Efficiency analysis of a multisectoral economic system

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Efficiency Analysis of a Multisectoral Economic System Mikuláš Luptáčik Vienna University of Economics and Business Vienna, Austria Bernhard Böhm Vienna University of Technology Vienna,Austria Paper prepared for the 7th International Conference on Data Envelopment Analysis, July 10 –July 12, 2009, Philadelphia “Recessions are easily recognizable from a decrease in GDP. What really should interest us, however, is the difference between the potential of an economy and its actual performance” (J. Stiglitz, 2002). 1. Introduction In the literature two approaches of productivity and efficiency analysis can be found, namely the neoclassical approach and a frontier approach know as data envelopment analysis (DEA). Under the neoclassical approach we refer to the seminal paper by DEBREU (1951), measuring the efficiency of the economy by a coefficient of resource utilization and to the books by ten RAA (1995) and (2005). DEA approach allows decomposing productivity growth into a movement of the economy towards the frontier and a shift of the latter. The neoclassical approach imputes productivity growth to factors, but cannot distinguish a movement towards the frontier and a movement of the frontier. In the paper by ten RAAMOHNEN (2002) a synthesis of both approaches is provided. The problem addressed in the present paper is concerned with efficiency analysis applied to a single economy represented by the Leontief input-output-model extended by the constraints for primary factors. First, the efficiency frontier is generated using a multi-objective optimization model instead of having to use data from different decision making units. The solutions of the multi-objective optimization problems define efficient virtual decision making units (DMUs). The efficiency of the given economy is defined as the difference between the potential of an economy and its actual performance and can be obtained as a solution of a DEA model. It can be shown that the solution of the above defined DEA model yields the same efficiency score and the same shadow prices as the models by ten RAA (1995), (2005), despite the different variables used in both models. Using duality theory of 1

linear programming the equivalence of the approaches permits a clear economic interpretation. In the second part of the paper this approach is extended to the augmented Leontief model including emissions of pollutants and abatement activities. In this way the eco-efficiency of an economy can be analyzed. 2. The production possibility set of a Leontief input-output model Leontief’s input-output model conveniently describes the production relations of an economy for a given nonnegative vector of final demands for n goods (y) produced in n interrelated sectors. Gross output of the sectors is denoted by the n-dimensional vector x. Production technology is given by a constant (n x n) input coefficient matrix A, which informs about the use of a particular good i required for the production of a unit of good j, together with primary factor requirements per unit of output as given by (m x n) matrix B. The use of primary input factors is restricted by the m-vector of available input quantities z. (1)

( I − A) x ≥ y Bx ≤ z

In order to model multi-output multi-input technologies the notion of input and output distance functions can be used. For a single output this corresponds to the concept of a production function. Distance functions are well suited to define input and output oriented measures of technical efficiency. To work out such efficiency measures and to derive the output potential of an economy with n outputs we face in principle a multi objective optimization problem. In many cases such problems are reduced to a single objective optimization problem by suitable aggregation (e.g. ten Raa (1995, 2005) uses world market prices for the n commodities employed in his model to reduce the optimization of n outputs to that of a single sum of values of the net products). Pursuing the multiple objective approach we propose to solve the following optimisation model where each net output y is maximised subject to restraints on the availability of inputs z0 :

Max y x



( I − A) x − y ≥ 0 Bx ≤ z 0 x, y ≥ 0 2

We use the notation “Max” for a vector optimization problem and “max” for a single objective problem. We solve, thus, n single objective problems where final demand for each commodity is maximized, i.e. (3) max yj (j = 1,…, n) subject to the constraints in (2). For each of the n solutions of (3) denote the (also ndimensional) solution vector x*j (j = 1,…,n) representing the gross productions of commoditites. The respective net-output column vectors are denoted y*j . Alternatively, for a given level of final demand y0 the use of inputs z is minimized: Min z (4)



