Discounting & the Ramsey Rule

Why the discount rate matters

The factor-of-20 problem

Remember: Weitzman called choosing the SDR for long-term CBA "one of the most critical problems in all of economics." Stern (SDR ~1.4%): invest 250 CHF today to avoid 1000 CHF of damage in 100 yrs. Nordhaus (SDR 4.5%): invest just 12 CHF. Same problem, ~20× different policy. With $\delta = 1\%$ (the Swiss and German SCC guidance value), the well-being of a person born in 100 years is worth only ~1/3 of the same well-being today.

The Simple Ramsey Rule

The formula

$$\mathrm{SDR} = r = \delta + \eta\, g$$

• $\delta$ = PRTP (Pure Rate of Time Preference) — how much we discriminate against the future just because it's the future (discounting future utilities).
• $\eta$ = elasticity of marginal utility = intergenerational inequality aversion.
• $g$ = expected per-capita real consumption growth.

Why: Operationalises the SDR from primitive preferences plus growth. Two ways to set it: a normative route (pick $\delta$, $\eta$, $g$) or a descriptive route (calibrate to market interest rates — e.g. ~100-year government bonds, or compare freehold vs. leasehold land via hedonic analysis to back out very-long-run rates, though these are special, non-representative assets).

Remember: The $\eta g$ term says: discount future consumption if the future will be richer. With $g \approx 2\%$, the generation in 100 yrs is ~7× richer; an $\eta$ of 1 (or 2) weights an extra future dollar by only ~1/7 (or ~1/50). UK guidance: similar magnitude to Switzerland but split into ~0.5% genuine utility discounting + ~0.5% explicit extinction risk. Lower SDR ⇒ higher SCC ⇒ more aggressive climate action.

Nordhaus vs. Stern in DICE

Parameter comparison

Remember:

Same model — they disagree about ethics. In Nordhaus's optimal path, industrial emissions keep rising to mid-century then decline but stay positive at 2100; SCC starts a bit above $30 and grows with the economy.

Expert survey through updated DICE (Drupp/Hänsel/expert study)

Remember: Each grey dot = one published expert's recommended ($\delta$, $\eta$); the median expert (blue) sits at smaller values on both dimensions than Nordhaus. Running views through an updated DICE (newer climate science + damage estimates): Nordhaus's own view now gives ~$90 in 2020 (up from ~$40) and ~2 °C (not 3.5 °C); the median-expert view ≈ $200+, landing below 1.5 °C but requiring carbon prices ~$200/tCO₂ now rising to ~$400 by 2100. Asking philosophers instead: they prefer a normative approach and more egalitarian values → lower PRTP (raises SCC) and higher inequality aversion (lowers SCC, since the future is assumed richer) — the two effects roughly cancel, so median-economist and median-philosopher SCC paths are virtually indistinguishable. Caveat (Wagner et al.): why ask only economists about ethics? Ideally a public vote.

The simple rule is rejected

Remember: For >80% of experts the simple Ramsey Rule does not hold. When you ask experts for the SDR directly vs. imputing it from their $\delta$, $\eta$, $g$, the directly-stated SDR is >1 percentage point lower — so governments using the simple rule discount the future more than experts think they should (Drupp et al. 2018). Four reasons for deviation follow.

Extension 1 — Limited substitutability of non-market goods (RPC)

Two arguments, two goods

What: Standard Ramsey is set in one comprehensive consumption good, but calibrations only use market consumption and ignore non-market goods (clean environment, ecosystem services) that also give utility. So utility is $u(C, E)$.

Why: Two ways to handle it: (a) differentiated / dual discount rates — one for market goods, one for environmental goods; or (b) keep a single discount rate but adjust the future relative (shadow) price of non-market goods first, building a true comprehensive-consumption good.

Dual discount rates

$$r_C = \delta + \eta_{CC}\,g_C + \eta_{CE}\,g_E, \qquad r_E = \delta + \eta_{EE}\,g_E + \eta_{EC}\,g_C$$

The cross-good elasticity terms capture how marginal utility of one good changes as the other good's availability changes. If $g_E < 0$ (environment shrinking), $r_C$ can fall below the simple-Ramsey value.

