Maximization of Technical Efficiency of a Normal- Half Normal Stochastic Production Frontier Model

Transcription

1 Global Journal of Pure and Applied Mathematic. ISS Volume 13, umber 1 (17), pp Reearch India Publication Maximization of Technical Efficiency of a ormal- Half ormal Stochatic Production Frontier Model Sangeethamani. S #1 and Mary Loui. L * #1 PhD Scholar, Department of Mathematic, Avinahilingam Univerity, Coimbatore-64118, Tamil adu, India * Aociate Profeor of Mathematic, Faculty of Engineering, Avinahilingam Univerity, Coimbatore-64118, Tamil adu, India Abtract The main apect of tochatic frontier analyi i to etimate the technical efficiency baed on the choice and price of input and the level of output produced. Further the rank of the firm in term of efficiency can be obtained. The derivation of the Stochatic Production Frontier Model play a vital role in meauring technical efficiency. The current tudy focue on the derivation and maximization of technical efficiency of ormal- Half ormal Stochatic Production Frontier Model. Keyword: Technical Efficiency, ormal-half ormal ditribution, Stochatic Production Frontier Model, Maximization. ITRODUCTIO In 5 Coelli et al aumed a functional form for the relationhip between input and an output in the Stochatic Frontier Analyi method of frontier etimation. The tochatic production function model wa propoed independently by Aigner, Lovel, Schmidt Meeuen and van den Broeck in the year 1977 propoed the tochatic production function model. Kumbhakar, Ghoh and McGuckin [1991 and Huang and Liu [1994 deigned tochatic production model for the parametric etimation of both the tochatic frontier and the inefficiency function. Aigner and Lovell (1976) worked on the production frontier model y i = f(x i, β) + ε i where ε i = v i u i

2 7354 Sangeethamani. S. and Mary Loui. L The key perpective of tochatic frontier analyi i the introduction of the compoite error term which contain two component : a technical inefficiency component and a noie component. A firm i aid to be efficient or inefficient with repect to it own production frontier baed on the compoite error term. The Maximum Likelihood Method or Method of Moment can be employed to obtain the technical efficiency of the firm with the ditributional aumption baed on the error term. Variou ditributional aumption have been ued in the literature for the two error term. In thi paper in the derivation of ormal-half ormal Stochatic Production Frontier Model the following ditributional aumption were made. (i) The error term repreent the tatitical noie v i ~iid (, σ v ) (ii) The error term repreenting the technical efficiency u i ~iid + (, σ u ) (i.e non-negative half normal). (iii) v i and u i are ditributed independently of each other and of the regreor. Parameter like α,, μ are etimated uing the method of maximum likelihood. Thi paper involve four major ection namely: Section I: Derivation of the ormal-half ormal Stochatic Production Frontier Model. Section II: Etimation of the parameter of the ormal-half ormal Stochatic Production Frontier Model. Section III: Meaurement of the technical efficiency of the ormal-half ormal Stochatic Production Frontier Model. Section IV: Maximization of the technical efficiency of the ormal-half ormal Stochatic Production Frontier Model. I. THE ORMAL-HALF ORMAL STOCHASTIC PRODUCTIO FROTIER MODEL The Probability denity function of u i given by f(u) = πσ u exp { u σ u } (1) The Probability denity function of v i given by f(u) = 1 πσ v exp { v σ v } ()

