Efficiency Analysis of a Efficiency Analysis of a Multisectoral Economic Multisectoral Economic

SystemSystem

Mikulas LuptáčikMikulas LuptáčikUniversity of Economics and Business Administration,University of Economics and Business Administration,

Vienna, AustriaVienna, Austria

Bernhard BöhmUniversity of Technology, Vienna, Austria

16th International Input-Output Conference 16th International Input-Output Conference Istanbul – TurkeyIstanbul – Turkey2 – 6 July 20072 – 6 July 2007

Efficiency of production Efficiency of production

Multi-input and multi-output Multi-input and multi-output production technologyproduction technology

Use distance functions to Use distance functions to characterise efficiency of productioncharacterise efficiency of production

Input distance function is reciprocal Input distance function is reciprocal of output distance functionof output distance function

A radial measure of efficiencyA radial measure of efficiency

Constant returns to scale Constant returns to scale (Input-Output model)(Input-Output model)

Consider Consider environmentenvironment::pollution and abatement in pollution and abatement in the „augmented“ Leontief modelthe „augmented“ Leontief model

Use suitable definition of „Use suitable definition of „eco-eco-efficiencyefficiency““

quantities of undesirable outputs quantities of undesirable outputs (pollutants) are treated like inputs (i.e. (pollutants) are treated like inputs (i.e. are minimised).are minimised).

four different eco-efficient models can four different eco-efficient models can be constructed be constructed

cf. Korhonen and Luptacik (2004) cf. Korhonen and Luptacik (2004)

The augmented Leontief model with The augmented Leontief model with n outputs (sectors), k pollutants, k abatement activities, m primary inputs represents the economic constraintsrepresents the economic constraints

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in the following LP – models (in the in the following LP – models (in the spirit of Debreu (1951) and Ten Raa spirit of Debreu (1951) and Ten Raa (1995):(1995):

1. 1. Minimise the use of primary factors for a Minimise the use of primary factors for a given level of final demand and tolerated given level of final demand and tolerated pollutionpollution

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(1)

2. Maximise the proportional expansion of final 2. Maximise the proportional expansion of final demand ydemand y11 for given levels of tolerated pollution for given levels of tolerated pollution

(environmental standards) and primary factors(environmental standards) and primary factors

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(2)

Due to the presence of the pollution Due to the presence of the pollution subsystem representing undesirable subsystem representing undesirable outputs, the optimal values of outputs, the optimal values of γγ and and αα are not the reciprocal of each other. are not the reciprocal of each other.

However, by treating these undesirable However, by treating these undesirable outputs like inputs in the model, i.e. by outputs like inputs in the model, i.e. by changing the problem formulation into a changing the problem formulation into a proportional reduction of primary inputs proportional reduction of primary inputs and undesirable outputs for given final and undesirable outputs for given final demanddemand, the reciprocal property of the , the reciprocal property of the distance function can be re-established.distance function can be re-established.

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3. Minimise the use of primary inputs and 3. Minimise the use of primary inputs and emission of pollutants for given levels of final emission of pollutants for given levels of final demanddemand

(3)

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4. Minimise the production of pollutants for given 4. Minimise the production of pollutants for given levels of primary inputs and final demandlevels of primary inputs and final demand

(4)

These models could formally be seen These models could formally be seen as data envelopment analysis (DEA) as data envelopment analysis (DEA) models when using sectors as DMU’s.models when using sectors as DMU’s.

But But application of DEA to the application of DEA to the Input-Output framework requires Input-Output framework requires some additional considerations:some additional considerations:

Because:Because: DEA uses inputs and outputs of different DEA uses inputs and outputs of different

independentindependent decision making units, the I-O decision making units, the I-O model uses data of usually only one country model uses data of usually only one country but disaggregated into but disaggregated into interrelated sectors interrelated sectors with different technologieswith different technologies..

Therefore: not meaningful to compare Therefore: not meaningful to compare sectors with respect to their relative sectors with respect to their relative efficiency. efficiency. Direct interpretation as DEA-model is Direct interpretation as DEA-model is economicallyeconomically not meaningful! not meaningful!

