Discussion Papers In Economics
And Business

Systematized and Path-independent Measurement of Biased Technical Change

Tsunehiro Otsuki

Discussion Paper 09-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSystematized and Path-independent Measurement of Biased Technical Change

Tsunehiro Otsuki

Discussion Paper 09-11

April 2009

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSystematized and Path-independent Measurement of Biased Technical Change
Tsunehiro Otsuki †Abstract
The sample-specificity and path-dependence of the data envelopment analysis (DEA) based
technical change index as a component of Malmquist indexes prevent us from obtaining
overall and systematic information on technical change. This paper develops a pathindependent method to estimate technical change using a systematized set of controlled input–
output vectors and visualization of the DEA frontiers. The application to the panel datasets ofagricultural production in the Brazil Amazon in 1975–1995 indicates non-Hicks-neutral
technical change, with crossings of frontiers in both the 1975–1985 and the 1985–1995
periods. The alternative measure of overall technical change shows that moderatetechnological progress may have occurred on the whole in 1975–1995. The results also show
heterogeneous trends across products. The mean of the sample-specific technical change
scores are found to be quite different from the overall technical change measure.

JEL classification: D24、O30
Key words: Data envelopment analysis, Path dependence, Agricultural intensification

Acknoledgement: This research is funded by the Japan Society for the Promotion of Science, Grant-in-Aid for
Scientific Research (C), 2007-2009.
†
Corresponding author: Tsunehiro Otsuki, Osaka School of International Public Policy, Osaka University, 1-31
Machikaneyama, Toyonaka, Osaka, 560-0043 JAPAN. E-mail : otsuki@osipp.osaka-u.ac.jp

I. Introduction
Data envelopment analysis (DEA) has been a widely used nonparametric method to estimate
productivity. The popularity of this DEA-based Malmquist productivity change index rests in
its decomposition into measures of technical change and change in technical efficiency inaddition to the common advantage of DEA to accommodate multiple-input and multipleoutput technology. As the DEA-based method calculates the components of Malmquist
indexes at the level of individual decision-making units (DMUs) (Färe et al. (FGNZ), 1994),
the contribution of each component can be investigated for each DMU. When overall
technical change, such as how the production frontier shifts over time, is of interest, however,the DMU-specific measures provide little information because information on each DMU ismutually-isolated. Also, in the case of non-Hicks-neutral technical change, the DMU-specifictechnical change indexes encounter the problem of path dependence, or non-transitivity, thatstems from the path dependence of Malmquist indexes, which causes the estimate of technicalchange to take different values depending on the input-output levels at which technicalchange is measured.
The DMU-specificity of DEA-based indexes prevents estimation of the production
frontier as an integrated set of best-practice DMUs and, hence, estimation of overall technicalchange. While technical efficiency is attributable to individual DMU’s ability, technicalchange is a shift of production frontier that occurs exogenously and indiscriminately to all
DMUs over the economy. Accordingly, the difference in technical change scores across
DMUs is simply accounted for by the difference in their input-output bundles. Therefore, amethod to estimate overall technical change would need to allow for the flexibility to measuretechnical change at arbitrary input-output bundles without being constrained to existing inputoutput bundles.

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Furthermore, path dependence of the technical change index makes it difficult toestimate overall technical change using actual data, when technical change is non-Hicksneutral. Färe et al. (FGR) (1998) showed that the Malmquist indexes are path dependent
except when technical change is Hicks-neutral. The DMU-specific technical change index cantake different values depending on which input-output levels are to be used between two
different points of time for the same overall technical change. Färe et al. (FGGL) (1997)
suggested further a decomposition of the technical change index into input and output bias
indexes. Managi and Karameri (2004) is among recent empirical studies that estimate
FGGL’s bias indexes, and they demonstrated using the US agricultural production data in
1960-1996 that the size of the biases is large enough to attract attention. Chen and Ali (2004)
indicated the possibility of having an opposite sign of technical change score for the same
DMU due to the path dependence of the technical change index when frontiers of two periods
cross each other.
This paper develops a DEA-based method to estimate overall technical change using
controlled input–output vectors while avoiding the DMU-specificity and the path-dependence
problems. An application of the method is undertaken to assess technical change in the
agricultural sector in the Brazil Amazon in 1975–1995.