3 Maximization of Technical Efficiency of a ormal-half ormal Stochatic 7355 Since u and v are independently ditributed, the joint denity function of u and v i the product of their individual probability denity function f(u, v) = f(u). f(v) = πσ u σ v exp { u σ u + v σ v } (3) Uing the tranformation, ε = v u, the joint denity function of u and ε i f(u, ε) = 1 πσ u σ v exp { u σ u (u+ε) σ v } (4) The marginal denity function of ε i given by f(ε) = f(u, ε)du πσ u σ v πσ u σ v πσ u σ v exp { u σ (u +ε +uε) u σ v } du exp { u σ v u σ u ε σ u uεσ u } du σ u σ v exp [ 1 (σ v +σ u )+ε σ u +uεσ u {u } du σ u σ v (5) (6) (7) (8) Let σ = σ u + σ v and μ = σ u σv πσ u σ v exp [ 1 (σ )+ε σ u +uεσ u {u } du σ u σ v exp [ 1 {σ πσ u σ v σ u σ (u + ε σ u +uεσ u )} du v σ exp [ πσ u σ v σ u σ {(u + ε σ u v σ exp [ πσ u σ v σ u σ {(u + ε σ u v σ exp [ πσ u σ v σ u σ {(u + uεσ u v σ exp [ { πσ u σ v σ u σ v + uεσ u )} du σ + uεσ u σ + ε σ 4 u σ 4 + εσ 4 u σ 4 ) + ε σ u σ ) + ε σ u ε σ 4 u σ 4 εσ 4 u σ 4 ε σ 4 u σ 4 } du )} du } du (9) (1) (11) (1) (13) (14) exp [ πσ u σ v σ u σ v σ ). exp [ σ u σ { v σ 4 } du (15) σ ε σ u ε σ u 4 exp [ πσ u σ v σ u σ v σ ). exp [ σ u σ {σ u v σ 4 } du ( (16) σ ( ε ε σ u ) exp [ πσ u σ v σ u σ v σ ). exp [ 1 exp [ πσ u σ v σ u σ v σ ). exp [ ε σ { ε ε σ u v σ σ { σ u v σ } du } du (17) (18)

4 7356 Sangeethamani. S. and Mary Loui. L exp [ πσ u σ v σ u σ v σ ). exp [ ε exp [ πσ u σ v σ u σ v σ ). exp [ ε σ {σ u +σ v σ u v σ σ {σ v v σ exp [ πσ u σ v σ u σ v σ ). exp [ ε du σ exp [ ε exp [ πσ u σ v σ σ u σ v σ ) du } du } du (19) () (1) () ) σ Let t = σ u σ v dt = σ u σ v du A u, t εμ πσ u σ v exp [ ε and a u, t exp ( t ) σ uσ v εμ dt (3) exp [ ε exp ( t εμ ) π σ dt (4) f(ε) = exp [ ε exp ( t εμ ) π σ dt (5) f(ε) = exp [ ε ( π) exp ( t εμ ) σ dt (6) f(ε) = exp [ ε 1 exp ( t εμ ) π σ π dt (7) f(ε) = exp [ ε π σ [1 Φ (εμ) (8) f(ε) = φ ( ε ) Φ ( εμ ) (9) Where φ i the denity function and Φ i the tandard normal cumulative ditribution. E(ε) = E(v u) = E(v) E(u) = E(u) = E(u) (3) E(ε) = u f(u) du = u exp ( u πσ u σ u ) du (31) Put t = dt = u σ u u du σ u

5 Maximization of Technical Efficiency of a ormal-half ormal Stochatic 7357 u du = σ u dt A u, t and a u, t E(ε) = σ u exp( t) dt πσ u (3) E(ε) = π σ u (33) V(ε) = V(v) V(u) (34) V(u) = E(u ) [E(u) (35) E(u ) = E(u ) = u Put t = dt = u du σ u u σ u u du = σ u dt u = tσ u u = tσ u u f(u) du πσ u exp ( u σ u ) du (36) (37) E(u ) = tσ u e t σ π u dt σ u (38) E(u ) = σ π u t e t dt E(u ) = σ π u t ( ) e t dt 3 E(u ) = σ π u Г ( 1 + 1) (41) E(u ) = π σ u Г ( 1 ) (4) E(u ) = π σ u 1 Г (1 ) (43) E(u ) = 1 π σ u π (44) E(u ) = σ u (45) V(u) = E(u ) [E(u) (46) (39) (4)

