| Rate | 0 | 10 | 20 | 30 | 50 | 75 | 100 |
|---|---|---|---|---|---|---|---|
| 1% | 100 | 90.53 | 81.95 | 74.19 | 60.80 | 47.41 | 36.97 |
| 2% | 100 | 82.03 | 67.30 | 55.21 | 37.15 | 22.65 | 13.80 |
| 3% | 100 | 74.41 | 55.37 | 41.20 | 22.81 | 10.89 | 5.20 |
| 4% | 100 | 67.56 | 45.64 | 30.83 | 14.07 | 5.28 | 1.98 |
| 5% | 100 | 61.39 | 37.69 | 23.14 | 8.72 | 2.58 | 0.76 |
| 6% | 100 | 55.84 | 31.18 | 17.41 | 5.43 | 1.26 | 0.29 |
| 7% | 100 | 50.83 | 25.84 | 13.14 | 3.39 | 0.63 | 0.12 |
| 8% | 100 | 46.32 | 21.45 | 9.94 | 2.13 | 0.31 | 0.05 |
| 9% | 100 | 42.24 | 17.84 | 7.54 | 1.34 | 0.16 | 0.02 |
| 10% | 100 | 38.55 | 14.86 | 5.73 | 0.85 | 0.08 | 0.01 |
Lecture 9
Balancing the Present and Future: The Discount Rate
October 24, 2025
The Discount Rate
Why Discount the Future?
- In most environmental policies, costs/benefits often occur at different times
- Converting everything to real (inflation-adjusted) dollars handles inflation—but not time preference
- People (and societies) often prefer benefits now (investment opportunity, uncertainty, impatience)
- Therefore we apply discounting to compare future and present values on a common basis
Note
- Idea: $100 today can be worth the same as $200 ten years from now at an appropriate discount rate.
Present Value (PV) Formula
Let \(X_t\) be a cost/benefit received in \(t\) years and \(r\) the discount rate:
\[ \text{PV}(X_t) \;=\; \frac{X_t}{(1+r)^{\,t}} \]
- Further in time (\(t\uparrow\)) → PV falls
- Higher discount rate (\(r\uparrow\)) → PV falls
Tip
- Examples (for \(X_t = \$100\))
- \(r=3\%, t=10 \Rightarrow \text{PV}=\$74.41\)
- \(r=7\%, t=10 \Rightarrow \text{PV}=\$50.83\)
- \(r=3\%, t=10 \Rightarrow \text{PV}=\$74.41\)
How Fast Does PV Decline?
- High \(r\) greatly devalues distant impacts
- \(r=7\%, t=50\Rightarrow \text{PV}=\$3.39\) for $100
- \(r=10\%, t=50\Rightarrow \text{PV}=\$0.85\)
- \(r=7\%, t=50\Rightarrow \text{PV}=\$3.39\) for $100
- Small changes in \(r\) matter a lot over long horizons
- After 100 years, PV at 1% is about seven times greater than the PV at 3%
PV of $100 by Discount Rates and Years
PV of $500 Billion — Comparing Discount Rates
- Question
- Discount rates play a critical role in evaluating long-term issues such as climate change.
- Suppose a climate policy implemented today will reduce future damages by $500 billion, occurring 50 years from now.
- Two present values of this damage reduction are shown below.
- Which value corresponds to a 2% discount rate, and which corresponds to a 5% discount rate?
- Discount rates play a critical role in evaluating long-term issues such as climate change.
- $185,763,941,063
- $ 43,601,863,486
- 💡 Why it matters: Even small differences in the discount rate can dramatically change how much we value the future — and therefore, how much we act today.
💡 Quick Detour: The Value of Manhattan Island
- The funds used to purchase Manhattan Island for $24 in 1626!
💡 Quick Detour: The Rule of 72
A simple way to estimate how long it takes for a value to double given a fixed annual growth (or interest) rate.
\[ \text{Years to Double} \approx \frac{72}{r} \]
where \(r\) is the annual percentage rate (not in decimal form).
