Cobb-Douglas Function of Production - Example

Outline and Introduction Name: University affiliation Abstract For a long time the Cobb-Douglas function of production has been used to measure economics. This function has faced a number of criticisms based on its approach. However, there is no sufficient research in the function to determine its accuracy on measuring financial data that varies over time. The main aim of this project was to evaluate the relevance and effectiveness of the function when approaching financial problems. A set of data from Chinese economy was used to test the function hypotheses using EViews. Various tests were carried out on the hypotheses. The results of the tests were noted down and analyzed. The ADF and Wald Tests showed that the unction hypotheses were true and there exists unit root. These results showed that the function was both relevant and accurate in measuring financial problems. This suggests that Cobb-Douglas function of production can be used effectively to estimate factors of production over a given time. Contents Abstract 2 Contents 3 Introduction 4 Literature Review 6 Data 7 Method 8 Results 10 Discussion 11 Conclusion 13 References 14 Appendices 15 Appendix 1 16 Appendix 2 18 Appendix 3 18 Appendix 4 19 Appendix 5 20 Appendix 6 20 Appendix 7 21 Introduction Economy can be thought as a financial system of a country while economics as fiscal matters. The method of approaching financial problems using statistical data is called econometrics. In this method, mathematical functions are used to evaluate financial problems. This approach of mathematical functions in solving financial problems has become popular in the world today. Many countries and organization use the method to estimate their production an handle problems related to production to boost their production over a given time. There is a well known approach in econometrics called Cobb-Douglas equation of production. This is an equation proposed by Charles Cobb and Paul Douglas (Tan, 2008). The equation says that total production of a country over a year varies directly as labor and capital. Mathematically, this can be expressed as: Product varies as labor and capital P α L. K Introducing constants to labor and capital; P = bLαKβ Introducing variables L and K in the form of function P(x,y); P (L, K) = bLαKβ Therefore the equation can be written as a mathematical function in the form: P (L, K) = bLαKβ This function can be easily plotted on a line graph to determine the values of P using known variables L and K. It therefore becomes easy to give values of P for a specified time using the L and K. The situation arises: is this function relevant to the current trends of production? This function has for many years been used in measuring economy of a country. It is the basic function in econometrics. However, the variables of this theory and the theory itself have often been criticized. This paper aims to find the basis to this criticism by testing the function through various test methods on its functions to determine the relevance of the function in measuring economy depending on the current production trends as compared to the time when it was first proposed. It tries to analyze whether the function is efficient in measuring production over a varied time series. This is by subjecting the function to different tests on its variables and analyzing the results. Thus this paper is addressing the following objectives in Cobb-Douglas function: Do the variables in the function have meaning or they are just assumed random values? Do the variables have a real value (i.e. constant value of 1) or do they have an absolute value? Is the function relevant in measuring production over a variable time t? Does the function hold true for production analysis in the current situation? Literature Review Economics has been a field of research for a long time. When people learnt to make income from their activities, people began investigating on various methods to improve their income and thus acquire greater wealth with lesser strains. Various researches and publications have therefore been made in the field of economics. This made economics a field of interest attracting renowned economists and researchers. However, less research has made on the discipline of econometrics. Econometrics is a branch of economics which deals with use of statistical methods to solve financial problems. (Brooks, 2008) where as a branch of economics has attracted few researchers. This is because the field has come up recently. The field is more inclined to mathematical function to find solutions to financial problems making it a more complex study. This keeps off researchers who are not fond of mathematical functions as compared to theoretical assumptions. The most basic function used in econometrics is the Cobb-Douglas function of production. The function is used to evaluate production over varying inputs of labor and capital for a given time t. The function was initially projected by Knut Wicksell. Later in 1928, Charles Cobb and Paul Douglas tested