Efficiency Analysis of a Multisectoral Economic System

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1 Efficienc Anali of a Multiectoral Economic Stem Mikuláš uptáčik Vienna Univerit of Economic and Buine Vienna, Autria Bernhard Böhm Vienna Univerit of Technolog Vienna,Autria Paper prepared for the 7 th International Conference on Data Envelopment Anali, Jul Jul, 9, Philadelphia Receion are eail recogniable from a decreae in GDP. What reall hould interet u, however, i the difference between the potential of an econom and it actual performance (J. Stiglit, ).. Introduction In the literature two approache of productivit and efficienc anali can be found, namel the neoclaical approach and a frontier approach know a data envelopment anali (DEA). Under the neoclaical approach we refer to the eminal paper b DEBREU (95), meauring the efficienc of the econom b a coefficient of reource utiliation and to the book b ten RAA (995) and (5). DEA approach allow decompoing productivit growth into a movement of the econom toward the frontier and a hift of the latter. The neoclaical approach impute productivit growth to factor, but cannot ditinguih a movement toward the frontier and a movement of the frontier. In the paper b ten RAA- MOHNEN () a nthei of both approache i provided. The problem addreed in the preent paper i concerned with efficienc anali applied to a ingle econom repreented b the eontief input-output-model etended b the contraint for primar factor Firt, the efficienc frontier i generated uing a multi-objective optimiation model intead of having to ue data from different deciion making unit The olution of the multi-objective optimiation problem define efficient virtual deciion making unit (DMU). The efficienc of the given econom i defined a the difference between the potential of an econom and it actual performance and can be obtained a a olution of a DEA model. It can be hown that the olution of the above defined DEA model ield the ame efficienc core and the ame hadow price a the model b ten RAA (995), (5), depite the different variable ued in both model Uing dualit theor of

2 linear programming the equivalence of the approache permit a clear economic interpretation. In the econd part of the paper thi approach i etended to the augmented eontief model including emiion of pollutant and abatement activitie In thi wa the eco-efficienc of an econom can be analed.. The production poibilit et of a eontief input-output model eontief input-output model convenientl decribe the production relation of an econom for a given nonnegative vector of final demand for n good () produced in n interrelated ector Gro output of the ector i denoted b the n-dimenional vector. Production technolog i given b a contant (n n) input coefficient matri A, which inform about the ue of a particular good i required for the production of a unit of good j, together with primar factor requirement per unit of output a given b (m n) matri B. The ue of primar input factor i retricted b the m-vector of available input quantitie. () ( I A) B In order to model multi-output multi-input technologie the notion of input and output ditance function can be ued. For a ingle output thi correpond to the concept of a production function. Ditance function are well uited to define input and output oriented meaure of technical efficienc. To work out uch efficienc meaure and to derive the output potential of an econom with n output we face in principle a multi objective optimiation problem. In man cae uch problem are reduced to a ingle objective optimiation problem b uitable aggregation (e.g. ten Raa (995, 5) ue world market price for the n commoditie emploed in hi model to reduce the optimiation of n output to that of a ingle um of value of the net product). Puruing the multiple objective approach we propoe to olve the following optimiation model where each net output i maimied ubject to retraint on the availabilit of input : Ma () ( I A) B,

3 We ue the notation Ma for a vector optimiation problem and ma for a ingle objective problem. We olve, thu, n ingle objective problem where final demand for each commodit i maimied, i.e. (3) ma j (j =,, n) ubject to the contraint in (). For each of the n olution of (3) denote the (alo n- dimenional) olution vector *j (j =,,n) repreenting the gro production of commoditite The repective net-output column vector are denoted *j. Alternativel, for a given level of final demand the ue of input i minimied: (4) Min ( I A) B, In thi cae, therefore, m ingle objective problem are olved (5) min i (i =,, m) ubject to the contraint in (4). The m olution vector *i (i =,, m) decribe the gro production value of commoditie for given final demand under the individual minimiation of the primar factor i =,,m. The optimal input vector are denoted b *i. Thee et of value of both problem defined above are arranged column-wie in a pa-off matri with the optimal value appearing in the main diagonal while the off-diagonal element provide the level of other ector net-output and input compatible with the individuall optimied one The paoff matri of dimenion (n+m n+m) i written * P = * *n n + * + m *m P P where i the vector of the lack variable of the n output and i the vector of the m input lack Thu, each column of the paoff matri containing either the maimal net output of a particular commodit or the minimal input ield an efficient olution (in the ene of Pareto- Koopman). In thi wa the efficienc frontier of the economic tem can be generated. In other word, the matri P relate the combination of output quantitie which are poible to produce for an given combination of input, In thi wa the macroeconomic production function for multi-input multi-output technologie can be decribed. A hown b Belton and Vicker(99, 993) conidering the input and output a aociated objective b minimiing input and/or maimiing output the approache of multiple 3

