359 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Using Restricted Least Squares Method to Estimate and Analyze the Cobb-Douglas Production Function with Application H. M. Gorgess, A. I. Abdul Naby Department of Mathematics , College of Education -Ibn-Al-Haitham -, University of Baghdad Received in : 8 September 2011 Accepted in : 18 October 2011 Abstract In this paper, the restricted least squares method is employed to estimate the parameters of the Cobb-Douglas production function and then analyze and interprete the results obtained. A practical application is performed on the state company for leather industries in Iraq for the period (1990-2010). The statistical program SPSS is used to perform the required calculations. Key Words: General linear model, Cobb-Douglas production function, OLS estimators, RLS estimators, multicollinearity. 1. Introduction A production function is a mathematical description of the various production activities faced by a firm. Algebraically, It can be written as: [1] q = f(x1,x2,…,xn) …(1) where q represents the flow of output produced and x1,x2,…,xn are the flows of inputs, each measured in physical quantities. Often production functions appear in literature written with two inputs as q = f(K,L) where K denotes the amount of capital and L denotes the amount of labor. Equation (1) is assumed to provide for any conceivable set of inputs, the solution to the problem of how to best (more efficiently) combine different quantities of those inputs to get the output. However, the key question from an economic point of view is how the levels of output and inputs are chosen by firms to maximize profits. Thus, economists use production function in conjunction with marginal productivity theory to provide explanations of factor prices and the levels of factor utilization. The marginal productivity of an input is the additional output that can be produced by employing one more unit of the input while holding all other inputs constant [2]. Algebraically, q K ∂ ∂ is the marginal productivity of capital and q L ∂ ∂ is the marginal productivity of labor. It is assumed that both marginal productivities are positive, that is q K ∂ ∂ > 0, q L ∂ ∂ > 0. The negative marginal productivity means that using more of the input results in less output being produced. It is also usually assumed that the production process exhibits diminishing marginal productivity. This means that successive additions of one factor while keeping the other one constant yields smaller and smaller increases of output, that is: 360 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 2 2 q K ∂ ∂ < 0 and 2 2 q L ∂ ∂ < 0. Factor elasticity is the percentage change in output in response to an infinitesimal percentage change in a factor given that all other factors are held fixed [2], that is: L q L e L q ∂ = ∂ , K q K e K q ∂ = ∂ where eL, eK represent the factor elasticity of labor and capital respectively. 2. The Cobb-Douglas Production Function In economics the Cobb-Douglas functional form of production function is widely used to represent the relationship of an output to inputs. It was proposed by Kunt Wicksell (1851- 1962), and tested against statistical evidence by Charles Cobb and Paul Douglas in 1928 [3]. In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American economy during the period 1899-1922. They considered a simplified view of the economy in which production output is determined by the amount of labor involved and the amount of capital invested. While there are many other factors affecting economic performance, their model proved to be remarkably accurate. The function they used to model production was of the form p(L,K) = b La Kb …(2) where p = total production (the monetary value of all goods produced in a year). L = labor input (the total number of persons-hours worked in a year) K = capital input (the monetary worth of all machinery, equipment, and buildings) b = total factor productivity a and b are the output elasticities of labor and capital, respectively. Each b, a, b are the parameters that must be estimated by using suitable method of estimation. The property of production that examines changes in output subsequent to a proportional change in all inputs (where all inputs increase by a constant factor) is refereed to as returns to