Data for a monopolist's total revenue (tr), total cost (tc), and output (q), for 48 consecutive months. Suppose that the monopolist's economic models for total revenue and total revenue and total cost are given respectively by

a. Show that marginal cost and marginal revenue are given by

b. Show that the profit maximizing quantity which equates marginal revenue and marginal cost is

c. Use the least squares estimator to estimate the total revenue and total cost functions. For what statistical models are these estimates appropriate? What do the least squares estimates suggest is the profit maximizing level of output?

d. After rounding the optimizing output to an integer, use that to predict total revenue, total cost, and profit for the next three months. (Continue to assume the least squares statistical assumptions are appropriate.)

e. Separately test the errors for each of the functions to see if these errors might be autocorrelated.

f. Where autocorrelation has been suggested by the tests in part (e), find generalized least squares estimates of the relevant function(s).

g. What is the profit-maximizing level of output suggested by the results in part (f)?