The duopoly model we have been working with views Örms as choosing their quantities of output, with the market then setting a common price for the two Örms. Here is an alternative model. Firms 1 and 2 set prices p1 and p2. If p1 < p2, Örm 1 sells quantity A Bp1 and Örm 2 sells nothing. Similarly, if p1 > p2, Örm 2 sells quantity ABp2 and Örm 1 sells nothing. If p1 = p2, each Örm sells half of the quantity ABp1 = ABp2. Suppose that each Örm has constant marginal cost c, so that the cost of producing output xi for Örm i is C(xi) = cxi . (a) Find the equilibrium prices p1 and p2 in this market. This is not a job for calculus, because the Örmsí payo§s are not di§erentiable functions of p1 and p2. Instead, try a few combinations of prices, and for each one, ask yourself whether each Örm is doing the best it can given the other Örmís price, or whether either Örm has an incentive to change its price. You have an equilibrium when each Örm is setting a price that maximizes its proÖts, given the price of the other Örm. (b) How does the outcome youíve found in [4a] compare to the equilibrium of the quantity-setting model, or to the competitive outcome in this market? Now suppose you are involved in a case before the 2 Justice Department, in which a merger is to be evaluated that will leave a market with only two Örms. The concern is that this merger will lead to higher consumer prices. If you represented one of the Örms 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?
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