Generate the production possibility setGenerate the production possibility set Each output is maximised subject to Each output is maximised subject to

restraints on the production of other restraints on the production of other outputs and available inputs outputs and available inputs (multiobjective optimisation problem). (multiobjective optimisation problem).

Measure distance of actual economy to Measure distance of actual economy to the production frontierthe production frontier

Application of DEA to the Application of DEA to the input-output frameworkinput-output framework

Optimisation modelOptimisation model

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ss11 is the vector of n slack variables of is the vector of n slack variables of the n sectorsthe n sectors

ss22 is the vector of slack variables of is the vector of slack variables of the k pollutantsthe k pollutants

ss33 are the slacks in the m inputs are the slacks in the m inputs Solve the model n+k+m times Solve the model n+k+m times

for given values of sector net-outputs for given values of sector net-outputs and inputs for the maximal values of and inputs for the maximal values of each slack variable seach slack variable sjj for for j=1,...,n,...,n+k,...,n+k+mj=1,...,n,...,n+k,...,n+k+m

Individually optimal desirable and Individually optimal desirable and undesirable outputs and input undesirable outputs and input values are calculated from values are calculated from y*y*1 1 = y= y11 + s + s11 y*y*2 2 = y= y22 – s – s22 z*z* = z – s= z – s33 and are arranged to form a and are arranged to form a pay-off matrixpay-off matrix P. P.

Pay-off matrixPay-off matrix

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s ys y s y

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Efficient envelopeEfficient envelope

P is used to establish the frontier of P is used to establish the frontier of the production possibility set (or the the production possibility set (or the input requirement set) input requirement set) i.e. the efficient envelope. i.e. the efficient envelope.

This efficient envelope is used to This efficient envelope is used to evaluate the relative inefficiency of the evaluate the relative inefficiency of the economy given by the actual output economy given by the actual output and input data (yand input data (y11

00, y, y2200, z, z00) e.g. in the ) e.g. in the

following input oriented DEA problemfollowing input oriented DEA problem

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(6)

The relationship between the DEA The relationship between the DEA model and the LP modelmodel and the LP model

To provide a clear economic interpretation we To provide a clear economic interpretation we consider LP-model (3) and DEA-model (6) consider LP-model (3) and DEA-model (6) without pollution and abatement sectorswithout pollution and abatement sectors

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(6')

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Proposition 1:Proposition 1:

The efficiency score θ of DEA The efficiency score θ of DEA problem (6') is exactly equal to the problem (6') is exactly equal to the radial efficiency measure radial efficiency measure γ γ of LP-of LP-model (3').model (3').

The dual solution of model (3') The dual solution of model (3') coincides with the solution of the coincides with the solution of the DEA multiplier problem (which is the DEA multiplier problem (which is the dual of problem (6'))dual of problem (6'))

Consider first LP-problem (1) and the Consider first LP-problem (1) and the corresponding DEA model (7)corresponding DEA model (7)

The analysis can be extended to the model The analysis can be extended to the model with pollution and abatement :with pollution and abatement :

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(7)

Proposition 2:Proposition 2:

The efficiency score θ of DEA The efficiency score θ of DEA problem (7) is exactly equal to the problem (7) is exactly equal to the radial efficiency measure radial efficiency measure γ γ of LP-of LP-model (1).model (1).

The dual solution of model (1) The dual solution of model (1) coincides with the solution of the coincides with the solution of the DEA multiplier problem (which is the DEA multiplier problem (which is the dual of problem (7))dual of problem (7))

Proposition 3:Proposition 3: The efficiency score θ of DEA problem The efficiency score θ of DEA problem

(6) is exactly equal to the radial (6) is exactly equal to the radial efficiency measure efficiency measure γ γ of LP-model (3).of LP-model (3).

The dual solution of model (3) The dual solution of model (3) coincides with the solution of the DEA coincides with the solution of the DEA multiplier problem (which is the dual multiplier problem (which is the dual of problem (6))of problem (6))

For other models similar propositions can be proved:

An application to the Austrian economy highly aggregated version of the Austrian input output

table 1995 and NAMEA data for air and water pollution (five sectors, two pollutants and two primary inputs).

