A firm practices third-degree price discrimination by charging different prices to two distinct consumer groups based on their willingness to pay. The firm faces the following demand functions: Group 1: Q1=50−2P1  Group 2: Q2=80−4P2   The firm's marginal cost is constant at $10. What are the optimal quantities for each group if the firm maximizes its profit? 

12. To find the optimal quantities for each group when a firm practices third-degree price discrimination and maximizes its profit, we need to follow these steps: Step 1: Determine the inverse demand functions for each group. · Group 1: Q1=50−2P1 2P1=50−Q1 P1=25−0.5Q1 · Group 2: Q2=80−4P2 4P2=80−Q2 P2=20−0.25Q2 Step 2: Calculate the Total Revenue (TR) for each group. · Group 1: TR1=P1×Q1 =(25−0.5Q1)Q1 =25Q1−0.5Q12 · Group 2: TR2=P2×Q2 =(20−0.25Q2)Q2 =20Q2−0.25Q22 Step 3: Calculate the Marginal Revenue (MR) for each group by taking the derivative of TR with respect to Q. · Group 1: MR1= d(TR1)/ dQ1=25−Q1 · Group 2: MR2= d(TR2)/ dQ2=20−0.5Q2 Step 4: Set Marginal Revenue (MR) equal to Marginal Cost (MC) for each group to find the optimal quantity for each group. The firm's marginal cost (MC) is constant at $10. · For Group 1: MR1=MC 25−Q1=10 Q1=25−10 Q1=15 · For Group 2: MR2=MC 20−0.5Q2=10 0.5Q2=20−10 0.5Q2=10 Q2=10/0.5 Q2=20 Step 5: Calculate the optimal price for each group using the inverse demand functions. · For Group 1: P1=25−0.5Q1 P1=25−0.5(15) P1=25−7.5 P1=17.5 · For Group 2: P2=20−0.25Q2 P2=20−0.25(20) P2=20−5 P2=15 Summary of Optimal Quantities and Prices: · Group 1: o Optimal Quantity (Q1): 15 units o Optimal Price (P1): $17.50 · Group 2: o Optimal Quantity (Q2): 20 units o Optimal Price (P2): $15.00 Therefore, the optimal quantities for each group if the firm maximizes its profit are 15 units for Group 1 and 20 units for Group 2.