A Note on Theory of Productive Efficiency and Stochastic Frontier Models

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1 European Research Studes, Volume XIII, Issue (4), 2010 A Note on Theory of Productve Effcency and Stochastc Fronter Models Akatern Kokknou 1 Abstract: Neoclasscal economcs assume that producers n an economy always operate effcently, however n real terms, producers are not always fully effcent. Ths dfference may be explaned both n terms of effcency, as well as unforeseen exogenous shocks outsde the producer control. Ths paper ams to analyse the productve effcency estmaton through a stochastc fronter analyss approach. Partcularly, ths paper attempts to examne systematcally the theoretcal background of stochastc fronter functon estmaton, focusng on the analyss of the effcency functon, n order to provde a sold background for productve effcency estmaton. JEL Classfcaton: C14, C23 Keywords: Stochastc Fronter Analyss, Productve Effcency Functon, Effcency 1. Introducton The man core of the modern economc theory s based on the assumpton of optmsng behavour, ether from a producer or a consumer approach. Economc theory assumes that producers optmse both from a techncal and economc perspectve: 1. From a techncal perspectve, producers optmse by not wastng productve resources. 2. From an economc perspectve producers optmse by solvng allocaton problem nvolvng prces. However, not all producers succeed n solvng both types of optmsaton problem n all crcumstances. Performance at frm or ndustry level, defned as the rato of output(s) a producton unt produces to the nput(s) that a producton unt uses, yeldng a relatve measure of performance appled to factors of producton 1 Department of Economcs, Unversty of Glasgow, Adam Smth Buldng, G12 8RT, Glasgow, Unted Kngdom, e-mal: A.Kokknou.1@research.gla.ac.uk The author should thank Prof. Dr. Rchard Harrs, Department of Economcs, Unversty of Glasgow for all the gudance and contrbutng remarks. However, any faults are of sole responsblty of the author.

2 110 European Research Studes, Vol XIII, Issue (4), 2010 (Fred et al., 1993, Lovell, 1993), may depend: a) on dfferences n producton technology, b) on dfferences n the effcency of the producton process, or, c) on dfferences n the envronment where producton occurs 2. However, at a gven moment of tme, even when technology and producton envronment are essentally the same, frms or ndustres may exhbt dfferent productvty levels due to dfferences n ther producton effcency (Korres, 2007). For ths reason t s mportant to have a way of analysng the degree to whch producers fal to optmse, the departures from full techncal and economc effcency. Based on ths general noton, one of the man analytcal approaches to effcency measurement s the analyss of producton fronters, a tool whch has expanded greatly n the last decades. However, even though the concept of producton effcency s central n producton performance, ts estmaton has been proved to be rather complex, wth relevant lterature provdng a range of dfferent methodologes and approaches (Lovell, 1993), wth one of the major approaches to be the stochastc fronter analyss. Ths paper examnes the man characterstcs of stochastc fronter models, especally the alternatve model specfcatons. 2. The Theoretcal Background The stochastc fronter model was orgnally developed by Agner, Lovell and Schmdt (1977). Typcally, the producton or cost model s based on a Cobb Douglas functon: log y x v u (1) where y s the observed outcome x v s the optmal producton fronter (e.g. maxmum producton output or mnmum cost), x s the determnstc part of the fronter and v ~ Ν(0, σ 2 v ) s the stochastc part, respectvely. The components of x are generally logs of nputs for a producton model or logs of output and nput prces for a cost model, or ther squares and/or cross products. These two parts consttute the stochastc fronter. The amount by whch the observed ndvdual fals to reach the optmum (the fronter) s u, namely neffcency, where u= U and U ~ Ν[0, σ 2 u ]. The stochastc fronter model becomes: y x v u, u U (2) 2 In early economc studes, productvty change was allocated exclusvely to shfts n producton technology (the magntude of neutral techncal change), eventually roles were also assgned to the bases of techncal change and the structure of the technology, namely scale economes (Kumbhakar and Lovell, 2000).

