Estimation of COBB-Douglas Production Function in Econometric Model

Transcription

1 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at Estimatio of COBB-Douglas Poductio Fuctio i Ecoometic Model R.V.S.S. Nagabhushaa Rao, V.Muaiah, J. Pabhakaa Naik 3, K.Vasu 4, G. Mokesh Rayalu 5 ssistat pofesso, Depatmet of Statistics, Vikama Simhapui Uivesity, Nelloe Lectue, Depatmet of Statistics, PVKN Govt College, Chittoo 3 Lectue i Statistics, SSBN Degee & PG College (utoomous), atapu 4 ssistat Pofesso, Vidya Educatioal Istitutios, adamai, Chittoo 5 ssistat Pofesso, Depatmet of Mathematics, School of dvaced scieces, VIT, Velloe bstact: impotat ifeetial phase i oliea poductio fuctio model poblems is the exploatio of the estimatio of paametes of elatioship betwee the output (depedet vaiable) ad iputs (idepedet vaiables). The most fequetly used oliea poductio fuctio models i the ecoometic aalysis ae the Cobb-Douglas ad Costat Elasticity of Substitutio (CES) ad Tascedetal logaithmic (Taslog) poductio fuctios. The pupose of this chapte is to descibe the vaious estimatio methods fo estimatig the paametes of these thee types of oliea poductio fuctio models. I. INTRODUCTION Besides the existig sigle equatio methods of estimatio, the detemiistic ad stochastic specificatios i the foms of simultaeous equatios have bee explaied fo Cobb-Douglas ad CES poductio fuctio models. The ifeetial poblems icludig two impotat methods of estimatio amely Geealized Least Squaes (GLS) ad Zele s Feasible estimatio have bee peseted fo estimatig the paametes of the flexible oliea poductio fuctio amely Taslog Poductio. Some special ifeetial poblems such as the estimatio of Cobb-Douglas poductio fuctio model with multiplicative ad additive distubaces; estimatio of Cobb-Douglas poductio fuctio by emovig the poblem of multicollieaity; ad oliea method of estimatio of Cobb-Douglas poductio fuctio, have bee discussed with efeece to the most fequetly used oliea poductio fuctio model amely the Cobb-Douglas poductio fuctio. II. NONLINER METHOD OF ESTIMTION OF COBB-DOUGLS PRODUCTION FUNCTION MODEL Coside the Cobb-Douglas poductio fuctio model as = (.) Whee, ad ae paametes By selectig some iitial values of the ukow paametes, 0, 0, ad 0, oe may fom the followig expessio by usig Taylo seies expasio i the eighbohood of ( 0 ), ( 0 ) ad ( 0 ) : 0 = [ ] + ( 0 ) + ( 0 ) ( 0 ) (.) ddig a eo tem to this expessio ad miimize the eo sum of squaes S = paamete estimates as, ad ; ad agai fom a lieaized expessio : i i oi, oe ca obtai the IJRSET (UGC ppoved Joual): ll Rights ae Reseved 780

2 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at = [ ] + ( ) + + ( - ) ( ) (.3) ddig agai a eo tem to this expessio ad the secod oud paamete estimates, ad ; ca be obtaied by miimizig eo sum of squaes S = i i. (.4) i Oe ca cotiue this pocedue iteatively util the paamete estimates appea to covege to a paticula set of values.othe appoach is to miimize the esidual sums of squaes S = i i ˆ ˆ i ˆ i obtai the followig system of thee equatios S = 0 (.5) ˆ ˆ with espect to S = 0 (.6) S = 0 (.7) ˆ ˆ, ˆ ad ˆ to These ae thee simultaeous oliea equatios i the thee ukow paametes. Each of these thee equatios ca be expaded about assumed iitial values of the paametes as follows : S S = 0 Simila expessios fo S, S + S S + 0 = 0 (.8) 0 0 S about 0, 0 ad 0 (whee 0, 0 ad 0 be the iitial values of paametes) yield thee simultaeous liea equatios i the thee ukow paametes, which ca the be solved fo, ad. We the poceed iteatively. This techique, with modificatios to appoach the miimum. S efficietly, is the pocedue suggested by Eisepess ad Geestadt (964). This pocedue guaatees covegece to a local extemum, but oe ca ot be cetai that oe foud the global miimum. The umbe of iteatios equied will deped upo how accuate the iitial guesses of the paamete values tu out to be i the iteative pocess. The Maximum likelihood estimatio of paametes i a system of equatios follows the same piciples. IJRSET (UGC ppoved Joual): ll Rights ae Reseved 78

