Cost, revenue and profit efficiency measurement in DEA: A directional distance function approach

Transcription

1 Cst, evenue and pft effcency measuement n DEA: A dectnal dstance functn appach Besh K. Sah a, Mahmd Mehdlzad b, Kau Tne c a Xave Insttute f Management, Bhubaneswa 75 03, Inda b Depatment f Mathematcs, Cllege f Scences, Shaz Unvesty, Shaz, Ian c Natnal Gaduate Insttute f Plcy Studes, Rppng, Mnat-ku, Tky , Japan Abstact Estmatn f effcency f fms n a nn-cmpettve maket chaactezed by hetegeneus nputs and utputs alng wth the vayng pces s questnable when fact-based technlgy sets ae used n data envelpment analyss (DEA). In ths scena, a value-based technlgy becmes an apppate efeence technlgy aganst whch effcency can be assessed. In ths cntbutn, the value-based mdels f Tne (2002) ae extended n a dectnal DEA set up t develp new dectnal cst- and evenue-based measues f effcency, whch ae then decmpsed nt the espectve dectnal value-based techncal and allcatve effcences. These new dectnal value-based measues ae me geneal, and nclude the exstng value-based measues as specal cases. These measues satsfy seveal desable ppetes f an deal effcency measue. These new measues ae advantageus ve the exstng nes n tems f ) the ablty t satsfy the mst mptant ppety f tanslatn nvaance; 2) chces ve the use f sutable dectn vects n handlng negatve data; and 3) flexblty n pvdng the decsn makes wth the ptn f specfyng pefeable dectn vects t ncpate the pefeences. Fnally, unde the cndtn f n p unt pce nfmatn, a dectnal value-based measue f pft neffcency s develped f fms whse undelyng bectves ae pft maxmzatn. F an llustatve empcal applcatn, u new measues ae appled t a eal-lfe data set f 50 US banks t daw nfeences abut the pductn cespndence f bankng ndusty. Keywds: Data envelpment analyss; Cst effcency; Revenue effcency; Pft effcency; Tanslatn nvaance; Dectnal dstance functn Cespndng auth. E-mal: m.mehdlzad@gmal.cm (M. Mehdlzad)

2 2. Intductn Snce ts nceptn by Chanes, Cpe, and Rhdes (978), data envelpment analyss (DEA) has been ganng nceasng ppulaty n the lteatue as a cnvenent tl f estmatng the effcences f fms chaactezed by mult-nput-mult-utput pductn technlges. The nnpaametc methdlgy f DEA has been used f measung and analyzng a numbe f effcency cncepts, ncludng cst effcency (CE) and evenue effcency (RE). One f the mst mptant aspects n appled pductn analyss f fms s the measuement f the cst and evenue effcences (Faell, 957), n whch we cncentate wthn the famewk f the dectnal dstance functn (DDF) by Chambes, Chung, and Fäe (996, 998). F the fst tme, Fäe, Gsskpf, and Lvell (985) develped pcedues f the empcal mplementatns f the CE and RE measues n DEA. Snce then, the aspect f measung cst and evenue effcences has been expled n many studes. See, e.g., Ray and Km (995), Cpe, Thmpsn, and Thall (996), Schaffnt, Rsen, and Paad (997), Sueysh (997), Pug-Juny (2000), Kusmanen and Pst (200, 2003), Tne (2002), Tne and Sah (2005), Tne and Sah (2006),Manadaks and Thanassuls (2004), Sengupta and Sah (2006), Jahanshahl, Sleman- Damaneh, and Mstafaee (2008), Mstafaee and Salgh (200), Sah, Kestens, and Tne (202), amng thes. Bth the CE- and RE-based DEA mdels develped by Fäe, Gsskpf, and Lvell (985) eque nt nly nput and utput quantty data but als the pces at each fm. These mdels can be f lmted use n actual applcatns when maket mpefectns exst (Camanh & Dysn, 2008; Pak & Ch, 20; Sah & Tne, 203). Ths s because these mdels ae based n a numbe f smplfyng assumptns. Fst, fact nputs ae hmgeneus acss fms; the pces ae exgenusly gven, and ae measued and knwn wth full cetanty. In eal-lfe applcatns, hweve, when pductn s expanded, fms expeence changes n the ganzatn f the pcesses n the chaactestcs f the nputs that ae ecnmcally me attactve than the eplcated altenatves f thse aleady n use. Theefe, the technques and nputs used at hghe scale ae vey dffeent fm thse used at lwe scale. Hence, fact nputs ae thus hetegeneus, and as a esult, the pces may vay acss fms. Snce nputs vay n the qualty, the cnstuctn f fact-based technlgy set n DEA becmes pblematc. Futhe, nput pces ae nt exgenus, but they vay n accdance wth the actns by fms (Chambeln, 933; Engel & Rges, 996; Rbnsn, 933). Als, fms ften face ex ante pce uncetanty whle makng the pductn decsns (McCall, 967; Sandm, 97; Camanh &

3 3 Dysn, 2005). Whle csts and evenues ae all well measued, physcal nput and utput quanttes and the pces ae ften nt. Ecnmc they suggests that fms enyng sme degee f mnply pwe shuld chage dffeent pces f thee s hetegenety n the pductvty f the nputs. Ths s empcally vald snce mst fms ae bseved facng an upwad-slpng supply cuve n the puchase decsns. Ths bsevatn als suggests that the assumptn f facng cmmn unt pces by fms,.e., the law f ne pce, whch has lng been mantaned as a necessay and suffcent cndtn f Paet effcency n cmpettve makets (Kusmanen et al., 2006), s nt at all ustfed n evealng the ppe CE behav f fms. Secnd, the CE measue by Fäe, Gsskpf, and Lvell (985) can be f lmted value n actual applcatns even when (physcal) nputs ae hmgeneus. Ths s because, as pnted ut by Camanh and Dysn (2008), the CE measue eflects nly nput neffcences (techncal neffcency and/ allcatve neffcency) but nt maket (pce) neffcences (devatn fm fully cmpettve settng leadng t pce dffeences between fms). Theefe, as a emedy, they suggested a cmpehensve CE measue that accunts f bth nputs and maket neffcences. Thd, n many eal-lfe applcatns the pce data n nputs and utputs ae synthetcally cnstucted, and hence, epesent aveage, athe than magnal pces. Snce manages make decsns at the magn, analyss f effcency usng aveage pces can dstt measues f allcatve effcency (Fukuyama & Webe, 2008). Theefe, when nputs/utputs ae hetegeneus, n de t accunt f stuatns whee the nput/utput pces vay between fms as a esult f negtatns t eflect the qualtatve dffeences n the esuces/pducts, the altenatve CE/RE mdel f Tne (2002) shuld be fllwed by settng up technlgy n a cst-utput/nput-evenue space. Usng the dectnal DEA stuctue, Fukuyama and Webe (2004) and Fäe and Gsskpf (2006) extended ths altenatve value-based CE mdel t develp the dectnal nput-cst dstance functn (DICDF), whch, n tun, pvdes a dectnal measue f value-based techncal neffcency. Usng the DICDF, we develp tw new dectnal cst- and evenue-based measues f effcency,.e., and DRE, whch all satsfy the ppety f tanslatn nvaance. Ths ppety s cnsdeed mst mptant f any effcency measue (Al & Sefd, 990; Cpe et al., 999; Lvell & Past, 995; Past, 996). Futheme, we develp tw new dectnal nput- and utput-ented value-based measues f techncal effcency (TE). We then decmpse u new and DRE measues nt the espectve dectnal value-based TE and allcatve effcency (AE) cmpnents. These new and DRE measues ae me geneal, and nclude the Tne (2002) s CE and RE measues as specal cases. Ou ppsed new measues satsfy seveal

