Question

Two profit-maximizing firms compete in a market. Firm 1 chooses quantity qı > 0 and Firm 2 chooses quantity 42 > 0. The market price is: p(91,92) = 8 - 2q1 - 42. The cost to Firm 1 of producing qi is C1 = 41. The cost to Firm 2 of producing 92 is C2 = 42 + 42. a.) * Calculate the best-response function for each firm. b.) Suppose the two firms choose their quantities simultaneously. What is the Cournot-Nash equilibrium? c.) * If the firms choose quantities sequentially and Firm 1 is the first mover, what quantities are produced in the subgame-perfect Nash equilibrium? d.) Does the equilibrium quantity of Firm 1 increase or decrease when she chooses first in part (c), compared to when the firms choose simultaneously in part (b)? Does the quantity of Firm 2 increase or decrease when she chooses second in part (c)? Explain intuitively. e.) Explain why Firm 1 cannot threaten to choose the equilibrium quantity from part (c) in any Nash equilibrium of the simultaneous-move game from part (b).

Answer 1

P=8-2q1-q2

C1=q12​​​​​​

C2=q2+q22​​​​​​

a) For best response curve:

TR1=P*q1=8q1-2q12-q1q2

MR1=8-4q1-q2

MC1=2q1

MR1=MC1

8-4q1-q2=2q1

8=6q1+q2 equation 1

q1=(8-q2)/6 Best response curve of firm 1

Now for firm 2

MR2=8-2q1-2q2

MC2=1+2q2

MR2=MC2

8-2q1-2q2=1+2q2

7=2q1+4q2 equation 2

q2=(7-2q1)/4 best response curve of firm 2

b) For nash equilibrium:

Solve equation 1 and 2

q1=25/22

q2=26/22

P=8-2q1-q2=8-2(25/22)-26/22=100/22

C) If they choose sequentially:

First step to calculate best response curve of firm 2 which alreasy done above in a).

Second step: Profit of firm 1= P*q1-TC1=8q1-2q12-q1q2 - q12​​​​​​

Put value of q2 from its BRS

Profit= 8q1-2q12-q1(7-2q1 /4)-q12​​​​​​

Differentiate with respect to q1

dprofit/dq1= 8-4q1-7/4+4q1/4-2q1=0

q1=5/4

Put this value in BRS of firm 2

q2=9/4

d) The quantity of firm 1 increases by 0.12 while quantity of firm 2 increases as well by 1.07