Bayesian Analysis of Economic Growth in Taiwan and
Mainland China via Dynamic Production Functions
Koki Kyo*
Hideo Noda†

Research Group of Economics and Management
No. 2008-E04
2008.10 (Revised 2009.1)*†

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

Bayesian Analysis of Economic Growth in Taiwan and
Mainland China via Dynamic Production Functions
Hideo Noda
Koki Kyo

Abstract
To analyze the dynamic structures of the economic growth of Taiwan and mainland
China, we propose a set of models of dynamic production functions that consider
total factor productivity (TFP) and output elasticities as time-varying parameters.
Bayesian method based on the smoothness priors approach is applied for parameter
estimation. A new approach, called the random grouping method, is proposed toovercome diﬃculties resulting from rapid changes in the TFP trend. According to
the estimation results, we ﬁnd that the growth rates of TFP in Taiwan have been
ﬂuctuated at very low levels since the late 1980s. On the other hand, TFP in mainland
China has shown a gentle upward tendency since the late 1970s. Consequently it would
appear that the reform and open-door policies since 1978 have contributed to TFP inmainland China.
Keywords: economic growth, total factor productivity, dynamic production function, Bayesian estimation, method of random grouping
JEL classification: C11, C22, C51, O40, O57

1

Introduction

Under the circumstances of the advance of globalization and growing international economic interdependence, the trends for growth of nations have come a major concern formany people such as entrepreneurs, and administration policy-makers. Especially, the highgrowth performance of China (hereinafter called mainland China) has recently received
much worldwide attention because it has achieved remarkable economic development since
beginning its economic reforms and opening-up policy in 1978. The process of mainland
China’s growth has often been compared with that of Taiwan (e.g. Chow (2002, 2007)).
Taiwan’s rapid growth occurred about two decades earlier than that of mainland China.
Among Asian newly industrializing economies (NIEs), the Taiwan economy has achieved
signiﬁcant successes.
1

The sources of economic growth are generally decomposed into capital, labor and totalfactor productivity (TFP). As it is obviously diﬃcult to perpetually increase capital orlabor, many researchers consider that from a long-term perspective an increase in TFP is
the most important factor to achieve sustained economic growth. Hence, grasping trends
related to TFP is a crucial issue in empirical research of economic growth. In this paper,
we propose a new approach based on Bayesian methodology to estimate the trends in TFP
for mainland China and Taiwan.
The two methodologies used in most earlier literature on the growth performance of
mainland China and Taiwan have been growth accounting and the econometric estimationof production functions or its variations. For example, reports using the growth accounting
approach include Young (1995), Borensztein and Ostry (1996), Singh and Trieu (1999),
Wu (2002, ch.5), Wang and Yao (2003), Young (2003), Bosworth and Collins (2008). On
the other hand, reports based on the econometric approach include Armer and Liu (1993),
Chow (1993), Chow and Li (2002), Chow and Lin (2002), Lin (2003, 2004).
Though earlier studies provide interesting results, we believe that there remains scopefor additional research in this ﬁeld. As pointed out by Barro (1999) and Kyo and Noda
(2006), one problem with growth accounting is the assumption that social marginal products can be measured by production factor prices. We believe that this is not necessarily
the case if social marginal products coincide with production factor prices. The econometric approach of the production functions or its variations does not rely on the abovementioned assumption. However, a disadvantage of the econometric approach is that it
fails to estimate well trends in TFP. As well, in the econometric approach it is usuallyassumed that the output elasticities of production factors are constant over time. Suchoutput elasticities may, however, vary over time. For this reason we consider a model inwhich the output elasticities vary over time. The foundation of the approach applied inthis paper is the methodologies for Bayesian linear modeling using the smoothness priors
approach, developed by Akaike (1980) and Kitagawa and Gersch (1996). In Jiang (1995)
these methods are applied to regression models with time-varying coeﬃcients.
The paper is organized as follows. In the next section, we present the basic models.
2The third section introduces a set of Bayesian methods for parameter estimation. Resultsof the estimation and discussions thereof are given in the fourth section. Finally, the ﬁfth
section concludes the paper.

