Consider a Cournot duopoly with the following inverse demand function: P = 100 − 2Q₁ − 2Q₂.The firms' marginal costs are identical and are given by MC_i(Q_i)= 2Q_i.Based on this information,firm 1 and 2's reaction functions are: A) r₁(Q₂) = 24.5 − 0.5Q₁ and r₂(Q₁) = 24.5 − 0.5Q₂. B) r₁(Q₂) = 24.5 − 0.5Q₂ and r₂(Q₁) = 24.5 − 0.5Q₁. C) Q₁ = 49 − 0.5Q₂ and Q₂ = 49 − 0.5Q₁. D) Q₁ = 49 − 0.25Q₂ and Q₂ = 49 − 0.25Q₁.

Question

Quizplus · Accepted Answer

To find the reaction functions, we need to use the profit maximization condition of each firm, which is their first order condition.

For firm 1:

πS1U1B1= (100 − 2(QS1U1B11S1U1B0 + QS1U1B12S1U1B0))QS1U1B11S1U1B0 − 2(QS1U1B11S1U1B0)²

Taking the derivative with respect to QS1U1B11S1U1B0 and solving for QS1U1B11S1U1B0, we get: 
MR1 = ∂πS1U1B1/∂ QS1U1B11S1U1B0 = 100 − 4QS1U1B11S1U1B0 − 2QS1U1B12S1U1B0 − 4(QS1U1B11S1U1B0) = 0

Simplifying gives us:

QS1U1B11S1U1B0 = 25 − 0.5QS1U1B12S1U1B0

Similarly, for firm 2:

MR2 = ∂πS1U1B2/∂ QS1U1B12S1U1B0 = 100 − 4QS1U1B12S1U1B0 − 2QS1U1B11S1U1B0 − 4(QS1U1B12S1U1B0) = 0

Simplifying gives us:

QS1U1B12S1U1B0 = 25 − 0.5QS1U1B11S1U1B0

Therefore, the reaction functions are:

rS1U1B11S1U1B0(QS1U1B12S1U1B0) = 25 − 0.5QS1U1B12S1U1B0

rS1U1B12S1U1B0(QS1U1B11S1U1B0) = 25 − 0.5QS1U1B11S1U1B0

Consider a Cournot Duopoly with the Following Inverse Demand Function