COBB-Douglas, Constant Elasticity of Substitution (CES) and Transcendental Logarithmic Production Functions in Non-linear Type of Special Functions

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1 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at COBB-Douglas, Constant Elasticity of Substitution (CES) and Transcendental Logarithmic Production Functions in Non-linear Tye of Secial Functions R.V.S.S. Nagabhushana Rao, J. Prabhakara Naik, V. Munaiah 3, K.Vasu 4, G. Mokesh Rayalu 5 Assistant rofessor, Deartment of Statistics, Vikrama Simhauri University, Nellore Lecturer in Statistics, SSBN Degree & PG College(Autonomous), Anantaur 3 Lecturer, Deartment of Statistics, PVKN Govt College, Chittoor 4 Assistant Professor, Vidya Educational Institutions, Yadamari, Chittoor 5 Assistant Professor, Deartment of Mathematics, School of Advanced sciences, VIT, Vellore Abstract: In the literature, many Statisticians and Econometricians have described the various nonlinear statistical models in the context of roduction function analysis. It is no exaggeration to say that in econometric analysis, the Cobb-Douglas, Constant Elasticity of substitution (CES) and transcendental logarithmic (known as Translog) roduction functions are the most frequently used nonlinear tye of secial functions. They have been construed as the simle tyes of meaningful functions in both theoretical and emirical studies of roduction functions and have received rivileged attention in the analysis of economic growth. I. INTRODUCTION The roduction analysis deals with the structural relationshis between inuts and oututs of Firms. A roduction function is a technique which indicates the technology involved in the rocess of roduction. Mathematically, a roduction function can be reresented by Y = f(x, X,, X n ), where Y is outut and X s are inuts such as labour, caital, land, raw material, energy etc. In other words the roduction function is a technological relationshi between outut and inuts by disaggregation. If outut is a flow variable, all inuts must be exressed in flow terms. It is not as easy and trivial as it sounds. Generally, a roduction function deicts a relationshi between a flow of outut over a defined time interval and a flow of inuts over the same or revious time. Nonlinear roduction function models have been used in a wide variety of contexts and the roblem of statistical inference in these models based on measurements arises in almost every corner of econometric research. The Cobb-Douglas roduction function model lasted 60 years before generalizations began to aear. A growing dissatisfaction with the descritive accuracy of the function and with its econometric imlications has sawned a multitude of alternative functional forms during the ast four decades. First came the CES functional form, followed by several Variable Elasticity of Substitution (VES) forms and more flexible forms such as Translog and Frontier functional forms. The Cobb-Douglas form has been generalized in other ways, as have the CES, VES and Translog generalizations themselves. II. CONCEPTS IN THE PRODUCTION FUNCTION ANALYSIS The various concets frequently used by Econometricians in the context of study of roduction function analysis are : A. Isoquants In the case of a firm roducing a single outut from two intus, the roduction function may be defined as y = f(x, x ) (.) Where y is the maximum ossible level of outut Y; x and x are the levels of the two inuts X and X resectively; and f(.,.) is a function that is generally assumed to be continuously differentiable, so that the artial derivatives are continuous Consider, f(x,x ) = y 0 = constant (.) Totally differentiating (.) gives

2 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at f f dx + dx = 0 (.3) x x dx f / x f = = dx f / x (.4) f f f Where f = and f = are called marginal roducts of inuts X and X resectively. x x x reresents the rate at which the entrereneur can substitute inut X for X ( or X for X ) in order to maintain secific x outut level y 0. The ratio of the marginal roducts of two inuts X and X is known as the Marginal Rate of Technical Substitution (MRTS) f between two inuts X and X. i.e., MRTS =. f B. Homogeneous Production Function And Euler s Theorem When the inuts are changed by the some roortion k, the function Y changes by some ower of k. This roerty renders the function is homogeneous roduction function. The ower of the multilicant k in the transformed function defines the degree of homogeneity of the function. It should be noted that the sum of the owers of the variables (inuts) in each term on the R.H.S is constant in a homogeneous function. If the sum is zero, then the degree of homogeneity of the function is zero.when the degree of homogeneity is one, then the function is known as the Linearly Homogeneous Function.. Suose Z = f(x, x ) is assumed to be a linearly homogeneous roduction function, then the following three roerties hold good for all the values of inuts x and x ) The given linearly homogeneous functions can be written in either of the following two forms : Z = x x x, or Z = x where and are some functions of a single variable. ) The artial derivatives, Z x and Z x x x are functions of the ratio of x to x 3) Linearly homogeneous functions satisfy Euler s theorem. C. Euler s Theorem The value of homogeneous function can always be written as a sum of terms, each of these being the roduct of the first artial derivative and the corresonding inut variable. If Z =f(x,x ) be a linearly homogeneous function, then it satisfies the Euler s theorem as follows. Z x Z x + y = Z x D. Elasticity Of Substitution Between Factors Of Production () Every firm is interested in knowing of the extent of substitutability between the factors of roduction in order to maximise rofit or minimise cost looking to the rices which the factors command in the market. The measure of substitution between the factors is known as Elasticity of Technical Substitution. This is a ure number and measures the extent to which the substitution between the factors can take lace. Since the substitution deends mainly on the sloe or MRTS, Elasticity of Substitution is defined as the roortionate change in the ratio between the factors due to roortionate change in MRTS. Consider the roduction function y = f(x,x ) The elasticity of substitution between the two factors is given by

