Bayesian Estimation of the CES Production
Function with Labor- and Capital-Augmenting
Technical Change

Hideo Noda * and Koki Kyo †Research Group of Economics and Management
No. 2011-E01
2011.6*†

Yamagata University
Obihiro University of Agriculture and Veterinary Medicine

YAMAGATA UNIVERSITY FLSS DISCUSSION PAPER SERIES

Bayesian Estimation of the CES Production Function
with Labor- and Capital-Augmenting Technical Change∗
Hideo Noda† and Koki Kyo‡
Discussion Paper 2011-E01
June 2011

Abstract
In this paper, we propose a new approach to time-series analysis of factor augmenting technical change based on a constant elasticity of substitution (CES) production function. To estimate trends in labor- and capitalaugmenting technical change, smoothness priors are introduced and Bayesian
linear models are constructed. Parameter estimations are then calculated using
the methods of maximum likelihood and Bayesian model averaging. As a practical application, we examine the technical changes in Taiwan and South Korea
at the macroeconomic level. The results of this analysis illustrate that ourBayesian approach can capture the movements of technical change rigorously
compared with conventional approaches.
Keywords: Factor augmenting technical progress; CES production functions;
Smoothness priors; Bayesian linear modeling; Bayesian model averaging
JEL classification: C11; C22; O30∗

An earlier draft of the paper was presented at the 2nd European Asian Economics, Finance,
Econometrics and Accounting Conference in 2010 and the 68th Annual Meeting of the Japan Economic Policy Association in 2011. We are grateful to Ernesto Lorenzo Felli, Koichi Yano, andTatsuyoshi Matsumae for their helpful comments and suggestions. This work is partially supportedby the Grant-in-Aid for Scientific Research (C) (21530193) from the Japan Society for the Promotion of Science, and by the Cooperative Research Program 2 (2017) of the Institute of Statistical
Mathematics.
†
Corresponding author. Faculty of Literature and Social Sciences, Yamagata University; 1-4-12
Kojirakawa-machi, Yamagata 990-8560, Japan; (e-mail): noda@human.kj.yamagata-u.ac.jp‡
Department of Agro-Environmental Science, Obihiro University of Agriculture and Veterinary
Medicine; Inada-cho, Obihiro, Hokkaido 080-8555, Japan; (e-mail): kyo@obihiro.ac.jp1

1

Introduction

Understanding trends in technical change has signiﬁcant implications when forecasting long-term economic performance. A measure of technical change is seen asa key issue in empirical studies of economic growth. However, even if we can deﬁnetechnical progress within the framework of a theoretical analysis, it is very diﬃcult
to quantify that actual aspect. Therefore, models and statistical methods for estimating factors that reﬂect technical progress become indispensable tools. The aimof this paper is to develop a Bayesian approach to time-series analysis for labor- andcapital-augmenting technical change.
The starting point for this type of econometric analysis is the construction of aframework based on production functions. Many researchers use the Cobb-Douglas
production function or the CES production function with Hicks neutral technicalchange. It should be noted, however, that there are some drawbacks with these.
For example, although the Cobb-Douglas production function assumes an elasticity
of substitution equal to one, its true value in an economy is not necessarily very
close to unity. Also, observations that the shares of the labor income and thecapital income change in an almost constant manner during a certain period, areoften considered as evidence supporting the use of the Cobb-Douglas production
function. However, as pointed out by Sato (1970), the invariance of each share does
not guarantee that the functional form is consistent with Cobb-Douglas type. As for
the CES production function with Hicks neutral technical change, the implication isthat the eﬃciencies in production of labor and capital are identical. This is a verystrong assumption, and obviously unusual: the statistical analyses based on such
CES production functions have a narrow focus and scope. Hence, any discussion
based on such analysis becomes extremely limited.
Considering these problems in statistical analyses that can be seen in many previous studies, we adopt a CES production function with factor-augmenting (biased)
technical change. The term biased technical change refers to a situation in which
the eﬃciency of capital and the eﬃciency of labor exhibit diﬀerent growth rates.
Hence, movements in the capital- and labor-augmenting technical progress can beclearly identiﬁed.
Recent studies related to the present paper include Antràs (2004), Klump et al.
(2007b) and Sato and Morita (2009).1 Antràs (2004) and Klump et al. (2007b)
examined the econometric estimations of production functions using data stemmingfrom the US economy. Sato and Morita (2009) estimated the growth rates of capitaland labor eﬃciencies for the Japanese and US economies according to equations
1

See Klump et al. (2007a) for a review article on empirical studies of CES production functions.

