Ans_HW11 - Answer to Practice Homework 11 13.18 a A...

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Answer to Practice Homework 11 13.18 a) A monopolist sets MR = MC (don’t forget to invert the demand curve first!) so 100 – Q m = 4. Thus, Q m = 96 and P m = 52. As a monopolist, Apple’s profits are π = (52 – 4)*96 = 4608. b) Each fringe firm maximizes profits by setting P = MC = 6 q + 20, so we can derive a single firm’s supply curve as q = ( P – 20)/6, so long as P > 20. With the fringe comprising 12 firms, total supply is Q fringe = 12 q , or - = 20 40 2 20 0 P P P Q fringe c) First, find Apple’s (denoted DF for “dominant firm”) residual demand, for P > 20: Q DF = Q mkt – Q fringe = 200 – 2 P – (2 P – 40) = 240 – 4 P . Inverting, this is P = 60 – 0.25 Q . So Apple sets MR = MC or 60 – 0.5 Q DF = 4 Q DF = 112 From the residual demand, Apple’s price is P = 60 – 0.25*112 = 32. At this price, the fringe supplies Q fringe = 2 P – 40 = 2*32 – 40 = 24. Apple’s profits are π = (32 – 4)*112 = 3136. d) On the graph, note that (i) residual demand begins at P = 60, precisely where fringe supply intersects market demand; (ii) residual demand connects back with market demand at P = 20, where the fringe supply curve intersects the P -axis; and (iii) the dominant firm’s equilibrium output occurs where its MR = MC.
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13.20 a) To find the supply curve of the competitive fringe, we proceed as follows. Each price-taking fringe firm produces to the point at which the market price equals marginal cost: P = 10 + 50q, or q = (P – 10)/50. This equation is valid only if the price is greater than or equal to 10. If the price is less than 10, a fringe firm produces nothing. In this problem, there is no loss of generality in assuming that the market price will exceed 10, and that a fringe firm’s supply curve is given by q = (P – 10)/50. This is because the marginal cost of the dominant firm is 10, and the dominant will not operate at a point at which its price is less than its marginal cost. Given this, the fringe’s overall supply curve is found by multiplying the individual fringe supply curve by the number of fringe firms (100): Q s = (100)(P – 10)/50 = 2P – 20. Thus, the overall fringe supply curve is Qs = 2P – 20. b) We find the residual demand curve by subtracting the overall fringe supply from the market demand curve. Letting Q r denote residual demand, we have: Q r = Q d – Q s = (400 – 8P)-(2P – 20) Q r = 420 – 10P. Fringe supply Q fringe = 24 Q DF = 112 Q mkt = 136 Market demand Residual demand MC MR
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c) To find the profit-maximizing quantity of the dominant firm, we first invert the residual demand (dropping the superscript r) to get: P = 42 – (1/10)Q. The corresponding marginal revenue curve is: MR = 42 – (1/5)Q. Equating marginal
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