Econ 414 Midterm Exam

Transcription

1 Econ 44 Midterm Exam Name: There are three questions taken from the material covered so far in the course. All questions are equally weighted. If you have a question, please raise your hand and I will come to your desk. Make sure that you defend your answers with economic reasoning mathematical arguments, and show that you are using the crect game theetic concepts by identifying the equilibria explicitly. Good luck.

2 . Consider the following game in strategic fm: B w x y z a 4,4 3,3 5,, b 3,6,5 6,-3,4 A c -,0,- 0,0, d,4,, 3,5 (i.) Perfm iterated deletion of strictly dominated strategies (Strike them out with a line, writing the letter of the dominant strategy beside it). (ii.) Find all pure-strategy Nash equilibria (Circle them clearly). (iii.) Is there a mixed strategy equilibrium in this game? If so, find the mixed strategy f the row player. i. Round : w strictly dominates x f the column player; y strictly dominated by z f the column player; a strictly dominates b f the row player. Round : Nothing else can be deleted; stop. ii. (a, w) and (d, z) are pure-strategy Nash equilibria. iii. Yes. First we do IDWDS and remove c f the row player, since it is weakly dominated by a. Then we choose (σ a,σ d ) to make the column player indifferent between w and z: σ a 4+σ d 4 = σ a +5σ d σ a = σ d and we know σ a + σ d =, so that 3σ a =. Then Row s equilibrium strategy is σ a = /3 and σ d = /3.

3 . Consider the following game in extensive fm: a b L e M f R c d,,5,, 3,3 3, (i.) Write all behavi strategies f all players. (ii.) Write the game in strategic fm and solve f all pure-strategy Nash equilibria. (iii.) Find all sub-game perfect equilibria. (iv.) If there are any Nash equilibria ( me than one) that fail to be sub-game perfect, choose one and briefly explain why this failure occurs. i. Player : L, M, R. Player : aec, aed, afc, afd, bec, bed, bfc, bfd ii. aec aed afc afd bec bed bfc bfd L -,- -*,- -,- -,-,5* *,5*,5* *,5* M -,* -*,*,- *,- -,* -,*,- *,- R 3*,3* -3,- 3*,3* -3,- 3*,3* -3,- 3*,3* -3,- So we have (R,aec), (M,aed), (R,afc), (R,bec), (L,bed), (R,bfc), and (L,bfd) as pure-strategy Nash equilibria. iii. Doing backwards induction, we find the only SPNE is (R,bec) iv. The Nash equilibrium (L,bed) is not subgame perfect because it relies on player believing that player would ever choose d if R was played; this is a non-credible threat. Since c gives a strictly higher payoff to player, player should conclude that player would always choose d, and that it is safe to player R and get 3 rather than from L. 3

4 3. Splitting the Gains from Merger There are two firms that compete in a Cournot market. In particular, they choose quantities q and q simultaneously to maximize their profits. The market price is P(q,q ) = A q q. They have no costs. (i.) Solve f the Nash equilibrium of the game. Sketch the firms reaction function/best-response functions. Does the game exhibit strategic complements substitutes? How do the equilibrium strategies change if A increases? (ii.) What are profits f one of the firms in this industry (call this π c )? How much would a monopolist produce, and what are its profits (call this π m )? Assume A = 6 (iii). Assume the owners of the two firms consider merging, and have to bargain over the gains from merging, π = π m π c. Assume firm one makes the first offer and both firms discount at rate δ. Firm one gets to make one offer, and firm two gets to make one counter-offer. After a rejection, discount all later pay-offs. If agreement isn t reached, they continue as Cournot duopolists. Fill in the following extensive fm with pay-offs and solve f the sub-game perfect Nash equilibrium:, x y,, i. (Step ) The players payoffs are π = (A q q )q and π = (A q q )q (Step ) Now we maximize with respect to what each player controls, and solve f best-response functions π = A q q = 0 q = A q q π = A q q = 0 q = A q q This is a game of strategic substitutes, because each player s strategy is decreasing in the other player s strategy: dq = dq = dq dq The graph is identical to the usual Cournot graph, with c = 0. 4

5 (Step 3) Solving simultaneously yields the Nash equilibrium q = q = A 3 The strategies are increasing in A, since it appears positively in the numerat. ii. The profits f a single firm are π c = (A (A/3))(A/3) = A /9 A monopolist would maximize (A q)q q m = A and monopoly profits are π m = A /4 iii. x x, A /4 x y δy, δ( A 4 y) δ A 9, δa 9 The SPNE is In the last period, firm accepts only if And rejects otherwise. Then player solves δy δ A 9 y A 9 maxδ(a /4 y) y subject to y A 9. So player wants to keep y as small as possible, y = A /9. 5

6 Then player accepts player s first offer only if A /4 x δ(a /4 A /9) and player rejects otherwise. Then player solves A 4 ( δ)+δa 9 x maxx x subject to A 4 ( δ)+δa x. Since player wants to maximize his payoff x, he picks the 9 largest x possible, x = A 4 ( δ)+δa 9 The above offers and counter-offers are the subgame perfect Nash equilibrium. 6