The ultimate cost of carbon

Abstract

We estimate the potential ultimate cost of fossil-fuel carbon to a long-lived human population over a one million–year time scale. We assume that this hypothetical population is technologically stationary and agriculturally based, and estimate climate impacts as fractional decreases in economic activity, potentially amplified by a human population response to a diminished human carrying capacity. Monetary costs are converted to units of present-day dollars by multiplying the future damage fractions by the present-day global world production, and integrated through time with no loss due from time-preference discounting. Ultimate costs of C range from 10kto10k to 750k per ton for various assumptions about the magnitude and longevity of economic impacts, with a best-estimate value of about $100k per ton of C. Most of the uncertainty arises from the economic parameters of the model and, among the geophysical parameters, from the climate sensitivity. We argue that the ultimate cost of carbon is a first approximation of our potential culpability to future generations for our fossil energy use, expressed in units that are relevant to us.

Full text

The ultimate cost of carbon
David Archer
1
&Edwin Kite
1
&Greg Lusk
2,3
Received: 16 March 2018 /Accepted: 2 July 2020/
#The Author(s) 2020
Abstract
We estimate the potential ultimate cost of fossil-fuel carbon to a long-lived human
population over a one million–year time scale. We assume that this hypothetical popu-
lation is technologically stationary and agriculturally based, and estimate climate impacts
as fractional decreases in economic activity, potentially amplified by a human population
response to a diminished human carrying capacity. Monetary costs are converted to units
of present-day dollars by multiplying the future damage fractions by the present-day
global world production, and integrated through time with no loss due from time-
preference discounting. Ultimate costs of C range from $10k to $750k per ton for various
assumptions about the magnitude and longevity of economic impacts, with a best-
estimate value of about $100k per ton of C. Most of the uncertainty arises from the
economic parameters of the model and, among the geophysical parameters, from the
climate sensitivity. We argue that the ultimate cost of carbon is a first approximation of
our potential culpability to future generations for our fossil energy use, expressed in units
that are relevant to us.
Keywords Climate impact .Carbon cycle .Deep future .Cost of carbon
1 Introduction
The social cost of carbon (SCC) is a concept that was formulated to account for climate change
in cost-benefit analysis (Nordhaus 1982, Pearce 2003, Stern 2006, Greenstone et al. 2013,
National Academies of Sciences 2017. Costs from climate change are imposed on models of
https://doi.org/10.1007/s10584-020-02785-4
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10584-020-
02785-4) contains supplementary material, which is available to authorized users.
*David Archer
d–archer@uchicago.edu
1
Department of the Geophysical Sciences, University of Chicago, Chicago, IL, USA
2
Department of Philosophy, University of Chicago, Chicago, IL, USA
3
Lyman Briggs College, Michigan State University, East Lansing, MI, USA
Published online: 15 July 2020
Climatic Change (2020) 162:2069–2086
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the human economy, assessed relative to a control case without climate change. The present-
day value of future costs is obtained by discounting (Ramsey 1928). For emissions in 2010 and
discount rates of 2–5%, the SCC is estimated to be about $20–50 per ton of CO2($70–180 per
ton of C) although higher values have been proposed (Ackerman and Stanton 2012,Koppand
Mignone 2012,Greenstoneetal.2013, Moyer et al. 2014, Moore and Diaz 2015,Nordhaus
2017).
Because future damages are discounted to calculate their present-day values, costs that
accrue more than a few centuries into the future become negligible to the present. However,
the time scales of the carbon cycle, sea level change, and soil responses to present emissions
are much longer than this and are mostly missed by the SCC (National Academies of Sciences
2017). This paper formulates and estimates an ultimate cost of carbon (UCC), integrated over
the entire geophysical response to the human climate perturbation. The formulations of the
UCC and the SCC are contrasted in Table 1. The two metrics are based on irreconcilably
different assumptions, and apply over different time frames, and are therefore not simply or
directly comparable.
2 Methods
2.1 Model formulation
It is impossible to predict the deep future of humanity, so our formulation leaves out any
consideration of human technological development. The UCC is based on climate impacts to a
hypothetical human population that is technologically stationary, agrarian, and in steady state
with the carrying capacity of the planet. This unrealistic formulation is not a prediction, but a
necessary idealization that makes the problem tractable to address, and the result easy to
visualize.
