Question

Two firms have the cost function C(y) = y2. The inverse demand function for the firms' output is pi = 120 – D, where D is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

Answer 1

Let firm 1's output be y1 and firm 2's output be y2.
So, D = y1 + y2
p = 120 - (y1 + y2) = 120 - y1 - y2
MC = dC/dy = 2y

Each firm maximizes profit where its MR = MC.

Firm 1: TR1 = p*y1 = (120 - y1 - y2)y1 = 120y1 - y1² - y2y1
MR1 = d(TR1)/dy1 = 120 - 2y1 - y2
Now, MR1 = MC1 gives,
120 - 2y1 - y2 = 2y1
So, 2y1 + 2y1 = 4y1 = 120 - y2
So, y1 = (120/4) - (y2/4)
So, y1 = 30 - 0.25y2
This is the best response function of firm 1. As the demand and cost fucntions are similar for both firms, so, we can write the best response function of firm 2 as
y2 = 30 - 0.25y1
Substituting the value of y1, we get,
y2 = 30 - 0.25(30 - 0.25y2) = 30 - 7.5 + 0.0625y2
So, y2 - 0.0625y2 = 0.9375y2 = 22.5
So, y2 = 22.5/0.9375 = 24

y1 = 30 - 0.25y2 = 30 - 0.25(24) = 30 - 6 = 24

So, each firm produces 24 units in a Nash equilibrium.