Let's break down the Cournot duopoly problem, given: - **Inverse demand:** \( P = 100 - 2Q \), where \( Q = Q_1 + Q_2 \) - **Firm 1 marginal cost:** \( MC_1(Q_1) = 2Q_1 \) - **Firm 2 marginal cost:** \( MC_2(Q_2) = 4Q_2 \) The question provides several pairs of reaction functions. We need to **find which pair is correct** for the given cost and demand functions. --- ## 1. **Profit Functions** For Firm 1: \[ \pi_1 = P Q_1 - \int_0^{Q_1} MC_1(x) dx = (100 - 2(Q_1+Q_2)) Q_1 - Q_1^2 \] But marginal cost is \( 2Q_1 \), so total cost is \( Q_1^2 \): \[ \pi_1 = (100 - 2Q_1 - 2Q_2) Q_1 - Q_1^2 = 100Q_1 - 2Q_1^2 - 2Q_1Q_2 - Q_1^2 = 100Q_1 - 3Q_1^2 - 2Q_1Q_2 \] For Firm 2: \[ \pi_2 = (100 - 2Q_1 - 2Q_2) Q_2 - 2Q_2^2 \] But marginal cost is \( 4Q_2 \), so total cost is \( 2Q_2^2 \): \[ \pi_2 = 100Q_2 - 2Q_1Q_2 - 2Q_2^2 - 2Q_2^2 = 100Q_2 - 2Q_1Q_2 - 4Q_2^2 \] --- ## 2. **Reaction Functions** ### Firm 1's Best Response Take derivative w.r.t. \( Q_1 \): \[ \frac{\partial \pi_1}{\partial Q_1} = 100 - 6Q_1 - 2Q_2 = 0 \] \[ 6Q_1 = 100 - 2Q_2 \] \[ Q_1 = \frac{100}{6} - \frac{1}{3}Q_2 \] \[ Q_1 = 16.67 - 0.33Q_2 \] --- ### Firm 2's Best Response \[ \frac{\partial \pi_2}{\partial Q_2} = 100 - 2Q_1 - 8Q_2 = 0 \] \[ 8Q_2 = 100 - 2Q_1 \] \[ Q_2 = \frac{100}{8} - \frac{1}{4}Q_1 \] \[ Q_2 = 12.5 - 0.25Q_1 \] --- ## 3. **Identify the Correct Pair** So, the correct pair of reaction functions is: \[ \boxed{ \begin{align*} Q_1 &= 16.67 - 0.33 Q_2 \\ Q_2 &= 12.5 - 0.25 Q_1 \end{align*} } \] Looking at your options, the correct pair is: > Q1 = 16.7 - 0.33 Q2 > Q2 = 12.5 - 0.25 Q1 --- ### **Summary Table** | Option | Q1 | Q2 | **Is this correct?** | |--------|----------------------------|---------------------------|----------------------| | 1 | 16 - 0.33 Q2 | 16 - 0.33 Q1 | No | | 2 | 16 - 0.5 Q2 | 16 - 0.5 Q1 | No | | 3 | 24.6 - 0.25 Q2 | 24.6 - 0.25 Q1 | No | | 4 | **16.7 - 0.33 Q2** | **12.5 - 0.25 Q1** | **Yes** | --- ## **Final Answer** The correct reaction functions are: \[ \boxed{ Q_1 = 16.7 - 0.33 Q_2 \\ Q_2 = 12.5 - 0.25 Q_1 } \]