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Climate as an Economic System

Volume VII — Planetary and Cosmic

THE PLATE

A timber room with a low fire and floor cushions, a wall of glass framing snow-capped mountains.
Plate VII.02The Thermometer on the Balance Sheet.Climate economics is not the study of a catastrophe. It is the study of a quantity that has a physical unit and no price, and of what happens to a balance sheet when somebody finally writes one in.

THE LETTER

You have almost certainly been handed a number for a tonne of carbon dioxide, and you have almost certainly been handed a different one by somebody else the same month. One of them was thirty-one dollars and one of them was three hundred and forty, and both were produced by serious people using published models. The natural conclusion is that the field is unsettled to the point of uselessness and that the number can therefore be chosen to suit.

That conclusion is wrong, and this chapter is about why — because the difference between those two figures is not disagreement about the climate. It is almost entirely one parameter, it is known, it is computable, and you can work it out yourself in about four minutes with the table on this page.

So here is what you are being offered. First, precisely what an integrated assessment model does: five modules, in order, each of which you could in principle audit. Second, the damage function at the heart of the most-used of them, stated with its actual coefficient and its actual empirical base, which is smaller than almost anyone outside the literature imagines. Third, the social cost of carbon with its real range across methods, and a sensitivity computed rather than asserted, so that you can see exactly how much of the range is physics and how much is ethics. Fourth, the record of carbon pricing as measured: real prices, real coverage, and the emissions effects the credible ex-post evaluations actually find — which are positive, which are smaller than advocates claim, and which are far larger than critics claim.

Then the part that is yours. Every figure in this chapter converges on one operating decision that any organisation can take this quarter without legislation, without a market, and without anybody's permission: choose the price you will use internally, write it down, and make one capital decision with it. The chapter ends with the instrument for doing that, at the level of a treasurer.

The discount rate itself — the Ramsey equation, its three terms, what each of them is a claim about — is Chapter III.05, and it is the best treatment in this edition. This chapter does not re-derive it. It does something III.05 leaves open on purpose: it shows what moving that one parameter does to the single most consequential number in environmental economics.

— The Editors

DISCOVERY

What is already working

Begin where the evidence is strongest, because the strongest evidence in climate economics is not a projection. It is a set of instruments that have been running for years, in public, with their effects measured by people who were not paid to find an effect.

Sweden has priced carbon since 1991. The tax stands at $144.6 per tonne of CO₂ equivalent as of 1 April 2025 — the second-highest direct carbon price in the world after Uruguay's $158.8 — and it covers 40 percent of Swedish emissions and raised $2,306 million in 2024. The important thing is not the level. It is that a price at that level has coexisted with three decades of growth, and that its effect has been measured cleanly. Julius Andersson's synthetic-control study in the American Economic Journal found transport CO₂ in Sweden fell 10.9 percent below its synthetic counterfactual after the reform, of which 6.3 points were attributable to the carbon tax alone. He also found something more useful than the headline: the carbon-tax elasticity of demand is roughly three times the ordinary price elasticity. A tonne priced by a government moves behaviour more than the same tonne priced by a market, because the tax is legible, announced and expected to persist.

Britain decarbonised its power sector with a top-up, not a cap. The Carbon Price Support, introduced in 2013 on top of the EU allowance price, is the most cleanly identified carbon-pricing result in the literature. Marion Leroutier's 2022 study in the Journal of Environmental Economics and Management built a synthetic Britain from other European systems and found power-sector emissions 20 to 26 percent lower per year than the counterfactual over 2013–2017 — 143 to 191 million tonnes of CO₂ avoided in five years, which is 28.6 to 38.2 million tonnes a year. The mechanism is worth carrying: not one channel but three. Less running of existing coal plant, closure of some coal plant, and a raised probability of closure for plant already marginal under air-quality rules. A price does not have one lever. It has as many levers as there are decisions the price enters.