II. The ModelLet x ( x1 ,..., xN ) ∈ℜ+N=
and y ( y1 ,..., yM ) ∈ℜ+M be the vectors of inputs and outputs,=
respectively. Let Y t ( x ) be the output set in period t that is producible by x . The output set isassumed to be closed, bounded, and convex and to exhibit strong disposability of all outputs.
The output-oriented Malmquist index under the Caves et al. (1982) (CCD) definition is a ratioof the following two distance functions:-2-=
D t ( y t , x t ) inf{θ : y t / θ ∈ Y t ( x t )}

(1)

t
D
=
( y t +1 , x t +1 ) inf{θ : y t +1 / θ ∈ Y t ( x t +1 )} .

(2)

and

Assuming constant returns to scale (CRS), the solution of DEA for the distancefunction in (1) is obtained for the kth DMU using the following linear programming:
JJ−1{D t ( yk t , xk t )=} sup{θ : s.t.∑ z j y t mj ≥ θ y t mk , ∑ z tj xnjt ≤ x t nk ,z j ,θ
=j 1 =j 1(3)=
m 1,..., M=, n 1,..., N , z j ≥ =0, k 1,..., J }The value of D t ( y t +1 , x t +1 ) is calculated by replacing y t mk and xt nk with y t +1mk and xt +1nk

D t ( y t +1 , x t +1 ). It isfrom (3), respectively. Thus, the Malmquist index is given as M CCD1 =Dt ( y t , x t )constructed based on the period t reference technology. Alternatively, the Malmquist index
can adopt the reference technology in period t+1, M CCD 2 =D t +1 ( y t +1 , x t +1 ).
D t +1 ( y t , x t )The CCD Malmquist index can be decomposed into technical change (TC) andefficiency change (EC) components:

 D t ( y t +1 , x t +1 )   D t +1 ( y t +1 , x t +1 ) 
M CCD1 =TCCCD1 ⋅ ECCCD1 =  t +1 t +1 t +1   , (4)ttt
D (y , x )  D (y , x ) Thus, technical change can be defined as a ratio of values of distance functions at periods t
and t+1 evaluated at the same input–output bundle. When the values of distance functions are

Dt ( y t , x t )based on the input–output bundle in period t, then TCCCD 2 = t +1 t t . Likewise, the FGNZ
D (y , x )-3-

Malmquist index can be decomposed into TC and EC while TC is a geometric mean of
TCCCD1 and TCCCD 2 .
The path dependence of the technical change index implies the difference between
TCCCD1 and TCCCD 2 , and vise versa, when the technology does not exhibit the Hicks-neutrality.

FGGL (1997) showed that the ratio of TCCCD1 to TCCCD 2 can be expressed as the product of an
input and an output bias indexes (IB and OB, respectively), where OB (IB) measures thedeparture from the Hicks-output(input)-neutrality of technical change. When the technology does
not exhibit the Hicks-output-neutrality, their Proposition 1 implies that OB equals unity (i.e.
there is no output bias) only if y t +1 = λ y t , λ > 0 (temporal constancy of the output mix). Whenthe technology does not exhibit the Hicks-input-neutrality, their Proposition 2 implies that IB
equals unity (i.e. no input bias) only if x t +1 = x t under the CRS assumption. Thus, pathdependence problem is inevitable in practice as the above conditions on input-output bundle are
seldom satisfied.
This path-dependence problem can be avoided by using common input–output bundles
at which technical change is measured. Such input-output bundles need not to coincide with thesample ones, and they can be controlled by the researcher. Thus, the input–output bundles can be
set systematically such that technical change is measured in overall directions. Let us denote acontrolled input–output vector (u, v ) , where u ∈ℜ+M , u = {u1 ,..., uM } and v ∈ℜ+N ,v = {v1 ,..., vN } . Distance functions at t and t+1 can be defined as=
D t (u, v ) inf{θ : u / θ ∈ Y t (v )}and =D t +1 (u, v ) inf{θ : u / θ ∈ Y t +1 (v )} . Then, accordingly:

D t +1 (u, v ).
TC = tD (u, v )

(5)

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The value of the distance function for lth controlled vector D t (ul , vl ) , where l = 1,..., L , canbe calculated byJJ{D t (ul , vl=) −1} su {θp : s.t.∑ z j y t mj ≥ θ uml , ∑ z tj xnjt ≤ vnl ,z j ,θ
=j 1 =j 1(6)=
m 1,..., M=, n 1,..., N , z j ≥=0, l 1,..., L}.
The value for D t +1 (ul , vl ) can be calculated by replacing y t mj and xt nj with y t +1mj and xt +1nj ,respectively.
By definition, the point on the period t frontier is given by:yl *t =uul
and yl *t +1 = t l.
D (ul , vl )D (ul , vl )t