2

Model construction

2.1

BackgroundThe structure of our model, introduced below, is related to that of Wan and Yao (2003),
but diﬀers from theirs in several respects. These diﬀerences are elaborated below. In thefollowing discussion, we use the symbol t as the time variable.
Wang and Yao (2003) used the growth accounting approach to examine the sources of
economic growth in China. They speciﬁed the aggregate production function as
Qt = At Ktθ (L∗t Ht )μ ,(1)

where Qt is the real gross domestic product (GDP), Kt is the stock of physical capital,L∗t is total employment, and Ht is the stock of human capital, measured as the number of
average eﬀective years of schooling per person in the 14-65 age group. Hence, Lt ≡ L∗t Ht
is a skill-adjusted measure of labor input. At represents the productivity of an economy
when all factor inputs are used. For this reason, it is called total factor productivity, orjust TFP. θ and μ are output elasticities with respect to physical capital and skill-adjusted
labor, respectively.
In this paper, we regard θ and μ as time-varying parameters together with At , andconsequently, henceforth, are replaced with θt and μt . In Wang and Yao (2003), a constantreturns to scale (CRS), that is, θ + μ = 1 is assumed. This assumption was also applied fora dynamic production function in Kyo and Noda (2008) to obtain stability of estimation.
In this paper we waive the assumption of CRS because it may cause a biased estimation
for the time-varying parameter case.

3

2.2

Models

To analyze economic growth in Taiwan and mainland China, with a few modiﬁcations to
the model in Eq. (1) we introduce the dynamic production function as
Qt = At Ktθt Lμt t .

(2)

As stated above, one notable feature of the production function in Eq. (2) is that θtand μt are considered time-varying parameters together with TFP At . Under logarithmictransformation, Eq. (2) can be rewritten as
qt = θt kt + μt lt + at ,where qt = ln Qt , kt = ln Kt , lt = ln Lt , at = ln At . Thus, we have a statistical model fora set of sample data of size n as follows:
qt = θt kt + μt lt + at + t(t = 1, 2, . . . , n),(3)where t is assumed to be a Gaussian white noise sequence with t ∼ N(0, σ 2 ), where σ 2
is a constant unknown parameter.
Clearly, we can not obtain valuable estimates of the parameters by a method such as
the method of least squares. In this paper, we regard θt , μt and at for t = 1, 2, . . . , n as
random variables from a Bayesian perspective, and apply the smoothness priors approach,introduced by Kitagawa and Gersch (1996), in setting up prior distributions. Smoothness
priors are introduced to these parameters based on the assumption that they vary smoothly
over time. A set of ﬁrst order stochastic diﬀerence equations is used as priors of the timevarying parameters as follows:θt − θt−1 = ψt ,(4)

μt − μt−1 = φt ,(5)

at − at−1 = νt .

(6)

Here we assume that ψt and φt are Gaussian white noise sequences with ψt ∼ N(0, σ 2 /d21 )and φt ∼ N(0, σ 2 /d21 ).
4

However, empirical knowledge obtained from Kyo and Noda (2008) implies that there
may be rapid changes in the trend of at , the logarithmic transformation of TFP. Thus, anadded relationship is introduced to the sequence νt as
νt = ct + ωt ,(7)

where ωt is a Gaussian white noise sequence with ωt ∼ N(0, σ 2 /d22 ). Here, ct is considered
an unknown parameter for the case that there is a rapid change on at and ct = 0 otherwise.
The constraint

nt=1 ct = 0 is introduced here to ensure thatnt=1 E{νt } = 0.

Moreover, in the above models, θ0 , μ0 , a0 , d1 , d2 and the parameters in the set{c1 , c2 , . . . , cn } are as parameters in the prior distributions of the random parametersθ = (θ1 , θ2 , . . . , θn )t , μ = (μ1 , μ2 , . . . , μn )t and a = (a1 , a2 , . . . , an )t ; thus they are called
hyperparameters in Bayesian modeling. Here, we regard the hyperparameters as unknown
constants. Further, we also assume that t , ψt , φt and ωt are independent of each other.
By combining Eqs. (3), (4), (5), (6) together with Eq. (7), a set of Bayesian linear models
for the parameters θ, μ and a can be constructed.

