Question

There are two Cournot competitors, A & B. Each has total cost of 12x, where x is a firm’s output,

and demand is X = 400 – P, where P is market price. What will market price be in equilibrium?

(a.) $239.8

(b.) $141.3

(c.) $130

(d.) $208

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 = 12X_A

C₂ = 12X_B

The profits of the duopolists are

Π_A = PX_A – C_A = [400 – (X_A + X_B)]X_A – 12X_A

Π_A = 400 X_A – X²_A – X_AX_B – 12X_A

Π_A = 388X_A – X²_A – X_AX_B

Π_B = PX_B – C_B = [400 – (X_A + X_B)]X_B – 12X_B

Π_B = 400X_B – X_AX_B – X²_B – 12X_B

Π_B = 388X_B – X_AX_B – X²_B

For profit maximization under the Cournot assumption we have

∂Π_A/∂X_A = 0 = 388 – 2X_A – X_B

∂Π_B/∂X_B = 0 = 388 – 2X_B – X_A

The reaction functions are

X_A = 194 – 0.5X_B

X_B = 194 – 0.5X_A

Replacing X_B into the X_A reaction function we get

X_A = 194 – 0.5(194 – 0.5X_A)

X_A = 194 – 97 + 0.25X_A

X_A = 129.33

And

X_B = 194 – 0.5X_A

X_B = 194 – 0.5(129.33)

X_B = 194 – 64.67

X_B = 129.33

Thus, the total output in the market is

X = X_A + X_B = 129.33 + 129.33 = 258.66

And the market price

P = 400 – X

P = 400 – 258.66

P = 141.3

Equilibrium price is $141.3