(7)This projection is repeated for all l = 1,..., L to obtain sufficient points to recover the frontiers
for all time points. The input vector v is kept constant to normalize the output set and to ruleout the presence of input bias.
Once y *t and y *t +1 are obtained for each controlled vector, those points are used toconstruct a convex hull for the each year’s frontier, and the frontiers are visualized using 2-D
or 3-D graphs as long as m ≤ 3 . Aoki et al.’s (2005) introduced a method of visualizing DEA
frontiers while their method used only the existing data points in a single time period. This
paper extends their method using controlled vectors to allow for comparison of frontiers in
multiple time points.
Technical change in individual sectors can be visually compared in 2-D or 3-D
space. Also, we can quantify technical change on the whole by comparing the space boundedby the frontier and the three planes that are formed by the coordinate axes. Let us denote thevolume of the bounded space by Vol (Y (⋅)) in the case m = 3 . The value of the ratio
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δ=

Vol (Y t +1 (v ))Vol (Y t (v ))greater (less) than one means positive (negative) technical change.

Assuming that technical change had occurred in a Hicks-neutral fashion, a homothetic ratio
1
3that is given by δ can be considered as an alternative measure of technical change.

III. Application to the Agricultural Sector in the Brazil Amazon
I apply the above method to estimating technical change in the agricultural sector in the Brazil
Amazon using a cross-county dataset for 255 counties in 1975, 1985 and 1995 that areavailable from the online data that were compiled by Instituto de Pesquisa Econômica
Aplicada (IPEA). The counties in this region are reasonably homogeneous in geoclimaticconditions and production profiles, which allows us to assume that they face the same
technology at a given point of time. Agricultural intensification through technologicalprogress is thought to contribute to both economic development and environmental
conservation. * Furthermore, this is one of the few datasets in developing countries that areavailable for a large number of cross-section units for multiple-year periods. †For analytical tractability, I aggregate four annual crops (rice, cassava, beans andmaize) into one category, “annual crop,” measured in 2000 US dollar. “Perennial crop”
includes only bananas, measured in tons. “Cattle” measures the number of head of cattle. The
two inputs comprise labor and land. ‡The input levels of the controlled vectors are fixed at the regions’ average in 1985.
Figures 1 and 2 show the visualized frontiers in 1975 and 1985, and in 1985 and 1995,*

See Lee et al. (2006) for extensive discussion on definitions for agricultural intensification.
IPEA established a geographical definition for the analytical units of the dataset that remained consistentthroughout the period despite the rearrangements of counties that occasionally took place.
‡
While it would be ideal to have capital data, they are not available. The major products in the region are notcapital intensive and, hence, it is unlikely that the omission of capital data will lead to large biases.
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respectively, where levels of outputs are normalized such that the intercepts of the 1975
frontier are unity.
Both figures suggest non-Hicks-neutrality in technical change. From 1975 to 1985,
crossing of the two frontiers is observed around the middle of the frontiers in Figure 1. The
1985 frontier dominates the 1975 frontier as we approach the cattle (CT) and the annual crop(AC) axes, which is likely to be the result of positive technical change in the CT and AC
sectors. On the other hand, the dominance of the 1975 frontier around the perennial crop (PC)axis implies a technological regress in that product sector. The first three columns of Table 1
show the degree of frontier shifts along each axis. The shift of the frontier along the CT, AC
and PC axes from 1975 to 1985 is found to be 33.4%, 21.3% and –14.6%, respectively.
From 1985 to 1995, the overall dominance of the frontier is not obvious as, again, thefrontiers cross in the middle. The frontier shift is –6.9%, 6.7% and –14.3% along the cattle,
annual crop and perennial crop axes, respectively. The Hicks-neutral equivalent of frontier
1

shift is calculated by δ 3 . The growth rates are found to be –3.1% (technological regress) in
1975–1985 and 4.8% (technological progress) in 1985–1995.
For comparison purposes, CCD and FGNZ’s DMU-specific technical change
measures are calculated and the descriptive statistics are presented in Table 2. The mean valueof TCFGLZ is 7.7 (28) percentage points smaller (greater) than the Hicks-neutral equivalentmeasure in 1975–1985 (1985–1995). Potentially, this may be because the DMU-specificmeasures tend to be more heavily weighted by the sectors with higher sample
concentrations—perennial crops for the former period and annual crops for the latter period.
Further, the last column of Table 2 shows that only 32.9% of the counties benefited fromtechnological progress, whereas 90.6% benefited from technological progress in 1985–1995
based on the FGNZ measure.