2that are derived in Sato (1970). These articles treat CES production functions withtechnical change that augment the eﬃciency of both capital and labor, and present
careful analyses. Nevertheless, we consider that there is still room for improvement.
For speciﬁcations for the trends in technical change, Antràs (2004) follows the bulkof the literature in assuming that the eﬃciencies in both labor and capital grow atconstant rates.2 Also, Klump et al. (2007b) assumes constant growth rates as well
as a form that includes exponential, logarithmic and hyperbolic growth as special
cases. It seems to give rise to inﬂexible models because the actual technical progressmight not necessarily show the trends that Antràs (2004) and Klump et al. (2007b)
assume. Moreover, even if the ﬁt of the model is suitable for the relevant period,
the forecast might be wide of the mark when the assumed trends are not forthcoming. For an estimation of the factor-augmenting technical changes, Sato and Morita(2009) did not a priori assume exponential, hyperbolic, or any other form. In thatsense, the method of Sato and Morita (2009) might be superior to those of Antràs(2004) and Klump et al. (2007b). However, we think that problems will arise as
their proposed method cannot calculate the elasticity of factor substitution directlyfrom a CES production function with factor-augmenting technical change. To estimate the capital- and labor-augmenting technical changes, they use the elasticity of
substitution derived from a CES production function with Hicks neutral technicalchange as a proxy.
Unlike Antràs (2004), Klump et al. (2007b) and Sato and Morita (2009), we
address the estimation of this factor-augmenting technical change from a Bayesian
perspective. Speciﬁcally, we combine the use of a Bayesian linear modeling approach in Akaike (1980) and a smoothness-priors approach in Kitagawa and Gersch(1996). By introducing the smoothness priors, we can set up the model with a sufﬁciently ﬂexible structure to capture the dynamic features of the economic system,
hence our model is very useful for estimating complicated behaviors of economic
variables that cannot be observed directly. Studies on the application of such an
approach to the analyses of dynamic structure of an economy are rare. The exceptions include Kyo and Noda (2008) and Noda and Kyo (2009a, 2009b). In Kyo andNoda (2008), smoothness priors were incorporated into a statistical model basedon a Cobb-Douglas production function, and addressed, by applying Bayesian techniques, the estimation of trends in China’s total factor productivity (TFP). Noda
and Kyo (2009a) also incorporated a smoothness prior in a statistical model basedon a Cobb-Douglas form for the production function, and estimated the trends in
TFP for the Japanese prefectural economies. Based on a CES production function
2

See, for example, Kotowitz (1968), Panik (1976), Kalt (1978), Jalava et al. (2006) and Masanjala
and Papageorgiou (2004).

3

with Hicks neutral technical change, Noda and Kyo (2009b) applied the smoothness
priors approach in a comparative analysis of the technical change in the Japaneseand U.S. macro-economies. Recall, however, that there are some drawbacks mentioned above in the Cobb-Douglas production functions and the CES production
functions with Hicks neutral technical change.
In our framework, the eﬃciencies of capital and labor are regarded as timevarying parameters, and thereby introduce dynamic features into capital- and laboraugmenting technical change. Parameter estimations are then carried out using
the methods of maximum likelihood and Bayesian model averaging. As a practicalapplication of the proposed Bayesian approach, we attempt to estimate the technicalchanges in the macro-economies of Taiwan and South Korea. The results of thisanalysis illustrate that our Bayesian approach can treat movements in technicalchange in more detail compared with the conventional production function approach,and therefore provide a useful insight into the empirical study of technical change.
The rest of the paper is organized as follows. In Section 2, we construct ananalytical framework based on Bayesian models via a CES production function withfactor augmenting technical change. Section 3 presents the basic scheme for parameter estimation and a procedure to apply Bayesian model averaging. In Section 4,
our proposed approach is applied to time-series analysis of the macro economies inSouth Korea and Taiwan. Finally, Section 5 concludes the paper.

2

Bayesian Modeling based on a CES Production Function

Consider an aggregative economy in which each year a single homogeneous good is
produced by two factors of production, capital and labor, under constant returns
to scale technology. We follow David and van de Klundert (1965) by writing the
production function[
Qt = (At Kt )−ρ

+ (Bt Lt )

−ρ

]− 1

ρ

,

(1)

where the subscript t indexes yearly increments, Qt the value added, Kt capital, andLt labor. Note that Kt and Lt are stocks, that is, amounts observed at one point in
time, while Qt is a ﬂow. The model in Eq. (1) is a speciﬁcation of the production
function with factor-augmenting technical change.3 The elasticity of factor substitution equals 1/(1 + ρ), hence ρ is termed the substitution parameter. The coeﬃcients
At and Bt indicate the eﬃciency in production of capital and labor, respectively.
3

See Sato and Beckmann (1968) for a derivation of the generalized form of such production
functions.

