Metafrontier Framework for the Study of
Firm-Level Efficiencies and Technology Gaps
D.S. Prasada Rao
Centre for Efficiency and Productivity Analysis
School of Economics
The University of Queensland. Australia
Joint research with George Battese, Chris O’Donnell and
Alicia Rambaldi
1
Outline
• Motivation
• Meta-frontiers for efficiency comparisons
across regions
– Conceptual framework
– Methodology
• DEA
• Stochastic Frontiers
– Application to global agriculture
• Metafrontiers and productivity growth
– Metatfrontier Malmquist Productivity Index
(MMPI)
– Decomposition of MMPI
• Catch-up and convergence term
– Cross-country productivity growth
2
Motivation
• Hyami (1969) introduced the concept of metaproduction function
• The metaproduction function can be regarded
as the envelope of commonly conceived
neoclassical production functions (Hyami and
Ruttan, 1971)
• Work on Indonesian Garment industry by
regions
• National and international benchmarking
studies – integrating a national study with data
from other countries
• Performance of globalised and non-globalised
economies
3
Basic Framework: Production Technology
• We assume that there is a production
technology that allows transformation of
a vector of inputs into a vector of
outputs
T = {(x,q): x can produce q}.
• It can be equivalently represented by
– Output sets – P(x); Input sets – L(y)
– Output and input distance functions
4
Basic Framework: Production Technology
• Properties of P(x)
– 0  P(x) (inactivity);
– If y P(x) then y* = y  P(x) for all 0 <  
1 (weak disposability);
– P(x) is a closed and bounded set; and
– P(x) is a convex set.
• Output distance function is defined as:
D (x, y)  inf    0: (y /  )P(x)
• In this paper we just focus on output
distance functions
5
Distance Functions
Output Distance Function
Input Distance Function
y2
x2

y2A
A
B
A
x2A

C
PPC-P(x)
L(y)
B
Isoq-L(y)
C
P(x)
0
y1A
y1
Do(x,y)
The value of the distance function
is equal to the ratio =0A/0B.
0
x1A
x1
Di(x,y)
The value of the distance function is
equal to the ratio =0A/0B.
6
Group frontier vs. metafrontier
• We assume that there are k groups of “firms” or
“DMUs” included in the analysis.
• The group specific technology, output sets and
distance functions can be defined, for each k=1,2,…K
as
T k  (x, y) : x  0; y  0; x can be used by firms in group k to produce y
P k ( x )  y : ( x , y )  T k 
D k (x, y )  inf    0: ( y /  )  P k ( x)
7
Group frontier vs. metafrontier
• The metafrontier is related to the group
frontiers as:
– If
–
(x, y) T k for any k then (x, y) T
T  convex hull T 1  T 2  ...  T K 
– If D(x,y) represents the output distance
function for the metafrontier, then
D (x, y)  D(x, y)
k
8
Metafrontiers
Output y
M’
3’
F
B
1’
2’
E
D
A
C
M
0
1
3
2
Input x
Figure 1: Technical Efficiencies and Technology gap ratios
9
Technology Gap Ratio
• The output-orientated Technology Gap Ratio (TGR):
TGRk ( x , y ) 
D ( x , y ) TE ( x , y )

k
D ( x , y ) TE k ( x , y )
Example:
Country i in region k, at time t
TE(x,y) = 0.6
TEk(x,y) = 0.8
Then, TGR = 0.6/0.8=0.75
The potential output vector for country i in region k technology is 75
per cent of that represented by the metatechnology.
10
Technology Gap Ratio (cont.)
y1
D (x , y )
TGR ( x , y )  k
D (x , y )
k
C
B
A
Metafrontier
TE ( x , y )

TE k ( x , y )
0 A / 0C 0 B


0 A / 0 B 0C
kth group
0
y2
11
Computation of TGR’s
• Using DEA:
– Run DEA for each group separately and
compute technical efficiency scores, TEk;
– Run DEA for all the groups together –
pooled data and compute TE scores;
– Compute TGR’s as the ratio of the scores
from the two DEA models; and
– Given that DEA uses LP technique it follows
that
TEk(x,y)  TE(x,y) for each firm or DMU
12
Computation of TGR’s
• Using SFA
– Estimate stochastic group frontiers using the
following specification
yit  f ( x1it , x2it ,..., xNit ;  ) e
k
Vitk Uitk
e
xit  k Vitk Uitk
which is a model that is linear in parameters; u’s
represent inefficiency and v’s represent statistical
noise.
• Meta frontier is defined as:
y  f ( x1it , x2it ,..., xNit ;  )  e
*
it
xit 
such that xit   xit  for all k =1,2,…K
k
13
Identifying the meta frontier
• Estimate parameters for each group frontier
and obtain ˆ .
• Identify the metafrontier, by finding a suitable
, that is closest to the estimated group
frontiers – need to solve the optimisation
problem (using method described in Battese,
Rao and O’Donnell, 2004).
k
L
min

