You are planning to estimate a short-run production function for your firm, and you have collected the following data on labor usage and output:
Labor Usage:
3
7
9
11
17
17
20
24
26
28
30
Output:
1
2
3
5
8
10
15
18
22
21
23
1) does a cubic equation appear to be a suitable specification, given these data? You may wish to construct a scatter diagram to help you answer this question.
2) Using a computer and software for regression analysis, estimate your firm's short run production function using the data given here. Do the parameter estimates have the appropriate algebraic signs? Are they statisically significant at the 5 percent level?
3) At what point do you estimate marginal product begins to fall?
4) Calculate estimates of tota, average, and marginal products when the firm employs 23 workers.
5) When the firm employes 23 workers, is short-run marignal cost (SMC) rising or falling? How can you tell?
1) To determine if a cubic equation is a suitable specification for the data, you can construct a scatter plot of labor usage on the x-axis and output on the y-axis. Plotting the data points will help you visualize the relationship between the two variables. If the relationship appears to be nonlinear and there is a curved pattern, a cubic equation might be appropriate.
2) To estimate the short-run production function using regression analysis, you will need to use statistical software like Excel, R, or Python. Input the labor usage as the independent variable (X) and the output as the dependent variable (Y). Run a regression analysis to obtain the parameter estimates. Check if the parameter estimates have the appropriate algebraic signs. For example, a positive coefficient on labor usage would indicate a positive relationship between labor and output. To determine if the coefficients are statistically significant at the 5% level, you can look at the p-values associated with each coefficient. If the p-value is less than 0.05, then the coefficient is considered statistically significant.
3) To estimate when marginal product begins to fall, you need to analyze the derivative of the production function with respect to labor usage. The point at which the derivative becomes negative indicates the beginning of diminishing marginal returns. In this case, you can estimate the labor usage value where the marginal product starts to decline.
4) To calculate the estimates of total, average, and marginal products when the firm employs 23 workers, you can use the estimated short-run production function from question 2. Plug in 23 workers into the equation to calculate the total product. Divide the total output by 23 to calculate the average product. To calculate the marginal product, take the derivative of the production function with respect to labor usage and evaluate it at 23 workers.
5) To determine whether the short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers, you need to analyze the relationship between marginal cost and output. If marginal cost is increasing with output, then SMC is rising. If marginal cost is decreasing with output, then SMC is falling. You can determine this by calculating the derivative of the cost function with respect to output and evaluating it at the level of output when the firm has 23 workers. If the derivative is positive, then SMC is rising. If the derivative is negative, then SMC is falling.