STOCHASTIC FRONTIER ANALYSIS OF PRODUCTION FUNCTION AND COST FUNCTION ESTIMATION METHODS. STUDY OF EFFICIENCY AT INDUSTRY LEVEL

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1 Abstract. Ths paper analyses the effcency or productvty at the level of a producton unt, but also at ndustry level, resortng for ths purpose to both parametrc and non-parametrc technques. Cost functon model specfcatons are descrbed heren, consderng that the techncal neffcency effects determne the companes to operate below the producton stochastc fronter. Also, the estmatons of the producton fronter are rendered by means of Cobb-Douglas, C.E.S. and translog functons, evdencng that the latter s the most flexble of them. On the other hand, n many respects, C.E.S. producton functon s more approprate to realty as compared to Cobb-Douglas functon. At the same tme, the estmaton of C.E.S. functon parameters s more dffcult, Cobb-Douglas functon beng, n ths respect, preferred. The last part of the paper s consecrated to the study of effcency at ndustry level and to conclusons. The Input varables used wthn the analyss are: fxed assets, nventores, number of employees, whle the Output ones are: operatng revenues and net proft. As for the conclusons, the results reveal that the effcency level for several companes s qute low. Therefore, a deeper nterest should be manfested n order to ncrease the effcency level n the constructon ndustry. STOCHASTIC FRONTIER ANALYSIS OF PRODUCTION FUNCTION AND COST FUNCTION ESTIMATION METHODS. STUDY OF EFFICIENCY AT INDUSTRY LEVEL Iulan LIŢĂ Academy of Economc Studes, Bucharest, Paţa Romană no. 6, Bucharest, Romana e-mal: ulan_lta_ct@yahoo.com Tănase STAMULE Academy of Economc Studes, Bucharest, Paţa Romană no. 6, Bucharest, Romana e-mal: tasestamule@yahoo.com Management & Marketng Challenges for the Knowledge Socety (011) Vol. 6, No. 1, pp Keywords: cost functon ndustry, Data Envelopment Analyss, producton functon, stochastc fronter.

2 Management & Marketng 1. Introducton In analysng the effcency or productvty of a producton unt, we may use the dstance functon for both nputs and outputs. Ths allows us to compute the radal dstance of the producton unt s relaton to the producton functon, the mportant ssue beng to estmate such producton fronter. For ths end, we may start from the dea that the producton theory has revealed classes of producton functons, dependng on many parameters, functons correspondng to the transformaton of nputs nto outputs. Cobb-Douglas producton functons wth three parameters, translog producton functon ncludng also the tme parameter etc. belong to such classes. Therefore, n case of a homogenous producton unt group, we should dentfy frst of all the class of producton functons correspondng to the nternal process of transformaton of nputs nto outputs and then to estmate the approprate parameters. As for the effcency measurement, ths could be done by appealng to both parametrc and non-parametrc technques. For the frst case, we could state that the unts for whch we do have observed values on nputs and outputs form a sample. By usng econometrc technques, we wll estmate all parameters of the selected model and, for each unt of the sample, we wll also estmate ts dstance to the producton fronter. For the non-parametrc technques measurng the dstance up to the producton fronter, whch approxmate the fronter by creatng an envelope of the nput and output varables correspondng to a scale yeld, lnear and/or non-lnear mathematc programmng models are used. The effcency analyss s not a recent topc of nterest for economsts, ts roots comng from Knght, n In 1951, Debreu and Koopmans have presented the results of ther studes regardng the effcency computng. Schmdt (1977), Olsen et al. (1980), Forsund et al. (1980), Forsund and Hjalmarsson (1987), Lovell ş Schmdt (1988), Greene (1993), Cooper et al. (007), hu (009) and others have brought mportant contrbutons to the effcency study, by usng both parametrc and nonparametrc methods.. Model specfcatons The stochastc fronter of the producton functon has been ndependently proposed by Agner, Lovell and Schmdt (1977) and Meeusen and van den Broeck (1977). The orgnal specfcaton mples a producton functon generated for a crosssectonal data set, wth a two-component error term, one relatng to stochastc effects and the other one to techncal neffcency. Ths model may be rendered under the followng form: Q u ( ), cu 1, n (1) 164