4In addition, variation in At with time is interpreted as capital-augmenting technicalchange and variation in Bt over time is interpreted as labor-augmenting technicalchange.
−ρ−ρ
We can rewrite Eq.(1) by setting at = A−ρt , bt = Bt , x1t = (−1/ρ)Kt ,x2t = (−1/ρ)L−ρand yt = (−1/ρ)Q−ρtt , in which notation Eq.(1) becomes
yt = at x1t + bt x2t .

(2)Furthermore, we augment Eq. (2) to include a random disturbance:
yt = at x1t + bt x2t + ϵt ,(3)where ϵt is taken to be Gaussian white noise.
Noda and Kyo (2009b) applied a Bayesian approach using smoothness priors of
Kitagawa and Gersch (1996) to estimate the Hicks neutral technical change. This
Bayesian approach can also be applied in estimating the time-varying parametersat and bt . However, it may be diﬃcult to obtain meaningful estimates for at andbt simultaneously, because there is a remarkable multicollinearity between the timeseries data for x1t and x2t .
To ameliorate this diﬃculty, we simplify the model in Eq. (3) by treating bt as aconstant eb, i.e., we put b1 = b2 = · · · = eb. Then, the model reduces to the following
form:
(1)
yt = at x1t + ebx2t + ϵt ,where eb is an unknown parameter, and ϵt

(1)

N(0,

σ 2 ).

(4)(1)

is a Gaussian white noise with ϵt∼

For the model in Eq. (4), the time-varying parameter at is treated as arandom variable from a Bayesian viewpoint. Thus, we use the following smoothness
prior:
at − 2at−1 + at−2 = ν1t ,(5)

where ν1t is a Gaussian white noise with ν1t ∼ N(0, σ 2 /d21 ), and d1 > 0 is an unknown(1)

parameter. It is also assumed that ϵtand ν1t are independent of each other. To

deﬁne a proper prior distribution for the at ’s it is necessary to set a0 and a−1 . Here,all of the parameters ρ, eb, σ 2 , a0 , a−1 and d1 are considered as unknown constants.
When the values of these parameters are given, a system of linear Bayesian equations
for at ’s can be constructed based on Eqs. (4) and (5). From this, we can obtain aposterior distribution for the at ’s by using the Bayesian linear modeling method.
Alternatively, in an analogous manner, the model in Eq. (3) can also be simpliﬁedby treating at as constant, i.e., we assume that a1 = a2 = · · · = ea. Thus, the model5

becomes the following form:(2)

yt = eax1t + bt x2t + ϵt ,
(2)

(6)
(2)

where ea is an unknown parameter, ϵt is a Gaussian white noise with ϵt∼ N(0, σ 2 ).

Here we treat bt as a random variable and use the smoothness prior as follows:bt − 2bt−1 + bt−2 = ν2t ,(7)

where ν2t is a Gaussian white noise with ν2t ∼ N(0, σ 2 /d22 ), and d2 > 0 is an unknown(2)

parameter. Again, we assume that ϵtand ν2t are independent of each other, andthat the parameters ρ, ea, σ 2 , b0 , b−1 and d2 are unknown constants. When thevalues of these parameters are given, a system of Bayesian linear equations for the
bt ’s can be constructed based on Eqs. (6) and (7). Then we can obtain a posterior
distribution for bt ’s by using the Bayesian linear modeling method.
Thus, we have two similar Bayesian models that simplify the model of Eq. (3),speciﬁcally, those deﬁned by the pair Eqs. (4) and (5) and the pair Eqs. (6) and (7).
By combining the estimates from the two Bayesian models and using the Bayesianmodel averaging (BMA) approach, we can obtain synthetic estimates for the modelin Eq. (3). Note that ρ and σ 2 are parameters common to the two models.

3

Parameter Estimation

3.1

Basic Scheme

We now present a procedure for estimating the time-varying parameter at , basedon the constant-b Bayesian model, Eqs. (4) and (5). Assume that we have a data
set for Qt , Kt and Lt for a period of n points as t = 1, 2, . . . , n. From Eq. (4), thelikelihood of a = (a1 , a2 , . . . , an )t and the other related parameters is given by:
(
)n
[
]
1
1
f (Q|a, ρ, eb, σ 2 ) = J √exp − 2 ||y − Xa − ebx||2 ,2σ2πσ 2where J is the Jacobian of the transformation from Q to y deﬁned by J = Πnt=1 (∂yt /∂Qt ).
The other symbols are deﬁned as follows:
Q = (Q1 , Q2 , . . . , Qn )t ,y = (y1 , y2 , . . . , yn )t ,X = diag(x11 , x12 , · · · , x1n ),x = (x21 , x22 , . . . , x2n )t ,and || ∗ || denotes the Euclidean norm.

