Monopoly Pricing with Fixed Costs
A pharmaceutical company has a patent on a new drug, giving it a monopoly. The firm's total cost function is TC = 1000 + 10Q, where Q is the number of doses produced. The market demand curve is given by P = 100 - Q. The company currently charges $55 per dose and sells 45 doses. A government agency is considering imposing a price ceiling equal to marginal cost.
1. Calculate the profit-maximizing quantity and price for the monopolist. Show your work.
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Profit maximization occurs where marginal revenue (MR) equals marginal cost (MC). Total revenue TR = P*Q = (100 - Q)*Q = 100Q - Q^2. MR = dTR/dQ = 100 - 2Q. MC = dTC/dQ = 10. Setting MR = MC: 100 - 2Q = 10 => 2Q = 90 => Q = 45. The profit-maximizing price is P = 100 - 45 = $55. So the firm is already maximizing profit at Q=45, P=$55.
2. If the government imposes a price ceiling equal to marginal cost, what price and quantity will result? Will the firm continue to operate in the short run? Explain.
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A price ceiling equal to marginal cost means P = MC = $10. At P=$10, quantity demanded is Q = 100 - 10 = 90. The firm will supply 90 doses if it operates. However, at Q=90, total revenue = $1090 = $900. Total cost = 1000 + 1090 = $1900. The firm incurs a loss of $1000, which equals its fixed cost. In the short run, the firm will continue to operate if price covers average variable cost (AVC). AVC = VC/Q = 10Q/Q = $10. Since P = $10 = AVC, the firm covers variable costs but not fixed costs. It is indifferent between operating and shutting down in the short run. In the long run, it will exit.
3. Calculate the deadweight loss if the firm continues to charge the monopoly price instead of the socially optimal price (where P = MC).
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Socially optimal quantity is where P = MC: 100 - Q = 10 => Q = 90. Monopoly quantity is 45. Deadweight loss is the area of the triangle between the demand curve and the MC curve from Q=45 to Q=90. The height of the triangle at Q=45 is the difference between price and MC: $55 - $10 = $45. The base is 90 - 45 = 45. DWL = 0.5 * 45 * 45 = $1012.50.