( I − A) x ≥ y 0 Bx − z ≤ 0 x, z ≥ 0

In this case, therefore, m single objective problems are solved (5) min zi (i = 1,…, m) subject to the constraints in (4). The m solution vectors x*i (i = 1,…, m) describe the gross production values of commodities for given final demand y0 under the individual minimization of the primary factors i = 1,…,m. The optimal input vectors are denoted by z*i . These sets of values of both problems defined above are arranged column-wise in a pay-off matrix with the optimal values appearing in the main diagonal while the off-diagonal elements provide the levels of other sector net-outputs and inputs compatible with the individually optimized ones. The payoff matrix of dimension (n+m x n+m) is written  y*1 P= 0 1 z − s z




z 0 − s z2 L z 0 − s zn

y 0 + s1y L y 0 + s ym   P1  ≡  z*1 z*m   P2  L

where sy is the vector of the slack variables of the n outputs and sz is the vector of the m input slacks. Thus, each column of the payoff matrix containing either the maximal net output of a particular commodity or the minimal input yields an efficient solution (in the sense of ParetoKoopmans). In this way the efficiency frontier of the economic system can be generated. In other words, the matrix P relates the combinations of output quantities which are possible to produce for any given combination of inputs, In this way the “macroeconomic production function” for multi-input multi-output technologies can be described. As shown by Belton and Vickers(1992, 1993) considering the inputs and outputs as associated objectives by minimizing inputs and/or maximizing outputs the approaches of multiple 3

criteria decision making and data envelopment analysis coincide (although their ultimate aims may still differ). Each of the points in the payoff-matrix P is constructed independently of the other points but taking account of the entire systems relations. Knowing the efficient frontier we can estimate the efficiency of the actual economy. Each of the columns of the pay-off matrix can be seen as a virtual decision making unit with different input and output characteristics which all are using the same production technique. The real economy as given by actual output and input data defines a new decision making unit whose distance to the frontier can be estimated. This frontier constitutes the standard envelope as proposed by Golany and Roll (1994) for the DEA model which we use for measuring the efficiency of the economy given by the actual output and input data (y0, z0) in the following input oriented DEA problem min θ



P1 µ ≥ y 0


− P2 µ + θz 0 ≥ 0

µ ≥ 0,θ ≥ 0 Now the question arises how this approach is related to the neoclassical one of ten Raa and Debreu. 3. The relationship between the DEA model and the LP - Leontief model

In the spirit of ten Raa (1995, 2005) and Debreu (1951) the Leontief-model (1) can be formulated as an optimization problem in the following way: minimize the use of primary inputs for given levels of final demand. min γ x


( I − A) x ≥ y 0 (7)

− Bx + γz 0 ≥ 0 x, γ ≥ 0

The parameter γ provides a radial efficiency measure. It records the degree by which primary inputs could be proportionally reduced but still capable of producing the required net outputs.


Taking into account the interpretation of the efficiency parameter θ in the DEA-model we see that despite of the different model formulations the objective functions are similar. They measure the efficiency of the economy by radial reduction of primary inputs for given amounts of net outputs. The relationships between (6) and (7) are given by the following proposition. Proposition 1:

The efficiency score θ of DEA problem (6) is exactly equal to the radial efficiency measure γ of LP-model (7). The dual solution of model (7) coincides with the solution of the DEA multiplier problem which is the dual of problem (6). Proof:

The dual model to (7) can be written p' y 0




p' ( I − A) − r ' B ≤ 0 r' z0 ≤ 1 p, r ≥ 0

where p are the shadow prices of the n commodities and r the shadow prices of the primary input factors. Assuming indecomposability of A it follows for the Leontief model that x > 0 and (I-A)-1 > 0 (cf. e.g. Nikaido (1968)). From the complementary slackness theorem follows p' ( I − A) − r ' B = 0 and thus p' = r ' B( I − A) −1 > 0 which has a clear economic interpretation. Matrix B(I-A)-1 contains the cumulative requirements of primary inputs. Therefore the total values of used primary inputs determine the shadow prices of commodities. The dual to (6) is max u ' y 0


s.t. u ' P1 − v' P2 ≤ 0 v' z 0 ≤ 1 u, v ≥ 0


Multiplying the Leontief inverse by the matrix of generated net outputs P1 we obtain the corresponding total gross output requirements, denoted by matrix T: (10)

(I-A)-1 P1 = T ≥ 0.

In other words T represents the total output requirements for each virtual decision making unit. Consequently BT = B (I-A)-1 P1 gives the necessary amount of primary inputs to satisfy the generated total output requirements. This coincides with the construction of matrix P2 describing the primary input requirements necessary to satisfy final demands P1. Therefore (11)

P2 = BT.

It follows from (10) that (12)

P1 = (I-A)T.

Multiplying the first constraint in (8) by T yields (13)

p ' ( I − A)T − r ' BT ≤ 0 .