Sectors:1. Agriculture, forestry, mining (mill. ATS) 2. Industrial production (mill. ATS) 3. Electricity, gas, water, construction (mill. ATS) 4. Trade, transport and communication (mill. ATS) 5. Other public and private services (mill. ATS)

Pollutants: 6. Air pollutant (NOx, tons per year) 7. Water pollutant (P, tons per year)

Primary Inputs: 8. Labour (total employment, 1000 persons) 9. Capital (gross capital stock, 1995, nominal, mill. ATS)

Experiment with simple models (no pollution)

with levels of capital and labour corresponding to a 5% underutilisation of both inputs.

As expected the proportional efficiency measure of α yields 1.05,(output could be expanded by 5% proportionally) and the minimum γ equals 0.952, the reciprocal value of α.

The λ values are the same for all sectors (λi = 1.05 for min-model and equal to one for the max-model (i.e. the same output can be produced by a 4.76189% reduction of both inputs).

Expanded model with pollutants and abatement

Assumption: levels of capital and labour correspond to a 5% underutilisation

Min-Model (inputs and undesirable outputs) yields a minimum value of γ equal to 0.953297 with λi slightly larger than 1.000 for i = 1, ..., 5 and λ6 = 1.084, λ7 = 1.049.

Calculating the output oriented model the maximum α = 1.04899 is the reciprocal value of min γ. Here again the intensities are almost the same for the outputs but different for the pollutants (λi = 1.0494 for i = 1, ..., 5 and λ6 = 1.137, λ7 = 1.1007).

We observe that the efficiency measure of the extended model gives a proportional factor of expansion or reduction of outputs (respectively inputs) while intensities reveal disproportionate abatement activities

Empirical eco-efficiency analysis with DEA

Construct the envelope: Pay-off tableConstruct the envelope: Pay-off table

Capital Labour P o l l 1 P o l l 2

max y 1 1 0 6 1 8 . 1 8 8 4 1 2 3 . 8 3 0 6 5 . 8 5 0 1 0 1 0 . 3 4

max y 2 1 0 7 0 5 . 3 1 2 4 1 2 3 . 8 3 0 6 5 . 8 5 0 1 0 1 0 . 3 4

max y 3 1 0 7 8 5 . 8 1 0 4 1 2 3 . 8 3 0 6 5 . 8 5 0 1 0 1 0 . 3 4

max y 4 1 0 7 0 4 . 5 7 1 4 1 2 3 . 8 3 0 6 5 . 8 5 0 1 0 1 0 . 3 4

max y 5 1 0 8 4 0 . 9 4 0 4 0 5 2 . 4 7 7 3 6 5 . 8 5 0 1 0 1 0 . 3 4

min P o l l 1 1 0 4 5 8 . 7 5 6 3 9 6 2 . 0 6 1 0 1 0 1 0 . 3 4

min P o l l 2 1 0 4 0 3 . 4 7 7 3 9 4 7 . 8 3 1 6 5 . 8 5 0 0

min K 1 0 3 2 4 . 7 0 5 4 8 4 1 2 3 . 8 3 0 6 5 . 8 5 0 1 0 1 0 . 3 4

min L 1 0 8 4 0 . 9 4 0 3 9 2 7 . 5 5 3 6 5 . 8 5 0 1 0 1 0 . 3 4

y 1 y 2 y 3 y 4 y 5

max y 1 7 1 . 4 0 9 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

max y 2 2 8 . 6 9 3 1 2 1 7 . 2 2 7 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

max y 3 2 8 . 6 9 3 1 0 3 7 . 5 1 4 4 2 4 . 7 8 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

max y 4 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 7 5 5 . 2 3 1 9 1 8 . 2 3 9

max y 5 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 1 0 1 1 . 1 0 3

min P o l l 1 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

min P o l l 2 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

min K 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

min L 2 8 . 6 9 3 1 0 3 7 . 5 1 4 2 8 7 . 5 3 1 6 3 8 . 2 5 1 9 1 8 . 2 3 9

DEA model with pollutantsDEA model with pollutants Using this pay-off table for the same

experiment as before (with 5% capital and labour surplus)(with 5% capital and labour surplus)

