A Note on Equivalences in Measuring Returns to Scale

Transcription

1 Internatnal Jurnal f Busness and Ecnmcs, 2013, Vl. 12, N. 1, A Nte n Equvalences n Measurng Returns t Scale Valentn Zelenuk Schl f Ecnmcs and Centre fr Effcenc and Prductvt Analss, The Unverst f Queensland, Australa Ke wrds: scale elastct; ecnmes f scale; dstance functns JEL classfcatn: D24 In ths nte we derve necessar and suffcent cndtn fr equvalence f nput rented and utput rented scale elastct measures fr mult-utput, multnput technlges and prvde a Lagrange multpler (r shadw prce) nterpretatn f these scale elastct measures. Fllwng Färe et al. (1986), recall that ne can measure lcal returns t scale va the utput rented measure f scale elastct, defned as: e ln ( x, ), such that D ( x, ) 1. (1) ln 1, 1 r, alternatvel, va the nput rented measure f scale elastct, defned as: ln e (, x), such that D (, x) 1. (2) ln 1, 1 where x and are vectrs f nputs and utputs, respectvel, whle D (, x ) and D ( x, ) are Shephard s (1957, 1970) nput and utput dstance functns, respectvel. T save space here, we refer t Färe and Prmnt (1995) fr the ntatn, defntns and prpertes f the functns nvlved. 1, 2 Meanwhle, nte that bth frmulas are trng t measure scale elastct b lkng at the relatnshp between equ-prprtnal changes n all nputs wth equ-prprtnal changes n all utputs, but the d t b usng dfferent characterzatns f technlg. Thus, several natural questns arse: What s a relatnshp between these tw alternatve measures f scale elastct? Are (1) and (2) equvalent? Alwas? Under what cndtns? Färe et al. (1986) were the frst t prvde a versn f ths result wth a prf. In the therem belw, we revst ths result and prvde a Lagrange multpler nterpretatn f t. Crrespndence t: Schl f Ecnmcs and Centre fr Effcenc and Prductvt Analss, The Unverst f Queensland, Cln Clark Buldng (39), St Luca, Brsbane Qld 4072, Australa. E-mal: v.zelenuk@uq.edu.au.

2 86 Internatnal Jurnal f Busness and Ecnmcs Therem. Gven defntns abve, standard regulart cndtns f prductn ther and assumng that, n a neghbrhd f a pnt f nterest, the dstance functns D (, x ), D (, x ), and D ( x, ), D ( x, ) are cntnuusl dfferentable wth respect t each element f ( x,, ) we have: e ( x, ) 1/ e (, x), (3) f and nl f D (, x) 0 and D ( x, ) x 0. (4) x Prf. Due t the mplct functn therem, we can rewrte (1) and (2) n a mre cmpact frm: e x D x x and x e x D x. (5) T prve necesst, assume that e ( x, ) 1/ e (, x), then (5) mples that: D ( x, ) x = 1/ D (, x), x whch can hld nl f (4) s true. T prve suffcenc, assume (4) s true. Further, nte that gven standard regulart cndtns, we can rewrte the utput dstance functn as fllws: D ( x, ) nf{ 0: D ( /, x) 1}. (6) The resultng Lagrangan functn fr ths ptmzatn prblem can be wrtten as: L(, x, ) ( D ( /, x) 1), (7) and s the sstem f equatns defned b the frst rder cndtn fr (7) s gven b: L 1 D ( /, x) ( 1/ ( ) ) 0, (8) 2 / and L D x, (9) ( /, ) 1 0 where ( x, ), ( x, ) are the ptmal slutns t (7). Thus, frm (8), we have:

3 Valentn Zelenuk 87 1/[ D ( /, x) / ( ) ]. (10) 2 / And, snce s a slutn t ptmzatn prblem (7), ts value must be equal t unt, and s: 1/ D (, x). (11) The rght-hand sde n (11) s nn-zer due t (4), s 0,.e., the cnstrant n (6) s bndng at the ptmal values,.e., D (, x) 1. Thus, due t (5), equatn (11) reduces t: 1/ e x. (12) On the ther hand, nte that the envelpe therem appled t (7) tells us that: D ( x, ) L(, x, ) D ( /, x). (13) x x x Pst-multplng bth sdes f (13) b x x x and usng agan that 1, we get: D ( x, ) x D (, x) x. (14) The left-hand sde n (14) s assumed t be nn-zer accrdng t (4) and equals e x accrdng t (5) snce D ( x, ) 1 at the ptmum. Because D (, x ) s hmgeneus f degree ne n x, Euler s rule tells us that xd(, x) x D (, x), whch s equal t unt at the ptmum. Thus, (14) becmes: e x. (15) Cmbnng ths last result n (14) wth (12), we get the desred expressn: 3 e ( x, ) 1/ e (, x). Intutvel, the therem abve tells us that the tw scale elastct frmulas measure the same prpert f technlg equvalentl and we can get ne frm the ther b just takng the recprcal, prvded that certan cndtn s satsfed. Imprtantl, ths cndtn s the necessar and suffcent cndtn fr ths equvalence result t hld and t states that, at the pnts where elastct s measured, the gradents f the nput and utput dstance functns shall nt be rthgnal t the utput and nput vectrs, respectvel. In turn, ths techncal requrement ensures nt runnng nt a stuatn f dvsn b zer (as can be seen amng the steps n the prf f the therem). B ths cndtn, we ensure that at a gven pnt f measurement, nether measure f scale elastct expldes r degenerates t zer, and then (and nl then) the gve equvalent nfrmatn abut the scale f technlg at that pnt. On the ther hand, the necessar and suffcent cndtn (4) als has ecnmc meanng: t requres that an ncrease n all nputs b the same

