Efficiency Analysis of a Multisectoral Economic System Efficiency Analysis of a Multisectoral Economic System Mikulas Luptáčik University of Economics and Business Administration, Vienna, Austria Bernhard Böhm University of Technology, Vienna, Austria 16th International Input-Output Conference Istanbul – Turkey 2 – 6 July 2007

Efficiency of production Multi-input and multi-output production technology Multi-input and multi-output production technology Use distance functions to characterise efficiency of production Use distance functions to characterise efficiency of production Input distance function is reciprocal of output distance function Input distance function is reciprocal of output distance function A radial measure of efficiency A radial measure of efficiency

Constant returns to scale (Input-Output model) Constant returns to scale (Input-Output model) Consider environment: pollution and abatement in the „augmented“ Leontief model Consider environment: pollution and abatement in the „augmented“ Leontief model Use suitable definition of „eco-efficiency“ Use suitable definition of „eco-efficiency“

quantities of undesirable outputs (pollutants) are treated like inputs (i.e. are minimised). quantities of undesirable outputs (pollutants) are treated like inputs (i.e. are minimised). four different eco-efficient models can be constructed four different eco-efficient models can be constructed cf. Korhonen and Luptacik (2004) cf. Korhonen and Luptacik (2004)

The augmented Leontief model with represents the economic constraints The augmented Leontief model with n outputs (sectors), k pollutants, k abatement activities, m primary inputs represents the economic constraints in the following LP – models (in the spirit of Debreu (1951) and Ten Raa (1995):

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

2. Maximise the proportional expansion of final demand y 1 for given levels of tolerated pollution (environmental standards) and primary factors (2)

Due to the presence of the pollution subsystem representing undesirable outputs, the optimal values of γ and α are not the reciprocal of each other. Due to the presence of the pollution subsystem representing undesirable outputs, the optimal values of γ and α are not the reciprocal of each other. However, by treating these undesirable outputs like inputs in the model, i.e. by changing the problem formulation into a proportional reduction of primary inputs and undesirable outputs for given final demand, the reciprocal property of the distance function can be re-established. However, by treating these undesirable outputs like inputs in the model, i.e. by changing the problem formulation into a proportional reduction of primary inputs and undesirable outputs for given final demand, the reciprocal property of the distance function can be re-established.

3. Minimise the use of primary inputs and emission of pollutants for given levels of final demand (3)

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

These models could formally be seen as data envelopment analysis (DEA) models when using sectors as DMU’s. These models could formally be seen as data envelopment analysis (DEA) models when using sectors as DMU’s. But application of DEA to the Input-Output framework requires some additional considerations: But application of DEA to the Input-Output framework requires some additional considerations:

Because: Because: DEA uses inputs and outputs of different independent decision making units, the I-O model uses data of usually only one country but disaggregated into interrelated sectors with different technologies. DEA uses inputs and outputs of different independent decision making units, the I-O model uses data of usually only one country but disaggregated into interrelated sectors with different technologies. Therefore: not meaningful to compare sectors with respect to their relative efficiency. Direct interpretation as DEA-model is economically not meaningful! Therefore: not meaningful to compare sectors with respect to their relative efficiency. Direct interpretation as DEA-model is economically not meaningful!

Generate the production possibility set Generate the production possibility set Each output is maximised subject to restraints on the production of other outputs and available inputs (multiobjective optimisation problem). Each output is maximised subject to restraints on the production of other outputs and available inputs (multiobjective optimisation problem). Measure distance of actual economy to the production frontier Measure distance of actual economy to the production frontier Application of DEA to the input-output framework

Optimisation model (5)

s 1 is the vector of n slack variables of the n sectors s 1 is the vector of n slack variables of the n sectors s 2 is the vector of slack variables of the k pollutants s 2 is the vector of slack variables of the k pollutants s 3 are the slacks in the m inputs s 3 are the slacks in the m inputs Solve the model n+k+m times for given values of sector net-outputs and inputs for the maximal values of each slack variable s j for j=1,...,n,...,n+k,...,n+k+m Solve the model n+k+m times for given values of sector net-outputs and inputs for the maximal values of each slack variable s j for j=1,...,n,...,n+k,...,n+k+m

Individually optimal desirable and undesirable outputs and input values are calculated from y* 1 = y 1 + s 1 y* 2 = y 2 – s 2 z* = z – s 3 and are arranged to form a pay-off matrix P. Individually optimal desirable and undesirable outputs and input values are calculated from y* 1 = y 1 + s 1 y* 2 = y 2 – s 2 z* = z – s 3 and are arranged to form a pay-off matrix P.

