ESTIMATION OF PARAMETERS OF GENERALISED COBB-DOUGLAS PRODUCTIONFUNCTIONAL MODEL

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1 Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology ISSN: CODEN: IJPTFI Available Online through Research Article ESTIMATION OF PARAMETERS OF GENERALISED COBB-DOUGLAS PRODUCTIONFUNCTIONAL MODEL Dr B.Mahaboob, Dr B.Venkateswarlu,, 3 Prof.P.Balasiddamuni, Department of Mathematics, Swetha Institute of Technology and Science,Tirupati. Department of Mathematics, VIT University, Vellore, Tamilnadu. Rtd.Professor, Department of Statistics, S.V.University, Tirupati. bmahaboob750@gmail.com Received on: Accepted on: Abstract: In Mathematical economics more general non-linear regression models are usedin the estimation of production functions and demand functions. The Cobb-Douglas [] production function involving an additive error term; constant Elasticity of Substitution (CES) and Variable Elasticity of Substitution (VES) production functions are more complicated and highly nonlinear models. This research paper gives a new method of estimation of generalised Cobb- Douglas production functional model. Keywords: Output, labour input, technological coefficient parameter, labour input elasticity parameter and capital input elasticity parameter.. Introduction: Mathematical Economic theory indicates certain concepts in nonlinear production function analysis, which have frequently been used in practice. For instance, some of them are economics of scale, the degree of substitutability of the factors, capital or labour intensives and the efficiency of the process. Among all nonlinear production functions existing in the literature, the best known and the most widely used nonlinear production function is the Cobb-Douglas production function. The mathematical form of the Cobb-Douglas [] production function was due to Cobb (98), a mathematician; but the economic and empirical properties of the Cobb-Douglas production function was due to Douglas, an economist, with his vast amount of empirical support for the function, it gained current popularity. The nonlinear Cobb-Douglas [] production function model has been used in a wide variety of contexts in social and behavioural sciences.. Specification and properties of nonlinear Cobb-Douglas production functional model: The general form of nonlinear Cobb-Douglas [] production function is given by IJPT April-07 Vol. 9 Issue No Page 98

2 Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology Y.. (.) 0 Where, Y-Output; -Labour Input; -Capital Input; β 0 -Technological Coefficient Parameter;β -Labour Input Elasticity Parameter and β -Capital Input Elasticity Parameter. Properties of Nonlinear Cobb-Douglas production function: (i) (ii) It is intrinsically linear model It is a homogeneous nonlinear function of degree (β + β ). If (β + β ) =, then this model turns to be linearly homogeneous functional model. (iii) Isoquants for the nonlinear Cobb-Douglas production function are negatively sloped and convex downwards. (iv) The marginal products of labour and capital inputs are respectively given by MP and MP. Y Y MP 0... (.) Y Y MP (.3) (v) The Marginal Rate of Technical Substitution of for MRTS of the Cobb-Douglas production functional model is given by MRTS MP. (.4) MP (vi) The Labour Input Elasticity of Output and Capital Input Elasticity of Output are given by: Y / Y (.5) Y / Y.. (.6) (vii) The expansion path generated by the Cobb-Douglas production functional model is a straight line and it passes through the origin. (viii) The elasticity of substitution between factors of the Cobb-Douglas production functional model is unity. IJPT April-07 Vol. 9 Issue No Page 98