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The last column of Table 2 also indicates that there is a large difference between
TCCCD1 and TCCCD 2 . This implies that one can reach quite different conclusions on technicalchange depending on the choice of the input–output bundle between two points in time.
IV. Conclusions
This paper develops a path-independent method to estimate technical change using asystematized set of controlled input–output vectors and visualization of the DEA frontiers.
The application to the panel datasets of agricultural production in the Brazil Amazon in 1975–
1995 indicates non-Hicks-neutral technical change, with crossings of frontiers in both the
1975–1985 and the 1985–1995 periods. The alternative measure of overall technical changeshows that moderate technological progress may have occurred in 1975–1995; thereby
suggesting that agricultural intensification was in progress. The results also show
heterogeneous trends across products—moderate progress in cattle and annual crops, and a
decline in perennial crops (bananas). The mean DMU-specific technical change scores arefound to be quite different from the overall technical change measure and, thus, one should
not rely solely on the DMU-specific measures.

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References
Aoki, S., Naito, Y. and Tsuji, H. (2005) DEA-based indicator for performance improvement,
The 2005 International Conference on Active Media Technology, 327-330.
Caves, D. W., Christensen, L. R. and Diewert, W. E. (1982) The economic theory of index
numbers and the measurement of input, output and productivity, Econometrica, 50(6),1393–414.
Chen, Y. and Ali, A. A. (2004) DEA Malmquist productivity measure: new insights with anapplication to computer industry, European Journal of Operational Research, 159,
239-49.
Färe, R., Grifell-Tatje, E., Grosskopf, S. and Knox Lovell, C. A. (1997) Biased technicalchange and the Malmquist productivity index, Scandinavian Journal of Economics,
99(1), 119-27.
Färe, R., Grosskopf, S., Norris, M. and Zhang, Z. (1994) Productivity growth, technicalprogress, and efficiency change in industrialized countries, American Economic
Review, 84(1), 66-83.
Färe, R., Grosskopf, S. and Roos, P. (1998) Malmquist productivity indexes: a survey oftheory and practice, in Färe, R., Grosskopf, S. and Russell, R.R. eds. Index numbers:
essays in honour of Sten Malmquist, Kluwer Academic Publishers,
Boston/London/Dordrecht.
Lee, D. R., Barrett, C. B. and McPeak, J. G. (2006) Policy, technology, and management
strategies for achieving sustainable agricultural intensification, Agricultural
Economics, 34, 123-27.
Managi, S. and Karemera, D. (2004) Input and output biased technological change in US
agriculture, Applied Economics Letters, 11(5), 283–86.

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Table 1. Technical change estimates along coordinate axes and volume-based indexes
CattleAnnualPerennial

Hicks-neutral

crop

crop

equivalent growthrate (based on
homothetic ratio)

1985 level (1975=1)

1.334

1.213

0.854

Growth in 1975–1985+33.4%+21.3%–14.6%

1995 level (1975=1)

1.242

1.294

0.732

Growth in 1985–1995

–6.9%

+6.7%

–14.3%

–3.1%

+4.8%

Table 2. FGNZ- and CCD-based technical change estimates
Obs

Mean

SD

Min

Max

MedianNumber of counties
with non-negative
growth ( TC ≥ 0 )1975–

TCFGLZ

255

0.892

0.227

0.432

1.986

0.930

84(32.9%)

TCCCD1

255

0.858

0.252

0.383

1.922

0.900

73(28.6%)

TCCCD 2

255

0.944

0.271

0.402

2.421

0.959

102(40.0%)

TCFGLZ

255

1.328

0.288

0.875

2.429

1.263

231(90.6%)

TCCCD1

255

1.300

0.299

0.576

2.055

1.250

220(86.3%)

TCCCD 2

255

1.380

0.425

0.839

4.348

1.290

221(86.7%)

1985

1985–
1995

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Figure 1. Visualized production frontiers in 1975 and 1985

Figure 2. Visualized production frontiers in 1985 and 1995

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