4 88 Internatnal Jurnal f Busness and Ecnmcs prprtn shall cause sme prprtnal nn-zer change n all utputs, and, naturall, t shuld nt be an nfnte ncrease. In turn, ths cndtn als mples that, fr the equvalence f the tw measures, the technlg must be s that D( x, ) 1 D(, x) 1 at a pnt where the elastct s measured, as s als seen frm the steps f the prf f the therem. It s als wrth ntng that a sde frut f the prf f the therem abve s an nterestng ecnmc ntutn f the scale elastct measures. Specfcall, (15) tells us that the utput rented scale elastct has Lagrange multpler meanng t s the shadw prce f relaxng the cnstrant, n the ptmzatn prblem (6). A smlar argument can be made abut the nput rented scale elastct measure beng the Lagrange multpler r the shadw prce f relaxng the cnstrant n ptmzatn f the nput dstance functn defned n terms f the cnstrant n the utput dstance functn, frmulated analgus t (6). Such Lagrange multpler nterpretatns f scale elastct measures als gve a smple wa f btanng and analzng a measure f scale elastct, wthut takng dervatves as n (1), (2), and (5). Ntes 1. See als Hanch (1975), Panzar and Wllg (1977), Banker (1984), Banker et al. (1984), Färe et al. (1986), Banker and Thrall (1992), Førsund (1996), Glan and Yu (1997), Fukuama (2000, 2003), Førsund and Hjalmarssn (2004), Krvnzhk et al. (2004), Hadjcstas and Steru (2006, 2010), and Pdnvsk et al. (2009), t mentn just a few. 2. It s wrth ntng that defntns n (1) and (2) relate the vectr f nputs t the vectr f utputs mplctl, and s we rel n the mplct functn therem and assumptns requred b t. 3. Nte that a smlar prf can be establshed b startng frm the defntn f D (, x) n (6). References Banker, R. D., (1984), Estmatng Mst Prductve Scale Sze Usng Data Envelpment Analss, Eurpean Jurnal f Operatnal Research, 17(1), Banker, R. D., A. Charnes, and W. W. Cper, (1984), Sme Mdels fr Estmatng Techncal and Scale Ineffcences n Data Envelpment Analss, Management Scence, 30(9), Banker, R. D. and R. M. Thrall, (1992), Estmatn f Returns t Scale Usng Data Envelpment Analss, Eurpean Jurnal f Operatnal Research, 62(1), Färe, R., S. Grsskpf, and C. A. K. Lvell, (1986), Scale Ecnmes and Dualt, Jurnal f Ecnmcs, 46(2), Färe, R. and D. Prmnt, (1995), Mult-Output Prductn and Dualt: Ther and Applcatns, Bstn, MA: Kluwer Academc Publshers. Hadjcstas, P. and A. C. Steru, (2006), One-Sded Elastctes and Techncal Effcenc n Mult-Output Prductn: A Theretcal Framewrk, Eurpean Jurnal f Operatnal Research, 168(2),

5 Valentn Zelenuk 89 Hadjcstas, P. and A. C. Steru, (2010), Dfferent Orders f One-Sded Elastctes n Mult-Output Prductn, Jurnal f Prductvt Analss, 33(2), Hanch, G., (1975), The Elastct f Scale and the Shape f Average Csts, Amercan Ecnmc Revew, 65(3), Krvnzhk, V. E., O. B. Utkn, A. V. Vldn, I. A. Sabln, and M. Patrn, (2004), Cnstructns f Ecnmc Functns and Calculatns f Margnal Rates n DEA Usng Parametrc Optmzatn Methds, Jurnal f the Operatnal Research Scet, 55(10), Panzar, J. C. and R. D. Wllg, (1977), Ecnmes f Scale n Mult-Output Prductn, The Quarterl Jurnal f Ecnmcs, 91(3), Shephard, R. W., (1953), Cst and Prductn Functns, Prncetn, NJ: Prncetn Unverst Press. Shephard, R. W., (1970), Ther f Cst and Prductn Functns, Prncetn, NJ: Prncetn Unverst Press. Pdnvsk, V. V., F. R. Førsund, and V. E. Krvnzhk, (2009), A Smple Dervatn f Scale Elastct n Data Envelpment Analss, Eurpean Jurnal f Operatnal Research, 197(1),