- The higher the rate of return, the faster the doubling.
- Works well for rates between 2% and 15%.
- Useful for understanding economic/financial growth, inflation, or discounting.
- Example: If the economy grows at 3% per year, its size doubles in about 24 years.
Choosing the Discount Rate: Two Approaches
Why Discount the Future?
The discount rate answers a fundamental question:
At what rate should society discount future benefits so that sacrificing one unit of consumption today is fairly balanced by gains to future well-being?
- This rate reflects how we value the future relative to the present, and it can be determined in two main ways:
- Through observed market outcomes → the market-based (descriptive) approach
- Through ethical principles about intergenerational welfare → the social (prescriptive) approach
Linking \(\color{#0072b2}{\eta}\), Diminishing Utility, and Fairness
- Higher \(\color{#0072b2}{\eta}\) → Faster Diminishing Marginal Utility
- The extra satisfaction from each additional dollar drops faster as consumption rises.
- Higher \(\color{#0072b2}{\eta}\) → Stronger Concern for Equality
- Because extra consumption benefits the poor more than the rich, a higher \(\color{#0072b2}{\eta}\) means society places greater weight on poorer or earlier generations when choosing how to allocate resources (\(C_{t}\) and \(I_{t}\)) over time.
⚖️ Dynamic Efficiency and Resource Allocation
- Resource Allocation Across Generations: In each period \(t\), total income (\(Y_t\)) can be used for consumption or investment:
- Consumption (\(C_t\)): provides immediate welfare for the current generation
- Investment (\(I_t\)): builds future productive capacity, benefiting future generations
- Consumption (\(C_t\)): provides immediate welfare for the current generation
- To maximize well-being across generations, dynamic efficiency seeks a path that balances present and future welfare:
- How much of today’s output should we consume now versus invest for the future (e.g., in renewable energy or infrastructure)?
- The \(\text{SDR}_{t} = \color{#d55e00}{\rho} + \color{#0072b2}{\eta}\cdot \color{#009e73}{g}_{t}\) provides the answer.
- How much of today’s output should we consume now versus invest for the future (e.g., in renewable energy or infrastructure)?
Discounting Climate Change — Two Lenses
| Feature | Market-based | Ethical |
|---|---|---|
| Approach | How markets actually value the future | How we ought to value the future |
| \(\rho\) | \(1\%\) | \(0.1\%\) |
| \(\color{#0072b2}{\eta}\) | \(1.5\) | \(1.0\) |
| \(\color{#009e73}{g}\) | \(2.25\%\) | \(1.3\%\) |
| Resulting rate | \(\approx 4.4\%\) | \(\approx 1.4\%\) |
| Policy implication | Supports gradual mitigation — high SDR discounts future damages more heavily, making near-term action seem less urgent | Strong case for immediate, large cuts — low SDR gives high present value to avoiding long-term damages |
1️⃣ Market-Based (Descriptive) Approach
Note
The Opportunity Cost of Capital is the return forgone when spending $1 today on a project, instead of investing that same $1 in the best available alternative (for example, in private capital markets or government bonds).
- Set SDR \(\approx\) low-risk market return (e.g., government bonds or average real return on capital)
- Logic: Funds spent today could have been invested — the opportunity cost of capital determines the appropriate SDR
- Idea: Society’s time preference between present and future consumption is revealed through saving, investment, and consumption decisions, as reflected in observed market returns and long-run economic growth.
- Caveat: Market rates change over time
- Very low in the 2020s; very high in the early 1980s
1️⃣ Market-Based (Descriptive) Approach
“What do markets reveal about valuing the future?”