3 A Note on Theory of Productve Effcency and Stochastc Fronter Models 111 In the stochastc fronter model, the error term ε s made up of two ndependent components, v - u, where u measures techncal neffcency, namely the shortfall of output y from ts maxmal possble value gven by the stochastc fronter gx0, v. When a model of ths form s estmated, the obtaned resduals ˆ y gx ˆ, may be regarded as estmates of the error term ε (Jondrow et al., 1982). The condtonal dstrbuton of u gven ε, E[u ε] s the mean productve effcency. Under each of the assumed possble dstrbutonal forms for the neffcency term n a model, ths mean hat ths dstrbuton contans whatever nformaton ε yelds about u. The predcted value s x. The resdual s computed by the Jondrow et al. (1982) formula: E[ u v u] or E[ u v u] (3) or ( z) (4) E ˆ[ u ] z, v u, z ( z) The margnal effects n the model are the coeffcents. Estmaton of the model parameters s usually of secondary nterest, whereas, estmaton and analyss of the neffcency of ndvduals n the sample and of the aggregated sample are of greater concern. The results obtaned are crtcally dependent on the model form and the assumptons set. Regardng ths, specal focus has been gven to panel data estmaton technque. 3. The Model Specfcatons The stochastc fronter model follows Battese and Coell (1995) and conssts of two equatons, one to represent the producton fronter and a second to measure techncal neffcency: Y t =exp(x jt β+v t -U t ) (5) and Ε t =exp(-u t )=exp(-z t δ W t ) (6) In the frst equaton, Y t represents output of the th frm at tme t. X jt s a vector of productve nputs and ndcator varables for the th frm at tme t. The parameter vectors β and δ are estmated together wth the varance parameters: v u and 2 2 (7) u Techncal effcency s measured usng the condtonal expectaton E t =exp(-u t ) gven the composed error term. The frst component, (v t ), accounts for random events. The second component, u t, s a non-negatve random varable whch captures unobservable neffcency effects relatve to the stochastc fronter. The random component, v s assumed to be ndependently and dentcally dstrbuted

4 112 European Research Studes, Vol XIII, Issue (4), 2010 wth Ν(0, σ 2 v ). The techncal neffcency component, u, s assumed to follow an arbtrary dstrbutonal form, n ths case a half-normal dstrbuton Ν(Z t δ, σ 2 u ) 2. The neffcency model random component, w, s not dentcally dstrbuted nor s t requred to be non-negatve (Battese and Coell, 1995) 3. Bascally, there are two methods of estmaton n the lterature. In the frst, the estmaton of the parameters of the producton fronter s done condtonally on fxed values of the u t s whch leads to the fxed effects model and the wthn estmator of the fronter coeffcents. In the second, the estmaton s carred out margnally on the frm specfc effects u t s whch leads to the random effects model and ether the Generalsed Least Squares (GLS) estmaton of the parameters. Although the fxed effects models have the advantage of followng correlaton between the neffcency term and the ndependent varables, and of allowng no dstrbutonal assumpton on effcency, the results should be nterpreted carefully. Smar (1992) has shown that the fxed effects model appears to provde a poor estmaton of the ntercepts and of the slope coeffcents of fronter producton functons and consequently unreasonable measures of techncal effcency. In the random effects model, the stochastc nature of the effcency effects s explctly taken nto account n the estmaton process. In the fxed effects model, the coeffcents of tme nvarant regressors, even though they may vary across frms, cannot be estmated because these tme nvarant regressors wll be elmnated n the wthn transformaton, as shown n the equaton: ( y y ) ( x x ) v (8) t t In ths case, the frm specfc techncal effcency effects wll nclude the nfluence of all varables that are tme nvarant at the frm level wthn the sample. Ths would make techncal effcency comparsons dffcult unless the excluded fxed regressors nfluence all frms n the sample equally. 4. Extended Model In the lterature there are several varants of the prevous model allowng for dfferent dstrbutons of the u and v term (see Kalrajan and Shand, 1999): t 3 Before Battese and Coell (1995), Jondrow et al.(1982) provded an ntal soluton by dervng the condtonal dstrbuton of [-u (v u )] whch contans all the nformaton (v u ) contans about u. Ths enabled to derve the expected value of ths condtonal dstrbuton, from whch they proposed to estmate the techncal effcency of each producer: ˆ TE (, ) exp ˆ 1 0 x y E u v u 1 whch s a functon of the MLE parameter estmates. Later, Batesse and Coell (1988) proposed to estmate the techncal ˆ TE (, ) [exp{ ˆ } ] 1 1 effcency of each producer from: 0 x y E u v u whch s slghtly dfferent functon of the same MLE parameter estmates.

