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. 1

The following questions deal with step 1. But the incumbents have two considerations in choosing output, deterring entry and maximising profits given F3’s entry decision. We will consider entry deterrence in question (g) and the most profitable output in questions (h),(i). Then we will combine both considerations in question (j).

g) Imagine that in step 1, F1 decided to produce just enough so that it would not be worthwhile for F3 to enter the industry. How much would he have to produce to deter entry? Your answer should be a function of q2 but not of q3, ie. of the form q1 = R(q2). Your answer to question (e) may be useful here. Sketch the function that you identify in q1, q2 space.

h) Now imagine that in step 1, F1 didn’t take into account the possibility that his output could deter entry and just assumed that F3 would enter. That is, he just chose q1 to maximise his profits for an assumed fixed level of q2 and also subject to q3 = B3(q1 +q2), the best-response function for F3 that you derived in (c). Solve this problem. You might wish to use the method of substitution. Just as in question (g), your answer should be a function of q2 but not of q3, ie. q1 = S(q2). Sketch the function that you identify in q1, q2 space.

i) Next, consider the opposite assumption. Under this assumption F1 predicts that F3 will not enter. What will F1’s best response be, if he assumes that he will be in a duopoly with F2?

j) Finally, put together the three components you found in the previous three questions. What is F1’s overall best response, if he realises that entry by F3 will be deterred only if q1 + q2 is high enough.

Answer 1