Problem 4. Bertrand Competition Instead of choosing quantity, assume firms 1 and 2 choose prices P and P2. If pi P2, firm 2 sells quantity a bp2 and firm 1 sells nothing. If p1 = P2, each firm sells half of the quantity a bp1 - bp2. Suppose that each firm has constant marginal cost c, so that the cost of producing output x; for firm i is C(21) = c.. = a - (a) Find the equilibrium prices p and p2 in this market. This is not a job for calculus, because the firms' payoffs are not differentiable functions of p and p2. Instead, try a few combinations of prices, and for each one, ask yourself whether each firm is doing the best it can given the other firm's 4 price, or whether either firm has an incentive to change its price. You have an equilibrium when each firm is setting a price that maximizes its profits, given the price of the other firm. (b) How does the outcome you've found in part (a) compare to the equi- librium of the quantity-setting model, or to the competitive outcome in this market? Now suppose you are involved in a case before the Jus- tice Department, in which a merger is to be evaluated that will leave a market with only two firms. The concern is that this merger will lead to higher consumer prices. If you represented one of the firms that wanted to merge, which model of the resulting duopoly market would you be inclined to use as the basis for your analysis? Which would you use if you worked for the Justice Department? As an outsider, which do you think is more appropriate? (c) Now suppose you have the price-setting market that you have analyzed in part (a), but the firms set their prices and collect their profits not just once, but every year. To make things simple and concrete, set a = 10, b = 1, c = 0, and let the interest rate be 10%. The firms would like to collude, and consider the following scheme. In each period, each of them will set a price p*, as long as both firms have done so in every period in the past. Should any firm ever set another price, then in every following period both firms behave as in part (a) (for the special case of the parameter values we have assumed, namely a = 10, b = 1, c = = 0). If you were advising these firms, what value of p* would you tell them to set? Suppose that in the first period firm 1 decides to set some other price. What is the most it could earn by doing so? Would this be better than always setting price p*? (d) Now suppose you work for the Justice Department, that you see the firms consistently setting the price p* calculated in part (c), and that you prosecute them for colluding. They respond that they are not colluding at all but are instead behaving just as in part (a), and that their cost level is in fact not 0 but is such that p* is the price that comes out of the analysis of part (a). Assuming that it is still the case that a = 10 and b = 1, what are they implicitly claiming c to be? What would be your response to their argument? Problem 4. Bertrand Competition Instead of choosing quantity, assume firms 1 and 2 choose prices P and P2. If pi P2, firm 2 sells quantity a bp2 and firm 1 sells nothing. If p1 = P2, each firm sells half of the quantity a bp1 - bp2. Suppose that each firm has constant marginal cost c, so that the cost of producing output x; for firm i is C(21) = c.. = a - (a) Find the equilibrium prices p and p2 in this market. This is not a job for calculus, because the firms' payoffs are not differentiable functions of p and p2. Instead, try a few combinations of prices, and for each one, ask yourself whether each firm is doing the best it can given the other firm's 4 price, or whether either firm has an incentive to change its price. You have an equilibrium when each firm is setting a price that maximizes its profits, given the price of the other firm. (b) How does the outcome you've found in part (a) compare to the equi- librium of the quantity-setting model, or to the competitive outcome in this market? Now suppose you are involved in a case before the Jus- tice Department, in which a merger is to be evaluated that will leave a market with only two firms. The concern is that this merger will lead to higher consumer prices. If you represented one of the firms that wanted to merge, which model of the resulting duopoly market would you be inclined to use as the basis for your analysis? Which would you use if you worked for the Justice Department? As an outsider, which do you think is more appropriate? (c) Now suppose you have the price-setting market that you have analyzed in part (a), but the firms set their prices and collect their profits not just once, but every year. To make things simple and concrete, set a = 10, b = 1, c = 0, and let the interest rate be 10%. The firms would like to collude, and consider the following scheme. In each period, each of them will set a price p*, as long as both firms have done so in every period in the past. Should any firm ever set another price, then in every following period both firms behave as in part (a) (for the special case of the parameter values we have assumed, namely a = 10, b = 1, c = = 0). If you were advising these firms, what value of p* would you tell them to set? Suppose that in the first period firm 1 decides to set some other price. What is the most it could earn by doing so? Would this be better than always setting price p*? (d) Now suppose you work for the Justice Department, that you see the firms consistently setting the price p* calculated in part (c), and that you prosecute them for colluding. They respond that they are not colluding at all but are instead behaving just as in part (a), and that their cost level is in fact not 0 but is such that p* is the price that comes out of the analysis of part (a). Assuming that it is still the case that a = 10 and b = 1, what are they implicitly claiming c to be? What would be your response to their argument