6 7358 Sangeethamani. S. and Mary Loui. L V(u) = σ u [ σ π u V(u) = σ u σ π u (47) (48) V(u) = ( π π ) σ u (49) V(ε) = σ v ( π π ) σ u (5) II. ESTIMATIO OF THE PARAMETERS The likelihood function of the ample i given by L = i=1 f(ε i ) (51) The log likelihood function for ε i = y i m(x i, α) i ln L = contant ln σ + [ln Φ ( ε iμ ) 1 i=1 ε σ i=1 i (5) ln L = ( ) (lnπ + lnσ ) + [ln Φ ( ε iμ ) ε i i=1 (53) L[α, μ, = ln L ( ) lnπ ( ) ln 1 (y i x i α ) i=1 + lnφ [ (y i x i α)μ i=1 (54) The firt order partial derivative with repect to α, μ, are obtained by differentiating (54) partially and equating them to zero a below P α = lnl = 1 (y α σ i x i=1 i α)x i + { φ[ (y i x i α)μ (y i x i α)μ ( x i μ i=1 )} (55) P μ = lnl = μ {φ[ ( y i x i α i=1 )} (56) P σ = lnl σ = + 1 (y σ σ 4 i x i α) + μ i=1 σ 3 { φ[ { φ[(y i x i α)μ Φ[ (y i x i ( y i x i α α)μ (y i x i α)μ i=1 (y i x i α)} (57) P μ = i=1 )} = (58) + 1 (y σ σ 4 i=1 i x i α) = (59)

7 Maximization of Technical Efficiency of a ormal-half ormal Stochatic 7359 = 1 (y σ σ 4 i=1 i x i α) (6) = 1 (y σ i=1 i x i α) (61) Where x i i a m 1 column vector. The likelihood etimator of i obtained a σ = 1 (y i=1 i x i α) (6) Let A m data matrix B 1 data vector ε 1 vector β = φ[(yi xi α)μ Uing the above aumption in (55),(56)and (57) we have P α = 1 (A y A Aα) = μa β = (63) P μ = 1 ε β = (64) P σ = εε + με β 3 = (65) Multiplying (63) by (AA ) (AA ) (A y A Aα) (AA ) μa β = (66) (AA ) A y α (AA ) μa β = (67) α = (AA ) A [y μβ (68) The likelihood etimator of α i α = p q (69) Define lope vector of the OLS etimate p = (AA ) A y OLS etimate q = (AA ) μa β From equation (63) 1 σ A ε μa β = (7) Multiplying (7) by α α A ε α μa β = (71)

8 736 Sangeethamani. S. and Mary Loui. L The likelihood etimator of μ i μ = α A ε α A β If u i ~(, σ u ), the conditional ditribution of u given ε i f(u ε) = f(u,ε) f(ε) (7) (73) f(u ε) = f(u ε) = [1 Φ( εμ 1 u exp { πσuσv σu (u+ε) σv } ε exp[ π [1 Φ(εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v f(u ε) = [1 Φ( εμ ) π σ u σ v ) (74) exp { 1 [ u + (u +ε +uε) σ u σ ε v σ } (75) exp { 1 σ v +(u +ε +uε)σ u [u ε } (76) σ u σ v σ exp { 1 (σ u +σ v )+(ε σ u +uεσ u ) [u σ u σ ε v σ } (77) exp { 1 σ +(ε σ u +uεσ u ) [u ε } (78) σ u σ v σ exp { 1 [ σ σ u σ v σ ) + ε ε σ σ } (79) exp { 1 [ σ (u + εσ u σ u σ v σ ) } (8) exp { 1 ( u+ εσu σ u σ ) v } (81) Let λ = εσ u and = σ u σ v f(u ε) = [1 Φ( λ π ) exp { 1 (u+λ ) } (8) III. MEASUREMET OF TECHICAL EFFICIECY A f(u ε)~ + (λ, σ ), the mean E(u ε) can be regarded a the point etimator of u i E(u ε) = u f(u ε) du (83)