Relative Price Change (RPC) via CES

What: Use a CES (Constant Elasticity of Substitution) utility function:

$$U = \big(\alpha C^{\rho} + (1-\alpha) E^{\rho}\big)^{1/\rho}, \quad \sigma = \tfrac{1}{1-\rho}$$

The shadow price of $E$ (in market-good units) = the marginal rate of substitution $U_E/U_C$. Its growth over time is the RPC, which for CES simplifies to:

$$\mathrm{RPC} = \xi\,(g_C - g_E), \qquad \xi = \frac{1}{\sigma} = \eta$$

Why: $\xi$ = elasticity of complementarity = the income elasticity of willingness-to-pay (WTP) for the (rationed/public) environmental good — how much WTP rises as income rises (NOT a demand elasticity — quantity is fixed). Effective SDR for environmental goods:

$$\mathrm{SDR}_E = \mathrm{SDR}_C - \mathrm{RPC} = \delta + \eta g_C - \xi(g_C - g_E)$$

Remember: Two effects bundled into RPC. (1) Relative scarcity / income effect — even if $E$ stays constant while GDP grows ~2%, a widening wedge means we value the now-scarcer environment more in monetary terms (same logic as rising VSL or value-of-travel-time as we get richer). (2) Absolute scarcity — most environmental indicators (forest area, IUCN Red List, LPI = Living Planet Index biodiversity) are declining, deepening the wedge. Drupp et al. (2024): at $\xi = 1$ (Cobb-Douglas), environmental stagnation ($g_E = 0$) → RPC ~2%/yr → PV of ecosystem services up ~130%; with LPI biodiversity decline ($g_E \approx -2.8\%$) → RPC ~4.8%/yr → PV up >1000% (~+1213%). Ignoring RPC makes a large CBA error.

CES nests weak vs. strong sustainability

Remember: $\sigma \to \infty$ ($\xi=0$) = perfect substitutes = weak sustainability (arithmetic mean). $\sigma = 1$ ($\xi=1$) = Cobb-Douglas (geometric mean). $\sigma \to 0$ ($\xi=\infty$) = Leontief / perfect complements = strong sustainability (minimum). Left of Cobb-Douglas → strong; right → weak. Problem: we don't know $\sigma$ — one elasticity decides which camp you're in.

Subsistence-threshold tweak (bridging weak ↔ strong)

Remember: Replace $E$ with $(E - \bar E)$, where $\bar E$ = environmental subsistence requirement (basic-needs threshold). Then the effective $\sigma$ is no longer constant: in a world of plenty ($E \to \infty$) you recover standard CES (substitutes); as $E$ approaches $\bar E$, preferences shift toward complementarity (strong sustainability). So one model spans both — substitutes when nature is abundant, complements when it's scarce. Drupp & Hänsel 2021: introducing limited substitutability (for environment and health) raises the SCC by >50% vs. assuming perfect substitutability.

Extension 2 — Intragenerational inequality & equity weighting

Whose growth matters: mean vs. median

What: The simple model assumes one infinitely-lived agent; really there are ~9 billion heterogeneous people. We may have inequality aversion across space, not just time.

Remember: US data — mean GDP/capita grew substantially (fueled by the very rich) while median household income stagnated (even real degrowth recently). That mean-median wedge is what makes the equity correction bite.

EDE and the inequality-adjusted Ramsey Rule

What: Atkinson's EDE = the lower, equal consumption level that society would judge equivalent (in welfare terms) to today's unequal average. Assuming a log-normal income distribution gives a clean extension:

$$\mathrm{SDR}^{IA} = \delta + \eta\, g_{\text{mean}} + \eta^2\,(g_{\text{med}} - g_{\text{mean}})$$

Remember: When median grows slower than mean, the third term is negative → a lower SDR. This adjusts the denominator (discount rate). It's real but not huge quantitatively; no government currently uses it.

Equity weighting (adjusting the numerator)

What: Climate damages fall hardest on already-poor regions (Africa, Latin America, Southern Asia hit hardest; Europe possibly benefits — two mechanisms: higher marginal damage from hotter baselines, and lower adaptive capacity at low income). Standard CBA implicitly does "$1 = 1 vote." Equity weighting corrects this with diminishing marginal utility:

$$w_i = \left(\frac{\hat y_i}{\tilde y}\right)^{-\eta}$$

where $\hat y_i$ = group income, $\tilde y$ = median income.