3

Parameter estimation

3.1

Basic scheme

Here we give a basic scheme for parameter estimation based on Jiang (1995), which applied
the methodology developed by Akaike (1980).
We assume that there are m non-zero elements, denoted by {c∗1 , c∗2 , . . . , c∗m }, in the set{c1 , c2 , . . . , cn }, and denote a vector of the non-zero elements by c∗ = (c∗1 , c∗2 , . . . , c∗m )t , so
we have m < n. We treat the elements of c∗ as unknown parameters. From Eq. (3), thelikelihood of θ, μ, a, and σ 2 is given by 1 n1exp − 2 ||y − X 1 θ − X 2 μ − a||2 ,f (y|θ, μ, a, σ 2 ) = √
2σ
2πσ 2

where y = (q1 , q2 , . . . , qn )t , X 1 = diag(k1 , k2 , · · · , kn ), X 2 = diag(l1 , l2 , · · · , ln ), and || ∗ ||denotes the Euclidian norm. Furthermore, from the assumption in Eqs. (4), (5), (6)together with Eq. (7) for a set of given values of σ 2 and the hyperparameters, the prior5densities for θ, μ, a are respectively given by
n


d
d2
√ 1
exp − 12 ||Dθ − θ0 u||2 ,
2σ
2πσ 2
 d
n


d2
√ 1
exp − 12 ||Dμ − μ0 u||2 ,π2 (μ|σ 2 , μ0 , d1 ) =
2σ
2πσ 2
 d
n


d2
√ 2
exp − 22 ||Da − a0 u − Gc∗ ||2 ,π3 (a|σ 2 , a0 , c∗ , d2 ) =
2σ
2πσ 2

π1 (θ|σ 2 , θ0 , d1 ) =

⎡

where



⎤

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

0

⎡

(8)

⎤

1
⎢ 0 ⎥
⎢
⎥
⎢
⎥
0 ⎥,
u=⎢
⎢
⎥
⎢ .. ⎥⎣ . ⎦0

and G = (gij ) is an n × m matrix with gij = 1 for c∗j = ci and gij = 0 otherwise.
Then, by putting bt = (θt , μt , at ), we obtain the marginal likelihood of σ 2 and thehyperparameters as
L(σ 2 , θ0 , μ0 , a0 , c∗ , d1 , d2 )f (y|θ, μ, a, σ 2 )π1 (θ|σ 2 , θ0 , d1 )π2 (μ|σ2 , μ0 , d1 )π3 (a|σ 2 , a0 , c∗ , d2 )dθdμda

=


= √

1

n

2πσ 2

n
t
−1/2
d2n
exp
1 d2 det(W W )

⎡

where

⎤

X1 X2
I
⎢ d D
O
O ⎥
⎢
⎥
W =⎢ 1⎥,⎣ O
d1 D O ⎦O
O d2 D



−


1
2

||W
b
−
h||
,
2σ2

⎡

⎤

(9)

y
⎢
⎥
d1 θ0 u
⎢
⎥
h=⎢
⎥
⎣
⎦
d1 μ0 ud2 a0 u + d2 Gc∗

with O being a zero-matrix, and b = (W t W )−1 W t h denoting the mode of a posterior
density,f (b|y; σ 2 , θ0 , μ0 , c∗ , d1 , d2 )=f (y|θ, μ, a, σ 2 )π1 (θ|σ 2 , θ0 , d1 )π2 (μ|σ 2 , μ0 , d1 )π3 (a|σ 2 , a0 , c∗ , d2 ),f (y|θ, μ, a, σ 2 )π1 (θ|σ 2 , θ0 , d1 )π2 (μ|σ 2 , μ0 , d1 )π3 (a|σ 2 , a0 , c∗ , d2 )dθdμdaof b given σ 2 and the hyperparameters. Maximizing the likelihood L(σ 2 , θ0 , μ0 , a0 , c∗ , d1 , d2 )
0 , 
a0 , c∗ ,in Eq. (9) with respect to the hyperparameters, we obtain their estimates θ0 , μd1 and d2 .
6Computationally, if the values of d1 and d2 are given, b together with θ0 , μ0 , a0 and c∗
can be estimated simultaneously using the least squares method (see Jiang (1995)). Thatis, if we set
⎡

