Question

one Japanese firm and one American firm dominate the US market of widgets. They share the same cost structure: TC = 251 + 40q. The only demand for widgets is in the US and is p = 100 – Q.

If these two firms compete in quantity at the same time, what is the Cournot equilibrium output, price, profit level by each firm?

Answer 1

We have the following information

P = 100 – Q where P is the price and Q is total quantity (q₁ + q₂). Here q₁ is output of American firm and q₂ is output of Japanese firm.

P = 100 – (q₁ + q₂)

Total Cost of American Firm: C₁ = 251 + 40q₁

Total Cost of Japanese Firm: C₂ = 251 + 40q₂

The profits of the duopolists are

Π₁ = pq₁ – C₁ = [100 – (q₁ + q₂)]q₁ – 251 – 40q₁

Π₁ = 100q₁ – q²₁ – q₁q₂ – 251 – 40q₁

Π₁ = 600q₁ – q²₁ – q₁q₂ – 251

Π₂ = pq₂ – C₂ = [100 – (q₁ + q₂)]q₂ – 251 – 40q₂

Π₂ = 100q₂ – q₁q₂ – q²₂ – 251 – 20q₂

Π₂ = 60q₂ – q₁q₂ – q²₂ – 251

For profit maximization under the Cournot assumption we have

∂Π₁/∂q₁ = 0 = 60 – 2q₁ – q₂

∂Π₂/∂q₂ = 0 = 60 – 2q₂ – q₁

The reaction functions are

q₁ = 30 – 0.5q₂

q₂ = 30 – 0.5q₁

Replacing q₂ into the q₁ reaction function we get

q₁ = 30 – 0.5(30 – 0.5q₁)

q₁ = 30 – 15 + 0.25q₁

0.75q₁ = 15

Output of the American Firm: q₁ = 20

And

q₂ = 30 – 0.5q₁

q₂ = 30 – 0.5(20)

Output of the Japanese Firm: q₂ = 20

Thus, the total output in the market is

Q = q₁ + q₂ = 20 + 20 = 40

And the market price

P = 100 – (q₁ + q₂)

P = 100 – (20 + 20)

P = 100 – 40

Market Price: P = 60

Π₁ = pq₁ – C₁

Π₁ = (60 × 20) – 251 – 40(20)

Π₁ = 1200 – 251 – 800

Π₁ = 149

And

Π₂ = pq₂ – C₂

Π₂ = (60 × 20) – 251 – 40(20)

Π₂ = 1200 – 251 – 800

Π₂ = 149