Practice question

Question 4 | 1 Consider the following market. In a first stage, a large (potentially infinite) number of identical firms decide whether or not to enter. If they enter, they each must pay an entry cost K > 0. In a second stage, if N firms have entered, they play a symmetric Cournot game in which the inverse demand is given by P(Q) = a - bQ, and all have a constant marginal cost c. In all this problem, please check for second-order conditions whenever necessary. (a) [10 points] Assuming that N firms enter the market, find the quantity that each will produce in the Cournot equilibrium, as well as the market quantity, market price, and individual profits. (b) [10 points] Find the welfare W(N) if N firms enter the market. (Welfare is the con- sumer surplus plus the firms' profit, which includes the fixed cost of entry.) (c) [10 points] Ignoring the integer constraint (that is, the number of firms does not need to be an integer), show that the number of firms N" that maximizes welfare satisfies (N* + 1)3_ (@-c)2 bK (d) [5 points] Ignoring the integer constraint (that is, the number of firms does not need to be an integer), show that the number of firms N that will enter in the first stage, if there is free-entry - that is, if firms enter as long as there is a positive profit to make by entering - satisfies (N* + 1)2 (@-c)2 bK (e) [5 points] Show that N* > N*. (f) [5 points] Show that if it is socially optimal for three firms to enter, then seven firms will actually enter