Suther Illiis Uiversity Carbdale OpeSIUC Discussi Papers Departmet f Ecmics 2004 Directial Duality Thery Daiel Primt Suther Illiis Uiversity Carbdale Rlf Fare Oreg State Uiversity Fllw this ad additial wrks at: http://pesiuc.lib.siu.edu/ec_dp Recmmeded Citati Primt, Daiel ad Fare, Rlf, "Directial Duality Thery" (2004). Discussi Papers. Paper 18. http://pesiuc.lib.siu.edu/ec_dp/18 This Article is brught t yu fr free ad pe access by the Departmet f Ecmics at OpeSIUC. It has bee accepted fr iclusi i Discussi Papers by a authrized admiistratr f OpeSIUC. Fr mre ifrmati, please ctact pesiuc@lib.siu.edu.
Directial Duality Thery Rlf Färe Oreg State Uiversity Daiel Primt Suther Illiis Uiversity Octber 24, 2003 Abstract Shephard (1953, 1970,1974)itrduced radial distace fuctis as represetatis f a firm s techlgy ad develped a umber f dual represetatis that have bee widely applied i empirical wrk. A systematic expsiti f Shephard s wrk ca be fud i Färe ad Primt (1995). Mre recetly, wrk by Lueberger (1992, 1995) has prvided sme ew techlgy represetatis, the beefit ad the shrtage fuctis. Explitig these results, Chambers, Chug, ad Färe (1996, 1998a) itrduced directial distace fuctis; these ca be thught f as additive alteratives t the crrespdig radial ccepts. I this paper, the radial apprach is further exteded by itrducig ad characterizig idirect directial distace fuctis; these are directial versis f their radial cuterparts. This, i tur, leads t a ew set f duality results that will be f use i applied wrk. 1 Itrducti Shephard (1953, 1970,1974) itrduced radial distace fuctis as represetatis f a firm s techlgy ad develped a umber f dual represetatis that have bee widely applied i empirical wrk. A systematic expsiti f Shephard s wrk ca be fud i Färe ad Primt (1995). Mre recetly, wrk by Lueberger (1992,1995) has prvided sme ew techlgy represetatis, the beefit ad the shrtage fuctis. Explitig these results, Chambers, Chug, ad Färe (1996, 1998a) itrduced directial distace fuctis; these ca be thught f as additive alteratives t the crrespdig radial ccepts. I this paper, the radial apprach is further exteded by itrducig ad characterizig idirect directial distace fuctis; these are directial versis f their radial cuterparts. This, i tur, leads t a ew set f duality results that will be f use i applied wrk. Befre prceedig we eed t ask the questi: Des the wrld eed mre duality therems? We thik that the aswer is Yes. I Färe ad Primt (1995) we made the argumet that a variety f mdels are useful i explaiig the behavir f firms. 1
This arises frm the csiderati that it is useful t study differet firms usig differet behaviral assumptis 1. These differet behaviral assumptis lead us t differet dual represetatis f the prducti techlgy. These dual fuctis have atural cuterparts, called distace fuctis, that represet either direct r idirect techlgies. Here is a list f these dual fucti-distace fuctis pairs alg with their related behaviral assumpti. Reveue fucti - Output distace fucti: the firm maximizes the reveue frm utputs give iputs ad utput prices. Cst fucti - Iput distace fucti: the firm miimizes the cst f