# Step-by-Step Solution Let's go through each part: --- ## **1. Draw a Graph of Marginal Costs for Each Firm** ### **Marginal Cost Equations** - For Firm A: \( MC_a = 900 - 15X_a \) - For Firm B: \( MC_b = 200 - 25X_b \) Where \( X \) is the pollution level. ### **To Plot:** - **X-axis:** Pollution level (\( X \)) - **Y-axis:** Marginal Cost (MC) #### **Firm A** - When \( X_a = \): \( MC_a = 900 \) - When \( MC_a = \): \( 900 - 15X_a = \implies X_a = 60 \) #### **Firm B** - When \( X_b = \): \( MC_b = 200 \) - When \( MC_b = \): \( 200 - 25X_b = \implies X_b = 80 \) **Graph:** - Firm A: Line from (, 900) to (60, ) - Firm B: Line from (, 200) to (80, ) --- ## **2. No Regulation Against Pollution** - **Firms will not abate (reduce) pollution** because it costs money. - **Each firm's MC = at max pollution.** ### **Firm A:** - \( MC_a = \implies 900 - 15X_a = \implies X_a = 60 \) ### **Firm B:** - \( MC_b = \implies 200 - 25X_b = \implies X_b = 80 \) ### **Total Pollution:** \[ X_{total} = X_a + X_b = 60 + 80 = 140 \] --- ## **3. Council Reduces Total Pollution by 40% and Assigns Equal Pollution Levels** ### **Step 1: Find Target Pollution Level** - Original total: 140 - 40% reduction: \( 140 \times .4 = 56 \) - New total: \( 140 - 56 = 84 \) ### **Step 2: Assign Equal Pollution** - Each firm: \( \frac{84}{2} = 42 \) ### **Step 3: Pollution per Firm** - \( X_a = 42, X_b = 42 \) - Total: \( X_{total} = 84 \) ### **Step 4: Is this cost-effective?** - Check marginal costs at \( X_a = X_b = 42 \): - \( MC_a = 900 - 15 \times 42 = 900 - 630 = 270 \) - \( MC_b = 200 - 25 \times 42 = 200 - 105 = 950 \) - **Not cost-effective:** Marginal costs are not equal, so total abatement cost is not minimized. --- ## **4. Cost-Effectiveness Rule** Set marginal costs equal for both firms to minimize total cost: \[ MC_a = MC_b \] \[ 900 - 15X_a = 200 - 25X_b \] And total pollution is 84: \[ X_a + X_b = 84 \] ### **Solve the System:** 1. \( 900 - 15X_a = 200 - 25X_b \implies 25X_b - 15X_a = 110 \) 2. \( X_a + X_b = 84 \implies X_b = 84 - X_a \) **Substitute \( X_b = 84 - X_a \) in the first equation:** \[ 25(84 - X_a) - 15X_a = 110 \] \[ 210 - 25X_a - 15X_a = 110 \] \[ 210 - 40X_a = 110 \] \[ 40X_a = 210 - 110 = 100 \] \[ X_a = \frac{100}{40} = 25 \] \[ X_b = 84 - 25 = 59 \] ### **Final Pollution Levels:** - \( X_a = 25 \) - \( X_b = 59 \) - \( X_{total} = 84 \) --- ## **Summary Table** | Scenario | \(X_a\) | \(X_b\) | \(X_{total}\) | Cost-Effective? | |--------------------------|---------|---------|---------------|-----------------| | No regulation | 60 | 80 | 140 | No | | Equal allocation | 42 | 42 | 84 | No | | Cost-effective allocation| 25 | 59 | 84 | Yes | --- ### **Explanation** - **Cost-effectiveness**: Achieved when marginal abatement costs are equal. Otherwise, total costs are not minimized. - **Equal allocation**: Not cost-effective, as marginal costs differ. - **Cost-effectiveness rule**: Ensures least-cost pollution reduction. --- Let me know if you need the graph or further explanation for any part!

