Consider a Cournot duopoly with the following inverse demand function: P = 50 - 0.2Q₁ - 0.2Q₂.The firms' marginal cost are identical and given by MC_i(Q_i) = 2.Based on this information firm 1 and 2's reaction functions are A)r₁(Q₂) = 4.8 - 0.5Q₁ and r₂(Q₁) = 4.8 - 0.5Q₂. B)r₁(Q₂) = 4.8 - 0.5Q₂ and r₁(Q₂) = 4.8 - 0.5Q₁. C)Q₁ = 25 - 0.75Q₂ and Q₂ = 25 - 0.75Q₁. D)Q₁ = 240 - 0.5Q₂ and Q₂ = 240 - 0.5Q₁.

Question

Quizplus · Accepted Answer

The reaction function for firm 1 is derived by taking the first derivative of their profit with respect to their quantity and setting it equal to zero: 
$\frac{\partial\pi_1}{\partial Q_{1}}=50-0.2Q_{1}-0.2Q_{2}-2Q_{1}=0$
$Q_{1}=\frac{1}{5}(50-0.2Q_{2})=10-0.04Q_{2}$
Similarly, the reaction function for firm 2 is: 
$Q_{2}=10-0.04Q_{1}$ 
Plugging in one reaction function into the other yields the following best-response function for each firm: 
$r_{1}(Q_{2})=10-0.04Q_{2}$ 
$r_{2}(Q_{1})=10-0.04Q_{1}$ 
Therefore, the correct answer is B.

Consider a Cournot Duopoly with the Following Inverse Demand Function