X1

⎢ d D
⎢
V =⎢ 1
⎣ O

O

⎤

X2
I
0
0
0
0
O
O −d1 u
0
0
0 ⎥
⎥
⎥,
d1 D O
0
−d1 u
0
0 ⎦
0
0
−d2 u −d2 G
O d2 D

⎡

y

⎤

⎢ 0 ⎥
⎢
⎥
z=⎢
⎥,
⎣ 0 ⎦

0

⎡

⎤

b
⎢ θ ⎥
⎢ 0 ⎥
⎢
⎥
β = ⎢ μ0 ⎥ ,⎢⎥⎣ a0 ⎦c∗t−1 t

the estimate, β(d1 , d2 ) = (V V ) V z, for β is obtained. Here, 0 denotes a vector of 1 , d2 )zeros. Thus, the partial likelihood of d1 and d2 can be obtained by substituting β(dinto Eq. (9), and then the log-likelihoods for σ 2 , d1 and d2 are given as 2 (d1 , d2 )1σ
− 2 ln(d21 ) − ln(d22 )) + ln(det(W t W ))} (10)(σ 2 , d1 , d2 ) = − {n(ln(2πσ2 ) +
2
σ2

with
 2 (d1 , d2 ) =σ

1 1 , d2 )||2 .
||z − V θ(dnSo the estimates d1 , d2 of d1 and d2 can be obtained by maximizing the function (σ 2 , d1 , d2 )in Eq. (10) with respect to d1 and d2 using a quasi-Newton method. Finally, the esti d , d ). Hence the ﬁnal estimates for all the parameters are = β(
mate of β is given by β
1 2

obtained.
 2 (d1 , d2 ), which is its maximum
It is well-known that σ 2 can be estimated with σlikelihood estimate, or, equivalently, the least squares estimate. However, for the data 2 (d1 , d2 ) may be very small, which may lead to unstable
analyzed here, the value of σ
 2 (d1 , d2 ) as the estimate of σ 2 . In thisestimates for the parameters when we apply σpaper, we give an estimate for σ 2 in a reasonable way.

3.2

Identifying rapid changes in TFP

Based on the above basic scheme for parameter estimation, we set out a procedure for
identifying rapid changes on {a1 , a2 , . . . , an }; that is, we propose a new approach for
identifying c∗ from {c1 , c2 , . . . , cn }. The newly-proposed approach is called the method of
random grouping.
7

First, for integers r and s that satisfy 3s + r = n and 1 ≤ r ≤ 3, we consider that there
are not changes on the ﬁrst r initial values of the sequence at , so we put c1 = c2 = · · · =cr = 0. Thus, it is only necessary to identify c∗ from {cr+1 , cr+2 , . . . , cn }. In the process ofidentifying c∗ , we assume, temporarily, that all of elements in {cr+1 , cr+2 , . . . , cn } are unknown parameters. Clearly, the elements in the set {cr+1 , cr+2 , . . . , cn } can be divided intos groups of three because the number of the elements is 3s. We haveassumption thatnnt=r+1 ct = 0 from the

t=1 ct = 0; that is, the mean of all the elements in {cr+1 , cr+2 , . . . , cn }is zero.
For three elements, say c1 , c2 and c3 , in a group, we consider an orthogonal transformation of (c1 , c2 , c3 )t as follows:H(c1 , c2 , c3 )t = (c1 , c2 , c3 )t ,where H = (hij ) is a 3-dimensional Helmert matrix with the elements h11 = h12 = h13 =√√√√1/ 3, h21 = −h22 = 1/ 2, h23 = 0, h31 = h32 = 1/ 6 and h33 = −2/ 6. The set{c1 , c2 , c3 }, can be considered a sample drawn from the set {cr+1 , cr+2 , . . . , cn }. Thus, it
can be seen that c1 is in proportion to the mean of the sample {c1 , c2 , c3 }, so c1 = 0 holdswith a larger probability. Therefore, we can set c1 = 0 and treat c2 and c3 as parameters.
The above consideration is applied as follows.
Let P denote a 3s-dimensinal permutation matrix and consider an orthogonal transformation of the vector (cr+1 , cr+2 , . . . , cn )t as follows:H ∗ P (cr+1 , cr+2 , . . . , cn )t = (c1 , c2 , . . . , c3s )t
with

⎡

H

⎢
⎢ O
⎢
H∗ = ⎢ .
⎢ .
⎣ .

