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**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

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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

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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

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**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.

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**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.

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**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.

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**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.

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**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.

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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

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**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!

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**Technology Gap Ratio (cont.)**

Metafrontier B A kth group y2

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**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

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**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

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**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).

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**Computation of TGR’s This is same as solvin**

We can decompose the frontier function as below:

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**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

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**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.

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**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

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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.

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**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.

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**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.

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**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 .

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**Malmquist Productivity Index (cont.)**

Decomposition of MPI into Technical Change, Technical Efficiency Change,

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**Graphical Representation**

y 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)

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GMPI and MMPI

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**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

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**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

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**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.

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**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

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**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

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**Empirical Application (cont.)**

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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

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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

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