3 Stochastc fronter analyss of producton functon and cost functon estmaton methods where: Q producton (or producton logarthm) of company ; u vector of type k 1; t represents the nput quanttes of company ; vector of unknown parameters; stochastc varables consdered N(0, ) and ndependent from. non-negatve stochastc varables relatng to producton techncal neffcency and consdered N(0, ). Varous authors have consstently contrbuted to ths area of nterest, among them: Forsund, Lovell and Schmdt (1980), Schmdt (1986), Bauer (1990) and Greene (1993), Cooper et al. (007), hu (009). FRONTIER 4.1 s a tool allowng maxmum probablty estmates of a subset of the stochastc fronter producton and of the cost functons proposed n the related lterature. FRONTIER 4.1 has been conceved to estmate the specfcatons of the model detaled n Battese and Coell (1988, 199 and 1995) and Battese, Coell and Colby (1989). Snce then, the specfcatons n Battese and Coell (1988) and Battese, Coell and Colby (1989) are partcular cases of the Battese ş Coell (199) specfcaton. Model 1: Battese and Coell (199) Specfcaton Battese and Coell (199) propose a stochastc fronter producton functon. The model can be rendered as follows: Qt u t (t t ), wth 1, n and t 1, T () where: Q t producton logarthm at the level of company at tme t; u t vector of type k 1; t represents the nput quanttes (transformatons) of company at tme t; prevously defned; t stochastc varables consdered N(0, ) and ndependent from ; (tt) e ; t non-negatve stochastc varables relatng to producton techncal neffcency and consdered truncated to zero at dstrbuton N(0, ) ; parameter to estmate. Battese ş Corra (1977) parametersaton s used; t replaces. and Parameter belongs to the nterval ( 0,1). and by 165

4 Management & Marketng Also, the stochastc character of the producton functon can be tested. If the null hypothess, when equals zero, s accepted, t wll ndcate that s zero, and thus the term t can be taken out of the model, leavng a specfcaton wth parameters to be compatbly assessed by resortng to smallest normal dfferences. Model : Battese and Coell (1995) specfcatons The emprcal studes of Ptt and Lee (1981) have estmated the stochastc fronter and the effcency at company level, by usng, to ths end, estmated functons. Such ssue has been also approached by Kumbhakar, Ghosh and McGukn (1991) and Refschneder and Stevenson (1991) who propose stochastc fronter models where the neffcent effects ( ) are expressed as an explct functon of a vector of specfc varables, at the company level. Battese and Coell (1995) propose a model equvalent to the specfcaton made by Kumbhakar, Ghosh ş McGukn (1991), but the dstrbuted effcency s mposed. The model specfed by Battese and Coell (1995) can be expressed as follows: Qt u t (t t ), cu 1, n and t 1, T (3) where : Q t, u t and are prevously defned; - stochastc varables consdered N(0, ) and ndependent from ; - non-negatve stochastc varables relatng to producton techncal neffcency and frequently consdered as N(mt, ) where: m t z t (4) where: z t - vector of type p1 that can nfluence the company, and - vector of type 1p of the parameters to be estmated; We wll resort agan to the parametersaton proposed by Battese and Corra (1977), by replacng and by ş. Ths specfcaton of the model cumulates a number of specfcatons from other models, as well as specal cases. If T 1 and z t takes value one and no other values, than the model may be reduced to the one gven by Stevenson (1980). 3. Cost functon analyss All the above-mentoned specfcatons have been expressed n the terms of a producton functon, wth beng construed as techncal neffcency effects, 166

