You are planning to estimate a short- run production function for your firm, and you have collected the following data on labor usage (L) and output (Q):


Labor usage 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’s short-run production function using the data given here. Do the parameter estimates have the 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 data, we can construct a scatter diagram. A scatter diagram is a visual representation of the relationship between two variables, in this case, labor usage (L) and output (Q). By plotting the data points on a graph, we can examine the pattern and shape of the relationship between the variables.

b. To estimate the short-run production function using regression analysis, we can use computer software such as Excel, R, or Python. Regression analysis helps us determine the relationship between the independent variable (labor usage) and the dependent variable (output) and estimate the parameters of the production function.

c. To determine at what point marginal product begins to fall, we can examine the estimated coefficients of the production function. In this case, the coefficients represent the effect of labor usage on output. If a coefficient is negative, it indicates that the marginal product is diminishing as labor usage increases. We can identify the point at which marginal product begins to fall by analyzing the change in coefficients.

d. To calculate estimates of total, average, and marginal products when the firm employs 23 workers, we can use the estimated production function from the regression analysis. By plugging in the value of 23 for labor usage in the production function, we can calculate the corresponding values of output (Q). Total product is the output value at a given level of labor usage, average product is total product divided by labor usage, and marginal product is the change in output when one additional unit of labor is employed.

e. To determine whether short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers, we can examine the relationship between marginal cost and output. Marginal cost represents the additional cost of producing one more unit of output. If the marginal cost is increasing, it indicates that the firm is experiencing diminishing returns to labor and the cost of producing additional output is rising. By analyzing the estimated values of marginal cost at different levels of output, we can determine whether SMC is rising or falling.