Remember: With $\eta = 1$, someone at half-median income gets weight 2; at $\eta = 1.4$ (the Biden-administration CBA default) → weight 2.6. Drupp/Brent et al. (Science): distribution-weighted vs. unweighted SCC ratio rises with $\eta$ — at $\eta = 1.4$ the SCC is ~7× the unweighted; at $\eta = 1$ ~3×. Swiss BAFU / Ecoplan (GIVE model), 2024: with $\delta = 1\%$, no equity weighting → 130 CHF/t; with equity weighting → ~430 CHF/t (~3×); with $\delta = 0$ + equity weighting → ~1370–1400 CHF/t. Equity weighting is used in Germany and Switzerland.

Should governments use equity-weighted SCC?

Remember: Efficiency only → no: freeze in the status-quo distribution using Negishi weights (which put lower weight on poorer countries) so you just internalise given today's distribution. Equity-minded → maybe: you do want active redistribution — but should it run through climate policy or through other instruments (international transfers)? A mix may be best. You can report two SCCs: a (lower) optimal-trajectory SCC assuming substantial transfers happen via other means, and a (higher) business-as-usual SCC to inform loss-and-damage / UN compensation.

Extension 3 — Risk, uncertainty, tipping points, fat tails

Knight's taxonomy

Remember: Certainty (one state, prob 1) → risk (many states, known objective probabilities) → uncertainty / ambiguity (many states, probabilities not objectively given) → deep uncertainty / ignorance (we don't even know the possible states). The simple Ramsey Rule and DICE are purely deterministic — they ignore all of this.

Precautionary (extended) Ramsey Rule

What: If growth is i.i.d. normal with variance $\sigma^2$:

$$\mathrm{SDR} = \delta + \eta\,\bar g - \tfrac{1}{2}\,\eta(\eta+1)\,\sigma^2$$

Remember: The last term is a prudence/precautionary effect — growth uncertainty lowers the risk-free SDR for a prudent planner (you save more to prepare for bad future states). Gollier: the global average precautionary effect could be a full percentage point. Lemoine (2021): considering growth risk + insurance effects likely raises the SCC by >20%.

Damage uncertainty and tipping points

Remember (damage uncertainty): IPCC estimates of GDP loss at 3 °C range from a few % to >20% of GDP — huge spread. Climate sensitivity (warming per CO₂ doubling) has a modal value ~3 °C but a fat tail — can't exclude 8–9 °C+.

Remember (tipping points — Dietz et al. 2021): modeled tipping points current evidence allows (permafrost-carbon feedback, Greenland ice sheet, AMOC/Gulf-Stream collapse, Amazon dieback). Permafrost feedback alone ≈ +8.4% to SCC; all tipping points combined ≈ +25% to SCC.

Weitzman's Dismal Theorem

Remember: A sufficiently fat-tailed climate-damage distribution + sufficient risk aversion ($\eta > 1$) can drive the SCC to infinity — a tail-hedge insurance approach dominates any standard CBA. Caveat (Martin & Pindyck 2015, "between Scylla and Charybdis"): even if averting each catastrophe individually passes CBA, you can't afford to avert all of them — budgets are finite, so we must balance across catastrophes. Upshot: discounted utilitarianism isn't fit for purpose here; we need alternative models/ethics.

Extension 4 — Alternative ethical approaches

Remember: Beyond standard discounted utilitarianism:

• Feasible-paths approach — instead of imposing one SWF and optimising, depict a set of feasible consumption/temperature/distribution paths and let policymakers choose.
• Tolerable windows — set physical limits that must not be exceeded (climate, biodiversity) for justice reasons (≈ planetary boundaries / doughnut), then optimise via discounted utilitarianism within the limits.
• Sustainable discounted utilitarianism — only discount future utility in periods where the future generation is richer than us.
• Maximin — maximise the worst-off.
• Safety-first — maximise $W$ subject to $P(\text{worst-off below threshold}) \le \epsilon$ (threshold ≈ subsistence requirement).
• Plus various non-consequentialist approaches. Most of these still need fleshing out; academic research remains dominated by standard discounted utilitarianism.

