Technology Gap, Efficiency, and a Stochastic Metafrontier Function

Transcription

1 Intrnatonal Journal of Busnss and Economcs, 00, Vol., No., Tchnology Gap, Effcncy, and a Stochastc Mtafrontr Functon Gorg E. Batts Unrsty of Nw England, Australa D. S. Prasada Rao Unrsty of Nw England, Australa Abstract Ths papr consdrs a stochastc mtafrontr functon to nstgat th tchncal ffcncs of frms n dffrnt groups that may not ha th sam tchnology. A dcomposton of output s prsntd nolng th tchnology gap and tchncal ffcncy ratos for frms n a group rlat to th bst practc n th ndustry. Ky word: JEL classfcaton: C5; D4. Introducton Th mtaproducton functon was frst ntroducd by Hayam (969) and Hayam and Ruttan (970, 97). As statd by Hayam and Ruttan (97, p. 8), Th mtaproducton functon can b rgardd as th nlop of commonly concd noclasscal producton functons. In thr dscusson of agrcultural productty across arous countrs, Ruttan t al. (978, p. 46) stat, W now dfn th mtaproducton functon as th nlop of th producton ponts of th most ffcnt countrs. Th concpt of a mtaproducton functon s thortcally attract bcaus t s basd on th smpl hypothss that all producrs n dffrnt groups (countrs, rgons, tc.) ha potntal accss to th sam tchnology. Followng th smnal work of Hayam and Ruttan (970), Mundlak and Hllnghausn (98) and Lau and Yotopoulos (989) mployd th approach to compar agrcultural productty across countrs. Som conomtrc adantags of applyng th mtaproducton functon ar dscussd by Lau and Yotopoulos (989), but th lack of comparabl data and th prsnc of nhrnt dffrncs across groups ar th major lmtatons of th ap- Rcd August 3, 00, accptd Octobr 3, 00. Corrspondnc to: School of Economcs, Unrsty of Nw England, Armdal NSW 35, Australa. Emal: gbatts@mtz.un.du.au. W ar gratful to th anonymous rfrs.

2 88 Intrnatonal Journal of Busnss and Economcs proach. Boskn and Lau (99) usd a nw framwork for th analyss of productty and tchncal progrss, basd on drct conomtrc stmaton of th aggrgat mtaproducton functon. Th concpt of a stochastc mtafrontr functon, usd n ths papr, opratonalss th standard mtaproducton functon approach. Th stochastc mtafrontr modl has an rror trm that comprss a symmtrc random rror and a non-ngat tchncal nffcncy trm, as n th stochastc frontr producton functon modl, orgnally proposd by Agnr, Loll, and Schmdt (977) and Musn and an dn Brock (977). Howr, a stochastc mtafrontr functon may not nlop th sparat producton frontrs for th dffrnt groups nold. It s possbl to us non-stochastc approachs to construct mtafrontr functons. A stochastc mtafrontr modl was adoptd by Gunaratn and Lung (00) and Sharma and Lung (000) n studs of th ffcncy of aquacultur farms n sral countrs. Sharma and Lung (000) usd th Batts and Coll (995) modl for th tchncal nffcncy ffcts n th stochastc mtafrontr functon n thr mprcal analyss of data on carp pond cultur n South-Asan countrs.. Stochastc Mtafrontr Modl Suppos that th nputs and outputs for frms n a gn ndustry ar such that stochastc frontr producton functon modls ar dfnd for dffrnt groups wthn th ndustry. Our analyss assums that thr ar sral wll-dfnd groups for th sam ndustry, such as dffrnt rgons wthn a country, dffrnt typs of ownrshp or thnc groups nold n producton, but not for dffrnt ndustrs wthn th sam country or among countrs. Suppos that, for th jth group, thr ar sampl data on N j frms that produc on output from th arous nputs and th stochastc frontr modl for ths group s dfnd by Y j j V j U j f ( x, β ), =,,, N j, () whr Y j dnots th output for th th frm n th jth group; x j dnots a ctor of functons of th nputs usd by th th frm n th jth group; th V j s ar assumd to b dntcally and ndpndntly dstrbutd as N( 0, ν ) -random arabls, ndpndnt of th U j s, whch ar dfnd by th truncaton (at zro) of th N( μ, j ) -dstrbutons, whr th j s ar dfnd by som approprat nffcncy modl,.g., on of th Batts and Coll (99, 995) modls. For smplcty, th subscrpt j s omttd hraftr, so that th modl for th jth group s gn by V x β V U U Y f x, β. () Th xprsson of quaton () assums that th xponnt of th frontr producton functon s lnar n th paramtr ctor, so that x s a ctor of functons of (logarthms of) th nputs for th th frm nold. Th Cobb-Douglas or translog