9 Maximization of Technical Efficiency of a ormal-half ormal Stochatic 7361 E(u ε) = u [1 Φ( λ π ) exp { 1 (u+λ ) } du (84) Define t = u+λ and dt = du A u, t λ and a u, t λ [1 Φ( ) E(u ε) = π ( t λ ) exp { 1 λ t } dt (85) E(u ε) = [1 Φ ( λ ) [ π t exp { 1 λ t } dt + λ exp { 1 π λ t } dt (86) Let = t, d = tdt A t, and t λ, λ E(u ε) = [1 Φ ( λ E(u ε) = [1 Φ ( λ E(u ε) = [1 Φ ( λ E(u ε) = [1 Φ ( εσ u ) { ) { π ). λ π e d + λ [1 Φ ( λ )} (87) exp ( λ σ ) + λ } (88) { φ ( λ ) + λ σ } (89) ) { φ ( εσ u. ) + λ σ u σ v σ σ u σ } (9) v E(u ε) = [1 Φ ( εμ ) {σ φ ( εμ ) + εσ u } (91) σ E(u ε) = { φ(εμ E(u i ε i ) = { ) [1 Φ( εμ φ(ε iμ ) ) + εμ [1 Φ( ε i μ + ε iμ ) Etimate of u i can be obtained from } (9) } (93) TE i = exp[ E(u i ε i ) (94) A propoed by Battee and Coelli(1988) an alternative point etimator for TE i can be preferred when u i i not cloe to zero which i defined a

10 736 Sangeethamani. S. and Mary Loui. L TE i = E[exp { u i } ε i (95) IV. MAXIMIZATIO OF TECHICAL EFFICECY The econd order partial derivative are obtained a follow α lnl = 1 ( x i α ) [ Φ[ ( x σ i ) + { φ[ i=1 i=1 (y i x i α)μ Let δ = φ[(yi xi α)μ φ [ (y i x i α)μ (y i x i α)μ ( x i ) + ( x i μ ) φ[(y i x i α)μ Φ [ (y i x i α)μ ( x i μ ) { } } (96) lnl = 1 ( x α σ i ) + {( x i ) δ + ( x i α ) i=1 i=1 δ } (97) lnl = [ 1 ( x α σ i ) + {( x i ) δ + ( x i α ) i=1 δ } (98) lnl = μ (y i x i α ) [ Φ[ ) i=1 (99) α (y i x i α)μ φ [ (y i x i α)μ ( x i μ ) φ[(y i x i α)μ Φ [ (y i x i α)μ ( x i μ { = μ (y i x i ) ( x i i=1 ) δ (1) α } lnl = Φ[ 1 [ σ 3 + σ 4 (1) ( (y i x i α)μ φ [ (y i x i α)μ ) (y σ 6 i x i α) + (y i x i α) μ φ[ i=1 i=1 { (y i x i α)μ (y i x i α) μ 3+φ[(y i x i α)μ Φ [ (y i x i α)μ (y i x i α) μ 3 { } ( 3 ) (σ ) ( 3 ) + } (11) lnl = 1 ( 1 σ 4 3 [ δ (y i x i α) i=1 i=1 6 ) (y i x i α) + (y i x i α) μ μ 3 {δ ( 3 5 ) + } (1) The econd order partial derivative calculated above uing equation (98), (1) and (1) are etimated at each of it critical point for maximization of the output or minimization of the input.