3 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at = x d ln x M ln x d M x = x d ln x d lnmrts x x (3.) Here, the numerator involves the ratio of X to X while the denominator involves the ratio of the Marginal roduct of X to that of X, ensuring that is nonnegative. It gives a measure of the curvature of the Isoquants.The magnitude of is an indication of the case with which roduct can be maintained by substituting one factor for another. Higher the value of, the higher the degree of substitutability between the factors. The value of lies between 0 and. = 0 two factors (or inuts) are incaable of substitution; = two factors are erfect substitutes; 0 the shae of Isoquant tend to be an angle of 90 ; the shae of Isoquant tends to be flatter. If y = f(x,x ) be the linear homogeneous function, then the elasticity of substitution between x and x is given by = y y x x y y. xx (3.) E. Returns To Scale Returns to scale may be constant, increasing or decreasing. If we increase all factors (i.e., scale) in a given roortion and the outut increases in the same roortion, returns to scale are said to be constant. Thus, if a doubling or trebling of all factors causes a doubling or trebling of oututs, returns to scale are constant. But if the increase in all factors leads to a more than roortionate increase in outut, returns to scale are said to be increasing. Thus, if all factors are doubled and outut increases by more than a double, then the returns to scale are increasing. On the other hand, if the increase in all factors leads to a less than roortionate increase in the outut, returns to scale are decreasing. The k th degree homogeneous roduction function satisfies the condition f(mx, mx ) = m k f(x, x ), m > 0, m = Constant returns to scale ; m> Increasing returns to scale ; m< Decreasing returns to scale. Thus, homogeneous roduction function of degree one or linearly homogeneous roduction function exhibits constant returns to scale in the economy. III. SOME NONLINEAR PRODUCTION FUNCTIONS Some imortant nonlinear roduction function models which are frequently used in ractice are : A. Cobb Douglas Nonlinear Production Function One of the most widely used nonlinear roduction functions for emirical estimation is the Cobb Douglas roduction function. Cobb and Douglas (98) roosed the general FORM of roduction function as Y = A X X (4.) Where Y is the value added bylabour (X ) and fixed caital (X ); A, are the fixed ositive arameters. Another form of Cobb-Douglas roduction function is given by

4 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at Y = A X X (4.) Where Y is the outut of the firm; X is labour inut; X is caital inut; A is technological coefficient arameter; islabour inut elasticity arameter; and is caital inut elasticity arameter. B. Constant Elasticity Of Substitution (Ces) Nonlinear Production Function The CES roduction function was first develoed by Arrow, Chenery, Minhas and Solow (96) which includes the Cobb-Douglas roduction function as a secial case. This function has constant elasticity of substitution between the factors of roduction. An essential form of CES roduction function is given by - y = A X ( ) X (4.3) Where y = outut of firm X, X = Two factors of roduction (may be caital and labour inuts resectively) A = Efficiency arameter or Scale arameter, A>0 = Distribution arameter, 0<< = Substitution arameter, - It can be shown that this roduction function is linearly homogeneous roduction function. It shows constant returns to scale and justifies Euler s theorem. Though this function is linearly homogeneous, the value of elasticity of substitution () is not unity. By introducing returns to scale arameter(), the general form of CES roduction function is given by Y = A - X ( ) X, >0 (4.4) C. Transcendental Logarithmic (Translog) Production Function The translog roduction function was initiated by Christensen, Jorgenson and Lau (973), which is quadratic in the logarithms of the variables. This function is quite flexible in aroximating arbitrary roduction technologies interms of substitution ossibilities. It rovides a local aroximation to any roduction frontier. For n inuts, the translog roduction functional form is given by n ln y = a 0 + i ln x n n i + ij ln x i lnx j (4.5) i i j Where x i is the i th inut and ij = ji. This function is not invariant to a change of units. The translog roduction function is neither additive nor homogeneous and it can easily include multile oututs and multile inuts. The translog roduction function which contains an outut (Y) and four inuts say Labour (X ), Caital (X ), Raw material (X 3 ) and Energy (X 4 ) is secified by y = f (x, x, x 3, x 4 ). Assuming constant returns to scale with exogenous factor rices,, 3 and 4, and imosing symmetry on the second order artial derivatives, gives the translog cost function as ln C = a 0 + ln Y + ln + ln + 3 ln ln 4 + (ln ) + (ln ) (ln ) + 3 (ln ) (ln 3 )

