Let's solve each part step by step: --- ## **Part (a): Optimal Markups and Prices** **Given:** - Elasticity for Group 1 (\(E_1\)) = −4 - Elasticity for Group 2 (\(E_2\)) = −6 - Marginal Cost (MC) = $50 ### **Step 1: Monopoly Pricing Rule (Inverse Elasticity Rule)** The monopoly price formula under third-degree price discrimination is: \[ P = \frac{MC}{1 + \frac{1}{E}} \] where \( E \) is the price elasticity of demand (always negative, so keep the sign). ### **Step 2: Calculate for Each Group** --- #### **Group 1** - \( E_1 = -4 \) \[ P_1 = \frac{MC}{1 + \frac{1}{E_1}} \] \[ P_1 = \frac{50}{1 + \frac{1}{-4}} \] \[ P_1 = \frac{50}{1 - 0.25} \] \[ P_1 = \frac{50}{0.75} \] \[ P_1 = 66.67 \] Markup for Group 1: \[ \text{Markup}_1 = P_1 - MC = 66.67 - 50 = 16.67 \] --- #### **Group 2** - \( E_2 = -6 \) \[ P_2 = \frac{50}{1 + \frac{1}{-6}} \] \[ P_2 = \frac{50}{1 - 0.1667} \] \[ P_2 = \frac{50}{0.8333} \] \[ P_2 = 60.00 \] Markup for Group 2: \[ \text{Markup}_2 = P_2 - MC = 60.00 - 50 = 10.00 \] --- ### **Summary Table** | Group | Elasticity | Price (\$) | Markup (\$) | |---------|------------|------------|-------------| | Group 1 | −4 | 66.67 | 16.67 | | Group 2 | −6 | 60.00 | 10.00 | --- ### **Final Answers for Part (a):** - **Markup for Group 1:** $16.67 - **Price for Group 1:** $66.67 - **Markup for Group 2:** $10.00 - **Price for Group 2:** $60.00 --- ## **Part (b): Conditions for Profit-Enhancing Price Discrimination** Let's analyze each statement: --- ### **(i) At least one group has elasticity of demand less than one in absolute value.** - **False.** - For price discrimination to be profitable, both groups must have elasticities (in absolute value) greater than one, i.e., demand must be elastic in each group. Otherwise, raising price would reduce total revenue. --- ### **(ii) We are able to prevent resale between the groups.** - **True.** - If consumers can resell the product between groups, arbitrage would eliminate the price difference, and price discrimination wouldn't be possible. --- ### **(iii) At least one group has elasticity of demand greater than 1 in absolute value.** - **False.** - Both groups must have elasticities greater than 1 in absolute value (i.e., demand is elastic). Having only one group with elastic demand is not sufficient. --- ### **(iv) There are two different groups with different (and identifiable) elasticities of demand.** - **True.** - The monopolist must be able to segment the market into groups with different price sensitivities for price discrimination to be profitable. --- ### **Final Answers for Part (b):** - **(ii) We are able to prevent resale between the groups.** - **(iv) There are two different groups with different (and identifiable) elasticities of demand.** --- ### **Reasoning Recap:** - (i) **False**: Both groups need elastic demand, not just one inelastic. - (ii) **True**: Preventing resale is essential. - (iii) **False**: Both, not just one, must have elastic demand. - (iv) **True**: Different, identifiable groups are required. --- ## **Summary Table** | Statement | True/False | Reason | |-----------|------------|--------| | (i) | False | Need both groups to be elastic (|E| > 1) for profitable discrimination. | | (ii) | True | Preventing resale is required for price discrimination. | | (iii) | False | Both groups must be elastic, not just one. | | (iv) | True | Must have different, identifiable groups for discrimination. | --- ## **Boxed Final Answers** ### **Part (a):** - **Markup for Group 1:** $16.67 - **Price for Group 1:** $66.67 - **Markup for Group 2:** $10.00 - **Price for Group 2:** $60.00 ### **Part (b):** - **(ii) and (iv) only are correct.** - (i) and (iii) are **false** as explained above.