The European system now clears at a real price, and it works at the level of the firm. The EU Emissions Trading System covers 40 percent of EU emissions, traded at $70.4 on 1 April 2025, and raised $41,703 million in 2024 — 41.7 percent of all carbon-pricing revenue on earth. Two literatures matter here. Patrick Bayer and Michaël Aklin, in PNAS in 2020, found the system cut about 1.2 gigatonnes of CO₂ between 2008 and 2016 — 3.8 percent of EU-wide emissions, or 133.3 million tonnes a year — in a period when the price was widely dismissed as too low to matter. And Jonathan Colmer, Ralf Martin, Mirabelle Muûls and Ulrich Wagner, in the Review of Economic Studies in 2024, matched regulated French manufacturing firms against comparable unregulated ones and found emissions 14 to 16 percent lower, with no detectable contraction in output or employment and no evidence of outsourcing. The abatement came from targeted investment in emissions intensity.

And the synthesis exists. Niklas Döbbeling-Hildebrandt and colleagues, in Nature Communications in 2024, ran a machine-assisted systematic review of 80 causal ex-post evaluations covering 21 schemes and 483 effect sizes — the first time the question "does carbon pricing reduce emissions" has been answered by the whole body of evidence rather than by a favourite case. The answer: statistically significant reductions of 5 to 21 percent, or 4 to 15 percent after correcting for publication bias, with immediate reductions in 17 of the 21 schemes. That is 81 percent of the schemes ever studied.

One more thing is working, and it is methodological. The damage modules behind the current United States estimates — GIVE and DSCIM — are open source, and two independent teams reached nearly the same answer with them. Kevin Rennert and colleagues, in Nature in 2022, computed $185 per tonne at a two percent near-term rate; the Environmental Protection Agency's 2023 report, on a different assembly of the same components, computed $190. A ratio of 0.97. In a field whose reputation is for irreconcilable numbers, two routes built by different people converged to within three percent. That is what a maturing empirical science looks like from the outside, and it is worth saying before the criticism starts.

THE ARITHMETIC

What works, what does not, and where the line sits

First: what an integrated assessment model actually does.

Strip the acronym and an IAM is five modules wired in a loop, each of which hands one number to the next.

  1  socioeconomics   population and output per head, to 2300
  2  emissions        output x carbon intensity, minus abatement
  3  carbon cycle     emissions -> atmospheric concentration
  4  climate          concentration -> radiative forcing -> temperature
  5  damages          temperature -> a loss of output, in money
        then: discount that loss back to today

The social cost of carbon is one specific operation on that loop: add one tonne today, run the whole thing twice, take the difference in discounted damages. Nothing mystical. An arithmetic difference between two runs of a spreadsheet with a climate model in it.

Second: the damage function, with its real coefficient.

In DICE-2016R, the module the field is built on, module five is one line:

        D(T) = a2 · T²          a2 = 0.236 % of global income per °C²
        Ω(T) = D / (1 + D)      damages as a share of output

Run it and you get Nordhaus's published figures exactly: 2.1 percent of global income lost at 3 °C, 8.5 percent at 6 °C. That is the whole of the damage side of the most influential model in the field: one coefficient, one exponent.

Now the base under that coefficient, because it is the part that is almost never stated. The 2016 revision fitted it to a survey of 26 studies, of which 16 contained independent damage estimates and were included, and of which 9 received full weight — 34.6 percent of the survey. A 25 percent upward adjustment was then added for omitted, non-market and catastrophic damages. Peter Howard and Thomas Sterner's competing meta-analysis, which finds a substantially steeper relationship, rests on a preferred dataset of 21 observations, and excludes estimates above 4 °C on the explicit ground that they are empirically shaky and rely on extrapolation.

Third: the discount rate, which is the chapter's central number.

Chapter III.05 sets out the Ramsey equation, its three terms, and which of them are ethical claims rather than measurements. Take it as read. What III.05 leaves for here is the magnitude. Nordhaus's own Table 1 gives the 2015 social cost of carbon under four constant discount rates, same model, same damage function, same climate module, in 2010 dollars:

  discount rate on goods      SCC, $/tCO2 (2010$)
  --------------------------------------------------
  2.5 %                                     128.5
  3   %                                      79.1
  4   %                                      36.3
  5   %                                      19.7

128.5 / 19.7 = 6.52× across two and a half points — 2.12× per percentage point. The Environmental Protection Agency's 2023 estimates, on wholly different damage modules and a Ramsey rather than constant formulation, give the same shape: $120 at 2.5 percent, $190 at 2.0 percent, $340 at 1.5 percent, in 2020 dollars for a tonne emitted in 2020. 340 / 120 = 2.83× across one point; the two half-point steps are 1.58× and 1.79×, a mean of 1.68×. Expressed as an elasticity: ln(340/120) / 0.01 = 104.1, so one basis point off the discount rate raises the social cost of carbon by 1.04 percent.

