A monopolist’s demand function is P = 1624 – 4Q, and its total cost fu

A monopolist’s demand function is P = 1624 – 4Q, and its total cost function is

TC = 22,000 + 24Q -4Q2 + 1/3 Q3, where Q is output produced and sold.

a. At what level of output and sales (Q) and price (P) will total profits be maximized?
b. At what level of output and sales (Q) and price (P) will total revenue be maximized?
c. At what price (P) should the monopolist shut down?

ANSWER

a. Total Profits are maximized where MR = MC, and MR = dTR/dQ, with TR = P(Q), and
MC = dTC/dQ. TR = 1624Q -4Q2, so MR = 1624 – 8Q. MC = 24 – 8Q + Q2.
MR = MC is 1624 – 8Q = 24 – 8Q + Q2, or 1600 = Q2, and Q = 40. With Q = 40, P = 1464.
b. Total Revenue is maximized when MR = 0, or 1624 – 8Q = 0, or Q = 203 with P = 203.
c. Shut down would occur whenever price(P) is less than average variable cost (AVC), or below P = AVC, or 1624 – 4Q = 24 – 4Q + 1/3 Q2, or 1600 =1/3 Q2, or Q2 = 4800, or Q = 69 (approximately). When Q = 69, P = 1348, so any price below 1348 would cause the firm to shut down since it is not covering its variable costs.