the function, modified it and published the results (Tan, 2008). The function has gone through various studies and research over the time. However, there is no significant research on the function in terms of its application on current trends of production. This has made the function to undergo a lot of criticism from various econometrists with others supporting it. It is due to this issue that this project has been carried out. Data The research on this function was based on data from China between 1980 and 2010. The data was chosen as it is easy to evaluate and work on therefore prove the accuracy of Cobb-Douglas function. For each period of the year, there are corresponding Capital, GDP and labor values. (See appendix 7). The first column shows time periods in terms of years. The second shows GDP output for the given year. The third shows capital input for that year and the fourth shows labor input for that year. These are the basic values to compute the function and test it. The data was then converted into a linear graph. This is because the data represented changes varying over a continuous time range hence a continuous data structure. This makes the data more realistic to its nature i.e. a line graph shows a continuous data. Each factor of the equation is represented as a linear graph with the values changing over a given time period. The data was in form of graphs (see appendix 1). The first graph shows variation of capital from the year 1980 to the year 2010. This is a continuous line graph. The second shows variation of GDP over a period from 1980 to 2010. It is also continuous. The third graph shows variation of labor for the period from 1980 to 2010. These three graphs represent the variables of the production function that will be tested. Therefore, they are the basic data values to be used in testing the function as the function is of the same nature as data. The data values are continuous in nature as the period from 1980 to 2010 is a continuous period rather than discrete time allocations. The limitations of the data were as follows: The data was not current as primary data was not used in evaluating the function. Our research was based on evaluating the available published data rather than carrying out the actual data collection in the field to obtain consistent current data. Therefore, the data might be different according to the current records of the data collection. There was no time to carry out an extensive research for the project. The time set for the project was limited. Therefore, the project was not as extensive as required. Little areas were covered in the study which included the essential and major areas. The minor areas were neglected making the research incomprehensive. The data was biased as only one country was used. The data for evaluation was based on Chinese data only. There were no comparisons with other data from various countries. This made the data inclined to a specific region where trends might have been different in other regions. Method The function was to be tested using EViews. This is because the program is easy to use and offers a wide analysis of hypothesis making it a more appropriate computer application for use in hypothesis testing. Also, it has a graphical user interface that is more user oriented than other applications. It is also not prone to errors as compared to human mathematical testing of the hypothesis. The function can be stated as: P (L, K) = bLαKβ This can be simplified as: P = bLαKβ The function can be broken down to individual functions using product rule of functions: P = (bLα) (K β) Breaking it further: (i) P = (bLα) (ii) P = (K β) Finding the log of (i) above: (iii) Α log bK Assuming the value of P to be Y (linear function): P = Y The derivative of the function is: (i) ΔYt = δYt-1 + Ut This becomes a retrogression equation without constant and trend (ii) ΔYt = α + δYt-1 + Ut Adding constant to the equation (i) (iii) ΔYt = α + βT + δYt-1 + Ut Adding trend to the equation (ii) The reason for adding trend and constant is to obtain a panel data. The equation is of time series hence has a trend. The hypothesis becomes: H0: δ = 0 H0: δ ≠ 0 The assessment imperative therefore becomes: If t* is greater than ADF critical value; do not reject null hypothesis as unit root exists. If t* is less than ADF critical value; reject null hypothesis as unit root is not present. The retrogressive equation ΔYt = α + βT + δYt-1 + Ut is submitted to ADF and Wald tests using EViews program. The variables are substituted with K, L and GDP for capital, labor and production respectively. The results are recorded for analysis. Results The following tables give the results of the Unit root test for each variable in the equation: (i) Capital – the results table of ADF test on capital are on appendix 2 (ii) Gross Domestic Product (GDP) – the results for GDP ADF test are on appendix 3 (iii) Labour – the results for labor ADF test are on appendix 4 (iv) The log of GDP results are on appendix 5 (v) The Walt test results are on appendix 6 Discussion In Table 1 