Consider two firms, F and F that compete by setting prices simultaneously. Production is costless and...

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Consider two firms, F and F that compete by setting prices simultaneously. Production is costless and demand for good j is given by Q=100-Pj+Bip-j where , (0,1) measures the degree of differentiability of each good which need not be symmetric 1 + 2. a) Write down the profit function of each firm and derive the best response functions of each firm. b) What is the pure strategy Nash equilibrium of the game? (3 marks) (2 marks) c) What is the effect on prices, quantities, and profits of increasing B,? Show that if each firm could choose its the degree of differentiability B, they would choose the maximum amount. Page 2 of 3 (5 marks) TURN OVER d) Now assume that the firms are forced by a regulator to charge a price that is determined by maximizing the joint profits. That is, they are asked to behave like a monopolist. What price would they charge? (5 marks) e) Assume that B = 32 = . Will the firms benefits from the regulators proposal? What if B = and B = . who benefits now? (5 marks) f) How is it possible that reducing competition by coordinating offers as monopolists makes one of the firms worse off? (5 marks) Question 1 (50 marks) General Equilibrium and Welfare Consider an economy with two goods, bread (1) and games (x2), and two consumers, Anik and Bibi. Anik's preferences are represented by the utility function: u = 2x1x2. Bibi's preferences are represented by the utility function: u = xx. Initial allocations are e = (1,2) and eB = (1,0). a) Set up the consumers' utility maximisation problems and find their Marshallian Demands. (10 marks) b) Determine the market equilibrium prices and allocation. (5 marks) c) Depict the endowments, equilibrium allocation, preferences and prices in an Edgeworth box. (5 marks) d) You work as an economic advisor for the government. Your manager has a social welfare function of the form W = u + u. Which feasible allocation would maximise this social welfare function? Comment on the merits and limitations of using such a social welfare function. (10 marks) Finally, your manager's manager decides that the allocation a*, with a 4* = (1.25, 1.25) and xB* = (0.75, 0.75) should be implemented. e) Your colleague suggests to fix prices to p = 1 for bread and p2 = for games. Comment on this suggestion. (10 marks) f) What policy would you suggest to achieve the desired allocation. Explain your reasoning. (10 marks) Question 3 Infinitely repeated game Consider the stage game described by the following payoff matrix with a >b>1. Column Player Left Right Top , 1,a+1 Row Player Bottom a+1,1 b,b (25 marks) The game is repeated infinite times and players discount factor is 8 (0, 1). Normalize payoff in the following way: T U(x1, x2, xT) = (1-8)*x Consider the following profile of strategies: t=0 In any history in which both players have always played (Top, Left), then play Top if you are the row player and play Left if you are the Column player. In any other history play Bottom if you are the row player and play Right if you are the Column player. a) Show that for any pair a, b that verifies a > > 1 there is some < 1 such that for all > the profile of strategies described above is a subgame perfect Nash equilibrium of the repeated game? (10 marks) b) How does & changes when a increases and when b increases? Explain. (5 marks) c) Comment on the following statement:" One of the strongest messages that Game Theory popularize, is the idea that changes in payoffs that are not realized should not affect the behavior of rational players in the game. This is because, rationality allows players to under- stand that our decisions should not be affected by events that will never become real." Do you agree? Explain your answer. (5 marks) d) Is the pair of strategies (Bottom, Right) a subgame perfect Nash equilibrium of the repeated game? Prove or explain. (5 marks) Consider two firms, F and F that compete by setting prices simultaneously. Production is costless and demand for good j is given by Q=100-Pj+Bip-j where , (0,1) measures the degree of differentiability of each good which need not be symmetric 1 + 2. a) Write down the profit function of each firm and derive the best response functions of each firm. b) What is the pure strategy Nash equilibrium of the game? (3 marks) (2 marks) c) What is the effect on prices, quantities, and profits of increasing B,? Show that if each firm could choose its the degree of differentiability B, they would choose the maximum amount. Page 2 of 3 (5 marks) TURN OVER d) Now assume that the firms are forced by a regulator to charge a price that is determined by maximizing the joint profits. That is, they are asked to behave like a monopolist. What price would they charge? (5 marks) e) Assume that B = 32 = . Will the firms benefits from the regulators proposal? What if B = and B = . who benefits now? (5 marks) f) How is it possible that reducing competition by coordinating offers as monopolists makes one of the firms worse off? (5 marks) Question 1 (50 marks) General Equilibrium and Welfare Consider an economy with two goods, bread (1) and games (x2), and two consumers, Anik and Bibi. Anik's preferences are represented by the utility function: u = 2x1x2. Bibi's preferences are represented by the utility function: u = xx. Initial allocations are e = (1,2) and eB = (1,0). a) Set up the consumers' utility maximisation problems and find their Marshallian Demands. (10 marks) b) Determine the market equilibrium prices and allocation. (5 marks) c) Depict the endowments, equilibrium allocation, preferences and prices in an Edgeworth box. (5 marks) d) You work as an economic advisor for the government. Your manager has a social welfare function of the form W = u + u. Which feasible allocation would maximise this social welfare function? Comment on the merits and limitations of using such a social welfare function. (10 marks) Finally, your manager's manager decides that the allocation a*, with a 4* = (1.25, 1.25) and xB* = (0.75, 0.75) should be implemented. e) Your colleague suggests to fix prices to p = 1 for bread and p2 = for games. Comment on this suggestion. (10 marks) f) What policy would you suggest to achieve the desired allocation. Explain your reasoning. (10 marks) Question 3 Infinitely repeated game Consider the stage game described by the following payoff matrix with a >b>1. Column Player Left Right Top , 1,a+1 Row Player Bottom a+1,1 b,b (25 marks) The game is repeated infinite times and players discount factor is 8 (0, 1). Normalize payoff in the following way: T U(x1, x2, xT) = (1-8)*x Consider the following profile of strategies: t=0 In any history in which both players have always played (Top, Left), then play Top if you are the row player and play Left if you are the Column player. In any other history play Bottom if you are the row player and play Right if you are the Column player. a) Show that for any pair a, b that verifies a > > 1 there is some < 1 such that for all > the profile of strategies described above is a subgame perfect Nash equilibrium of the repeated game? (10 marks) b) How does & changes when a increases and when b increases? Explain. (5 marks) c) Comment on the following statement:" One of the strongest messages that Game Theory popularize, is the idea that changes in payoffs that are not realized should not affect the behavior of rational players in the game. This is because, rationality allows players to under- stand that our decisions should not be affected by events that will never become real." Do you agree? Explain your answer. (5 marks) d) Is the pair of strategies (Bottom, Right) a subgame perfect Nash equilibrium of the repeated game? Prove or explain. (5 marks)