iputs give utputs ad iput prices. Idirect reveue fucti - Idirect utput distace fucti: the firm maximizes the reveue frm utputs give iput prices, ttal iput cst, ad utput prices. Idirect cst fucti - Idirect iput distace fucti: the firm miimizes the cst f iputs give utput prices, ttal utput reveue, ad iput prices. Prfit fucti: the firm maximizes prfit give utput prices ad iput prices. (The prfit fucti is dual t the techlgy set; a distace fucti was t used t represet this set.) Nw that Lueberger (1992,1995) ad Chambers, Chug, ad Färe (1996, 1998a) have itrduced the directial techlgy distace fucti we are mtivated t update ur 1995 bk t accut fr directial distace fuctis. The argumets fr why we shuld use directial distace fuctis has bee advaced i the abve refereces ad w t be repeated here. Additial refereces with applicatis iclude Chambers ad Färe (1998b), Färe ad Grsskpf (2000), Färe ad Primt (2003), ad Hudgis ad Primt (2003). 2 Sme Basic Ccepts Let x R N + be the iput vectr ad let y R M + be the utput vectr. The techlgy T is give by T = {(x, y) : x ca prduce y}. (1) 1 Sice the differet dual fuctis have differet argumets they dictate the use f differet data. I practice this may be reversed; the availability f data may dictate the mdel chice. 2
Certai assumptis abut the techlgy eable us t establish duality prperties fr fuctial represetatis f T.Theyare 2 T is a empty, clsed, cvex set ad bth iputs ad utputs are strgly dispsable. (T ) Whe wrkig with duality 3 relatiships i utput price/quatities spaces it is smetimes cveiet t represet the techlgy with direct utput sets defied by We assume that P (x) ={y :(x, y) T }. (2) Fr all x i R+ N,P(x) is a empty, cmpact, cvex set ad utputs are strgly dispsable. (P ) At ther times it is cveiet t represet the techlgy by idirect utput sets defied by IP(w/C) ={y : y P (x),wx C}, (3) where w 0 N is a vectr f iput prices ad C > 0 is the ttal iput cst. We assume that Fr all w/c i R+ N,IP(w/C) is a empty, cmpact, cvex set ad utputs are strgly dispsable. (IP) Fr duality relatiships i iput spaces it is smetimes cveiet t represet techlgy with direct iput sets defied by Duality thery requires that L(y) ={x :(x, y) T }. (4) Fr all y i R+ M,L(y) is a empty, clsed, cvex set ad iputs are freely dispsable. At ther times it is cveiet t wrk with idirect iputs sets defied by (L) IL(p/R) ={x : x L(y),py R}, (5) where p is a vectr f utput prices ad R is ttal reveue. We als assume that Fr all p/r i R+ M,IL(p/R) is a empty, clsed, cvex set ad iputs are freely dispsable. I clsig, we te that the abve defiitis f (2) ad (4) imply that (IL) x L(y) (x, y) T y P (x). (6) 2 Iputs are strgly dispsable if (x 0,y) T wheever x 0 x ad (x, y) T. Outputs are strgly dispsable if (x, y 0 ) T wheever y 0 y ad (x, y) T. 3 Duality will t be defied i this paper. 3