Hold that beside the other lever. Replacing Nordhaus's damage function with Howard and Sterner's raises the 2015 figure three- to four-fold, from $31.2 to $93.6–$124.8; including catastrophic impacts, four- to five-fold, to $124.8–$156.0. So: the discount rate moves the answer by about 6.5× and the entire damage literature moves it by about 3× to 5×. Both are large. Only one of them is a forecast rather than an ethic, and it is not the one that gets argued about in public.

Fourth — and this is the cut — the dispute is settled inside one model, by changing the question.

Everyone knows the DICE baseline number: $31.2 a tonne in 2015. Almost nobody quotes the other row of the same table. Ask that model not what is the marginal damage along the path we are on but what does it cost to hold 2.5 °C — same damage function, same author, same page — and the answer is $184.4. A hard cap; $106.7 if the cap is a hundred-year average rather than a peak. And running the same model with the Stern Review's discounting gives $197.4.

  DICE-2016R, 2015, 2010 US$
  baseline marginal damage                  31.2
  price consistent with a 2.5 °C cap       184.4        5.91x
  Stern Review discounting                 197.4        6.33x

The famous gulf between the "low" and "high" camps is reproducible inside a single model without touching the science. One camp is pricing damage along a path; the other is pricing a constraint. They are different questions with different correct answers, and the argument between them has been conducted for twenty years as though they were the same question answered differently. If your organisation has a temperature commitment, the relevant figure was never the marginal-damage estimate at all. It is the shadow price of your own constraint, and it is roughly six times larger.

Fifth: the honest negative, stated at full strength.

The damage functions under almost every published social cost of carbon are calibrated on a narrow empirical base, and their own authors say so. Three measurements, not opinions:

  1. The coefficient's own dispersion. Nordhaus assigns the damage parameter a standard deviation of 0.118 against a mean of 0.236 — exactly half, chosen, in his words, to reflect "the great divergence today among different studies."
  2. The t statistic is a trap. The estimated coefficient has a t of −7.8 and is, as he writes, "extremely well determined" — and one sentence later, "this estimate does not reflect specification uncertainty, parameter uncertainty, or study dependence." A tight standard error on a fitted line says nothing about whether the line is the right shape.
  3. The dispersion of the output exceeds the output. In Nordhaus's own Monte Carlo over climate sensitivity, productivity growth and the damage coefficient, the standard deviation of the 2015 social cost of carbon is $31.5 against a mean of $31.2. A ratio of 1.01. The spread is larger than the estimate.

Add the structural point. The estimates are fitted mostly below 3 °C and are then evaluated at temperatures where no data exist. Howard and Sterner exclude the high-temperature estimates because extrapolation makes them unreliable; Nordhaus's survey has few of them to begin with. And the EPA states the consequence plainly in its own report: data and modelling limits "restrain the ability of SC-GHG estimates to include all physical, ecological, and economic impacts of climate change, implicitly assigning a value of zero to the omitted climate damages."

So what should be done with the number? Not ignored — that is the wrong inference and it is the one most often drawn. A quantity whose omissions are all of one sign is not an unknown. It is a bound with a known direction. Three consequences follow, and they are operational:

Sixth: where the price actually is.

As of 1 April 2025 there are 80 carbon taxes and emissions trading systems in operation, covering 28 percent of global greenhouse gas emissions — against about 5 percent in 2005 — and raising over $100 billion in 2024. That is roughly 15 gigatonnes priced out of 52. The emissions-weighted average price across covered emissions is $19 a tonne, up from just above $10 ten years earlier. Across all emissions, priced and unpriced, the global average is $5.