5 A Note on Theory of Productve Effcency and Stochastc Fronter Models The half normal model 2. The fxed effects model 3. The random effects model 4. The latent class model 4.1 The Half Normal Model The essental form of the stochastc producton fronter model [see Agner et al. (1977)] s: (9) where y t x v t t u v ~ N[0,σ v 2 ], u = U, and U ~ N[0,σ u 2] As descrbed n Greene (2003), ths s the canoncal half normal model. A central parameter n the model s the asymmetry parameter, λ = σ u /σ v ; the larger s λ, the greater s the neffcency component n the data. Parameters are estmated by maxmum lkelhood, rather than least squares. Estmaton of u s the central focus of the analyss. Wth the model estmated n logarthms, u would correspond to 1 - TE. Indvdual specfc effcency s typcally estmated wth exp( uˆ ). Alternatvely, (10) û provdes an estmate of proportonal neffcency. Wth parameter estmates n hand, one can only obtan a drect estmate of v u. Ths s translated nto an estmate of u usng Jondrow et al. (1982) formula: z Eu ( ) (11) z z ( z, 2 1 ) and φ(z) and Φ(z) are the densty and CDF of the standard where 2 u v normal dstrbuton, respectvely. The narrow assumpton of half normalty s a vewed as sgnfcant drawback n ths model. Ths feature leads to the extenson of the model to a truncated normal model by allowng the mean of U to be nonzero (Stevenson, 1980). The major shortcomng here s that the strct assumpton suppresses ndvdual heterogenety n neffcency that s allowed, for example, by the fxed effects formulaton. Lettng h denote a set of varables that measure the group heterogenety, we wrte: ' E U h (12)

6 114 European Research Studes, Vol XIII, Issue (4), 2010 The Jondrow et al. (1982) result s now changed by replacng z wth z * = z - µ /(σλ) representng a sgnfcant extenson of the model. 4.2 The Fxed Effects Model The fxed effects model s based on the Schmdt and Sckles (1982) formulaton: ' ' yt ( u ) xt vt xt v (13) t where uˆ max ( ˆ ) ˆ 0 (14) Ths varaton has two mportant restrctons. Frst, any tme nvarant heterogenety wll be pushed nto α and ultmately nto û. Second, the model assumes that neffcency s tme nvarant. For short tme ntervals, ths may be a reasonable assumpton. But, ths s to be questonable. Both of these restrctons can be relaxed by placng country specfc constant terms n the stochastc fronter model we call ths a true fxed effects model: ' (15) y t x v t t u where u t has the stochastc specfcatons noted earler for the stochastc fronter model. The model s stll ft by maxmum lkelhood, not least squares. 4.3 The Random Effects Model As referred n Greene (2003), the random effects model s obtaned by assumng that u s tme nvarant and also uncorrelated wth the ncluded varables n the model: ' (16) y t x v t t t u In the lnear regresson case, the parameters are estmated by two step generalzed least squares (Green, 2003). The regresson based random effects model has a sgnfcant drawback: there s no mpled estmator of neffcency n ths model, that s, no estmator of techncal effcency TE as n the fxed effects case. So, the model would not have been useful n any event. Wth ths extenson, the Jondrow et al. (1982) estmator becomes:

7 A Note on Theory of Productve Effcency and Stochastc Fronter Models 115 where Z E u,,..., h 1 2 T Z 1, T Z Z 2 2, u, t 1 T (17) (18) The tme nvarance of the neffcency component of the random effects model s a potental drawback n the random effects model. Battese and Coell (1988, 1995) have proposed a modfcaton of the model that allows some systematc varaton n 2 the followng model: u U where 1 ( t T ) ( t T and U ~ t t t 1 2 ) N[0,σ u 2]. A random effects counterpart to the true fxed effects model would be a ' true random effects stochastc fronter model: yt ( w ) xt vt ut. Ths form of the model overcomes both of the drawbacks noted earler. As broadly presented n Greene (2003), the precedng has suggested varous ways to accommodate both cross country heterogenety and tme varaton n neffcency n the stochastc fronter model. Tme varaton n neffcency s acheved by removng restrctons on u t and allowng t to vary unsystematcally through tme. model: 4.4 The Latent Class Model Presented n Greene (2003) s also another model varaton, the latent class ' y class j x v j u j (19) t j t j t and a model for the mxng probabltes: country s a member of class j F h,, 0 F 1 Pr ob j j (20) Heterogenety enters ths model through the pror mxng probabltes. As before, t can also enter through the dstrbuton of u t. The latent class model s an alternatve to the random parameters model descrbed n the precedng secton. Wth a suffcent number of classes, the fnte mxture can provde a good approxmaton to contnuous parameter varaton. In practcal terms, ths model s somewhat less flexble than the random parameters model dscussed above. Greene (2003) has extended t to the most general varant of the Battese and Coell formulaton of the

