DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYSTEMS

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1 DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYSTEMS EKONOMIKA A MANAGEMENT Tao Zhang Introducton Data envelopment analyss (DEA, the non-parametrc approach to measurng effcency, was frst ntroduced n the lterature as a lnear programmng model by Charnes et al. [1], followng on Farrell s [3] posng of the queston of relatve techncal effcency n the form of a unt soquant model. Generally, the DEA approach defnes the techncal effcency n terms of a mnmum set of nputs needed to produce a gven output known as nput-orentated model or maxmum output obtanable from a gven set of nputs known as output-orentated model. [2] However, snce the DEA s non-parametrc lnear program model, the estmated effcency mght be based f there s data aggregaton n DEA. A seres of artcles debates on the techncal effcency bas caused by data aggregaton n DEA. Fare et al. [4, 5] propose that both the nter-nput aggregaton and nter-output aggregaton wll make the estmated techncal effcency based downwardly. Barnum et al. [6, 7] dscuss the ntra-nput aggregaton and ntra-output aggregaton caused by the lnear aggregaton of the same types of nputs and outputs. Barnum et al. [8] propose the ntra-nput allocatve effcency whch measures the effcency n allocatng each type of nput among outputs usng nput orented DEA. But they do not study the normal allocatve effcency bas caused by nter-output or nter-nput aggregaton. Ths paper wll concern the nter-output allocatve effcency bas whch comes from output aggregaton. However, nearly all the observed studes are focused on the effcency bas caused by data aggregaton and blame the DEA method for ts bad performance n front of data aggregaton. Htherto, we have not observed any studes on consderng how to utlze ths character of DEA n methodology extensons for mult-output producton system. As presented n the exstng lterature, the reason of the effcency bas estmated from DEA s that one part of allocatve effcency wll be ncorporated nto the estmated techncal effcency because of output aggregaton. Then, we can use ths character to decompose the allocatve effcency accordng to emprcal requrements. Ths paper hopes to shed new lght on the methodology extensons of DEA for decomposng allocatve effcency usng output aggregaton. In fact, the meanng of results from ths paper s outsde the DEA method, and the generalty of the fndngs n ths paper can provde useful nformaton for researchers who concern the decson-makng process n allocatng resources for mult-product producton system. 1. Decomposng Allocatve Effcency To explan the theoretcal underpnnng for decomposng allocatve effcency by data aggregaton, we use output orented technology wth (=1,,I observatons. Suppose that for each observaton there s M nputs X =( X 1,, X M + M and J outputs Y = ( Y 1, J + J wth correspondng output prces P = ( P 1,, P J J. + The output orented techncal effcency DEA wth fully dsaggregated outputs and nputs can be wrtten as: A = max { X Łm X, m m = 1,...,M; Y, j Y, j j = 1,...,J; (1 = 1, 0 = 1,...,I } A s pure techncal neffcency. The pure techncal effcency (TE can be computed by TE A = 1/ A. Then, consder the followng lnear program to maxmze revenue A : E + M EKONOMIE A MANAGEMENT 2 / 2010 strana 71

