A firm's profit ?(Q) is the difference between its revenue R(Q) and costs C(Q)..
(a) What condition holds when the firm is maximizing profits? Explain in words what the math of the answer means.
(b) What conditions do the functions R(Q) and C(Q) have to satisfy for you to know that the quantity Q* that satisfies the condition in (a) in fact maximizes profit? Why might you expect this to be true most of the time?
(c) Derive the condition that holds when average cost C(Q)/Q is minimized. What condition on C(Q) ensures that the average cost is minimized where this condition holds?
(d) In general, will the condition in (a) hold when the firm is minimizing average cost? Why or why not? From (c), what would marginal revenue R'(Q*) have to equal at Q* for average cost to be minimized for the same quantity Q* where profits are maximized?