3 Gorg E. Batts and D. S. Prasada Rao 89 producton functons ar connnt for th prsntaton of th dcomposton to b prsntd n Scton 3 blow. Th stochastc mtafrontr modl for frms n all groups of th ndustry s xprssd by x β V U (, x β ) V U Y f, =,,, N, (3) R whr N N j s th total numbr of sampl frms n all R groups and th assumptons for th V s and th U s ar analogous to thos for th V s and th U s, r- j spctly. If th assumptons for th stochastc frontrs for th dffrnt groups assocatd wth quatons () and () ar rasonabl for gn sampl data, thn th corrspondng assumptons assocatd wth th stochastc mtafrontr modl of quaton (3) may not b approprat (.g., th V s may not b dntcally dstrbutd or all groups). Th paramtrs of a gn frontr for a group ar stmatd usng data from frms n that group. Th paramtrs of th mtafrontr modl ar stmatd usng data from frms n all groups (n th combnd data st). Th mtafrontr of quaton (3) s consdrd to b an nlop functon of th stochastc frontrs of th dffrnt groups such that t s dfnd by all obsratons n th dffrnt groups n a way that s consstnt wth th spcfcatons of a stochastc frontr modl. Obsratons on nddual frms n th dffrnt groups may b gratr than th dtrmnstc componnt of th stochastc frontr modl, but datons from th stochastc frontr outputs ar du to nffcncy of th frms n th dffrnt groups. Th stochastc frontrs for th dffrnt groups and that of th mtafrontr would gnrally b assumd to b of th sam functonal form (.g., Cobb-Douglas or translog), but thr ar no problms of aggrgaton as wth th rlatonshp btwn frm and ndustry functons. In th dscusson blow, th paramtrs (for th frontr for th jth group) and (for th mtafrontr) ar assumd known. Th productty and tchncal ffcncy of frms n th jth group can b nstgatd usng thr th frontr for th jth group or th mtafrontr. Th maxmum-lklhood stmats of th paramtrs of th stochastc mtafrontr modl (3) do not ncssarly rsult n th stmatd functon bng an nlop of th stmatd producton frontrs for th dffrnt groups. That s, th maxmsaton of th lklhood functon for all obsratons dos not guarant that th stmatd mtafrontr nlops th stmatd frontrs for th dffrnt groups. Thus, for som groups, th mtafrontr functon could ha alus lss than th corrspondng frontr for a gn group. Howr, t s possbl to constran th stmaton of th mtafrontr, such that t s an nlop of obsratons for ffcnt frms n all groups. A constrand mathmatcal programmng algorthm, such as n data nlopmnt analyss (DEA) could b usd n th stmaton of a mtafrontr. Howr, such non-stochastc frontr mthods do not adquatly account for th prsnc of tradtonal random rrors and assum that all datons from th frontr ar du to nffcncy. W ar workng on a possbl mthod of mposng