4 4 desable ppetes, such as unt nvaance (Cpe, et al., 999; Lvell & Past, 995) and stng mntncty (Blackby & Russell, 999; Cpe, et al., 999). Nte that u value-based measue s develped based n the assumptn that physcal utputs ae hmgenus, but nt physcal nputs. Smlaly, the value-based DRE measue s develped based n the assumptn that physcal nputs ae hmgenus, but nt physcal utputs. Hweve, when bth physcal nputs and utputs ae hetegeneus, u and DRE measues cannt be appled t measue the espectve cst and evenue effcences. T deal wth ths stuatn, we develp a dectnal value-based measue f pft (n)effcency that s based n a technlgy set cmpsng f all feasble nput-cst (nput-spendng) and utput-evenue (utputeanngs) by bseved fms. Ths measue wll be me meanngful f a fm when ts undelyng behaval bectve s pft maxmzatn. Whle nne f the exstng CE, RE, and AE measues s tanslatn-nvaant, u ppsed new measues satsfy ths ppety that enables them t effectvely deal wth negatve data. These new measues ae flexble n the sense that they pvde the decsn makes wth the ptn f specfyng pefeable dectn vects t ncpate the decsn-makng pefeences. Specally, they can deal wth value udgments (pefeence) as t whch specfc nput-cst t educe whch specfc utput-eanngs t ncease by a fm t mpve ts veall pefmance. Thugh the cntbutn f ths pape s manly theetcal, t demnstate ts eady applcablty n empcal wk, we cnduct an llustatve empcal analyss based n a data set f 50 US banks. The emande f the pape unflds as fllws. Sectn 2 gves a bef evew f methds amed at measung CE and RE. Sectn 3 epesents the man cntbutn f the pape, whee we pesent u new dectnal CE, RE and pft (n)effcency measues and then dscuss the ppetes. Sectn 4 demnstates the eady applcablty f u ppsed measues n a eal-lfe data set f 50 US banks f the yea 996. Fnally, Sectn 5 cncludes wth emaks. 2. Pelmnaes Thughut ths pape, we assume t deal wth n bseved decsn makng unts (DMUs); each uses m nputs t pduce s utputs. Let x (,..., ) T m = x xm R 0 and y (,..., ) T s = y ys R 0 be, espectvely, the nput and utput vects f DMU, J {,..., n} =. Let c (,..., ) T m = c cm R 0 and p (,..., ) T s = p ps R 0 be, espectvely, the nn-negatve pce vects f nput and utput f DMU. The supescpt T stands f a vect tanspse. Let the nput-

5 5 spendng and utput-eanngs f DMU be x and y espectvely, whee x = c x and y = p y. Hee, dentes the cmpnent-wse multplcatn f vects. We futhe assume as the ndex f DMU unde evaluatn. We nw defne fu pductn technlges dependng upn data avalablty. If bth physcal nput and utput data ae bseved, and ae hmgeneus, we epesent technlgy as m + s m s {(, ) can pduce } T = x y R x R y R. () x, y If physcal utputs ae bseved (and ae hmgeneus) but nt physcal nputs, then we can epesent the technlgy by cnsdeng all feasble nput-spendng and physcal utput vects as m + s m s {(, ) can pduce } T = x y R x R y R. (2) x, y If physcal nputs ae bseved (and ae hmgeneus) but nt physcal utputs, then we can epesent the technlgy by cnsdeng all feasble physcal nput and utput-eanngs vects as m + s m s {(, ) can pduce } T = x y R x R y R. (3) x, y Fnally, f bth physcal nputs and physcal utputs ae nt bseved, we can epesent the technlgy by cnsdeng all feasble nput-spendng and utput-eanngs vects as m + s m s {(, ) > can pduce } T = x y R x R y R. (4) x, y If each f these technlges satsfes ppetes such as n fee lunch, fee (stng) dspsablty, clsue, cnvexty and bundedness, fllwng Banke, Chanes, and Cpe (984), the nn-paametc DEA epesentatns f these technlges can, espectvely, be epesented unde vaable etuns t scale (VRS) as DEA, (, ) m + T s x y = x y R 0 x λ x, y λ y, λ =, λ 0,, (5) J J J DEA, (, ) m + T s x y = x y R 0 x λ x, y λ y, λ =, λ 0,, (6) J J J DEA, (, ) m + T s x y = x y R 0 x λ x, y λ y, λ =, λ 0,, (7) J J J DEA, (, ) m + T s x y = x y R 0 x λ x, y λ y, λ =, λ 0,. (8) J J J The assumptn f VRS s mantaned n u DEA technlgy cnstucts ((5)-(8)) f thee easns: () assumng that sme data can be negatve, ne may be nt able t defne an effcent fnte passng thugh the gn, as s assumed unde cnstant etuns t scale (CRS). Thus, an

6 6 assumptn f CRS beaks dwn f negatve data (Slva Ptela and Thanassuls, 200), (2) the assumptn f CRS s nt cnsstent wth sme dectnal DEA mdels based n specfc dectn vects, whch all dectly deal wth bth pstve and negatve data (Sah et al., 20), and (3) the eal-lfe stuatns d nt always exhbt CRS. 2. Cst effcency measuement If the undelyng behaval bectve f a DMU s cst mnmzatn, ts CE measue,, can be btaned as the ptmal bectve value f the fllwng lnea pgammng (LP) pblem (Fäe et al., 985): = Mn m λ, x C = s. t. λ x x, =,..., m, λ y y, =,..., s, λ =, λ 0, J, c x (9) whee C m = c x = s the bseved cst, and C = s the mnmum cst, f DMU. m c x = Hee, s defned as the at f the mnmum cst t the bseved cst, Hence, 0 <. DEA The mdel (9) that s based n T x, y n (5) s cnstucted n the assumptn that nput pces ae avalable, and physcal nputs and utputs ae bseved. Hweve, as agued eale, when (physcal) nputs ae hetegeneus, n de t accunt f stuatns whee nput pces vay between fms as a esult f negtatns t eflect the qualtatve dffeences n the esuces, the DEA altenatve value-based CE mdel f Tne (2002) that s based n T x, y n (6) shuld be used. Ths altenatve CE measue, CE, can be epesented as CE = Mn λ, x C m = s. t. λ x x, =,..., m, λ y y, =,..., s, λ =, λ 0, J, x (0) whee C m = x = s the bseved cst f DMU.