Table 1 Comparison of the social and ultimate costs of carbon
Social cost of carbon (SCC) Ultimate cost of carbon (UCC)
Goal Present-day net value of present and future
costs, which are discounted due to assumed
future growth and preference for
present-day welfare
Equivalent value of cumulative future costs, in
present-day units, based on extrusion of our
present-day world into the indefinite future.
With no growth, the discount rate = 0%.
Characteristic
time scale
30 years, set by discount rate 10 to 200 kyr, set by landscape processes or the
geological carbon cycle
Economic
model
Extrapolated labor and predicted capital are
combined using a specified or predicted
productivity coefficient
Human activity, population, and economy are
equal to present day, scaled by decreases in
the planetary carrying capacity
Climate costs Fractional decrease in economic production
relative to a no-warming control
same
Utility
function
Average (per capita), taken as the log of
consumption (GWP/cap).
Near-future population changes are usually
specified or strongly influenced by inertia
of the present day.
Total utility (GWP). Costs accrue linearly,
in units of present-day dollars, rather than
logarithmically. Human population in the
distant future is determined by steady-state
carrying capacity.
Cost basis Marginal cost of emitting one additional ton of
C, given what has already been emitted
Average cost of all carbon ever emitted
(total cost/cumulative emission)
Value, $/ton C Typically < $100 $10,000–$600,000
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2.2 Economic formulation
Climate damages in the model are cast in the form of a fractional decrease of economic activity
between a warming case and a no-warming control. The control case is assumed to have a
steady, persistent economic productivity equal to the present-day gross world production
(GWP) of $100 trillion/year. Future fractional changes in human activity are multiplied by
the present-day GWP to tabulate costs in units of present-day dollars.
If in an alternate scenario the GWP were to grow before plateauing at, for example, a higher
steady GWP than we have today, the actual dollar cost from climate damages would be higher
than we calculate. A UCC defined in actual dollars, including growth, would be impossible to
constrain, because the long-term economic and technological future of humanity is impossible
to reliably predict, and it is beyond the scope of this paper to try. In our formulation, scaling
future damage fractions by present-day GWP, we are valuing the fractional climate damages to
future generations as though they applied to our own world: valuing the existence of future
generations, rich or poor, as much as we value ours. Normalizing to present-day GWP puts the
result in units of present-day dollars, making it easy to visualize.
Uncertainty in future growth undermines the use of the UCC in any calculation of
optimality. Comparing it with the SCC or costs of mitigation also requires relating costs
through deep time, which remains a question for economics, futurology, and moral philoso-
phy, and is also beyond the scope of this paper. The UCC is cast in units of present-day dollars,
but it cannot be taken to the bank, exactly. It is a scale bar for climate damages, presented in
modern-day terms.
It seems likely that the dominant impact of our fossil fuel use on the world and human
activity 100,000 years from now will be through persistent changes in climate, rather than by
our fossil-fuel-based construction of some long-lived economic infrastructure or persistent
wealth. If this is the case, the UCC can be seen as a first approximation of our potential
culpability to future generations, expressed in units that are meaningful to us.
The costs are accrued linearly, even though the welfare benefit of consumption is typically
treated as logarithmic (for example, Golosov et al. 2014), for clarity and because the econo-
mies of the hypothetical worlds differ by order tens of percent, a relative range over which the
log function is approximately linear. The scaled costs from all future years are accumulated
through time, with no loss due to discounting, analogously to how one would time-integrate a
CO2source to get cumulative emissions that drive climate change. In the exponentially
growing world of economics and finance, relating value across time requires consideration
of how future values should be discounted. The discount rate, r, used to calculate the present-
day value of a future cost (Ramsey 1928), is calculated as
r¼δþηg
where δis a pure rate of time preference (concern for the future vs. eating dessert first), gis the
growth rate of per capita income, and ηis the elasticity of the marginal utility of consumption
(expressing the different impacts of giving an extra dollar to a rich versus a poor person). The
pure time preference may be larger than 0% to reflect a selfish interest in the present day, but to
express a particular ethical position, or if we assume that the representative individuals in the
scenario live forever, δmight be set to 0% (Stern 2006).
The growth term (ηg) expresses the idea that a cost, fixed in dollars, would be relatively
smaller in a wealthier future. In the formulation for the UCC, the assumption of steady state
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requires that the growth rate g= 0% under conditions of constant planetary habitability. Under
these idealized conditions, values are combined through time using a discount rate requal to
0%.
