Question

Two firms compete with quantities as in Cournot. Each firm has a marginal cost of $12. The industry demand is P=48-2Q. How much output will each firm produce individually?

Answer 1

Given that MC₁ = MC₂ = $12 and demand function is  P=48-2Q where Q = Q₁ + Q₂

The demand curve is P = 48 - 2(Q₁ + Q₂) = 48 - 2Q₁ - 2Q₂

Profit function of firm 1 is given by

π₁ = TR₁ - TC₁

Here, TR₁ = P * Q₁ = (48 - 2Q₁ - 2Q₂) * Q₁ = 48Q₁ - 2Q₁² - 2Q₁Q₂

TC₁ = MC₁* Q₁ = 12Q₁

So, π₁ = TR₁ - TC₁ = 48Q₁ - 2Q₁² - 2Q₁Q₂ - 12Q₁

The best response function of firm 1 is given by

∂π₁ / ∂Q₁ = 0

48 - 4Q₁ - 2Q₂ - 12 = 0

4Q₁ + 2Q₂ = 36

or 2Q₁ + Q₂ = 18 ......(1)

Profit function of firm 2 is given by

π₂ = TR₂ - TC₂

Here, TR₂ = P * Q₂= (48 - 2Q₁ - 2Q₂) * Q₂= 48Q₂ - 2Q₁Q₂ - 2Q₂²

TC₂ = MC₂* Q₂ = 12Q₂

So, π₂ = TR₂ - TC₂ = 48Q₂ - 2Q₁Q₂ - 2Q₂² - 12Q₂

The best response function of firm 2 is given by

∂π₂ / ∂Q₂ = 0

48 - 2Q₁ - 4Q₂ - 12 = 0

2Q₁ + 4Q₂ = 36

or Q₁ + 2Q₂ = 18 .......(2)

Solving equations (1) and (2) simultaneously, we get

Q₁ = 6 units and Q₂ = 6 units

Thus, each firm will produce 6 units of output individually.