CHAPTER 3 LITERATURE REVIEW ON EFFICIENCY, MEASUREMENT AND EMPIRICAL APPLICATIONS

Transcription

1 CHAPTER 3 LITERATURE REVIEW ON EFFICIENCY, MEASUREMENT AND EMPIRICAL APPLICATIONS 3.1 Introducton The objectve of ths chapter s to gve an overvew of the concept of effcency and fronter models, the dfferent approaches to ts measurement n the context of fronter models and emprcal studes on effcency. Approaches to effcency measurement are broadly specfed nto parametrc and non parametrc approaches. Gven the large volume of theoretcal and emprcal lterature n the feld of effcency measurement, the revew of emprcal studes s further subdvded nto three namely: a revew of emprcal comparatve studes n agrculture, a revew of emprcal comparatve studes n other sectors where the dstance functon approach was used and fnally a revew of emprcal studes n Ngeran agrculture. The revew s ntended not only to provde a proper understandng of the specfc area of research but t also helps the researcher to establsh a vvd framework to be employed for analyss. 3.2 The Concept of Effcency and Fronter Models In mcroeconomc theory a producton functon s defned n terms of the maxmum output that can be produced from a specfed set of nputs, gven the exstng technology avalable to the frms nvolved (Battese, 1992). The maxmum possble output becomes relevant n order to answer certan economc questons such as the measurement of effcency of frms, hence the ntroducton of fronter producton functons whch estmates the maxmum output as functon of nputs. Smlarly, a cost fronter functon would gve the mnmum cost as a functon of output quantty and nput prces. The papers by Debreu (1951) and Koopmans (1951) mark the orgn of dscusson on the measurement of productvty and effcency n the economc lterature. The work 44

2 of Debreu and Koopmans was frst extended by Farrell (1957) n order to perform the measurement of productvty and effcency. The productvty of an economc agent can be measured smply as a scalar rato of outputs to nputs that the agent uses n ts producton process. Productvty could be measured ether as partal productvty such as yeld per hectare (land productvty) or output per person (labour productvty) or more approprately as total factor productvty (TFP) whch s defned as rato of aggregate outputs to aggregate nputs. An economc agent s productvty may vary based on dfferences n producton technology, n the effcency of the producton process, n the envronment n whch producton occurs, and fnally n the qualty of nputs used by the agent (Haghr, 2003). On the other hand, effcency s measured by comparng observed and optmal values of the agent s outputs and nputs. Pror to Farrell s work, efforts were made to measure effcency by nterpretng the average productvty of nputs, then to constructon of effcency ndexes. However, these methods were found unsatsfactory by economsts and agrcultural economsts as the methods suffered from one shortcomng to another. The use of the tradtonal least squares methods for estmatng the producton functon has been crtqued as ths s not consstent wth the defnton of the producton functon. The estmated functons could at best be descrbed as average or response functons because such regresson estmates the mean output (rather than the maxmal output) gven quanttes of nputs (Schmdt, 1986). Ths led to the development of a better-founded theoretcal method for measurng effcency,.e. the fronter method. Fronters models are descrbed as boundng functons (Coell, 1995b). The fronter approach holds a number of advantages over average or response functons as well as over non-fronter models. There are two man benefts that result from estmatng fronter functons, as compared to estmatng average functons usng ordnary least squares (OLS) approach. Frst, when a fronter functon s estmated, the result s strongly nfluenced by the best performng frm, and therefore the fronter reflects the technology set that the most effcent frm employs. However, the estmaton of an average functon only reflects the technology set employed by an average frm. Second, fronter functons provde a useful performance benchmark. These functons normally represent best practce technology, aganst whch the effcency of other frms wthn the ndustry can be measured. Fronter models also provde a number of advantages over non-fronter models lke the one proposed by 45

3 Lau and Yotopoulos (1971). A non-fronter model yelds effcency measures for groups of frms, whereas a fronter model can provde frm specfc effcency measures to the researcher. Another advantage of the fronter methodology s that the word fronter s consstent wth the theoretcal defnton of a producton, cost, and proft functon,.e., a soluton to a maxmum and mnmum problem. These advantages make the fronter methodology popular n appled economc research (Forsund et al., 1980; Bravo-Ureta and Pnhero, 1993; Haghr, 2003; Alene, 2003). Fronter functons can be classfed based on certan crtera. Frst, based on the way the fronter s specfed, fronters may be specfed as parametrc functon of nputs or non-parametrc. Second, t may be specfed as an explct statstcal model of the relatonshp between observed output and the fronter or t may not. Fnally, a fronter functon can be classfed accordng to how one nterprets the devaton of a group of agents or frms from the best performng agents n the sample. In ths sense, fronter functons can be ether determnstc or stochastc. In the sub-sectons that follow, we broadly classfy the fronter models nto parametrc or non-parametrc fronters. 3.3 Non-Parametrc Fronter Approach A non-parametrc approach nether specfes a functonal form for the producton technology nor makes an assumpton about the dstrbuton of the error terms. In other words t s robust wth respect to the partcular functonal form and to the dstrbuton assumptons. The non-parametrc approach s manly determnstc n nature. In a determnstc producton fronter model, output s assumed to be bounded from above by a determnstc (non-stochastc) fronter. However, the possble nfluence of measurement errors and other statstcal nose upon the shape and postonng of the estmated fronter s not accounted for. The orgnal work of Farrell (1957) serves an mportant startng pont for dscusson of non-parametrc fronters. Farrell llustrated the measurement of effcency usng an nput-orented approach. Hs argument s emboded n fgure 3.1. Ths llustraton was done by consderng a frm usng two nputs x 1 and x 2 to produce output y, such that the producton fronter s y = f x 1, x ) Assumng constant returns to scale, then one ( 2 46

