# Net Benefits of a Pollution-Control Program Suppose: - **Annual cost**: $80 (each year, forever) - **Annual benefit**: $50 (years 1–50), $150 ( 51 onward) - **Time horizon**: Perpetuity - **Discount rates**: 4% and 2% We will calculate the **present value (PV) of costs and benefits**, then compute the **net present value (NPV)** for each discount rate. --- ## 1. Present Value Calculations ### Present Value of Costs Costs are constant at $80 per year, forever: \[ PV_{\text{cost}} = \frac{80}{r} \] where \( r \) is the discount rate. --- ### Present Value of Benefits Benefits change after 50 years, so calculate in two parts: - **Years 1–50:** $50 per year - **Year 51 onward:** $150 per year #### PV of Years 1–50 (\( PV_{B1} \)): This is a 50-year ordinary annuity: \[ PV_{B1} = 50 \times \frac{1 - (1 + r)^{-50}}{r} \] #### PV of Years 51 Onward (\( PV_{B2} \)): From year 51, $150 per year in perpetuity. The present value at year 50 is: \[ PV_{B2,50} = \frac{150}{r} \] To bring this back to present (year ): \[ PV_{B2} = PV_{B2,50} \times (1 + r)^{-50} \] --- ## 2. Calculating Net Present Value (NPV) \[ NPV = PV_{\text{benefits}} - PV_{\text{costs}} \] where \[ PV_{\text{benefits}} = PV_{B1} + PV_{B2} \] --- ## 3. Calculations for Discount Rates ### At 4% Discount Rate (\( r = .04 \)) #### PV of Costs \[ PV_{\text{cost}} = \frac{80}{.04} = \$2{,}000 \] #### PV of Benefits - \( (1 + .04)^{-50} \approx .1407 \) \[ PV_{B1} = 50 \times \frac{1 - .1407}{.04} = 50 \times \frac{.8593}{.04} = 50 \times 21.48 = \$1{,}074 \] \[ PV_{B2,50} = \frac{150}{.04} = \$3{,}750 \] \[ PV_{B2} = 3{,}750 \times .1407 = \$527.6 \] **Total PV of Benefits:** \( 1,074 + 527.6 = \$1{,}601.6 \) \[ NPV_{4\%} = 1{,}601.6 - 2{,}000 = -\$398.4 \] --- ### At 2% Discount Rate (\( r = .02 \)) #### PV of Costs \[ PV_{\text{cost}} = \frac{80}{.02} = \$4{,}000 \] #### PV of Benefits - \( (1 + .02)^{-50} \approx .3642 \) \[ PV_{B1} = 50 \times \frac{1 - .3642}{.02} = 50 \times \frac{.6358}{.02} = 50 \times 31.79 = \$1{,}589.5 \] \[ PV_{B2,50} = \frac{150}{.02} = \$7{,}500 \] \[ PV_{B2} = 7{,}500 \times .3642 = \$2{,}731.5 \] **Total PV of Benefits:** \( 1,589.5 + 2,731.5 = \$4,321 \) \[ NPV_{2\%} = 4,321 - 4,000 = \$321 \] --- ## 4. Summary Table | Discount Rate | PV of Benefits | PV of Costs | Net Present Value | |---------------|----------------|-------------|-------------------| | 4% | \$1,601.6 | \$2,000 | -\$398.4 | | 2% | \$4,321 | \$4,000 | \$321 | --- ## 5. Commentary - At **4%** discount rate, **net benefits are negative**: the program is not justified on economic grounds. - At **2%** discount rate, **net benefits are positive**: the program is justified. - **Lower discount rates** place more value on future benefits, especially those occurring after 50 years. This can flip the decision from negative to positive net benefits, highlighting the importance of the discount rate in policy evaluation for long-term projects. --- ## 6. Visual Summary  *Alt text: Timeline showing $80/year costs forever, $50/year benefits for 50 years, then $150/year benefits thereafter.* --- **Conclusion:** The net benefits of the pollution-control program are highly sensitive to the discount rate, primarily because the bulk of benefits occur in the distant future. Lower discount rates make long-term projects more attractive.

# Understanding Present Value Calculations and Discount Rates ## Present Value Calculations Present Value (PV) allows us to assess the value of future cash flows in today's terms. It is crucial in financial decision-making, especially for long-term projects like an environmental pollution-control program. ### Components of Present Value 1. **Present Value of Costs**: - Costs are constant and calculated as follows: \[ PV_{\text{cost}} = \frac{C}{r} \] where \( C \) is the annual cost and \( r \) is the discount rate. 2. **Present Value of Benefits**: - Benefits vary over time, requiring separate calculations: - **Years 1–50**: Ordinary annuity: \[ PV_{B1} = n \times \frac{1 - (1 + r)^{-n}}{r} \] where \( n \) is the number of years. - **After Year 50**: Perpetuity: \[ PV_{B2,50} = \frac{B}{r} \] The present value at year 0 is then adjusted to present value terms: \[ PV_{B2} = PV_{B2,50} \times (1 + r)^{-n} \] ## Calculating Net Present Value (NPV) Net Present Value (NPV) quantifies the difference between the present value of benefits and costs: \[ NPV = PV_{\text{benefits}} - PV_{\text{costs}} \] ## Discount Rates The discount rate reflects the time value of money. Different rates can significantly influence NPV results, particularly for long-term benefits. ### Impact of Discount Rates 1. **Higher Discount Rate (e.g., 4%)**: - Future benefits are discounted more heavily, making them less valuable today. - Often results in negative NPV when costs exceed discounted benefits. 2. **Lower Discount Rate (e.g., 2%)**: - Future benefits retain more value, making the project more appealing. - Can lead to positive NPV if benefits significantly outweigh costs over time. ## Conclusion Understanding present value calculations and the impact of discount rates is essential for evaluating projects with long-term benefits. The choice of discount rate can drastically change the economic justification for such projects.