7 7 IVTE In addtn, the nput-ented value-based TE measue, ρ, can be set up as ρ IVTE = Mn λ, θ θ s. t. λ x θ x, =,..., m, λ y y, =,..., s, λ =, λ 0, J. () Obvusly, ne can have the fllwng elatnshp: CE IVTE ρ. (2) Usng (2) ne can defne the nput-ented AE (pce effcency) as α IAE CE =. (3) IVTE ρ CE Fm (3), the CE measue can be expessed as the pduct f the (nput-ented) valuebased AE and TE,.e., = α ρ. (4) CE IAE IVTE Thus, t s clea that a DMU wll be cst-effcent (.e., CE =) f t s bth value-based CE techncal- and allcatve-effcent. If <, t ncus hghe csts due t nt beng able t use, the mst effcent technlgy (.e., techncal neffcency) and/ the cst-mnmzng nput mx (.e., allcatve neffcency). Fukuyama and Webe (2004) pesented an altenatve appach f estmatng the value-based techncal neffcency usng the cncept f DDF f Chambes, Chung, and Fäe (996, 998), whch genealzes the Shephad s dstance functns (Shephad, 953, 970). The DDF seeks t smultaneusly mnmze nputs and maxmze utputs f a gven DMU usng a pe-specfed dectn vect, g = ( g, g + ). Applyng the nput-ented DDF ( g = ( g, 0) ) elatve t T, n (6), the dectnal nput-cst dstance functn (DICDF) can be set up as DEA x y β = Max λ, β β s. t. λ x x β g, =,..., m, λ y y, =,..., s, λ =, λ 0, J. (5)

8 8 β can be ntepeted as the dectnal value-based measue f techncal neffcency. Futheme, the Nelvan cst neffcency measue s defned as m m x x = = m g =, (6) whee x s the ptmal slutn vect f the mdel (0). F futhe detals cncenng ths cst neffcency measue, nteested eades may efe t Fukuyama and Webe (2004). 2.2 Revenue effcency measuement If the undelyng behaval bectve f a DMU s evenue maxmzatn, ts RE measue, η, can be btaned as the nvese f the ptmal bectve value f the fllwng LP pblem: = Max η λ, y R s = s. t. λ x x, =,..., m, λ y y, =,..., s, λ =, λ 0, J, p y (7) whee R s = p y = s the bseved evenue, and R = s the maxmum evenue f s p y = DMU. Hee, η s defned as the at f the bseved evenue t the maxmum evenue. Hence, 0 < η. DEA The mdel (7) that s based n T x, y n (5) s cnstucted n the assumptn that utput pces ae avalable, and physcal nputs and utputs ae bseved. Hweve, as agued n Sah and Tne (203), when (physcal) utputs ae hetegeneus, n de t accunt f stuatns whee utput pces vay between fms t eflect the qualtatve dffeences n the pducts, the altenatve value-based RE mdel f Tne (2002) that s based n T, RE value-based RE measue, η can be set up as DEA x y n (7) shuld be used. Ths altenatve

9 9 = Max RE η λ, y R s = s. t. λ x x, =,..., m, λ y y, =,..., s, λ =, λ 0, J, y (8) whee R s = y = s the bseved evenue f DMU. whee RE Smla t the elatnshp n (4), η can be decmpsed as OAE α and η = α ρ, RE OAE OVTE OVTE ρ epesent, espectvely, the utput-ented valued-based AE and TE. 3. Ou ppsed appach 3. Dectnal nput-ented value-based TE, nput-ented AE, and CE measues In de t develp a dectnal value-based TE measue, cnsde the mdel (5) whse ptmal slutn vect s (, ) β λ. Snce λ =, ne can deduce fm the th nput-cst cnstant the fllwng elatn: λ x x β g β x λ x g { } { } x Max x x Mn x g g g Max x Mn x g { } { } (9) T guaantee that β, the dectn vect g satsfes the fllwng pmay cndtn: x Max =,..., m g { } Mn x. (20) F example, cnsde the fllwng dectn vects, whch all fulfll the pmay cndtn (20). g = x, g + = 0, =,..., m, =,..., s (2)

10 0 g { } = Max x, g + = 0, =,..., m, =,..., s (22) { } g = x Mn x, g + = 0, =,..., m, =,..., s { } { } g = Max x Mn x, g + = 0, =,..., m, =,..., s 2 (23) (24) Hweve, the fxed dectn vect des nt satsfy the pmay cndtn (20). g =, g + = 0, =,..., m, =,..., s (25) It s thus clea hw the ptmal slutn t (5) s dependent n the dectn vect g. Because ths explct dependence was nt cnsdeed n the DICDF appach, the slutns d nt necessaly pvde a dect measue f the value-based TE. F any pe-specfed dectn vect g satsfyng the pmay cndtn (20), ne can defne the dectnal nput-ented value-based TE (DIVTE) measue as DIVTE ρ β =. (26) DIVTE ρ genealzes the nput-ented value-based TE measue. Specfcally, f ne chses the DIVTE IVTE dectn vect (2), ρ tuns t ρ. DIVTE Based n Theems 2 and 4 n Fukuyama and Webe (2004), ρ s weakly mntnc wth espect t vaatn n nput-csts, and s tanslatn-nvaant wth espect t utputs. Hweve, unlke what was clamed n Theem 2 n Fukuyama and Webe (2004), DIVTE ρ s nt always tanslatn-nvaant wth espect t nput-csts f the nn-fxed (dependent n the nput-utput data) dectn vects, because any change n the nput-utput data may affect such dectn vects. F nstance, t can be seen that the mdel (5) s nt tanslatn-nvaant f the dectn vects (2) and (22), but t s f the dectn vects (23) and (24). Nw we have the fllwng theem n tanslatn nvaance. DIVTE Theem 3.. (Tanslatn nvaance). ρ s tanslatn-nvaant wth espect t nput-csts f the tanslatn f nput-csts data has n effect n the pe-assgned dectn vect g. als tanslatn-nvaant wth espect t utputs. DIVTE ρ s Pf. Let xɶ = x + τ ( =,..., m ) be the tanslated th nput-cst f DMU, J. Because these tanslatns have n effect n g, the cnstants f mdel (5) based n the tanslated data can be epesented as