6

When the values of σ 2 , c = (a−1 , a0 )t , and d1 are given, from Eq. (5) the priordensity for a is given by:(π(a|σ 2 , c, d1 ) =


√

)n

d1
2πσ 2

[

]
d21
exp − 2 ||Da + Bc||2 ,2σ··· ··· ··· 0..  −2 1 . . .
. 



.. 
.
.
 1 −2 1
.
. 

D=

..  ,
.
.
.
.
.
.
.
.
 0
.
.
.
. . 


 ..

.. .. .. ..
 .
.
.
.
. 0 
0 ··· 01 −2 1

where

1

0



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

Next, we obtain the marginal likelihood of the related parameters as∫∫∫
2e
L(ρ, b, σ , c, d1 ) =· · · f (Q|a, ρ, eb, σ 2 )π(a|σ 2 , c, d1 )da
(8)
[
)n
]
(
1
1
2
n
t
− 12
b || ,
d1 det(W W ) exp − 2 ||h − W a
= J √
2σ
2πσ 2
[

where
W =

X
−d1 D

]

[
,

h=

y + ebx
d1 Bc],

b = (W t W )−1 W t h denotes the mean of a posterior densityand af (a|Q; ρ, eb, σ 2 , c, d1 ) = ∫∫f (Q|a, ρ, eb, σ 2 )π(a|σ 2 , c, d1 ),∫· · · f (Q|a, ρ, eb, σ 2 )π(a|σ 2 , c, d1 )da

(9)

b 2
for a. Therefore, the estimates, ρb, eb, σb ,bc, and db1 for the related parameters canbe obtained by maximizing the likelihood L(ρ, eb, σ 2 , c, d1 ) in Eq. (8).
Computationally, if the values of ρ and d1 are given, then a together with cand eb can be estimated simultaneously by using the least squares method (see forexample, Jiang, 1995). That is, when we put
[
z=

y
0n

]

[
,

V =

X
O
x
d1 D d1 B 0n



]
,


a
β= c eb

with O denoting a zero matrix, and 0n being an n-dimensional vector of zeros, theestimate,b d1 ) =
β(ρ,

(

)t
b
b t (ρ, d1 ), bact (ρ, d1 ), eb(ρ, d1 )= (V t V )−1 V t z,7bfor β is obtained. Thus, by substituting eb = eb(ρ, d1 ) and c = bc(ρ, d1 ) into L(ρ, eb, σ 2 , c, d1 )in Eq. (8), the partial likelihood of ρ, σ 2 and d1 can be obtained asb
L1 (ρ, σ 2 , d1 ) = L(ρ, eb, σ 2 , bc, d1 ).

(10)

Correspondingly, a conditional posterior density, g(a|Q; ρ, σ 2 , d1 ), for a can be derived from f (a|Q; ρ, eb, σ 2 , c, d1 ) in Eq. (9) by:()
bg(a|Q; ρ, σ 2 , d1 ) = f a|Q; ρ, eb(ρ, d1 ), σ 2 , bc(ρ, d1 ), d1 .
Moreover, the estimate db1 of the parameter d1 is obtained by maximizing themarginal likelihood L1 (ρ, σ 2 , d1 ) in Eq. (10) for given ρ and σ 2 . Thus, we obtain
the partial likelihood of ρ and σ 2 asL∗1 (ρ, σ 2 ) = L1 (ρ, σ 2 , db1 ).

(11)

Note that the procedure for estimating the parameter b = (b1 , b2 , . . . , bn )t in theconstant-a Bayesian model, Eqs. (6) and (7), is the same, and is omitted here forbrevity. For this model, we can obtain the partial likelihood L2 (ρ, σ 2 , d2 ) of ρ, σ 2and d2 by a procedure similar to the above basic scheme. Then the partial likelihoodof ρ and σ 2 can be obtained by using the maximum likelihood estimate db2 of d2 asL∗2 (ρ, σ 2 ) = L2 (ρ, σ 2 , db2 ).

