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. Give all the Nash Equilibria for the following games. For each game, explain whether the NE (if any) is socially 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 di±erent 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 ²rms 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 ²rm are given by c ( q ) = q 2 (so that mc = 2 q with no ²xed costs), create a payo± matrix for the game, and determine the Nash equilibrium(a). (a) Compare the Cournot Nash equilibrium outcome to the ²rst best and to the monopoly (collusive) outcome. [You may reproduce a partial payo± matrix if you wish; i.e., you only need to show enough to calculate the equilibria and the other relevant quantities. Hint: ²gure out the ²rst best ²rst 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 payo±s of (32,32). The collusive outcome would be (3,3) for total payo±s of 66 (no other cell has a higher total payo±, although it’s equivalent to 4,3 and 3,4). The ²rst best is where price equals marginal cost for each ²rm, so the ²rst best is at (5,5): price is 10 and mc is 10. (a) Explain how a cartel between the two ²rms would break down. Explain how your answer involves an externality. 1
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Beginning at (3,3), each frm has an incentive to cheat. Given that frm 2 is playing “3”, then frm 1’s best response is “4.” Ditto For frm 2. By cheating on the agreement, each frm can increase its profts by 3. But, this cheating leads to a negative externality: the other frm’s profts decrease. It’s an externality because frm 1’s private costs and benefts don’t take into account frm 2’s loss. (b) Explain why frms do not make zero proft at the frst best. This is because we’ve limited entry to two frms. IF we allowed For entry, frms would enter. But, there are no fxed 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 infnite number oF frms each producing an infnitesimal 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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ps5f06s - Economics 101, Problem Set 5 Solutions Alan C....

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