O

O

(11)

⎤

··· O
. ⎥
.
H . . .. ⎥
⎥
⎥
.. ..
.
. O ⎥
⎦
··· O H

being a 3s-dimensional orthogonal matrix. Suppose that in the set {c1 , c2 , . . . , c3s } there
are s zero elements and 2s non-zero elements. We treat the non-zero elements in {c1 , c2 , . . . , c3s }c∗ . Thus,
as unknown parameters and express the vector of these unknown parameters by 
8Eq. (11) can be rewritten asc∗ ,H ∗ P (cr+1 , cr+2 , . . . , cn )t = U or equivalently,(cr+1 , cr+2 , . . . , cn )t = P (H ∗ )t U 
c∗

(12)

with U being a properly-designed 3s × 2s matrix.
From Eq. (6) together with Eqs. (7) and (12), we can set up an another prior density
for a as follows:
 d
n


d2
2
c∗ , d2 ) = √exp − 22 ||Da − a0 u − e||2 ,π3∗ (a|σ 2 , a0 , 
2σ
2πσ 2

(13)

where et = (c1 , c2 , . . . , cr , (U c∗ )t H ∗ P ) with c1 = c2 = · · · = cr = 0. By replacing thec∗ , d2 ) and using the basic
prior density π3 (a|σ 2 , a0 , c∗ , d2 ) in Eq. (8) with π3∗ (a|σ 2 , a0 , c∗ together with other parametersscheme in Section 3.1, we can obtain the estimate of and hyperparameters. Therefore, all the estimates {c1 , c2 , . . . , c3s } for the elements in{c1 , c2 , . . . , c3s } can be obtained. Hence, we can an estimate (cr+1 , cr+2 , . . . , cn )t for thevector (cr+1 , cr+2 , . . . , cn )t using(cr+1 , cr+2 , . . . , cn )t = P H ∗t (c1 , c2 , . . . , c3s )twhich is derived from Eq. (11).
Furthermore, we can obtain N sets of estimates for (cr+1 , cr+2 , . . . , cn )t , each using a
(j)

(j)

(j)

permutation matrix constructed randomly. Let (cr+1 , cr+2 , . . . , cn )t be the j-th estimatefor (cr+1 , cr+2 , . . . , cn )t , we can use
1

ci =

N

(j)
ci ,

N j=1

(i = r + 1, r + 2, . . . , n)

(14)

as the ﬁnal estimates for {cr+1 , cr+2 , . . . , cn }. Then, we select m elements, that have larger
absolute values estimated, as unknown parameters, hence the unknown parameters in theset {cr+1 , cr+2 , . . . , cn } are identiﬁed.

9

4

Results and discussions

4.1

Results for Taiwan

To estimate the model for Taiwan, we use real GDP (Qt ), the number of hours worked
adjusted by using the number of years of schooling of the working population as a skilladjusted labor (Lt ), and physical capital (Kt ) from Chow and Lin (2002). Hence, we canregard the skill-adjusted labor (for descriptive purposes, hereinafter called labor) data of
Chow and Lin (2002) as a proxy of human capital. The data of Chow and Lin (2002)
cover the period 1951-1999.
Chen (2003) divides the stages of postwar economic development of Taiwan into four
periods as: the reconstruction period from 1945 to 1952, the self-sustained period from1953 to 1963, the high-growth period from 1964 to 1973, and the medium growth period
after 1974. The reconstruction period has also been called the economic system reformatting period. In this period, agricultural goods were mainly exported. The self-sustained
period is marked by import substitution industrialization, which meant switching policy
from imports to domestic production. In the high-growth period, an export-oriented industrialization policy is implemented. Trade during this period is characterized by importing
raw materials and exporting manufactured products. In the early 1970s, the Taiwan government adopted a heavy and chemical industry policy to promote the advancement of
Taiwan’s industrial structure, and since the 1980s has developed industrial policy programs
focused on the high-tech industry..
Figures 1 and 2 show the estimated trends in the output elasticity with respect tot ) in Taiwan, respecphysical capital (θt ) and the output elasticity with respect to labor (μtively. The estimated trend in the output elasticity with respect to the physical capitalvaries between 0.613 and 0.616. On the other hand, the estimated trend in the output
elasticity with respect to labor varies between 0.414 and 0.416. As well, because the sumof the output elasticities of physical capital and labor exceeds one, the Taiwan economy
is regarded as showing increasing returns to scale in the period 1951-1999.