4 can wrte = f ( x / y, x / ), that s the fronter technology can be characterzed by a y unt soquant and ths s denoted S S n fgure 3.1. Knowledge of the unt soquant of a fully effcent frm permts the measurement of techncal effcency. For a gven frm usng x, *) defned by pont A x / y, x * / ) to produce a unt of output ( 1 x2 ( 1 2 y y *, the rato OQ/OA measures techncal effcency and t defnes the ablty of a frm to maxmze output from a gven set of nputs. The rato measures the proporton of ( 1 x2 x, ) needed to produce y *. Techncal effcency takes a value between zero and one and therefore provdes an ndcaton of techncal neffcency. Thus, the techncal neffcency of the frm, 1-OQ/OA, measures the proporton by whch x, *) could ( 1 x2 be reduced (holdng the nput rato x 1 / x2 constant) wthout reducng output. A frm that s fully techncally effcent would le on the effcent soquant (example, pont Q) and t takes a value of 1. x 2 /y S A P R Q Q* S 0 P x 1 /y Fgure 3.1: Techncal, Allocatve and Economc Effcency Further, Farrell demonstrated that the unt soquant can provde a set of standards for measurng allocatve (referred to as prce effcency by Farrell) effcency. Let represent the rato of nput prces. Then the rato OR/OQ measures the allocatve effcency (the ablty of a frm to use nputs n optmal proportons, gven the respectve prces at pont A). Correspondngly, allocatve neffcency s 1- OR/OQ. The dstance RQ s the reducton n producton costs whch would have been acheved had producton occurred at Q*- the allocatvely and techncally effcent pont, rather PP ' 47

5 than Q- the techncally effcent, but allocatvely neffcent pont. Fnally, the rato OR/OA measures the economc effcency (referred to as overall effcency by Farrell) and correspondngly 1-OR/OA measures the total neffcency. The dstance RA s the cost reducton achevable whch s obtaned from movng from A (the observed pont) to Q* (the cost mnmzng pont). In ths approach, the effcent unt soquant s not observable; t must be estmated from a sample of observatons. The approach s non-parametrc because Farrell smply constructs the free dsposal convex hull of the observed nput-output ratos by lnear programmng technques whch are supported by a sub-set of the sample, wth the rest of the sample ponts lyng above t. Accordng to Forsund et al. (1980), the major advantage of non-parametrc approach s that no functonal form s mposed on the data. One dsadvantage of the approach s that the fronter s computed from a supportng subset of observatons, and s therefore partcularly susceptble to extreme observatons and measurement error. A second dsadvantage s that the estmated functons have no statstcal propertes upon whch nferences can be made; however, recent developments are attemptng to overcome ths drawback. Farrell s approach has been extended by Charnes et al. (1978) gvng rse to what s known as data envelopment analyss (DEA). The technque envelopes observed producton possbltes to obtan an emprcal fronter and measures effcency as the dstance to the fronter. Effcent frms are those that produce a certan amount of or more outputs whle spendng a gven amount of nputs, or use the same amount of or less nputs to produce a gven amount of outputs, as compared wth other frms n the test group. Ths approach generalzes Farrell s approach of computng the effcency fronter as a pecewse-lnear convex hull n the nput coeffcent space to multple outputs. Charnes et al. (1978) reformulated Farrell s approach nto calculatng the ndvdual nput savng effcency measures by solvng a lnear programmng problem for each unt under the constant returns to scale (CRS) assumpton whle Banker et al. (1984) extended t to the case of varable returns to scale (VRS) snce mperfect competton, fnancal constrants may cause a frm not to be operatng on an optmal scale, the assumpton upon whch CRS s approprate. Charnes et al. (1978) proposed 48

6 a model whch had an nput-orentaton. The DEA can be consdered as a nonparametrc approach to estmaton of dstance functons (Färe et al., 1985; 1994). Assumng there s data on K nputs and M outputs on each of N frms. For the th frm, these are represented by the vectors x and y, respectvely. The K x N nput matrx, X and the M x N output matrx, Y, represent the data of all N frms. The purpose of the approach s to construct a non-parametrc envelopment fronter over the data ponts such that all observed ponts le on or below the producton fronter. The nput-orented constant returns to scale DEA fronter s defned by the soluton to N lnear programs of the form: mn θ, θ,λ subject to + Yλ 0, y θx Xλ 0, (3.1) λ 0 where θ s a scalar and λ s an Nx1 vector of constants. The value of θ s an ndex of techncal effcency for the th frm and wll satsfy 0 θ 1, wth value of 1 ndcatng a pont on the fronter and hence a techncally effcent frm, accordng to Farrell (1957) defnton. Thus, proportonally reduced wthout any loss n output. 1 θ measures how much a frm s nputs can be However, the assumpton of CRS s correct only as long as frms are operatng at an optmal scale (Coell et al, 2002). Usng the CRS DEA model when frms are not operatng at ther optmal scale wll cause the techncal effcency measures to be nfluenced by scale effcences and thus the measure of techncal effcency wll be ncorrect. The CRS lnear programmng problem can easly be modfed to account for varable returns to scale by addng the convexty constrant: N1 ' λ = 1to equaton (3.1) to provde an nput-orented VRS model: mn θ θ,λ 49

7 subject to + Yλ 0, y θx Xλ 0, (3.2) N 1 ' λ = 1 λ 0 where N1 s an Nx1 vector of ones. Ths approach forms a convex hull of ntersectng planes whch envelope the data ponts more tghtly than the CRS concal hull and thus provde techncal effcency scores whch are greater than or equal to those obtaned usng the CRS model. The output-orented models are very smlar to ther nput-orented counterparts. For nstance, the output-orented VRS model s defned by soluton to N lnear programs of the form: max φ φ,λ subject to φ + Yλ 0, y x Xλ 0, (3.3) N 1 ' λ = 1 λ 0 where 1 φ <, and φ s the proportonal ncrease n output that could be acheved by the th frm, wth nput held constant. 1 / φ defnes a techncal effcency score whch vares between zero and one. The CRS output-orented model can be defned smlarly by removng the convexty constrant, N 1 ' λ = 1 from equaton (3.3). In the nput-orented models, the method sought to dentfy techncal neffcency as a proportonal reducton n nput usage. They are nput-orented because they try to fnd out how to mprove the nput characterstcs of the frm concerned so as to become effcent. The output-orented measure sought to dentfy techncal neffcency as a proportonal ncrease n output producton. The nput and output orentatons provde the same value under CRS but are unequal under the assumpton of a VRS. Thus, the nput- and output-orented models wll estmate exactly the same fronter and 50

