# Supplementary Material: Climate uncertainty impacts on optimal mitigation
pathways and social cost of carbon

Christopher J. Smith  Priestley International Centre for Climate, University
of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom  Energy, Climate and
Environment Program, International Institute for Applied Systems Analysis,
Schloßplatz 1, 2361 Laxenburg, Austria  Alaa Al Khourdajie  Energy, Climate
and Environment Program, International Institute for Applied Systems Analysis,
Schloßplatz 1, 2361 Laxenburg, Austria  Centre for Environmental Policy,
Imperial College, South Kensington Campus, London, SW7 2AZ, United Kingdom  Pu
Yang  Energy and Power Group, University of Oxford, OX5 1PF, United Kingdom
Bartlett School of Sustainable Construction, University College London, Gower
Street, London, WC1E 6BT, United Kingdom  Doris Folini  Institute for Climate
and Atmospheric Science, ETH Zürich, Universitätstrasse 16, 8092 Zürich,
Switzerland

##  1 Modifications to DICE-2016R

Our starting point for analysis in this paper is the DICE-2016R integrated
assessment model (IAM). DICE-2016R is a well-known cost-benefit IAM and is
well-documented in the literature [1, 2]. Recently, a beta version of
DICE-2023R has been submitted as an NBER report which incorporates FaIR v1.0
as its carbon cycle and climate module components [3]. We comment extensively
on DICE-2023R in section 6, and compare both DICE-2016R and DICE-2023R to the
version presented in this study which we name FaIR-DICE.

###  1.1 Time step and time horizon

We reduce the model time step in DICE-2016R from 5 years to 3 years. The
benefits are that this allows for an integral number of periods from the
nominal pre-industrial start year to the present-day (2023 minus 1750 is
divisible by 3), while maintaining an integral number of periods to 2050 (2050
minus 1750 is also divisible by 3). It also allows model-derived mitigation
trajectories to be more responsive. This is important given the proximity to
1.5°C in the present day and limited headroom available to stay under this
limit.

###  1.2 Population

The projections of world population from DICE-2016R are updated with the
median projection of 10,000 scenarios from the Resources For the Future
Socioeconomic Pathways (RFF-SPs) [4, 5].

The RFF-SPs run to 2300, which we extend to 2500 by taking the average growth
rate over 2250–2300 in each projection and linearly declining this growth rate
to zero by 2500. This produces a very wide range of potential population
scenarios ranging from 200 million to 200 billion people in 2500 (fig. 1 grey
lines). The median pathway used for our ensemble is depicted in fig. 1 by the
black curve.

In the median of the RFF-SP projections, global population peaks in 2116 at
11.2bn, declining to 7.3bn in 2300 and 4.9bn in 2500. From 2200 onwards this
population trajectory is substantially different to DICE-2016R (orange line)
and DICE-2023R (green line), which assume asymptotic convergences to 11.5bn
and 10.8bn respectively by 2500.

Figure 1: Population projections from the 10,000 RFF-SPs from 2020 to 2300,
extended to 2500 using the average growth rate of the last fifty years,
declining to zero growth over 2300–2500 (coloured lines). The thick black line
is the ensemble median, used for the population projection in this study.

###  1.3 Model for future land-use CO2 emissions

The DICE-2016R and DICE-2023R models use exogenous time series of CO2
emissions from agriculture, forestry and other land use (AFOLU). These
emissions do not result from fossil fuel combustion and have differing
economic and social drivers to fossil fuel emissions that are not modelled
explicitly in DICE. Nevertheless, the AFOLU sector will play a vital role in
climate change mitigation, and land-based carbon dioxide removal strategies
(afforestation and reforestation) are featured heavily in process-based (PB-)
IAM scenarios that limit global mean temperatures to 1.5°C or 2°C [6].
Therefore, it could be reasonably expected that a mitigation scenario that
took strong action on fossil emissions would also address the AFOLU sector,
even if the causal drivers are not strongly linked.