above, the results show that unit root exists. The computed t (2.652961) is greater than absolute critical t values at 1% (-4.296729), 5% (-3.568379) and 10% (-3.218382). Also, the coefficient of D (CAPITAL (-1)) is zero. This means that the first difference of capital with a lag of 1 is coefficient 0. There is therefore presence of a unit root. Hence, we cannot eject null hypothesis. R-squared is 0.95. This means that the regression fits the data by 95% hence the low ratio between SE of regression and SD dependant var. Therefore it nearly fits perfectly. The p-value of the results is 0.00. This means that the variable Capital is significant in the hypothesis. Durbin-Watson stat of 1.98 shows that there is no serial correlation in the Capital’s data. In table 2, the results also show existence of a unit root. The computed t (2.236450) is greater than absolute critical t values at 1% (-4.394309), 5% (-3.612199) and 10% (-3.243079). The coefficient of 1st difference GDP with a lag of 1 i.e. D (GDP (-1)) is 0. This illustrates that there is presence of unit root. We cannot ignore the null hypothesis as it is present. R-squared is 0.98 meaning that the regression fits the data by 98%. This is a more accurate fit compared to that of capital with a lesser ratio between SE of regression and SD dependant var. The p-value is also 0.00 meaning that GDP is significant in the hypothesis. Durbin-Watson stat of 2.41 shows that there is no serial correlation in the GDP’s data. In Table 3, labor has a unit root. The computed t (-2.543349) is greater than absolute critical t values at 1% (-4.571559), 5% (-3.690814) and 10% (-3.286909). The coefficient of 1st difference LABOUR with a lag of 1 i.e. D (LABOUR (-1)) is 0. These occurrences prove the existence of unit root. Therefore, we cannot forego a null hypothesis in this case too. R-squared is 0.57 meaning that the regression roughly fits the data with 57%. This is further accounted by a wider ratio between SD dependent var and SE of regression. The p-value is 0.046. this value is less than 0.05 meaning that LABOUR is significant in the hypothesis. Durbin-Watson stat of 2.47 shows that there is no serial correlation in the Labor’s data. In Table 4, GDP is significantly dependent on Capital (K) and labor (LABOUR). Both have high values of computed t i.e. 8.633901 and 2.808397 respectively. R-squared is 0.996 meaning the regression almost fits perfectly. The ratio between SD dependent var and SE of regression is very low. This shows that data is fitting the linear line. Durbin-Watson stat of 0.496 shows that there is probably serial correlation of the variables. In Table 5, there is Wald Test results on the hypothesis. The coefficients of capital (C2) and labor (C3), when summed tend to 1 but not 0. The probability is not 0.0000. This shows that α + β =1 meaning the constants have a return to scale. An increase in the inputs (capital and labor) leads to an increase in the output (GDP).These test results therefore proves the Cobb-Douglas function in the following manner: P (L, K) = bLαKβ When L and K which are labor and capital increase, there is an increase in production. Therefore the equation holds. α + β =1 When elasticity of labor and capital are added, they tend to 1 rather than 0. This means that they result to 1. This shows that they return to scale hence are real. This makes the function to be true in production. Where; P is total production (GDP), L is labor input, K is capital input, b is total factor of production and, β and α are elasticity of capital and labor respectively. Conclusion The aim of the project was to test the relevance of the production function with modern trends. This aim has been successfully met. Therefore, it is evident that the variables are not random numbers but calculated values. The results have shown existence of unit root proving the function relevant to the current production trends. The study had a limited time frame hence could not explicitly expound the subject area. The data used was also secondary making the research rather passive than active as there was no first hand information gained. The function is still a complex area of study. Ore research should be done to improve it and make it more relevant. References 1. Brooks, C. (2008). Introductory Econometrics for Finance. New York: Cambridge University Press. 2. Tan, B.H. (2008). Cobb-Douglas Production Function. Retrieved on 2 October 2014 from http://docentes.fe.unl.pt/~jamador/Macro/cobb-douglas.pdf Appendices Appendix 1 Appendix 2 Null Hypothesis: CAPITAL has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic - based on SIC, maxlag=7) t-Statistic   Prob.* Augmented Dickey-Fuller test statistic  2.652961  1.0000 Test critical values: 1% level -4.296729 5% level -3.568379 10% level -3.218382 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(CAPITAL) Method: Least Squares Date: 10/02/14 Time: 20:38 Sample (adjusted): 1982 2011 Included observations: 30 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   CAPITAL(-1) 0.114113 0.043013 2.652961 0.0134 D(CAPITAL(-1)) 0.478485 0.220945 2.165635 0.0397 C -2.292671 115.1845 -0.019904 0.9843 @TREND("1980") -1.619070 9.998714 -0.161928 0.8726 R-squared 0.950117     Mean dependent var 758.2407 Adjusted R-squared 0.944361     S.D. dependent var 999.0920 S.E. of regression 235.6645     Akaike info criterion 13.88626 Sum squared resid 1443981.     