T
 ln f ( x
i 1 t 1
s.t.
1it
, x2it ,..., xNit ;  )  ln f ( x1it , x2it ,..., xNit ; ˆ k ) 
ln f ( x1it , x2it ,..., xNit ;  )  ln f ( x1it , x2it ,..., xNit ; ˆ k ) , for all i and t;
14
Computation of TGR’s
L
min

T
 ln f ( x
i 1 t 1
s.t.
, x2it ,..., xNit ;  )
ln f ( x1it , x2it ,..., xNit ;  )  ln f ( x1it , x2it ,..., xNit ; ˆ k ) , for all i and t.
1it
This is same as solvin
min
x
s.t.
xit   xit ˆ k for all i and t

We can decompose the frontier function as below:
yit  e
Uitk
TEitk
xit  k
e
xit  Vitk
 xit   e
.
e
TGRitk
15
Computation of TGR’s
Thus we have:
yit
TE 
k
it
TEit 
e
e
xit  k Vitk
yit
e
xit  Vitk
U itk
.
ˆ k
TEˆit  TEˆitk  TGR
it
TE of i-th firm in k-th group frontier
TE of i-th firm from the metafrontier
Estimated TGR for each firm
These estimates are based on the estimated
coefficients from the fitted SF models
16
SF Approach – further work
• The SF approach described here can be
applied only for single output firms.
• For multi-output firms currently we use DEA
approach.
• Work on the use of multi-output distance
functions for the purpose of identifying the
meta-frontier is in progress.
• Weighted optimisation in identifying the
metaftontier: firm-specific weights
• Possibility of a single-step estimation of group
and meta-frontiers using a possible seemingly
unrelated regression approach.
17
An empirical application
•
•
•
•
•
Inter-regional comparisons of agricultural efficiency
Coelli and Rao (2005) data set
97 countries and five-year period 1986-1990
Pool 5-year data for all the countries
Four groups of countries:
–
–
–
–
Africa: 27 countries
The Americas: 21 countries
Asia: 26 countries
Europe: 23 countries
• agricultural output – expressed in common
1990 prices
• Five inputs: land; labour; tractors; fertiliser;
livestock
18
Results
• DEA and SF results are presented for selected
countries and regional groupings.
– Results are presented as an average over the 5year period with min. and max values reported.
– For each country TE levels with respect to the
group-frontier as well as TGR’s are reported.
• DEA results:
– TE of South Africa is 0.964 relative to its group (Africa) frontier
but it is only 0.610 when measured against metafrontier
showing a TGR of 0.633;
– Average TGR for Asian countries is 0.925
– DEA-MF values with maximum equal to 1 indicate that some
countries from those regions were on the metafrontier at least
in one year.
19
Results
• SFA is based on translog specification
• Pooled translog model is also presented
• The Likelihood-ratio test rejects the null hypothesis of
identical group frontiers – shows that metafrontier
framework is appropriate
• Some major differences between SFA and DEA results
• SFA efficiency scores are typically lower than those
under DEA
• Indonesia, for example, has an efficiency score of
0.563 under SFA compared to 0.997 using DEA.
• SFA-MF efficiency estimates appear to be more
plausible than SFA-POOL efficiency estimates –
suggests the use of metafrontiers.
20
Metafrontier Malmquist Productivity
Index
• Measuring productivity growth over time for
different countries.
• Extension of metafrontier work to panels
• Quantification of relative technological
progress and “technology gap” between
economies and its’ evolution through time.
• Concept of Malmquist Productivity index is
used along with metafrontiers
21
Malmquist Productivity Index
• MPI. Caves, Christensen and Diewert
(1982).
• Two technologies and two observed
points, t and t+1
D (x , y )
M t  t t 1 t 1
Dt ( xt , yt )
M t 1 
Dt 1 ( x t 1 , y t 1 )
Dt 1 ( x t , y t )
• MPI is geometric mean
 Dt ( x t 1 , y t 1 ) Dt 1 ( x t 1 , y t 1 ) 
M t , t 1 ( x t , y t , x t 1 , y t 1 )  


D
(
x
,
y
)
D
(
x
,
y
)
t 1
t
t
 t t t

1/ 2
22
Malmquist Productivity Index (cont.)
• Decomposition of MPI into
– Technical Change, TCt ,t 1 
– Technical Efficiency Change, TECt ,t 1 
1/ 2
Dt 1 ( xt 1 , yt 1 )  Dt ( xt 1 , yt 1 )
Dt ( xt , yt ) 
M t ,t 1 ( xt , yt , xt 1 , yt 1 ) 


