Economics 3382A-001 Problem Set 7
Due on Thursday December 2 at 4:00pm
经济Problem代写 Find the equilibrium Örmsíquantities (q1; q2) and the Örmsíexpected proÖts (1; 2).Now suppose Örm 1 Örst learns whether
Problem 1 (Two-period game). 经济Problem代写
Consider a two-player two-period game where each period the players play the following simultaneous move stage game:
After the Örst period the players observe what was played in the Örst period. The payo§ of each player is the sum of the payo§s from the two periods. Throughout the question restrict attention to pure strategies.
(a) Find all Nash equilibria in the stage game.
(b) Show that there exists a subgame-perfect equilibrium of the entire game such that the players play (B; a) under all circumstances: in the Örst period, and in the second period regardless of what happened in the Örst period.
(c) Can you Önd a subgame-perfect equilibrium of the entire game such that players play (A; a) in the Örst period? Explain your answer.
(d) Can you Önd a Nash equilibrium of the entire game that is not a subgame-perfect equilibrium? Explain your answer
Problem 2 (Entry game with incomplete information).
Consider the following of the Golden Balls game (from Problem 3 in Problem set 3) with incomplete information about player 2. Player 1 is altruistic. Player 2 is equally likely to be selÖsh (S) and altruistic (A). SpeciÖcally, the payo§s of the players depending on the type of player 2 are given in the two matrices:
Find all (pure strategy) Nash equilibria of this game.
Problem 3 经济Problem代写
(Value of information in Cournot duopoly model) Let the inverse demand is P (Q) = 120 Q, where Q is the total production. It is known
that Örm 1ís cost is equally likely to be high, CH (q) = 24q, or low, CL (q) = 12q. It is also known that Örm 2 has low cost, CL (q) = 12q. Suppose that Örm 1 has to choose its production q1 before it learns whether it has high or low cost. Firm 2 is aware of this fact and cannot observe the cost of Örm 1 either. Throughout the problem you can assume that each Örm will produce positive quantity.
(a) Find the equilibrium Örmsíquantities (q1; q2) and the Örmsíexpected proÖts (1; 2).Now suppose Örm 1 Örst learns whether its cost is high or low and after that chooses its production: q1H if its cost is high, and q1L if its cost is low. Suppose that Örm 2 is aware of this fact but still cannot observe the cost of Örm 1.
(b) Find the equilibrium Örmsí quantities (q 1H; q1L; q2), and the Örmsí expected proÖts(1,2).
(c) Find how much Örm 1 is willing to pay to Önd out whether its cost is high or low before the choice of production (suppose that Firm 2 can observe whether Örm 1 has learned its cost type or not). Now suppose that both Örms Örst learn the type of Örm 1 and after that choose their
production: q1H and q2H if Örm 1 has high cost, and q1L and q2L if Örm 1 has low cost. Suppose that both Örms are aware that everyone observes whether Örm 1 has low or high cost.
(d) Find the equilibrium Örmsí quantities (q1H; q1L; q2H; q2L), and the Örmsís expected proÖts (1,2).
(e) Find how much Örm 2 is willing to pay to Önd out the type of Örm 1 before the choice of production (suppose that Örm 1 can observe whether Örm 2 has learned Örm 1ís cost type or not).