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ps5f06s - Economics 101 Problem Set 5 Solutions Alan C...

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Economics 101, Problem Set 5 Solutions Alan C. Marco December 7, 2006 1. GivealltheNashEquilibriaforthefollowinggames. Foreachgame, explainwhethertheNE(ifany)issocially optimal. Explain your answer. (a) Drop Dead Fred Equilibrium: (Ouch, Sleep) Fred Phoebe Laugh Cry Sleep Fly 1 , 1 6 , - 1 7 , 0 Swim 2 , 0 5 , 0 8 , - 1 Ouch 3 , - 1 4 , 1 9 , 1 (b) Quarks [Express your answer for different values of x, i.e., “If x < blah, then the NE are blank. If x = blab, then...”] If x > 6: (Strange, Bottom) If x 6: (Up, Charm), (Strange, Bottom) Amber Buffy Top Charm Bottom Up 6 , 1 6 , 6 3 , 4 Strange x, 2 x, 0 4 , 3 Down 2 , 7 5 , 9 4 , 4 2. Suppose two firms compete according to Cournot competition. They can produce in discrete units only. If demand is given by p =20 - Q and total costs for each firm are given by c ( q )= q 2 (so that mc =2 q with no fixed costs), create a payoff matrix for the game, and determine the Nash equilibrium(a). (a) Compare the Cournot Nash equilibrium outcome to the first best and to the monopoly (collusive) outcome. [You may reproduce a partial payoff matrix if you wish; i.e., you only need to show enough to calculate the equilibria and the other relevant quantities. Hint: figure out the first best first so that you know the maximum quanitity. Show at least one less than the monopoly output as well, in order to show that it really is the monopoly outcome.] 2 0 1 2 3 4 5 6 0 0,0 0,18 0,32 0,42 0,48 0,50 0,48 1 18,0 17,17 16,30 15,39 14,44 13,45 12,42 2 32,0 30,16 28,28 26,36 24,40 22,40 20,36 1 3 42,0 39,15 36,26 33,33 30,36 27,35 24,30 4 48,0 44,14 40,24 36,30 32,32 28,30 24,24 5 50,0 45,13 40,22 35,27 30,28 25,25 20,18 6 48,0 42,12 36,20 30,24 24,24 18,20 12,12 The Cournot equilibrium is given by (4,4) for payoffs of (32,32). The collusive outcome would be (3,3) for total payoffs of 66 (no other cell has a higher total payoff, although it’s equivalent to 4,3 and 3,4). The first best is where price equals marginal cost for each firm, so the first best is at (5,5): price is 10 and mc is 10. (a) Explain how a cartel between the two firms would break down. Explain how your answer involves an externality. 1
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Beginning at (3,3), each firm has an incentive to cheat. Given that firm 2 is playing “3”, then firm 1’s best response is “4.” Ditto for firm 2. By cheating on the agreement, each firm can increase its profits by 3. But, this cheating leads to a negative externality: the other firm’s profits decrease. It’s an externality because firm 1’s private costs and benefits don’t take into account firm 2’s loss. (b) Explain why firms do not make zero profit at the first best. This is because we’ve limited entry to two firms. If we allowed for entry, firms would enter. But, there are no fixed costs, and MC is upward sloping. This means that AC is at its minimum at q =0 . So entry will occur until we have an infinite number of firms each producing an infinitesimal amount, so MC and price will be driven to zero, and quantity will be driven to 20. Not really realistic is it?
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