Suyanto Malmquist Productivity Index Idea, Framewok
.
Volume 16 Nomor 2 , Mei 2011
ISSN 1410-9204
Berbla Publikasi Gagasan Konseptua l, Hasil Penetitian, Kajian, dan Terapan Teori
Suyanto
Malmquist Productivity Index: Idea, Ftamework, and Its
Extensiong on Parametric Approache
I.rza Meinginclra Putri Radjamin Peranan Perbankan Dalam Rangka Menunjang
Industrialisasi di lndonesiaa
Prospek Usaha KreatifBidang Kesehatan di Indonesia Pada
Era Konseptual
Lucia E. Wuryaningsih
Bam bang Budiarto
A. Hery Pratono
Membangun BUMN Ber-Compedtive Advantage di Era
Global
Reinventing The International Trade Theory
ＮＡＭｾ］
Ekon. Bisnis
H
I Vol.16 '
No.2
I H""alaman 1-44 f
Surabaya, Mei 2011 j iSSN 1410-9204
ISSN 1410-920
EKONOMI DAN BISNIS
ｕｮｩｶ･ｲｳｴ｡ｾ＠
Diterbitkan okh Program Studi Ilmu Ekonmni dan Studi Pcmbangunan, Fakultas Ekonom
Surabaya, Jalan Rayn Kalirungkut Surahaya 60293.
Kebijakan Pcnyuntingan: Berkala EKONOMI dan BISNIS diterhitbn sehagai mt.'dia puhlikas
yani
hasil penelitian, bjian, dan rerap _v; = 0. ln a practical sense. this zero distance
fun ction i s unlikely to occur since a producer may not produce at zero output given an avai labi lity
of inputs. Hence, empirical studies, such as the one conducted in this thesis, hardly consider the
zero distance function.
Ill. T he Original Decomposition of MPI
The decomposition of MPI was origi nated by Caves, Christensen, and Diewert ( 1982).
Using the output-oriented Shephard's ( 1970) distance function, as described above, Caves et a/.
( 1982) shows that a change in a firm 's technology fronLier between two consecutive periods can be
decomposed into two components (technological change and cfliciency change).
Suppose that firm i's technology is observed in two periods, r = I. 2. The tt! ( x:, _v, • x,
2
I
ＱＩ］Ｈ
X,.J,,X, ,J,
0
ｄ ｾ Ｈｸ ［ ＮｬＩ ｸ＠
ｄ ｴｾＨ ｸＬ Ｒ Ｎ ｬＩＲ＠
D0 x, ,Y;
D0 x,,_v,
I ( I I)
l ) is a MPI for period t= 1.2 ,
,
I
ｄ ｾ＠ ( x,2 ,
yn
compares second period fi rms to first period technology, ｄ ｾ＠ (
at the first period technology. ｄ ｾ＠
technology, and ｄ ｾ＠ (
x.', y,
1
(5)
2( I I)
represents a distance function that
x;, y}) is a distance function lo r lirm i
(x?,l ) denotes a distance function for firm i at the second period
i s a distance functi on that compares first period firms to the second
)
period technology.
The right-hand side of equation (5) can be rewritten as:
12
I
I
2
2
D0I ( x,1• Y,1)
Mo (x, .y,, x, .y, )= ( D2(
1 1(
o x, • Y;
4
I
2)l"i
0 01 ( X;2 , Y,
D2( z 2)
o x, · .Y,
D2
(} ( .x,.2 ' Y,2)
x Dl { 1 1)
o x, • Y,
(6)
-TCI.2
(,.1
Yl; , ...t,.2 ' Y,2)xTEC1.2
()
""-j '
. 0 ( X;1' Y;1'x,2' Y;2)
where TC is technological change (i.e. , the shift in the technology frontier between the two periods)
and TEC is technical efficiency change (i.e., the movement to the technology frontier).
After the original decomposition of MPl , as in equation (6), researchers then develop
various possible decompositions of MPI by taking into account the scale efficiency change. T he
rationale behind including the scale effi ciency is to relax the constant return to scale assumption.
