winter problem set 2 solutions

winter problem set 2 solutions - Problem Set 2 Solutions...

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Problem Set 2 Solutions Chapter 13 13.1 a) P = MC implies 70 – 2 Q = 10, or Q = 30 and P = 10. b) A monopolist produces until MR = MC yielding 70 – 4 Q = 10 so Q m = 15 and P m = 40. Thus π m = (40 – 10)*15 = 450. c) For Amy, MR A = MC implies 70 – 4 q A 2 q B = 10. We could either calculate Beau’s profit-maximization condition (and solve two equations in two unknowns), or, inferring that the equilibrium will be symmetric since each seller has identical costs, we can exploit the fact that q A = q B in equilibrium. (Note: You can only do this after calculating marginal revenue for one Cournot firm, not before.) Thus 70 – 6 q A = 10 or q A = 10. Similarly, q B = 10. Total market output under Cournot duopoly is Q d = q A + q B = 20, and the market price is P d = 70 – 2*20 = 30. Each duopolist earns π d = (30 – 10)*10 = 200. 13.2 a) With two firms, demand is given by . If , then or . Setting implies If , then . Setting implies
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b) For Firm 1, . Setting implies Since the marginal costs are the same for both firms, symmetry implies . Graphically, these reaction functions appear as c) Because of symmetry, in equilibrium both firms will choose the same level of output. Thus, we can set and solve Since both firms will choose the same level of output, both firms will produce 22.22 units. Price can be found by substituting the quantity for each firm into market demand. This implies price will be . d) If this market were perfectly competitive, then equilibrium would occur at the point where . e) If the firms colluded to set the monopoly price, then
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At this quantity, market price will be . f) If the firms acted as Bertrand oligopolists, the equilibrium would coincide with the perfectly competitive equilibrium of . g) Suppose Firm 1 has and Firm 2 has . For Firm 1, . Setting implies For Firm 2, . Setting implies Solving these two reaction functions simultaneously yields and . With these quantities, market price will be .
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13.3 For Zack, MR Z = MC Z implies 100 – 2 q Z – q A = 1, so Zack’s reaction function is q Z = ½*(99 – q A ). Similarly, MR A = MC A implies 100 – 2 q A – q Z = 10 so Andon’s reaction function is q A = ½*(90 – q Z ). Solving these two equations in two unknowns yields q Z = 36 and q A = 27. The market price is P = 100 – (36 + 27) = 37. Zack earns π Z = (37 – 1)*36 = 1296 and Andon earns π A = (36 – 10)*27 = 702. 13.4 a) The inverse market demand curve is P = 100 – ( Q /40) = 100 – ( Q 1 + Q 2 )/40. Firm 1’s reaction function is found by equating MR 1 = MC 1 : MR 1 = [100 – Q 2 /40] – Q 1 /20 MR 1 = MC 1 [100 – Q 2 /40] – Q 1 /20 = 20 Solving this for Q 1 in terms of Q 2 gives us: Q 1 = 1,600 – 0.5 Q 2 Similarly, Firm 2’s reaction function is found by equating MR 2 = MC 2 : MR 2 = [100 – Q 1 /40] – Q 2 /20 MR 2 = MC 2 [100 – Q 1 /40] – Q 2 /20 = 40 Solving this for Q 2 in terms of Q 1 gives us: Q 2 = 1,200 – 0.5 Q 1 b) The two reaction functions give us two equations in two unknowns. Using algebra we can
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winter problem set 2 solutions - Problem Set 2 Solutions...

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