Consider an industry that consists of two firms, A and B. They face a demand curve q = qA + qB = 14 - p, where p is the industry price of output. Both firms have constant average and marginal cost of $2. (a) Suppose they form a cartel and choose the price that maximizes the sum of their profits. Show that they will choose p = $8. (b) Now suppose that instead of forming a cartel, they choose prices simultaneously. If they choose different prices, the firm that chooses the lower price captures the entire market; if they set the same price they split the market evenly. Suppose they play this game once. Show that in a Bertrand equilibrium, both firms will charge $2. (c) Suppose they play this game an infinite number of times. Consider the following grim trigger strategy. Choose the cartel price (i.e., $8) in the first period. Continue to choose the cartel price in subsequent periods if your opponent has always chosen the cartel price up to that point. If your opponent ever chooses a price other than $8, choose the Bertrand price (i.e., $2) from that point forward. For what range of values of the discount factor do these trigger strategies constitute a subgame perfect equilibrium? (d) Now change this game so that there are N ≥ 2 oligopolists; thus if they all charge the same price, each will sell a proportion 1/N of the market demand at that price. Express the critical discount factor as a function of N. Does your answer suggest that it will be easier to sustain cooperation when N is small or when N is large? What is the intuition behind this result?