Set that beside the EPA's central figure and the gap is the whole chapter:

  instrument                     price    % of $190      gap $/t
  ------------------------------------------------------------------
  Uruguay CO2 tax                158.8       83.6 %         31.2
  Sweden carbon tax              144.6       76.1 %         45.4
  Switzerland carbon tax         136.0       71.6 %         54.0
  EU ETS                          70.4       37.1 %        119.6
  Canada federal OBPS             66.2       34.8 %        123.8
  UK ETS                          57.2       30.1 %        132.8
  California cap and trade        29.3       15.4 %        160.7
  China national ETS              11.8        6.2 %        178.2
  Mexico carbon tax                2.8        1.5 %        187.2
  average, covered emissions      19.0       10.0 %        171.0
  average, all emissions           5.0        2.6 %        185.0

The comparison is undeflated — the prices are nominal April 2025 and the benchmark is in 2020 dollars — which makes every gap above a floor, since carrying the benchmark forward would raise it. At the global average, the unpriced marginal damage runs at $9.62 trillion a year: (190 − 5) × 52 billion tonnes. The largest priced market on earth stands at 37.1 percent of the modelled social cost.

There is a second gap inside the first. Revenue of $100 billion across 15 gigatonnes of covered emissions is $6.67 a tonne collected — 35.1 percent of the $19 posted average. Free allocation, exemptions and output-based rebating take the rest. That is not necessarily wrong: rebating the average cost while preserving the marginal price is exactly what the competitiveness evidence supports. But it means the fiscal size of carbon pricing is about a third of its apparent size, and any plan that spends the revenue should start from $6.67, not $19.

Seventh: the honest sum.

Take the meta-analysis midpoint after publication-bias correction — 9.5 percent — and apply it to the 15 gigatonnes actually covered. That is 1.43 gigatonnes a year, 2.74 percent of global emissions. The IPCC's 1.5 °C pathway requires 43 percent below 2019 levels by 2030 — about 22.36 gigatonnes a year. Delivered against required: 6.4 percent, or one part in 15.7.

Carbon pricing works and is too small. Both halves are measured, and a chapter that gives you only one of them is selling something.

DREAM

What becomes ordinary

In the economy that has absorbed this, the price of a tonne is not a political event. It is a line in a standard cost model, the way a wage rate or a freight rate is, and it is revised on a schedule rather than in a crisis.

Every capital paper that crosses a board carries three columns where it used to carry one: the cash cost, the cost at the compliance price the asset will actually face over its life, and the cost at the house's shadow price. Nobody finds the third column ideological. It is simply the column that has been right before — the one that would have flagged the asset the firm wrote down in year six, and the one that priced the retrofit nobody could get approved in year two.

The shadow price is published. Firms state theirs the way they state their cost of capital, and analysts compare them, and a firm using a number a quarter of its peers' is asked why in the same tone it would be asked about an unusual depreciation schedule. The disclosure is not a virtue claim. It is a parameter, and parameters are comparable.

Inventory and ledger have merged. The tonnage column and the money column sit in one system, reconciled monthly, audited annually, with the same controls as revenue — because the moment a tonne has a price that governs a decision, an unaudited tonne is an unaudited cost. The question "how many tonnes did that plant emit last month" is answered as quickly as "what did that plant cost to run last month", and by the same person.

The argument about the discount rate has not been resolved and does not need to be. It has been located. Everyone involved understands that a large part of the range in the number comes from two parameters that are ethical rather than empirical, that this is a matter for the board to decide once and record, and that the deciding is a governance act with minutes, not a modelling act. A firm that has written down its own rate has converted a permanent argument into a settled policy, and it can now spend its attention on the abatement curve instead.

And the curve is where the attention goes. Firms know their own marginal abatement costs the way they know their own unit costs, tranche by tranche, and the interesting question in the room is no longer what is the price of carbon but at what price does our own next tranche clear, and how do we move that. That is an engineering question with a capital budget attached, and it is the one that actually reduces emissions.

DESIGN

The structure that gets there

Five parts, in order, and the order matters because each earns the next.

One: the inventory before the price. A price applied to an unreliable tonnage produces confident nonsense. Scope 1 and 2 first, monthly, reconciled to meters and invoices rather than to estimates, with a named owner and the same close discipline as the management accounts. This takes a quarter and it is not glamorous. Everything downstream is a multiplication by this number.

Two: the price, decided once, in the open. The board picks a shadow price and records the basis. Three defensible bases, and the choice among them is a real decision:

Three: the price enters exactly one decision at first. Not a strategy, not a report — one capital approval threshold. Every project above a size threshold is appraised twice, once on cash and once at the shadow price, and both appear in the paper. The second number is not binding in year one. It is visible, which is enough: within four quarters the firm has a file of projects whose ranking changed, and that file is the whole argument for making it binding.

