Question

The market demand curve is given by

p = 100 - Q

Two firms, A and B, are competing in the Cournot fashion. Both firms have the constant marginal cost of 70. Suppose firm A receives a new innovation which reduces its marginal cost to c. Find the cutoff value of c which makes this innovation "drastic".

Answer 1

Each firm’s marginal cost function is MC = 70 and the market demand function is P = 100 – (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 = (100 – (q1 + q2))*q1 = 10q1 – q12 – q1q2.

Firm 1 has the following marginal revenue and marginal cost functions:

MR1 = 100 – 2q1 – q2

MC1 = 70

Profit maximization implies:

MR1 = MC1

100 – 2q1 – q2 = 70

which gives the best response function:

q1 = 15 - 0.5q2.

By symmetry, Firm 2’s best response function is:

q2 = 15 - 0.5q1.

Cournot equilibrium is determined at the intersection of these two best response functions:

q1 = 15 - 0.5(15 - 0.5q1)

q1 = 7.5 + 0.25q1

This gives q1 = q2 = 10 units This the Cournot solution. Price is (100 – 20) = $80. Profit to each firm = (80 – 70)*10 = $100

Now the value of c should be such that q2 = 0. In that case only firm A produces monopoly output. It will therefore charge a price of $70 which is the marginal cost of 2nd firm so that firm B is not able to produce anything

MR = 100 - 2Q

MC = c

100 - 2Q = c

This gives Q = (100 - c)/2 = 50 - 0.5c

Price = 100 - Q = 100 - 50 + 0.5c or P = 70

50 + 0.5c = 70

c = 20/0.5 = 40

Hence the cut off value of c is 40.