Question

There are two incumbent firms, F1,F2 and also a potential entrant, F3. The steps of the game are:

1. F1 and F2 simultaneously choose outputs q1 ∈ R+ and q2 ∈ R+ respectively.
2. F3 observes q1,q2 and then chooses whether to enter the industry. If she does not, then q3 = 0 and she gets a payoff of zero, but...

3. if she has entered the industry, F3 chooses her own output level, q3 ∈ R+.

Inverse demand is p = 16−q1−q2−q3. Production costs for incumbent firms are cj(qj) = 4qj,j = 1,2. The entrant has the same production costs, but would also incur a fixed investment cost of 4 in order to enter the industry; c3(q3) = 4 + 4q3 if q3 > 0 and c3(0) = 0.

a) In step 3, F3 chooses q3 to maximise her profits, given an observed level of q1 +q2. Write down a mathematical expression for her profits, price times quantity less entry costs. Your expression should involve q1,q2,q3, but price should have been substituted out.
b) Take Firm 3’s first-order condition.
c) Solve your condition from question (b) for q3. The result should be F3’s best response function, q3 = B3(q1 + q2).
d) How high would F3’s payoff be if she does enter? Your answer should be a function of q1 + q2, but q3 should have been substituted out.
e) In step 2, F3 decides whether to enter the industry. How high would q1 + q2 have to be, in order to persuade her not to enter? Assume that F3 will not enter if she would get exactly the same payoff from entering and staying out.
f) Write down F1’s payoff.

Answer 1