Please Help
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 Output
3 1
7 2
9 3
11 5
17 8
17 10
20 15
24 18
26 22
28 21
30 23
a.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.
b.Using a computer and software for regression analysis, estimate your firm short-run production function using the data given here. Do the parameter estimates have appropriate algebraic signs? Are they statistically significant at the 5 percent level?
c.At what point do you estimate marginal product begins to fall?
d.Calculate estimates of total, average, and marginal products when the firm employs 23 workers.
e.When the firm employs 23 workers, is short-run marginal cost (SMC) rising or falling? How can you tell?
a. To determine whether a cubic equation is a suitable specification for the short-run production function, you can construct a scatter diagram of the labor usage and output data points. This will help you visually observe the relationship between the variables and identify any patterns or trends that may suggest a cubic relationship.
b. To estimate the short-run production function using regression analysis, you can use computer software such as Excel, Python, or statistical software like SPSS. Input the labor usage as the independent variable (X) and the output as the dependent variable (Y). Fit the data to a cubic regression model and obtain the parameter estimates.
To check if the parameter estimates have the appropriate algebraic signs, you need to interpret the coefficients of the cubic equation. For example, if the coefficient of the cubic term is positive, it suggests an increasing concavity, while a negative coefficient indicates a decreasing concavity.
To determine if the parameter estimates are statistically significant at the 5 percent level, you can look at the p-values associated with the coefficients. If the p-value is less than 0.05, it suggests that the coefficient is statistically significant and not due to random chance.
c. To estimate the point at which marginal product begins to fall, you can analyze the trend in the data or calculate the derivative of the production function. As labor usage increases, observe if the rate of increase in output (marginal product) starts to decline. The point at which this decline begins can be considered as the point where marginal product starts to fall.
d. To calculate the estimates of total, average, and marginal products when the firm employs 23 workers, you will need to use the estimated short-run production function. Plug in 23 as the labor input and solve for the corresponding output. Total product can be directly calculated as the output. Average product can be calculated by dividing the total product by the number of workers, and marginal product can be calculated as the change in total product when one additional unit of labor is added.
e. To determine whether short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers, you need to analyze the trend in the marginal product of labor. If marginal product is increasing, it suggests that each additional worker is adding more output than the previous worker, indicating falling SMC. Conversely, if marginal product is decreasing, it suggests diminishing returns to labor, indicating rising SMC.