3 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at III. ESTIMTION OF COBB-DOUGLS PRODUCTION FUNCTION USING LOGRITHMIC LINER TRNSFORMTION Coside the Cobb-Douglas poductio fuctioal model is of the fom i = i i e i, i =,,, (3.) Whee : Output of the fim : Labou iput : Capital iput : Techological Coefficiet : Labou iput elasticity paamete : Capital iput elasticity paamete : Classical distubace vaiable To estimate the ukow paametes, ad, oe may take logaithmic tasfomatio o both sides of the model (3.), which gives i = + i + i + i (3.) o Whee, = ; i = + i + i = i ; i + i (3.3) i =,,, i = i, Model (3.3) ca be cosideed as a thee vaiable liea model. The least squaes estimates of, ad ca be obtaied as follows : ˆ x x x x y ad Whee, ˆ = x x x  = x ˆ = x x y (3.4) ˆ (3.5) = = x x x y = IJRSET (UGC ppoved Joual): ll Rights ae Reseved 78

4 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at x y = Â. ad  = ti The least squaes estimated Cobb-Douglas poductio fuctioal model is give by (3.6) ˆ i =  ˆ i ˆ, i =,,, i IV. ESTIMTION OF COBB-DOUGLS PRODUCTION FUNCTION THROUGH COST FUNCTION Coside the Cobb-Douglas poductio fuctio with labou ( ) ad capital ( ) iputs as = Let the pices of output, labou ad capital be P, P = = = (4.) ad = P By cosideig wage ate (w) ad ate of iteest () as poxies fo espectively. Wite the magial poductivity coditios as P (4.) P (4.3) P ad P ; ad substitutig the magial poductivity coditios (4.) ad (4.3) ito the poductio fuctio model (4.), oe ca have the cost fuctio coespods to the Cobb-Douglas poductio fuctio as C = w. (4.4) By itoducig multiplicative eo tem ito the cost fuctio ad usig logaithmic liea tasfomatio. The cost fuctio ca be estimated. Usig the estimates of the paametes of the cost fuctio, the Cobb-Douglas poductio fuctio ca be estimated. I pactice, it is moe coveiet to estimate the cost fuctio athe tha to estimate the Cobb-Douglas poductio fuctio. V. ESTIMTION OF COBB-DOUGLS PRODUCTION FUNCTION MODEL WITH MULTIPLICTIVE ND DDITIVE ERRORS Coside a geeal Cobb-Douglas poductio fuctio model i which thee ae both additive ad multiplicative eos as y = 0 x x k x e k + u (5.) Whee y is output; x i s ae iputs; ad u ae eos which ae distibuted idepedetly ad omally with u E() = E (u) = 0; E( ) =, E(u ) = ; E( i j ) = E (u i u j ) = 0, i j ad E( i u j ) = 0 i, j. lso x i s ae ostochastic ad j s ae paametes. Let Z be the adom vaiable such that IJRSET (UGC ppoved Joual): ll Rights ae Reseved 783

5 If ~ N (0, the p.d.f. of u is give by Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at Z = 0 x x k k x e = Q e (5.) ) the the pobability desity fuctio (p.d.f) of Z is give by f(z) = g(u) = z u exp exp z log Q u u Now, the p.d.f. of y is the covolutio of (5.3) ad (5.4) which is give by h(y) = u 0 log Q exp y - u d (5.3) (5.4) (5.5) Hee is simply a dummy vaiable of itegatio. The age of itegatio is (0, ) as the logomally distibuted vaiable is defied oly fo positive values. Fo a sample of obsevatios o y ad x i s, the likelihood fuctio may be witte as L = h(y i ) (5.6) i Istead of maximizig (5.6), it is coveiet to maximize log L as Log L = i log u exp - 0 i log Q i yi u d (5.7) With espect to 0,,,, k ; ad u. This gives the maximum likelihood estimates of the paametes. Howeve, it is ot possible to fid aalytic expessios fo the solutios of maximizatio of (5.7). The maximizatio of (5.7) equies o algoithm fo umeical maximizatio of a fuctio of may vaiables. Goldfeld ad Quadt (970) used Powell s (964) cojugate Gadiet method to obtai solutios. Kelejia (97) studied futhe the Cobb-Douglas model with multiplicative ad additive eos suggested by Goldfeld ad Quadt (970). He assumed that the eos may follow ay pobability distibutio ot ecessaily the omal distibutio. Let E(e ) = such that e may be expessed as e = + (5.8) IJRSET (UGC ppoved Joual): ll Rights ae Reseved 784