# Microeconomic Analysis of Pollution Control This analysis focuses on the marginal costs of pollution reduction for two firms in a hypothetical setting. We will explore the impact of regulation on pollution levels, cost-effectiveness, and the implications of different regulatory approaches. --- ## **1. Graph of Marginal Costs for Each Firm** ### **Marginal Cost Equations** - **Firm A:** \( MC_a = 900 - 15X_a \) - **Firm B:** \( MC_b = 2000 - 25X_b \) ### **To Plot the Graph:** - **X-axis:** Pollution level (\( X \)) - **Y-axis:** Marginal Cost (MC) #### **Key Points for Plotting:** - **For Firm A:** - When \( X_a = 0 \): \( MC_a = 900 \) - When \( MC_a = 0 \): \( 900 - 15X_a = 0 \implies X_a = 60 \) - **For Firm B:** - When \( X_b = 0 \): \( MC_b = 2000 \) - When \( MC_b = 0 \): \( 2000 - 25X_b = 0 \implies X_b = 80 \) **Graph Construction:** - Plot \( MC_a \) from (0, 900) to (60, 0). - Plot \( MC_b \) from (0, 2000) to (80, 0). --- ## **2. Pollution Levels Without Regulation** ### **Assumption:** - Firms will not reduce pollution since there are no abatement costs. ### **Pollution Levels:** - **Firm A:** - \( X_a = 60 \) - **Firm B:** - \( X_b = 80 \) ### **Total Pollution Level:** \[ X_{total} = X_a + X_b = 60 + 80 = 140 \] --- ## **3. Regulation: 40% Reduction in Total Pollution** ### **Step 1: Calculate Target Pollution Level** - Original total pollution: 140 - 40% reduction: \[ 140 \times 0.4 = 56 \] - New total pollution level: \[ 140 - 56 = 84 \] ### **Step 2: Assign Equal Pollution Levels** - Each firm: \[ \frac{84}{2} = 42 \] ### **Pollution Levels After Regulation:** - \( X_a = 42, \, X_b = 42 \) - Total: \[ X_{total} = 84 \] ### **Cost-Effectiveness Check** - **Marginal Costs at \( X = 42 \):** - \( MC_a = 900 - 15 \times 42 = 270 \) - \( MC_b = 2000 - 25 \times 42 = 950 \) - **Conclusion:** Not cost-effective as marginal costs are not equal. --- ## **4. Cost-Effectiveness Rule Implementation** ### **Objective:** Achieve the pollution target (84) while equalizing marginal costs. ### **Equation Setup:** 1. Set \( MC_a = MC_b \): \[ 900 - 15X_a = 2000 - 25X_b \] 2. Total pollution constraint: \[ X_a + X_b = 84 \] ### **Solving the System of Equations:** 1. Rearranging: \[ 25X_b - 15X_a = 110 \] 2. Substitute \( X_b = 84 - X_a \): \[ 25(84 - X_a) - 15X_a = 110 \] \[ 2100 - 25X_a - 15X_a = 110 \] \[ 2100 - 40X_a = 110 \] \[ 40X_a = 1990 \] \[ X_a = 49.75 \] \[ X_b = 84 - 49.75 = 34.25 \] ### **Final Pollution Levels:** - **Firm A:** \( X_a \approx 49.75 \) - **Firm B:** \( X_b \approx 34.25 \) - **Total Pollution:** \( X_{total} = 84 \) --- ## **Summary of Results** | Scenario | \(X_a\) | \(X_b\) | \(X_{total}\) | Cost-Effective? | |--------------------------|---------|---------|---------------|-----------------| | No regulation | 60 | 80 | 140 | No | | Equal allocation | 42 | 42 | 84 | No | | Cost-effectiveness rule | 49.75 | 34.25 | 84 | Yes | --- ### **Conclusion** - The effectiveness of pollution control measures can be assessed through the marginal costs associated with each firm. - Cost-effectiveness ensures that pollution reduction is achieved at the lowest overall cost, which is demonstrated through the final calculations. --- Feel free to ask for further details or clarifications on any part of this analysis!

# Marginal Cost Graph Explanation The marginal cost (MC) equations for the two firms are: - **Firm A:** \( MC_a = 900 - 15X_a \) - **Firm B:** \( MC_b = 2000 - 25X_b \) ### Plotting Points 1. **For Firm A:** - At \( X_a = 0 \), \( MC_a = 900 \). - At \( MC_a = 0 \), solving \( 900 - 15X_a = 0 \) gives \( X_a = 60 \). 2. **For Firm B:** - At \( X_b = 0 \), \( MC_b = 2000 \). - At \( MC_b = 0 \), solving \( 2000 - 25X_b = 0 \) gives \( X_b = 80 \). ### Graph Features - The graph has pollution levels (\( X \)) on the X-axis and marginal costs (MC) on the Y-axis. - **Firm A's line** descends from (0, 900) to (60, 0). - **Firm B's line** descends from (0, 2000) to (80, 0). This visual representation illustrates how each firm's marginal costs of emission reduction decline as pollution levels increase.

# No Regulation Against Pollution: Explanation In the absence of regulation, firms A and B will not reduce pollution due to the associated costs. - **Firm A's maximum pollution level:** \( X_a = 60 \) (when \( MC_a = 0 \)) - **Firm B's maximum pollution level:** \( X_b = 80 \) (when \( MC_b = 0 \)) ### Total Pollution Level \[ X_{total} = X_a + X_b = 60 + 80 = 140 \] Thus, the total pollution produced without regulation is 140 units.

