Consider the following market game: An incumbent firm, called firm 3, is already

Question

Consider the following market game: An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10. First, firm 1 decides whether to enter or not. Then, after observing firm 1’s choice, firm 2 decides whether or not to enter. Every firm, including firm 3, observes the choices of firm 1 and 2. After this all of the firms in the industry (including firm 3) compete in a Cournot Oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve p = 12 –Q, where Q is the total quantity produced in the industry. Assume the firms produce at no cost in this cournot game. Thus if firm i is in the industry and produces qi, then it earns a gross profit of (12-Q)qi in the cournot phase. (remember that firms 1 and 2 have to pay fixed cost 10 to enter.) Compute the equilibrium of this market game.

Explanation / Answer

Firm 1's profit if it enters, (12-q1-q2-q3)q1 - 10

maximizing it, 12 - 2q1 - q2 - q3 =0 -----(1)

If firm1 enters, firm 2's profit = (12 - (-q2-q3+12/2) - q2 -q3)q2 - 10

maximizing it, 12 + q2 +( q3)/2 -6 - 2q2 -q3 =0

6 = q2 + (q3)/2 ------(2)

q2 = 6-(q3)/2

Maximize firm 3's profit, (12 -  (-q2-q3+12/2) - q2 - q3) q3

(12 +6 - (q3)/2 + q3 -6 - (6-(q3)/2) -q3) q3

Maximize it w.r.t q3,

6 - q3 +2q3 +q3 + 2q3 = 0

q3 = 6/4

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