2 A = max { P j Y j X, m X, m m = 1,...,M; Y, j Y j j = 1,...,J; (2 = 1, 0 = 1,...,I } Specfcally EE = P j Y j /A s economc effcency (EE or aggregate techncal and al- j locatve effcency. Normally, allocatve effcency (AE s calculated by A AE = EE /TE (3 when there s no output aggregaton. Ths allocatve effcency calculated from economc effcency and pure techncal effcency can be defned as whole allocatve effcency whch measures the effcency n allocatng all resources among all fully dsaggregated and undvdable outputs. But, f the outputs (or nputs n nput orented DEA are not fully dsaggregated and estmated techncal effcency s based, then the allocatve effcency calculated by Equaton (3 s also based. We frst consder a sub-vector of output whch s lnearly aggregated wth prces as: C, j = P j, j = 1,...I ˆj J (4 When some outputs are aggregated usng Equaton (4, the output orented techncal neffcency DEA can be expressed as: B,C, j,y, j +1,...,Y, J (5.0 c j j=1 j = max { X, m X, m m =1,...,M; (5.1 Y, j Y, j j = ˆj +1,...,J; (5.2 C, j C, j (5.3 =1, =1,...,I } (5.4 and those obtaned from the same measure but f all outputs are aggregated nto one output varable use Equaton (4 C,C, j (6.0 c = max { X, m X, m m =1,...,M; (6.1 C, j C, j (6.2 =1, =1,...,I } (6.3 The techncal effcency for aggregated data can be computed by TEc B = 1/ c B and TEc C = 1/ c C. Accordng to Fare et al. [4][5], t s obvous that c B and c C are based. Therefore, the techncal effcences computed by them are also downwardly based because the allocatve effcences are ncorporated n the techncal effcency scores. We start to answer the questons proposed n the Introducton from explorng the bas bounds of allocatve effcency. As showed by Fare et al.[5], the bas bounds of techncal effcency can be gven as: TE C B, C TE c, C, j,y, j +1,...,Y, J TE A (7 Because normal allocatve effcency s calculated by dvdng economcs effcency by techncal effcency, f economc effcency s fxed, then we can gve: AE c, C AE c, C, j,y, j +1,...,Y, J AE (8 Banker et al. [9] propose and proof that the estmated techncal effcency TE C c calculated usng Equaton (6 s dentcal to economc effcency EE calculated usng Equaton (2. Then, AE c, C s equal to 1, because TE C, C = EE, C when all the outputs are aggregated nto one varable. In addton, AE c s the whole allocatve effcency whch s calculated from the pure techncal effcency TE. Accordng to the above proofed proposton, t s ntutvely to know that ncorporatng the lnearly aggregated output usng Equaton (4 n techncal effcency DEA wll ncorporate the allocatve effcency (relatve to the aggregated outputs nto the techncal effcency. Here, the ncorporated allocatve effcency only measures the effcency n allocatng resources among those outputs whch are aggregated usng Equaton (4. In other words, the estmated techncal effcences usng Equaton (5 nclude the allocatve effcences for the aggregated outputs n Equaton (5.3. As a result, the estmated allocatve effcency AE c, C, j, j +1,..., J only measures the effcency n allocatng resources among the outputs n Equaton (5.2. Here, t should be noted that AE c also ncludes the allocatve effcency for Equaton (5.3 as a whole output choce but not the ndvdual outputs aggregated n Equaton (5.3. Consequently, the whole allocatve effcency s decomposed nto two components. It s also easy to fnd the ndvdual allocatve effcency AE c, strana 72 2 / 2010 E + M EKONOMIE A MANAGEMENT c

3 C,j,j+1,...,J for the aggregated outputs n Equaton (5.3 by dvdng the estmated techncal effcency TE B by pure techncal effcency TE A. c The relatonshp of these allocatve effcency components and techncal effcency can be expressed as: EE (X AE =, C A TE EE (X =, C TE c, C, j, j +1,...,Y, J TE B (X, C, Y,...,Y c, j, j +1, J B TEA,Y =AE c, C, j, j +1,...,Y, J AE c, C, j, j +1,...,Y, J (9 and: B A AE c, C, j, j +1,...,Y, J = TE c, C, j, j +1,...,Y, J /TE (10 Above functons can be used n the specfc applcaton for measurng allocatng resources. For example, f we focus on the allocatve effcency component for some specfc outputs whch we are nterested n, we can aggregate all the other outputs and then calculate the allocatve effcency component whch we want. 2. An Emprcal Example for the Composton of Allocatve Effcency Accordng to the prevous secton, the allocatve effcency can be composed through functons (9 and (10 n the emprcal analyss. Ths paper provdes an emprcal example to show how to use the above method n allocatng resources for mult-product producton process. Normally, when measurng the allocatve effcency for mult-output producton, the tradtonal DEA only gves one allocatve effcency score ndcatng the effcency for whole producton. However, wth the ncreasng complexty of producton system, the whole AE can not provde enough or clear nformaton n decson-makng process for allocatng resources. For example, n the agrcultural sector, snce the relance of farm households on non-farm ncome s an ncreasng phenomenon n transformng economes, the effects of off-farm job on the farms and farm households become a growng area of research. These studes should manly concern the effcency n allocatng household EKONOMIKA A MANAGEMENT resources between on-farm work and off-farm job whle excludng the AE for on-farm outputs. However, the tradtonal allocatve effcency only gves the AE for whole outputs ncludng all farm outputs and off-farm ncome. In ths case, the method developed n ths paper can gve us the allocatve effcency component only for on-farm and off-farm choce. The logc routne for the above method can be explaned easly as follows: The ndvdual farmer frstly has to decde whether he wll take off-farm job or not and f yes, how much tme he wll nput n the off-farm job; Then, he wll allocate the household resources for on-farm nputs for dfferent farm products. Ths process, n fact, gves two stages n allocatng household resources. If we want to know the allocatve effcences n the frst stage and the second stage respectvely, the method n the prevous secton can satsfy us. As for the ndustral sector, ths paper wll show an emprcal example n detal wth a data set and the applcaton result. The data set comes from 20 bo-chemcal companes n Chna s Jangsu provnce n The data are collected by Jangsu Statstcs Bureau for small and medum sze enterprses (SMEs. The man objectve of ths applcaton s to act as an example for the method developed n ths paper. The data set ncludes two nputs and 3 outputs. The nputs are the captal nput measured by 100 thousand Yuan CNY (Chnese Yuan, and labour nput measured by the number of employees. The data of nputs and outputs are lsted n the Table 1. The chosen companes have the smlar outputs. The man outputs can be classfed nto two types: One s the pestcde; the other s the anmal pharmaceutcals. In addton, wth the mprovement of bo-chemcal technologes, the pestcde can be further dvded nto two categores: tradtonal chemcal pestcde and bo-pestcde. Therefore, the two-stage allocaton process ncludes: The frst, allocatng the resources between the pestcde and anmal pharmaceutcals; the second, allocatng the resources, whch have been decded to be allocated n the pestcde producton, between the chemcal pestcde and bo- -pestcde. Ths paper wll provde the tradtonal whole allocatve effcency, allocatve effcency component for allocatng all resources between the pestcde and anmal pharmaceutcals, and the allocatve effcency component for allocatng the frst-stage-decded resources between the E + M EKONOMIE A MANAGEMENT 2 / 2010 strana 73

