Q2 Consider the following demand function, p= 1-Q/2, Q = 91 +92, where qi denotes firm...

 

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Q2 Consider the following demand function, p= 1-Q/2, Q = 91 +92, where qi denotes firm i's output, i=1,2. Assume that the marginal cost of each firm is equal to c> 0. Firms choose quantities simultaneously and non cooperatively (Cournot competition). The game described above is infinitely repeated. Firms use grim trigger strategies. Firms discount future profits at a rate r > 0 (the discount factor is 8 = 1/(1+r)). a) Derive the critical discount factor above which full cartelization (joint profit maximization) is sustainable as a Subgame Perfect Nash Equilibrium (SPNE) of the infinitely repeated game. Consider now the following prisoner's dilemma game: с N C 3,3 0,4 N 4,0 1,1 The above game is repeated a random number of times, potentially infinitely many times. After each stage is played, the game continues with probability q. This probability is independent from one round to the next. The probability that the game is still being played T rounds from now is q. 1 b) Assume that players do not discount future payoffs (i.e. r = 0). Compute the threshold of q above which (C,C) is a SPNE of the repeated game that ends after a random number of repetitions. Comment. c) Assume now that players discount future payoffs at a rate r > 0. Compute the critical discount factor above which (C, C) is a SPNE of the re- peated game that ends after a random number of repetitions. Comment. d) In repeated games in general, if players discount future payoffs at a different rate, is it possible for cooperation to be sustainable? Explain. Q2 Consider the following demand function, p= 1-Q/2, Q = 91 +92, where qi denotes firm i's output, i=1,2. Assume that the marginal cost of each firm is equal to c> 0. Firms choose quantities simultaneously and non cooperatively (Cournot competition). The game described above is infinitely repeated. Firms use grim trigger strategies. Firms discount future profits at a rate r > 0 (the discount factor is 8 = 1/(1+r)). a) Derive the critical discount factor above which full cartelization (joint profit maximization) is sustainable as a Subgame Perfect Nash Equilibrium (SPNE) of the infinitely repeated game. Consider now the following prisoner's dilemma game: с N C 3,3 0,4 N 4,0 1,1 The above game is repeated a random number of times, potentially infinitely many times. After each stage is played, the game continues with probability q. This probability is independent from one round to the next. The probability that the game is still being played T rounds from now is q. 1 b) Assume that players do not discount future payoffs (i.e. r = 0). Compute the threshold of q above which (C,C) is a SPNE of the repeated game that ends after a random number of repetitions. Comment. c) Assume now that players discount future payoffs at a rate r > 0. Compute the critical discount factor above which (C, C) is a SPNE of the re- peated game that ends after a random number of repetitions. Comment. d) In repeated games in general, if players discount future payoffs at a different rate, is it possible for cooperation to be sustainable? Explain.

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