10

0.617
0.616
0.615
0.614
0.613
0.612
1951

1955

1959

1963

1967

1971

1975

1979

1983

1987

1991

1995

1999

Figure 1: The estimated trend in the output elasticity of physical capital (θt ) in Taiwan
0.417
0.416
0.415
0.414
0.413
0.412
1951

1955

1959

1963

1967

1971

1975

1979

1983

1987

1991

1995

1999

Figure 2: The estimated trend in the output elasticity of labor (μt ) in Taiwan
From Figures 1 and 2, we ﬁnd that the estimated trend in both output elasticities
with respect to physical capital and labor indicate a similar movement during the period
1951-1999. That is, there are some peaks and troughs from the 1950s to the 1980s. In the
1990s, we show that there is little change in the estimates of both output elasticities. As
well, it is found that the estimates of both output elasticities are relatively high comparedwith those in the previous periods. Therefore, it would appear that the contributions perunit of physical capital and labor to economic growth from the 1990s onward were greaterthan in previous periods.
t )) in Taiwan. As is clear
Figure 3 shows the trend for estimated TFP (At = exp(afrom Figure 3, we ﬁnd that there is a pronounced drop in TFP from the mid-1970s to themid-1980s. That is, we conﬁrm that there was a precipitous fall in the mid-1970s, andthen an accelerated rise in the mid-1980s. Therefore, in the mid-1980’s, the TFP had amarked eﬀect on economic growth in the Taiwanese economy. Since the late 1980s, the11growth rates of TFP in Taiwan have ﬂuctuated, but at very low levels.
4.2
4
3.8
3.6
3.4
3.2
1951

1955

1959

1963

1967

1971

1975

1979

1983

1987

1991

1995

1999

Figure 3: The estimated trend in TFP (At ) in Taiwan

4.2

Results for mainland China

Our sample for mainland China covers a 54-year period (1952-2005). Data for real GDP(Qt ), the stock of physical capital (Kt ) and the number of employed persons (L∗t ) are takenfrom Chow (2008), while data for the average number of schooling years of the working
population as a proxy of human capital (Ht ) is taken from Minami, Makino and Luo(2008). As in Wang and Yao (2003), we use a skill-adjusted measure of labor, Lt ≡ L∗t Ht ,as the data for labor.
First, noting as a background to the social economy of mainland China, political movements may have aﬀected the economic growth in China during the period studied. During
the period 1952-2005,there were four main political movements as social campaigns. The
ﬁrst was the Great Leap Forward campaign of 1958-1962. Although Chairman Mao’s hopefor economic growth was well directed, the Great Leap Forward caused many people tostarve to death, and so was a monumental failure. The second was the Cultural Revolutionof 1966-1976, which climaxed around 1968. It also caused extensive damage to the Chinese
economy, hence it was also a total failure. The third was the reform and opening-up from1978. This led to the restoration of growth in the mainland Chinese economy. The fourth
was the demonstration at Tian’anmen Square in Beijing in the spring of 1989. This caused
economic stagnation because for a few years there was a slump in trade and tourism to12mainland China.
t ) are
The estimated trends in output elasticities of physical capital (θt ) and labor (μshown in Figures 4 and 5, respectively. We can see that the trend in output elasticities ofphysical capital nearly ﬂatten out to a very high level (an average of about 0.887) over thet parallels that in θt at a very low level (an average of
period studied, and the trend in μ

about 0.036). From these Figures, we can see also that these two output elasticities changeover time with similar patterns. The output elasticities decrease in the periods 1958-1962(Great Leap Forward), 1966-1968 (the climax of Cultural Revolution), and 1988-1990(demonstration at Tian’anmen Square), while they increase from 1978 (beginning of thereform era).
0.890
0.888
0.886
0.884
0.882
0.880

1952

1956

1960

1964

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004

Figure 4: The estimated trend in the output elasticity of physical capital (θt ) in mainland China
0.040
0.038
0.036
0.034
0.032
0.030