scale. If output increases by the same proportional change in all inputs, then there are constant returns to scale (CRTS). If output increases by less than that proportional change, there are decreasing returns to scale (DRTS). If output increases by more than that proportion there are increasing returns to scale (IRTS). However, if a + b = 1 the production function has constant returns to scale. If a + b < 1, returns to scale are decreasing, and if a + b > 1 returns to scale are increasing [4]. In our study we focus our attention on the Cobb-Douglas production function with (CRTS). The assumptions made by Cobb and Douglas can be stated as follows: 1- If either labor or capital vanishes, then so will production, that is p(K,0) = p(0,L) = 0. 2- The marginal productivity of labor is proportional to the amount of production per unit of labor, that is p p L L ∂ µ ∂ . 3- The marginal productivity of capital is proportional to the amount of production per unit of capital, that is p p K K ∂ µ ∂ . 4- b j > 0, j = 1, 2. 3. Deriving the Cobb-Douglas Production Function 361 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Since the production per unit of labor is p L , then according to assumption 2 we have p p L L ∂ = a ∂ for some constant a. If we keep K constant (K = K0) then this partial differential equation will become an ordinary first order separable differential equation d p p d L L = a . Re-arranging the terms and integrating both sides we obtain: 1 1 d p d L p L = aÚ Ú and this yields: ln p = a ln L + c1(K0), then p(L,K0) = c1(K0)La …(3) where c1(K0) is the arbitrary constant of integration and we write it as a function of K0 since it could depend on the value of K0. Similarly, assumption 3 says that: p p K K ∂ = b ∂ for some constant b , keeping L constant (L = L0), this differential equation can be solved to get: p(L0,K) = c2(L0)Kb …(4) And finally, combining equations (3) and (4) to obtain equation 2 which is, [2]: p(L,K) = b La Kb where b is a constant that is independent of both L and K. Notice from equation (2) that if labor and capital are both increased by a factor m, then: p(mL,mK) = b(mL)a(mK)b = ma + b b La Kb = ma + b p(L,K). If a + b = 1, then p(mL,mK) = m p(L,K), which means that production is also increased by a factor m, as discussed earlier. 4. The Case of Multicollinearity In the general linear regression model y = Xb + e where y is (n¥1) vector of response variables, X is (n¥p) matrix, (n > p) of explanatory variables, b is (p¥1) vector of unknown parameters and e is an (n¥1) vector of unobservable random errors, where E(e) = 0 , var(e) = s2In. The problem of multicollinearity exists, when there exists a linear relationship or an approximate linear relationship among two or more explanatory variables. Multicollinearity can be thought of as a situation where two or more explanatory variables in the data set move together, as a consequence it is impossible to use this data set to decide which of the explanatory variables is producing the observed change in the response variable. Some multicollinearity is nearly always exist, but the important point is whether it is serious enough to cause appreciable damage to the regression analysis. The best way to deal with this problem may be to find a different data set, simplify the model by using variable selection techniques or using additional data to break the association between the related variables. Some indicators of multicollinearity include a low determinant of the information 362 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 matrix (X'X), the smallest eigenvalue of the information matrix is very close to zero, a very high correlation among two or more explanatory variables, [5]. However, Farrar-Glauber test can be used to detect multicollinearity, where the null hypothesis to be tested is: H0: xj's are orthogonal against the alternative hypothesis H1: xj's are not orthogonal, j = 1, 2,…,p. The test statistic is: 2 0 1 [n 1 (2p 5)]ln D 6 c = - - - + …(5) where, n is the sample size, p is the number of explanatory variables and D is the determinant of the correlation matrix of explanatory variables. The calculated value of 20c from equation (5) will be compared with the theoretical value obtained from the chi square table with p(p – 1)/2 degrees of freedom and specified level of significant. The null hypothesis H0 will be rejected when the calculated value is more than the tabulated value, which means that the explanatory variables are not orthogonal and hence the multicollinearity problem is presented. 