In addition to direct climate impacts on the economy due to impacts on worker productivity,
storm damage, etc. (Weitzman 2009, Howard and Sterner 2017,Hsiangetal.2017,National
Academies of Sciences 2017, Burke et al. 2018), in the long-term steady state, there is a
potential population feedback that could impact the scale of the human enterprise. In economic
models of climate change, GDP does not respond strongly to changes in agricultural land
(Nordhaus 2017). On the longer time scales of the UCC, a steady-state population declines by
definition in response to a decrease in Earth’s carrying capacity or agricultural potential, just as
wildlife populations are declining now, in response to decreasing undisturbed range. This
might not affect the well-being of individuals (GWP per capita), but its impact on the total
utility (GWP) counts as a cost in the UCC. From the perspective of people living in the no-
warming control case, the loss of some fraction of their world would seem costly, as it would
to us if we were to sacrifice some fraction of our world today. Decreasing the scale of a future
human population would increase their risk of extinction and decrease their scope for creativity
(Parfit 2017).
2.3 Biophysical formulation
The analysis in this paper is based on a simple numerical model, coded in python, of
temperature, sea level, and economic response to global warming (see supplemental
materials). In order to capture the entire climate perturbation until the eventual natural uptake
of the fossil fuel carbon, the model is run one million years into the future (Walker and Kasting
1992, Archer et al. 1997,Archer2005, Archer and Brovkin 2008, Archer et al. 2009, Eby et al.
2009, Clark et al. 2016).
2.3.1 Evolution of atmospheric CO2
The scenario begins with the instantaneous release of 1000–5000 Gton C to the atmosphere.
The high end of this range is commonly taken as complete fossil fuel utilization (Sundquist
1985;Archeretal.1997; Caldeira and Wickett 2005). Most of the available fossil carbon is in
the form of coal, the extractable inventory of which has at least a factor of three uncertainties
(Rogner 1997; Rutledge 2011; Ritchie and Dowlatabadi 2017). Business as usual projections,
without mitigation, result in the release of about 1300 Gton C by the year 2100, with an
additional few centuries required to emit the full 5000 Gton C. The model does not attempt to
resolve the dynamics of the CO2release and equilibration with the ocean, on time scales of
decades to centuries, because the costs accruing on these time scales are negligible compared
with those accruing over millions of years. The first parameter in the model is the amount of
CO2release (Table 2).
Natural carbon-cycle feedbacks may amplify or ameliorate the anthropogenic carbon
release. The land surface biosphere has been taking up anthropogenic CO2(Tans 2009), for
reasons that are not well known but might be a response to a longer growing season, or
fertilization due to the rise in atmospheric CO2or nitrate deposition. However, a dominant
response looking forward is a release of carbon from the decomposition of thawing permafrost
soils, which contain thousands of gigatons of C (Lawrence and Slater 2005, Zimov et al.
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2009). The lack of runaway carbon greenhouses in the paleoclimate record suggests the
magnitude of this feedback is probably less than 1.5 (Archer and Buffett 2005).
The CO2perturbation relaxes back to zero on two time scales, representing different stages
of the response. First, on a shorter time scale, acidification of the ocean drives an imbalance in
the weathering and burial of CaCO3, which ultimately restores the pH of the ocean (Broecker
and Peng 1987; Archer et al. 1997; Caldeira and Wickett 2005), replenishing its buffer
capacity to absorb CO2. Second, a longer time scale, the excess CO2from the surface “fast”
carbon cycle (atmosphere, ocean, land surface) is removed by enhanced weathering of
calcium-bearing igneous rocks (Berner 2006), which carry CO2back to the deep Earth as
CaCO3. This process is known as the rock weathering “thermostat.”Each exponential decay is
parameterized by an initial “airborne”component of the released CO2(residing in the
atmosphere), and a time scale, as
CAtm ¼CAtmInit þCRelease AOcnEquil e−λAcidtþANeut e−λThermostatt

where Cdenotes a mass of carbon in gigatons as CO2,Aan initial airborne fraction, and λa
time scale in years. The airborne fractions, AOcnEquil and ANeut, are taken from the carbon cycle
model results from the “long tail”model intercomparison project (Archer et al. 2009)(Fig.1).