Grifcll-T atje and Lovell ( 1995) argue that a decomposition of MPI without taking into account scale
efficiency may not measure productivity change as the change in average productivity. Sim ilarl y,
Ray and D esli ( 1997) show that the efficiency change components in MPl consist not on ly or
technical effi ciency change but also scale efficien cy change if rirms operate under variable returns
to scale. In a more formal argument, Hire et al. ( 1994) and ｇｲｩｬｾ
ｬ Ｍ ｔ｡ｴｪ
｣＠ and Lovell ( 1997) indicate
that the constant returns to scale assumption in the original MPl is not properly applied for firms
under a competitive environment. They suggest generalizing the MPI with a scale component that
takes into account the contribution of returns to scale.
In recent yt:ars, various possible ways have been proposed to develop a generalized MPI
(i.e. , a measure of productivity change allributable to scale economics). The non-parametric
techniques (DEA) for decomposing a generalized MPI are proposed by Fare et a/. ( 1994) and
Grifell-T atje and Lovel l ( 1997), and the parametric (SFA) decompositions are addressed by Balk
(200 I ) and Orca (2002). In a non-parametric context, the scale efficiency change is measured hy
comparing the scale efficiency level between two periods. The level of scale efficiency is calculated
using the ratio of distance function values corresponding to constant and variablt: returns to scalt:
technology. I n a parametric context, the scale effi ciency change is direclly measured from the
output-oriented translog distance function with variable returns to scale.
Both paramt:tric and non-parametric techniques have their own merits and demerits in
decomposing a generalized MPI. The debate over which one is the mort: appropriate technique
continues. This thesis adopts a parametric technique for a consistency wi th the stochastic fron tier
approach (SFA) employed in the previous section. From the estimates or SFA and the technical
efficiency scores in the previous section, a generalized MPI aml its components can be calculated.
The next sub-section explains the parametric decomposition of a generalized MPI as proposed by
Ort:a (2002).
IV. Parametric Decomposition of MPI with Scale Efficiency Cha nge
Under a parametric decomposition of a general ized MPI, the distance funct ion is represented by a
specilic functional form. Suppose that finn i's technology in time t is represented by a
transcendental logarithmic (translog) output-ori t:nted distance fun ction, In D0 ( y.,, x;, ,1) . By
applying Diewert's ( 1976) Quadratic I dentity Lemma, Orea (2002) shows that the logarithm of a
generali zed output-oriented MPI hetwecn time period t and t+ I. c;;'+1 , can he decomposed into
technical efficiency change (T EC), technological change (TC), and scale eiTiciency change (SEC),
as expressed below:
c;;;+•=TEe;·"· + rc;·•+l +SEc;•+•
(7)
5
Ekonomi dan Bisnis Vol. 16 No. 2, Mei 2011
where
For the purpose of this paper, it is necessary to assume that the output is only one. 3 Hence, the
econometric version of a stochastic translog output-oriented distance function for a firm panel data
can be represented by:
where y1, represents output of tlrm i at time period t, x,1, represents input n for firm i at time period
t, v1, is a stochastic error component, and u1, represents the error component related to technical
inefficiency.
The technology frontier of the distance function (i.e., D0
In yft =
I
N
yi,
K
I
N
2
nz l
/3 + "L.-!3, ln x,1, +- LLP.k lnx, ,1n.x. + /3,t+-P,/ + "L.-P, 1nx,1,t
0
2 Ｌ ｾ Ｑ＠ ｫ ｾ ｬ＠
n -1
where
N
(y1, ,x1,,t) =I ) is expressed as
1
1,
( 12)
represents the potential maximum output that can be achieve given a set of inputs.
Given equations (I I) and ( I2), the distance to the technology front.ier can be calculated from
In D0 (y,,x1,,t) =In y1, - In y;- v1,
( 13)
which is equi valent to technical inefficiency, u1,. Following Coelli et al. (2005), the technical
efficiency change (TEC) from Equation (8) can be measured by:
ｔｅｃ［ﾷＧ
ｾ Ｇ＠ =In TEu+1 - In TE1,
( 14)
The technological change (TC) index can be obtained from Equations (9) and (II) as follows:
The original output-oriented translog distance function In o0 (Yir • .xit , 1) is expressed in a multi-outputs a nd multiinputs function. The complete translog distance function is given in Orca (2002). In this paper, the o utput is assumed as
only one. and the translog distance function for the econometric version is given in equation ( II ). An assumption of one
output in this thesis is related to the availability of data.
