2.8 Suppose that a firm has a marginal cost function given by MC(q) = q+ 1. a. What is this firm's total cost function? Explain why total costs are known only up to a constant of integration, which represents fixed costs. b. As you may know from an earlier economics course, if a firm takes price (p) as given in its decisions, then it will produce that output for which p = MC(q). If the firm follows this profit-maximizing rule, how much will it produce when p = 15? Assuming that the firm is just breaking even at this price, what are fixed costs? c. How much will profits for this firm increase if price increases to 20? d. Show that, if we continue to assume profit maximization, then this firm's profits can be expressed solely as a func- tion of the price it receives for its output. e. Show that the increase in profits from p = 15 to p = 20 can be calculated in two ways: (i) directly from the equa- tion derived in part (d) and (ii) by integrating the inverse marginal cost function [MC ' (p) = p - 1] from p = 15 to p = 20. Explain this result intuitively using the envelope theorem