5. Restricted Least Squares Estimator: The restricted least squares (RLS) method of estimation is used when one or more equality restrictions on the parameters of the model are available, [6]. Suppose the general linear model y = xb + e is subject to j equality restrictions represented by the matrix equation Rb = r …(6) where R is (j¥p) matrix of restrictions, b is a p¥1 vector of parameters and r is a j¥1 vector of values of the restrictions. In order to minimize the error sum of squares we have to minimize the Lagrangean function e'e = (y – xb )'(y – xb ) – 2l'(Rb – r) …(7) where l is a j¥1 vector of Lagrange multipliers. Equation (8) can be written as: e'e = y'y – 2b x'y + b 'x'xb – 2l'(Rb – r) Differentiating partially with respect to b and l then set the derivatries equal to zero we obtain, [3]: bRLS = bOLS + (x'x)– 1R'[R(x'x) – 1R'] – 1(r – RbOLS) …(8) where bOLS = (x'x) – 1x'y is the ordinary least squares estimator. The efficiency of the restricted least squares estimator is RLS i RLS i OLS i var(b ) eff (b ) , i 0,1, 2,..., p var(b ) = = In matrix terms we have, [3]: eff(bRLS) = Ip – R'[R(x'x) – 1R'] – 1R(x'x) – 1 …(9) The matrix in equation (9) is a p¥p square matrix; the elements on the main diagonal represent the relative efficiency of the parameters estimated by using the RLS method, these 363 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 diagonal elements must be less than or equal to one, the off-diagonal elements are meaningless. 6. The Practical Application: In this section we try to estimate, analyze and then interpret the Cobb-Douglas production function for the state company for leather industries in Iraq by employing the data obtained from the company for the period (1990-2010). In our study we propose an extension of the usual Cobb-Douglas production function to include the technological progress represented by the time Tt as well as the capital Kt and labor Lt as input variables. Thus our proposed function is: t t1 2 CT U t t tP L K e e b b= a …(10) where C represent the rate of annual growth in production as a consequence to technological progress, moreover we suppose that the production function is CRTS that is b 1 + b 2 = 1. The linear from of this equation is: ln Pt = ln a + b 1 ln Lt + b 2 ln Kt + CTt + Ut setting yt = ln Pt, b 0 = ln a, x1 = ln Lt, x2 = ln Kt, x3 = Tt, b 3 = C, e = Ut we get: yt = b 0 + b 1x1 + b 2x2 + b 3x3 + e …(11) The first four columns of table (1) below represent the values of production Pt, labor Lt, capital Kt and time Tt, the natural logarithms of these quantities are presented in the next three columns. 6.1 The OLS Estimators: According to the principles of ordinary least squares estimation method [6]. The following results were obtained: 21.00 169.23 409.81 231.00 169.23 1364.99 3306.52 1880.32 x ' x 409.81 3306.52 8058.14 4695.67 231.00 1880.32 4695.67 3311.00 È ˘ Í ˙ Í ˙= Í ˙ Í ˙ Î ˚ ( ) 1 119.844 11.984 1.561 0.658 11.984 1.423 0.053 0.048 x ' x 1.561 0.053 0.068 0.018 0.658 0.048 0.018 0.007 - - -È ˘ Í ˙- -Í ˙= Í ˙- - Í ˙ - -Î ˚ 461.93 3724.35 x ' y 9055.24 5263.94 È ˘ Í ˙ Í ˙= Í ˙ Í ˙ Î ˚ 1 OLS 57.8153 4.0659 b (x ' x) x ' y 0.4012 0.4341 - È ˘ Í ˙-Í ˙= = Í ˙- Í ˙ Î ˚ 364 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 The regression equation is: t 1 2 3ŷ 57.8 4.07x 0.401x 0.434x= - - + The analysis of variance calculations are summarized in table (2). The hypothesis H0:b j = 0, j = 0, 1, 2, 3 is tested by comparing the calculated F from the ANOVA table with an appropriate percentage point of the F(v1,v2) distribution where v1, v2 are degrees of freedom due to regression and error respectively. If the calculated value is more than the theoretical value obtained from the F table then we reject H0. At the basis of this test we have: F(3.17,0.05) = 3.59. Since 19.10 > 3.59, then we reject H0. This means that on the basis of this test at least, we have no reason to doubt the adequacy of our model. The variance-covariance matrix of ordinary least square estimators is given as, [3]: ( ) 12OLS 116.859 11.686 1.522 0.642 11.686 1.387 0.052 0.047 var cov(b ) s x ' x 1.522 0.052 0.066 0.017 0.642 0.047 0.017 0.007 - - -È ˘ Í ˙- -Í ˙- = = Í ˙- - Í ˙ - -Î ˚ The coefficient of determination R2 is a convenient measure of the success of the regression equation in explaining the variation of the data. 2 SS (Re gression)R 0.77123 SS (tatal) = = which means that 77.123 % of variations in the data can be explained by the regression equation. 