4 Tab. 1: The data for the emprcal example Captal Labour Chemcal Pestcde Bo-pestcde Anmal Pharmaceutcals Data source: Jangsu Statstcs Bureau chemcal pestcde and bo-pestcde. The GAMS software (please refer to s used to estmate the emprcal example. The Table 2 depcts the whole and composed allocatve effcences, techncal effcences and economc effcences for 20 companes. In the Table 2, EE s the economc effcency estmated usng functon (2 or (6. Pure TE s the tradtonal techncal effcency estmated by functon (1 usng fully dsaggregated outputs. Whole AE s the tradtonal whole allocatve effcency estmated by dvdng EE by pure TE. TE- -aggregated s the techncal effcency estmated by functon (5 where some of outputs are aggregated. (Here, we aggregate the bo-pestcde and chemcal pestcde. AE for the frst-stage allocaton ndcates the allocatve effcency measurng the effcency n allocatng all resources between the pestcde producton and anmal pharmaceutcals producton. AE for the second-stage allocaton measures the effcency n allocatng the determned resources between the chemcal pestcde and bo-pestcde. The estmated economc effcences change from to 1 wth the geometrc mean at 0.629, ndcatng a relatvely low economc effcency as a whole. The pure techncal effcences range from to 1 wth a geometrc mean at The techncal effcences estmated from aggregated outputs range from to 1 wth the geometrc mean at Snce the second- -stage AE s ncorporated nto the aggregated techncal effcency, the aggregated TE s probably based from pure techncal effcency. For example, the pure TE of company 19 s 1 whle ts aggregated TE s whch s equal to the strana 74 2 / 2010 E + M EKONOMIE A MANAGEMENT

5 Tab. 2: The estmated effcences for the emprcal example EKONOMIKA A MANAGEMENT EE Pure TE Whole AE TEaggregated AE for the frst- -stage allocaton AE for the second- -stage allocaton Geomean Data source: Jangsu Statstcs Bureau product of pure TE and the second-stage AE. But, for the company 1, 3, 6, 7, and 17, because the second-stage AEs of them are 1, the pure TEs and the aggregated TEs are the same. The frst-stage allocatve effcences range from 0.43 to 1 wth the mean at The second-stage allocatve effcences range from to 1 wth the mean at It s clear that there are dfferent effcences n allocatng all resources between the pestcde producton and anmal pharmaceutcals producton and allocatng determned resources between the chemcal pestcde producton and bo-pestcde producton for each company. Except for the company 1, 7, and 17, all other companes have the dfferent values for the whole AE, the frst-stage AE, and the second-stage AE. Ths further suggests that decomposng the whole allocatve effcency for mult-product system s necessary. There are two nterestng propostons of decomposed AE components and whole AE as follows: Proposton 1, f whole AE s 1 (full effcent, then all the decomposed AE components must be 1. Proposton 2, whole AE must be lower or equal to each decomposed AE component. The proof of above propostons s smple. Accordng to the functon (9, whole AE s the product of decomposed AE components. Snce E + M EKONOMIE A MANAGEMENT 2 / 2010 strana 75