1952

1956

1960

1964

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004

Figure 5: The estimated trend in the output elasticity of labor (μt ) in mainland China
Moreover, it is found that the estimates of the output elasticities of both physicalcapital and labor, diﬀer from the values of capital share and labor share computed from13

national accounts data. In the conventional growth accounting approach, the output
elasticities of physical capital and labor is frequently approximated using capital share andlabor share. Hence, Our estimation results suggest that researches using the conventional
growth accounting approach may lead to the erroneous conclusions.
Figure 6 shows the estimated trend in TFP in the mainland China economy. This result
suggests that the trend in TFP may be inﬂuenced by the political movements mentioned
above. For example, there was a sharp decline around the time at the end of the period
1958-1962 (Great Leap Forward), while there is a full turnaround around the climax of
the Proletarian Cultural Revolution. It is interesting that the trend in TFP is mostly
downwards during the period 1966-1976 (Cultural Revolution), especially it undergoes
a rapid change during the period 1966-1968 (the climax of Cultural Revolution), andthen turns slightly upwards during the post-reform period. However, there is a minor
trough around 1990, which may be the result of the shock caused by the demonstration
at Tian’anmen Square in 1989.
0.6
0.5
0.4
0.3
0.2
1952

1956

1960

1964

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004

Figure 6: The estimated trend in TFP (At ) in mainland China
Overall, mainland China’s TFP has shown a gentle upward tendency since the late1970s. Consequently it would appear that the reform and open-door policies since 1978
have contributed to mainland China’s TFP.

14

4.3

Discussions

By comparing the above results for Taiwan with those for mainland China, we can see thatthe estimates of the output elasticities of physical capital and labor are not so diﬀerent in
Taiwan, but they are very diﬀerent in mainland China. Speciﬁcally, the output elasticityof labor is very low in mainland China. As stated in Chow and Lin (2002), the rationale
for the low estimate for the output elasticity of labor may be to the result of an abundanceof labor in mainland China. Thus, surplus labor may yield almost zero marginal output.
On the other hand, Interestingly, the trends in the output elasticities of both physicalcapital and labor for Taiwan and mainland China show similar patterns. This is useful
information for improving modeling in the future studies.

5

Conclusions

To analyze the dynamic structure of the economic growth of Taiwan and mainland China,
we have proposed a set of models of dynamic production functions that consider TFP and
output elasticities as time-varying parameters. We use the smoothness priors approachfor parameter estimation. The new approach, called the random grouping method, wasproposed to overcome diﬃculties resulting from rapid changes in the TFP trend. The
results illustrated good performance of the newly-proposed method.
The main results can be summarized as follows. According to the estimation resultsof Taiwan’s aggregate production function, in the period 1951-1999, we found that output
elasticity with respect to physical capital changed values from around 0.613 to 0.616, while
output elasticity with respect to labor has changed values from around 0.414 to 0.416. Wealso conﬁrmed that the contributions of per unit of physical capital and labor to economic
growth from the 1990s onward were greater than in previous periods. As well, because thesum of output elasticities of physical capital and skill-adjusted labor exceeds one, Taiwan’s
aggregate production function is regarded as increasing returns to scale during the period
1951-1999. As for TFP in Taiwan, we found that there was a precipitous fall in the mid1970s, and then an accelerated rise in the mid-1980s. Therefore, in the mid-1980’s, the15

TFP had a marked eﬀect on economic growth in the Taiwanese economy. Since the late1980s, Taiwan’s TFP growth rates have ﬂuctuated at very low levels.
On the other hand, the estimation results of mainland China’s aggregate production
function for the period 1952-2005 indicated that output elasticity with respect to physicalcapital has had values change from around 0.885 to 0.888, while output elasticity withrespect to labor has changed in value from around 0.033 to 0.039. Therefore, in China,
output elasticity with respect to physical capital was relatively high, whereas output elasticity with respect to labor was very low. Since 2000, both output elasticities have showna gradual upward tendency. Also, because the sum of output elasticities of physical capitaland skill-adjusted labor was below one, mainland China’s aggregate production function
is regarded as decreasing returns to scale during the period 1952-2005. As for mainland
China’s TFP trend, it was conﬁrmed that the sharp declines from the late 1950s to theearly 1960s and from the mid-1960s to the late 1960s, corresponded to the periods of Great
Leap Forward and the climax of Cultural Revolution, respectively. Since the late 1970s,
TFP has shown a gentle upward tendency. This suggests that the reform and open-door
policies implemented since 1978 have had a positive eﬀect on mainland China’s TFP.

Acknowledgments
We gratefully acknowledge the valuable comments of Professor Genshiro Kitagawa at theInstitute of Statistical Mathematics, Japan. This study was carried out under the auspicesof the ISM Cooperative Research Program (2008-ISM·CRP-2013).

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