The airborne fraction values from the models at 1000 years are taken to be representative of the
stage in which the ocean is acidified, after CO2invasion but mostly before CaCO3dissolution
neutralizes the acidity of the fossil CO2. The airborne fractions at 10,000 years are taken to
represent neutralized ocean conditions, which will subside on the rock weathering “thermo-
stat”time scale. The simulations show higher airborne fractions for 5000-Gton C release
experiments than for 1000 Gton C, due to exhaustion of the buffer chemistry of the ocean
before its restoration by CaCO3compensation. For release between 1000 and 5000 (for
marginal cost estimation), the airborne fraction values are interpolated.
The time scale for ocean pH neutralization has been estimated to range between 1000 and
10,000 years (Broecker and Peng 1987;Archeretal.1997;Archeretal.1998; Caldeira and
Wickett 2005). The decrease in the ocean concentration of carbonate ion (CO32−
) causes a
Table 2 Parameters to the geophysical component of the model
Parameter name Description Value note
CReleaseGton Gigatons of anthropogenic carbon The buffer chemistry in the model is intended for the
range 1000–5000 Gton C
CFeedbackFactor Fraction by which the natural carbon
cycle amplifies the human source
Presumably < 1.5, or else the natural carbon cycle
would be observably tippier
(Archer and Buffett 2005)
oceanAcidTime Time scale for pH recovery of the
ocean, years
Probably a few thousand years
(Broecker and Peng 1987) (Archer et al. 1997)
thermostatTime Time scale for atmospheric CO2
recovery, years
About 200 kyr from models (Berner 2006), probably
longer than the 100-kyr glacial interglacial cycles
warmingTime Time scale for temperature
equilibration
Governed by ocean circulation, in reality a range of
time scales up to about 1000 years
(Bala et al. 2005).
iceMeltingTime Time scale for collapsing ice sheets,
years
Models find time scales of a few thousand years
(Clark et al. 2016), but could be centuries
(Hansen et al. 2016)
dt2X The equilibrium climate sensitivity,
degrees C per doubling of CO2
Range from IPCC Fifth Assessment Report is
1.5–4.5 °C (IPCC 2014)
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decrease in CaCO3production (Riebesell et al. 2000), and an increase in sedimentary CaCO3
dissolution (Archer et al. 1998), while the increase in atmospheric CO2and the increase in
temperature lead to an increase in terrestrial CaCO3weathering (Liu et al. 2018), all of which
contribute to the neutralization response. The ocean pH evolution through the glacial cycles
serves as a constraint on this response time, but it is complicated by a significant deglacial
increase in shallow-water CaCO3deposition on flooding continental shelves (Milliman 1993).
The time scale for ultimate CO2removal by the weathering thermostat is set by the rates of
geochemical weathering reactions, combining a CaO component of igneous rocks with CO2to
produce CaCO3, the pathway for carbon return to the deep Earth. These reactions, balanced
against natural CO2degassing from the Earth, comprise a planetary carbon cycle thermostat
system (Walker et al. 1981,Berneretal.1983). The time constant for reaching the equilibrium
CO2concentration is estimated from the Geocyc model (Archer et al. 2009,
http://climatemodels.uchicago.edu/geocyc/, based on Berner et al. 1983) to be about 200
,000 years.
The weathering thermostat time scale is informed by reconstructions of the past. The
Paleocene-Eocene thermal maximum event can be taken as an analog to global warming,
with a fast release of isotopically light carbon, a resulting climate warming, and a gradual
recovery. The event as recorded in ocean sediments has a smooth exponential recovery with a
time scale of ~ 100,000 years (Zachos et al. 2001). The more recent glacial cycles included
large swings in atmospheric CO2, the 100-kyr durations of which set a lower limit to the
response time, consistent with our best estimate of a 200-kyr response time.
The anthropogenic impact on climate will take place within the context of natural climate
variability, driven through glacial/interglacial cycles resulting, in part, from wobbles in Earth’s
orbit around the sun. However, Earth’s orbit around the sun is currently approaching circu-
larity, as it does every 400 kyr, minimizing variations in sunlight intensity and leading to an
expected long interglacial interval like Marine Ice Sheet Stage 11 about 400 kyr ago (Paillard
2001). The rhythm of the ice ages complicates the impact of global warming, but less so than if
we had started industrial activity 200 kyr ago when orbital variations in sunlight intensity were
Fig. 1 Atmospheric CO2evolution in the model. Results from 1000 and 5000 Gton C anthropogenic CO2
releases are plotted on a linear time scale on the left, and a log scale on the right
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more intense (Archer and Ganapolski 2005). It would no doubt be costly to descend into
another ice age, but if this was a motivating factor in fossil energy use, our carbon stock could
be deployed more strategically by waiting until the descent into an ice age is imminent (Shaffer
2009).