8 therefore, by defnton, dentfy the same set of frms as beng effcent. It s only the effcency measures assocated wth the neffcent frms that may dffer between the two methods. Gven that lnear programmng cannot suffer from such statstcal problems as smultaneous equaton bas, the choce of an approprate orentaton s not very crucal. Essentally, one should select an orentaton accordng to whch quanttes (nputs or outputs) the managers have most control over. In many nstances, the choce of orentaton wll have only mnor nfluences upon the scores obtaned (Coell, 1995b, Coell and Perelman, 1999). Wth avalablty of prce nformaton, t s possble to consder a behavoural objectve, such as cost mnmzaton or revenue maxmzaton so that both techncal and allocatve effcency can be measured. For the case of a VRS cost mnmzaton, one would run the nput-orented DEA model set out n equaton (3.2) to obtan techncal effcency (TE). One would then run the followng cost mnmzaton DEA mn λ, x * ' x * w, subject to + Yλ 0, y x * Xλ 0, (3.4) N 1 ' λ = 1 λ 0 where w s a vector of nput prces for the th frm and x * s the cost mnmzng vector of nput quanttes for the th frm gven the nput prces w and the output levels y and ths s calculated by the lnear programmng. The total cost effcency (CE) or economc effcency of the th frm would be calculated as CE w ' x * = (3.5) w ' x That s, the rato of mnmum cost to observed cost. One can then use equaton (3.5) to calculate the allocatve effcency resdually as 51

9 CE AE = (3.6) TE Ths procedure wll nclude any slacks nto the allocatve effcency measure. Ths s often justfed on the grounds that slack reflects an napproprate nput mx (Ferrer and Lovell, 1990). The am of DEA analyss s not only to determne the effcency rate of the unts revewed, but also to fnd target values for nputs and outputs for an neffcent unt. After reachng these values, the unt would arrve at the threshold of effcency. The major dsadvantage of the determnstc DEA approach s that t takes no account of possble nfluence of measurement error and other nose n the data and as such t has been argued that t produces based estmates n the presence of measurement error and other statstcal nose. However, t has the advantage of removng the necessty to make arbtrary assumptons about the functonal form of the fronter and the dstrbutonal assumpton of the error term. Wth DEA, multple output technologes can be examned very easly wthout aggregaton. As t has been stated earler, one of the man drawbacks of non-parametrc technques s ther determnstc nature. Ths s what tradtonally has drven specalsed lterature on ths ssue to descrbe them as non-statstcal methods. Nevertheless, recent lterature has shown that t s possble to defne a statstcal model allowng for the determnaton of statstcal propertes of the non-parametrc fronter estmators (Murllo-Zamorano, 2004). For nstance, DEA models wth stochastc varatons have recently receved attenton (Banker, 1993; Land et al., 1993; Sengupter 2000a; Smar and Wlson, 1998, 2000a, 2000b; Huang and L, 2001; Kao and Lu, 2009; Shang et al., 2009). Smar and Wlson (1998, 2000a, 2000b) for example, methodcally studed statstcal propertes of DEA models, and developed bootstrap algorthms whch can be used to examne the statstcal propertes of effcency scores generated through DEA. Therefore, one mght conclude that today statstcal nference based on nonparametrc fronter approaches to the measurement of economc effcency s avalable ether by usng asymptotc results or by usng bootstrap. However, a couple of man ssues stll reman to be solved, namely the hgh senstvty of non-parametrc approaches to extreme values and outlers, and also the way for allowng stochastc 52

10 nose to be consdered n a non-parametrc fronter framework (Murllo-Zamorano, 2004). 3.4 Parametrc Fronter Approach The parametrc approach nvolves a specfcaton of a functonal form for the producton technology and an assumpton about the dstrbuton of the error terms. The major advantage of the parametrc approach compared to the non-parametrc approach s the ablty to express the fronter technology n a smple mathematcal form. However, the parametrc approach mposes structure on the fronter that may be unwarranted. The parametrc approach often mposes a lmtaton on the number of observatons that can be techncally effcent. For example, n the case of homogeneous Cobb-Douglas form, when the lnear programmng algorthm s used, there wll n general be only as many techncally effcent observatons as there are parameters to be estmated (Forsund et al, 1980). Ths approach can be subdvded nto determnstc and stochastc fronters. The parametrc determnstc approach s further subdvded nto statstcal and non-statstcal methods Determnstc Non-Statstcal Fronters Few people adhered to the non-parametrc approach by Farrell (1957). Almost as an after thought, Farrell (1957) proposed a second approach. In ths approach, Farrell proposed computng a parametrc convex hull of the observed nput-output ratos. He recommended the Cobb-Douglas producton functon for ths purpose gven the lmted selecton of functonal form then. He acknowledged the undesrablty of mposng a specfc (and restrctve) functonal form on the fronter but also noted the advantage of beng able to express the fronter n a smple mathematcal form. Ths suggeston was however not followed up by Farrell. Agner and Chu (1968) were the frst to follow Farrell s suggeston. In order to express the fronter n a mathematcal form, they specfed a Cobb-Douglas producton fronter, and requred all observatons to be on or beneath the fronter. Ther model may be wrtten as: 53