Substituting (11) and (12) for P2 and P1 respectively into (13) we obtain exactly the constraints as of the dual problem (9): p'P1 – r'P2 ≤ 0 Now we have two problems with the same constraints and the same coefficients y0 of the objective functions. Therefore the optimal values of the objective functions must be the same: p'y0 = u'y0. Consequently p' = u' and according to the duality theorem of linear programming γ = θ. Since γ > 0 implies r’z0 = 1 and θ > 0 implies v’z0 = 1 we have r' = v'. 4. Extension to the augmented Leontief model

The analysis can be extended to the model versions including pollution generation and abatement activities. The well known augmented Leontief model (Leontief, 1970) is written as


 I − A11 −A 21 


− A12   x1   y1  = I − A22   x2  − y2 

x  B2 ] 1  = z  x2 

where the following notation is used x1 is the n-dimensional vector of gross industrial outputs; 6

x2 is the k-dimensional vector of anti-pollution activity levels; A11 is the (nxn) matrix of conventional input coefficients, showing the input of good i per unit

of the output of good j (produced by sector j); A12 is the (nxk) matrix with aig representing the input of good i per unit of the eliminated

pollutant g (eliminated by anti-pollution activity g); A21 is the (kxn) matrix showing the output of pollutant g per unit of good i (produced by sector

i); A22 is the (kxk) matrix showing the output of pollutant g per unit of eliminated pollutant h

(eliminated by anti-pollution activity h); I is the identity matrix; y1 is the n-dimensional vector of final consumption demands for economic commodities; y2 is the k-dimensional vector of the net generation of pollutants which remain untreated after

abatement activity. The g-th element of this vector represents the pollution standard of pollutant g and indicates the tolerated level of net pollution. In addition the relation for primary inputs contains B1 the (m x n) matrix of primary input coefficients for production activities (denoted matrix B in the previous section), and B2 the (m x k) matrix of primary input coefficients for abatement activities. Formulating the Leontief model as an LP-problem by minimising primary inputs for given levels of final demand y10 and environmental standards y20 we get min γ x



( I − A11 ) x1 − A12 x2 ≥ y10 − A21 x1 + ( I − A22 ) x2 ≥ − y20 − B1 x1 − B2 x2 + γz ≥ 0 x1 , x2 , γ ≥ 0

In analogy to section 2 we formulate the multiobjective optimization problem as follows Max y ( I − A) x − y ≥ 0


Bx ≤ z 0 x, y ≥ 0 where A A =  11  A21

A12  x  , x =  1 ,  A22   x2 

 y  y =  1 , B = [B1  − y2 


B2 ]

We again solve n single objective problems maximizing final demand for each commodity separately: (17)

max y1j (j = 1, …, n)

s.t. the constraints in (16). Minimisation of net pollution under the constraints (16) yields the trivial solution where all variables are zero. Alternatively for given final demand y10 and environmental standard y20 (the tolerated level of net-pollution) the inputs z are minimised. Min z ( I − A) x ≥ y 0 Bx − z ≤ 0 x, y ≥ 0


Solving the m separate single objective problems (19)

min zi (i = 1, …, m)

we can derive the payoff matrix of dimension (n+k+m) x (n+m) for the augmented model partitioned in the following way  y1*1 y1*n L Q   1 y n2 L  Z  =  y2   z 0 − s1 L z 0 − s n z z 

y10 + s1y1 y02 − s1y 2 z*1

L y10 + s ym1   Q1   L y02 − s ym2  ≡ Q2  z*m   Z  L

The notation corresponds to that of section 2, the sji (j = y1, y2, z) represent the respective vectors of slack variables in the optimisation of variable i (i = 1,…,n, n+1,…, n+m). The DEA model related to the optimisation problem (15) is now min θ µ



Q1µ ≥ y10 − Q2 µ + y20 ≥ 0 − Zµ + θz 0 ≥ 0 µ ≥ 0,θ ≥ 0

As in the previous section we can prove the following proposition relating the models (15) and (20). Proposition 2:

The dual solution of model (15) coincides with the solution of the DEA multiplier problem (which is the dual of problem (20)).