5 Stochastc fronter analyss of producton functon and cost functon estmaton methods determnng the company to operate below the producton stochastc fronter. If the specfcaton of cost functon stochastc fronter s wanted, the error term specfcaton wll be changed from ( ) to ( ). For nstance, ths substtuton wll transform the functons defned n (1) nto a cost functon: Q u ( ), wth 1, N (5) where: Q - producton logarthm at the level of company ; u - vector of type k 1; t represents the nput and outputs prces (transformatons) of company ; - vector of unknown parameters; - stochastc varables consdered N(0, ) and ndependent from ; U - non-negatve stochastc varables relatng to producton techncal neffcency and consdered N(0, ). Ths cost functon defnes heren how far downward the cost fronter the company operates. If the allotted effcency s presumed, s very close to the techncal neffcency cost. If such presumpton s not undertaken, construng as a cost functon s less clear, wth the two (techncal and allotment) neffcences possbly nvolved. The cost fronter (5) s dentcally proposed also by Schmdt and Lovell (1979). The log-probablty functon of the cost fronter s smlar to the cost fronter one, save for several dfferent sgns. The log-probablty functon for the cost functon s analogue to that of the models of Battese and Coell (199, 1995). 4. Producton fronter estmaton by means of Cobb-Douglas, CES and translog functons Ths approach starts from the assumpton of a Cobb-Douglas producton functon: f (L,K) AL K. Ths functon f (, ) s a power functon wth three parameters A, α and β; therefore, t s log-lnear (lner n the logarthm of the varables nvolved). Here, A s a scalng factor, and α and β are the elastctes correspondng to the two nputs consdered. For the Cobb-Douglas functon, the scale yeld type s determned by the sum of the parameters representng elastctes, that s by α + β, and the substtuton elastcty s equal to 1. Cobb-Douglas producton functon s used under an equvalent form, obtaned by logarthmc transformaton, that s: ln Y ln A ln L ln K. Parameters α and β may be also construed as costs of the two producton factors. If we denote by w the labour force unt prce and by e the captal unt prce, we could mnmse the total 167

6 Management & Marketng producton cost, dependng on L and K nputs, for a producton process descrbed by the producton functon f. Mathematcally, ths could be rendered as follows: [mn] wl ek L,K Y f (L,K) AL K The assocated Lagrangan s: wl ek f (L, K), and the necessary optmum condtons are: f f w 0 and e 0 L L K K w e By elmnatng λ, we obtan: f f L K f f As we have, for Cobb-Douglas producton functon, L and L f f Lw Ke, the optmum necessary condton becomes. K K Ths relaton expresses the fact that n the producton functon the cost of the two producton factors (labour force cost Lw and captal cost Ke) are proportonal to Cobb-Douglas functon parameters. If we denote by p the producton unt prce, the whole producton value resulted Y s P = Yp. Therefore, another proportonalty relaton between the producton value and the labour force (respectvely captal) cost can be wrtten down, Lw P P for nstance: or Cw, where C s a constant (respectvely L P ' C e ). K Under economc equlbrum condtons (mnm cost), the raton P/L should be proportonal to the labour force cost (cost of producton factor L). Yet, econometrc researches performed, along years, for varous ndustres, have nfrmed the prevous statement. On the contrary, an approprate adjustment of the raton P/L s gven by the P relaton: log C d log w, where parameter d s sgnfcantly from zero. L Startng wth expermental results, a producton functon compatble wth them has been searched. A homogenous frst degree producton functon has been looked for, resultng n: CES (Constant Elastcty of Substtuton), gven by the expresson: 168