11 ln y = ln f ( x ; α ) u ; u 0 (3.7) where y s the output of the th sample frm; x s the nputs of the th frm, u s a one-sded non-negatve random varable assocated wth frm-specfc factors that contrbute to the th frm nablty to attan maxmum effcency of producton. The one sded error term, u forces y f (x). The elements of the parameter vector, α, may be estmated ether by lnear programmng (mnmzng the sum of the absolute values of the resduals subject to the constrant that each resdual s non-postve) or by quadratc programmng (mnmzng the sum of squared resduals, subject to the same constrant). Although Agner and Chu (1968) dd not do so, the techncal effcency of each observaton can be computed drectly from the vector of resduals, snce u represents techncal effcency. A major problem wth ths approach s that t produces estmates that lack statstcal propertes. That s, the programmng procedure produces estmates wthout standard errors, t-ratos, etc. Ths s because no statstcal assumptons are made about the regressors or the dsturbance term n equaton (3.7) and therefore nferences cannot be obtaned Determnstc Statstcal Fronters The prevous models were crtqued on ther lack of statstcal propertes. Ths problem can be addressed by makng some assumptons about the dsturbance term. The model n equaton (3.7) can be wrtten as u ln y = f ( x) e, (3.8) or [ f ( x u] ln y = ln ), (3.9) where u 0, mplyng 0 u 1 e, ln[ ( x) ] f s lnear n the Cobb-Douglas case presented n equaton (3.7). Some assumptons are usually made about u and x and that s, that u are ndependently and dentcally dstrbuted (d), wth mean µ and 54

12 fnte varance and that x s exogenous and ndependent of u. Any number of dstrbutons for u (or u e ) could be specfed. Agner and Chu (1968) dd not explctly assume such a model though t seems clear t was assumed mplctly. However, the frst to explctly propose ths type of model was Afrat (1972), who proposed a two-parameter beta dstrbuton for u e, and that the model be estmated by maxmum lkelhood method. Ths amounts to gamma dstrbuton for u, as consdered further by Rchmond (1974). On the other hand Schmdt (1976) has demonstrated that f u s exponental, then Agner and Chu s lnear programmng procedure s maxmum lkelhood, whle ther quadratc programmng procedure s maxmum lkelhood f u s half-normal. In the fronter settng, there are some problems wth maxmum lkelhood. Frst, maxmum lkelhood estmates (MLE) depend on the choce of dstrbuton for u such that dfferent assumptons yeld dfferent estmates. Ths s a problem because there are no good a pror arguments for choce of any partcular dstrbuton. Second, the range of the dependent varable (output) depends on the parameters to be estmated (Schmdt, 1976). Ths s because y f (x) and f (x) nvolves the parameters whch are to be estmated. For any one-sded error dstrbuton, y f (x) volates one of the usual regularty condtons for consstent and asymptotc effcency of maxmum lkelhood estmators (namely, that the range of the random varable should not depend on the parameters). Thus, the statstcal propertes of the MLE s are n general uncertan. Greene (1980a) fnds suffcent condtons on the dstrbuton of u for the MLE s to have ther usual desrable asymptotc propertes: () f g s the densty of u, g(0) = 0,.e. the densty of u s zero at u = 0 and ( ) g ( u) 0 as u 0,.e. the dervatve of the densty of u wth respect to ts parameters approaches zero as u approaches zero. However, as Schmdt (1986) noted, t s clearly not desrable that one s assumptons about the error term be governed by the need to satsfy such condtons. An alternatve method of estmaton based on ordnary least squares was frst proposed by Rchmond (1974) and s called corrected OLS or COLS. Suppose equaton (3.9) s assumed to be lnear (Cobb-Douglas) and lettng µ be the mean of u, then 55

13 n ln y = ( α 0 µ ) + α ln x ( u µ ) (3.10) = 1 where the new error term has zero mean. Snce the error term satsfes all the usual deal condtons except normalty, equaton (3.10) can be estmated by OLS to obtan best lnear unbased estmates of ( α 0 µ ) and of α. If a specfc dstrbuton s assumed for u, and f the parameters of the dstrbuton can be derved from hgherorder (second, thrd, etc.) central moments, then these parameters can be consstently estmated from the moments of the OLS resduals. Snce µ s a functon of these parameters, t can also be estmated consstently, and ths estmate can be used to correct the OLS constant term, whch s consstent estmate of ( α µ ). Thus, COLS provdes consstent estmates of all the parameters of the fronter. However, ths technque poses some dffcultes. Frst, some of the resduals may stll have wrong sgns after correctng the constant term so that these observatons end up above the estmated producton fronter. Ths makes COLS seem not to be a very good technque for computng techncal effcency of ndvdual observatons. There are two ways of resolvng ths problem namely, by use of stochastc fronter approach or to estmate equaton (3.10) by OLS, then correct the constant term not as above, but by shftng t up untl no resdual s postve, and one s zero. Another dffculty wth COLS technque s that the correcton to the constant term s not ndependent of the dstrbuton assumed for u. That s, dfferent assumptons yelds systematcally dfferent correctons for the constant term, and systematcally dfferent estmates of techncal effcency, except for the specal case var (u ) =1. However, ths problem agan can be resolved by shftng the functon upward untl no resdual s postve, and one s zero Stochastc Fronters They emerged as an mprovement over average functons and determnstc fronters. In the determnstc fronters, all varatons n the frm performance are attrbuted solely to varaton n frm effcences relatve to the common famly of fronters, be t producton, cost or proft fronters. Thus, the dea of a determnstc fronter shared by all frms gnores the very real possblty that a frm s performance may be affected by 56