Dt ( xt , yt )  Dt 1 ( xt 1 , yt 1 ) Dt 1 ( xt , yt ) 
TEC
t, t  1
TCt ,t 1
23
Graphical Representation
y


D
C
k1,t+1
(xt+1, yt+1)

C*
B
A


A*
 (xt, yt)
0
Mt+1
Mt
k1,t
x
24
GMPI and MMPI
kth Group MPI (GMPI )
M tk,t 1 ( xt , yt , xt 1 , yt 1 ) 
1/ 2
D ( xt 1 , yt 1 )  D ( xt 1 , yt 1 ) D ( xt , yt ) 


k
Dt ( xt , yt )  D ( xt 1 , yt 1 ) D ( xt , yt ) 
k
t 1
k
t
k
t 1
k
t
k
t 1
 TECtk,t 1  TCtk,t 1
Metafrontier MPI ( MMPI )
M t*,t 1 ( xt , yt , xt 1 , yt 1 ) 
1/ 2
D ( xt 1 , yt 1 )  D ( xt 1 , yt 1 ) D ( xt , yt ) 


*
Dt ( xt , yt )  D ( xt 1 , yt 1 ) D ( xt , yt ) 
*
t 1
*
t
*
t 1
*
t
*
t 1
 TECt*,t 1  TCt*,t 1
25
GMPI and MMPI Decompositions
– TEC* and TECK
k
k
D
(
x
,
y
)
TGR
*
t 1
t 1
t 1
t 1 ( xt 1 , yt 1 )
TECt ,t 1 

k
Dt ( xt , yt )
TGRtk ( xt , yt )
k
TGR
k
t 1 ( xt 1 , yt 1 )
 TECt ,t 1 
TGRtk ( xt , yt )
TGR _ GR
TGR_GR is a relative technological gap change of the specific region from
period t to t+1 evaluated at each period’s specified input-output mix
26
GMPI and MMPI Decompositions (cont.)
–TC* and TCk
*
t ,t 1
TC
1/ 2
 TC
 TGR ( xt , yt ) TGR ( xt 1 , yt 1 ) 



TGR
(
x
,
y
)
TGR
(
x
,
y
)
t
t
t 1
t 1 

 TC
 TGR ( xt , yt )
TGR ( xt 1 , yt 1 ) 



k
k
TGR
(
x
,
y
)
TGR
(
x
,
y
)
t 1
t 1
t 1
t 1
t
t


k
t ,t 1
k
t ,t 1
k
t
k
t 1
k
t
k
t 1
k
t
k
t
1/ 2
TGR 1
TGR-1 can be interpreted as the inverse of the relative technological
gap change, which is “benchmark time period” invariant
27
GMPI and MMPI Decompositions (cont.)
• MMPI can then be expressed as:
M
*
t ,t 1
 M
k
t ,t 1
GMPI
1/ 2
 TGR ( xt 1 , yt 1 ) TGR ( xt 1 , yt 1 ) 



k
k
TGR
(
x
,
y
)
TGR
(
x
,
y
)
t
t
t
t 1
t
t


k
t 1
k
t
( catch upt ,t 1 )1
If the second term is not equal to 1, a single frontier approach will
under/over estimate productivity change.
28
Empirical Application
•
•
•
69 Countries
1982 – 2000
Four Geographical Regions
– The Americas (AM) - 18 countries
– Europe (EU) - 19 countries
– Africa and the Middle East (AF) - 18
countries
– Asia-Pacific (AP) - 14 countries
29
Empirical Application
• Variables:
– Real GDP (a chain index in 1996
international dollars)
– Capital Stock (constructed from PWT using
the perpetual inventory method)
– Total Labour Force (World Development
Indicators)
• Estimated with DEA
– (see O’Donnell et al (2005))
– 19 periods
30
Empirical Application (cont.)
31
MMPI-GMPI Results
• MMPI is generally higher than GMPI with the
exception of the Americas during 1998-2000;
• Metafrontier technical change seems to be
only marginally higher than the group-specific
technical change estimates – no evidence that
any particular region is falling behind;
• African region has shown some signs of catchup;
• There are few instances of “technological
regression” – a phenomenon that is generally
seen when DEA is applied.
• Need to replicate these using SF models
32
Conclusions
• Metafrontier concept is very useful in
international benchmarking studies
• Choice of country or firm groupings is dictated
by the particular problem under consideration
• Analysis is sensitive to the choice of groupings
• The basic framework has been developed, but
further work needs to be focused on:
– The estimation of metrafrontiers for multioutput/multi-input firms;
– Efficient estimation of metafrontiers: possibility of a
single-step estimator of the metafrontiers;
– Estimation of MMPI using SF approach
33