J
6
TCi.rH.r =2 ｾＬＸ＠
J[
N
ln xu ..1 , + ｾｬｮｸ
N
［Ｌ＠ +2,8, +2,8,, ((t+l)+t)
.
]
( 15)
From Equations (I I ), the scale elasticity is expressed as
I
E.,;, =
K
L {J.,k X
2
/3, + -
nil
+ fJ111 t
(16)
k; J
The index of scale efficiency change can then be calculated by using Equations (I 0) and (16).
V. Conclusion
This paper expl ores the theoretical framework and the development of the Malmquist Productivity
I ndex (MPf). The quantity index of Stan Malmquist is discussed in the beginning of this paper as a
basic fundamental for the recent developed MPT. The output-distance function of Shephard (1970)
is presented as the framework within the theoretical analysis of MPJ. The original decomposition of
Caves, Christensen, and Diewert ( I 982) is then explored under the framework of the outputdistance function. This paper presents the parametric decomposition of MPI with scale eiTiciency
change, to show the application of MPl on the productivity measures.
References
Balk, B. M . 200 I. Scale Efficiency and Productivity Change. .lou mal of Productivity Analysis 15
(1): 159-1 83.
Caves, R. E.. L. R. Christensen, and W. E. Diewert. 1982. The Economic Theory oflnd ex Numbers
and the Measurement of Input. Output, and Productivity. Econometrica 50 (6): 1393- 1414.
Coelli, T. J ., D. S. P. Rao, C. J. O'Donnell, and G. E. Battese. 2005. An Introduction. to Efficiency
and Productivity Analysis. 2nd cd. New York: Springer.
Diewert, W. E. 1976. Exact and Superlati ve I ndex Numbers. Journal of Econometrics 4 (2): 115155.
Fare, R., and S. Grosskopf. 1996. lntertemporal Production Frontiers: with Dynamic DEA. Boston,
MA: Kluwer Academic Publishers.
Fare, R., S. Grosskopf. M. Norris, and Z. Z hang. 1994. Productiv ity Growth, Technical Progress,
and Efficiency Change in I ndustrialized Cou ntries. American Economic Review 84 ( I ): 6683.
Fare, R., and D. Primont. 1995. Multi-output Production and Duality: Theory and Applications.
Boston and Dordrencht: Kluwer Academic.
Grifcii -Tatje, E., and C. A. K. Lovell. 1995. A Note on Lhc Malmquist Productivity Index.
Economics Letters 47 (2): 169- 175.
Grifeii-Tatje, E., and C. A. K . Lovell. 1997. T he Sources of Producti vity Change in Spanish
Banking. European Journal of Operational Research 98 (2): 364-380.
Lovell , C. A. K. 2003. The Decomposition
Malmquist Productivity Indexes. Journal of
Productivity A1wlysis 20 (J): 437-458.
M almquist, S. 1953. I ndex numbers and ind ifference surfaces. Trahajos de Estadistica 4: 209-42.
or
7
Ekonomi dan Bisnis Vol. 16 No. 2, Mei 201 I
Nishimizu, M., and 1. Page. 1982. Total Factor Productivity Growth, Technological Progress and
Technical Efficiency Change: Dimensions of Productivity Change in Yugoslavia, 1965-78.
Economic Jouma/92 (368): 920-935.
Orca, L. 2002. Parametric Decomposition of a Ge nerali7.cd Malmquist Productivity Index. Journal
of Productivity Analysis 18 ( I): 5-22.
Ray, S. 1998. Measuring Scale Efficiency from a Translog Production Function. Journal of
Productivity Analysis II (2): 183-1 94.
Ray. S., and E. Dcsli. 1997. Productivity Growth, Technical Progress, and Efliciency Change in
Industrialized Countries: Comment. American Economic Review 87 (5): 1033- 1039.
Shephard. R. W. 1970. Theory of Cost and Production Function. New Jersey: Princeton University
Press.
Zo11o, J. L. 2007. Malmquist Productivity Index Decompositions: A Unifying Framework. Applied
Economics 39 ( 18): 2371-2387.