4 criteria deciion making and data envelopment anali coincide (although their ultimate aim ma till differ). Each of the point in the paoff-matri P i contructed independentl of the other point but taking account of the entire tem relation Knowing the efficient frontier we can etimate the efficienc of the actual econom. Each of the column of the pa-off matri can be een a a virtual deciion making unit with different input and output characteritic which all are uing the ame production technique. The real econom a given b actual output and input data define a new deciion making unit whoe ditance to the frontier can be etimated. Thi frontier contitute the tandard envelope a propoed b Golan and Roll (994) for the DEA model which we ue for meauring the efficienc of the econom given b the actual output and input data (, ) in the following input oriented DEA problem (6) min θ P P + θ, θ Now the quetion arie how thi approach i related to the neoclaical one of ten Raa and Debreu. 3. The relationhip between the DEA model and the P - eontief model In the pirit of ten Raa (995, 5) and Debreu (95) the eontief-model () can be formulated a an optimiation problem in the following wa: minimie the ue of primar input for given level of final demand. (7) min γ ( I A) B + γ, γ The parameter γ provide a radial efficienc meaure. It record the degree b which primar input could be proportionall reduced but till capable of producing the required net output 4

5 Taking into account the interpretation of the efficienc parameter θ in the DEA-model we ee that depite of the different model formulation the objective function are imilar. The meaure the efficienc of the econom b radial reduction of primar input for given amount of net output The relationhip between (6) and (7) are given b the following propoition. Propoition : The efficienc core θ of DEA problem (6) i eactl equal to the radial efficienc meaure γ of P-model (7). The dual olution of model (7) coincide with the olution of the DEA multiplier problem which i the dual of problem (6). Proof: The dual model to (7) can be written (8) ma p' p'( I A) r' B p, r r' where p are the hadow price of the n commoditie and r the hadow price of the primar input factor Auming indecompoabilit of A it follow for the eontief model that > and (I-A) - > (cf. e.g. Nikaido (968)). From the complementar lackne theorem follow p'( I A) r' B = and thu p' = r' B( I A) > which ha a clear economic interpretation. Matri B(I-A) - contain the cumulative requirement of primar input Therefore the total value of ued primar input determine the hadow price of commoditie The dual to (6) i (9) ma P v' P u, v v' 5

6 Multipling the eontief invere b the matri of generated net output P we obtain the correponding total gro output requirement, denoted b matri T: () (I-A) - P = T. In other word T repreent the total output requirement for each virtual deciion making uni Conequentl BT = B (I-A) - P give the necear amount of primar input to atif the generated total output requirement Thi coincide with the contruction of matri P decribing the primar input requirement necear to atif final demand P. Therefore () P = BT. It follow from () that () P = (I-A)T. Multipling the firt contraint in (8) b T ield (3) p '( I A) T r' BT. Subtituting () and () for P and P repectivel into (3) we obtain eactl the contraint a of the dual problem (9): p'p r'p Now we have two problem with the ame contraint and the ame coefficient of the objective function Therefore the optimal value of the objective function mut be the ame: p' =. Conequentl p' = and according to the dualit theorem of linear programming γ = θ. Since γ > implie r = and θ > implie v = we have r' = v'. 4. Etenion to the augmented eontief model The anali can be etended to the model verion including pollution generation and abatement activitie The well known augmented eontief model (eontief, 97) i written a (4) I A A A I A [ B B ] = = where the following notation i ued i the n-dimenional vector of gro indutrial output; 6

7 i the k-dimenional vector of anti-pollution activit level; A i the (nn) matri of conventional input coefficient, howing the input of good i per unit of the output of good j (produced b ector j); A i the (nk) matri with aig repreenting the input of good i per unit of the eliminated pollutant g (eliminated b anti-pollution activit g); A i the (kn) matri howing the output of pollutant g per unit of good i (produced b ector i); A i the (kk) matri howing the output of pollutant g per unit of eliminated pollutant h (eliminated b anti-pollution activit h); I i the identit matri; i the n-dimenional vector of final conumption demand for economic commoditie; i the k-dimenional vector of the net generation of pollutant which remain untreated after abatement activi The g-th element of thi vector repreent the pollution tandard of pollutant g and indicate the tolerated level of net pollution. In addition the relation for primar input contain B the (m n) matri of primar input coefficient for production activitie (denoted matri B in the previou ection), and B the (m k) matri of primar input coefficient for abatement activitie Formulating the eontief model a an P-problem b minimiing primar input for given level of final demand and environmental tandard we get (5) min γ,, γ ( I A ) A A + ( I A) B B + γ In analog to ection we formulate the multiobjective optimiation problem a follow (6) Ma ( I A) B, where A A = A A A, =, =, B = [ B B ] 7