Synthesising the SCC

Why estimates differ + the piecemeal problem

Remember: Published SCCs span ~$30 (Nordhaus) to >$1000 depending on ethical choices — differences come from discounting, damage estimates, model structure, and whether equity weighting is used. The literature is piecemeal: "one paper, one idea" — each study adds one extension to a standard model, so no single paper gives the best overall estimate, and many extension combinations have never been studied jointly (lots of grey zeros in the interaction matrix).

Moore et al. / Drupp et al. (2024) synthesis — 3 steps

Remember: (1) Meta-analysis of ~150 studies (2000–2020), ~1800 SCC estimates — biased downward because most papers don't aim for a best estimate. (2) Survey experts for the distribution of the literature vs. the true best estimate, and what explains the wedge. (3) Train a random-forest model on the literature, then re-weight by whether experts think an extension (distribution weighting, updated Earth system, limited substitutability) should be featured and by their discounting views → synthetic SCC ≈ $283 (Drupp/Moore; ~$280–300), 95% range roughly $32–$874.

The wedge experts cite: literature uses too-low damages, too few tipping points, ignores limited substitutability, discounts future utilities too much, ignores persistent growth damages, uses too little distribution weighting / alternative ethics — partly offset because models also under-model endogenous adaptation / technological progress.

Bottom line for policy

Remember: Synthesis SCC ≈ 300 CHF/ton ≈ 75 rappen/litre of gasoline ≈ the magnitude of the Swiss mineral-oil tax — but that tax funds roads (no environmental motivation), so a Pigouvian carbon tax would come on top. Switzerland prices CO₂ for fuels/industry at ~120 CHF/ton (not transport). So despite a large uncertainty range, we should very likely do more ambitious climate policy.

Key formulas & one-line takeaways

Key formulas

Simple Ramsey Rule: $\mathrm{SDR} = \delta + \eta g$.

Dual rates: $r_C = \delta + \eta_{CC} g_C + \eta_{CE} g_E$; $r_E = \delta + \eta_{EE} g_E + \eta_{EC} g_C$.

RPC (CES): $\mathrm{RPC} = \xi(g_C - g_E)$, $\xi = 1/\sigma = \eta$; $\;\mathrm{SDR}_E = \delta + \eta g_C - \xi(g_C - g_E)$.

CES utility: $U = (\alpha C^\rho + (1-\alpha)E^\rho)^{1/\rho}$, $\sigma = 1/(1-\rho)$; subsistence tweak $E \to E - \bar E$.

Inequality-adjusted SDR: $\mathrm{SDR}^{IA} = \delta + \eta g_{\text{mean}} + \eta^2(g_{\text{med}} - g_{\text{mean}})$.

Equity weight: $w_i = (\hat y_i/\tilde y)^{-\eta}$.

Precautionary Ramsey: $\mathrm{SDR} = \delta + \eta\bar g - \tfrac12\eta(\eta+1)\sigma^2$.

One-line takeaways

• $\mathrm{SDR} = \delta + \eta g$ — lower SDR ⇒ higher SCC ⇒ more action; the rate alone swings policy ~20×.
• Nordhaus ($\delta$ 1.5%, $\eta$ 1.45) → 3.5 °C, SCC $36; Stern ($\delta$ 0.1%, $\eta$ 1) → 2 °C, SCC $300 — pure ethics.
• >80% of experts reject the simple Ramsey Rule; their imputed SDR is ~1.2 pp lower (Drupp et al. 2018).
• RPC: falling/scarcer environment ⇒ rising relative price ⇒ lower effective SDR for env goods (Drupp et al. 2024: +130% to >+1200% on ecosystem-service PV).
• CES nests weak ↔ strong sustainability; a subsistence threshold $(E-\bar E)$ bridges them; limited substitutability raises SCC >50%.
• Equity weighting adjusts the numerator (damages to the poor count more): Swiss BAFU 130 → 430 → 1370 CHF/t; use Negishi weights to freeze distribution if you only want efficiency.
• Uncertainty: precautionary term lowers SDR (~1 pp, Gollier); Lemoine +20% SCC; Dietz et al. tipping points +25% SCC; Weitzman's dismal theorem can send SCC → ∞ ($\eta > 1$, fat tails), tempered by Martin & Pindyck.
• Alternatives: tolerable windows, sustainable discounted utilitarianism, maximin, safety-first.
• Literature is piecemeal and biased down; synthesis SCC ≈ $283 (~$32–$874) ≈ 300 CHF/t ⇒ do more.