3.2

(12)

Bayesian Model Averaging Approach

To obtain the estimates for ρ and σ 2 and the ﬁnal estimates for parameters a and b of
the model in Eq. (3), we use the BMA approach as follows. Here, we reconsider thetwo Bayesian models with constant-b and constant-a; these models are identiﬁed hereas M1 and M2 respectively. ¿From the BMA perspective (see for example, Claeskens
and Hjort, 2008), it is assumed that all models {M1 , M2 } are uncertain and all haveequal uncertainties. Thus, we use a uniform prior for the models {M1 , M2 } as follows:1q(M1 ) = q(M2 ) = ,2
where q(Mi ) is the prior probability for the i-th model Mi (i = 1, 2). As shown in
the preceding subsection, a partial likelihood for the model Mi is L∗i (ρ, σ 2 ), as given
in Eqs. (11) and (12). Therefore, a conditional posterior probability for model Mi
is given byp(Mi |Q; ρ, σ 2 ) ==L∗i (ρ, σ 2 )q(Mi )L∗1 (ρ, σ 2 )q(M1 ) + L∗2 (ρ, σ 2 )q(M2 )L∗i (ρ, σ 2 ).
L∗1 (ρ, σ 2 ) + L∗2 (ρ, σ 2 )8Moreover, from the basic estimation scheme just mentioned, we can see that forb (ρ, d1 ) and the estimate ofmodel M1 given ρ and d1 , the posterior mean of a is abb = eb1n is eb(ρ, d1 )1n . Similarly, for model M2 given ρ and d2 , the posterior meanbof b is bb(ρ, d2 ) and the estimate of a = e
a1n is ea(ρ, d2 )1n . Therefore, based on theBayesian model averaging approach, the synthetic estimates for a and b are givenby (see, Draper, 1995):bb (ρ, σ 2 ) = p(M1 |Q; ρ, σ 2 )baa(ρ, db1 ) + p(M2 |Q; ρ, σ 2 )bea(ρ, db2 )1n ,bbbb(ρ, σ 2 ) = p(M2 |Q; ρ, σ 2 )bb(ρ, db2 ) + p(M1 |Q; ρ, σ 2 )eb(ρ, db1 )1n .
Furthermore, we consider the quantity
L(ρ, σ 2 ) = L∗1 (ρ, σ 2 )q(M1 ) + L∗2 (ρ, σ 2 )q(M2 ))1( ∗=L1 (ρ, σ 2 ) + L∗2 (ρ, σ 2 )2(13)

to be a measure of the ﬁtting reliability for the model in Eq. (3) with respect
to the parameters ρ and σ 2 . Thus, the estimates ρb and σb2 of ρ and σ 2 can beobtained by maximizing the mean likelihood L(ρ, σ 2 ) in Eq. (13) with respect tothese parameters. Hence, the ﬁnal estimates for a and b are obtained as
b
b
b=a
b (b
a
ρ, σ
b2 ),

4

Application

4.1

Data Sources

b
b
b
b=b
b(b
ρ, σ
b2 ).

To estimate the CES production function for Taiwan, we use real GDP (Qt ) andnon-land ﬁxed physical capital (Kt ) from Chow and Lin (2002).4 In addition, labor
data (Lt ) are calculated according to the product of the number of employed workers
from Mizoguchi (2008) and the quality of human capital per worker from Chow and
Lin (2002). The annual time-series data for Taiwan cover the period from 1951 to1999.
Also, we conducted parameter estimations of the CES production function forSouth Korea using macro-data taken from Pyo (2001), which are real GDP (Qt ),physical capital (Kt ), and the number of employed workers (Lt ). The annual timeseries data for South Korea covers a similar period, from 1946 to 1999.

4

The non-land fixed capital comprises fixed assets excluding land values. Chow and Lin (2002)
excluded land values and cumulative depreciation from total gross fixed assets to obtain a net value
for non-land fixed assets.

9

4.2

Results and Discussion

When we calculated the estimation procedure presented in Section 3, we used yearlydata with values from each year divided by the value in the initial year, that is,
Qt /Q1 , Kt /K1 and Lt /L1 for t = 1, 2, . . . , n, where t = 1 denotes the initial year
and t = n denotes the ﬁnal year in our sample. Considering this data, we follow de
La Grandville (2009) by rewriting Eq. (1) as
Qt
Q1

[
=

] 1
( L )−ρ − ρKt )−ρtδ ht+ (1 − δ) gt
.
K1
L1
(

(14)

In Eq. (14), we call ht and gt the modiﬁed eﬃciencies in production of capital andlabor, respectively. Under the assumptions h1 = g1 = 1 and competitive factor
imputation, δ denotes the share of capital income from the total income at baseyear t = 1 and (1 − δ) denotes the corresponding share of labor income (see, de La

Grandville, 2009 for details). Note that we have relations at = At−ρ = δh−ρandtbt = Bt−ρ = (1 − δ)gt−ρ .