- Sets parameters based on observed market behavior, not ethical judgments
- \(\color{#d55e00}{\rho}\): higher (≈ 1%) — reflects society’s revealed time preference in saving and investment decisions
- \(\color{#0072b2}{\eta}\) (Higher, ≈ 1.5) — means society places greater weight on poorer or earlier generations
- \(\color{#009e73}{g}\): assumes stronger long-run growth (e.g., 2.25%), consistent with historical productivity trends
- \(\color{#d55e00}{\rho}\): higher (≈ 1%) — reflects society’s revealed time preference in saving and investment decisions
- Result: Higher SDR (≈ 4.4%) → less weight on distant future impacts and supports relatively gradual, cost-efficient mitigation of GHG
2️⃣ Ethical (Prescriptive) Approach
“How should markets actually value the future?”
- Parameters are chosen based on ethical reasoning about intergenerational fairness, rather than market behavior
- \(\color{#d55e00}{\rho}\): very low (≈ 0–0.1%) → future welfare is valued almost equally to present welfare
- \(\color{#0072b2}{\eta}\): low (≈ 1) → society gives less additional value to income gains for richer future generations
- \(\color{#009e73}{g}_{t}\): assumes modest long-run growth (≈ 1.3%), which reflects:
- Gradual improvements in living standards without relying on unsustainably rapid innovation or excessive resource use
- \(\color{#d55e00}{\rho}\): very low (≈ 0–0.1%) → future welfare is valued almost equally to present welfare
- Result: Low SDR (≈ 1.4%) → places greater weight on future generations and stronger justification for early climate action
📊 Policy Relevance & Ethics
- 2018 Survey of Economists on the SDR for Intergenerational Global Projects:
- Mean = 2.3%, Median = 2.0%
- 68% of economists recommend a rate between 1–3%
- Trend: Steady shift toward lower SDR since 2001, reflecting greater concern for long-term welfare
- High SDR → prioritizes the present, discounting future impacts heavily
- Low SDR → gives greater weight to future generations
- In environmental cost–benefit analysis (e.g., climate policy):
- Costs occur now, benefits arise later → a lower SDR supports stronger environmental protection
- Costs occur now, benefits arise later → a lower SDR supports stronger environmental protection
- There is no single “correct” rate — analysts should be transparent, justify their assumptions, and test results through sensitivity analysis.
Cost–Benefit Analysis
What Is Cost–Benefit Analysis (CBA)?
CBA evaluates whether the present value of benefits exceeds the present value of costs for a project or policy.
- All costs and benefits are expressed in today’s dollars using a discount rate (\(r\))
- Enables meaningful comparison of outcomes occurring at different times
- Helps allocate scarce resources efficiently across competing alternatives
CBA in Government Decision-Making
Governments use CBA to decide:
- Which projects should be funded under budget constraints
- How to balance economic, health, and environmental goals
- Whether policy benefits justify the present value of implementation costs
Example: Regulating Ozone Pollution
The U.S. Environmental Protection Agency (EPA) must decide:
Should the national standard for ground-level ozone be tightened?
- Ozone affects:
- Human health (asthma, emphysema)
- Crop and vegetation damage
- Stricter standards → higher abatement costs now, health benefits later
Example: Regulating Ozone Pollution
Hypothetical Scenario (Present Value Terms)
| Option | PV of Costs (Billion $) | PV of Benefits (Billion $) |
|---|---|---|
| Maintain 75 ppb | — | — |
| Tighten to 70 ppb | 16 | 24 |
\[ PV_{NB} = PV_B - PV_C = 24 - 16 = 8 \text{ billion} \]
\[ PV_{B/C} = \frac{PV_B}{PV_C} = \frac{24}{16} = 1.5 \]
✅ Positive PV net benefits and \(PV_{B/C} > 1\) → Economically justified.
Example: Regulating Ozone Pollution
The Real EPA Decision
- EPA lowered ozone standard from 75 → 70 ppb
- Industry: “Too costly — burdensome for economy”
- Environmentalists: “Too weak — thousands of preventable deaths remain”
Example: Regulating Ozone Pollution
EPA’s Present Value Estimates
| Standard (ppb) | PV of Costs (Billion $) | PV of Benefits (Billion $) | PV of Net Benefits (Billion $) |
|---|---|---|---|
| 70 | ~5 | 10–22 | +5–17 |
| 65 | ~15 | 18–35 | +3–20 |
| 60 | ~40 | 25–50 | –15–10 |
Note
Interpretation:
The 70 ppb standard produced positive PV net benefits.