6 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at whee E() = 0. By substitutig (5.8) ito (5.9), oe ca have y = x x k k x + (5.9) Whee = 0 ad = 0 x x k k =,,, k. VI. ESTIMTION OF COBB-DOUGLS PRODUCTION FUNCTION B REMOVING MULTICOLLINERIT Coside the Cobb-Douglas poductio fuctio as y t = t x t x t x + u.sice ad u ae idepedet of x i s, it follows that E[/x i ] = 0, i t e, t =,,, (6.) Whee y t = output fo the time peiod t ; t = Labou iput fo the time peiod t ; t = Capital iput fo the time peiod t ;, ae the elasticity paametes of labou ad capital espectively; t measues the level of techology o techological coefficiet; ad t eo tem. Takig logaithms o both sides of (6.), gives y t = t + x t + x t + t (6.) By assumig a costat ate of techological chage t = 0 e t, (6.) may be witte as y t = 0 + t + x t + x t + t (6.3) Hee, is a time ted paamete. Most of the empiical studies idicate vey high coelatio betwee x t ad x t, i fact all the estimates ae suspect due to the multicollieaity poblem. To ovecome the multicollieaity poblem. Oe may usually assume the costat etus to scale, i.e., + = ad efomulate the Cobb-Douglas poductio fuctio as y t = 0 + t + () x t + x t + t o Whee x t xt y t xt = 0 + t + is the capital labo atio. x t xt + t (6.4) Some empiical studies [ fo istace, Owyog ad Bhaoji Rao (998)] exhibited that still the multicollieaity peset due to the coelatio betwee x t xt ad the time ted t. Oe emedy to this poblem is to omit the time ted vaiable t fom the model (6.4). But, this itoduces the poblem of omitted vaiable bias ito the estimates. To ovecome this poblem, coside the fist diffeece of the equatio (6.4), ad ewite the models as yt yt - x t x t - u t x t xt x t xt IJRSET (UGC ppoved Joual): ll Rights ae Reseved 785

7 whee u t = t - t-. Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at y t xt = + x t xt + u t (6.6) (6.5) Thus, the fist diffeece of the atual logaithme output labou atio may be egessed o the fist diffeece of the atual logaithmic capital labou atio without multicollieaity poblems. Futhe, the ate of gowth of Total Facto Poductivity (TFP) say, may be obtaied simply fom the itecept estimate. VII. DETERMINISTIC SIMULTNEOUS EQUTIONS MODEL FOR COBB-DOUGLS PRODUCTION FUNCTION Coside the Cobb-Douglas poductio fuctio model as = (7.) Defie the pofit fuctio as R = P w (7..) Whee R : Pofit, P : Pice of Output w : Wage ate (chose as poxy fo the pice of Labou) : ate of iteest (chose as poxy fo the pice of capital) Usig the magial poductivity coditios, the maximizatio of pofit fuctio (7.) subject to the poductio fuctio (7.) gives the followig elatioships = P w = P P = Fom (7.), (7.3) ad (7.4), oe ca have The equatios (7.5) may be ewitte as w w = o = P Pw w o = Pw w P P (7.3) (7.4) (7.5) IJRSET (UGC ppoved Joual): ll Rights ae Reseved 786 w P P (7.6)

8 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at IJRSET (UGC ppoved Joual): ll Rights ae Reseved = P P w o = P P w = + P w + P = + P w + P = + P w + P (7.7) = a 0 + a P w + a P = b 0 + b P w + b P (7.8) = c 0 + c P w + c P Whee a 0 = = b0 = c 0

9 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at a = a = = c = b b = c = Sice diffeet fims have diffeet outputs ad iputs eve whe cofoted with the same set of pices, the detemiistic model will be too estictive fo its estimatio. Maschak ad dews (944) have itoduced stochastic distubaces ito model ad estimated the paametes by usig the least squaes estimatio. Howeve these estimates ae biased ad icosistet. May Statisticias ad Ecoometicias have tied to obtai at least cosistet estimates of the paametes of this simultaeous equatios model. VIII. CORRECTED LEST SQURES ESTIMTES FOR THE PRMETERS OF COBB-DOUGLS PRODUCTION FUNCTION By itoducig stochastic distubaces ito the detemiistic simultaeous equatios model, (7.5) may be witte as = (8..) w = + (8.) = P P + (8.3) Hoch (958) specified the model (8.) as model by assumig the ucoelatio betwee the ecoomic distubaces ad : Whee = (8.4) 0 = ; =, = = ; = ad = By applyig Least Squaes estimatio, oe ca estimate the tasfomed model ad hece obtai the estimates fo ad as ˆ ad ˆ espectively. Hoch (958) poposed the geealized fomula fo the coected least squaes estimates of the paametes of the Cobb-Douglas poductio fuctio model as = ˆ i i M 00 M M M M M M Q00 Mij M M M M i =,, j =, ad i j. (8.5) IJRSET (UGC ppoved Joual): ll Rights ae Reseved 788