As mentioned earlier, the CES production function with Hicks neutral technicalchange is a special case of the CES production function with factor augmenting
technical change. To conﬁrm the superiority of the factor augmenting type as astatistical model, we estimated not only the CES production function with factor
augmenting technical change in the present paper, but also the CES production
function with Hicks neutral technical change, such as Noda and Kyo (2009b) using
the data set mentioned above. As a result, when we use the Taiwan data, the value ofthe Akaike information criterion (AIC), given Hicks neutral type is −17.052, whilethat for the factor augmenting type is −398.025. Also, when we use the South
Korea data, the value of AIC, given Hicks neutral type is −0.017, while that for
the factor augmenting type is −367.431. Consequently, from the viewpoint of theminimum AIC model selection procedure, it is obvious that the factor augmenting
type is better model. Thus, we present only the results of the estimation for the
CES production function with factor augmenting technical change.
For Taiwan, the estimated results of the constant parameters are as follows:bd1 = 3.4924, db2 = 4.4152, ρb = −0.09418, and σb2 = 4.0896 × 10−5 . For South Korea,
these are: db1 = 10.7064, db2 = 14.3917, ρb = −0.1504, and σb2 = 4.6191 × 10−4 .
The estimate for the elasticity of substitution can be calculated from the estimate
ρb. Speciﬁcally, these estimates for Taiwan and South Korea are 1.104 and 1.177,
respectively. De La Grandville (2009) derived and proved two remarkable theoremson the relationship between the elasticity of substitution and performance of macroeconomy. These can be stated as follows:

10(i) If an economy is driven by a CES production function exhibiting capital- andlabor-augmenting technical progress, then at any stage of its development percapita income is an increasing function of its elasticity of substitution.
(ii) If an economy is driven by a CES production function exhibiting capital- andlabor-augmenting technical progress, then at any stage of its development thecost of producing one unit of national product is a decreasing function of theelasticity of substitution.
According to these two theorems, we can conjecture that the long-term per capita
income in South Korea may be greater than that in Taiwan, and the long-term costof producing one unit national product in South Korea may be lower than in Taiwan.
Tables 1 and 2 show the estimates b
at and bbt for Taiwan and South Korea respectively.
Table 1: Estimated results of the time-varying parameters for Taiwan
b
at
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975

0.69462
0.69677
0.698650.700210.701400.702370.703240.70405
0.705020.706090.70730
0.70868
0.710290.71209
0.713630.71484
0.715810.716650.71763
0.718910.720440.72178
0.722440.72249
0.72311

bbt

b
at
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999

0.724440.72551
0.726710.72757
0.728290.728840.729180.729840.73075
0.731480.73257
0.733840.734900.73579
0.736520.73730
0.73801
0.738710.739390.740080.740810.741450.74187
0.74237

1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975

11

0.864210.87066
0.87635
0.881160.884860.88795
0.89080
0.893440.896700.90033
0.904510.90933
0.91501
0.921470.927100.93169
0.93555
0.939070.943130.948390.95466
0.96023
0.963290.964420.96774

bbt
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999

0.97354
0.97838
0.983640.98788
0.99168
0.99478
0.99699
1.00015
1.00422
1.00776
1.012541.018171.02337
1.02806
1.032221.036471.04056
1.04472
1.04880
1.05300
1.05735
1.06128
1.06438
1.06763

Table 2: Estimated results of the time-varying parameters for South Korea
b
at
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972

bbt

b
at

0.498440.497120.495770.494140.49235
0.490990.490240.490180.490190.489900.48915
0.48792
0.48625
0.48428
0.48235
0.48095
0.480290.48040
0.48106
0.48189
0.48280
0.48366
0.48457
0.48552
0.48632
0.48697
0.48748

1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999

0.48799
0.48837
0.48861
0.48860
0.488100.486990.485520.48420
0.483890.48486
0.486860.48930
0.49176
0.494190.49632
0.49792
0.49882
0.499290.499510.49960
0.499830.500210.500450.500200.499470.49894
0.49917

1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972

0.54203
0.541070.540070.53876
0.537210.53604
0.535450.53556
0.53577
0.53573
0.535290.534450.533210.53166
0.530100.52904
0.52864
0.52894
0.529710.53063
0.53161
0.53256
0.53360
0.53471
0.535700.53656
0.53731

bbt
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1991
1993
1994
1995
1996
1997
1998
1999