3 Fucti Characterizati f Techlgy The five differet techlgy sets, T,P(x),IP(w/C),L(y), ad IL(p/R) ca all be represeted by the directial distace fuctis that are defied i this secti. Their derivati ivlves sme frm f techical efficiecy, i.e., mvemets t frtiers f techically efficiet iput ad/r utput vectrs. As we will s see, tw f these five represetatis als have sme ecmic efficiecy embedded i them. I additi t these five directial distace fuctis, e ca defie five dual fuctis that are explicitly derived by sme frm f ecmic ptimizati, e.g., reveue maximizati, cst miimizati, ad prfit maximizati. 3.1 Directial Distace Fuctis We start with T. We must first chse a directial vectr, g =( g x,g y ) where g x R N +,g y R M +, ad g 6= 0 M+N. The directial techlgy distace fucti T is defied by 4 DT (x, y; gx,gy) =sup{ :(x g x,y+ g y ) T }. (7) We illustrate this distace fucti fr the e-iput, e-utput case. Output ( x gx, y + g y ) ( g x, g y ) ( x, y) Iput Figure 1 The directial techlgy distace fucti has a umber f useful prperties. Tw f them are: T Idicati: D T (x, y; g x,g y ) 0 (x, y) T. (8) 4 This fucti was itrduced by Lueberger (1992) wh amed it the shrtage fucti. We fllw Chambers, Chug, ad Färe (1998). 4
Traslati: D T (x αg x,y+ αg y ; g x,g y )= D T (x, y; g x,g y ) α fr all α R. (9) The idicati prperty fllws directly frm the strg dispsability assumptis. The traslati prperty fllws directly frm the defiiti. T see this te that D T (x αg x,y+ αg y ; g x,g y ) = sup{ :(x αg x g x,y+ αg y + g y ) T } = sup{ :(x (α + ) g x,y+(α + ) g y ) T } = sup{α + :(x (α + ) g x,y+(α + ) g y ) T } α α+ = D T (x, y; g x,g y ) α. I additi, it has bee shw that D T ( ) is hmgeeus f degree mius e i ( g x,g y ), decreasig i x, icreasig i y, ad ccave i (x, y). See Chambers, Chug, ad Färe (1998). We w mve t P (x). The directial utput distace fucti, defied the utput sets, i the directi g x,isgiveby D (x, y; g y )=sup{ :(y + g y ) P (x)} (10) The fllwig prperty is easily established usig strg dispsability f utputs. D -Idicati Thus, Nw te that Hece, we have D (x, y; g y ) 0 y P (x). (11) (y + g y ) P (x) D (x, y + g y ; g y ) 0. (12) (y + g y ) P (x) (x, y + g y ) T by (6) D (x, y; g y ) = sup{ :(y + g y ) P (x)} = sup{ :(x, y + g y ) T } = sup{ :(x 0 N,y+ g y ) T } = D T (x, y;0 N,g y ) 5
This gives us the relatiship betwee D T (x, y; g x,g y ) ad D (x, y; g y ), viz., D (x, y; g y )= D T (x, y;0 N,g y ) (13) Mrever, usig (11) we may recver the directial techlgy distace fucti D T frm the utput directial distace fucti D by D T (x, y; g x,g y )=sup : D (x g x,y+ g y ; g y ) 0 (14) It is imprtat t stress that the pair f equatis (13) ad (14) d t cstitute a duality relatiship. Istead, we refer t this pair f equatis as a iverse relatiship. We w tur ur atteti t L(y). The directial iput distace fucti fr the directi g x is defied by D i (x, y, g x )=sup{ :(x g x ) L(y)}. (15) Strg dispsability f iputs implies D i - Idicati: D i (x, y, g x ) 0 x L(y). (16) Nte that ad thus (x g x ) L(y) (x g x,y) T by (6) D i (x, y; g x ) = sup{ :(x g x ) L(y)} = sup{ :(x g x,y) T } = sup{ :(x g x,y+ 0 M ) T } = D T (x, y; g x, 0 M ). i.e., we ca cmpute D i frm D T by D i (x, y; g x )= D T (x, y; g x, 0 M ). (17) Mrever, usig (16), we may cmpute D T frm D i by D T (x, y, g x,g y )=sup : D i (x g x,y+ g y, g x ) 0 (18) Itisimprtattstressthatthepairfequatis(17)ad(18)d t cstitute a duality relatiship. It is, rather, a iverse relatiship. 6