11 ( x + ) x + g, =,..., m, λ τ τ β λ y y, =,..., s, (27) Snce λ =, we elmnate τ n bth sdes f the nput-cst cnstants, and btan the same cnstants as n (5). Theefe, the slutn set emans unchanged, thus ndcatng the ppety f tanslatn nvaance. Ths ppety wth espect t utputs can als be pved n an analgus manne. We nw have the fllwng theem that pvdes a cndtn f the unt ndependency f the DIVTE effcency measue ρ n (26). Theem If the dectn vect g s chsen n such a way that each cmpnent g ( DIVTE =,..., m ) has the same unts f measuement as the th nput-cst, then ρ wll be untnvaant. Pf. Let us escale the th nput-cst and the th utput by the scalas τ > 0 and w > 0, espectvely. Snce bth g and x have the same unts f measuement, the cespndng cmpnent f the new dectn vect s τ g. Hence, we have By elmnatng τ and ( ) ( ) ( ) λ τ x τ x β τ g, =,..., m, ( ) ( ) λ w y w y, =,..., s. w n bth sdes f these expessns, ne can btan the same cnstants as n (5). Thus, the slutn set emans unchanged, whch cmpletes the pf. F example, the dectn vect (2) pvdes the ppety f unt nvaance. DEA In de t evaluate the CE sce, we develp, based n T x, y n (6), a new dectnal CE () measue as = Mn λ, β m = s. t. λ x x β g, =,..., m, λ y y, =,..., s, λ =, λ 0, J, g G β (28)

12 2 whee the dectn vect g satsfes the pmay cndtn (20), and epesents the ate f mpvement n the th nput-cst f DMU. G m = g. Hee, β = Because the dectn vect g satsfes (20), t can be easly establshed that β, =,..., m, f each ptmal slutn (, ) equement cndtn,.e., 0 <. λ β t (28). Ths mples that Nte that s a genealzatn f s cnsdeed, then the mdel (28) can be tansfmed as By lettng x ( β ) Mn λ, β C CE satsfes the effcency. Specfcally, f x > 0 and the dectn vect (2) m ( β ) = ( β ) s. t. λ x x, =,..., m, λ y y, =,..., s, λ =, λ 0, J. x (29) = x, the mdel (29) can be easly tansfmed t the mdel (0). In addtn, f nput pces ae cmmn acss DMUs, then can be easly deved fm. The ptmal slutn t (28) f DMU yelds the fllwng expessn: λ x = x β g, =,..., m, λ y y, =,..., s. Nte that all the nput-cst cnstants n (28) ae actve (bndng) at the ptmum; thewse the ptmalty wuld be vlated. Based n (30), we defne the -based pectn f DMU by The pectn pnt ( ˆ ˆ, ) the ttal cst. Thus, ( ˆ ˆ, ) ˆ = λ, =,...,, x x x m y yˆ = y, =,..., s. (30) (3) x y s btaned by bngng n the maxmum pssble mpvement n x y epesents the levels f peatn f nput-csts and utputs that wuld make DMU cst-effcent,.e., -effcent.

13 3 Smla t Theem 6 n Tne (2002), t can be demnstated that the -effcent pectn pnt ( xˆ, ˆ y ) s als DIVTE-effcent. Despte ths esult, ( ˆ ˆ, ) x y may nt be stngly techncaleffcent n a Paet-Kpmans sense due t the exstence f pssble utput slacks. Hweve, t s stngly techncal-effcent wth espect t nputs as n Tne (2002). x, y T Theem Let ( ), whee DEA x y x s the ptmal slutn t (0). Pf. Let (, ) x λ and ( β ), ˆ λ and (28). Then, ( ) ( ) x, λ x, ˆ β g λ. Then, gven the dectn vect g, we have m m ( x β g ) = x (32) = = be, espectvely, the ptmal slutn vects f the mdels (0) = s a feasble slutn t (0). Fm the ptmal slutn m m vect ( x, λ ) t (0), we have x ( x β g ). At the same tme, x x β =,, λ = λ, g = = s a feasble slutn t (28). It then fllws that m m x ( x β g ), whch cmpletes the pf. = = whee As an mmedate cnsequence f Theem 3..3, we btan the fllwng esult: m m = x x G = =, (33) x s the ptmal slutn vect t (0). Ths pves that u defntn f the dectnal CE measue s cnsstent wth that f the Nelvan cst neffcency measue (6). It can nw be easly shwn that the value f the dectnal CE measue s n me than that f the dectnal nput-ented value-based TE,.e., DIVTE ρ. (34) DIVTE Hence, we defne a new dectnal nput-ented AE by the at f t ρ,.e., α DIAE =. (35) ρ DIVTE Based n (35), we expess the -based measue as the pduct f the dectnal (nputented) AE ( α ) and value-based TE ( ρ DIAE DIVTE ) cmpnents,.e.,

14 4 = α ρ. (36) DIAE DIVTE Wth egad t the ppety f unt nvaance f, we have a esult smla t that as stated n Theem Specfcally, f the dectn vect g and the nput-csts f all the DMUs have the same unts f measuement (e.g., dlla, pund), then wll be unt-nvaant. Theem 3..4 (Mntncty). s stngly mntnc n nput-csts. Pf. Cnsde tw dffeent unts, namely DMU and DMU p, wh have all the same nputcst/utput data exceptng the hth nput-cst;.e., assume xhp = xh + δ g h whee δ > 0. Let ( β, λ ) be an ptmal slutn vect f DMU n (28) and (, ) pectn pnt. It s easy t shw that ( ˆ β, ˆ λ ), whee β = β, h, ˆ x y be the cespndng = +, ˆh β β h δ ˆλ λ =, s a feasble slutn f DMU p n (28) wth the fllwng bectve functn value: ( G m ) β ( β δ h ) / + + =, h m ( G ) ( ) ( G ) m β, h β h δ β = = / + + < / whch cmpletes the pf.. Snce the mdel (28) s a mnmzatn pblem, and, ne can cnclude that p <, Theem 3..4 shws that an ncease n an nput-cst wth all the the nput-csts and utputs beng cnstant wll educe the sce. Hweve, t s nt always s n case f ncement n utputs, f whch we have the fllwng emak. CE Remak. Lke, des nt take nt accunt the nn-ze utput slacks f nn-bndng utput cnstants, whch may esult n assgnng dentcal CE sces t the DMUs wth dffeent utputs (Fukuyama & Webe, 2009). T get d f ths pblem n a way dffeent fm the ne ppsed n Fukuyama and Webe (2009), we slve the fllwng pblem: whee ( β, λ ) fm the utput slacks as Max + λ, β + s s = + s t x x g m.. λ = β, =,...,, + + λ y y + β g, =,..., s, λ =, λ 0, J, β (37) s the ptmal slutn t (28). Then, we defne the dectnal CE bas asng