11 Maximization of Technical Efficiency of a ormal-half ormal Stochatic 7363 COCLUSIO If the econd order partial derivative are le than zero then the technical efficiency i aid to be maximum. Alo the inefficiency u i can be obtained following the etimation of the parameter. If ε i <, then the producer i aid to be technically efficient and alternatively if ε i >, then the producer i technically inefficient. REFERECES [1 Alvarez A, Aria C., 4, Technical Efficiency and Farm Size:a Conditional Analyi, Agricultural Economic, 3, [ Aigner D.J, Knox Lovell C.A, Schmidt P., 1977, Formulation and Etimation of Stochatic Frontier Production Function, Journal of Econometric, 6, [3 Battee G.E, Coelli T.J., 1995, Frontier Production Function, Technical Efficiency and Panel Data: With Application to Paddy Farmer in India, Journal of Productivity Analyi, 3, [4 Caudill S.B, Ford J.M., 1993, Biae in Frontier Etimation Due to Heterokedaticity, Economic Letter, 41, 17-. [5 Caudill S.B, Ford J.M, Gropper D.M., 1995, Frontier Etimation and Firm Specific Inefficiency Meaure in the Preence of Heterokedaticity, Journal of Buine and Economic Statitic,13, [6 Hazarika G, Alwang J.,3, Acce to Credit, Plot Size and Cot Inefficiency Among Smallholder Tobacco Cultivator in Malawi, Agricultural Economic, 9, [7 Huang C.J, Liu J.T., 1994, Etimation of a on-eutral Stochatic Frontier Production Function, Journal of Productivity Analyi, 5, [8 Kumbhakar S, Ghoh S, McGuckin J., 1991, A Generalized Production Frontier Approach for Etimating Determinant of Inefficiency in US Dairy Farm, Journal of Buine and Economic Statitic, 9, [9 Meeuen W, van den Broeck J., 1977, Efficiency Etimation from Cobb- Dougla Production Function with Compoed Error, International Economic Review,18, [1 Reifchneider D, Stevenon R., 1991, Sytematic Departure from the Frontier: A Framework for the Analyi of Firm Inefficiency, International Economic Review,3, [11 Stevenon RE.,198, Likelihood Function for Generalized Stochatic Frontier Etimation, Journal of Econometric, 13, [1 Wang HJ, Schmidt P.,, One-Step and Two-Step Etimation of the Effect of Exogenou Variable on Technical Efficiency Level, Journal of Productivity Analyi, 18,

12 7364 Sangeethamani. S. and Mary Loui. L [13 Coelli, T.J., Rao, D.S.P., O'Donnell, C.J., Battee, G.E., 5, An Introduction to Efficiency and Productivity Analyi, nd Edition, Springer. [14 Farrell, M., 1957, The Meaurement of Productive Efficiency, Journal of the Royal Statitical Society Serie A (General), 1 (3), [15 H. Fried, C.A.K. Lovell, S. Schmidt (ed)., 8, The Meaurement of Productive Efficiency and Productivity Change, ew York, Oxford Univerity Pre, [16 Greene, W. H., 199, A Gamma-Ditributed Stochatic Frontier Model, Journal of Econometric. 46, [17 Jondrow, J., C. A. Knox Lovell, Ivan S. Materov, and Peter Schmidt., 198, On the etimation of technical inefficiency in the tochatic frontier production function model, Journal of Econometric 19, [18 Kalirajan, K. P. and Shand, R. T., 1999, Frontier Production Function and Technical Efficiency Meaure, Journal of Economic Survey, vol. 13(), page [19 Kalirajan, K. P. and Shand, R. T., 199, Cauality between technical and allocative efficiencie: an empirical teting, Journal of Economic Studie, 19, [ Kumbhakar, S. C. and C.A. Knox Lovell.,, Stochatic Frontier Analyi, Cambridge Univerity Pre, Cambridge. [1 Meeuen, W. and van Den Broeck,. J., 1977, Efficiency etimation from Cobb Dougla Production Function with Compoed Error, International Economic Review, 18, : [ Schmidt, P., 1985, Frontier production function, Econometric Review, 4:, [3 Schmidt, P. and Lovell, C. A. K.,1979, Etimating technical and allocative inefficiency relative to tochatic production and cot frontier, Journal of Econometric, 9, pp [4 Schmidt, P. and Sickle, R. C., 1984, Production frontier and panel data, Journal of Buine and Economic Statitic,, [5 Simar, L.,199, Etimating efficiencie from frontier model with panel data: A comparion of parametric, non-parametric and emi-parametric method with boottrapping, The Journal of Productivity Analyi, 3, [6 Stevenon, R. E., 198, Likelihood function for generalized tochatic frontier etimation, Journal of Econometric, 13, [7 Chritopher F. Parmeter and Subal C. Kumbhakar., 14, Efficiency Analyi:A Primer on recent advance, Univerity of Miami, State Univerity of ew York at Binghamton.14.