3 dln dln MRTS Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology (.7) 3. Estimation of parameters of CES production functional model: The constant elasticity of substitution (CES) production functional model form was first given by Dickinson (954) and later popularized by Arrow, Chenery, Minhas and Solow [5]. This function cannot be linearized by logarithmic transformation and its estimation is more complex than the Cobb-Douglas production functional model.the general form of CES production functional model is given by Y, 0.(3.) Where, Y: output; : Capital Input; : Labour Input; : Efficiency parameter : Distribution parameter; : Substitution parameter; : Returns to Scale parameter The first order partial derivatives of Y with respect to inputs and yield, Y Y (3.) Y Y.... (3.3) Here, Y and Y give the marginal products of capital and labour inputs respectively. In practice, the prices of labour and capital are not directly measurable; the wage rate(w) and the Rate of Interest (r) may be taken as their corresponding prices respectively. From the Mathematical Economics, under Theory of Firms, the marginal productivity conditions give Y Y w and r w Y... (3.4) And r Y. (3.5) w r..... (3.6) IJPT April-07 Vol. 9 Issue No Page 983

4 Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology By taking logarithms on both sides of the equations (3.4), (3.5) and (3.6) and introducing the classical error variables, one may obtain Ln w Ln Ln Y Ln u... (3.7) Ln r Ln Ln Y Ln u.. (3.8) w Ln Ln Ln u r 3 (3.9) The equations (3.7), (3.8) and (3.9) may be written as w Y u (3.0) 0 r Y u... (3.) 0 Q Z u... (3.) Ln ; Where, ; ; 0 Ln ; ; 3 ; 0 Ln ; ; w Ln w ; r w Ln r ; Q Ln r ; Z Ln ; Y LnY, Ln ; Ln ; u, u and u 3 are classical error variables. By considering (3.0), (3.) and (3.) multiple linear regression models and applying ordinary least squares (OLS) method of estimation, one can obtain the OLS estimators of parameters in the equations respectively as ˆ, ˆ and ˆ, (ii) ˆ 0, ˆ ˆ and and(iii) ˆ ˆ 0, (i) 0 Now, the estimators of,, and are given by ˆ ˆ... (3.3) ˆ ˆ ˆ (3.4) ˆ Anti Ln ˆ 0... (3.5) 4. Estimation of parameters of CES production functional model by Taylor series expansion method: Consider the constant elasticity of substitution (CES) production functional model as IJPT April-07 Vol. 9 Issue No Page 984

5 Y e, 0 Here, is a classical error variable By taking logarithms on both sides of equation (5.9.) gives Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology... (4.) Ln Y Ln Ln (4.) Expanding Ln Y by Taylor series formula for expansion about 0, after discarding the terms of third and higher orders, the expansion yields Ln Y Ln Ln Ln Ln Ln... (4.3) Y.. (4.4) Where, Y Ln Y, Ln, 0 Ln,, Ln, Ln Ln 3 and 3 Thus, the nonlinear CES production functional model (5.9.) reduces to a multiplelinear regression model (4.4).The application of Ordinary Least Squares (OLS) method of estimation to the model (4.4) yields the OLS estimators of parameters 0,, and 3 as 0 ˆ, ˆ, ˆ and 3 ˆ respectively. Now the estimators of the parameters of the CES production functional model are given by ˆ ˆ ˆ Anti Ln 0, ˆ ˆ ˆ ˆ ˆ ˆ and, ˆ ˆ ˆ ˆ 3 ˆˆ 5. Specification and Estimation of Parameters of generalised Cobb-Douglas Production Functional model with additive error term: A generalized Cobb-Douglas production functional model involving additive error variable is specified by 3 k Y..., i,,...,n... (5.) i i 3i ki i Where, Y is output; s are inputs; s are parameters, is error variable and n is number of observations j j on each variable. Without loss of generality, ignoring suffixes, the model may be re-written as IJPT April-07 Vol. 9 Issue No Page 985

