Quiz 7: Production Economics

Business

The production function is given as: img We need to calculate the parameters img and img . We will use the regression analysis. img We denote img by img , ln img by img and img by img . Therefore, img In the table below, we will calculate the values of img , img , img , img , img , img , img and img . img img img img img img img img img img img The Cobb-Douglas production function can be written as: img

a) In this table, we will calculate the marginal product and average product. The total product is given. img The capital is fixed at 500-bhp rating. The total product of labor img is output. The marginal product of labor img is img . The average product of labor img is total product divided by labor. In this question, we were given the total product. We were to calculate the average product and the marginal product. For instance, the total product at img is 16. Hence the average product at img is img . In a similar way, we can calculate the average product for all labor inputs. The total product at img is 6 and at img is 16. The marginal product at img is img . Similarly, we can calculate the marginal product for other labor inputs. b) We will draw the total product, average product and marginal product curve here. img The TP, MP and AP are shown in the diagram above. The point to note is that when average product is at its maximum point, it is equal to the marginal product curve. c) In stage 1, both AP and MP are rising. MP reaches its maximum point and starts falling. The point to note is that MP is always greater than AP in this stage. In stage 2, AP reaches its maximum point and starts falling. MP also falls and becomes zero. Here MP is always less than AP. In stage 3, MP becomes negative and AP decreases but remains positive.

The cobb-Douglas production function takes the form img Where L is the labor input and K is the capital input used in producing Q units of output, and img are the constants. Take log on both sides of the equation to derive following. img The table below shows the output levels with varying units of labor and capital and their logs. • Calculate img by entering the formula img in cell E2 and stretching it through cell E16. • Calculate img by entering the formula img in cell F2 and stretching it through cell F16. • Calculate img by entering the formula img in cell G2 and stretching it through cell G16. img To run regression, • Select the "Data" tab and click "Data Analysis" in the "Analysis" grouping. • Input the data of dependent variable img in the "Input Y-Range" field, and then enter data of independent variables img in the "Input X-Range" field for multiple columns. • Click "OK" to create the analysis as below. img The intercept is calculated as img . The coefficients of labor and capital are img and img , respectively. Thus, the regression equation estimated using OLS is img . Since img , thus, img Thus, the estimated equation in its multiplicative form is img . We want to test the hypothesis that whether units of labor and capital are useful in predicting the output at 0.05 level of significance. That is, we would perform a statistical test to determine whether the sample values img and img are significantly different from zero. For Labor: The null hypothesis is img , no relationship between output and labor. And, the alternative hypothesis is img , no relationship between output and labor. With 15 observations, a t -distribution for the sample statistic img will have img degrees of freedom. From Table 2 of Appendix B, the t-value is obtained as img The calculated value of img is greater than img , we reject null hypothesis img . Thus, we conclude relationship exists between output and labor at the 5 percent level of significance. For Capital: The null hypothesis is img , no relationship between output and capital. And, the alternative hypothesis is img , no relationship between output and capital. With 15 observations, a t -distribution for the sample statistic img will have img degrees of freedom. From Table 2 of Appendix B, the t-value is obtained as img The calculated value of img is greater than img , we reject null hypothesis img . Thus, we conclude relationship exists between output and capital at the 5 percent level of significance.