Pay-off matrix

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

This efficient envelope is used to evaluate the relative inefficiency of the economy given by the actual output and input data (y 1 0, y 2 0, z 0 ) e.g. in the following input oriented DEA problem This efficient envelope is used to evaluate the relative inefficiency of the economy given by the actual output and input data (y 1 0, y 2 0, z 0 ) e.g. in the following input oriented DEA problem (6)

The relationship between the DEA model and the LP model To provide a clear economic interpretation we consider LP-model (3) and DEA-model (6) without pollution and abatement sectors To provide a clear economic interpretation we consider LP-model (3) and DEA-model (6) without pollution and abatement sectors (6')(3')

Proposition 1: The efficiency score θ of DEA problem (6') is exactly equal to the radial efficiency measure γ of LP-model (3'). The efficiency score θ of DEA problem (6') is exactly equal to the radial efficiency measure γ of LP-model (3'). The dual solution of model (3') coincides with the solution of the DEA multiplier problem (which is the dual of problem (6')) The dual solution of model (3') coincides with the solution of the DEA multiplier problem (which is the dual of problem (6'))

Consider first LP-problem (1) and the corresponding DEA model (7) Consider first LP-problem (1) and the corresponding DEA model (7) The analysis can be extended to the model with pollution and abatement : (7)

Proposition 2: The efficiency score θ of DEA problem (7) is exactly equal to the radial efficiency measure γ of LP-model (1). The efficiency score θ of DEA problem (7) is exactly equal to the radial efficiency measure γ of LP-model (1). The dual solution of model (1) coincides with the solution of the DEA multiplier problem (which is the dual of problem (7)) The dual solution of model (1) coincides with the solution of the DEA multiplier problem (which is the dual of problem (7))

Proposition 3: The efficiency score θ of DEA problem (6) is exactly equal to the radial efficiency measure γ of LP-model (3). The efficiency score θ of DEA problem (6) is exactly equal to the radial efficiency measure γ of LP-model (3). The dual solution of model (3) coincides with the solution of the DEA multiplier problem (which is the dual of problem (6)) The dual solution of model (3) coincides with the solution of the DEA multiplier problem (which is the dual 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 (λ i = 1.05 for min-model and equal to one for the max-model (i.e. the same output can be produced by a % 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 with λ i slightly larger than for i = 1,..., 5 and λ 6 = 1.084, λ 7 =

Calculating the output oriented model the maximum α = is the reciprocal value of min γ. Here again the intensities are almost the same for the outputs but different for the pollutants (λ i = for i = 1,..., 5 and λ 6 = 1.137, λ 7 = ). 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 table Construct the envelope: Pay-off table

DEA model with pollutants (with 5% capital and labour surplus) Using this pay-off table for the same experiment as before (with 5% capital and labour surplus) Solve: Solve:

Achieve a minimum  = for the economy Achieve a minimum  = for 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 )

DMUData ProjectionDifference % Capital % Labour % Poll % Poll % y % y % y % y % y % CCR-I θ =

This frontier is constructed from the following optimal weights: µ 1 = µ 2 = µ 3 = µ 4 = µ 5 = µ 6 = µ 7 = 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 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.

DMUData ProjectionDifference% Capital % Labour % Poll % Poll % y % y % y % y % y % CCR-O 1/α = θ

Slack based measures of eco-efficiency To avoid limitation of efficiency indicators assuming unchanged proportions of inputs and outputs To avoid limitation of efficiency indicators assuming unchanged proportions of inputs and outputs Formulate a goal-programming model (treat pollutants as inputs) Formulate a goal-programming model (treat pollutants as inputs) Minimise a scalar which is unit invariant and monotone function of slacks in the fractional program: Minimise a scalar which is unit invariant and monotone function of slacks in the fractional program:

Linearisation: Redefine S j = t s j (j = 1,2,3) x i = t x i * (i=1,2) Redefine S j = t s j (j = 1,2,3) x i = t x i * (i=1,2) yields: yields:

Linearised problem:

Empirical results all output slacks except of sector 1 are zero all output slacks except of sector 1 are zero pollution slacks and capital slack are positive pollution slacks and capital slack are positive inefficient economy produces too much agric. output generating more pollution requiring higher abatement intensities λ 6 and λ 7 inefficient economy produces too much agric. output generating more pollution requiring higher abatement intensities λ 6 and λ 7 too much pollution generated, too much capital available, labour used efficiently too much pollution generated, too much capital available, labour used efficiently