10 Whee, M ii = V 00 + V ii Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at V o, i =, i M = V 00 + V Vo V o Q 00 = M 00 = V 00 + ˆ Vo ˆ V o M 00 MM M M M M M M M M 00 (8.6) Hee, V 00 is the sample vaiace of V ii is the sample vaiace of V is the sample covaiace of i, i =,, ad, V o is the sample covaiace of i ad i, i =, By substitutig Q 00 i (8.6) ito (8.5) oe ca get the fomula fo Hoch s (958) coected least squaes estimates of ad as = i ˆ i MMM M 00Mij M M M M M M M M 00 i, j =,, ; i j (8.7) I. CONCLUSIONS I this pape we discuss about diffeet estimatio methods fo estimatig the paametes of thee fequetly used oliea poductio fuctios amely Cobb-Douglas, CES ad Taslog poductio fuctio models. The detemiistic ad stochastic specificatios of system of simultaeous equatios fo Cobb-Douglas ad CES poductio fuctio models have bee explaied besides the existig sigle equatio methods of estimatio. REFERENCES [] ige, D.J. (974), symptotic Miimum-MSE Pedictio i the Cobb-Douglas Model with a Multiplicative Distubace Tem, Ecoometica, 4, [] Bad,. (974), Noliea Paamete Estimatio, cademic Pess, New ok. [3] Cavaagh, C.L. (98), Hypothesis Testig i Noliea Models Pape Peseted at the Ecoometic Society Euopea Meetig, Dubli. [4] Davidso, R. ad MacKio, J.G. (98), Seveal Tests fo Model Specificatio i the Pesece of lteative Hypothesis, Ecoometica, 49, [5] Eisepess, H. ad Geestadt, J. (966), The Estimatio of Noliea Ecoometic Systems, Ecoometica, 34, [6] Goldbege,.S. (968), The Itepetatio ad Estimatio of Cobb-Douglas Fuctios, Ecoometica, 35, [7] Jeich, R.I. (969), symptotic Popeties of Noliea Least Squaes Estimatos, The als of Statistics, [8] Lu,.C. ad Fletche, L.B. (968), Geealizatio of the CES Poductio Fuctio, Review of Ecoomics ad Statistics, 50, [9] Nelove, M. (965), Estimatio ad Idetificatio of Cobb-Douglas Poductio Fuctios, Noth-Hollad, mstedam. [0] Olive, F.R. (966), spects of Maximum Likelihood Estimatio of the Logistic Gowth Fuctio, Joual of the meica Statistical ssociatio, 6, [] Owyog, D.T. ad Bhaogi Rao, v.v. (998), Total Facto Poductivity Gowth i the Sigapoe Ecoomy : Some Ecoometic Systems, The Sigapoe Ecoomic Review, 43, [] Pedda Reddeppa Reddy, M. (99), Some spects of Noliea Ecoometic Models, Upublished M.Phil., Thesis, S.V. Uivesity, Tiupati, Idia. IJRSET (UGC ppoved Joual): ll Rights ae Reseved 789

11 Iteatioal Joual fo Reseach i pplied Sciece & Egieeig Techology (IJRSET) ISSN: ; IC Value: 45.98; SJ Impact Facto :6.887 Volume 5 Issue II Decembe 07- vailable at [3] Peasa, M.H. ad Deatio,.S. (978), Testig No-Nested Noliea Regessio Models, Ecoometica, 46, [4] Stephe, M.G. ad Quadt, R.e. (970), The Estimatio of Cobb-Douglas Type Fuctios with Multiplicative ad dditive Eos, Iteatioal Ecoomic Review,, [5] Zele,. (96), Efficiet Method of Estimatig Seemigly Uelated Regessios ad Tests fo ggegatio Bias, Joual of meica Statistical ssociatio, 57, IJRSET (UGC ppoved Joual): ll Rights ae Reseved 790