Question

Two Cournot competitors, named firm A & B, each have total cost of 2x^2, where x is a firm’s
output choice. Say total demand is X = 400 – P, where P is price. If firm B produces 60 units of
output, what is firm A’s optimal output amount? What is the Nash equilibrium pair (xA, xB)?

(a.) (45.5, 45.5)

(b.) (57.1, 57.1)

(c.) (64.5, 64.5)

(d.) (69.9, 69.9)

Graph it with labels, thanks!

Answer 1

The market demand and costs of the duopolists are the following

P = 400 – (X_A + X_B)

C₁ = 2X²_A

C₂ = 2X²_B

The profits of the duopolists are

Π_A = PX_A – C_A = [400 – (X_A + X_B)]X_A – 2X²_A

Π_A = 400 X_A – X²_A – X_AX_B – 2X²_A

Π_A = 400X_A – 3X²_A – X_AX_B

Π_B = PX_B – C_B = [400 – (X_A + X_B)]X_B – 2X²_B

Π_B = 400X_B – X_AX_B – X²_B – 2X²_B

Π_B = 400X_B – X_AX_B – 3X²_B

For profit maximization under the Cournot assumption we have

∂Π_A/∂X_A = 0 = 400 – 6X_A – X_B

∂Π_B/∂X_B = 0 = 400 – 6X_B – X_A

The reaction functions are

X_A = 66.67 – 0.167X_B

X_B = 66.67 – 0.167X_A

Replacing X_B into the X_A reaction function we get

X_A = 66.67 – 0.167(66.67 – 0.167X_A)

X_A = 66.67 – 11.13 + 0.0278X_A

0.972X_A = 55.54

X_A = 57.1

And

X_B = 66.67 – 0.167X_A

X_B = 66.67 – 0.167(57.1)

X_B = 57.1

Thus, the total output in the market is

X = X_A + X_B = 57.1 + 57.1 = 114.2

And the market price

P = 400 – X

P = 400 – 114.2

P = 285.8

Equilibrium price is $285.8