4 90 Intrnatonal Journal of Busnss and Economcs such rstrctons n th contxt of a stochastc mtafrontr modl, but ths cannot b ncludd n ths papr bcaus t s n th dlopmnt stag. Clarly, ths s an mportant ssu that rqurs furthr rsarch. Th comparson of tchncal ffcncs of frms n dffrnt groups s a common problm that has bn of concrn to many rsarchrs. 3. Tchnology Gap and Effcncy Lls Th obsrd output for th th frm n th jth group can b xprssd by x β V U x β V U Y or Y, as spcfd by quatons () or (3), rspctly, from whch t follows that th rlatonshp x β V U x β V U s satsfd. It s xpctd that th dtrmnstc alus x and x β satsfy th nqualty x β x β bcaus x β s from th mtafrontr. If th mtafrontr wr stmatd to b an nlop functon for ffcnt frms, thn th rlatonshp would b satsfd by th stmatd functons. Ths rlatonshp can b rwrttn as x x V V U. U (4) Th thr ratos on th rght-hand sd of ths quaton ar calld th tchnology gap rato (TGR), th random rror rato (RER), and th tchncal ffcncy rato (TER),.., TGR x β V x ( β β ), RER V V x β V, and TER U U TE TE. Th tchnology gap rato ndcats th tchnology gap for th gn group accordng to currntly aalabl tchnology for frms n that group, rlat to th tchnology aalabl n th whol ndustry. Ths rato and th tchncal ffcncs (and, hnc, th tchncal ffcncy rato) can b stmatd for ach nddual frm. Th tchncal ffcncy of frm rlat to th frontr for ts group TE U can b stmatd by TE EU E V U. Th tchncal ffcncy of frm can b stmatd rlat to th mtafrontr by TE U E E V U. Clarly, th dntty E ( E x β β ) s satsfd. Consdr th tchncal ffcncy rato TER U U TE TE. Gnrally ths rato s xpctd to b gratr than or qual to unty. Bcaus U and U ar random arabls, thr s a non-zro probablty that th rato TER s lss than unty. Now, TE TE f and only f U U, or U U 0. But U U x ( β β ) V V. Th probablty that U s no gratr than U s

5 Gorg E. Batts and D. S. Prasada Rao 9 PU U 0 P( V V x ( β β )) ( x( β β ), f V and V ar ndpndnt normal random rrors, whr (.) rprsnts th dstrbuton functon for th standard normal dstrbuton. Clarly, th gratr x β xcds x β, th smallr th probablty that U s lss than U. Furthr, t can b shown that EY x x x β EV EU xβ μ μ μ. Anothr dntty rlatonshp, basd on th xpctd output undr th partcular group frontr and th mtafrontr, s drd as follows: xβ xβ E E U U xβ whr TGR = xβ, (5) x( β β ) s th man tchnology gap rato; RER s th man random rror rato; and EU TER s th man tchncal ffcncy rato. EU Clarly, from quaton (5), only two of ths ratos nd to b ndpndntly stmatd n any mprcal applcaton. 4. Conclusons Wth th man objct of prodng comparabl tchncal ffcncy scors for frms across groups, th tchncal ffcncs of frms can b stmatd usng a stochastc mtafrontr modl. In addton, w prsnt a mor transparnt analyss of th tchnology gap of dffrnt groups and thr ffcncy lls by usng a dcomposton rsult. Th man tchnology gap, random rror, and tchncal ffcncy ratos g addtonal xplanaton compard wth th analyss basd only on stochastc frontr functons for th dffrnt groups. Th tchnology gap rato plays an mportant part n xplanng th ablty of th frms n on group to compt wth othr frms from dffrnt groups wthn th ndustry. Ths rato prods an stmat of th tchnology gap btwn th groups and th ndustry as a whol. Th analyss of tchncal ffcncy usng a stochastc mtafrontr modl also