14 factors that are entrely outsde ts control such as bad weather, nput supply breakdowns etc as well as factors under ts control (neffcency). To lump these effects of exogenous shocks, both fortunate and unfortunate, together wth the effects of measurement error and neffcency nto a sngle one-sded error term, and to label the mxture neffcency s questonable and s a major weakness of determnstc fronters. Forsund et al. (1980) noted that ths concluson s renforced f one consders also the statstcal nose that every emprcal relatonshp contans. The standard nterpretaton s that frst, there may be measurement error on the dependent varable. Second, the equaton may not be completely specfed wth the omtted varables ndvdually unmportant. Both of these arguments hold just as well for producton functons as for any knd of equaton, and t s dubous at best not to dstngush ths nose from neffcency, or to assume that nose s one-sded. It s on ths bass that the stochastc fronter (composed error) model was ndependently proposed by Agner et al. (1977) and Meeusen and van den Broeck (1977). The vtal dea behnd the stochastc fronter model s that the error term s composed of two parts. A symmetrc component permts random varaton of the fronter across frms, and captures the effects of measurement error, other statstcal nose, and random shocks outsde the control of the frm. A one-sded component captures the effects of neffcency relatve to the stochastc fronter. The stochastc fronter functon may be defned accordng to Battese (1992) as: y = f (, α) exp ε ), = 1,... N (3.11) x ( where ε = v u. (3.12) The stochastc fronter s f (, α) exp v ), x ( y s the output of the th frm and s bounded above by the stochastc quantty, x are the nputs of the th frm. ε s a random varable. v s the random error havng zero mean, and s assocated wth random effects of measurement errors and exogenous shocks that cause the 57

15 determnstc kernel f (, α) to vary across frms. Techncal neffcency s captured x by the one-sded error component exp u ), where u 0 mplyng that all ( observatons must le on or beneath the stochastc producton fronter. The random errors, v were assumed to be ndependently and dentcally dstrbuted as 2 N(0, σ v ) random varables and ndependent of the u s, whch were assumed to be non-negatve truncatons of the half-normal dstrbuton.e., N(0, σ ) or exponental 2 σ u dstrbuton.e. EXP ( µ, ). Agner et al. (1977) consdered half-normal and exponental dstrbutons but Meeusen and van den Broeck (1977) consdered exponental dstrbuton only. Stevenson (1980) has shown how the half-normal and exponental dstrbutons can be generalzed to truncated normal ( N( µ, σ u ) ) and gamma dstrbutons, respectvely. There was a tendency for researchers to use the half-normal and truncated normal dstrbutons probably because of ease of estmaton and nterpretaton and more so, as there were no standard tests for dstrbuton selecton. However, Lee (1983) proposed a Lagrange-Multpler test to assess dfferent dstrbutons for the neffcency term. Gven the assumptons of the stochastc fronter model (3.11), nference about the parameters of the model can be based on the maxmum lkelhood estmators because the standard regularty condtons are satsfed. 2 u 2 Techncal effcency of an ndvdual frm s defned n terms of the rato of the observed output to the correspondng fronter output, condtonal on the levels of nputs used by that frm. Thus, the techncal effcency of frm n the context of the stochastc producton functon expressed n equatons (3.11) and (3.12) s gven as y y TE. = = = exp( u ) * y f ( x ; α)exp( v ) * (3.13) The predcton of techncal effcences of ndvdual frms assocated wth the stochastc fronter producton functon (3.11) was consdered mpossble untl the appearance of Jondrow et al. (1982). Followng Jondrow et al. (1982) and Battese and 58

16 Corra (1977) reparameterzaton, the frm specfc techncal effcency can be predcted by the condtonal expectaton of the non-negatve random varable, gven that the random varable, ε, s observable. The techncal effcency of the th frm s then gven by: u, 1/ 2 σ uσ v f ( ) ε γ E ( u / ε ) = (3.14) σ 1 F( ) σ 1 γ where ε are the estmated resduals for each frm, f ( ) and F( ) are the values of the standard normal densty functon and standard normal dstrbuton functon, respectvely, evaluated at 1/ 2 ε γ. The parameters of the model,.e. α, σ 1 γ 2 2 σ + = σ v σ u and 2 2 /σ 2 u γ = σ can be obtaned from the maxmum lkelhood estmaton of equaton (3.11). γ s bounded between zero and one and t explans the total varaton of output from the fronter whch can be attrbuted to techncal neffcency. The estmates of v and u can be obtaned by substtutng the estmates of ε, γ, and σ. Thus, the techncal effcency of ndvdual frms can be measured as TE = exp( E( / ε ) whch represents the level of techncal effcency of the th u frm relatve to the fronter frm. However Battese and Coell (1988) derved the best predctor of TE gven as 1 F( σ A + γε / σ A) 2 E( u / ε ) = exp( γε + σ A / 2). 1 F( γε / σ A) One can test whether any form of stochastc fronter producton s needed at all by testng the sgnfcance of the γ parameter. If the null hypothess, that γ equals zero, 2 s accepted, ths would ndcate that σ u s zero and hence that the u should be removed from the model, leavng a specfcaton wth parameters that can be consstently estmated usng ordnary least squares (Coell, 1996a). There are two approaches to estmatng the neffcency effect models, that s, the second part of the stochastc fronter models that provdes explanaton for varaton n effcency of frms. These may be estmated wth ether a one step procedure or a two 59

17 step procedure. In a one step procedure estmates of all the parameters are obtaned n one step. The neffcency effects are defned as a functon of the frm specfc factors (as n the two-stage approach) but they are then ncorporated drectly nto the MLE. That s, both the producton fronter and the neffcency effect models are estmated smultaneously. For the two-step procedure, the producton fronter s frst estmated and the techncal effcency of each frm s derved. These are subsequently regressed aganst a set of varables, z, whch are hypotheszed to nfluence the frms' effcency. The two-stage procedure has been crtqued of nconsstency n the assumptons about the dstrbuton of the neffcences. Ths s because n the frst stage, the neffcences are assumed to be ndependently and dentcally dstrbuted (d) n order to estmate ther values. However, n the second stage, the estmated neffcences are assumed to be a functon of a number of frm specfc factors, and hence are not dentcally dstrbuted unless all the coeffcents of the factors are smultaneously equal to zero (Coell, et al. 1998, Herrero and Pascoe, 2002). Thus, the dstrbutonal assumptons used n ether step contradct each other (Coell, et al, 2005). Kumbhakar et al. (1991) argued that the estmated techncal coeffcents and techncal effcency ndces are based when the determnants of techncal effcency are not ncluded n the frst step of the regresson. They provded a one-step procedure whch determnes the nfluence of socoeconomc varables on techncal effcency whle estmatng techncal coeffcents of the producton fronter. Kalrajan (1991), on the other hand, has defended the practce of the two-step regresson on the bass that socoeconomc varables have a roundabout effect on producton. Although the two-step procedure s crtqued of producng based results, there seems to be lttle evdence on the severty of ths bas. For example, Caudll and Ford (1993) provde evdence on the bas of the estmated technologcal parameters, but not on the effcency levels or ther relatonshp to the explanatory varables. However, Wang and Schmdt (2002) dentfed two sources of bas namely, that the frst step of the two-step procedure s based for the regresson parameters f the z and the nputs, x are correlated. Secondly, that even f z and x are ndependent, the estmated neffcences are under-dspersed when the effect of z on neffcency s gnored. Ths causes the second-step estmate of the effect of z on neffcency to be based downward (toward zero). Therefore, they suggested that a one step procedure be employed to overcome ths problem. There appear to be no consensus n the 60