7 Stochastc fronter analyss of producton functon and cost functon estmaton methods v f (L,K) A[ L (1 )K ] Here, A s a scalng factor that could be deemed as effcency factor as, for gven L and K, the producton obtaned s proportonal to t. Parameter v measures the scale yeld, and (0,1) s a parameter for revenue dstrbuton between the two nputs. As for ρ, ths s a substtuton parameter, because 1 where s the substtuton elastcty. In many respects, CES producton functon corresponds much better to the realty than Cobb-Douglas functon. At the same tme, the estmaton of CES functon parameters s more dffcult, Cobb-Douglas functon beng, n ths regard, preferred. A more general producton functon than CES s VES producton functon (Varable Elastcty of Substtuton), gven by: v(1) v 1, x ) Ax1 [x ( 1)x 1] f (x Here, the parameters are: A 0, 0, (0,1) (the latter measurng the ax1 soquant convexty). For ths functon, the substtuton elastcty s 1 and x obvously depends on the two nputs (from here comes also the functon name). Translog producton functon s used n practcal applcatons due to ts complex propertes. It has the followng form: 1 lny b0 b1 ln x1 b ln x [ b11 ln x1 b ln x b1 ln x1 ln x ] and gves a second order local approxmaton, beng ft for use n varous stuatons. From ths pont of vew, t has a flexble form. Consderng ths latter ssue, the resdual factor has a very heterogeneous content; t mght contan the effect of technologcal evoluton, scale economy, neffcency etc. 5. Cost fronter ut The techncal effcency for company durng t s defned by: TE t e, and the results of ths value are programmed n Fronter. The whole economc effcency of company s gven by the followng u formula: EE e, where u s the effect of a non-negatve neffcent cost. Ths value s comprsed between zero and one, and a smlar modalty can be predcted for descrbng the techncal effcency for the producton stochastc fronter. The whole economc effcency of cost EE may be decomposed nto ts techncal and allotment components, f the producton functon gven by the estmated 169

8 Management & Marketng cost functon can be explctly derved (ths can be done when Cobb-Douglas formula s used, as t s dual n form). For a smple example of such system, let s consder cost-translog functon usng one output and two nputs: ln c 0 1 ln w 1 ln w 3 ln y 1 ln w 1 ln w 13 ln w 1 ln y 1 3 ln w ln y [b11 ln w1 b ln w b3 ln y ] v u 6. Study of effcency at ndustry level 6.1. Data source The data set contans nformaton taken from the accountng balance sheet and the proft and loss account of 0 companes operatng n the constructon feld, for the perod Ths nformaton has been taken by means of the ste from where the currently tradable companes have been selected. The European currency deprecaton and the ncrease of the prce of utltes strongly affect the Romanan constructon ndustry. Out of more than companes operatng n the constructon ndustry, just 70 are large companes and only these ones have chances to extend ther lfetme on the market. The busness n the constructon ndustry wll be of about 8,5 bllon Euro n 011, two bllon Euro less as compared to Descrpton of varables The Input varables used n ths analyss are: - fxed assets, expressed n RON; - nventores, RON; - number of employees, expressed n persons, representng the number of employees of these companes, per year. The Output varables used n ths analyss are: - operatng revenues; - net proft Descrpton of data All data are expressed n real tme, for ths purpose beng used, as deflator, the Consumpton Prce Index relatng to

9 Stochastc fronter analyss of producton functon and cost functon estmaton methods 6.4. DEA results The elements of Data Envelopment Analyss methods are estmated by usng DEAP.1 software, programmng tool conceved by Tm Coell (1996a). The company effcency scores are computed by usng the two hypotheses: scale constant return CRS and scale varable return VRS. In order to analyse the above-mentoned data, by means of DEAP software, a data fle and an nstructon fle have been constructed. All fles contanng data, nstructons and results are text-type fles Complex analyss for the case wth two outputs and three nputs VRS Input Orentaton The data fle for ths case was named OOIII.DTA. Ths fle contans fve observatons of the two outputs and three nputs. The Output quanttes are lsted n the frst two columns and the Inputs n the next three columns. The fle wth nstructons, OOIII.INS, contans the names of the nstructons and data fles. In the next four lnes, the followng are rendered: number of companes (0); number of tme perods (5); number of outputs () due to the ncluson of a new output n the analyss; and number of nputs (3). The followng three lnes contan the specfcaton «1» for VRS method; «0» for nput orentaton and «0» for DEA standard model estmaton. After havng created the two fles, DEAP programme has been run. The name of the fle wth nstructons OOIII.INS was ntroduced. The programme centralsed the results n a fle named OOIII.OUT. Interpretaton of results DEA results VRS Input Orented are presented n the followng table: DEA results VRS Input Orentaton Company CRS TE VRS TE Scale E drs drs rs rs drs 171 Table 1