In this regard, we seek a regression relationship predicting CO2 AFOLU
emissions $E_{\mathrm{AFOLU}}$ from FFI emissions $E_{\mathrm{FFI}}$ and
period $t$, derived from 1202 PB-IAM scenarios from the IPCC WG3 database from
2030 to 2100 [7]:

$E_{\mathrm{AFOLU}}=\alpha+\beta_{1}E_{\mathrm{FFI}}+\beta_{2}t.$ (1)

Figure 2(a) shows there is a positive, time-dependent relationship between
fossil emissions and AFOLU emissions from PB-IAM scenarios justifying the form
of eq. 1 ($r^{2}=.47$). We derive $\alpha=1.54$ GtCO2 yr-1,
$\beta_{1}=0.0464$, $\beta_{2}=-0.189$ GtCO2 yr-1 period-1. Equation 1 is only
defined to 2100. To extend to 2500, we follow the assumption of a phase down
to zero CO2 AFOLU emissions by 2150 used for the SSP scenario extensions in
[8]. This is achieved by multiplying eq. 1 by a logistic curve $l(t)$ centred
around $t=35$ (model year 2125):

$l(t)=\left(1-\frac{1}{1+e^{-(t-35)}}\right).$ (2)

The comparison of our ensembles to the exogenous CO2 AFOLU emissions in
DICE-2016R and DICE-2023R is shown in fig. 2(b). All of our scenarios have
negative AFOLU emissions during the later 21st century, which is typical of
PB-IAM scenarios (fig. 2(a)). This is in contrast to the DICE time series
where AFOLU emissions converge to zero from above. The impact of the logistic
phase out can be seen in the 2100–2150 period. Our year-2023 CO2 AFOLU
emissions value, derived solely from the regression relationship, is also
slightly closer to the historical best estimate from the Global Carbon Project
(GCP; black line near the left axis in fig. 2(b)) than in DICE-2023, though it
should be cautioned that the 2022 land use emissions from GCP is preliminary,
and land use emissions have large uncertainty.

(a) Comparison of $E_{\textrm{AFOLU}}$ with $E_{\textrm{FFI}}$ for 1202 PB-IAM
scenarios submitted to the IPCC AR6 WG3 database. Regression lines drawn at 10
year intervals from 2030 to 2100.

(b) Comparison of $E_{\textrm{AFOLU}}$ from the three scenario ensembles in
this study (coloured lines and uncertainty ranges) with DICE-2016 (dotted) and
DICE-2023 (dashed) exogenously supplied time series. Historical best estimate
emissions from Global Carbon Project are shown in black as a comparison.

Figure 2: AFOLU emissions derivation for the scenario pathways in this study.

##  2 Reduced version of FaIR v2.1.0 in DICE

The Finite-amplitude Impulse Response (FaIR) model is a reduced-complexity
climate model used to project global mean surface temperatures from emissions
of greenhouse gases and short-lived climate forcers, and other anthropogenic
and natural influences on the climate system [9, 10, 11]. We implement a
reduced version of FaIR in DICE for future projections (2023–2500) that is
based on FaIR v2.1.0. A model description paper for FaIR v2.1.0 is not yet in
the scientific literature but it is similar to v2.0.0 [9]. The specific
implementation for DICE is described below.

DICE calculates CO2 emissions from fossil fuel and industry $E_{\mathrm{FFI}}$
from economic activity, abatement level and energy intensity (eq. 2, main
manuscript). Our treatment of CO2 emissions from agriculture, forestry and
other land use (AFOLU) are described in section 1.3. FaIR-DICE, based on
DICE-2016R, does not model any other anthropogenic greenhouse gas or short-
lived climate forcer explicitly. Therefore our reduced version of FaIR is
limited to the carbon cycle module and the temperature response module (both
described below) with non-CO2 forcing introduced exogenously (described in the
next section).