15 5 B DOS = + s s β + =, (38) whee β + s the ptmal slutn t (37). We can nw defne the dectnal utput-slack-adusted CE ( DOSACE ) as = B. (39) DOSACE DOS Wth egad t the tanslatn nvaance ppety f, we pesent the fllwng theem, whse pf s smla t that f Theem 3... Theem 3..5 (Tanslatn nvaance). s tanslatn-nvaant wth espect t nput-csts f the tanslatn f nput-cst data des nt affect the dectn vect g. It s als tanslatn-nvaant wth espect t utputs. If the dectn vect g n (28) s nt affected by the tanslatn f the nput-cst data, then wll be tanslatn-nvaant wth espect t nput-csts. A smple nstance f such dectn vect s the fxed dectn vect cnsdeed n (25). The the nstances ae the nn-fxed dectn vects n (23) and (24). F expstn, let us cnsde (23), and let xɶ = x + τ ( =,..., m ) be the th tanslated nput-cst f DMU. We shw hw the dectn vect (23) f the tanslated data ( gɶ ) s the same dectn vect f the gnal data: { ɶ } ( τ ) { τ } τ { } τ,,...,. gɶ = xɶ Mn x = x + Mn x + = x + Mn x = g = m 3.2. Dectnal utput-ented value-based TE, utput-ented AE, and RE measues DEA We nw pesent belw the utput-ented DDF-based DEA mdel elatve t T x, y as β = Max λ, β β s. t. λ x x, =,..., m, + λ y y + β g, =,..., s, λ =, λ 0, J. (40) Based n (40), we nw defne the new dectnal utput-ented value-based TE (DOVTE) measue as DOVTE ρ =. (4) + β

16 6 Smla t Theems 2 and 4 n Fukuyama and Webe (2004) and Theems 3.. and 3..2, we pesent seveal esults that ae lnked t ppetes such as mntncty, unt nvaance and DOVTE tanslatn nvaance. F example, ρ s unt-nvaant f the fllwng dectn vects: Futheme, dectn vects: g = 0, g + = y, =,..., m, =,..., s. (42) g + = 0, g = Max{ y}, =,..., m, =,..., s. (43) DOVTE ρ s bth unt-nvaant and tanslatn-nvaant f each f the fllwng g + = 0, g = Max{ y} y, =,..., m, =,..., s. (44) g + = 0, g = Max{ y} Mn{ y}, =,..., m, =,..., s. (45) We nw ntduce, based n T, DEA x y, a new dectnal RE (DRE) measue as η DRE = Max + + λ, β s = s. t. λ x x, =,..., m, + + λ y y + β g, =,..., s, λ =, λ 0, J, g G β (46) whee G =, and β + epesents the ate f mpvement n the th utput-eanngs f DMU. s + + g = DRE η s a genealzatn f cnsdeed, then the mdel (46) s tansfmed nt the fllwng: Lettng y ( β + ) Max + λ, β η. Specfcally, f y > 0 and the dectn vect (42) s R RE s + ( + β ) = s. t. λ x x, =,..., m, + ( β ) λ y + y, =,..., s, λ =, λ 0, J. = + y n (47) yelds exactly the mdel (8). Slvng the mdel (46) yelds the fllwng expessn f DMU : y (47)

17 7 λ x x, =,..., m, λ y = y + β g, =,..., s. + + Based n (48), we defne the DRE-based pectn f DMU by (48) The pectn pnt ( ˆ ˆ, ) eanngs f DMU t be DRE-effcent. x xˆ = x, =,..., m, ˆ = λ, =,...,. y y y s (49) x y epesents the maxmum pssble mpvements n the ttal It can be shwn that the pectn pnt ( ˆ ˆ, ) x y s als DOVTE-effcent, but nt stngly techncal-effcent n a Paet-Kpmans sense due t the exstence f pssble nput slacks. It can be als shwn that ( ˆ ˆ, ) whee x y s stngly techncal-effcent wth espect t utputs. Smla t Theem 3..3, we have that η = DRE s s + y + G = = y s the ptmal slutn vect t (8). Futhe, y, (50) DRE η s stngly mntnc n utputeanngs and s tanslatn-nvaant wth espect t utput-eanngs f the tanslatn f the utputeanngs data has n effect n the assgned dectn vect. F nstance, the dectn vects - (44) and (45), pvde ths ppety. We nw have the fllwng elatnshp: η DRE One can thus defne the dectnal utput-ented AE ( α DOVTE ρ. (5) DOAE ) as α DOAE η =. (52) ρ DRE DOVTE DRE Reaangng (52) yelds the fllwng decmpstn f η nt dectnal (utput-ented) AE ( α ) and value-based TE ( ρ ): DOAE DOVTE η = α ρ. (53) DRE DOAE DOVTE 3.3 Dectnal pft neffcency measue

18 8 In many eal-lfe applcatns bth physcal nputs and physcal utputs ae hetegeneus acss fms, n whch case u and DRE measues cannt be appled t evaluate the espectve cst and evenue effcences. In ths scena, a dectnal measue f pft DEA (n)effcency based n T x, y needs t be develped. Ths measue wll be me meanngful f fms when the undelyng behaval bectves ae pft maxmzatn athe than cst mnmzatn evenue maxmzatn. DEA Cnsde the fllwng dectnal pft-maxmzatn pblem based n T x, y n (8): κ g β g β s + m + = Max λ, β, β = G = G s. t. λ x x β g, =,..., m, + + λ y y + β g, =,..., s, λ =, λ 0, J, (54) DMU s dectnal pft-effcent f κ = 0. Hweve, f pft-neffcent n whch case the degee f pft neffcency s defned as: D Ineff π π DIneff s m + + β g + β g = = + =. G + G s m s m + + ( y + β g ) ( x β g ) y x = = = = = + G + G can be estmated f each f the fllwng dectn vects: g κ > 0, then DMU s dectnal (55) g =, g + =, =,..., m, =,..., s. (56) x =, g + = y, =,..., m, =,..., s. (57) + = { }, g Max{ y} g Max x J =, =,..., m, =,..., s. (58) + = { }, { } g x Mn x J g = Max y y, =,..., m, =,..., s. (59) + = { } { }, g Max{ y} Mn{ y} g Max x Mn x J J Nte that β + ( ) and β =, =,..., m, =,..., s. (60) ( ) n (54) ae nt all necessaly nn-negatve. It mples that the mdel (54) may pemt a DMU t becme pft-effcent by educng ts spendng n sme nputs whle nceasng n the the nput csts; and by nceasng ts eanngs n sme utputs whle