5 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at (ln ) (ln 4 ) + (ln ) + 3 (ln ) (ln 3 ) + 4 (ln ) (ln 4 ) + 33 (ln 3 ) + 34 (ln 3 ) (ln 4 ) + 44 (ln 4 ) (4.6) The cost share equations are given by lnc ln = S = + ln + ln + 3 ln ln 4 lnc ln = S = + ln + ln + 3 ln ln 4 lnc ln 3 = S3 = ln + 3 ln + 33 ln ln 4 (4.7) lnc ln 4 = S4 = + 4 ln + 4 ln + 34 ln ln 4 Since, the shares must sum to unity, = (4.8) and the s sum to zero in each column (and row). i.e., By imosing the row-wise constraints on the first three share equations gives the system. S 3 ln ln 3 ln S 3 ln ln 3 ln S ln 3 ln 33 ln (4.9) (4.0) IV. CONCLUSIONS In this aer, we rooses certain nonlinear regression models. A method of estimation has been roosed to a general nonlinear model with auto correlated disturbances. A model selection criterion has been derived for testing the no nested nonlinear hyotheses. A traditional simultaneous equations model for a Cobb-Douglas roduction function is secified and iterative indirect least squares and two stage least squares estimation methods have been develoed in this aer

6 ISSN: ; IC Value: 45.98; SJ Imact Factor :6.887 Volume 5 Issue XII December 07- Available at REFERENCES [] Aigner, D.J. (974), Asymtotic Minimum-MSE Prediction in the Cobb-Douglas Model with a Multilicative Disturbance Term, Econometrica, 4, [] Amemiya, T. and Powell, J.L. (98), A Comarison of the Box-Cox Maximum Likelihood Estimator and the Non-Linear Two-Stage Least Squares Estimators, Journal of Econometrica, 7, [3] Burguete, J.F., Gallant, A.R. and Souza, G. (98), On the Unfication of the Asymtotic Theory of Nonlinear Econometric Models, Econometric Reviews,, [4] Christensen, L.R., Jorgenson, D.W. and Lau, L.J. (973), Transcendental Logarithmic Production Frontiers, Review of Economics and Statistics, 55, [5] Cook, R.D., Tsai, C.L. and Wei, B.C. (986), Bias in Nonlinear Regression, Biometrica, 73, [6] Davidson, R. and MacKinnon, J.G. (98), Several Tests for Model Secification in the Presence of Alternative Hyothesis, Econometrica, 49, [7] De-Min Wu (975), Estimation of the Cobb-Douglas Production Function, Econometrica, 43, [8] Dhrymes, P.J. (965), Some Extensions and Tests for the CES Class of Production Functions, Review of Economics and Statistics, 47, [9] Dhrymes, P.J. (967), Adjustment Dynamics and the Estimation of the CES Class of Production Functions, International Economic Review, 8, [0] Dhrymes, P.J. (97), A Simlified Structural Estimator for Large Scale Econometric Methods, The Australian Journal of Statistics, 3, [] Draer, N.R. and Smith, H. (966), Alied Regression Analysis, First Edition, John Wiley and Sons, Inc., New York. [] Fisk, P.R. (966), The Estimation of Marginal Product from a Cobb-Douglas Production Function, Econometrica, 34, 6-7. [3] Fletcher, R. and Reeves, C.M. (964), Function Minimization by Conjugate Gradients, Comuter Journal, 7, [4] Frydman, R. (980), A Proof of the Consistency of Maximum Likelihood Estimators of Nonlinear Regression Models with Auto Correlated Errors, Econometrica, 48, [5] Gallant, A.R. (975a), Testing a Subset of the Parameters of a Nonlinear Regression Model, Journal of the American Statistical Association, 70, 35, [6] Oliver, F.R. (964), Methods of Estimating The Logistic Growth Function, Alied Statist, 3, [7] Zellner, A.Z. and Richard, J.F. (973), Use of Prior Information in the Analysis and Estimation of Cobb-Douglas as Production Function Models, International Economic Review, 4,