Four: the revenue design, if you are designing policy rather than a firm. The evidence is specific here and it cuts against instinct. Output-based rebating — returning the average cost while preserving the marginal price — is what the firm-level evidence supports: Colmer and colleagues found 14 to 16 percent abatement with no output or employment effect under exactly that structure. Free allocation is not a loophole if it is allocated on output rather than on history. It is the design that lets the marginal signal survive politically, and the durability is worth more than the foregone revenue, because a repealed price abates nothing.

Five: the escalator and the floor. Two design features carry most of the measured effect. A price floor — Britain's top-up is the cleanest natural experiment in the literature, at 20 to 26 percent a year — and an announced escalation path, which is what makes the tax elasticity three times the ordinary price elasticity in Sweden. A price that might vanish is discounted by the people it is meant to move. Announce the path and you get the abatement before the price arrives, which is the cheapest abatement there is.

DESTINY

How it holds when nobody is pushing

Three things make a carbon price durable, and their absence is the whole failure literature.

It survives a change of government. Australia priced carbon in 2012 and repealed it in 2014. Two years is shorter than the payback on almost any abatement investment, so the price changed announcements and not assets. The durable instruments are the ones whose revenue has a constituency: over half of the $100 billion raised in 2024 was earmarked to environment, infrastructure and development spending, and earmarking is what turns a tax into a programme with defenders.

Its cap adjusts to its own errors. The European system's first phase over-allocated and the price fell to near zero. What fixed it was not enforcement but a supply mechanism that withdraws allowances when the surplus grows. A cap without an adjustment rule is a forecast pretending to be a constraint, and forecasts of demand over a decade are wrong.

It does not lean on offsets. This is where the failures are largest and best measured, so name them exactly. The European Commission's 2016 study of the Clean Development Mechanism found 85 percent of projects — and 73 percent of potential credit supply — had a low likelihood that their reductions were additional and not overestimated; only 2 percent of projects and 7 percent of supply had a high likelihood. Thales West and colleagues, in Science in 2023, examined 18 forest-conservation projects that issued 62 million credits, of which 14.6 million had already been used to offset emissions, and found roughly 6 percent represented real additional reductions. That is 3.72 million genuine tonnes among the credits issued, and 13.72 million tonnes of claimed offsetting that did not occur.

An offset that is not additional is not cheap abatement. It is an emission with a receipt, and a system that accepts them converts its own price into a fee for paperwork. The design rule that follows is blunt and it holds: offsets may fund abatement outside your boundary; they may not discharge a tonne inside it.

The fourth failure mode is the quiet one. A price below the level that changes an operating decision buys the appearance of action at the cost of the argument for the real thing. Mexico's tax is $2.8 a tonne — 1.5 percent of the modelled social cost. A quarter of firms with an internal carbon price set it below $20, which is 10.5 percent of the shadow price and below the month-to-month movement in the fuel prices they already manage. Such a price changes nothing and answers the question, which is worse than leaving the question open.

DELIGHT

What it feels like

There is a specific pleasure in the moment the two columns line up. You have had a tonnage report and a management account in the same building for years, in different systems, owned by different people, arriving on different dates. The morning they reconcile — the same period, the same boundary, the same closing discipline — something settles that had been quietly unsettled for a long time. The physical world and the financial one stop being two accounts of the same firm and become one.

Then a smaller, better pleasure: the first project whose ranking changes. Not a project anybody argued about — a routine retrofit that sat at position eleven on a list of eight-funded items, and at the shadow price sits at position three. Nobody has to be persuaded. The arithmetic simply moved it, and the person who built the model watches it move and says nothing, which is the correct response and also the most satisfying one available in corporate life.

And the argument gets quieter. The room stops holding a position on climate and starts holding a number, with a date on it and a note of who approved it. Positions are exhausting because they must be defended every time. A parameter is restful. It is written down, it is revisable on a schedule, and between revisions everybody can get on with the work.

OPERATIONALIZE THIS

At the level of finance

The instrument: a carbon liability reserve with a price-linked drawdown.

It converts an unpriced physical exposure into a funded, governed and auditable one, without waiting for a regulator and without a promise anyone has to believe.

