Question

Demand in a market dominated by two firms (a Cournot duopoly) is determined according to: P = 300 – 4(Q₁ + Q₂), where P is the market price, Q₁ is the quantity demanded by Firm 1, and Q₂ is the quantity demanded by Firm 2. The marginal cost and average cost for each firm is constant; AC=MC = $77.

The cournot-duopoly equilibrium profit for each firm is _____.

Answer 1

P = 300 – 4(Q₁ + Q₂) = 300 - 4Q₁ - 4Q₂

Each firm maximizes profit according to the rule, MR = MC

Firm 1: Total revenue, TR1 = P*Q₁ = (300 - 4Q₁ - 4Q₂)*Q₁ = 300Q₁ - 4Q₁² - 4Q₁Q₂
So, Marginal Revenue, MR1 =

Now, MR1 = MC gives,
300 - 8Q₁ - 4Q₂ = 77
So, 8Q₁ = 300 - 77 - 4Q₂ = 223 - 4Q₂
So, Q₁ = (223/8) - (4Q₂/8)
So, Q₁ = 22.875 - 0.5Q₂
This is the best response function of firm 1. As demand function and MC is same for both firms, so best response funtion of firm 2 can be written as:
Q₂ = 22.875 - 0.5Q₁

Now, we substitute Q1 into Q2. We get,
Q₂ = 22.875 - 0.5(22.875 - 0.5Q₂) = 22.875 - 11.4375 + 0.25Q₂
So, Q₂ - 0.25Q₂ = 0.75Q₂ = 11.4375
So, Q₂ = 11.4375/0.75
So, Q₂ = 15.25

Now, Q₁ = 22.875 - 0.5Q₂ = 22.875 - 0.5(15.25) = 22.875 - 7.625 = 15.25
So, Q₁ = 15.25

P = 300 – 4(Q₁ + Q₂) = 300 - 4(15.25+15.25) = 300 - 122 = 178
So, P = 178

Profit for firm 1 = Profit for firm 2 = Total revenue - Total cost = P*Q₁ - AC*Q₁ = (P - AC)*Q₁ = (178-77)*15.25 = 101*(15.25) = 1,540.25

The cournot-duopoly equilibrium profit for each firm is $1,540.25