###  2.1 Carbon cycle model

We retain the four time-constant model of atmospheric CO2 decay to a pulse
emission from FaIR v2.0.0. Written in forward difference mode, this is

$R_{i}(t+1)=a_{i}E_{\textrm{FFI}}(t)\alpha(t)\tau_{i}\left(1-\exp\left(-\frac{\Delta
t}{\alpha\tau_{i}}\right)\right)+R_{i}(t)\exp\left(-\frac{\Delta
t}{\alpha\tau_{i}}\right)$ (3)

where $R_{i}$ is the carbon content in pool $i$, $\Delta t$ is the model
timestep (3 yr), $a_{i}$ is the amount of fresh emissions allocated to each
pool, $\tau_{i}$ is the e-folding lifetime of each pool, and $\alpha$ is the
lifetime scaling factor that models the effect of carbon cycle feedbacks. The
values of $a_{i}$ and $\tau_{i}$ are derived from Earth System and
intermediate complexity models in [12] (table 1). In FaIR v2.1.0 a user-
definable $\Delta t$ is incorporated, partly for applications including IAM
coupling. Previously, the model timestep was hard-coded to be one year.

Table 1: Partition fractions and lifetimes of the four carbon box impulse-response model in FaIR Pool $i$ | Partition fraction $a_{i}$ | Lifetime $\tau_{i}$ (yr)
---|---|---
1 | 0.2173 |  $10^{9}$ (essentially $\infty$)
2 | 0.2240 | 394.4
3 | 0.2824 | 36.54
4 | 0.2763 | 4.304

Carbon uptake efficiency reduces with increasing warming and accumulated CO2
in land and ocean sinks [13, 14]. In FaIR, the increase in atmospheric
lifetime of CO2 is parameterised by $\alpha$. $\alpha$ is determined from the
100-year time-integrated airborne fraction from a pulse of CO2 emissions,
$I_{100}$:

$\alpha(t)=g_{0}\exp\left(\frac{I_{100}(t)}{g_{1}}\right)$ (4)

where the airborne fraction is the proportion of CO2 remaining in the
atmosphere following a pulse emission, and follows the empirically derived
equation [11]

$I_{100}(t)=r_{0}+r_{u}(1-f_{a}(t))G(t)+r_{t}T_{1}(t)+r_{a}f_{a}(t)G(t).$ (5)

In eq. 5, $r_{0}$ is the pre-industrial value of $I_{100}$, $r_{u}$ is the
sensitivity of $I_{100}$ to accumulated carbon in land and ocean sinks,
$r_{t}$ is sensitivity of $I_{100}$ to global mean surface temperature anomaly
$T_{1}$, and $r_{a}$ is the sensitivity of $I_{100}$ to airborne CO2. $G$
represents cumulative CO2 emissions since pre-industrial and $f_{a}$ is the
airborne fraction of CO2 since pre-industrial (total increase of CO2 in the
atmosphere divided by $G$). $I_{100}$ has units of yr.

In eq. 4, $g_{0}$ and $g_{1}$ are normalisation constants defined as

$g_{1}=\sum_{i}a_{i}\tau_{i}\left(1-\left(1+\frac{H}{\tau_{i}}\right)\exp\left(-\frac{H}{\tau_{i}}\right)\right)$
(6)

$g_{0}=\exp\left(-\frac{\sum_{i}a_{i}\tau_{i}\left(1-\exp\left(-\frac{H}{\tau_{i}}\right)\right)}{g_{1}}\right)$
(7)

with $H=100$ yr. The atmospheric CO2 concentration $C_{\mathrm{CO}_{2}}$ is
the sum of the four boxes added to the pre-industrial CO2 atmospheric
concentration ($C_{\mathrm{CO}_{2},\mathrm{ref}}$), i.e.

