, respectively. On the other hand, for
the United States, the average annual growth rates of the eﬃciency index in the four
relevant decades are: 1.97 percent for 1960-1969; -0.56 percent for 1970-1979; 1.46
percent for 1980-1989; and 1.65 percent for 1990-2001. Taking a look at the average
growth rates of the eﬃciency index in both countries since the 1980s, the eﬃciency
performance in Japan has declined, while in the United States it has risen.
Although an eﬃciency index includes various components, as mentioned earlier,
it may be inﬂuenced by the technological factors of industries in an economy. A key
constituent of the technological factors is innovation. In connection with the innovative performance and quality of institutions in Japan, for example, Goto (2006)

14

emphasizes not only the research and development of individual companies, but also
the importance of reforming the national systems of innovation, such as national
institutions, customs, and policies, which together form the basis of research anddevelopment. As well, OECD (2006) points out that Japan is lagging behind the
U.S. and Europe in the reformation of innovation systems that correspond to current changes. Our empirical evidence supports a policy proposal by OECD (2006)
that the reforms of national innovation systems is an important issue for restoring
Japan’s economic growth.

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