Suppose that two duopolists are producing given differentiated products (like Hyundai and GM cars in the automobile industry). The market demands for each firm's products are given by = 12 - 2p + P and Y = 12 - 2p + P, respectively. Assume, for simplicity, that each firm does not pay any cost to produce any positive quantity of output. (1) Suppose two firms compete in the Bertrand way. That is, two firms choose their prices simultaneously and independently. Write down the normal form of the game. (2) Find the Nash equilibrium for the game. (3) Now suppose firm 1 is the price leader of the industry. Find the subgame perfect equilibrium of the price leadership game. (4) Suppose that two firms compete in Cournot fashion. That is, two firms choose their quantities simultaneously and independently. Write down the normal form of the new game. (5) Find the Nash equilibrium for the new game. (6) Now suppose two firms compete in Stackelberg fashion and firm 1 is the leader of the industry. Find the subgame perfect equilibrium of the quantity leadership game. (7) Now suppose two firms decide to collude and form a cartel. Find the joint-profit maximizing price-quantity combination for the cartel. (8) What is the market equilibrium in this industry if two firms behave like firms in the competitive market? (9) Compare all the equilibria you found above. Is it always better for a firm to be an industry leader than to be a follower? (10) Now suppose two firms have to play the Bertrand duopoly game in part (1) for ten periods without discounting. What is the subgame perfect equilibrium outcome in the new game? How many subgame perfect equilibria are there in this game? (11) Now suppose two firms have to play the Bertrand game repeatedly infinitely many times. Can they support the monopoly allocation found in (7) by choosing an appropriate subgame perfect equilibrium? If so, write down a trigger strategy equilibrium which supports it. What is the minimal discount factor (6) in this case? (12) Answer question (11) again when firms play the repeated Cournot game.