The efficiency score θ of DEA problem (20) is exactly equal to the radial efficiency measure γ of LP-model (15). Proof:

We start with the dual problem to (15). p' y 0


p ' ( I − A) − r ' B ≤ 0


r' z0 ≤ 1 p' , r ' ≥ 0

where p’= (p1’ , p2’) with p1’ the (1xn) vector (shadow) prices of commodities, p2’ the (1xk) vector of shadow prices for abating pollutants, and r the (1 x m) shadow prices of the primary input factors. Multiplying the augmented Leontief inverse by Q we obtain the gross production vectors augmented by the anti-pollution activity levels corresponding to the individually optimal outputs and primary inputs. (I-A)-1 Q = T ≥ 0. The total primary inputs required by maximized outputs are given by BT = Z The multiplier DEA model is max u ' y 0 s.t. u ' Q − v' Z ≤ 0 v' z 0 ≤ 1 u, v ≥ 0

where u’ is a 1 x (n+k) vector and v’ a (1 x m) vector. In analogy to the proof of proposition 1 substituting (I-A)T = Q and BT = Z the constraints of the multiplier problem and the dual to (15) are the same. Since p’y0 = u’y0 the dual solutions coincide p’ = u’ and the efficiency scores as well. To investigate the relations between the output and input oriented model we rewrite the augmented Leontief model as a maximization model of final demand with respect to constraints on primary inputs max α x



− ( I − A) x + αy 0 ≤ 0 Bx ≤ z 0 x, α ≥ 0 9

The dual model is min r ' z 0 s.t. − p ' ( I − A) + r ' B ≥ 0 p' y 0 ≥ 1 p' , r ' ≥ 0

Using the same payoff matrix (Q, Z)’ as derived above the output oriented DEA model is formulated as max ϕ µ



− Q1  0  Q µ + ϕ y ≤ 0  2  Zµ ≤ z0

µ ≥ 0,ϕ ≥ 0 The multiplier DEA model has the following form min v' z 0 − Q  s.t. u '  1  + v' Z ≥ 0  Q2  u' y 0 ≥ 1 u, v ≥ 0 Similar to the proof of the previous proposition it can be shown that the efficiency score α from (21) coincides with the efficiency score φ of the DEA model (22). Because for the efficiency score of the output and input oriented DEA model under constant returns to scale φθ = 1 the following proposition holds: Proposition 3

The efficiency score α for the model (21) is the reciprocal value of the efficiency score γ of model (15). It is quite obvious that the same result holds true also for the Leontief optimisation model without pollution and abatement activities. 5. Conclusion

The equivalence of the different approaches to efficiency measurement of an economy provides us with a deeper insight into the processes ongoing within an economy, not usually 10

visible when using standard DEA models. The construction of the efficiency frontier permits an assessment with respect to the own potential of an economy (even in the case of multiple outputs and inputs) without the need to compare it with other economies possessing possibly different technologies and obvious mutual interdependencies due to international trade. Due to our results the relative merits of both approaches can be used. For intertemporal comparisons of productivity growth the movement of the economy towards the frontier and its shift can be obtained by using the DEA formulation which in our formulation provides the same imputations of productivity growth to individual factors as the neoclassical model. A further important feature of our investigation is the inclusion of pollution and abatement activities in the (eco)-efficiency analysis of an economy. References

Belton V., Vickers S.P. (1992), “VIDEA: Integrated DEA and MCDA – A visual interactive approach” , in Proceedings of the 10th International Conference on MCDM, Vol.II, 419-429 Belton V., Vickers S.P. (1993), “Demystifying DEA – A visual interactive approach based on multi criteria analysis”, Journal of the Operational Research Society 44, 883-896 Debreu G. (1951), ”The coefficient of resource utilization”, Econometrica 19, No. 3, 273-292 Golany B., Y. Roll (1994), ''Incorporating Standards via DEA'', ch. 16 in Charnes A. et al. (eds.), Data Envelopment Analysis: Theory, Methodology, and Application, Kluwer, Boston - Dordrecht - London Leontief W. W. (1970), Environmental repercussions and the economic structure: An inputoutput approach, The Review of Economics and Statistics 52, 262-271 Luptáčik M., Böhm B. (2005), ”The Analysis of Eco-efficiency in an Input-Output Framework”, paper presented at the Ninth European Workshop on Efficiency and Productivity Analysis (EWEPA IX), Brussels, June 29th to July 2nd, 2005 Nikaido H. (1968), Convex Structure and Economic Theory, Academic Press Stewart Th. J. (1996), “Relationships between Data Envelopment Analysis and Multicriteria Decision Analysis”, Journal of the Operational Research Society 47, 654-665 ten Raa Th. (1995), Linear Analysis of Competitive Economics, LSE Handbooks in Economics, Harvester Wheatsheaf, New York-London-Amsterdam ten Raa Th. (2005): The Economics of Input-Output Analysis, Cambridge University Press. ten Raa, Th., Mohnen, P. (2002): “Neoclassical growth accounting and frontier analysis: a synthesis”, Journal of Productivity Analysis, Vol. 18/2, p. 111-128


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