8 8 We again olve n ingle objective problem maimiing final demand for each commodit eparatel: (7) ma j (j =,, n) the contraint in (6). Minimiation of net pollution under the contraint (6) ield the trivial olution where all variable are ero. Alternativel for given final demand and environmental tandard (the tolerated level of net-pollution) the input are minimied. (8) Solving the m eparate ingle objective problem (9) min i (i =,, m) we can derive the paoff matri of dimenion (n+k+m) (n+m) for the augmented model partitioned in the following wa The notation correpond to that of ection, the j i (j =,, ) repreent the repective vector of lack variable in the optimiation of variable i (i =,,n, n+,, n+m). The DEA model related to the optimiation problem (5) i now () A in the previou ection we can prove the following propoition relating the model (5) and (). Propoition : The dual olution of model (5) coincide with the olution of the DEA multiplier problem (which i the dual of problem ()).,.. min + + θ θ θ Z Q Q t *m * n *n * + + = Z Q Q Z Q n m m, ) ( B A I Min

9 The efficienc core θ of DEA problem () i eactl equal to the radial efficienc meaure γ of P-model (5). Proof: We tart with the dual problem to (5). ma p' p'( I A) r' B r' p', r' where p = (p, p ) with p the (n) vector (hadow) price of commoditie, p the (k) vector of hadow price for abating pollutant, and r the ( m) hadow price of the primar input factor Multipling the augmented eontief invere b Q we obtain the gro production vector augmented b the anti-pollution activit level correponding to the individuall optimal output and primar input (I-A) - Q = T. The total primar input required b maimied output are given b BT = Z The multiplier DEA model i ma Q v' Z v' u, v where u i a (n+k) vector and v a ( m) vector. In analog to the proof of propoition ubtituting (I-A)T = Q and BT = Z the contraint of the multiplier problem and the dual to (5) are the ame. Since p = u the dual olution coincide p = u and the efficienc core a well. To invetigate the relation between the output and input oriented model we rewrite the augmented eontief model a a maimiation model of final demand with repect to contraint on primar input () ma α ( I A) + α B, α 9

10 The dual model i Uing the ame paoff matri (Q, Z) a derived above the output oriented DEA model i formulated a () min r' p'( I A) + r' B ma ϕ, ϕ p', r' Q + ϕ Q Z p' The multiplier DEA model ha the following form min v' Q + v' Z Q u, v Similar to the proof of the previou propoition it can be hown that the efficienc core α from () coincide with the efficienc core φ of the DEA model (). Becaue for the efficienc core of the output and input oriented DEA model under contant return to cale φθ = the following propoition hold: Propoition 3 The efficienc core α for the model () i the reciprocal value of the efficienc core γ of model (5). It i quite obviou that the ame reult hold true alo for the eontief optimiation model without pollution and abatement activitie 5. Concluion The equivalence of the different approache to efficienc meaurement of an econom provide u with a deeper inight into the procee ongoing within an econom, not uuall

11 viible when uing tandard DEA model The contruction of the efficienc frontier permit an aement with repect to the own potential of an econom (even in the cae of multiple output and input) without the need to compare it with other economie poeing poibl different technologie and obviou mutual interdependencie due to international trade. Due to our reult the relative merit of both approache can be ued. For intertemporal comparion of productivit growth the movement of the econom toward the frontier and it hift can be obtained b uing the DEA formulation which in our formulation provide the ame imputation of productivit growth to individual factor a the neoclaical model. A further important feature of our invetigation i the incluion of pollution and abatement activitie in the (eco)-efficienc anali of an econom. Reference Belton V., Vicker S.P. (99), VIDEA: Integrated DEA and MCDA A viual interactive approach, in Proceeding of the th International Conference on MCDM, Vol.II, Belton V., Vicker S.P. (993), Demtifing DEA A viual interactive approach baed on multi criteria anali, Journal of the Operational Reearch Societ 44, Debreu G. (95), The coefficient of reource utiliation, Econometrica 9, No. 3, 73-9 Golan B., Y. Roll (994), ''Incorporating Standard via DEA'', ch. 6 in Charne A. et al. (ed), Data Envelopment Anali: Theor, Methodolog, and Application, Kluwer, Boton - Dordrecht - ondon eontief W. W. (97), Environmental repercuion and the economic tructure: An inputoutput approach, The Review of Economic and Statitic 5, 6-7 uptáčik M., Böhm B. (5), The Anali of Eco-efficienc in an Input-Output Framework, paper preented at the Ninth European Workhop on Efficienc and Productivit Anali (EWEPA IX), Bruel, June 9th to Jul nd, 5 Nikaido H. (968), Conve Structure and Economic Theor, Academic Pre Stewart Th. J. (996), Relationhip between Data Envelopment Anali and Multicriteria Deciion Anali, Journal of the Operational Reearch Societ 47, ten Raa Th. (995), inear Anali of Competitive Economic, SE Handbook in Economic, Harveter Wheatheaf, New York-ondon-Amterdam ten Raa Th. (5): The Economic of Input-Output Anali, Cambridge Univerit Pre ten Raa, Th., Mohnen, P. (): Neoclaical growth accounting and frontier anali: a nthei, Journal of Productivit Anali, Vol. 8/, p. -8