Schwarz criterion 14.07309 Log likelihood -204.2939     Hannan-Quinn criter. 13.94603 F-statistic 165.0732     Durbin-Watson stat 1.978573 Prob(F-statistic) 0.000000 Appendix 3 Null Hypothesis: GDP has a unit root Exogenous: Constant, Linear Trend Lag Length: 7 (Automatic - based on SIC, maxlag=7) t-Statistic   Prob.* Augmented Dickey-Fuller test statistic  2.236450  1.0000 Test critical values: 1% level -4.394309 5% level -3.612199 10% level -3.243079 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(GDP) Method: Least Squares Date: 10/02/14 Time: 20:38 Sample (adjusted): 1988 2011 Included observations: 24 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   GDP(-1) 0.208112 0.093055 2.236450 0.0421 D(GDP(-1)) 0.684006 0.298225 2.293593 0.0378 D(GDP(-2)) -1.226995 0.426655 -2.875851 0.0122 D(GDP(-3)) 1.385425 0.768476 1.802823 0.0930 D(GDP(-4)) -0.673285 1.062629 -0.633603 0.5366 D(GDP(-5)) -2.591491 1.156789 -2.240246 0.0418 D(GDP(-6)) 2.959928 1.070616 2.764697 0.0152 D(GDP(-7)) -2.119496 0.712020 -2.976736 0.0100 C -85.61980 501.7659 -0.170637 0.8670 @TREND("1980") 22.32436 41.50300 0.537897 0.5991 R-squared 0.975611     Mean dependent var 1920.098 Adjusted R-squared 0.959933     S.D. dependent var 1959.829 S.E. of regression 392.2953     Akaike info criterion 15.07624 Sum squared resid 2154539.     Schwarz criterion 15.56710 Log likelihood -170.9149     Hannan-Quinn criter. 15.20647 F-statistic 62.22605     Durbin-Watson stat 2.405439 Prob(F-statistic) 0.000000 Appendix 4 Null Hypothesis: LABOUR has a unit root Exogenous: Constant, Linear Trend Lag Length: 3 (Automatic - based on SIC, maxlag=4) t-Statistic   Prob.* Augmented Dickey-Fuller test statistic -2.543349  0.3062 Test critical values: 1% level -4.571559 5% level -3.690814 10% level -3.286909 *MacKinnon (1996) one-sided p-values. Warning: Probabilities and critical values calculated for 20 observations         and may not be accurate for a sample size of 18 Augmented Dickey-Fuller Test Equation Dependent Variable: D(LABOUR) Method: Least Squares Date: 10/02/14 Time: 20:38 Sample (adjusted): 1994 2011 Included observations: 18 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   LABOUR(-1) -0.503935 0.198139 -2.543349 0.0258 D(LABOUR(-1)) 0.550336 0.221352 2.486251 0.0286 D(LABOUR(-2)) 0.585562 0.219512 2.667569 0.0205 D(LABOUR(-3)) 0.114572 0.191566 0.598080 0.5609 C 282478.6 109327.2 2.583791 0.0239 @TREND("1980") 3873.847 1543.737 2.509395 0.0274 R-squared 0.571420     Mean dependent var 7360.506 Adjusted R-squared 0.392845     S.D. dependent var 1200.755 S.E. of regression 935.6301     Akaike info criterion 16.78152 Sum squared resid 10504844     Schwarz criterion 17.07831 Log likelihood -145.0337     Hannan-Quinn criter. 16.82244 F-statistic 3.199884     Durbin-Watson stat 2.474616 Prob(F-statistic) 0.045844 Appendix 5 Dependent Variable: LNGDP Method: Least Squares Date: 10/02/14 Time: 20:41 Sample (adjusted): 1990 2011 Included observations: 22 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C -41.61265 15.35110 -2.710728 0.0139 LNK 0.702846 0.081405 8.633901 0.0000 LNLABOUR 3.333540 1.186990 2.808397 0.0112 R-squared 0.996005     Mean dependent var 9.259715 Adjusted R-squared 0.995584     S.D. dependent var 0.934226 S.E. of regression 0.062082     Akaike info criterion -2.594582 Sum squared resid 0.073230     Schwarz criterion -2.445804 Log likelihood 31.54041     Hannan-Quinn criter. -2.559535 F-statistic 2368.189     Durbin-Watson stat 0.496267 Prob(F-statistic) 0.000000 Appendix 6 Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic  2.743406  19  0.0129 F-statistic  7.526277 (1, 19)  0.0129 Chi-square  7.526277  1  0.0061 Null Hypothesis: C(2)+C(3)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(2) + C(3)  3.036387  1.106794 Restrictions are linear in coefficients. Appendix 7 Year GDP Capital Labor 1980 454.56 159.97 1981 489.16 163.02 1982 532.34 178.42 1983 596.27 203.9 1984 720.81 251.51 1985 901.60 345.75 1986 1027.52 394.19 1987 1205.86 446.2 1988 1504.28 570.02 1989 1699.23 633.27 1990 1866.78 674.7 643079.6 1991 2178.15 786.8 656387.2 1992 2692.35 1008.63 666602.7 1993 3533.39 1571.77 673537.3 1994 4819.79 2034.11 681385.8 1995 6079.37 2547.01 687745 1996 7117.66 2878.49 695301.8 1997 7897.30 2996.8 702371.6 1998 8440.20 3131.42 708819.6 1999 8967.70 3295.15 716426.3 2000 9921.46 3484.28 724480.3 2001 10965.52 3976.94 732251.4 2002 12033.27 4556.5 741524.8 2003 13582.28 5596.3 749937.8 2004 15987.83 6916.84 758907.4 2005 18493.74 7785.68 767143.2 2006 21631.44 9295.41 775312.5 2007 26581.03 11094.32 782455 2008 31404.54 13832.53 786794.6 2009 34090.28 16446.32 793808.9 2010 40151.28 19360.39 799470 2011 47288.20 22910.24 806026.4     (1) China: Gross domestic product (GDP) Read More