19 9 deceasng n the thes. Hweve, f ne expects a DMU t becme pft-effcent by educng ts spendng n all ts nputs, and by nceasng ts eanngs n all utputs, then ne need t mpse the nn-negatvty estctns n the vaables n utput-eanngs and nput-spendng,.e., β + 0 ( ) and β 0 ( ) n (54). Futheme, t mght als be pssble t dscuss the value udgments (pefeences) n u ppsed measues as t whch csts t educe whch evenues t ncease. The fm manage may be wllng t make the fm cst-, evenue- pft-effcent wthut educng ts spendng n sme specfc nputs and/ nceasng ts eanngs n sme specfc utputs. In such stuatns, (s)he may chse a dectn vect whse cmpnents asscated wth these nputs/utputs ae ze. (S)he may als lke t set the uppe lwe bunds f the equed nput-spendng utput eanngs changes by addng cnstants L β g U ( ) and/ L β g U ( ) whee L and U ae espectvely the lwe and uppe bunds n the th nput-spendng, and L + and U + ae espectvely the lwe and uppe bunds n the th utput-eanngs. Ths wll enable eseaches t explan the ecnmc meanng f the pefeences,.e., why t educe sme specfc nput-csts me than thes; why t utlze capacty bette hw t educe the capacty csts whle pducng stll the same evenue as eale. Remak 2. Nte that the appach pesented n Remak s a tw-stage appach cnsstng f slvng the mdels (28) and (37), whch can be fmulated as a sngle mdel: Max + λ, β, β s m + g ε + β β s = = G s. t. λ x x β g, =,..., m, + + λ y y + β g, =,..., s, λ =, λ 0, J, (6) whee ε s a small nn-achmedean nfntesmal quantty. The mdel (6) can be equvalently wtten as:

20 20 Max + λ, β, β s m + β + = s = G s. t. λ x x β g, =,..., m, ε β + + λ y y + β g, =,..., s, λ =, λ 0, J. g (62) Cmpang (62) wth (54), we fnd ut that the abve mdel s a specal case f mdel (54) pvded that (a) the utput pces ae cmmn between fms, and (b) nstead f the utput weghts + + ( ) w = g G,,..., s =, n (54), w ( ε s) =, =,..., s, ae used. Nte that the pft neffcency measues based n the dect vects (56) (60) ae all nnadal (nn-pptnal) n natue. Thus, these measues have the flexblty t bette eflect the tade-ffs n the effcency estmates between nputs and/ utputs due t the dffeed pptunty csts. See, e.g., Sah and Tne (2009a), Sah and Tne (2009b), Sah and Achaya (200), Sah, Luptack, and Mahlbeg (20), Mahlbeg and Sah (20), amng thes, f the dscussn f the debate n the chce f nn-adal measues f effcency ve the adal nes Dealng wth negatve nput-utput data In the pesence f negatve data, the and DRE measues can be supe t the cuntepats,.e., the CE and RE measues, f apppate dectn vects ae cnsdeed. F example, the fllwng dectn vects can be chsen f the and DRE measues: g = x, g + = 0, =,..., m, =,..., s (63) and g = 0, g + = y, =,..., m, =,..., s. (64) Smlaly, the fllwng dectn vect enables the pft neffcency measue (54) t deal wth negatve data: g = x, g + = y, =,..., m, =,..., s 3. (65) The secnd pssble chce f the dectn vect can be agued based n the ppety f tanslatn nvaance, whch enables analyss f data sets cntanng negatve values. Snce the dectnal CE, RE and pft (n)effcency measues can be all tanslatn-nvaant, addng any abtay cnstant t the nput-spendng and utput-eanng cnstants wll nt affect the ptmal slutns. One can add sutable lage pstve cnstants t the nput-spendng and utput-eanngs

21 2 s as t make them all nn-negatve. F nstance, f the measue ne can use the fllwng tansfmatn: x + τ 0, =,..., m; y + σ 0, =,..., s. (66) Futheme, ne can tun t the dectn vects such as (23), (44) and (59) that cnsde takng the ange f pssble mpvements n the nput-csts and/ the utput-eanngs. Snce the dectnal CE, RE and pft (n)effcency measues ae all tanslatn-nvaant f (23), (44) and (59) espectvely, the use f these dectn vects makes the effcency assessment pssble n the pesence f negatve data. 4. An empcal llustatn T demnstate the eady applcablty f u ppsed measue, we cnduct an llustatve empcal analyss based n a eal-lfe data set f 50 US banks f the yea 996 (whch was taken fm Ray (2005)), wheen we make meanngful cmpasn between u ppsed appach and Tne s (2002) appach by cnsdeng hw the chce f dectn vect plays a sgnfcant le n effcency analyss. The data cnssts f fve utputs cmmecal and ndustal lans, cnsume lans, eal estate lans, nvestments and the ncme, and fu nputs tansactn depsts, nntansactn depsts, lab and captal. We fst make u cst and evenue effcency assessments usng the dectn vects (24) and (45), and then d the same f Tne s (2002) measues, whch ae n fact deved fm u measues by cnsdeng the dectn vects (2) and (42). We use GAMS (Geneal Algebac Mdelng System) sftwae cde t d the cmputatns. Table exhbts such detaled esults. We nw summaze the esults f Table as fllws.. Banches, 3, 5, 3, 24, 45, 48, and 49 ae all dentfed t be fully effcent by all f the effcency measues; and the emanng 42 banches ae declaed neffcent by at least ne f these effcency measues. 2. The cst neffcences f the banches - 2, 6, 7, 9,, 4 6, 2, 33, 34 and 47 ae all due t the allcatve neffcences. Hweve, the cst neffcences f the the banches ae due t bth techncal and allcatve neffcences. Smla esults hld f the evenue effcency as well. See the evenue effcency sces f banches 7, 9, 0, 7 2, 23, 28 30, 38 39, 4 44, 46 and 50.

22 22 Table. Cst and evenue effcences f 50 bank banches unde analyss Dectn vects (2) and (42) Dectn vects (24) and (45) CE Assessment RE Assessment CE Assessment RE Assessment DMU IVTE ρ CE IAE α OVTE ρ RE η OAE α DIVTE ρ DIAE α DOVTE ρ DRE η DOAE α DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU

23 23 DMU DMU DMU DMU DMU DMU DMU Aveage Thee s n pstve celatn between cst and evenue effcences f banks. On aveage, the cst effcency s fund t be hghe cmpaed t the evenue effcency, whch means that bank management has t take udcus decsns t enhance the pfts by pducng the ght ptfls f pduct-mxes. As agued eale, the cst- and evenue-based effcency esults ae nt vey meanngful when the undelyng bectve f a fm s pft maxmzatn. Theefe, we nw tun t exhbt the esults n pft neffcences f banks n Table 2 based n the fu dectn vects (57) (60). One can bseve that pft neffcency sces depend n the chce f the dectn vect used. Whle, ne banch,.e., 50, s fund pft-effcent f all the fu cnsdeed dectn vects, eght banches (, 3, 5, 3, 43, 45, 46 and 49) ae nly effcent f the ange dectn vect (60). The pft neffcency measue based n the dectn vect (57) yelds hghest level f aveage neffcency as cmpaed t pft neffcency measues based n the the dectn vects. As egads the taget settng (benchmakng execse) by fms as t whch csts t educe whch evenues t ncease, ne can see n Table 3 hw an aveage DMU can attan ts full pft effcency by changng ts nput-spendng and utput-eanngs. The mpvements n nput-spendng and utputs-eanngs f a banch depend pecsely n the chce f the dectn vect used. F example, the pft neffcency mdel based n the dectn vect (57) eques an aveage banch t attan ts full effcency by, educng ts spendng n tw nputs - tansactn cst and nntansactn depsts cst, nceasng ts spendng n the tw nputs - lab cst and captal cst, educng ts eanngs n ne utput,.e., cmmecal and ndustal lans, and nceasng ts eanngs n the the fu utputs - cnsume lans, eal estate lans, nvestments and the ncme. Smlaly, the pft neffcency mdels based n the the dectn vects can be ntepeted n an analgus manne. F lack f space, the detal esults n the mpvements n nput-spendng and utput-eanngs by the ndvdual banches based n dffeent dectn vects ae nt epted hee, but ae pvded as supplementay mateals (see Tables 4-7).