6.2 The Farrar-Glauber Test: In order to perform the farrar-Glauber test of multicollinearty we have to compute the determinant of the correlation matrix of explanatory variables which has the form: 1 2 1 3 2 1 2 3 3 1 3 2 x x x x x x x x x x x x 1 r r D r 1 r r r 1 = In our case of consideration. 1 0.427 0.622 D 0.427 1 0.867 0.867 0.622 1 È ˘ Í ˙= Í ˙ Í ˙Î ˚ and ÔDÔ = 0.147718. Applying the formula in equation (5) we obtain: 2 0c = –(20 – 1 6 (11)) ( – 1.91245) = 34.74289. 365 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 The theoretical value obtained from the Chi-square table with 3 degrees of freedom and 5% level of significant is equal to 7.816. Since 34.74289 > 7.816 we conclude that the explanatory variables are not orthogonal and hence we face the multicollinearty problem. 6.3 The RLS Estimators: In order to remove the effects of multicollinearty we propose using the restricted least squares method of estimation. In this case where the Cobb-Douglas production function assumed to be CRTS we have to write the restriction b 1 + b 2 = 1 in matrix equation as follows: Let R = [0 1 1 0], 0 1 2 3 bÈ ˘ Í ˙bÍ ˙b = Í ˙b Í ˙ bÎ ˚ , r = [1]. Hence: b 1 + b 2 = 1 implies that [ ] 0 1 2 3 0 1 1 0 [1] bÈ ˘ Í ˙bÍ ˙ = Í ˙b Í ˙ bÎ ˚ . 1 13.545 1.476 (x ' x) R ' 0.121 0.066 - -È ˘ Í ˙ Í ˙= Í ˙ Í ˙ -Î ˚ . R(x'x) – 1R' = 1.597 (R(x'x) – 1R') – 1 = 0.626174 [ ]OLS 57.8153 4.0659 Rb 0 1 1 0 [ 4.4671] 0.4012 0.4341 È ˘ Í ˙-Í ˙= = - Í ˙- Í ˙ Î ˚ r – RbOLS = 1 + 4.4671 = 5.4671. (R(x'x) – 1R') – 1(r – RbOLS) = 3.4233 (x'x) – 1R'(R(x'x) – 1R') – 1(r – RbOLS) = 46.36935 5.05287 0.414226 0.22594 -È ˘ Í ˙ Í ˙ Í ˙ Í ˙ -Î ˚ . BRLS = bOLS + (x'x) – 1R'(R(x'x) – 1R') – 1(r – RbOLS) 366 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 RLS 11.446595 0.9869731 b 0.013026 0.208158 È ˘ Í ˙ Í ˙= Í ˙ Í ˙ Î ˚ . Accordingly, the regression equation is: t 1 2 3ŷ 11.4465 0.9869x 0.0130x 0.2081x= + + + and hence, the Cobb-Douglas production function is: t0.2081T0.9869 0.0130 t t tP 93573.266 L K e= . To determine the efficiency of the restricted least squares estimators we apply the formula in equation (9) and setting Ip to be a (4¥4) identity matrix hence we obtain: RLS 1 0 0 0 8.4815 0.0752 0.0801 0.0413 eff (b ) 8.4815 0.9248 0.9192 0.0413 0 0 0 1 È ˘ Í ˙-Í ˙= Í ˙- Í ˙ Î ˚ . 7. Conclusions Applying the RLS method to estimate the parameters of Cobb-Douglas production function leads to the following conclusions: 1. In contrast with the ordinary least squares estimators, the restricted least squares estimators of b 1 and b 2 are with positive signs. Moreover b1 + b2 = 1 the fact that agrees with the assumptions of Cobb-Douglass production function. 2. The efficiency matrix of bRLS reveals that the restricted least squares estimators are more efficient than the ordinary least squares estimators that is because the second and third elements on the main diagonal which represents eff(b1) and eff(b2) respectively are less than one. 3. The values of estimated parameters means that a 100% increase in labor would lead to approximately a 98% increase in production, on the other hand, a 100% increase in capital would lead to approximately a 1.3% increase in production. Also the production satisfies an annual increment of 20.8% during the period of study as a result of technological progress. References 1. Border, K.C., (2004), On the Cobb-Douglas Production Function, Internet Reference, California Institue of Tegnologe, Division of the Humanities and Social Sciences. 2. Lovell, C.A.K., (2007), Production Function, International Encyclopedia of the Social Sciences, 2nd Eddition. رية والتطبيق، مطبعة الطيف، بغداد، ، القياس االقتصادي المتقدم: النظ)2002(كاظم، اموري هادي، مسلم، باسم شليبه، .3 العراق. 4. Hajkora, D. and Hurnik, J., (2007), Cobb-Douglas Production Function: The Case of Converging Economy, Journal of Economics and Finance, Vol. 57, No.9-10, pp.465-468. 