In summary, the carbon drawdown dynamics are idealized but well understood relative
to other parts of this analysis. Atmospheric pCO2values are plotted as a function of time in
Fig. 1.
2.3.2 Global mean temperature anomaly
The equilibrium temperature for a given atmospheric CO2concentration is calculated from the
climate sensitivity as
Tequil ¼ΔT2x
ln pCO2
pCOinit

ln 2ðÞ
where ΔT2xis a parameter in the model and is thought to be in the range of 1.5–4.5 °C for
doubling CO2(IPCC 2014).
The temperature of the Earth relaxes toward Tequil on a time scale specified as a model
parameter, warmingTime. The initial Tevolving is taken to be 80% of the initial Tequil value, as an
approximation to the present day, and it relaxes toward Tequil as
ΔTevolving ¼Tequil−Tevolving

Δt
warming Time
After the initial temperature transient, Tevolving is taken as equal to Tequil as a numerically stable
approximation. The evolution of temperature for the two values of CO2releases is shown in
Fig. 2.
Fig. 2 Model global temperature anomalies from releases of 1000 and 5000 Gton C
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2.3.3 Sea level
The time scale for the ice sheet response is thought to be a few millennia, based on ice sheet
modeling (Alley et al. 2005) and the general time scale of sea level change recorded in oxygen
isotopic composition of ice cores (Hays et al. 1976). However, past intervals of sea level rise
have been faster than this, including the Heinrich events (Broecker et al. 1992) and deglaci-
ation (Fairbanks 1989). A positive feedback between ice sheet melting, meltwater, and ocean
stratification has been proposed to explain the fast sea level rise events of the past, suggesting a
potential sea level rise time scale of centuries, rather than millennia, in our future (Hansen et al.
2016). Also, a real ice sheet grows until it melts at its base, at which point it collapses rapidly
(Macayeal 1993; Paillard 1998). The simple exponential-relaxation-to-equilibrium sea level
model used here is simpler than we expect from reality, and if anything conservative in
response time and possible tipping behavior.
Ultimate sea level rise in the model (Fig. 3) is driven by the temperature anomaly, based
on observed covariation of sea level and global temperature from paleoclimate reconstruc-
tions (Archer and Brovkin 2008). The forecast for the year 2100 is for about 1 m of sea level
rise (Rahmstorf 2007,Koppetal.2017), but a full climate/ice sheet model run for thousands
of years found sea level rise of 50 m (Alley et al. 2005; Clark et al. 2016). The discrepancy
is due to timing, in that ice sheets will presumably not have time to equilibrate by the year
2100.
The ice sheets in the model recover on the time scale of the ultimate CO2drawdown by the
weathering thermostat. This simple one-way forcing (CO2drives ice) is an idealization and
neglects whatever mechanism or (more likely) mechanisms (Archer et al. 2000, Ganopolski
and Brovkin 2017) were at work through the glacial cycles, when changes in orbital forcing
and ice sheet dynamics drove a feedback atmospheric CO2response. Because atmospheric
CO2is driven by human sources today, it would be speculative to predict how the natural CO2
ice sheet deglaciation feedback might operate in the future.
Fig. 3 Sea level responses of the model
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2.4 Impacts
In our formulation, costs accrue due to two factors: directly due to an altered climate and due to
sea level rise. Both are calculated based on coefficients in % GWP/°C and % GWP/meter
(Table 3).
2.4.1 Costs due to climate
Economics models have been used to predict the impact of warming on the economic
productivity of the near future. From the perspective of deep time, this scenario consists of a
short-term impact of a sudden shift in climate on an existing population and infrastructure.
Remembering that our formulation is an idealization in which technological innovation is not
allowed, we project these global costs forward into the deep future as a provisional starting
point.
For relatively conservative temperature changes of 3 °C, the models that are used in SCC
calculations predict 0–1%/°C (Greenstone et al. 2013). Studies cited in the IPCC scientific
assessments, for temperature changes of 2.5 °C, had mean values of about 0.6%/°C (Tol
2016). Values from Burke et al. (2015), for 3 °C of warming, range between 2.6 and 20%/°C.