24 24 Table 2. Pft effcency f 50 bank banches unde analyss Pft neffcency sces DMUs Dectn vect (57) Dectn vect (58) Dectn vect (59) Dectn vect (60) DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU

25 25 DMU DMU DMU DMU DMU DMU DMU DMU Avg Futheme, t s appaent that the ates f mpvements n ndvdual nput-spendng and ndvdual utput-eanngs ae dffeent, mplyng that a banch need nt educe ts nputs, and ncease ts utputs, equpptnately s as t mpve ts pft effcency. Ths establshes the empcal ealty that the pptunty csts f cnsumptn (pductn) f ne nput (utput) ve the the ae nt the same. Table 3. Aveage mpvements n nput-spendng and utput-eanngs acss mdels Dectn vect β β 2 β 3 β 4 β + β + 2 β + 3 β + 4 β + 5 (57) (58) (59) (60) Cncludng emaks Unde the cndtn f n p unt pce nfmatn, the cuent study develps new dectnal measues f value-based cst and evenue effcences, whch all satsfy seveal desable ppetes f an deal effcency measue. These new value-based cst/evenue effcences apply when utput quanttes/nput quanttes ae hmgenus and knwn. Ths study then decmpses these new value-based effcences nt the espectve value-based TE and AE cmpnents. Ou ppsed effcency measues have seveal advantages ve the exstng nes: ) they all satsfy the mst mptant ppety f tanslatn nvaance; 2) they all allw feedm f chce ve the use f sutable dectn vects n handlng negatve data; 3) they can be dectly appled as they can be easly mplemented n any standad DEA sftwae due t flexble cmpute pgammng; 4) they pvde the decsn makes wth the ptn f specfyng the pefeable

26 26 dectn vects t ncpate the decsn-makng pefeences; and fnally, 5) they nclude the Tne s (2002) value-based measues as specal cases. Fnally, unde the cndtn that when bth physcal nput and utput quanttes ae hetegeneus acss fms, when the pecse nput and utput pce data ae fequently unavalable, a dectnal measue f pft neffcency s develped f fms whse undelyng bectves ae pft maxmzatn. Ths measue allws empcal eseaches t ptentally deal wth value udgments (pefeences) as t whch specfc nput-cst t educe whch specfc utput-eanngs t ncease t mpve the pft effcency f a fm. Nte that the dectnal value-based measues develped n ths pape ae based n the assumptn that the fms hetegenety s eflected thugh the unt pces. Hweve, when pces vay accdng t the actn by fms,.e., when a fm wth hghe maket pwe exhbts hghe pces elatve t a fm wthut maket pwe, effcency estmates ae exaggeated. Theefe, when pce dffeences ae due t dffeent cmpettve envnments, t seems that these envnmental facts shuld be taken nt accunt nstead f tyng t dectly cmpae the effcences f fms. Ths cncens especally the case when we ty t defne the effcency mpvement ptental wth espect t each the. We suggest ths as a futue eseach subect. Acknwledgement We ae gateful t Rbet G Dysn (Edt) and fu annymus evewes f the Junal eve f the mst cnstuctve cmments and valuable suggestns. We ae thankful t Sanda Ban f he caeful pfead f the manuscpt. The usual dsclame apples. Refeences Al, A. I., & Sefd, L. M. (990). Tanslatn nvaance n data envelpment analyss. Opeatns Reseach Lettes, 9, Banke, R. D., Chanes, A., & Cpe, W. W. (984). Sme mdels f the estmatn f techncal and scale neffcences n data envelpment analyss. Management Scence, 30, Blackby, C., & Russell, R. R. (999). Aggegatn f effcency ndces. Junal f Pductvty Analyss, 2, Camanh, A. S., & Dysn, R. G. (2005). Cst effcency measuement wth pce uncetanty: A DEA applcatn t bank banch assessments. Eupean Junal f Opeatnal Reseach, 6,

27 27 Camanh, A. S., & Dysn, R. G. (2008). A genealsatn f the Faell cst effcency measue applcable t nn-fully cmpettve settngs. Omega: The Intenatnal Junal f Management Scence, 36, Chambeln, E. H. (933). The they f mnplstc cmpettn: a eentatn f the they f value. Cambdge: Havad Unvesty Pess. Chambes, R. G., Chung, Y., & Fäe, R. (996). Beneft and dstance functns. Junal f Ecnmc They, 70, Chambes, R. G., Chung, Y., & Fäe, R. (998). Pft, dectnal dstance functns, and Nelvan effcency. Junal f Optmzatn They and Applcatns, 98, Chanes, A., Cpe, W. W., & Rhdes, E. (978). Measung the effcency f decsn makng unts. Eupean Junal f Opeatnal Reseach, 2, Cpe, W. W., Pak, K. S., & Past, J. T. (999). RAM: A ange adusted measue f neffcency f use wth addtve mdels and elatns t the mdels and measues n DEA. Junal f Pductvty Analyss,, Cpe, W. W., Thmpsn, R. G., & Thall, R. M. (996). Intductn: Extensns and new develpments n DEA. Annals f Opeatns Reseach, 66, Emuznead, A., Anuze, A. L., & Thanassuls, E. (200). A sem-ented adal measue f measung the effcency f decsn makng unts wth negatve data, usng DEA. Eupean Junal f Opeatnal Reseach, 200, Engel, C., & Rges, J. H. (996). Hw wde s the bette? Amecan Ecnmc Revew, 86, Fäe, R., & Gsskpf, S. (2006). Reslvng a stange case f effcency. Junal f the Opeatnal Reseach Scety, 57, Fäe, R., Gsskpf, S., & Lvell, C. A. K. (985). The measuement f effcency f pductn Bstn: Kluwe Nhff. Faell, M. J. (957). The measuement f pductve effcency. Junal f the Ryal Statstcal Scety, Sees A, 29, Fukuyama, H., & Webe, W. L. (2004). Ecnmc neffcency measuement f nput spendng when decsn-makng unts face dffeent nput pces. Junal f the Opeatnal Reseach Scety, 55, Fukuyama, H., & Webe, W. L. (2008). Pft neffcency f Japanese secutes fms. Junal f Appled Ecnmcs,,