18 lteratures on the use of ether one step or two step procedure and the choce may be solely that of the analyst. The Cobb-Douglas functonal form s the commonly used n estmatng the stochastc producton fronter. Although ts most attractve feature s smplcty, but ths s assocated wth a number of restrctons. Most notably the returns to scale are restrcted to take the same value across all frms n the sample, and elastctes of substtuton are assumed equal to one. However, more flexble functonal forms lke the translog producton functon have also receved attenton. The translog form mposes no restrcton upon returns to scale or substtuton possbltes, but has the drawback of beng susceptble to multcollnearty and degrees of freedom problems (Coell, 1995b). In any case, the choce of approprate functon form can be made by conductng a lkelhood rato test between competng models. Stochastc fronter analyss (SFA) has both advantages and dsadvantages. The advantages nclude frst, t controls for random unobserved heterogenety among the frms. The neffcency effect can be separated from statstcal nose. Wth nonparametrc methods, any devaton of an observaton from the fronter must be attrbuted to neffcency, whch makes the results very senstve to outlers or measurement errors and uncertanty. Second, by usng SFA, the statstcal sgnfcance of the varables determnng effcency can be verfed usng statstcal tests, though ths s also true for recent bootstrapped DEA models. Thrd, the frm specfc neffcency s not measured n relaton to the best frm, as t s done n nonparametrc approaches. Hence, SFA s agan less senstve to outlers n the sample. Dsadvantages of the SFA approach consst of the need for dstrbutonal assumptons for the two error components as well as the assumpton of ndependence between the error terms and the regressors. Further, mplementaton of the model requres the choce of an explct functonal form, the approprateness of whch rases questons. The stochastc fronter specfcaton has been altered and extended n a number of ways. These extensons nclude: consderaton of panel data and tme-varyng techncal effcences, the extenson of the methodology to cost, revenue and proft fronters, estmaton of stochastc nput and output dstance functons, the estmaton of systems of equatons, the decomposton of the cost fronter to account for both 61

19 techncal and allocatve effcency. A revew of most of these extensons s provded by Forsund et al. (1980), Schmdt (1986), Bauer (1990), and Coell (1995b). However, n the subsequent sub-sectons bref explanatons of some these extensons are gven Panel Data Cross sectonal data provdes a snapshot of producers and ther effcency. Panel data provdes more relable evdence on ther performance, because they enable one to track the performance of each producer through sequence of tme perods. In the Panel data model, a tme varyng or tme nvarant neffcent effect may be specfed. Also, the model may assume ether a fxed or random effect. A sgnfcant advantage of panels s that gven consstently large tme perods, they permt consstent estmaton of the effcency of ndvdual producers, whereas the Jondrow et al. (1982) technque does not generate consstent estmators n a cross-sectonal context (Kumbhakar and Lovell, 2000). Another advantage of the panel data s that the dstrbutonal assumptons about the effcency term upon whch stochastc fronter rely s no longer necessary. Also the assumpton of ndependence between the neffcency term and nput levels s unnecessary wth panel data. Agan, panel data ncreases degrees of freedom for estmaton of parameters and t permts the smultaneous estmaton of techncal change and techncal neffcency changes over tme. However, the dearth of panel data on farmers especally n developng country agrculture has constraned the use of panel data methodologes Dualty Consderatons and Cost System Approaches The consderaton of dualty extends not only to cost mnmzaton but also proft maxmzaton, though cost mnmzaton s often made n the dual fronter lteratures. Thus, the dscusson here s bascally on cost mnmzaton behavour. It s very smple to change the sgn of the neffcency error component u and convert the stochastc producton fronter model to a stochastc cost fronter model such that we have: 62

20 C = c y, w ; β ).exp( v + u ) (3.15) ( where C s the cost of producton of the th frm, c y, w ; β ).exp( v ) s the stochastc ( cost fronter, w s a vector of nput prces of the th frm, y s output of the th frm; β s an vector of unknown parameters; v are random varables whch are assumed to be ndependently and dentcally dstrbuted N(0, σ ) and ndependent of, u, whch are non-negatve random varables whch are assumed to account for the cost of neffcency n producton, whch are often assumed to be d N(0, σ ). In ths cost 2 v 2 u functon, the u now defnes how far the frm operates above the cost fronter. If allocatve effcency s assumed, then u s closely related to the cost of techncal effcency. If ths assumpton s not made, the nterpretaton of the u n a cost fronter s less clear, wth both techncal and allocatve neffcences possbly nvolved (Coell, 1996a). The Jondrow et al. (1982) technque may be used to provde an estmate of the overall cost neffcency, but the dffcult remanng problem s to decompose the estmate of u nto estmates of the separate costs of techncal and allocatve neffcency. Schmdt and Lovell (1979) accomplshed the decomposton for the Cobb-Douglas case whle Kopp and Dewert (1982) obtaned the decomposton for the more general translog case based on determnstc fronter. Accordng to Coell (1995b) there are bascally three reasons for consderng the alternatve of dual forms of the producton technology, such as the cost or proft functon. Frst, s to reflect alternatve behavoural objectves such as cost mnmzaton. Second s to account for multple outputs. Thrd, s to smultaneously predct both techncal and allocatve effcency. The choce of whether to estmate a producton or cost fronter may be based on exogenety assumptons. It s more natural to estmate a producton fronter f nputs are exogenous and a cost fronter f output s exogenous (Schmdt, 1986). Schmdt and Lovell (1979) suggested a maxmum lkelhood system estmaton of ther Cobb-Douglas fronter, nvolvng the cost functon and k-1 factor demand equatons as ths s expected to mprove the precson of the parameter estmates. Such a system can be specfed as follows: 63