6 the AE components are all < or = to 1, the product of these AE components must be lower or equal to any of these AE components whch are the multplcand or multpler. Therefore, f the whole AE s 1, then ts AE components must be 1, such as company 1, 7, and 17. If the whole AE and ts components are all lower than 1, then the whole AE must be lower than ts multplcand and multpler snce they are all lower than 1. If the whole AE as well as one of ts components are lower than 1 whle the other AE component s equal to 1, the whole AE must be equal to the non-one component, such as company 3, 6, 11, and 16. Conclusons Ths study manly concerns the method to decomposng allocatve effcency usng the nfluence of data aggregaton on DEA measurement. Although most of other papers before ths study focused on the estmaton bas n techncal effcency, we analyze the relatonshp between data aggregaton and allocatve effcency and fnally gve a method for decomposng allocatve effcency for mult-output producton system. An emprcal example to show how to use the method n decomposng allocatve effcency for mult-product producton system s also presented n the paper. In addton, some emprcal stuatons (n both the agrcultural sector and the ndustral sector are provded to tell us when we should use ths method. Although ths paper only provdes a two-stage allocatng process, the method developed here can be easly extended for the three-stage allocatng process or even more complcated producton system. The method n ths paper and ts applcaton can provde useful nformaton for researchers who concern the decson-makng process for mult-product producton systems. References [1] CHARNES, A., W.W. COOPER and E. RHODES. Measurng the Effcency of Decson Makng Unts. European Journal of Operatons Research, 1978, Vol. 46, pp ISSN [2] CHARNES, A., W. W. COOPER, A. Y. LEWIN and L. M. SEIFORD. Data Envelopment Analyss: Theory, Methodology, and Applcatons. Boston/ Dordrecht/London: Kluwer Academc Publshers, ISBN [3] FARRELL, M. J. The Measurement of Producton Effcency. Journal of the Royal Statstcal Socety Seres A, 1957, Vol. 120, pp ISSN [4] FARE, R. and ZELENYUK, V. Input aggregaton and techncal effcency. Appled Economcs Letters, 2002, Vol. 9, pp ISSN [5] FARE, R., GROSSKOPF, S. and ZELENYUK, V. Aggregaton bas and ts bounds n measurng techncal effcency. Appled Economcs Letters, 2004, Vol. 11, pp ISSN [6] GLEASON, J. M. and BARNUM, D. T. Techncal effcency bas caused by ntra-nput aggregaton n data envelopment analyss. Appled Economcs Letters, 2005, Vol. 12, pp ISSN [7] GLEASON, J. M. and BARNUM, D. T. Techncal effcency bas n data envelopment analyss caused by ntra-output aggregaton. Appled Economcs Letters, 2007, Vol. 14, pp ISSN [8] GLEASON, J. M. and BARNUM, D. T. Measurng effcency n allocatng nputs among outputs wth DEA. Appled Economcs Letters, 2006, Vol. 13, pp ISSN [9] NATARAJAN RAM; BANKER RAJIV D. and CHANG HSIHUI. Estmatng DEA techncal and allocatve neffcency usng aggregate cost or revenue data. Journal of Productvty Analyss, 2007, Vol. 27, pp ISSN X. Tao Zhang Natonal Unversty of Ireland, Galway Department of Economcs (School of Publc Admnstraton Macao Polytechnc Insttute Rua Lus Gonzaga Gomes Macao Doručeno redakc: Recenzováno: ; Schváleno k publkování: strana 76 2 / 2010 E + M EKONOMIE A MANAGEMENT

7 ABSTRACT DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYS- TEMS Tao Zhang Data envelopment analyss (DEA, the non-parametrc approach to measurng effcency, was wdely used n the lterature as a lnear programmng model. Snce the DEA s non-parametrc lnear program model, the estmated effcency mght be based f there s data aggregaton n DEA. It s proposed that both the nter-nput aggregaton and nter-output aggregaton wll make the estmated techncal effcency based downwardly. Followng some dscussons on the techncal effcency bas caused by data aggregaton n data envelopment analyss, ths study presents the up-ward bas n the allocatve effcency caused by nter-output aggregaton. However, htherto, we have not observed any studes on consderng how to utlze ths character of DEA n methodology extensons for mult-output producton system. Therefore, ths paper orgnally proposes that the tradtonal allocatve effcency can be decomposed for mult-product system. Then, the method to obtan decomposed allocatve effcency components s provded. In fact, the meanng of results from ths paper s outsde the DEA method, and the generalty of the fndngs n ths paper can provde useful nformaton for researchers who concern the decson-makng process n allocatng resources for mult-product producton system. Fnally, an emprcal example to show how to use the method n decomposng allocatve effcency for mult-product producton system s also presented n the paper. In addton, some emprcal stuatons (n both the agrcultural sector and the ndustral sector are provded to tell us when we should use ths method. Although ths paper only provdes a two-stage allocatng process, the method developed here can be easly extended for the three-stage allocatng process or even more complcated producton system. Key Words: mult-product producton systems, aggregate, DEA, techncal effcency, decomposng, allocatve effcency, economc effcency, component. JEL Classfcaton: C61, C67. E + M EKONOMIE A MANAGEMENT 2 / 2010 strana 77

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