We may als csider the relatiship betwee D ad D i. We cmpute D i several steps, D (x, y; g y ) = sup{ :(y + g y ) P (x)} (19) { : x L(y + g y )} by (6) (20) = sup = sup ad we cmpute D i i several steps, : D i (x, y + g y ; g x ) 0 by (16). (21) D i (x, y; g x ) = sup{ :(x g x ) L(y)} (22) { : y P (x g x )} by (6) (23) = sup = sup We get the pair f relatiships: D (x, y; g y )=sup : D (x g x,y; g y ) 0 : D i (x, y + g y ; g x ) 0 by (11) (24) ad D i (x, y, g x )=sup : D (x g x,y; g y ) 0. (26) Equatis (25) ad (26) frm a iverse relatiship. 5 We w take up the fuctial represetati f the idirect utput sets: ad the idirect iput sets: IP(w/C) ={y : y P (x),wx C}, IL(p/R) ={x : x L(y),py R}. They are give by idirect directial utput distace fucti: I D (w/c, y; g y )=sup{ :(y + g y ) IP(w/C)}, (27) ad the idirect directial iput distace fucti: (25) I D i (x, p/r; g x )=sup{ :(x g x ) IL(p/R)}. (28) 5 The reader may be bthered by the fact that the righthad side f (25) seemigly depeds g x while the lefthad side des t. Hwever, the effect f g x is elimiated by the ptimizati ver. Put ather way, sice (19) ad (20) d t deped g x the either des (21). A aalgus argumet ca be made fr equati (26). 7
Nte that I D (w/c, y + αg y ; g y ) = sup{ :(y + αg y + g y ) IP(w/C)} = sup{α + :(y +(α + ) g y ) IP(w/C)} α = sup{α + :(y +(α + ) g y ) IP(w/C)} α α+ = I D (w/c, y; g y ) α. Hece, we have the traslati prperty: Similarly, e ca shw that I D (w/c, y + αg y ; g y )=I D (w/c, y; g y ) α. (29) I D i (x αg x,p/r; g x )=I D i (x, p/r; g x ) α (30) This cmpletes ur catalgue f directial distace fuctis. 3.2 Dual Fuctis Tw rather stadard dual fuctis are the reveue fucti, defied by ad the cst fucti, defied by: R(x, p) =sup{py : y P (x)}, (31) y C(y, w) =if x {wx : x L(y)}. (32) I additi, there are idirect versis; the first is the idirect reveue fucti defied by IR(w/C, p) =sup{py : y IP(w/C)}, (33) y ad the secd is the idirect cst fucti defied by IC(p/R, w) =if x Fially, we defie the prfit fucti: {wx : x IL(p/R)}. (34) Π(p, w) =sup{py wx :(x, y) T }. (35) x,y 8
4 The Duality Diamd The te fuctial represetatis f techlgy ca be displayed i a three dimesial figure that has the shape f a diamd. It is give belw. Duality Diamd Π( p, w) IC( p / R, w) IR( w / C, p) C( y, w) R( x, p) r IDi ( x, p / R; g x ) ID r ( p / C, y; g y ) r Di ( y, x; g x ) r DT ( x, y; g x, g y ) D r ( x, y; g ) 0 y 4.1 Iverse Relatiships Mvig frm e de t ather i the abve figure ivlves either a iverse perati r a dual perati. The iverse peratis are thse betwee twmember subsets (pairs) f DT (x, y; g x,g y ), D (x, y; g y ), D i (x, y; g x ). These represetatis are fuctis f quatities (ad directial vectrs) ly. We have already itrduced the iverse relatiships fr the three pssible pairs 9