Based on these results, our analysis assumes 1% GDP/°C in damages directly from climate
change. The linear dependence of the model will be conservative in that it excludes the
possible catastrophic changes at high degrees of warming (Weitzman 2009).
On time scales longer than a few human generation times, we might expect a human
population response to changes in agricultural potential. Crop yields decrease significantly at
temperatures above 30–34 °C (Porter et al. 2014), and continental interiors are expected to
systematically dry in a warming world (Byrne and O'Gorman 2015). These constraints suggest
that the agricultural capacity of a 3 °C warmer world could be decreased by 10% relative to the
no-warming control case (Hsiang et al. 2017), or about 3%/°C.
2.4.2 Costs due to sea level rise
The long-term costs of sea level rise become significant if we assume a human population
response to the loss of present-day agricultural land.
Figure 4a shows the cumulative fraction of land area, crop and pasture land, and population
as a function of elevation above sea level. These curves were generated from digital gridded
density fields for population (SEDAC, NASA, 2.5 min grid), and crop and pasture land
(Ramankutty et al. 2008), interpolated into a gridded field of elevation (ETOPO2, National
Table 3 Parameters to the economic component of the model
Parameter name Description Value note
climCostFactor Economic penalty due to climate change,
in percent GDP per degree C
1%/°C is in the range of IAM results
seaLevelCostFactor Economic penalty due to sea level rise,
in percent GDP for complete melting
(70 m)
15% of present-day cropland could be
flooded (Fig. 4)
econRecoveryTime Economic recovery time scale for both types
of cost.
This couldbe a soil generation time scale of
10 kyr or a CO2thermostat time of
200 kyr
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Geophysical Data Center 2006). The cumulative areal loss of cropland increases more quickly
with sea level rise than total land area loss, reflecting higher than average agricultural
productivity in low-lying soils, especially fertile river deltas near sea level.
A sea level rise of 70 m would inundate about 15% of present-day cropland (and 5% of
pasture land). We postulate that a decrease in agricultural land would translate eventually into
a decrease in human population and economic activity, with 70 m thus reducing GWP by
15%.
2.4.3 Mechanism and time scale for economic recovery
A major uncertainty in the model is how long the climate impacts would persist as costs (as per
the formulation, in the absence of technology change or adaptation to climate). Both types of
costs, due to climate change and sea level rise, might be ameliorated by human migration to
higher latitudes. It is possible that some low-latitude regions might become uninhabitable
wastelands in a much warmer world (Sherwood and Huber 2010), but there is a lot of currently
uninhabited land area in the high latitudes that could be more hospitable to human societies in
a warmer world (Fig. 5). A limitation on how quickly humankind could move to high latitudes
might arise from the time scales of soil formation, for example, in the Canadian Shield region.
If the opening of new landscapes for human habitation compensates for loss of habitability in
the topics, then the time scale for economic recovery might be set by the soil formation time
scale, rather than that for ultimate CO2drawdown.
It has been argued that the abundance and quality of the soil have been a strong determinant
in the rise and fall of civilizations through the past 5000 years (Montgomery 2007). Soil is
derived from weathering of primary bedrock reacting with water, catalyzed biologically by
bacteria, plant roots, and deposit-feeding animals. The rate of production of soil is slow by
human standards; accumulation rates of a few centimeters per century or less are common,
Fig. 4 aCumulative area fractions lost of present-day land surface, cropland, pasture land, and population, as a
function of sea level rise. Generated from digital gridded density fields for population (SEDAC, NASA, 2.5-min
grid), and crop and pasture land (Ramankutty et al. 2008), interpolated into a gridded field of elevation
(ETOPO2, National Geophysical Data Center, 2006). bMean land surface slope as a function of elevation
above present-day sea level (ETOPO2, National Geophysical Data Center, 2006)
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resulting in an overall soil formation time scale that might be of order 10 kyr (Minasny and
McBratney 2001).