The mechanics. Take a firm emitting 250,000 tonnes of CO₂e in scope 1, of which 60 percent falls under a compliance scheme at the EU price of $70.4. Its cash carbon cost is 250,000 × 0.60 × 70.4 = $10.56 million. At a shadow price of $190 its full exposure is 250,000 × 190 = $47.50 million. The unpriced exposure is $36.94 million a year.

Now put both against earnings. On $120 million of EBITDA, the cash cost is 8.8 percent — a line item, invisible, managed by procurement. The unpriced exposure is 30.8 percent. That is the number for the board, and the difference between the two is the entire reason the exposure is not currently managed by anybody senior.

The structure.

The counterparty. Internal first: treasury to business unit, documented in a week. Once two cycles are complete, the same structure supports an external sustainability-linked facility, where the margin ratchet is written against verified tonnes rather than against a rating. You are now presenting a mechanism with a track record rather than a commitment with a target date, and the pricing reflects it.

The number that decides it. One figure, on the front page, and it is not the exposure. It is where your own abatement curve clears.

        tonnes abatable below the shadow price
    ------------------------------------------------   and the cost of the tranche
        tonnes abatable below the market price

For the worked firm: 45,000 tonnes clear below $70.4 and 115,000 clear below $190, so the shadow price puts an incremental tranche of 70,000 tonnes a year into the money. At an average abatement cost of $130 that tranche costs $9.10 million a year and releases $4.20 million a year of value at the shadow price — 70,000 × (190 − 130). Note what the ratio says: the tranche clears at 1.85× today's market price and 0.68× the shadow price. The firm does not need the market to move. It needs to decide which of the two prices governs its own capital, and that decision is entirely inside the building.

The first ninety days.

DayActionArtifact
1–20Close scope 1 and 2 monthly, reconciled to meters and invoicesA tonnage close, with an owner
21–35Build the marginal abatement curve, tranche by trancheThe curve, costed
36–50Board sets the shadow price and the basis; minute itThe price memo, signed
51–65Auditors on the disclosure treatmentA one-page accounting note
66–80Appraise every live capital paper twiceTwo columns in every paper
81–90Fund the reserve; approve the first trancheThe first verified tonne

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

  1. Where have we already made a decision that priced carbon implicitly — an energy retrofit, a fuel switch, a site closure — and what number was actually in someone's head when they made it?
  2. Which of our measurements of physical quantities is already good enough to put a price against tomorrow, and who built it?
  3. When has a long-dated number — a cost of capital, a residual value, a discount rate — settled an argument here rather than starting one? What made that one stick?

Dream — what becomes possible

  1. If our capital papers carried a carbon column as routinely as they carry a payback, what would have been approved last year that was not?
  2. Imagine our shadow price published beside our cost of capital. What would we want it to be, and what would we want people to conclude from it?
  3. If our tonnage closed monthly with the same discipline as our revenue, what question could we answer that we currently cannot?

Design — what we build

  1. Which of the three bases for a price — regulatory expectation, published social cost, or the shadow price of our own target — best fits what we have actually committed to, and who decides?
  2. What is the smallest capital threshold above which every paper gets the second column, starting this quarter?
  3. Where does our own abatement curve clear today, and what would move the next tranche below the line?

Destiny — how it holds

  1. What would have to be true for our shadow price to survive a change of chief executive, and which of those things could we put in place this year?
  2. Who, outside the sustainability function, would notice first if the price quietly stopped being applied — and what would they be looking at?
  3. What would we want our successors to find written down about how we chose this number, so that they can revise it well rather than start again?

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West, T. A. P., Wunder, S., Sills, E. O., Börner, J., Rifai, S. W., Neidermeier, A. N., Frey, G. P. and Kontoleon, A. (2023). "Action Needed to Make Carbon Offsets from Forest Conservation Work for Climate Change Mitigation." Science, 381(6660), 873–877.

World Bank (2025). State and Trends of Carbon Pricing 2025. World Bank, Washington DC.

Note on figures. Every figure in this chapter is computed in lib/verify/VII_02.py and printed with its inputs by python3 lib/verify.py VII.02. Dollar years differ between sources and are never silently reconciled: Nordhaus (2017) is in 2010 international dollars, EPA (2023) and Rennert et al. (2022) in 2020 dollars, and World Bank (2025) prices are nominal as of 1 April

  1. The price-against-social-cost gaps are therefore undeflated and are

floors. The Ramsey machinery behind the discount rate is Chapter III.05 and is not re-derived here.