0.53809
0.538800.539420.539860.539910.539430.538630.53793
0.538100.539400.54167
0.544400.547220.55005
0.552630.55478
0.556360.55757
0.558540.559370.560210.56109
0.56175
0.56192
0.56164
0.56145
0.56177−ρ
Because, by deﬁnition, at = A−ρand bt = Bt−ρ = (1 − δ)gt−ρ , we obtaint = δht

−1/ρ

at

−1/ρ

= At = δ −1/ρ ht and bt= Bt = (1 − δ)1/ρ gt . Furthermore, the relations

At /A1 = ht /h1 and Bt /B1 = gt /g1 hold. In other words, the index of eﬃciency
of capital (At /A1 ) synchronizes with the index of the modiﬁed eﬃciency of capital(ht /h1 ). A similar remark holds between (Bt /B1 ) and (gt /g1 ). We can obtain the
−1/b
ρ
−1/b
ρ
bt and B
bt from b
estimates Aaand bb, respectively.
t

t

12

bt /Ab1 ) × 100 and (Bbt /Bb1 ) × 100 for Taiwan.
Figure 1 presents graphs for (A
1000
900
800

Index of capital's efficiency

700
600

Index of labor's efficiency

500
400
300
200
100
0
1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999

Figure 1: The indices of capital eﬃciency and labor eﬃciency for Taiwan

From Figure 1, we ﬁnd clearly that the eﬃciencies of labor and capital in Taiwan
have changed at diﬀerent growth rates. In particular, the rapid growth in labor eﬃciency is remarkable. This result suggests that labor-augmenting technical progress
had contributed much to Taiwan’s economic growth during the ﬁve-decade period.
bt /Ab1 ) × 100 and (Bbt /Bb1 ) × 100 for South Korea.
Figure 2 shows the graphs of (A
130

120

Index of capital's efficiency
110

Index of labor's efficiency
100

90

80

70
1946 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998

Figure 2: The indices of capital eﬃciency and labor eﬃciency for South Korea

The following features are seen from Figure 2. First, movements in labor andcapital eﬃciencies for South Korea are more complex than those of Taiwan. Moreover, we ﬁnd a similar pattern between the eﬃciencies, although the curves diﬀer inmagnitude. An upward tendency is seen in both over the period from the 1980s to
the early 1990s. However, this movement stagnated during the late 1990s.

13

The following conclusions are obtained from a comprehensive viewpoint. First,
there is a possibility that capital eﬃciency and labor eﬃciency have distinctly different trends. Figure 1 for Taiwan is one such example. Moreover, as shown in
Figure 2 for South Korea, movements of eﬃciencies of capital and labor may be
non-monotonic. Thus, our practical application suggests that modeling with a highdegree of ﬂexibility is indispensable in capturing the trends in technical change. In
other words, the model should be able to deal with the diversity and complexity inthe behavior of technical change. However, many earlier studies impose stringent
constraints on the trends in eﬃciencies of capital and labor, such as the eﬃciencies being exponentially dependent on time. Thus, conventional approaches thatlack the necessary ﬂexibility within the model cannot estimate trends in technicalchange precisely, and might be misleading.

5

Conclusion

Recently, the underlying role of technical progress in economic growth has become
increasingly important. However, statistical methods for estimating technical change
have so far not been suﬃciently developed. We improve the conventional production function based approaches by using Bayesian modeling, and propose a new
methodology in the time-series analysis of capital- and labor-augmenting technicalchange.
As an illustration, we applied our proposed approach to an analysis of data obtained for Taiwan and South Korea. Our main results can be summarized as follows:
The Taiwanese economy experienced rapid growth in labor eﬃciency from 1950s to
1990s, while capital eﬃciency showed a modest growth during the same period.
This implies that the labor-augmenting technical progress had contributed much to
Taiwan’s economic growth over the period. For South Korea, a similar pattern wasseen for labor and capital eﬃciencies, although each developed at diﬀerent levels andgrowth rates. In particular, both had shown an upward tendency from the 1980s to
the early 1990s, but movements stagnated during the late 1990s. Our estimations
suggest that to sustain future economic growth in South Korea, policies need to beestablished to promote both capital- and labor-augmenting technical progress.
To our knowledge, attempts at analyzing concepts like technical progress as in thepresent study have not yet been reported. The Bayesian perspective as contributed
here sheds light on an approach to capture such concepts as the factor augmenting
technical change. In particular, because our approach enables a statistical analysisof trends in technical change in a rigorous manner by a ﬂexible structure of themodels introducing smoothness priors, it has deﬁnite superiority over conventional

14

approaches. Therefore, our approach can be regarded as a promising tool for anempirical study of technical change.

