Question

Total market demand for pillows in San Francisco is given by P = 125 - 0.5Q. There are 2 suppliers of pillows in the market, who each have a constant marginal cost of $5 per pillow. If the 2 firms compete against each other in a Cournot duopoly, how many pillows will each firm produce?

Answer 1

Let supplier 1's output be q1 and supplier 2's output be q2. So, Q = q1 + q2
So, P = 125 - 0.5(q1+q2) = 125 - 0.5q1 - 0.5q2
MC = 5

Each supplier maximizes profit according to the rule: MR = MC

Supplier 1: Total revenue, TR1 = P*q1 = (125 - 0.5q1 - 0.5q2)*q1 = 125q1 - 0.5q1² - 0.5q2q1
So, Marginal revenue, MR1 = d(TR1)/dq1 = 125 - 2(0.5q1) - 0.5q2 = 125 - q1 - 0.5q2
So, MR1 = MC gives,
125 - q1 - 0.5q2 = 5
So, q1 = 125 - 5 - 0.5q2
So, q1 = 12 - 0.5q2
(This is the best response function of supplier 1).

As the demand and cost functions are same, so, best response function of supplier 2 will be,
q2 = 12 - 0.5q1 = 12 - 0.5(12 - 0.5q2) = 12 - 6 + 0.25q2
So, q2 - 0.25q2 = 0.75q2 = 6
So, q2 = 6/0.75
So, q2 = q1 = 8

Each firm will produce 8 pillows.