10 Management & Marketng Company CRS TE VRS TE Scale E rs rs rs rs drs rs drs rs Mean It can be seen that companes 4, 6, 7, 10, 14, 15 and 0 are the only effcent companes, when CRS method s appled, and companes 3, 4, 6, 7, 10, 14, 15, 16 and 0, when VRS method s appled. 5 companes regster scale decreasng return, 8 companes regster scale ncreasng return and 7 companes are effcent. Dfferent effcency value computaton can be llustrated by resortng to companes 1,, 5, 8, 9, 11, 1, 13, 17, 18, 19, companes neffcent n both methods: CRS and VRS. For nstance, for company 8, CRS techncal effcency s 0.503; VRS techncal effcency s and scale effcency s 0.659, computed as rato between the two terms. The techncal effcency shows us that the company may reduce the nput level by 3.70% and may obtan the same level of output. As t can be seen, company 8 regsters scale decreasng return. If we do compare the two analyses, we observe just small varatons n results, they beng, n essence, the same. Thus, the nfluence of the second output s not sgnfcant. The nformaton regardng the values of nputs and outputs slacks represent the ponts of projecton on the effcency fronter, and ndcate how much the output should ncrease so that the nput value mght reman the same The only dfference from the analyss correspondng to one output s the occurrence of the second one VRS Output Orentaton The data fle for ths case was named OOIIIo.DTA. Ths fle contans fve observatons of the two outputs and three nputs. The Output quanttes are lsted n the frst two columns and the Inputs n the next three columns. In the fle wth nstructons, OOIIIo.INS, the only modfcaton s gven by value «1» ndcatng an output orentaton. After havng created the two fles, DEAP programme has been run.the name of the fle wth nstructons OOIIIo.INS was ntroduced. The programme centralsed 17

11 Stochastc fronter analyss of producton functon and cost functon estmaton methods the results n a fle named OOIIIo.OUT. It s to be noted that, when VRS opton s selected, DEAP programme computes the techncal effcency correspondng to the CRS and VRS methods and the scale effcency. Interpretaton of results DEA results VRS Output Orented are presented n the followng table: DEA results VRS Output Orentaton Company CRS TE VRS TE Scale E drs drs rs drs drs drs rs drs drs drs drs drs rs Mean Table Table 3 centralses the data obtan n the two cases: nput orentaton, usng the nputs and outputs of the 0 companes all over fve perods ( ). The frst column ndcates the results obtaned after havng appled the scale constant return method (CRS), the second column presents the results obtaned after havng appled the scale varable return method and the last column centralses the scale effcency data. The effcency mean, usng CRS, s 0.975, the scale effcency mean s 0.89 and the effcency mean, usng VRS dffers a lttle bt between the two orentatons, amountng to 0.741, respectvely

12 Management & Marketng DEA Multstage Input Output Orentaton Table 3 Input Orentaton Output Orentaton CRS VRS SE CRS VRS SE Mean Table reveals that companes 4, 6, 7, 10, 14, 15 and 0 are the only effcent companes, when CRS method s appled, and companes 3, 4, 6, 7, 10, 14, 15, 16 and 0 when VRS method s appled. 10 companes regster scale decreasng return, 3 companes regster scale ncreasng return and 7 companes are effcent. Dfferent effcency value computaton can be llustrated by resortng to companes 1,, 5, 8, 9, 11, 1, 13, 17, 18, 19, companes neffcent n both methods: CRS and VRS. For nstance, for company 8, CRS techncal effcency s 0.503; VRS techncal effcency s and scale effcency s 0.536, computed as rato between the two terms. The techncal effcency shows us that the company may reduce the output level by 6.3% and may produce the same level of nput. As t can be seen, company 8 regsters scale decreasng return. The nformaton regardng the values of nputs and outputs slacks represent the coordnates of the ponts of projecton on the effcency fronter, and ndcate how much the output should ncrease so that the nput value mght reman the same CRS VRS Input Orentaton by years Herenafter, we wll analyse the effcency ndcators, by usng CRS and VRS methods, foe each and every year, for the perod To ths end, DEA multstage opton wth nput orentaton was selected. A DTA type fle was created, contanng data correspondng to the 0 companes, foe each of the fve analysed years. The results are centralsed and rendered n table 4. As t can be seen, wth CRS assumpton, the effcency level remaned constant durng the frst two years, than t ncreased n 008 from to Snce 008, there was a decrease, reachng n 009 a value of and n 010 a value of The scale effcency ncreases from 0.89 n 006 to n 009, then t decreases to n 010. The property of scale decrease return changes n tme, so that n companes show ths property, n companes, n 008 companes, n companes, and, n companes. The number of companes wth scale ncrease return also changes. Thus, durng the frst year 8 companes show ths property, the next year 10 companes, n companes, n companes and n companes. In 006, companes are effcent, n companes, n companes, decreasng n 010 to 7 effcent companes. It can be also seen that companes 4, 14, 15, 0 are effcent all over the fve years, company 16 manfested a scale decrease return durng the frst two years, becomng effcent the next three years. 174