References[1] Akaike, H. (1980), Likelihood and the Bayes Procedure, in Bayesian Statistics,
edited by J. M. Bernardo, M. H. de Groot, D. V. Lindley and A. F. M. Smith,
University Press, pp.143-166.
[2] Antràs, P. (2004), “Is the US Aggregate Production Function Cobb-Douglas?,”Contribution to Macroeconomics, 4, Article 4.
[3] Chow, G. C. and A. L. Lin (2002), “Accounting for Economic Growth in Taiwan and Mainland China: A Comparative Analysis,” Journal of Comparative
Economics, 30, pp.507-530.
[4] Claeskens, G. and N. L. Hjort (2008), Model Selection and Model Averaging,
Cambridge University Press.
[5] David, P. A. and Th. van de Klundert (1965), “Biased Eﬃciency Growth andCapital-Labor Substitution in the U.S., 1899-1960,” American Economic Review, 55, pp.357-394.
[6] de La Grandville, O. (2009), Economic Growth: A Unified Approach, Cambridge University Press.
[7] Draper, D. (1995), “Assessment and Propagation of Model Uncertainty,” Journal of the Royal Statistical Society, B-57, 45-97.
[8] Jalava, J., M. Pohjola, R. Matti, A. Ripatti and J. Vilmunen (2006), “Biased
Technical Change and Capital-Labour Substitution in Finland, 1902-2003,”
Topics in Macroeconomics, 6, Article 8.
[9] Jiang, X. Q., (1995), “A Bayesian Method for the Dynamic Regression Analysis,” Transactions of the Institute of Systems Control and Information Engineers, 8, pp.8-16.
[10] Kalt, J. P. (1978), “Technological Change and Factor Substitution in the UnitedStates: 1929- 1967 ,” International Economic Review, 19, pp.761-775.
[11] Kitagawa, G. and W. Gersch (1996), Smoothness Priors Analysis of Times
Series, Springer-Verlag.

15[12] Klump, R., P. McAdam, and A. Willman (2007a), “The Long-Term SucCESs
of the Neoclassical Growth Model,” Oxford Review of Economic Policy, 23,
pp.94-114.
[13] Klump, R., P. McAdam, and A. Willman (2007b), “Factor Substitution
and Factor-Augmenting Technical Progress in the United States: A Normalized Supply-Side System Approach,” Review of Economics and Statistics, 89,
pp.183-192.
[14] Kotowitz, Y. (1968), “On the Estimation of a Non-Neutral CES Production
Function,” Canadian Journal of Economics, 1, pp.429-439.
[15] Kyo, K. and H. Noda (2008), “Bayesian Analysis of the Dynamic Structure
in China’s Economic Growth,” in Bayesian Inference and Maximum Entropy
Methods in Science and Engineering, edited by M. S. Lauretto, C. A. B. Pereira,and J. M. Stern, American Institute of Physics, pp.309-316.
[16] Masanjala, W. H. and C. Papageorgiou (2004), “The Solow Model with CES
Technology: Nonlinearities and Parameter Heterogeneity,” Journal of Applied
Econometrics, 19, pp.171-201.
[17] Mizoguchi, T. (2008), Asian Histrical Statistics: Taiwan, Toyo Keizai (inJapanese).
[18] Noda, H. and K. Kyo (2009a), “Bayesian Methods for TFP Analysis of a MultiRegion Economy with Dynamic Structure and Application to Japan,” Journalof Economic Research, 14, pp.151-195．
[19] Noda, H. and K. Kyo (2009b), “Bayesian Estimation of the CES Production
Function with a Time-Varying Eﬃciency Parameter,” The Empirical Economics
Letters, 8, pp.949-960.
[20] Panik, M. J. (1976), “Factor Learning and Biased Factor-Eﬃciency Growth in
the United States, 1929-1966,” International Economic Review, 17, pp.733-739.
[21] Pyo, H. K. (2001), “Economic Growth in Korea (1911-1999): A Long-term
Trend and Perspective,” Seoul Journal of Economics, 14, pp.59-125.
[22] Sato, R. (1970), “The Estimation of Biased Technical Progress and the Production Function,” International Economic Review, 11, pp.179-208.
[23] Sato, R. (2006), Biased Technical Change and Economic Conservation Laws,
Springer.

16

[24] Sato, R. and M. J. Beckmann (1968), “Neutral Inventions and Production Functions,” Review of Economic Studies, 35, pp.57-66.
[25] Sato, R. and T. Morita (2009), “Quantity or Quality: The Impact of Labour
Saving Innovation on US and Japanese Growth Rates, 1960-2004,” Japanese
Economic Review, 60, pp.407-434．

17