331 AEM Suggested Answers to Problem Set #2 1. a. From the two points on the demand curve that we're given, we know inverse demand is: P = 300 4*Q. Under competition, consumer surplus is (50*200)/2 = 5000. Under monopoly, consumer surplus is (45*180)/2 = 4050. So, competition's consumer surplus is 950 higher. b. Under competition, producer surplus would be zero. Under monopoly, producer surplus is (120-60)*45 = 2700. c. The Planning Commission would choose monopoly since total surplus is higher (compare 6750 to 5000). 2. As we discussed in class, this is really ambiguous. Breaking it down between the two extremes of perfect competition and monopoly, we could argue that incentive is higher for either depending on circumstances. For monopoly, the incentive can be higher if a fringe firm can take control of the industry by innovating before the monopolist. For competition, the short-term gain is higher from jumping ahead of opponents if the monopoly is only effectively competing with itself. 3. a. P = MC, so 70 2Q = 10. Therefore, Q = 30, P = 10. b. MR = MC, so 70 4Q = 10. Therefore, Qm = 15, Pm = 40. Thus m = (40 10)*15 = 450. c. For Jack, MRA = MC, so 70 4qA 2qB = 10. Knowing Jill has a symmetric problem, we have for Jill, MRB = MC, so 70 4qB 2qA = 10. Using these two equations, we have qA = qB = 10, so total market output under Cournot duopoly is Qd = qA + qB = 20, and the market price is Pd = 70 2*20 = 30. Each duopolist earns d = (30 10)*10 = 200. d. If they collude successfully, Jack and Jill would each produce half the monopoly output, and the price would be Pm = 40, with Qm = 15 total units sold. If we assume Cournot behavior in the absence of collusion, the market price would have been Pd = 30. The overcharge is thus (40 30)*15 = $150. 4. a. The lower diagonal both charging the lower price. b. The upper diagonal both charging the higher price. Seems Crazy Eddie's name is just a decoy. 5. a. Solve for the reaction functions. To maximize your profit, you want to choose a quantity, q1 to maximize: (100 - q1 - q2 - 20) * q1 . Setting the q derivative equal to zero, we get: q1 = 40 - 2 . For Mr. Sobchak, he wants 2 to choose q2 to maximize: (100 - q1 - q2 - 12)* q2 . Setting the derivative q equal to zero, we get: q2 = 44 - 1 . Solving both reaction functions 2 simultaneously, we have: q1 = 24 and q2 = 32 .
b. Profit maximum is where MR = MC. MR for both 100 is 2*Q. For Mr. Sobchak, MC is 12, so his ideal quantity is 44 both of you producing 22. For you, MC is 20, so your ideal quantity is 40 both of you producing 20. c. Mr. Sobchak would actually prefer the non-cooperative outcome. When you split, he gets (56 12)*22 = 968. In the non-cooperative situation, he gets (44 12)*32 = 1024. Therefore, the problem isn't just coordinating the price, but also the market split. 6. Rachel pR = $30 pL = $30 Laura pL = $40 2000, 1500 1600, 1600 1800, 1800 pR = $40 1500, 2000
a. If both choose p = 30, then P = 60 for a pair of shoes. Then Q = 120 P = 60, so payoffs are 30*60 = 1800 apiece. If both choose p = 40, then P = 80, Q = 40, = 3200. If one chooses p = 30 and the other p = 40, then P = 70, Q = 50, and the first firm earns 1500 while the other gets 2000. In this game, the only NE is for each player to choose p = 40. b. The thrust of the Theatre Enterprises (1954) case was that conspiracy cannot be inferred from parallel behavior alone. In this case, Laura and Rachel are each maximizing her own payoff, given what her rival has chosen that is, they're playing NE strategies. Although each player has chosen the higher of two possible prices, it involved no explicit communication or otherwise aggressive tactic from which the Court could infer a "conspiracy in restraint of trade" in violation of section 1 of the Sherman Act. 7. a. Jack's profits are: (200 - 2 pJ + pD ) *( pJ - 0) Diane's profits are: (200 - 2 pD + pJ ) *( pD - 0) Let the prices be the same, so now we need a price to maximize: (200 - 2 p + p )*( p - 0) + (200 - 2 p + p )*( p - 0) = (400 - 2 p )* p Taking the derivative, we get: 400 - 4 p = 0 So, p = 100 b. Solve for Jack. Profits are (200 - 2 pJ + pD ) *( pJ - 0) . Take the derivative with respect to pJ to get: 200 - 4 pJ + pD = 0 . So, his reaction
p pD . Diane's looks the same: pD = 50 + J . 4 4 Plugging one into the other, we get that each charges 66.67. c. If Jack is charging 100, Diane's profits are: (200 - 2 pD + 100) *( pD ) . Her optimal price is then 75, yielding a profit of 11250. d. Consider the tit-for-tat strategy where each charges 100, but if the other doesn't, they'll charge 66.67 from then on. Sticking with the strategy gives: 10000 + 10000*(1/(1+r)) + ... = 200000. Deviating gives: 11250 + 8889*(1/(1+r)) + ... = 180141. So, both will stick with the strategy. function is: pJ = 50 +