i the previus secti. Frm equatis (13) ad (14) abve we have D (x, y; g y )= D T (x, y;0 N,g y ) D T (x, y; g x,g y )=sup : D (x g x,y+ g y ; g y ) 0 (I) frm equatis (17) ad (18) we have D i (x, y; g x )= D T (x, y; g x, 0 M ) D T (x, y; g x,g y )=sup : D i (x g x,y+ g y ; g x ) 0 (II) ad frm equatis (25) ad (26) we have D (x, y; g y )=sup : D i (x, y + g y ; g x ) 0 D i (x, y; g x )=sup : D (x g x,y; g y ) 0 (III) 4.2 Duality Relatiships The duality relatiship betwee the techlgy directial distace fucti ad the prfit fucti ivlves the bttm ad the tp f the duality diamd. It is give by the pair f ptimizati prblems: Π(p, w) =sup py wx + D T (x, y; g x,g y )(pg y + wg x ) x,y D T (x, y; g x,g y )=if p,w ½ Π(p, w) (py wx) pg y + wg x ¾. (IV) Sketch f Prf: 6 Frm the defiiti f D T, ( x, ȳ) = ³x D T (x, y; g x,g y )g x,y+ D T (x, y; g x,g y )g y T 6 See Lueberger (1992) fr a cmplete prf. 10
ad D T ( x, ȳ; g x,g y )=0. (36) The cditi (36) is clearly ecessary fr ay prfit maximizig chice f a ipututput vectr. Thus Π(p, w) = sup x,y = sup x,y = sup x,y This yields the fllwig iequality: {py wx :(x, y) T } p hy + D i T (x, y; g x,g y )g y py wx + D T (x, y; g x,g y )(pg y + wg x ) w hx D i T (x, y; g x,g y )g x Π(p, w) py wx + D T (x, y; g x,g y )(pg y + wg x ),. fr all (x, y) R N + R M + ad fr all (p, w) R M + R N +. Rearragig, D T (x, y; g x,g y ) Π(p, w) (py wx) pg y + wg x fr all (x, y) R N + R M + ad fr all (p, w) R+ M R+ N.Thus, ½ Π(p, w) (py wx) D T (x, y; g x,g y ) if p,w pg y + wg x ¾. (37) The rest f the prf simply shwig that (37) hlds with equality. This is de by usig strg dispsability f iputs ad utputs, cvexity f T, ad a separatig hyperplae therem. We w csider the duality relatiship betwee the directial iput distace fucti ad the cst fucti. It is give by the fllwig pair f ptimizati prblems. C(y, w) =if x D i (x, y; g x )=if w Frm the first ptimizati prblem we see that wx D i (x, y; g x ) wg x ½ ¾ wx C(y, w) wg x (V) C(y, w) wx D i (x, y; g x ) wg x (38) 11
fr all x R N +,w R N + sice C(y, w) wx fr all x L(y) ad Rearragig (38) yields x D i (x, y; g x ) g x L(y). D i (x, y; g x ) wx C(y, w) wg x, fr all x R+ N,w R+ N, a iequality that leads us t the secd ptimizati prblem. 7 A aalgus result hlds fr the directial utput distace fucti ad the reveue fucti. R(x, p) =sup py + D (x, y; g y ) pg y y ½ ¾ (VI) R(x, p) py D (x, y; g y )=if p pg y The first ptimizati prblem is justified by startig with the fllwig tw cditis: R(x, p) py fr all y P (x), ad y + D (x, y; g y ) g y P (x), which lead us t the iequality R(x, p) py + D (x, y; g y ) pg y, fr all y R M +,p R M +. The, rearragig this iequality we get D (x, y; g y ) R(x, p) py pg y, which leads us t the secd ptimizati prblem. We w csider dualities betwee direct ad idirect directial distace fuctis. O the utput side we start with sme particular iput vectr, x 0 R+ N,ad write the defiiti f the directial utput distace fucti: D x 0,y; g y =sup : y + gy P (x 0 ) ª. (39) 7 Fr a cmplete prf see Lueberger (1992) ad Chambers, Chug, ad Färe (1996). 12
Nw, chse ay rmalized iput price vectr, w/c, such that (w/c) x 0 1. If we elarge the feasible set i (39), the supremal value cat decrease. Hece, sup : y + gy P (x 0 ) ª sup { : y + g y P (x), (w/c) x 1} (40),x sice x 0 is i the elarged feasible set but it is t ecessarily ptimal. Mrever, sup,x { : y + g y P (x), (w/c) x 1} = sup{ : y + g y IP(w/C)} (41) The (39) - (42) imply the iequality: D (x, y; g y ) I D (w/c, y; g y ), = I D (w/c, y; g y ) (42) fr all x R+ N, (w/c) R+ N relatiship: such that (w/c) x 1. This leads us t the duality ID (w/c, y; g y )=sup D (x, y; g y ):(w/c) x 1 x D (x, y; g y )=if I D (w/c, y; g y ):(w/c) x 1 w/c (VII) Prf: The prf is similar t the prf f (IV) i Färe ad Primt (1995, page 88). I Färe ad Primt (1995, page 97) it is first prved that Hece IP(w/C) = y R M + : C(y, w) C ª. I D (w/c, y; g y ) = sup{ : y + g y IP(w/C)} = sup{ : C (y + g y,w) C} = sup{ : C (y + g y,w/c) 1}, where the last equality fllws frm the hmgeeity f C i w. Thus,if I D (w/c, y; g y )= 0 the C (y + g y,w/c) 1 ad hece C (y, w/c) 1 sice C is decreasig i y. (Lwerig utputs cat icrease cst sice utputs are strgly dispsable.) Cversely, if C (y + g y,w/c) 1 the I D (w/c, y; g y ). But the 13
ID (w/c, y; g y ) 0 ID (w/c, y + g y ; g y ) 0 usig the traslati prperty (29). Settig equaltzerwegettheresultthatifc (y, w/c) 1 the I D (w/c, y; g y ) 0. Hece we have shw that I D (w/c, y; g y ) 0 if ad ly if C (y, w/c) 1. (43) Nw, sice D (x, y; g y ) I D (w/c, y; g y ), fr all x R+ N, (w/c) R+ N such that (w/c) x 1 itmustbethecasethat D (x, y; g y ) if I D (w/c, y; g y ):(w/c) x 1. (44) w/c We wat t shw that (44) hlds with equality. Suppse it des t, i.e., suppse that D (x, y; g y ) < =if w/c ID (w/c, y; g y ):(w/c) x 1. The D (x, y; g y ) < 0 which implies that, usig the traslati prperties f D ad ID, D (x, ȳ; g y ) < 0= if I D (w/c, ȳ; g y ):(w/c) x 1 w/c where ȳ = y + g y.thus, (w/c) x 1 I D (w/c, ȳ; g y ) 0, which is equivalet t (w/c) x 1 C (ȳ, w/c) 1 (45) because f (43). Nw, D (x, ȳ; g y ) < 0 implies that x / L(ȳ) by (6) ad (11). Sice L(ȳ) is clsed ad cvex ad satisfies strg dispsability, the strgly separatig hyperplae therem (see ()) implies that there³ is a iput price vectr, ŵ>0, ³ suchthat ŵx < C(ȳ, ŵ). Let Ĉ = ŵx. The 1= ŵ/ĉ x<c(ȳ, ŵ/ĉ), i.e., ŵ/ĉ x =1 ad C(ȳ, ŵ/ĉ) > 1. This ctradicts (45). QED O the iput side we prceed i a similar fashi. Fr ay y 0 R+ M (p/r) y 0 1 we have D i x, y 0 ; g x = sup : x gx L(y 0 ) ª such that sup { : x g x L(y), (p/r) y 1} (sice (p/r) y 0 1),y = sup{ : x g x IL(p/R)} = I D i (x, p/r; g x ). 14
Hecewegettheiequality, D i (x, y; g x ) I D i (x, p/r; g x ) fr all y R M +,p/r R M + such that (p/r) y 1. Weccludethat I D i (x, p/r; g x )=sup D i (x, y; g x ):(p/r) y 1 y D i (x, y; g x )=if I D i (x, p/r; g x ):(p/r) y 1 p/r (VIII) The dualities betwee the idirect iput ad utput directial distace fuctis ad the idirect cst ad reveue fuctis are w csidered. Frm (34) we have This implies that Sice we have r IC(p/R, w) =if x {wx : x IL(p/R)}. IC(p/R, w) wx fr all x IL(p/R). x I D i (x, p/r; g x ) g x IL(p/R), IC(p/R, w) w ³x ID i (x, p/r; g x ) g x IC(p/R, w) wx I D i (x, p/r; g x ) wg x, fr all x R N +,w R N +. This iequality leads us t the duality result: IC(p/R, w) =if x I D i (x, p/r; g x )=if w wx I D i (x, p/r; g x ) wg x ½ ¾ wx IC(p/R, w) wg x (IX) O the utput side, the aalgus iequality is: IR(w/C, p) py + I D (w/c, y; g y ) pg y, 15