8
Volume 16 Nomor 2 , Mei 2011
ISSN 1410-9204
Berbla Publikasi Gagasan Konseptua l, Hasil Penetitian, Kajian, dan Terapan Teori
Suyanto
Malmquist Productivity Index: Idea, Ftamework, and Its
Extensiong on Parametric Approache
I.rza Meinginclra Putri Radjamin Peranan Perbankan Dalam Rangka Menunjang
Industrialisasi di lndonesiaa
Prospek Usaha KreatifBidang Kesehatan di Indonesia Pada
Era Konseptual
Lucia E. Wuryaningsih
Bam bang Budiarto
A. Hery Pratono
Membangun BUMN Ber-Compedtive Advantage di Era
Global
Reinventing The International Trade Theory
ＮＡＭｾ］
Ekon. Bisnis
H
I Vol.16 '
No.2
I H""alaman 1-44 f
Surabaya, Mei 2011 j iSSN 1410-9204
ISSN 1410-920
EKONOMI DAN BISNIS
ｕｮｩｶ･ｲｳｴ｡ｾ＠
Diterbitkan okh Program Studi Ilmu Ekonmni dan Studi Pcmbangunan, Fakultas Ekonom
Surabaya, Jalan Rayn Kalirungkut Surahaya 60293.
Kebijakan Pcnyuntingan: Berkala EKONOMI dan BISNIS diterhitbn sehagai mt.'dia puhlikas
yani
hasil penelitian, bjian, dan rerap _v; = 0. ln a practical sense. this zero distance
fun ction i s unlikely to occur since a producer may not produce at zero output given an avai labi lity
of inputs. Hence, empirical studies, such as the one conducted in this thesis, hardly consider the
zero distance function.
Ill. T he Original Decomposition of MPI
The decomposition of MPI was origi nated by Caves, Christensen, and Diewert ( 1982).
Using the output-oriented Shephard's ( 1970) distance function, as described above, Caves et a/.
( 1982) shows that a change in a firm 's technology fronLier between two consecutive periods can be
decomposed into two components (technological change and cfliciency change).
Suppose that firm i's technology is observed in two periods, r = I. 2. The tt! ( x:, _v, • x,
2
I
ＱＩ］Ｈ
X,.J,,X, ,J,
0
ｄ ｾ Ｈｸ ［ ＮｬＩ ｸ＠
ｄ ｴｾＨ ｸＬ Ｒ Ｎ ｬＩＲ＠
D0 x, ,Y;
D0 x,,_v,
I ( I I)
l ) is a MPI for period t= 1.2 ,
,
I
ｄ ｾ＠ ( x,2 ,
yn
compares second period fi rms to first period technology, ｄ ｾ＠ (
at the first period technology. ｄ ｾ＠
technology, and ｄ ｾ＠ (
x.', y,
1
(5)
2( I I)
represents a distance function that
x;, y}) is a distance function lo r lirm i
(x?,l ) denotes a distance function for firm i at the second period
i s a distance functi on that compares first period firms to the second
)
period technology.
The right-hand side of equation (5) can be rewritten as:
12
I
I
2
2
D0I ( x,1• Y,1)
Mo (x, .y,, x, .y, )= ( D2(
1 1(
o x, • Y;
4
I
2)l"i
0 01 ( X;2 , Y,
D2( z 2)
o x, · .Y,
D2
(} ( .x,.2 ' Y,2)
x Dl { 1 1)
o x, • Y,
(6)
-TCI.2
(,.1
Yl; , ...t,.2 ' Y,2)xTEC1.2
()
""-j '
. 0 ( X;1' Y;1'x,2' Y;2)
where TC is technological change (i.e. , the shift in the technology frontier between the two periods)
and TEC is technical efficiency change (i.e., the movement to the technology frontier).
After the original decomposition of MPl , as in equation (6), researchers then develop
various possible decompositions of MPI by taking into account the scale efficiency change. T he
rationale behind including the scale effi ciency is to relax the constant return to scale assumption.
