1. 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

2. 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

3. Motivation
Hyami (1969) introduced the concept of meta-production 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

4. 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
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

5. 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:
In this paper we just focus on output distance functions
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

6. Distance Functions Output Distance Function Input Distance Function
y1A
y2A B C A y1 y2
P(x) 
PPC-P(x)
The value of the distance function for the firm using input level x to produce the outputs, defined by the point A, is equal to the ratio =0A/0B.
The value of the distance function for the point, A, which defines the production point where firm A uses x1A of input 1 and x2A of input 2, to produce the output vector q, is equal to the ratio =0A/0B.
Output oriented TE = Do
Input oriented TE = 1/Di
Do(x,y)
The value of the distance function is equal to the ratio =0A/0B.
Di(x,y)
The value of the distance function is equal to the ratio =0A/0B.

7. 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
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

8. Group frontier vs. metafrontier
The metafrontier is related to the group frontiers as: If
If D(x,y) represents the output distance function for the metafrontier, then
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

9. Metafrontiers
The metafrontier concept is analogous to the concept of a metaproduction function, utilised in comparing agricultural productivity across countries which was first introduced by Hayami (1969),
Figure 1: Technical Efficiencies and Technology gap ratios

10. Technology Gap Ratio The output-orientated Technology Gap Ratio (TGR):
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.
In the example above, if the technical efficiency of (x, y) with respect to the
metatechnology is 0.6 then the technology gap ratio is 0.75 (= 0.6/0.8). This means
that, given the input vector, the potential output vector for region k technology is 75
per cent of that represented by the metatechnology.
Drawing!

11. Technology Gap Ratio (cont.)
Metafrontier B A
kth group y2

12. 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

13. Computation of TGR’s Using SFA
Estimate stochastic group frontiers using the following specification
which is a model that is linear in parameters; u’s represent inefficiency and v’s represent statistical noise.
Meta frontier is defined as:
such that for all k =1,2,…K

14. Identifying the meta frontier
Estimate parameters for each group frontier and obtain .
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).

15. Computation of TGR’s This is same as solvin
We can decompose the frontier function as below:

16. Computation of TGR’s Thus we have:
These estimates are based on the estimated coefficients from the fitted SF models
TE of i-th firm in k-th group frontier
TE of i-th firm from the metafrontier
Estimated TGR for each firm

17. 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.

18. An empirical application
Inter-regional comparisons of agricultural efficiency
Coelli and Rao (2005) data set
97 countries and five-year period
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

19. Results
DEA and SF results are presented for selected countries and regional groupings.
Results are presented as an average over the 5-year 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 relative to its group (Africa) frontier but it is only 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.

20. 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 under SFA compared to using DEA.
SFA-MF efficiency estimates appear to be more plausible than SFA-POOL efficiency estimates – suggests the use of metafrontiers.

21. 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
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

22. Malmquist Productivity Index
MPI. Caves, Christensen and Diewert (1982).
Two technologies and two observed points, t and t+1
MPI is geometric mean
Estimation of Distance functions will require panel data .

23. Malmquist Productivity Index (cont.)
Decomposition of MPI into
Technical Change,
Technical Efficiency Change,

24. Graphical Representationy 
k1,t+1
k1,t
(xt+1, yt+1)
(xt, yt)
Mt+1 Mt A* A B C* C D x
TE(K)(T)=OA(S)/OA; TE(S)(T)=OA(S)/OB; TGR(T)=OA/OB
TE(K)(T+1)=OC(S)/OC; TE(S)(T)=OC(S)/OD; TGR(T)=OC/OD
TEC(S)/TEC(K)=(OC/OD)/(OA/OB)

25. GMPI and MMPI

26. GMPI and MMPI Decompositions
TEC* and TECK
Show on the graph..
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

27. GMPI and MMPI Decompositions (cont.)
TC* and TCk
TGR-1 can be interpreted as the inverse of the relative technological gap change, which is “benchmark time period” invariant

28. GMPI and MMPI Decompositions (cont.)
MMPI can then be expressed as:
If the second term is not equal to 1, a single frontier approach will under/over estimate productivity change.

29. 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

30. 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

31. Empirical Application (cont.)

32. MMPI-GMPI Results
MMPI is generally higher than GMPI with the exception of the Americas during ;
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 catch-up;
There are few instances of “technological regression” – a phenomenon that is generally seen when DEA is applied.
Need to replicate these using SF models

33. 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 multi-output/multi-input firms;
Efficient estimation of metafrontiers: possibility of a single-step estimator of the metafrontiers;
Estimation of MMPI using SF approach