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Metafrontier Framework for the Study of Firm-Level Efficiencies and Technology Gaps

D.S. Prasada RaoCentre 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

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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

( , ) inf 0: ( / ) ( )D P x y y x

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Distance FunctionsOutput Distance Function

Do(x,y)The value of the distance function is equal to the ratio =0A/0B.

Input Distance Function

y1A

y2A

B

CA

y10

y2

P(x)

PPC-P(x)

Di(x,y)The value of the distance function is

equal to the ratio =0A/0B.

Isoq-L(y)

x1A

x2A

B

C

A

x1 0

x2

L(y)

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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

( , ) : ; ; can be used by firms in group to producekT k x y x 0 y 0 x y

( ) : ( , )k kP T x y x y

( , ) inf 0: ( / ) ( )k kD P x y y x

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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

( , ) for any then ( , )kT k T x y x y

1 2 ... KT convex hull T T T

( , ) ( , )kD Dx y x y

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Metafrontiers

Output y

Input x

1

0

C

D

F

A

1’ 2’

2

M

M’ 3’

3

B

E

Figure 1: Technical Efficiencies and Technology gap ratios

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Technology Gap Ratio• The output-orientated Technology Gap Ratio (TGR):

( , ) ( , )( , )

( , ) ( , )k

k k

D x y TE x yTGR x y

D x y TE 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.

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A

C

B

0

Technology Gap Ratio (cont.)

( , )( , )

( , )

( , )

( , )

0 / 0 0

0 / 0 0

kk

k

D x yTGR x y

D x y

TE x y

TE x y

A C B

A B C

y1

y2

kth group

Metafrontier

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• 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

thatTEk(x,y) TE(x,y) for each firm or DMU

Computation of TGR’s

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• 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

Computation of TGR’s

1 2( , ,..., ; )k k k k k

it it it it itV U V Ukit it it Nity f x x x e e x

*1 2( , ,..., ; ) it

it it it Nity f x x x e x

kit it x x

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• 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).

Identifying the meta frontier

ˆ k

min

1 2 1 2

1 1

ˆln ( , ,..., ; ) ln ( , ,..., ; )L T

kit it Nit it it Nit

i t

f x x x f x x x

s.t. 1 2 1 2ˆln ( , ,..., ; ) ln ( , ,..., ; )k

it it Nit it it Nitf x x x f x x x , for all i and t;

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Computation of TGR’s

.k

itk kit it it

it

U Vit

ey e e

e

x

xx

kitTGRk

itTE

min

1 2

1 1

ln ( , ,..., ; )L T

it it Niti t

f x x x s.t. 1 2 1 2

ˆln ( , ,..., ; ) ln ( , ,..., ; )kit it Nit it it Nitf x x x f x x x , for all i and t.

min

x

s.t. ˆ kit it x x for all i and t

This is same as solvin

We can decompose the frontier function as below:

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Thus we have:

These estimates are based on the estimated coefficients from the fitted SF models

Computation of TGR’s

kit

k kit it

Uk itit V

yTE e

e

xTE of i-th firm in k-th group frontier

.kit it

itit V

yTE

e

xTE of i-th firm from the metafrontier

ˆˆ ˆ k kit it itTE TE TGR Estimated TGR for each firm

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• 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.

SF Approach – further work

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• 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

An empirical application

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• 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 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.

Results

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• 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.

Results

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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

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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

),(

),( 11

ttt

tttt yxD

yxDM

),(

),(

1

1111

ttt

tttt yxD

yxDM

2/1

1

11111111, ),(

),(

),(

),(),,,(

ttt

ttt

ttt

ttttttttt yxD

yxD

yxD

yxDyxyxM

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• Decomposition of MPI into– Technical Change, – Technical Efficiency Change,

Malmquist Productivity Index (cont.)

, 1t tTEC

1, ttTC

1/ 2

1 1 1 1 1, 1 1 1

1 1 1 1

, 1, 1

( , ) ( , ) ( , )( , , , )

( , ) ( , ) ( , )t t t t t t t t t

t t t t t tt t t t t t t t t

TCTEC t tt t

D x y D x y D x yM x y x y

D x y D x y D x y

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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

0 x

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GMPI and MMPI

1/ 2

1 1 1 1 1, 1 1 1

1 1 1 1

, 1 , 1

** 1, 1 1 1

( )

( , ) ( , ) ( , )( , , , )

( , ) ( , ) ( , )

( )

( , , , )

k k kk t t t t t t t t tt t t t t t k k k

t t t t t t t t t

k kt t t t

tt t t t t t

kthGroup MPI GMPI

D x y D x y D x yM x y x y

D x y D x y D x y

TEC TC

Metafrontier MPI MMPI

DM x y x y

1/ 2* *

1 1 1 1* * *

1 1 1 1

* *, 1 , 1

( , ) ( , ) ( , )

( , ) ( , ) ( , )t t t t t t t t

t t t t t t t t t

t t t t

x y D x y D x y

D x y D x y D x y

TEC TC

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– TEC* and TECK

GMPI and MMPI Decompositions

* 1 1 1 1 1 1, 1

1 1 1, 1

_

( , ) ( , )

( , ) ( , )

( , )

( , )

k kt t t t t t

t t k kt t t t t t

kk t t tt t k

t t t

TGR GR

D x y TGR x yTEC

D x y TGR x y

TGR x yTEC

TGR x y

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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1/ 2

* 1 1, 1 , 1

1 1 1 1

1/ 2

1 1, 1

1 1 1 1

1

( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

k kk t t t t t t

t t t t k kt t t t t t

k kk t t t t t tt t k k

t t t t t t

TGR x y TGR x yTC TC

TGR x y TGR x y

TGR x y TGR x yTC

TGR x y TGR x y

TGR

–TC* and TCk

GMPI and MMPI Decompositions (cont.)

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.)

1, 1

1/ 2

* 1 1 1 1 1, 1 , 1

1

( )

( , ) ( , )

( , ) ( , )

t t

k kk t t t t t t

t t t t k kt t t t t t

catch up

TGR x y TGR x yM M

TGR x y TGR x yGMPI

• 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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• 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

Empirical Application

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Empirical Application (cont.)

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• 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 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

MMPI-GMPI Results

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• 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

Conclusions