28 28 Fukuyama, H., & Webe, W. L. (2009). Output slacks-adusted cst effcency and value-based techncal effcency n DEA mdels. Junal f the Opeatns Reseach Scety f Japan, 52, Jahanshahl, G. R., Sleman-Damaneh, M., & Mstafaee, A. (2008). A smplfed vesn f the DEA cst effcency mdel. Eupean Junal f Opeatnal Reseach, 84, Kestens, K., & Van de Westyne, I. (20). Negatve data n DEA: A smple pptnal dstance functn appach. Junal f the Opeatnal Reseach Scety, 62, Kusmanen, T., Chechye, L., & Splanen, T. (2006). The law f ne pce n data envelpment analyss: Restctng weght flexblty acss fms. Eupean Junal f Opeatnal Reseach, 70, Kusmanen, T., & Pst, T. (200). Measung ecnmc effcency wth ncmplete pce nfmatn: Wth an applcatn t Eupean cmmecal banks. Eupean Junal f Opeatnal Reseach, 34, Kusmanen, T., & Pst, T. (2003). Measung ecnmc effcency wth ncmplete pce nfmatn. Eupean Junal f Opeatnal Reseach, 44, Lvell, C. A. K., & Past, J. T. (995). Unts nvaant and tanslatn nvaant DEA mdels. Opeatns Reseach Lettes, 8, Mahlbeg, B., & Sah, B. K. (20). Radal and nn-adal decmpstns f Luenbege pductvty ndcat wth an llustatve applcatn. Intenatnal Junal f Pductn Ecnmcs, 3, Manadaks, N., & Thanassuls, E. (2004). A cst Malmqust pductvty ndex. Eupean Junal f Opeatnal Reseach, 54, McCall, J. J. (967). Cmpettve pductn f cnstant sk utlty functns. Revew f Ecnmc Studes, 34, Mehdlzad, M., Sah, B. K., Rshd, I. (204), A genealzed multplcatve dectnal dstance functn f effcency measuement n DEA, Eupean Junal f Opeatnal Reseach, 24, Mstafaee, A., & Salgh, F. H. (200). Cst effcency measues n data envelpment analyss wth data uncetanty. Eupean Junal f Opeatnal Reseach, 202, Pak, K. S., & Ch, J-W. (20). P-effcency: Data speak me than techncal effcency. Eupean Junal f Opeatnal Reseach, 25, Past, J. T. (996). Tanslatn nvaance n DEA: A genealzatn. Annals f Opeatns Reseach, 66,

29 29 Pug-Juny, J. (2000). Pattnng nput cst effcency nt ts allcatve and techncal cmpnents: An empcal DEA applcatn t hsptals. Sc-Ecnmc Plannng Scences, 34, Ray, S. C. (2005). Data Envelpment Analyss (Pat 2), Effcency analyss wth maket pces. Pesented at the cnfeence Fnancal Ecnmetcs and Bankng, IGIDR, Mumba, Inda. Ray, S. C., & Km, H. J. (995). Cst effcency n the US steel ndusty: A nnpaametc analyss usng data envelpment analyss. Eupean Junal f Opeatnal Reseach, 80, Rbnsn, J. (933). The ecnmcs f mpefect cmpettn. Lndn: Macmllan. Sah, B. K., & Achaya, D. (200). An altenatve appach t mnetay aggegatn n DEA, Eupean Junal f Opeatnal Reseach, 204, Sah, B. K., Kestens, K., & Tne, K. (202). Retuns t gwth n a nn-paametc DEA appach. Intenatnal Tansactns n Opeatnal Reseach, 9, Sah, B. K., Luptack, M., & Mahlbeg, B. (20). Altenatve measues f envnmental technlgy stuctue n DEA: An applcatn. Eupean Junal f Opeatnal Reseach, 25, Sah, B. K., & Tne, K. (2009a). Decmpsng capacty utlzatn n data envelpment analyss: An applcatn t banks n Inda. Eupean Junal f Opeatnal Reseach, 95, Sah, B. K., & Tne, K. (2009b). Radal and nn-adal decmpstns f pft change: Wth an applcatn t Indan bankng. Eupean Junal f Opeatnal Reseach, 96, Sah, B. K., & Tne, K. (203). Nn-paametc measuement f ecnmes f scale and scpe n nn-cmpettve envnment wth pce uncetanty. Omega: The Intenatnal Junal f Management Scence, 4, 97. Sandm, A. (97). On the they f the cmpettve fm unde pce uncetanty. Amecan Ecnmc Revew, 6, Schaffnt, C., Rsen, D., & Paad, J. C. (997). Best pactce analyss f bank banches: An applcatn f DEA n a lage Canadan bank. Eupean Junal f Opeatnal Reseach, 98, Sengupta, J. K., & Sah, B. K. (2006). Effcency mdels n data envelpment analyss: Technques f evaluatn f pductvty f fms n a gwng ecnmy. Lndn: Palgave Macmllan. Shephad, R. W. (953). Cst and Pductn Functns. New Jesey: Pncetn Unvesty Pess.

30 30 Shephad, R. W. (970). They f cst and pductn functn. Pncetn: Pncetn Unvesty Pess. Slva Ptela, M. C. A., & Thanassuls, E. (200). Malmqust-type ndces n the pesence f negatve data: An applcatn t bank banches. Junal f Bankng & Fnance, 34, Slva Ptela, M.C.A, Thanassuls, E., & Smpsn, G. (2004). Negatve data n DEA: A dectnal dstance appach appled t bank banches. Junal f the Opeatnal Reseach Scety, 55, -2. Sueysh, T. (997). Measung effcences and etuns t scale f Nppn telegaph & telephne n pductn and cst analyses. Management Scence, 43, Tne, K. (2002). A Stange case f the cst and allcatve effcences n DEA. Junal f the Opeatnal Reseach Scety, 53, Tne, K., & Sah, B. K. (2005). Evaluatng cst effcency and etuns t scale n the lfe nsuance cpatn f Inda usng data envelpment analyss. Sc-Ecnmc Plannng Scences, 39, Tne, K., & Sah, B. K. (2006). Re-examnng scale elastcty n DEA. Annals f Opeatns Reseach, 45, End Ntes Nte that Slva Ptela, Thanassuls, and Smpsn (2004) used the dectn vect (23) t develp the ange dectnal dstance functn mdel t measue the TE n the pesence f negatve data n DEA. 2 Mehdlzad, Sah, and Rshd (204) als used ths dectn vect n lg fm t develp a genealzed multplcatve dectnal measue f effcency. 3 In the pesence f negatve data, Kestens and Van de Westyne (20) used ths dectn vect n fact fm t develp a genealzed pptnal dstance functn measue f effcency.