Answer Happy

There are n > 2 profit-maximising firms producing a homogeneous good, competing in quantity, and facing the inverse demand function P(Q) = 10 - Q, where Q = 2:19 is the total quantity produced in the market. Each firm i faces the same linear cost function: C(q) = 24i.
Suppose that the n firms form a cartel that chooses a total quantity Q to produce. To avoid lengthy court proceedings, the competition authority has decided to allow cartels to be formed but issue them with a tax equal to 50% of the revenue that firms make above the competitive price. That is, letting P* be the competitive price (found in (a)) and letting be the quantity that the cartel chooses, the cartel has to pay max{0,Q(P(Q) - P*)/2}. (We assume that the competition commission knows P* and is able to collect the tax.) (b) Write down the formula for the profit of the cartel as a function of Q. (c) If the cartel maximises its profit, what will be the total quantity produced and the equilibrium price? (d) For what values of n are consumers strictly worse off under the cartel than under competition? (e) Comparing the competitive price (found in (a)), the cartel price (found in (c)), and the price that the cartel would set if there were no competition authority, how effective is the competition authority at keeping the price low?