# Explanation of Part 4: Cost-Effectiveness Rule ## Objective The goal of the Cost-Effectiveness Rule is to achieve the desired pollution reduction (in this case, a total of 84 units) while minimizing the total abatement costs by ensuring that the marginal costs of pollution reduction are equal for both firms. ## Process ### Setting Marginal Costs Equal To find the optimal pollution levels for both firms, we set their marginal cost equations equal to each other: \[ MC_a = MC_b \] \[ 900 - 15X_a = 2000 - 25X_b \] ### Total Pollution Constraint We also need to respect the total pollution limit: \[ X_a + X_b = 84 \] ### System of Equations 1. Rearranging the marginal cost equality gives: \[ 25X_b - 15X_a = 110 \] 2. Substitute \( X_b = 84 - X_a \) into this equation: \[ 25(84 - X_a) - 15X_a = 110 \] \[ 2100 - 25X_a - 15X_a = 110 \] \[ 2100 - 40X_a = 110 \] ### Solving for \( X_a \) Now, isolate \( X_a \): \[ 40X_a = 2100 - 110 = 1990 \] \[ X_a = \frac{1990}{40} = 49.75 \] ### Calculate \( X_b \) Using the total pollution constraint: \[ X_b = 84 - X_a = 84 - 49.75 = 34.25 \] ### Final Pollution Levels - **Firm A:** \( X_a \approx 49.75 \) - **Firm B:** \( X_b \approx 34.25 \) - **Total Pollution Level:** \( X_{total} = 84 \) ## Conclusion By applying the Cost-Effectiveness Rule, the firms reach a total pollution level of 84 units while ensuring that their marginal costs of abatement are equal, which minimizes the overall cost of pollution reduction. This method effectively balances the burden of reduction between the two firms, making it a more efficient regulatory approach compared to equal allocation or no regulation.

# Summary Table of Pollution Scenarios | Scenario | \(X_a\) (Firm A) | \(X_b\) (Firm B) | \(X_{total}\) | Cost-Effective? | |------------------------------|------------------|------------------|---------------|-----------------| | No regulation | 60 | 80 | 140 | No | | Equal allocation (40% reduction) | 42 | 42 | 84 | No | | Cost-effectiveness allocation | 49.75 | 34.25 | 84 | Yes | ### Key Insights - **No Regulation:** Firms produce at maximum pollution levels, resulting in high total emissions. - **Equal Allocation:** While it meets the reduction target, it is not cost-effective as marginal costs differ significantly. - **Cost-Effectiveness Rule:** Achieves the same pollution reduction at minimized costs by equalizing marginal costs, demonstrating a more efficient regulatory strategy.

# Final Answer Summary ## Overview This analysis examines pollution levels and costs for two firms (A and B) under different regulatory scenarios in the hypothetical region of Pastgreen. ## Key Findings ### 1. **Marginal Cost Equations** - **Firm A:** \( MC_a = 900 - 15X_a \) - **Firm B:** \( MC_b = 2000 - 25X_b \) ### 2. **Pollution Levels Without Regulation** - **Firm A Pollution Level:** \( X_a = 60 \) - **Firm B Pollution Level:** \( X_b = 80 \) - **Total Pollution Level:** \[ X_{total} = X_a + X_b = 140 \] ### 3. **Equal Pollution Allocation (40% Reduction)** - **Target Pollution Level:** \( 84 \) - **Each Firm's Pollution Level:** \( X_a = 42, X_b = 42 \) - **Total Pollution Level:** \[ X_{total} = 84 \] - **Cost-Effectiveness:** No, as marginal costs differ significantly. ### 4. **Cost-Effectiveness Rule Implementation** - **Final Pollution Levels:** - **Firm A:** \( X_a \approx 49.75 \) - **Firm B:** \( X_b \approx 34.25 \) - **Total Pollution Level:** \[ X_{total} = 84 \] - **Cost-Effectiveness:** Yes, by equalizing marginal costs. ## Summary Table | Scenario | \(X_a\) (Firm A) | \(X_b\) (Firm B) | \(X_{total}\) | Cost-Effective? | |------------------------------|------------------|------------------|---------------|-----------------| | No regulation | 60 | 80 | 140 | No | | Equal allocation (40% reduction) | 42 | 42 | 84 | No | | Cost-effectiveness allocation | 49.75 | 34.25 | 84 | Yes | ### Conclusion The Cost-Effectiveness Rule provides the most efficient method for reducing pollution, achieving the target level while minimizing overall costs by ensuring that both firms' marginal costs of abatement are equal. This structured approach highlights the importance of regulatory frameworks in environmental economics.

