Quiz 7: Production Economics
The production function is given as: We need to calculate the parameters and . We will use the regression analysis. We denote by , ln by and by . Therefore, In the table below, we will calculate the values of , , , , , , and . The Cobb-Douglas production function can be written as:
a) In this table, we will calculate the marginal product and average product. The total product is given. The capital is fixed at 500-bhp rating. The total product of labor is output. The marginal product of labor is . The average product of labor 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 is 16. Hence the average product at is . In a similar way, we can calculate the average product for all labor inputs. The total product at is 6 and at is 16. The marginal product at is . 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. 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 Where L is the labor input and K is the capital input used in producing Q units of output, and are the constants. Take log on both sides of the equation to derive following. The table below shows the output levels with varying units of labor and capital and their logs. • Calculate by entering the formula in cell E2 and stretching it through cell E16. • Calculate by entering the formula in cell F2 and stretching it through cell F16. • Calculate by entering the formula in cell G2 and stretching it through cell G16. To run regression, • Select the "Data" tab and click "Data Analysis" in the "Analysis" grouping. • Input the data of dependent variable in the "Input Y-Range" field, and then enter data of independent variables in the "Input X-Range" field for multiple columns. • Click "OK" to create the analysis as below. The intercept is calculated as . The coefficients of labor and capital are and , respectively. Thus, the regression equation estimated using OLS is . Since , thus, Thus, the estimated equation in its multiplicative form is . 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 and are significantly different from zero. For Labor: The null hypothesis is , no relationship between output and labor. And, the alternative hypothesis is , no relationship between output and labor. With 15 observations, a t -distribution for the sample statistic will have degrees of freedom. From Table 2 of Appendix B, the t-value is obtained as The calculated value of is greater than , we reject null hypothesis . Thus, we conclude relationship exists between output and labor at the 5 percent level of significance. For Capital: The null hypothesis is , no relationship between output and capital. And, the alternative hypothesis is , no relationship between output and capital. With 15 observations, a t -distribution for the sample statistic will have degrees of freedom. From Table 2 of Appendix B, the t-value is obtained as The calculated value of is greater than , we reject null hypothesis . Thus, we conclude relationship exists between output and capital at the 5 percent level of significance.