Question

Consider the following 2 period sequential game. There are two players, Firm 1 and Firm 2. They pro- duce identical goods and these goods are perfect substitutes. The inverse demand function in this market is given by P = 12 − (q1 + q2). Firm 1 moves first and choose its output q1. Firm 2 observes Firm 1’s decision of q1 and then chooses its output q2.\

1. Suppose that the cost function of both Firm 1 and 2 is given by C_i(q_i) = 8q_i. Find all subgame perfect Nash equilibria.
2. Now suppose that the cost function of both Firm 1 and 2 is given by C_i(q_i) = 8q_i+ 3 when q_i > 0. So both firms face a variable cost of 8 and a fixed cost of 3 if they choose to produce. If a firm chooses to produce nothing (q_i= 0) then its total cost is 0. Find all subgame perfect Nash equilibria.

Answer 1

A) P=12-q1-q2

A firm Profit Maximize at where MR= MC

MR2=12-2q2-q1

MC2==∆TC/∆Q=8

MR2=MC2

12-2q2-q1=8

Q2=2-0.5q1{ best response function of firm 2}

Putting Q2 into demand,

P=12-q1-(2-0.5q1)=10-0.5q1

MR1=10-q1

MC1=8

10-q1=8

Q1=10-8=2

Q2=2-0.5q1=2-0.5*2=1

P=12-2-1=9

B) Because fixed cost doesn't effect Profit Maximizing quantity, so the profit Maximizing ( or loss minimization) quantity of both firm will be same as before.

But Profit of firm 1=2-3=-1

Firm 2=1-3=-2

So both firm will be better off by not producing.but firm 1 moves first ,he knows if he choose q1=2, then firm 2 won't enter.

If q1=2 and q2=0( won't enter)

P=12-2-0=10

And firm 1 Profit

Profit of firm 1=(10-8)*2-3=1

Profit of firm 2=0( better than if it Produce)

So Subgame perfect nash equilibrium, Q1=2 and q2=0