Question

Q.

Two rival companies sell software packages that are perfect substitutes. The software is sold over the web as a download, so the marginal cost is zero. The demand for the software is Q = 900 – P. Assume that these firms are Cournot duopolists. Derive the reaction functions for the two firms, the quantity each will produce, and the market price that will be charged

Answer 1

Demand function is as follows -

Q = 900 - P

Inverse demand function is as follows -

P = 900 - Q

Where,

Q is the market demand

Q = q1 + q2

q1 = Output of Firm 1

q2 = Output of Firm 2

Determining market demand by equating inverse demand function and marginal cost

P = MC

900 - Q = 0

Q = 900

Reaction function of firm 1 is as follows -

q1 = 1/2(Market demand - Output produced by Firm 2) = 1/2(900 - q2) = 450 - 0.5q2        ---- Equation(1)

Reaction function of firm 2 is as follows -

q2 = 1/2(Market demand - Output produced by Firm 1) = 1/2(900 - q1) = 450 - 0.5q1

Put value of q2 in Equation 1

q1 = 450 - 0.5q2

q1 = 450 - [0.5(450 - 0.5q1)]

q1 = 450 - [225 - 0.25q1]

q1 - 0.25q1 = 225

0.75q1 = 225

q1 = 225/0.75 = 300

The Firm 1 will produce 300 software packages.

q2 = 450 - 0.5q1 = 450 - (0.5*300) = 450 - 150 = 300

The Firm 2 will produce 300 software packages.

P = 900 - Q = 900 - (q1 + q2) = 900 - (300+300) = 900 - 600 = 300

The market price that will be charged is $300 per software package.