21 ln y = A + α ln x + v u (3.16) j j ln x 1 j 1 ln xj = ln pj ln p 1 + lnα lnα j + ε j, j =2,.k (3.17) lnc k 1 α j = K + ln y + r r j= 1 lnp j 1 ( v r u ) + ( E ln r) (3.18) where y s output, x s nputs, p s are prces, ndexes frms and j ndexes nputs. Equaton (3.16) s a stochastc producton fronter, whle equaton (3.17) s the set of frst order condtons for cost mnmzaton. Equaton (3.18) s the cost functon. ε j represents allocatve effcency. r = k j= 1 α s the returns to scale, j E (equaton 3.19) s gven as a functon of ε s and the parameters. The cost of techncal neffcency s 1 r u, whle the cost of allocatve neffcency s ( E Inr). The latter s non-negatve, and zero f ε = 0 for all j. j E n α n j = ε j + In α 1 + α je j= 2 r j= 2 ε j (3.19) Ths approach faces two serous draw backs. Frst, n some cases t may not be practcal or approprate to estmate a cost fronter. For nstance, t wll not be practcal to estmate a cost functon when nput prces do not vary among frms and t wll not be approprate when there s a systematc devaton from cost-mnmsng behavour n an ndustry. Second, Schmdt and Lovell (1979) systems estmaton and the techncal and allocatve effcency measurement are lmted to self-dual functonal forms lke the Cobb-Douglas. Once one specfes a more flexble functonal form lke the translog forms whch are not self-dual, a problem arses. The major problem wth employng a translog form s assocated wth how to model the relatonshp between the allocatve neffcency error whch appears n the nput share equatons and that whch appears n the cost functon (sometmes referred to as the Green Problem because t was frst noted by Green (1980b). Although a number of approaches have been suggested and appled n modellng the Greene problem rangng from analytc soluton (e.g. Kumbhakar, 1989), approxmate soluton (e.g. Schmdt, 1984) to 64

22 qualtatve soluton (e.g. Greene 1980b), debate stll contnues on how best to address ths problem. Coell (1995b) noted that a sound approach to take (gven that the cost mnmzng assumpton s approprate and sutable prce data are avalable) s to estmate the cost functon usng sngle equaton maxmum lkelhood method and then use the method proposed by Kopp and Dewert (1982), and refned by Zeschang (1983) for determnstc fronter case or that extended by Bravo-Ureta and Reger (1991) for stochastc fronter case followng the prmal route, to decompose the cost effcences nto ther techncal and allocatve components. If the Cobb-Douglas functonal form s consdered approprate, then the procedure nvolved smplfy to those whch are outlned n Schmdt and Lovell (1979). Berger (1993) found that effcency estmates usng no cost share equatons, partally restrcted share equatons, and fully restrcted share equatons gave very smlar effcency results Producton Fronter and Effcency Decomposton Gven that t may not be approprate to estmate a cost functon when there s lttle or no varaton n prces among sample frms, Bravo-Ureta and Reger (1991) developed an alternatve approach to decompose the cost effcency nto techncal and allocatve effcences. They followed a prmal route n ther methodology. The methodology nvolved usng the level of output of each frm adjusted for statstcal nose, the observed nput rato and the parameters of the stochastc fronter producton functon (SFPF) to decompose economc effcency nto techncal and allocatve effcency. Then the cost functon s analytcally derved from the parameters of the SFPF. To llustrate the approach, a stochastc fronter producton functon s gven as: Y = f ( ; β ) + ε (3.20) X ε = v u (3.21) where ε s the composed error term. The two components v and u are assumed to be ndependent of each other, where v s the two-sded, normally dstrbuted random error and u s the one-sded effcency component wth a half normal dstrbuton. Y s the observed output of the th frm, X s the nput vectors of th frm and β s 65

23 unknown parameters to be estmated. The parameters of the SFPF were estmated usng the maxmum lkelhood method. Subtractng v from both sdes of the equaton (3.20) results n Y * = Y v = f ( X ; β ) u (3.22) where * Y s the observed output of the th frm adjusted for statstcal nose captured by v. From equaton (3.22), the techncally effcent nput vector, level of T X, for a gven * Y s derved by solvng smultaneously equaton (3.22) and the nput ratos, X / X k = ρ k ( k 1), where ρ k s the rato of the observed nputs. 1 > Assumng the producton functon s self-dual functon lke the Cobb-Douglas producton functon, the correspondng dual cost fronter can be derved and wrtten n a general form as: C = h( W, Y ; ) (3.23) * δ where C s the mnmum cost of the th frm assocated wth outputy ; * W s a vector of nput prces of the th frm; and δ s a vector of parameters whch are functons of the parameters n the producton functon. The economcally effcent (cost mnmzng) nput vector, E X, s derved by usng Shephard s Lemma and then substtutng the frm s nput prces and adjusted output quantty nto the system of demand equatons: C = W X E ( W, Y ; * δ ) (3.24) For a gven level of output, the correspondng techncally effcent, economcally effcent and actual costs of producton are equal to W X T, W X E and W X, 66