8 116 European Research Studes, Vol XIII, Issue (4), 2010 random effects model. Snce ths approach s new to the lterature, ts usefulness as an emprcal tool remans to be establshed. A number of studes have also attempted to estmate tme-varyng neffcency. Cornwell, Schmdt and Sckles (1990) replaced the frm effect by a squared functon of tme wth parameters that vary over tme. Kumbhakar (1990) allowed a tme-varyng neffcency measure assumng that t was the product of the specfc frm neffcency effect and an exponental functon of tme. Ths allows flexblty n neffcency changes over tme, although no emprcal applcatons have been developed usng ths approach (Coell, Rao and Battese 1998). 5. Concludng remarks In summary, n stochastc fronter model t s acknowledged that the estmaton of producton functons must respect the fact that actual producton cannot exceed maxmum possble producton gven nput quanttes. There s an extensve body of lterature on factors nfluencng productvty growth. In ths lterature body, t s wdely accepted that decson makng unts are not homogeneous producng unts and, they are not therefore, all operatng at the same level of effcency. On the contrary, n the real world some producers are more effcent than others. Emprcal analyses have shown that productvty level may consderably dffer even f they operate n the same market. Whle some unts operate at the technologcal fronter and earn hgh profts, others lag consderably behnd. As a management tool, stochastc fronter analyss focuses on productve effcency analysng varables under or beyond decson-makers control. These factors are nether nputs to the producton process nor outputs of t but nonetheless exert an nfluence on producer performance. 4. In ths context, the term envronment s used to descrbe factors that could nfluence the effcency of a frm, where such factors are not tradtonal nputs and are not under the control of management. In other words, some exogenous varables may nfluence the productve effcency wth whch nputs are converted nto outputs. In partcular, to nvestgate the determnants of the productve effcency we dstngush between frm / ndustry -specfc and external factors. External factors are not under drect control of the frm, at least n the short-run, as ndustry afflaton and frm locaton. Frm-specfc factors, on the other hand, refer to characterstcs that can be nfluenced by the frm n the shortrun, as frm sze, R&D ntensty and degree of outsourcng. These factors may express socal aspects, geographcal or clmatc condtons, as well as regulatory and nsttutonal constrants. 4 In many cases, the dstncton between decson-maker controlled and external varables s not always dstnct. As n McMllan and Chan (2006), external varables here nclude purely exogenous varables as well as frm-specfc varables representng producton methods and output characterstcs.

9 A Note on Theory of Productve Effcency and Stochastc Fronter Models 117 It becomes apparent that n the area of stochastc fronter models, estmaton of the model parameters s usually of secondary nterest, whereas, estmaton and analyss of the neffcency are of greater concern. The mportant task s to relate neffcency to a number of factors that are lkely to be determnants, and measure the extent to whch they contrbute to the presence of neffcency. Stochastc fronter approach has found wde acceptance wthn the producton economcs lterature, because of ther consstency wth theory, versatlty and relatve ease of estmaton. Some lterature focused on stochastc fronter model wth dstrbutonal assumptons by whch effcency effects can be separated from stochastc element n the model and for ths reason a dstrbutonal assumpton has to be made. However, these are computatonally more complex, there are no pror reasons for choosng one dstrbutonal form over the other, and all have advantages and dsadvantages. Wthn ths framework, the analyss so far provdes a sold background for further development of the model. References 1. Agner, D. J. & Chu, S. F. (1968), `On estmatng the ndustry producton functon', Amercan Economc Revew, 58(4), Agner D.J., Lovell C.A.K., Schmdt P. (1977) Formulaton and estmaton of stochastc fronter producton functons. Journal of Econometrcs 6: Battese, G.E. and G.S. Corra (1977) Estmaton of a Producton Fronter Model wth Applcaton to the Pastoral Zone of Eastern Australa. Australan Journal of Agrcultural Economcs, 21: Coell, T., Rao, D. S. P. & Battese, G. (1998) An Introducton to Effcency and Productvty Analyss (Boston, MA, Kluwer). 5. Coell, T.J., Rao, D.S.P., O'Donnell, C.J., Battese, G.E. (2005) An Introducton to Effcency and Productvty Analyss, 2nd Edton, Sprnger 6. Fare, R., Grosskopf, S. and Lovell, C. A. K. (1985) The Measurement of Effcency of Producton. Boston: Kluwer-Njhoff. 7. Fare, R., Grosskopf, S. Norrs, M. and Zhang, Z. (1994) Productvty Growth, Techncal Progress, and Effcency Change n Industralzed Countres. The Amercan Economc Revew. 84 (1) Farrell, M. (1957). The Measurement of Productve Effcency, Journal of the Royal Statstcal Socety Seres A (General), 120 (3), Fred, H.O., C.A.K. Lovell, S.S. Schmdt (2008) Effcency and Productvty, n: H. Fred, C.A.K. Lovell, S. Schmdt (eds) The Measurement of Productve Effcency and Productvty Change, New York, Oxford Unversty Press, Fred, H.; Lovell, C. and Schmdt, S. (eds.) (1993) The Measurement of Productve Effcency: Technques and Applcatons. New York: Oxford Unv. Press. 11. Greene, W. (2003) Dstngushng Between Heterogenety and Ineffcency: Stochastc Fronter Analyss of the World Health Organzaton s Panel Data on Natonal Health Care Systems, Workng Paper 03-10, Department of Economcs, Stern School of Busness, New York Unversty, New York.

