Question

There is a Cournot duopoly competition between Firm 1 and Firm 2. The inverse demand function is given by P(Q)=100-q, where Q=q1+q2 and qi denotes the quantity produced by firm i for all iÎ {1, 2} and the cost function is given by ci(qi)=10qi. Describe this problem as a normal-form game. Find pure-strategy Nash Equilibria for both firms.

Answer 1

P(Q)=100-q

P(Q)=100- q1 - q2

MC1 = MC2 = 10

Profit of firm 1:

1 = (P - MC1)*q1 = (100- q1 - q2 - 10)*q1

1 = (90 - q2)*q1 - q1^2

FOC: d1 / dq1 = 0

90 - q2 = 2q1

q1^BR = 45 - 0.5q2 ---------equation 1

Similarly, q2^BR = 45 - 0.5q1 ---------equation 2

Solving equation 1 and 2

q1 = 45 - 0.5*(45 - 0.5q1)

q1 = 45 - 22.5 + 0.25q1

0.75q1 = 22.5

q1* = q2* = 30

Pure Strategy Nash Equilibrium (q1*, q2*) = (30, 30)