24 respectvely. These three cost measures are then used as the bass for calculatng the techncal and economc (cost) effcency ndces for the th frm : W X T TE = W X (3.25) and W X E EE = W X (3.26) Followng Farrel (1957), allocatve effcency can be calculated by dvdng economc effcency (EE) by techncal effcency (TE): W X E AE = T W X (3.27) Dstance Functons and Effcency Decomposton The producton, cost, proft and perhaps revenue functons are well known alternatve methods of descrbng a producton technology. These functons have been used by economsts to measure effcences. Of recent the applcaton of dstance functons s growng. The majorty of recent dstance functon studes have been motvated by a desre to calculate techncal effcences or shadow prces. The prncple advantage of the dstance functon representaton s that t allows the possblty of specfyng a multple-nput, multple-output technology when prce nformaton s not avalable or alternatvely when prce nformaton s avalable but cost, proft or revenue functon representatons are precluded because of volatons of the requred behavoural assumptons (Coell and Perelman 2000). The dstance functon contans the same nformaton about technology as does the cost functon but may have some advantages econometrcally over the cost functon f, for example, nput prces are the same for frms, but nput quanttes vary across frms (Bauer, 1990). 67

25 The output dstance functon measures how close a partcular level of output s to the maxmum attanable level of output that could be obtaned from the same level of nputs f producton s techncally effcent. In other words, t represents how close a partcular output vector s to the producton fronter gven a partcular nput vector (Mawson et al., 2003). The defnton of an output-dstance functon starts wth a defnton of the producton technology of the frm usng the output set, P (x), whch represents the set of all output vectors, M y R+, whch can be produced usng the nput vector, K x R+. That s, P (x) = y R M : x can produce y } (3.28) { + The output-dstance functon s then defned on the output set, P (x), as D O { : ( y / ) P( )} ( x, y) = mn θ θ x (3.29) D O ( x, y) s non-decreasng, postvely lnearly homogeneous and convex n y, and decreasng n x (Lovell et al., 1994). The dstance functon, ( x, y), wll take the value whch s less than or equal to one f the output vector, y, s an element of the feasble producton set, P (x). That s, D O ( x, y) 1 f y P(x) D O. Furthermore, the dstance functon wll take the value of unty f y s located on the outer boundary of the producton possblty set. That s, D O ( x, y) =1 f y Isoq P (x) = { : y P( x), ωy P( x), ω > 1} y ; (3.30) A Stochastc Output Dstance Functon (SODF) s not the same as a Stochastc Fronter Producton Functon (SFPF). Both consder the maxmum feasble output from a gven set of nputs. The dfference s that SODF s defned n a set theoretc framework whch nvolves vector of outputs and nputs and can only be mplemented emprcally by normalzng usng one of the outputs whereas SFPF s smply defned for the case of one output or aggregated outputs and does not requre normalzaton. 68

26 An nput-dstance functon s defned n a smlar manner as the output dstance functon. However, rather than lookng at how the output vector may be proportonally expanded wth the nput vector held fxed, t consders by how much the nput vector may be proportonally contracted wth the output vector held fxed. The nput-dstance functon may be defned on the nput set, L (y), as D I { : ( x / p) L( )} ( x, y) = max ρ y (3.31) where the nput set, L (y), represents the set of all nput vectors, K x R+, whch can produce the output vector, M y R+. That s, L( y) = { K x R+ : x can produce y} (3.32) D I ( x, y) s non-decreasng, postvely lnearly homogenous and concave n x, and ncreasng n y. The dstance functon, ( x, y), wll take a value whch s greater D I than or equal to one f the nput vector, x, s an element of the feasble nput set, L ( y). That s, D I ( x, y) 1 f x L(y). Furthermore, the dstance functon wll take a value of unty f x s located on the nner boundary of the nput set. Under the assumpton of constant returns to scale (CRS), the nput dstance functon s equvalent to the nverse of the output dstance functon (.e., D O = 1/D I ) (Färe et al. 1993, 1994). That s, the proporton by whch one s able to radally expand output (wth nput held fxed), wll be exactly equal to the proporton by whch one s able to radally reduce nput usage (wth output held constant). However, under varable returns to scale (VRS) ths condton need not hold. The dstance functon can be llustrated graphcally. For nstance, the nput dstance functon as exemplfed n Coell et al. (2003) s shown n fgure 3.2. Here two nputs, x 1 and x 2, are used to produce output y. The soquant S S, s the nner boundary of the nput set, reflectng the mnmum nput combnatons that may be used to produce a gven output vector. In ths case, the value of the dstance functon for a frm 69

27 producng output y, usng the nput vector defned by pont P, s equal to the rato, OP/OQ. x 2 S P Q L(y) S 0 x 1 Fgure 3.2: The nput dstance functon and the nput set In emprcal lteratures on effcency measurement nvolvng dstance functons, dfferent methods have been employed to estmate the functon. These nclude the constructon of parametrc fronter usng lnear programmng methods (Färe et al., 1994; Coell and Perelman, 1999; Alene and Manfred, 2005); the constructon of nonparametrc pece-wse lnear fronter usng the lnear programmng method known as data envelopment analyss (DEA) (e.g. Färe et al., 1989; Färe et al., 1994; Coell and Perelman, 1999; Alene and Manfred, 2005); estmaton of parametrc fronter usng corrected ordnary least square (COLS) (e.g. Lovell et al., 1994; Grosskopf et al., 1997; Coell and Perelman, 1999) and maxmum lkelhood estmaton (MLE) of a parametrc stochastc dstance fronter (e.g. Coell et al., 2003; Irz and Thrtle, 2004; Sols et al., 2009). However all of these studes have bascally focused on analysng techncal effcency except that of Coell et al. (2003) that appled the cost decomposton approach to estmate both techncal and allocatve effcency. Decomposton of cost effcency n a sngle equaton stochastc nput dstance fronters framework was frst developed by Coell et al. (2003) to overcome the problems that arse when one ether tres to estmate a cost fronter and then use dualty to derve the mplct producton fronter as n Schmdt and Lovell (1979) or 70