On time scales of 10 to 100 millennia, the global soil reservoir will have time to evolve
through a series of steady states where soil loss must be balanced by soil production. The
impact of human soil stewardship will be a primary driver, but it is impossible to predict, so
we are left to consider long-term changes in the potential of the land surface to create new
soil. This might be affected by the changes in climate (in particular the drying of continental
interiors) and by the change in the landscape due to sea level rise. Soil naturally travels
downslope, accumulating in flat lowland valleys. Figure 6shows the average landscape
grade derived from ETOPO2 plotted as a function of the elevation above sea level. The
average grade at an elevation of 50 m is twice as steep on average as it is a sea level. The
sediment transport in riverine flow that currently deposits in low-lying river deltas (at 50-m
elevation or less) would be lost to the ocean in a higher sea level world. This effect would
subside as flooded valleys become filled with sediment, as is happening today after
deglacial sea level rise in, for example, Chesapeake Bay, on a time scale of up to tens of
thousands of years.
Leaving unresolved the question of whether the pace of economic recovery will be set by
the time scales of soil development or ultimate carbon cycle, our analysis posits a range of
possibilities from 10 to 200 kyr (Table 4).
2.5 Sensitivity analysis
Uncertainty ranges for the parameters are estimated in Table 5and propagated to cost estimates
using a Monte Carlo method (Suppl. Fig. 1). The model is run, varying one parameter at a
time, with log-uniform probability across a specified range, holding all other parameter values
unvarying at the mean between their minimum and maximum values. Then the model is run
varying all parameters simultaneously but independently, first for just the geophysical param-
eters, then for all parameters, economic and geophysical. The total amount of C emitted is
varied along with the rest of the parameters, through a range of 1000–5000 Gton C, but the
results are normalized into units of $/Gton C for all statistical treatment. Increasing the number
of realizations beyond 10,000 has only a small impact on the results.
Fig. 5 The distribution of land surface area, croplands, pasture lands, and population as a function of latitude
(data from SEDAC, NASA, 2.5-min grid and Ramankutty et al. 2008)
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3Results
Projected costs are plotted in Fig. 6andsummarizedinTable4, for two values of the carbon release
magnitude (1000 and 5000 Gton C) and two values of the time scale for economic recovery (10,000
Fig. 6 The time trajectory of economic costs from the model
Table 4 Costs, in $1000/ton C
Carbon release, Gton C Recovery time scale, years Direct climate cost Costs assuming a population
feedback
Climate Sea level Total
1000 10 kyr 8.3 33.1 25.8 59.0
200 kyr 75.2 301.0 180.3 481.3
5000 10 kyr 10.2 40.8 27.5 68.3
200 kyr 121.2 484.9 265.1 750.1
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Table 5 Monte Carlo uncertainty analysis
Parameter range Climate cost $k/ton Sea level cost $k/ton Total cost
Low High Base Median Sigma pct Median Sigma pct Median Sigma pct
Base 70.9 27.3 98.2
CReleaseGton 1000 5000 2236 70.8 7.5 11% 27.3 2.7 10% 98.1 10.1 11%
CFeedbackFactor 0.1 0.5 0.2 71.4 3.1 4% 27.5 1.2 4% 98.9 4.3 4%
oceanAcidTime 2000 8000 4000 70.7 1.4 2% 27.2 0.6 2% 97.9 2.0 2%
thermostatTime 100,000 400,000 200,000 69.2 4.3 6% 26.1 3.1 12% 95.4 7.4 8%
warmingTime 100 1000 316 70.9 0.0 0% 27.3 0.0 0% 98.2 0.1 0%
iceMeltingTime 300 3000 949 70.9 0.0 0% 27.4 0.7 3% 98.2 0.7 1%
dT2x 1.5 4.5 2.6 70.2 21.1 29% 27.1 8.1 29% 97.3 29.2 29%
climCostFactor 1 4 2 68.6 30.1 40% 27.3 0.0 0% 95.9 30.1 29%
SLCostFactor 1 15 3.9 70.9 0.0 0% 35.0 29.7 72% 105.9 29.7 27%
econRecoveryTime 10,000 200,000 44,721.4 71.8 55.2 64% 27.6 16.5 54% 99.5 71.7 61%
AllGeo 66.4 23.1 33% 26.4 8.7 32% 92.6 31.6 32%
All 87.1 76.1 75% 29.7 57.1 105% 135.5 120.3 77%
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and 200,000 years). The coefficient for direct climate costs is 1% GWP/°C, while accounting for a
human population feedback raises the value to 4%/°C, and that for 70 m sea level rise to 15% GWP.
