Help here4 tutors
4. Two firms compete to sell a good. Total costs of the two firms respectively satisfy Ci(qi) = 201 and Co(q) = $3. The total output produced in the economy is Q = q1 + 92. The inverse demand for the total output produced by the two firms in this market satisfies P(Q) = 10-20 if Q5 5 0 if Q>5 Remember that the inverse demand curve identifies the highest price for which all the units supplied to the market are purchased. (a) First assume that firms compete on quantities. Find the Cournot equilibrium output at the two firms, the equilibrium price, and the profits at each of the two firms. (20 marks) (b) Then suppose that the two firms form a cartel. Find the cartel price and the output produced at the two firms? Compare your results with part (a). Does the cartel make higher profits than the two firms in part (a)? Does it produce more output on aggregate? (15 marks) (c) Finally assume that firms choose their output taking prices as given. Find the perfect competition price and the output produced at the two firms? Again, compare your results with parts (a) and (b). (14 marks)Problem 1 (Uncertainty) Rick is considering whether to spend 5 dollars betting on Republicans winning the next election. If Republicans were to win the election, Rick would be paid 4 dollars for any dollar that he has bet. The utility that Rick derives from a (positive or negative) cash transfer of a dollars is determined by the following utility function, u(r) = (475 + 75x)1/2. Rick believes that the probability of republicans winning the next election is 1/3. 1. Find the expected value of such a lottery. Intermediate Microeconomics F. Nava 2. Find Rick's expected utility of taking such a gamble. Would he accept it? Or would he reject it and get r = 0? 3. What's the certainty equivalent of such a lottery. Problem 2 (Static Games) Consider the following static two-player game: 1\\2 A B A 3, 7 7,3 2,2 Player 1'is the row player, and his payoff is the first to appear in each entry. Player 2 is the column player and his payoff is the second to appear in each entry. 1. Find the pure strategy Nash equilibria of the game, and show that they are equilibria. 2. Find the mixed strategy Nash equilibrium of the game. 3. Derive the mixed strategy best responses. Problem 3 (Bayesian Games) Consider the following Bayesian game played by two players (1 and 2) who are deciding whether to cooperate, C, or defect, D. Two states are possible, Good and Bad. Suppose that Player 2 knows the state, while Player 1 thinks that the state is Good with probability p. Payoffs in each state respectively satisfy 12 C D 1\\2 C D State Good: 0 0.0 1, 1 State Bad: C 0.0 0. 1 . D | 1,1 0.0 D 1,0 3,3 Player 1 is the row player, and his payoff is the first to appear in each entry. Player 2 is the column player and his payoff is the second to appear in each entry. 1. What is the set of possible strategies for the two players in this game? 2. Find the pure strategy Bayes-Nash equilibria for all values of pe (0, 1). Problem 4 (Cournot Uncertainty) Two firms compete to sell a good. Firm I has total costs of production Ci(ni) = (qi)" + 21 and its costs are known to Firm 2. The total costs of Firm 2 depends on its type. If Firm 2 is of type L, its costs are Cr(q1) = 2qr. If Firm 2 is of type H, its costs are Cu(qn) = 2 (qu). Firm 2 knows its type. But Firm I only knows that Firm 2 can have either cost structure with equal probability. The inverse demand for the output produced by the two firms in this market satisfies: p(gi + 92) = 10 - 2(q1 + 42) Firms choose how much output to produce in order to maximize their profits. Find the Bayes-Nash equilibrium of this game. Characterize the equilibrium output strategies for both firms. Find the market price for each of the two possible cost configurations. Problem 5 (Repeated Games) Consider the following asymmetric Prisoner's Dilemma: 1\\2 C D 3.4 1,6 4,0 2.2 1. Find the minmax values of this game. 2. Then, consider the following "trigger" strategy: any player chooses C provided that no player ever played D; otherwise any player chooses D. Write the two incentive constraints that if satisfied would make such a strategy a NE. Then, write the two additional incentive constraints that if satisfied would make such a strategy a SPE. What is the lowest discount rate for which such strategy satisfies all the constraints.4. (30 points) Consider the following game. There are ten dollars to divide. Two players are each required to simultaneously name an integer between 0 and 10. The player who names the higher number gets to keep the money. If they name the same number, the money is equally shared between them. (a) Describe the set of players /, the set of strategies {S,lien, and the payoff function fuiliEN. (b) Are there strategies that are strictly dominated? Demonstrate your reasoning. What are the resulting strategies after iterated elimination of strictly dominated strategies? (c) Find the best responses (correspondence) for each player. That is, find the strategies that maximize a player's payoff given what the other player does. (d) Find the Nash equilibria of the game. (e) Suppose now the game is changed. Whenever there is a tie, each player receives nothing. Answer the same questions in parts (b) and (c). Find the pure-strategy Nash equilibria of the game.1. (20 points) Consider the Bertrand duopoly model with homogeneous products, Suppose that the quantity that consumers demand from i is a - p, when pi p,. and (a-p.)/2 when p, = p, Suppose also that there are no fixed costs and that marginal costs are constant at e, where c