# Final Summary Table (Copyable) ```markdown | Scenario | \(X_a\) (Firm A) | \(X_b\) (Firm B) | \(X_{total}\) | Cost-Effective? | |------------------------------|------------------|------------------|---------------|-----------------| | No regulation | 60 | 80 | 140 | No | | Equal allocation (40% reduction) | 42 | 42 | 84 | No | | Cost-effectiveness allocation | 49.75 | 34.25 | 84 | Yes | ``` You can copy and paste this table directly into any Markdown-compatible editor!

### Final Answer Summary 1. **Marginal Cost Equations**: - Firm A: \( MC_a = 900 - 15X_a \) - Firm B: \( MC_b = 2000 - 25X_b \) 2. **Pollution Levels Without Regulation**: - Firm A Pollution Level: \( X_a = 60 \) - Firm B Pollution Level: \( X_b = 80 \) - Total Pollution Level: \( X_{total} = 140 \) 3. **Equal Pollution Allocation (40% Reduction)**: - Target Pollution Level: \( 84 \) - Each Firm's Pollution Level: \( X_a = 42, X_b = 42 \) - Total Pollution Level: \( X_{total} = 84 \) - Cost-Effectiveness: No (marginal costs differ). 4. **Cost-Effectiveness Rule Implementation**: - Final Pollution Levels: - Firm A: \( X_a \approx 49.75 \) - Firm B: \( X_b \approx 34.25 \) - Total Pollution Level: \( X_{total} = 84 \) - Cost-Effectiveness: Yes (marginal costs equalized). ### Conclusion The Cost-Effectiveness Rule is the most efficient method for reducing pollution, achieving the target while minimizing costs.

### Final Answer Summary 1. **Marginal Cost Equations**: - Firm A: \( MC_a = 900 - 15X_a \) - Firm B: \( MC_b = 2000 - 25X_b \) 2. **Pollution Levels Without Regulation**: - Firm A Pollution Level: \( X_a = 60 \) - Firm B Pollution Level: \( X_b = 80 \) - Total Pollution Level: \[ X_{total} = X_a + X_b = 140 \] 3. **Equal Pollution Allocation (40% Reduction)**: - Target Pollution Level: \( 84 \) - Each Firm's Pollution Level: \( X_a = 42, X_b = 42 \) - Total Pollution Level: \( X_{total} = 84 \) - Cost-Effectiveness: No (marginal costs differ). 4. **Cost-Effectiveness Rule Implementation**: - Final Pollution Levels: - Firm A: \( X_a \approx 49.75 \) - Firm B: \( X_b \approx 34.25 \) - Total Pollution Level: \( X_{total} = 84 \) - Cost-Effectiveness: Yes (marginal costs equalized). ### Conclusion The Cost-Effectiveness Rule provides the most efficient method for reducing pollution, achieving the target while minimizing costs by ensuring that both firms' marginal costs of abatement are equal.

# Graph of Marginal Costs for Firms A and B To visualize the marginal costs of pollution reduction for both firms, we will plot the equations given: ### Marginal Cost Equations - **Firm A:** \( MC_a = 900 - 15X_a \) - **Firm B:** \( MC_b = 2000 - 25X_b \) ### Key Points for Plotting #### Firm A - **When** \( X_a = 0 \): - \( MC_a = 900 \) - **When** \( MC_a = 0 \): - \( 900 - 15X_a = 0 \implies X_a = 60 \) #### Firm B - **When** \( X_b = 0 \): - \( MC_b = 2000 \) - **When** \( MC_b = 0 \): - \( 2000 - 25X_b = 0 \implies X_b = 80 \) ### Graph Construction - **X-axis:** Pollution Level (\( X \)) - **Y-axis:** Marginal Cost (MC) ### Graph Data Points - **Firm A:** (0, 900) to (60, 0) - **Firm B:** (0, 2000) to (80, 0) ### Graph Below is a representation of the graph based on the equations and points derived: ```plaintext Y-axis (MC) | | Firm B | / | / | / | / | / | / | / | / | / | / | / | / | / | / | / | / | / | / Firm A | / | / | / | / | / |/ ------------------------------------------------- X-axis (Pollution Level) 0 10 20 30 40 50 60 70 80 ``` ## Description - **Firm A's curve** starts at a high marginal cost of 900 when no pollution is reduced and decreases to 0 at \( X_a = 60 \). - **Firm B's curve** starts at 2000 and decreases to 0 at \( X_b = 80 \). ### Conclusion The graph visually demonstrates how marginal costs of emission reduction vary with pollution levels for both firms, indicating the cost implications of pollution regulation strategies. If you need a digital graph or further modifications, please let me know!