The costs per ton C are relatively similar between 1000 and 5000 Gton C scenarios, compared
with much larger differences between the cases of slow versus fast economic recovery. For direct
climate costs only, the total cost is around $10k per ton if the recovery rate is set by soil formation in
high latitudes, or roundly $100k per ton if the economic recovery has to await the ultimate
drawdown of CO2. The costs that presume a population feedback come to roundly $60k/ton and
$600k/ton for the fast and slow recovery rates, respectively.
Using the median values of all input parameters, the costs are about $98k/ton C. The 1σ
uncertainty due to the geophysical parameters is about $34k, and due to geophysical plus economic
parameters, about $116k, with the distribution skewed toward higher costs in all cases. Costs are
simply linearly proportional to the cost coefficients, and nearly linearly dependent on the economic
recovery time scale (Suppl Fig. 2). Among the geophysical parameters, the climate sensitivity
contributes the most overall uncertainty, followed by the amount of carbon released and the time
scale for ultimate CO2drawdown.
4Discussion
The cost of abating our ongoing CO2emissions has been estimated to be in the range $10–100 per
ton of C (Enkvist et al. 2009). Failing that, the cost of chemically removing a ton of C as CO2from
the atmosphere using industrial chemical engineering methods has been estimated to be $2200
(Socolow et al. 2011) or $360 (Keith et al. 2018). These estimates are much lower than the ultimate
cost of carbon and would be a bargain if the UCC was directly fungible with present-day mitigation
costs. However, it might not in the self-interest of any single generation to pay to clean it up, since no
single generation would pay the entire ultimate climate costs.
The total cleanup costs would be enormous. A return to a “safe”atmospheric CO2concentration
of 350 ppm (Hansen et al. 2005) within a few decades would require removing about 440 Gton C
(using the ISAM carbon cycle/integrated assessment model at http://climatemodels.uchicago.
edu/isam/(Cao and Jain 2005)). At a cost of $360 per ton of C (Keith et al. 2018), the total cleanup
debt today is about $160 trillion, about 1.6 years of present-day GWP. The cleanup debt is
accumulating at a rate of about $3.6 trillion per year, about 3.6% of GWP.
Humanity could defend some fraction of the inundating land surface using dikes. Based on a
contemporary Dutch cost of about $10/m3of dam, assuming triple dike walls 60 m high and 120 m
wide at the base, we calculate a cost of about $100,000 per meter of coastline, coming to $1014 ($100
trillion, 1-year GWP) for the entire million-kilometer coastline of the world. Dividing by the amount
of carbon emitted results in a cost of about $5/ton of CO2emitted. This effort would ameliorate land
loss by sea level rise, but it would obviously not change the impacts of a warmer climate. A
downside is that this strategy would commit future generations to ongoing maintenance of the dikes.
The almost absurd global scale of the proposal, and its relatively inexpensive price tag, is another
demonstration of the massive scale of the ultimate damage from carbon energy.
5Conclusions
This paper presents a formulation for assessing deep-time climate change in units of present-
day dollars per ton of carbon. The scope of the definition for such a cost is limited by what is
2082 Climatic Change (2020) 162:2069–2086
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tractable, and the result is only valid under particular, idealized circumstances including zero
growth or technology change, and human population in steady state with Earth’s carrying
capacity. The UCC is unrealistic as a forecast for the future, in and of itself, but it is easy to
visualize and provides a scale bar for ultimate climate damage in present-day terms.
The UCC in our formulation ranges from $10k to $750k per ton of C, with a central value
about $100k per ton. The 1σuncertainty due to geophysical parameters is about $34k, and due
to geophysical plus economic, about $116k, skewing to higher values in all cases. Among
geophysical parameters to the model, the climate sensitivity contributes the most to the
uncertainty in the result.
The UCC is not directly comparable with the social cost of carbon, or costs of mitigation,
because of fundamental differences in their assumptions and formulations, and the fact that
they apply over different time scales. Assuming that the climate impacts from fossil carbon use
will persist much longer than any economic infrastructure that we use fossil fuels today to
build, the UCC can be seen as a first approximation of our culpability to future generations,
expressedintermsthatarerelevanttous.
Acknowledgments This paper benefitted immensely from comments and suggestions by Soren Anderson, Jim
Franke, Philippe Tortelle, Detlef van Vuuren, David Weisbach, and several anonymous reviewers.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and
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Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included
in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or
exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy
of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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