fr all y R M +,p R M +.Wearethusleadt IR(w/C, p) =sup py + I D (w/c, y; g y ) pg y y I D (w/c, y; g y )=if p ½ ¾ IR(w/C, p) py pg y (X) Oe ca als fid duality relatiships betwee the prfit fucti ad the idirect directial utput ad iput distace fuctis. Fr example, we may cmpute the prfit fuctiby Π(p, w) = sup y,c This leads t the iequality = sup y,c = sup y,c py C : ID (w/c, y; g y ) 0 p ³y + ID (w/c, y; g y ) g y C py C + I D (w/c, y; g y ) pg y Π(p, w) py C + ID (w/c, y; g y ) pg y fr all y R+ M,p R+ M,C >0. Hece, ID Π(p, w) (py C) (w/c, y; g y ) pg y fr all y R+ M,p R+ M,C >0. This suggests that ½ ¾ Π(p, w) (py C) I D (w/c, y; g y )=if. p,c pg y We cclude that Π(p, w) =sup py C + I D (w/c, y; g y ) pg y y,c I D (w/c, y; g y )=if p,c ½ ¾ Π(p, w) (py C) pg y (XI) I a aalgus fashi, Π(p, w) =sup x,r R wx + I D i (x, p/r; g x )wg x 16
leads t the iequality We cclude that Π(p, w) R wx + I D i (x, p/r; g x )wg x. Π(p, w) =sup R wx + I D i (x, p/r; g x )wg x x,r I D i (x, p/r; g x )=if w,r ½ ¾ Π(p, w) (R wx) wg x (XII) The remaiig duality relatiships i the duality diamd d t ivlve D, D i,i D r I D i. Therefre, they are the same as thse preseted i Färe ad Primt (1995) ad will t be repeated here. 5 Ccludig Remarks The te represetatis f the techlgy illustrated i the duality diamd ca be separated it tw types, thse that pssess a hmgeeity prperty ad thse that pssess the traslati prperty. This bservati has implicatis fr the apprpriate parametric frms f these represetatis. The prperty f hmgeeity is easily accmdated by traslg fuctis while the traslati prperty is easily mdelled with quadratic fuctial frms. Refereces [1] Chambers, Rbert G, Chug, Yagh, Färe, Rlf (1996). Beefit ad Distace Fuctis.. Jural f Ecmic Thery. Vl. 70 (2). pp. 407-19. August 1996. [2] Chambers, Rbert G, Chug, Yagh, Färe, Rlf (1998a), Prfit, Directial Distace Fuctis, ad Nerlvia Efficiecy," Jural f Optimizati Thery ad Applicatis, Vlume 98, N. 2, 351-364, August, 1998. [3] Chambers, Rbert G. ad Rlf Färe (1998b), Traslati Hmtheticity," Ecmic Thery, Vlume 11, N. 3, pp. 629-41. [4] Färe, Rlf ad Shawa Grsskpf (2000), Thery ad Applicati f Directial Distace Fuctis, Jural f Prductivity Aalysis, Vl. 13, pp. 93-103. 17
[5] Fare, Rlf, ad Daiel Primt (1995), Multi-utput Prducti ad Duality: Thery ad Applicatis, Kluwer Academic Publisher, Bst, Massachusetts, 1995. [6] Färe, Rlf ad Daiel Primt (2003), Lueberger Prductivity Idicatrs: Aggregati Acrss Firms, frthcmig, Jural f Prductivity Aalysis. [7] Hudgis, Lae Blume ad Daiel Primt (2003), Directial Techlgy Distace Fuctis: Thery ad Applicatis," preseted at the Cferece Aggregati, Efficiecy, ad Measuremet I Hr f R. Rbert Russell, Uiversity f Califria, Riverside, May 23-25, 2003 [8] Lueberger, David G. (1992), New Optimality Priciples fr Ecmic Efficiecy ad Equilibrium," Jural f Optimizati Thery ad Applicatis, Vlume 75, pp. 221-264, 1992. [9] Lueberger, David G. (1995), Micrecmic Thery, McGraw-Hill, New Yrk, New Yrk, 1995. [10] Shephard, Rald W. (1953), Cst ad Prducti Fuctis, Pricet Uiversity Press, Pricet, New Jersey, 1953. [11] Shephard, Rald W. (1970), Thery f Cst ad Prducti Fuctis, Pricet Uiversity Press, Pricet, New Jersey, 1970. [12] Shephard, Rald W. (1974), Idirect Prducti Fuctis, Mathematical Systems i Ecmics, N. 10, Meiseheirm am Gla: Verlag At Hai, 1974 18