Lower standards (65 or 60 ppb) have higher potential PV benefits, but with more uncertainty.
Steps in Conducting a PV-Based CBA
- List all costs and benefits over time
- Quantify each stream in monetary terms
- Discount each future value to its present value using rate \(r\)
- Sum PVs of costs and benefits:
\[ PV_B = \sum_t \frac{B_t}{(1 + r)^t}, \qquad PV_C = \sum_t \frac{C_t}{(1 + r)^t} \]
- Compute PV Net Benefits and PV Benefit–Cost Ratio
- Compare across policy alternatives, including a baseline (no action)
🧮 Decision Criteria (Using Present Value)
1️⃣ PV of Net Benefits
\[ PV_{NB} = PV_B - PV_C \]
- \(PV_{NB}\) (Present Value of Net Benefits) measures total welfare change in today’s dollars.
- Adds up all future benefits (\(B_t\)) and costs (\(C_t\)), converted into present value using the discount rate (\(r\)).
- If \(PV_{NB} > 0\), benefits exceed costs → policy improves total welfare and is worth undertaking.
2️⃣ PV Benefit–Cost Ratio
\[ PV_{B/C} = \frac{PV_B}{PV_C} \]
- Compares the relative size of discounted benefits and costs.
- If \(PV_{B/C} > 1\), benefits outweigh costs in present-value terms.
- Useful for comparing multiple projects when budgets are limited.
Opportunity Cost and Dynamic Efficiency
Even if \(PV_{NB} > 0\):
- Another policy might yield a higher \(PV_{NB}\)
- Always consider the opportunity cost of choosing one project over another
- Economics seeks to maximize \(PV_{NB}\) — the condition for dynamic efficiency
- Dynamic efficiency is achieved when the allocation of resources over time maximizes the present value of net benefits to society.
- Dynamic efficiency helps decide how much to use today and how much to save or invest for the future (e.g., in renewable energy, conservation, or infrastructure), since resources spent now can’t be used later.
⚖️ Distributional and Ethical Issues
- CBA aggregates all PV benefits and costs across society
→ ignores who gains and who bears the costs - Example:
- PV benefits → wealthier households (longer lifespan, better access)
- PV costs → lower-income communities (energy price hikes)
- Even with positive \(PV_{NB}\), may still be inequitable
- Efficiency alone is not sufficient — a policy can maximize total welfare (\(PV_{NB} > 0\)), while still being unfair or socially unacceptable.
⚖️ Why Do Equity Concerns Matter?
- Efficiency focuses only on the size of the total pie, not on how the pie is divided.
- Without considering equity, benefits may concentrate among the wealthy, while costs fall on vulnerable or marginalized groups.
- Because market power, wealth, and access to information often allow higher-income groups to capture more of the benefits and avoid much of the burden.
- Ignoring fairness can lead to social resistance, political backlash,
and reduced legitimacy of otherwise efficient policies.
- Equity ensures inclusiveness — that economic progress also supports
justice, opportunity, and shared well-being.
Note
- ✅ Efficiency tells us how to grow the pie; equity tells us how to share it fairly.
💡 How Can We Address Equity Concerns?
Economists can address equity concerns by:
- Applying distributional weights — give higher social value to welfare gains for lower-income groups
- Reporting disaggregated impacts — show who gains and who loses (by income, region, or demographic)
- Designing compensatory measures — use transfers or tax rebates to offset burdens on disadvantaged groups
- Embedding ethical judgment — recognize that efficiency alone does not ensure fairness or justice
Social Discount Rate (SDR)
\[ \text{SDR}_{t} = \color{#d55e00}{\rho} + \color{#0072b2}{\eta}\cdot \color{#009e73}{g}_{t} \]