Solve:Solve:

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Achieve a minimum Achieve a minimum = 0.953297 = 0.953297 for the economyfor the economy

The θ value indicates the inefficiency in the use of primary factors and excess pollution. In other words, both primary factors and both pollution levels should be reduced by 4.7% in order for the economy to become efficient.

This is the same efficiency measure as in the LP model (i.e. = = γ)

DMU Data  Projection Difference  %

Capital 10840.94 10334.6447 -506.295271 -4.67%

Labour 4123.937 3930.111 -193.825998 -4.70%

Poll1 65.85 62.774663 -3.07533697 -4.67%

Poll2 1010.34 963.154944 -47.1850563 -4.67%

y1 28.693 28.693 0 0.00%

y2 1037.514 1037.514 0 0.00%

y3 287.531 287.531 0 0.00%

y4 638.251 638.251 0 0.00%

y5 918.239 918.239 0 0.00%

CCR-I θ = 0.95329784

This frontier is constructed from the following optimal weights: µ1 = 0.0279 µ2 = 0.2399 µ3 = 0.0871 µ4 = 0.2267 µ5 = 0.4109 µ6 = 0.0074 µ7 = 0.0074

This is DEA-model B of Korhonen and Luptácik

DEA-model D

If we calculate the output oriented model we obtain the efficiency score of 1.04899

This is exactly the reciprocal of the input oriented value (and equal to α in the LP model).

For given levels of primary factors and net-pollution the net output (i.e. final demand) of all sectors could be increased by 4.9% to make the economy efficient.

DMU Data  Projection Difference %

Capital 10840.94 10840.94 0 0.00%

Labour 4123.937 4122.64753 -1.28946974 -0.03%

Poll1 65.85 65.85 0 0.00%

Poll2 1010.34 1010.34 0 0.00%

y1 28.693 30.0986729 1.40567292 4.90%

y2 1037.514 1088.34191 50.8279139 4.90%

y3 287.531 301.617172 14.0861722 4.90%

y4 638.251 669.51898 31.2679799 4.90%

y5 918.239 963.22362 44.9846198 4.90%

CCR-O1/α = θ 0.95329784

Slack based measures of Slack based measures of eco-efficiencyeco-efficiency

To avoid limitation of efficiency To avoid limitation of efficiency indicators assuming unchanged indicators assuming unchanged proportions of inputs and outputsproportions of inputs and outputs

Formulate a goal-programming Formulate a goal-programming model (treat pollutants as inputs)model (treat pollutants as inputs)

Minimise a scalar which is unit Minimise a scalar which is unit invariant and monotone function of invariant and monotone function of slacks in the fractional program:slacks in the fractional program:

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Linearisation:Linearisation:

Redefine Redefine SSjj = t s = t sjj (j = 1,2,3) (j = 1,2,3)

xxii = t x = t xii* (i=1,2)* (i=1,2) yields:yields:

Linearised problem:Linearised problem:

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Empirical resultsEmpirical results

all output slacks except of sector 1 are zeroall output slacks except of sector 1 are zero pollution slacks and capital slack are positivepollution slacks and capital slack are positive inefficient economy produces too much agric. inefficient economy produces too much agric.

output generating more pollution requiring output generating more pollution requiring higher abatement intensities λhigher abatement intensities λ66 and λ and λ77

too much pollution generated, too much capital too much pollution generated, too much capital available, labour used efficientlyavailable, labour used efficiently

1 1. 017 s1 25366. 251

2 0. 834 s2 s3 s4 s5 s8 0

3 0. 832 s6 54206. 968

4 0. 831 s7 831. 700 32

5 0. 827 s9 75486. 407

6 2. 454 t 0. 823

7 1. 995 0. 409