367 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 5. Kubokawa, T. and Srivastava, M.S., (2003), Improved Empirical Bayes Ridge Regression Estimators Under Multicollinearity, Internet Reference, CIRJE-F-190. 6. Drappers, N.R. and Smith, H., (1981), Applied Regression Analysis, Second Edition, John Wiley and Sons, New York. 7. Graybill, F.A., (1976), Theory and Application of the Linear Model, Wadsworth Publishing Company, INC.USA. Table (1) Pt Lt Kt Tt yt = ln Pt x1 = ln Lt x2 = ln Kt 77187839 3597 50000000 1 18.16175 8.18786 17.72753 45565859 2601 50000000 2 17.63467 7.86365 17.72753 159555332 2636 50000000 3 18.88790 7.87702 17.72753 416864650 2897 50000000 4 19.84827 7.97143 17.72753 1307335567 2928 50000000 5 20.99126 7.98207 17.72753 3518767839 3182 50000000 6 21.98138 8.06527 17.72753 4973225270 2957 50000000 7 22.32733 7.99193 17.72753 6260702000 2689 50000000 8 22.55756 7.89692 17.72753 10104612000 2376 50000000 9 23.03626 7.77317 17.72753 6451468000 2460 50000000 10 22.58757 7.80792 17.72753 12147023000 2451 1450000000 11 23.22035 7.80425 21.09483 17559452000 2628 1543829187 12 23.58886 7.87398 21.15753 27728280000 2699 1520000000 13 24.04572 7.90064 21.14198 7731060000 2727 1520000000 14 22.76851 7.91096 21.14198 4449594000 2788 1520000000 15 22.21608 7.93308 21.14198 3451738000 3687 1520000000 16 21.96214 8.21257 21.14198 2939702000 4680 1520000000 17 21.80157 8.45105 21.14198 7093426000 4925 1520000000 18 22.68243 8.50208 21.14198 8423365000 4561 1520000000 19 22.85428 8.42530 21.14198 38318430000 4434 1520000000 20 24.36920 8.39706 21.14198 39984891000 4471 1520000000 21 24.41177 8.40537 21.14198 Table (2) ANOVA Source DF SS MS F Regression 3 55.884 18.628 19.10 368 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Error 17 16.577 0.975 Total 20 72.460 -Cobbإستعمال طريقة المربعات الصغرى المقيدة لتقدير وتحليل معلمات دالة االنتاج لـ Douglas مع تطبيق عملي حازم منصور كوركيس ، أحمد عيسى عبد النبي جامعة بغداد -كلية التربية ابن الهيثم -قسم الرياضيات 2011تشرين األول 18قبل البحث في : 2011أيلول 8 استلم البحث في : الخالصة (Cobb-Douglas)فــي هــذا البحــث تــم اســتعملت طريقــة المربعــات الصــغرى المقيــدة لتقــدير معلمــات دالــة االنتــاج لــــ ومـن ثــم تحليــل وتفســير النتــائج التــي تــم التوصــل إليهــا مــع تطبيــق عملــي لتقــدير دالــة االنتــاج فــي الشــركة العامــة للصــناعات الجـــراء الحســـابات (SPSS)) وقــد قمنـــا باالســـتعانة بالبرنــامج االحصـــائي الجــاهز 2010-1990الجلديــة فـــي العــراق للمـــدة ( المطلوبة ، مقدرات المربعات الصغرى Cobb-Douglasطي العام ، دالة االنتاج لـ األنموذج الخالكلمات المفتاحية: . مقدرات المربعات الصغرى المقيدة، التعدد الخطي االعتيادية ، A production function is a mathematical description of the various production activities faced by a firm. Algebraically, It can be written as: [1] q = f(x1,x2,…,xn) …(1) where q represents the flow of output produced and x1,x2,…,xn are the flows of inputs, each measured in physical quantities. Often production functions appear in literature written with two inputs as q = f(K,L) where K denotes the amount of ... Thus, economists use production function in conjunction with marginal productivity theory to provide explanations of factor prices and the levels of factor utilization. The marginal productivity of an input is the additional output that can be produced by employing one more unit of the input while holding all other inputs constant [2]. Algebraically, is the marginal productivity of capital and is the marginal productivity of labor. It is assumed that both marginal productivities are positive, that is > 0, > 0. The negative marginal productivity means that using more of the input results in less output being produced. It is also usually assumed that the production process exhibits diminishing marginal productivity. This means that successive additions of one factor while keeping the other one constant yields smaller and smaller increases of output, that is: 3. Deriving the Cobb-Douglas Production Function 4. The Case of Multicollinearity قسم الرياضيات - كلية التربية ابن الهيثم - جامعة بغداد استلم البحث في : 8 أيلول 2011 قبل البحث في : 18 تشرين الأول 2011 في هذا البحث تم استعملت طريقة المربعات الصغرى المقيدة لتقدير معلمات دالة الانتاج لـ (Cobb-Douglas) ومن ثم تحليل وتفسير النتائج التي تم التوصل إليها مع تطبيق عملي لتقدير دالة الانتاج في الشركة العامة للصناعات الجلدية في العراق للمدة (1990-2010)...