Answers to Selected Problems

More documents

Recommendations

Info

we learn that p J = $2,000 and p A = $2,500. As the chapter shows, the elasticities of demand are ε J = p/(MC - p) = 2,000/(500 - 2,000) = - 4 3 and ε A = 2,500/(500 - 2,500) = - 5 4 tion 12.9, we find that p J p A = 2,000 2,500 5 1 + 1/1 - 42 = 0.8 = 1 + 1/1 - 4 . Using Equa- 32 = 1 + 1/εA . 1 + 1/εJ The profit in Japan is (pJ - m)QJ = ($2,000 - $500) * 3,000 = $4.5 million, and the U.S. profit is $4 million. The deadweight loss is greater in Japan, $2.25 million 1 = 1 2 * $1,500 * 3,0002, than in the United States, $2 million 1 = 1 2 * $2,000 * 2,0002. 3.6 By differentiating, we find that the American marginal revenue function is MRA = 100 - 2QA , and the Japanese one is MRJ = 80 - 4QJ. To determine how many units to sell in the United States, the monopoly sets its American marginal revenue equal to its marginal cost, MRA = 100 - 2QA = 20, and solves for the optimal quantity, QA = 40 units. Similarly, because MRJ = 80 - 4QJ = 20, the optimal quantity is QJ = 15 units in Japan. Substituting QA = 40 into the American demand function, we find that pA = 100 - 40 = $60. Similarly, substituting QJ = 15 units into the Japanese demand function, we learn that pJ = 80 - (2 * 15) = $50. Thus, the price-discriminating monopoly charges 20% more in the United States than in Japan. We can also show this result using elasticities. Because dQA /dpA = -1 the elasticity of demand is εA = -pA /QA in the United States and εJ = - 1 2 PJ /QJ in Japan. In the equilibrium, εA = -60/40 = -3/2 and εJ = -50/(2 * 15) = -5/3. As Equation 12.9 shows, the ratio of the prices depends on the relative elasticities of demand: pA /pJ = 60/50 = (1 + 1/εJ )/(1 + 1/εA ) = (1 - 3/5)/(1 - 2/3) = 6/5. 3.8 From the problem, we know that the profitmaximizing Chinese price is p = 3 and that the quantity is Q = 0.1 (million). The marginal cost is m = 1. Using Equation 11.11, (pC - m)/pC = (3 - 1)/3 = -1/εC , so εC = -3/2. If the Chinese inverse demand curve is p = a - bQ, then the corresponding marginal revenue curve is MR = a - 2bQ. Warner maximizes its profit where MR = a - 2bQ = m = 1, so its optimal Q = (a - 1)/(2b). Substituting this expression into the inverse demand curve, we find that its optimal p = (a + 1)/2 = 3, or a = 5. Substituting that result into the output equation, we have Q = (5 - 1)/(2b) = 0.1 (million). Thus, b = 20, the inverse demand function is p = 5 - 20Q, and the marginal revenue function is MR = 5 - 40Q. Using this information, you can draw a figure similar to Figure 12.3. Answers to Selected Problems E-45 3.11 If a monopoly manufacturer can price discriminate, its price is p i = m/(1 + 1/ε i ) in Country i, i = 1, 2. If the monopoly cannot price discriminate, it charges everyone the same price. Its total demand is Q = Q 1 + Q 2 = n 1 p ε 1 + n 2 p ε 2. Differentiating with respect to p, we obtain dQ/dp = ε 1 Q 1 /p + ε 2 Q 2 /p. Multiplying through by p/Q, we learn that the weighted sum of the two groups’ elasticities is ε = s 1 ε 1 + s 2 ε 2 , where s i = Q i /Q. Thus, a profitmaximizing, single-price monopoly charges p = m/(1 + 1/ε). Chapter 13 1.1 The payoff matrix in this prisoners’ dilemma game is Larry Squeal Silent –2 –5 Duncan Squeal Silent –2 –5 If Duncan stays silent, Larry gets 0 if he squeals and -1 (a year in jail) if he stays silent. If Duncan confesses, Larry gets -2 if he squeals and -5 if he does not. Thus, Larry is better off squealing in either case, so squealing is his dominant strategy. By the same reasoning, squealing is also Duncan’s dominant strategy. As a result, the Nash equilibrium is for both to confess. 1.3 No strategies are dominant, so we use the bestresponse approach to determine the pure-strategy Nash equilibria. First, identify each firm’s best responses given each of the other firms’ strategies (as we did in Solved Problem 13.1). This game has two Nash equilibria: (a) Firm 1 medium and Firm 2 low, and (b) Firm 1 low and Firm 2 medium. 1.8 Let the probability that a firm sets a low price be θ1 for Firm 1 and θ2 for Firm 2. If the firms choose their prices independently, then θ1θ2 is the probability that both set a low price, (1 - θ1 )(1 - θ2 ) is the probability that both set a high price, θ1 (1 - θ2 ) is the probability that Firm 1 prices low and Firm 2 prices high, and (1 - θ1 )θ2 is the probability that Firm 1 prices high and Firm 2 prices low. Firm 2’s expected payoff is E(π2 ) = 2θ1θ2 + (0)θ1 (1 - θ2 ) + (1 - θ1 )θ2 + 6(1 - θ1 )(1 - θ2 ) = (6 - 6θ1 ) - (5 - 7θ1 )θ2 . Similarly, Firm 1’s expected payoff is E(π1 ) = (0)θ1θ2 + 7θ1 (1 - θ2 ) + 2(1 - θ1 )θ2 + 6(1 - θ1 )(1 - θ2 ) = (6 - 4θ2 ) - (1 - 3θ2 )θ1 . Each firm forms a 0 0 –1 –1
