Suppose two firms compete by selecting quantities q1 and q2, respectively, with

Question

Suppose two firms compete by selecting quantities q1 and q2, respectively, with the market price given by p = 1000 – 3q1 -3q2. Firm 1 (the incumbent) is already in the market. Firm 2 (the potential entrant) must decide whether or not to enter, and if she enters, how much to produce. First the incumbent commits to its production level q1. Then the potential entrant having seen q1, decides whether or not to enter the industry. If firm 2 chooses to enter, then it selects its production level q2. Both firms have the cost function c(qi) = 100qi + F, where F is a constant fixed cost. If firm 2 decides not to enter, then it obtains a payoff of zero. Otherwise it pays the cost of production, including the fixed cost. Note that firm i in the market earns a profit

of pqi-c(qi).

a) What is firm 2’s optimal quantity as a function of q1, conditional on entry?

b) Suppose F=0. Compute the subgame perfect equilibrium of this game. Report equilibrium

strategies as well as output, profit and price realized in equilibrium.

c) Now suppose F>0 . compute as a function of F, the level of q1 that would make entry

unprofitable for firm 2.

d) Find incumbent’s optimal choice of output and the outcome of the game in the following

cases: (i) F = 18,723, (ii) F = 8,112 (iii) F=1,728 and (iv) F=108

Explanation / Answer

P= 1000-3q1-3q2

C= 100qi+F

Profit (PF) = p*q1 – c

a) Now, for firm 2 q1 is a constant

Profit for firm 2,

PF2 = (1000-3q1-3q2)q2-100q2-F =1000q2-3q1q2-3q2^2-100q2-F

= 900q2-3q1q2-3q2^2-F

Profit maximizing condition, d(PF2)/d(q2)=0

Thus, 900-3q1-6q2=0 or q2=150- 1/2q1

b) Profit for firm 1,

PF1 = (1000-3q1-3q2)q1-100q2 =1000q-3q1q2-3q1^2-100q1

= 900q1-3q1q2-3q1^2

In eq, q2=150- 1/2q1

Thus, PF1= 900q1-3q1(150- 1/2q1)-3q1^2= 900q1-450q1+3/2q1^2-3q1^2

=450q1-3/2q1^2

Profit maximizing condition, d(PF1)/d(q1)=0

450-3q1=0 or q1=150

Now, q2=150- 1/2q1. Thus, q2= 150-75=75

P= 1000-3q1-3q2 = 1000-3.450-3.75= 325

c) Profit for firm 2,

PF2 = (1000-3q1-3q2)q2-100q2-F =1000q2-3q1q2-3q2^2-100q2-F = 900q2-3q1q2-3q2^2-F

In order to be unprofitable for entry PF2<0

Thus, 900q2-3q1q2-3q2^2-F <0

900q2--3q2^2-F < 3q1q2 or q1> 300q2/q1-q2-F/3q2

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