ch05solutionssolved edit.docx

1、ch05solutionssolved editSolutions to Chapter 5 ExercisesSOLVED EXERCISES S1. (a) Rs best-response rule is given by y = 10x x. L spends $16 million, so x = 16. Then Rs best response is y = 1016 16 = 10(4) 16 = 40 16 = 24, or $24 million. (b) Rs best response is y = 10x x, and Ls best response is x =

2、10y y. Solve these simultaneously: x = 10(10x x)1/2 10x + x x = (10x x)1/2 x = 10x x 2x = 10x x = 5 x = 25 y = 1025 25 = 25S2. (a) Xaviers costs have not changed, nor have the demand functions, so Xaviers best-response rule is still the same as in Figure 5.1: Px = 15 + 0.25Py. Yvonnes new profit fun

3、ction is By = (Py 2)Qy = (Py 2)(44 2Py + Px) = 2(44 + Px) + (4 + 44 + Px)Py 2(Py)2. Rearranging or differentiating with respect to Py leads to Yvonnes new best-response rule: Py = 12 + 0.25Px. Solving the two response rules simultaneously yields Px = 19.2 and Py = 16.8. (b) See the graph below. Yvon

4、nes best-response curve has shifted down; it has the same slope but a new, lower intercept (12 rather than 15). Yvonne is able to charge lower prices due to lower costs. The new intersection point occurs at (19.2, 16.8) as calculated above. S3. (a) La Boulangeries profit is: Y1 = P1Q1 Q1 = P1 (14 P1

5、 0.5P2) (14 P1 0.5P2) = P12 + 15P1 0.5P1P2 + 0.5P2 14. To find the optimal P1 without using calculus, we refer to the result in the Appendix to Chapter 5, remembering that P2 is a constant in this situation. Using the notation of the Appendix, we have A = 0.5P2 14, B = 15 0.5P2, and C = 1, so the so

6、lution is: P1 = B/(2C) = 15 0.5P2/(2), or P1 = 7.5 0.25P2. This is La Boulangeries best-response function. You get the same answer by setting = 2P1 + 15 0.5P2 = 0 and solving for P1. Similarly, La Fromageries profit is: Y2 = P2Q2 2Q2 = P2(19 0.5P1 P2) 2(19 0.5P1 P2) = P22 + 21P2 0.5P1P2 + P1 38. Aga

7、in, using the notation in the Appendix, A = P1 38, B = 21 0.5P1, and C = 1, which yields: P2 = B/(2C) = 21 0.5P1/(2), or P2 = 10.5 0.25P1. This is La Fromageries best-response function. You get the same answer by setting = 2P2 + 21 0.5P1 = 0 and solving for P1. To find the solution for the equilibri

8、um prices analytically, substitute La Fromageries best-response function for P2 into La Boulangeries best-response function. This yields P1 = 7.5 0.25(10.5 0.25P1), or P1 = 5.2. Given this value for P1, you can find P2 = 10.5 0.25(5.2) = 9.2. The best-response curves are shown in the diagram below.

9、(b) Colluding to set prices to maximize the sum of profits means that the firms maximize the joint-profit function: Y = Y1 + Y2 = 16P1 + 21.5P2 P12 P22 P1P2 52. To answer without calculus, use the result in the Appendix. Using that notation to solve for P1, A = 21.5P2 P22 52, B = 16 P2, and C = 1, s

10、o the solution is: P1 = B/(2C) = 16 P2/(2), or P1 = 8 0.5P2. Similarly, solving for P2, A = 16P1 P12 52, B = 21.5 P1, and C = 1, so the solution is: P2 = B/(2C) = 21.5 P1/(2), or P2 = 10.75 0.5P2. Solving these two equations simultaneously yields the solution P1 = 3.5 and P2 = 9. You can get the sam

11、e answer by partially differentiating the joint-profit function with respect to each price. Profits must be maximized with respect to both P1 and P2, so we need = 16 2P1 P2 = 0 and = 21.5 P1 2P2 = 0. (c) When firms choose their prices to maximize joint profit, they act as a single firm and ignore an

12、y individual incentives that they might have to deviate from the joint profit goal. However, given their partners collusive price, each company can reap more profit individually by charging more. For instance, plugging the joint-profit-maximizing value of La Boulangeries price into La Fromageries in

13、dividual best-response rule will not yield La Fromageries joint profit-maximizing price: P2 = 10.5 0.25(3.5) = 9.625 9.Likewise, plugging the joint-profit-maximizing value of La Boulangeries price into La Fromageries individual best-response rule gives: P1 = 7.5 0.25(9) = 4.75 3.5.Thus, the two join

14、t profit-maximizing prices are not best responses to each other; that is, they do not form a Nash equilibrium. (d) When firms produce substitutes, a drop in price at one store hurts the sales of the other. Thus, as your rival drops her price, you also want to drop yours to attempt to maintain sales

15、(and profits). In the bistro example in the text and in Exercise 1 above, this result led to best-response curves that were positively sloped and Nash equilibrium prices that were lower than the joint-profit-maximizing prices. Here, the firms produce complements, so a drop in price at one store lead

16、s to an increase in sales at the other. In this case, as one store drops its price, the other can safely increase its price somewhat and still maintain sales (and profits). Thus, the best-response curves are negatively sloped, and the Nash equilibrium prices are higher than the joint-profit-maximizi