6 Y Z Z Z 3 k Where Z 3... k Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology... (5.)... (5.3) By taking logarithms on both sides of (5.) yields, Log Y Log Z Log Z..... (5.4) expressed as Using Logarithmic series expansion and discarding the terms of second andhigher orders (5.4) may be Log Y Log Z Z.... (5.5) ~ N O, and E O Z, the transformed model maybe written as i.i.d Assuming that Log Y Log Log Log... Log (5.6) 3 3 k k Where, such that Z E O, i,3,...,k i By ignoring the heteroscedasticity of and applying the OLS method of estimation givesthe estimators of parameters,,..., k. Conclusion: One of the most common situations of statistical analysis in business, economics, engineering, physical sciences is likely to lead to a nonlinear regression model. Generally the models to be obtained as solutions of differential equations arising in physical sciences, engineering, and Ecology and business economics are some examples of nonlinear models. In the present work one of the important applications of nonlinear regression models namely estimation of generalised Cobb-Douglas production functional model has been discussed. Though there are various methods of estimation of this model are available in the literature a new method of estimation using OLS(Ordinary Least Squares) method of estimation has been proposed. References:. Cobb, C. and Douglas, P.98. A Theory of Production American Economic Review, Vol.8, pp Cobb CW and Douglas PH (98) A Theory of Production, American Economic Review, 8 (Supplement), IJPT April-07 Vol. 9 Issue No Page 986

7 Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology 3. Douglas P H (934) The Theory of Wages,, New York.: Macmillan. 4. Douglas P H (948) Are There Laws of Production? American Economic Review, 38, Arrow,K; Chenery, H; Minhas, B; and Solow,R.96. Capital Labour Substitution and Economic Efficiency. Review of Economics and statistics, Vol.63, pp Goldfeld, S.M., and Quandt R.E. (97), Nonlinear Methods in Econometrics, North Holland Publishing Co., Amsterdam 7. Britt, H.I., and Lueke, R.H. (973), The Estimation of Parameters in Nonlinear ImplicitModels, Technometrics, 5, Amemiya, T. (973), Regression Analysis when the Dependent Variable is Truncated Normal, Econometrica, 4, Nakatani, I. (973), Production Functions with variable Elasticity of Substitution: A Comment, Review of Economics Statistics 55, pp Christensen.L, Jorgenson. Dand Lau, L.973. Transcendental Logarithmic Production Frontiers. Review of Economics and statistics, Vol.55, pp Gallant, A.R. (977), Three-stage Least Squares Estimation for a System of Simultaneous, Nonlinear, Implicit Functions, Journal of Econometrics, 5, Amemiya, T. (980), Lecture Notes in Advanced Theory of Econometrics, Department of Economics, Stanford University 3. White.H, (980), Nonlinear Regression on Cross-Section Data, Econometrica, 48, Robinson, P.M. (988), The Stochastic Difference between Econometrics and Statistics, Econometric a, 56, Cohen A.and Harcourt G.003. Whatever happened to the Cambridge Capital Controversies? Journal of Economics Perspectives, Vol.7, pp Miyagiwa, K. and C. Papa Georgiou (003), Elasticity of Substitution and Growth: Normalized CES in the Diamond Model, Economic theory, Md.Moyazzemhossain, An Application of Non-Linear Cobb-Douglas Production Function to selected Manufacturing Industries in Bangladesh, Open Journal of Statistics, 0,, IJPT April-07 Vol. 9 Issue No Page 987

8 Dr B.Mahaboobet al. /International Journal of Pharmacy & Technology 8. Giannis Karagiannis, Theodore Palivos, Chris Papa Georgiou, Variable Elasticity of Substitution and Economic Growth: Theory and Evidence, june Ashfaq Ahmad, Estimating the Cobb-Douglas Production Function, International Journal of Research in Business Studies and Management, Volume.Issue 5,May 05,pp 3 33,ISSN: Dr B.venkateswarlu, et.al. (05) Efficiency evaluation of total manufacturing sectors of India- DEA approach, Global Journal of Pure and Applied Mathematics, ISSN NO: Volume, Number 5 (05), pp Dr B.Mahaboob,.et.al. (06), A New Iterative Method of Estimation for Non-linear Regression Model, Indian Streams Research Journal, ISSN ,Vol.6(4). IJPT April-07 Vol. 9 Issue No Page 988