Question

Suppose the demand for pizza in a small isolated town is p = 10 - Q. There are only two firms, A and B, and each has a cost function TC = 2 + q. Determine the Cournot equilibrium

Answer 1

Each firm’s marginal cost function is MC = 1 and the market demand function is P = 10 – (q1
+ q2) where Q is the sum of each firm’s output q1
and q2.
Find the best response functions for both firms:
Revenue for firm 1
R1 = P*q1 = (10 – (q1 + q2
))*q1 = 10q1 – q1
2 – q1q2
.
Firm 1 has the following marginal revenue and marginal cost functions:
MR1 = 10 – 2q1 – q2
MC1 = 1
Profit maximization implies:
MR1 = MC1
10 – 2q1 – q2 = 1
which gives the best response function:
q1 = 4.5 - 0.5q2
.
By symmetry, Firm 2’s best response function is:
q2 = 4.5 - 0.5q1
.
Cournot equilibrium is determined at the intersection of these two best response functions:
q1
= 4.5 - 0.5(4.5 - 0.5q1
)
q1
= 2.25 + 0.25q1
This gives q1 = q2 = 3 units This the Cournot solution. Price is (10 – 6) = $4. Profit to each
firm = 4*3 – 2 – 3 = $7.