Grifcll-T atje and Lovell ( 1995) argue that a decomposition of MPI without taking into account scale
efficiency may not measure productivity change as the change in average productivity. Sim ilarl y,
Ray and D esli ( 1997) show that the efficiency change components in MPl consist not on ly or
technical effi ciency change but also scale efficien cy change if rirms operate under variable returns
to scale. In a more formal argument, Hire et al. ( 1994) and ｇｲｩｬｾ
ｬ Ｍ ｔ｡ｴｪ
｣＠ and Lovell ( 1997) indicate
that the constant returns to scale assumption in the original MPl is not properly applied for firms
under a competitive environment. They suggest generalizing the MPI with a scale component that
takes into account the contribution of returns to scale.
In recent yt:ars, various possible ways have been proposed to develop a generalized MPI
(i.e. , a measure of productivity change allributable to scale economics). The non-parametric
techniques (DEA) for decomposing a generalized MPI are proposed by Fare et a/. ( 1994) and
Grifell-T atje and Lovel l ( 1997), and the parametric (SFA) decompositions are addressed by Balk
(200 I ) and Orca (2002). In a non-parametric context, the scale efficiency change is measured hy
comparing the scale efficiency level between two periods. The level of scale efficiency is calculated
using the ratio of distance function values corresponding to constant and variablt: returns to scalt:
technology. I n a parametric context, the scale effi ciency change is direclly measured from the
output-oriented translog distance function with variable returns to scale.
Both paramt:tric and non-parametric techniques have their own merits and demerits in
decomposing a generalized MPI. The debate over which one is the mort: appropriate technique
continues. This thesis adopts a parametric technique for a consistency wi th the stochastic fron tier
approach (SFA) employed in the previous section. From the estimates or SFA and the technical
efficiency scores in the previous section, a generalized MPI aml its components can be calculated.
The next sub-section explains the parametric decomposition of a generalized MPI as proposed by
Ort:a (2002).
IV. Parametric Decomposition of MPI with Scale Efficiency Cha nge
Under a parametric decomposition of a general ized MPI, the distance funct ion is represented by a
specilic functional form. Suppose that finn i's technology in time t is represented by a
transcendental logarithmic (translog) output-ori t:nted distance fun ction, In D0 ( y.,, x;, ,1) . By
applying Diewert's ( 1976) Quadratic I dentity Lemma, Orea (2002) shows that the logarithm of a
generali zed output-oriented MPI hetwecn time period t and t+ I. c;;'+1 , can he decomposed into
technical efficiency change (T EC), technological change (TC), and scale eiTiciency change (SEC),
as expressed below:
c;;;+•=TEe;·"· + rc;·•+l +SEc;•+•
(7)
5
Ekonomi dan Bisnis Vol. 16 No. 2, Mei 2011
where
For the purpose of this paper, it is necessary to assume that the output is only one. 3 Hence, the
econometric version of a stochastic translog output-oriented distance function for a firm panel data
can be represented by:
where y1, represents output of tlrm i at time period t, x,1, represents input n for firm i at time period
t, v1, is a stochastic error component, and u1, represents the error component related to technical
inefficiency.
The technology frontier of the distance function (i.e., D0
In yft =
I
N
yi,
K
I
N
2
nz l
/3 + "L.-!3, ln x,1, +- LLP.k lnx, ,1n.x. + /3,t+-P,/ + "L.-P, 1nx,1,t
0
2 Ｌ ｾ Ｑ＠ ｫ ｾ ｬ＠
n -1
where
N
(y1, ,x1,,t) =I ) is expressed as
1
1,
( 12)
represents the potential maximum output that can be achieve given a set of inputs.
Given equations (I I) and ( I2), the distance to the technology front.ier can be calculated from
In D0 (y,,x1,,t) =In y1, - In y;- v1,
( 13)
which is equi valent to technical inefficiency, u1,. Following Coelli et al. (2005), the technical
efficiency change (TEC) from Equation (8) can be measured by:
ｔｅｃ［ﾷＧ
ｾ Ｇ＠ =In TEu+1 - In TE1,
( 14)
The technological change (TC) index can be obtained from Equations (9) and (II) as follows:
The original output-oriented translog distance function In o0 (Yir • .xit , 1) is expressed in a multi-outputs a nd multiinputs function. The complete translog distance function is given in Orca (2002). In this paper, the o utput is assumed as
only one. and the translog distance function for the econometric version is given in equation ( II ). An assumption of one
output in this thesis is related to the availability of data.