17、ng prices. S4. To rationalize the nine possible outcomes, you need a separate argument for each one. We offer just one example, leaving you to construct the rest. Note that you need not consider the strategy combination (A, A) since that is a Nash equilibrium and therefore rationalizable. Consider (

18、A, C) leading to the payoffs (0, 2). C is a possible best response for Column if he thinks that Row is playing A. Why does Column believe this? Because he believes that Row believes that Column is playing B. Column justifies this belief by thinking that Row believes that Column believes that Row is

19、playing C. The beliefs in this chain are all perfectly rational because each strategy of either player is a best response (or among the best responses) to some strategy of the rival player. S5. No matter what beliefs Column might hold about what Row is playing, South is never Columns best response.

20、Therefore South is not a rationalizable strategy for Column. Since Row recognizes this, and since Earth is Rows best response only against Columns South, Row does not play Earth. Since North is Columns best response only against Earth, Column will not play North. Since Column will never play North o

21、r South, Wind is never a best response for Row. The remaining strategiesWater and Fire for Row and East and West for Columnare used in the two pure-strategy Nash equilibria, so they are certainly rationalizable. S6. Using the third-round range of X, we have that Y = 12 X/2 must be at most 12 9/2 = 1

22、2 4.5 = 7.5 (nothing new here) and at least 12 12.75/2 = 12 6.375 = 5.625 (a narrowing of the range: after the third round, the lower bound was 4.5). Similarly, using the third-round range of Y, we find that X = 15 Y/2 must be at most 15 4.5/2 = 15 2.25 = 12.75 (nothing new) and at least 15 7.5/2 =

23、15 3.75 = 11.25 (a narrowing of the range: in the third round, the lower bound was 9). We see that in even rounds the lower bounds get tighter, and in odd rounds the upper bounds get tighter. S7. (a) Cart 0 serves x customers and Cart 1 serves (1 x), where x is defined by the equation p0+ 0.5x2 = p1

24、 + 0.5 (1 x)2. Expanding this equation yields p0 + 0.5x2 = p1 + 0.5 x +0.5x2, and solving for x yields x = p1 p0 + 0.5. Thus Cart 0 serves p1 p0 + 0.5 customers, and Cart 1 serves 1 x, or p0 p1 + 0.5, customers. (b) Profits for Cart 0 are (p1 p0 + 0.5)(p0 0.25). Profits for Cart 1 are symmetric: (p0

25、 p1 + 0.5)( p1 0.25). Expanding the expression for Cart 0 profits yields (p1 p0 + 0.75)p0 (0.25 p1 + 0.125). Solving for the profit-maximizing value of p0 by completing the square or differentiating with respect to p0 yields p0 = 0.5p1 + 0.375. Cart 1s best-response rule is symmetric: p1 = 0.5p0 + 0

26、.375. (c) The graph is shown below. The Nash equilibrium prices are the values of p0 and p1 that solve simultaneously the two best-response rules found in part (b). Substituting Cart 1s best-response rule into that for Cart 0, we find: p0 = 0.5(0.5p0 + 0.375) + 0.375 = p0 + 0.5625. Solving for p0 yi

27、elds p0 = 0.75 (75); Cart 1s price is p1 = 0.75 (75) also. S8. (a) South Koreas profit is YKorea = qKorea*P cKorea*qKorea = qKorea(180 Q) 30qKorea = qKorea(180 qKorea qJapan) 30qKorea = qKorea2 + (180 30)qKorea qKorea*qJapan. Using the notation in the Appendix yields A = 0, B = 150 qJapan, and C = 1

28、, so South Koreas best response is: qKorea = B/(2C) = 150 qJapan/(2), or qKorea = 75 0.5qJapan. This is South Koreas best-response function. You get the same answer by setting = 2qKorea + 150 qJapan = 0 and solving for qKorea. Since Japan has the same price and cost per ship as South Korea, Japans p

29、rofit is YJapan = qJapan2 + (180 30)qJapan qKorea*qJapan. Similarly, Japans best-response function is: qJapan = 75 0.5qKorea. (b) To find the solution for the equilibrium prices, substitute Japans best-response function for qJapan into South Koreas best-response function. This yields: qKorea = 75 0.

30、5qJapan = 75 0.5(75 0.5qKorea) = 37.5 + 0.25qKorea qKorea = 50Therefore: qJapan = 75 0.5(50) = 50. The price of a VLCC is given by the expression P = 180 Q where Q = qKorea + qJapan = 50 + 50 = 100. Therefore P = 180 100 = 80, or $80 million. South Koreas profit is YKorea = qKorea2 + (180 cKorea)qKo

31、rea qKorea*qJapan = (50)2 + (180 30)(50) (50)(50) = 2500 + 7500 2500 = 2500. Likewise, Japans profit is YJapan = qJapan2 + (180 cJapan)qJapan qKorea*qJapan = 2500. Therefore both countries make $2.5 billion in profits. (c) South Koreas new best-response function is: qKorea = 90 0.5cKorea 0.5qJapan = 90 0.5(20) 0.5qJapan = 80 0.5qJapan. Japans new best-response function is: qJapan = 90 0.5cJapan 0.5qKorea =