10 118 European Research Studes, Vol XIII, Issue (4), Greene W.H. (1993) The econometrc approach to effcency analyss. n: Fred HO, Lovell CAK, Schmdt SS (Eds) The measurement of productve effcency: Technques and applcatons. Oxford Unversty Press New York Greene, W. H. (1990) A GAMMA-Dstrbuted Stochastc Fronter Model. Journal of Econometrcs Grosskopf, S. (1993): Effcency and Productvty, In H.O. Fred, C.A.K. Lovell, and S.S. Schmdt (eds.), The Measurement of Productve Effcency: Technques and Applcatons, New York: Oxford Unversty Press, pp Jondrow, J., C. A. Knox Lovell, Ivan S. Materov, and Peter Schmdt (1982) On the estmaton of techncal neffcency n the stochastc fronter producton functon model. Journal of Econometrcs 19, Kalrajan, K. P. and Shand, R. T. (1999) Fronter Producton Functons and Techncal Effcency Measures, Journal of Economc Surveys, vol. 13(2), pages Kalrajan, K. P. and Shand, R. T. (1992) Causalty between techncal and allocatve effcences: an emprcal testng. Journal of Economc Studes, 19, Kalrajan, K.P. and J.C. Flnn, The Measurement of Farm Specfc Techncal Effcency. Pakstan, Journal of Appled Economcs, 11(2): Korres, G. (2007) Techncal Change and Economc Growth: Insde to the Knowledge Based Economy. Recent Evdence on European Perspectves, Avebury-Ashgate, London. 20. Kumbhakar, S. C. and Lovell K. C.A. (2000) Stochastc Fronter Analyss, Cambrdge Unversty Press, Cambrdge 21. Lovell Knox, C.A. (1993) Producton Fronters and Productve Effcency, n The measurement of productve effcency : Technques and applcatons, ed. Fred, H.O., Knox Lovell, C.A., Schmdt, S. S., Oxford Unversty Press, Meeusen, W. and van Den Broeck,. J. (1977) Effcency estmaton from Cobb Douglas Producton Functons wth Composed Error. Internatonal Economc Revew, 18, 2: Schmdt, P. (1985) Fronter producton functons, Econometrc Revews, 4:2, Schmdt, P. and Lovell, C. A. K. (1979) Estmatng techncal and allocatve neffcency relatve to stochastc producton and cost fronters, Journal of Econometrcs, 9, pp Schmdt, P. and Sckles, R. C. (1984) Producton fronters and panel data. Journal of Busness and Economc Statstcs, 2, Smar, L. (1992) Estmatng effcences from fronter models wth panel data: A comparson of parametrc, non-parametrc and sem-parametrc methods wth bootstrappng. The Journal of Productvty Analyss, 3, Stevenson, R. E., (1980). Lkelhood functon for generalzed stochastc fronter estmaton, Journal of Econometrcs, 13,