E-46 Answers to Selected Problems belief about its rival’s behavior. For example, suppose that Firm 1 believes that Firm 2 will choose a low price with a probability θn 2 . If θn 1 2 is less than 3 (Firm 2 is relatively unlikely to choose a low price), it pays for Firm 1 to choose the low price because the second term in E(π1 ), (1 - 3θn 2 )θ1 , is positive, so as θ1 increases, E(π1 ) increases. Because the highest possible θ1 is 1, Firm 1 chooses the low price with certainty. Similarly, if Firm 1 believes θn 2 is greater than 1 3 , it sets a high price with certainty (θ1 = 0). If Firm 2 believes that Firm 1 thinks θn 2 is slightly below 1 3 , Firm 2 believes that Firm 1 will choose a low price with certainty, and hence Firm 2 will also choose a low price. That outcome, θ2 = 1, however, is not consistent with Firm 1’s expectation that θn 2 is a fraction. Indeed, it is only rational for Firm 2 to believe that Firm 1 believes Firm 2 will use a mixed strategy if Firm 1’s belief about Firm 2 makes Firm 1 unpredictable. That is, Firm 1 uses a mixed strategy only if it is indifferent between setting a high or a low price. It is indifferent only if it believes θn 1 2 is exactly 3 . By similar reasoning, Firm 2 will use a mixed strategy only if its belief is that Firm 1 chooses a low price with probability θn 5 1 = 7 . Thus, the only possible Nash equilibrium is θ2 = 5 7 and θ 1 2 = 3 1.9 We start by checking for dominant strategies. Given the payoff matrix, Toyota always does at least as well by entering the market. If GM enters, Toyota earns 10 by entering and 0 by staying out of the market. If GM does not enter, Toyota earns 250 if it enters and 0 otherwise. Thus, entering is Toyota’s dominant strategy. GM does not have a dominant strategy. It wants to enter if Toyota does not enter (earning 200 rather than 0), and it wants to stay out if Toyota enters (earning 0 rather than -40). Because GM knows that Toyota will enter (entering is Toyota’s dominant strategy), GM stays out. Toyota’s entering and GM’s not entering is a Nash equilibrium. Given the other firm’s strategy, neither firm wants to change its strategy. Next, we examine how the subsidy affects the payoff matrix and For Chapter 13, Exercise 2.9 First stage Incumbent Do not invest Invest Second stage Entrant Entrant dominant strategies. The subsidy does not affect Toyota’s payoff, so Toyota still has a dominant strategy: It enters the market. With the subsidy, GM’s payoff if it enters increases by 50: GM earns 10 if both enter and 250 if it enters and Toyota does not. With the subsidy, entering is a dominant strategy for GM. Thus, both firms’ entering is a Nash equilibrium. 2.3 If the airline game is known to end in five periods, the equilibrium is the same as the one-period equilibrium. If the game is played indefinitely but one or both firms care only about current profit, then the equilibrium is the one-period one because future punishments and rewards are irrelevant to it. 2.9 The game tree illustrates why the incumbent may install the robotic arms to discourage entry even though its total cost rises. If the incumbent fears that a rival is poised to enter, it invests to discourage entry. The incumbent can invest in equipment that lowers its marginal cost. With the lowered marginal cost, it is credible that the incumbent will produce larger quantities of output, which discourages entry. The incumbent’s monopoly (no-entry) profit drops from $900 to $500 if it makes the investment because the investment raises its total cost. If the incumbent doesn’t buy the robotic arms, the rival enters because it makes $300 by entering and nothing if it stays out of the market. With entry, the incumbent’s profit is $400. With the investment, the rival loses $36 if it enters, so it stays out of the market, losing nothing. (If the rival were to enter, the incumbent would earn $132.) Because of the investment, the incumbent earns $500. Nonetheless, earning $500 is better than earning $400, so the incumbent invests. 2.10 The incumbent firm has a first-mover advantage, as the game tree illustrates. Moving first allows the incumbent or leader firm to commit to producing a relatively large quantity. If the incumbent does not make a commitment before its rival enters, entry occurs and the incumbent earns a relatively low Profits (πi , πe ) Do not enter ($900, $0) Enter Do not enter Enter ($400, $300) ($500, $0) ($132, –$36)
Page 1 and 2: E-32 Answers to Selected Problems I
Page 3 and 4: E-34 Answers to Selected Problems w
Page 5 and 6: E-36 Answers to Selected Problems l
Page 7 and 8: E-38 Answers to Selected Problems C
Page 9 and 10: E-40 Answers to Selected Problems p
Page 11 and 12: E-42 Answers to Selected Problems F
Page 13: E-44 Answers to Selected Problems 2
Page 17 and 18: E-48 Answers to Selected Problems 3
Page 19 and 20: E-50 Answers to Selected Problems C

Answers to Selected Problems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?