13 Stochastc fronter analyss of producton functon and cost functon estmaton methods Analysng the slacks contaned n table 5, assumng the two methods, the hghest values are reflected for the frst two nputs: fxed assets and nventores. Therefore, the companes could reach effcency by decreasng the level of nputs by the values rendered n the table. DEA Multstage by years Input Orentaton Model Years Mean CRS VRS SE Table 4 CRS VRS Slacks Input Orentaton Table 5 Input varables Fxed assets Inventores No. of employees

14 Management & Marketng 7. Conclusons Ths study renders the effcency analyss performed by usng Data Envelopment Analyss (DEA) - nput orentaton method, analyss made for each of the fve years consdered, from 006 to 010. The techncal effcency estmaton has been also appled n the followng cases: nput-output orentaton, by resortng to VRS method, all over the perod , wth two outputs and three nputs. The results reveal that the effcency level s qute low for certan companes. Therefore, there should be a hgher nterest for ncreasng the effcency level n the constructon ndustry. Besdes, there s a dfference between the techncal effcency values for the perods and If durng the frst perod, there s an ncrease of the effcency level, the followng perod, a decreasng trend s regstered. At company level, the number of the effcent ones s qute low, out of 0, only 7 beng effcent. Ths result reflects the current stuaton - out of companes actvatng n the ndustry feld, just 70 are large companes and only these ones have chances to extend ther lfetme on the market. The resultng effcent companes are: 4, 6, 7, 10, 14, 15 and 0. References Anderson, L.J., Bogetoft, P., Frost, H. (003), The applcaton of producton functons n boeconomc models, Dansh Research Insttute of Food Economcs Andre, T., Bourbonnas, R. (008), Econometre, Edtura Economcă, Bucureşt Battese, G.E., Coell, T.J. (1995), A model for techncal neffcency effects n a stochastc fronter producton functon for panel data, Emprcal Economcs, 0, pp Cambell, R., Rogers, K., Rezek, J. (006), Effcency fronter estmaton: A maxmum entropy approach, Department of Fnance and Economcs Msssspp State Unversty Coell, T., Perelman, S. (1999), A comparaton of paramterc and non-parametrc dstance functon wth applcaton to European ralways, European Journal of Operatonal Research Coell, T.J., Prasada Rao, D.S. (003), Total factor Productvty Growth n Agrculture: A Malmqust Index Analyss of 93 countres, , Centre for Effcency and Productvty Analyss, Unversty of Queensland, Australa, pp. -6 Cooper, W.W., Seford, L.M., Tone, K. (007), Data envelopment analyss, Sprnger, A Comprehensve Text wth Models, Applcatons, References and DEA-Solver Software, Second Edton Eurostat: Marn, D., Sprcu, L. (005), Analze economce canttatve, Edtura Independenţa Economcă, Pteşt Natonal Insttute of Statstcs: Scence Drect ste: Tmmer, C.P. (1971), Usng a probablstc fronter producton functon to measure techncal effcency, J. Polt. Econ hu, J. (009), Quanttatve Models for Performance Evaluaton and Benchmarkng, Data Envelopment Analyss wth Spreadsheets, Sprnger, Second Edton 176