6 9 Intrnatonal Journal of Busnss and Economcs gs a bttr orw of th comparablty of tchncal ffcncy scors across groups. How tchncal nffcncy changs or tm s obously assocatd wth th modl that s assumd for th nffcncy ffcts. Emprcal analyss wth altrnat stochastc frontr modls ar clarly dsrabl. Batts, Rao, and Walujad (00) apply th mthods of ths papr to nstgat th tchnology gap and tchncal ffcncs of frms n th garmnt ndustry n dffrnt rgons of Indonsa. Furthr thortcal rsarch s dsrabl for th stmaton of stochastc mtafrontr modls. Rfrncs Agnr, D., C. A. K. Loll, and P. Schmdt, (977), Formulaton and Estmaton of Stochastc Frontr Producton Functon Modls, Journal of Economtrcs, 6, -37. Batts, G. E. and T. J. Coll, (99), Frontr Producton Functons, Tchncal Effcncy and Panl Data: Wth Applcaton to Paddy Farmrs n Inda, Journal of Productty Analyss, 3, Batts, G. E. and T. J. Coll, (995), A Modl for Tchncal Inffcncy Effcts n a Stochastc Frontr Producton Functon for Panl Data, Emprcal Economcs, 0, Batts, G. E., D. S. P. Rao, and D. Walujad, (00), Tchncal Effcncy and Productty of Garmnt Frms n Dffrnt Rgons n Indonsa: A Stochastc Frontr Analyss Usng a Tm-aryng Inffcncy Modl and a Mtaproducton Functon, CEPA Workng Paprs, No. 7/00, Cntr for Effcncy and Productty Analyss, School of Economcs, Unrsty of Nw England, Armdal, pp. 6. Boskn, M. J. and L. J. Lau, (99), Captal, Tchnology and Economc Growth, n Tchnology and th Walth of Natons, N. Rosnbrg, R. Landau, and D. C. Mowry, ds., Stanford: Stanford Unrsty Prss. Gunaratn, L. H. P. and P. S. Lung, (00), Asan Black Tgr Shrmp Industry: A Productty Analyss, Chaptr 5 n Economcs and Managmnt of Shrmp and Carp Farmng n Asa: A Collcton of Rsarch Paprs Basd on th ADB/NACA Farm Prformanc Sury, Lung, P. S. and K. R. Sharma, ds., Bangkok: Ntwork of Aquacultur Cntrs n Asa-Pacfc (NACA), pp. 40. Hayam, Y., (969), Sourcs of Agrcultural Productty Gap among Slctd Countrs, Amrcan Journal of Agrcultural Economcs, 5, Hayam, Y. and V. W. Ruttan, (970), Agrcultural Productty Dffrncs Among Countrs, Amrcan Economc Rw, 60, Hayam, Y. and V. W. Ruttan, (97), Agrcultural Dlopmnt: An Intrnatonal Prspct, Baltmor: John Hopkns Unrsty Prss. Lau, L. J. and P. A. Yotopoulos, (989), Th Mta-Producton Functon Approach to Tchnologcal Chang n World Agrcultur, Journal of Dlopmnt Economcs, 3, 4-69.

7 Gorg E. Batts and D. S. Prasada Rao 93 Musn, W. and J. an dn Brock, (977), Effcncy Estmaton from Cobb-Douglas Producton Functons wth Composd Error, Intrnatonal Economc Rw, 8, Mundlak, Y. and R. Hllnghausn, (98), Th Intrcountry Agrcultural Producton Functon: Anothr Vw, Amrcan Journal of Agrcultural Economcs, 64, Ruttan, V. W., H. P. Bnswangr, Y. Hayam, W. W. Wad, and A. Wbr, (978), Factor Productty and Growth: A Hstorcal Intrprtaton, n Inducd Innoaton: Tchnology, Insttuton, and Dlopmnts, Bnswangr, H. P. and V. W. Ruttan, ds., Baltmor: Johns Hopkns Unrsty Prss. Sharma, K. R. and P. S. Lung, (000), Tchncal Effcncy of Carp Pond Cultur n South Asa: An Applcaton of Stochastc Mta-Producton Frontr Modl, Aquacultur Economcs and Managmnt, 4,