J
6
TCi.rH.r =2 ｾＬＸ＠
J[
N
ln xu ..1 , + ｾｬｮｸ
N
［Ｌ＠ +2,8, +2,8,, ((t+l)+t)
.
]
( 15)
From Equations (I I ), the scale elasticity is expressed as
I
E.,;, =
K
L {J.,k X
2
/3, + -
nil
+ fJ111 t
(16)
k; J
The index of scale efficiency change can then be calculated by using Equations (I 0) and (16).
V. Conclusion
This paper expl ores the theoretical framework and the development of the Malmquist Productivity
I ndex (MPf). The quantity index of Stan Malmquist is discussed in the beginning of this paper as a
basic fundamental for the recent developed MPT. The output-distance function of Shephard (1970)
is presented as the framework within the theoretical analysis of MPJ. The original decomposition of
Caves, Christensen, and Diewert ( I 982) is then explored under the framework of the outputdistance function. This paper presents the parametric decomposition of MPI with scale eiTiciency
change, to show the application of MPl on the productivity measures.
References
Balk, B. M . 200 I. Scale Efficiency and Productivity Change. .lou mal of Productivity Analysis 15
(1): 159-1 83.
Caves, R. E.. L. R. Christensen, and W. E. Diewert. 1982. The Economic Theory oflnd ex Numbers
and the Measurement of Input. Output, and Productivity. Econometrica 50 (6): 1393- 1414.
Coelli, T. J ., D. S. P. Rao, C. J. O'Donnell, and G. E. Battese. 2005. An Introduction. to Efficiency
and Productivity Analysis. 2nd cd. New York: Springer.
Diewert, W. E. 1976. Exact and Superlati ve I ndex Numbers. Journal of Econometrics 4 (2): 115155.
Fare, R., and S. Grosskopf. 1996. lntertemporal Production Frontiers: with Dynamic DEA. Boston,
MA: Kluwer Academic Publishers.
Fare, R., S. Grosskopf. M. Norris, and Z. Z hang. 1994. Productiv ity Growth, Technical Progress,
and Efficiency Change in I ndustrialized Cou ntries. American Economic Review 84 ( I ): 6683.
Fare, R., and D. Primont. 1995. Multi-output Production and Duality: Theory and Applications.
Boston and Dordrencht: Kluwer Academic.
Grifcii -Tatje, E., and C. A. K. Lovell. 1995. A Note on Lhc Malmquist Productivity Index.
Economics Letters 47 (2): 169- 175.
Grifeii-Tatje, E., and C. A. K . Lovell. 1997. T he Sources of Producti vity Change in Spanish
Banking. European Journal of Operational Research 98 (2): 364-380.
Lovell , C. A. K. 2003. The Decomposition
Malmquist Productivity Indexes. Journal of
Productivity A1wlysis 20 (J): 437-458.
M almquist, S. 1953. I ndex numbers and ind ifference surfaces. Trahajos de Estadistica 4: 209-42.
or
7
Ekonomi dan Bisnis Vol. 16 No. 2, Mei 201 I
Nishimizu, M., and 1. Page. 1982. Total Factor Productivity Growth, Technological Progress and
Technical Efficiency Change: Dimensions of Productivity Change in Yugoslavia, 1965-78.
Economic Jouma/92 (368): 920-935.
Orca, L. 2002. Parametric Decomposition of a Ge nerali7.cd Malmquist Productivity Index. Journal
of Productivity Analysis 18 ( I): 5-22.
Ray, S. 1998. Measuring Scale Efficiency from a Translog Production Function. Journal of
Productivity Analysis II (2): 183-1 94.
Ray. S., and E. Dcsli. 1997. Productivity Growth, Technical Progress, and Efliciency Change in
Industrialized Countries: Comment. American Economic Review 87 (5): 1033- 1039.
Shephard. R. W. 1970. Theory of Cost and Production Function. New Jersey: Princeton University
Press.
Zo11o, J. L. 2007. Malmquist Productivity Index Decompositions: A Unifying Framework. Applied
Economics 39 ( 18): 2371-2387.
8