$C_{\mathrm{CO}_{2}}(t)=C_{\mathrm{CO}_{2},\mathrm{ref}}+\sum_{i}R_{i}(t).$
(8)

###  2.2 Effective radiative forcing

The total effective radiative forcing (ERF) $F$ is given by

$F(t)=F_{\mathrm{CO}_{2}}(t)+F_{\mathrm{ext}}(t)$ (9)

where $F_{\mathrm{CO}_{2}}$ is the CO2 component and $F_{\mathrm{ext}}$ is the
and the exogenous non-CO2 component. In this reduced version, the CO2 forcing
is computed from concentrations as [15]

$F_{\mathrm{CO}_{2}}(t)=F_{2\times\mathrm{CO}_{2}}\frac{\log(C_{\mathrm{CO}_{2}}(t)/C_{\mathrm{CO}_{2},\mathrm{ref}})}{\log
2}.$ (10)

In eq. 10, $F_{2\times\mathrm{CO}_{2}}$ is the effective radiative forcing
from a doubling of CO2 above pre-industrial concentrations with a best
estimate value of 3.93 W m-2 [16]. More complex relationships exist for CO2
forcing that incorporate band overlaps between different greenhouse gases [17,
8], however, as concentrations of other gases are not book-kept in DICE, the
simpler formula in eq. 10 is used in FaIR-DICE.

###  2.3 Temperature response module

For computing the temperature response to effective radiative forcing, FaIR
uses an impulse-response formulation of the well-known $n$-layer energy
balance model [18]. The implementation here very closely follows the model of
Cummins et al. [19]. FaIR v2.1.0 allows for autocorrelated stochastic internal
variability in temperature and forcing following [19], but we do not use this
in the present study. There is evidence that inclusion of internal variability
increases the social cost of carbon (SCC) [20] and this would be a valuable
additional study in this setup.

We use $n=3$ layers, expected to be sufficient to capture short- and long-term
climate responses to forcing [9, 19]. The impulse-response model in forward
timestepping mode can be written as

$\mathbf{T}(t+1)=\mathbf{A_{d}T}(t)+\mathbf{b_{d}}F(t)$ (11)

where $\mathbf{T}=[T_{1}(t),T_{2}(t),T_{3}(t)]$ is a column vector of
temperature anomalies representing the near-surface ocean layer and
atmosphere, middle ocean, and deep ocean respectively. $T_{1}$ is used as
global mean surface temperature anomaly throughout this study.

$\mathbf{A_{d}}$ and $\mathbf{b_{d}}$ are computed using the discretisation
method of Cummins et al. [19] with one modification in that we discard the
first column and top row of $\mathbf{A_{d}}$ and first entry of
$\mathbf{b_{d}}$ as calculated in [19], as we do not use stochastic
variability in forcing or temperature. The reader is referred to sections 4a
and 4b of Cummins et al. [19] for the calculation. The Cummins et al. method
takes a matrix of energy balance model parameters $\mathbf{A}$ where

$\mathbf{A}=\frac{1}{\Delta t}\begin{bmatrix}-\gamma\Delta t&0&0&0\\\
1/C_{1}&-(\kappa_{1}+\kappa_{2})/C_{1}&\kappa_{2}/C_{1}&0\\\
0&\kappa_{2}/C_{2}&-(\kappa_{2}+\epsilon\kappa_{3})/C_{2}&\epsilon\kappa_{3}/C_{2}\\\
0&0&\kappa_{3}/C_{3}&-\kappa_{3}/C_{3}\end{bmatrix}$ (12)

and a column vector $\mathbf{b}=[\gamma,0,0,0]$ as inputs. Here, $\kappa_{j}$
[W m-2 K-1] are heat transfer coefficients between adjacent ocean layers (with
$-\kappa_{1}$ equivalent to the climate feedback parameter more commonly
denoted as $\lambda$ in climate science literature), $C_{j}$ are heat
capacities of each ocean layer [W m-2 yr K-1], and $\epsilon$ is the deep
ocean efficacy parameter incorporated to simulate a forced pattern effect
present in many Earth system models [21, 22]. $\gamma$ is a parameter
describing autocorrelation in the radiative forcing time series but is
redundant if internal variability is not used. These parameters are easily
calibrated to Earth System models (ESMs) as demonstrated in [18, 19].

##  3 Constrained probabilistic ensemble of FaIR v2.1.0

A semi-automated software package _fair-calibrate_ exists for producing
calibrated, constrained probabilistic ensembles from the FaIR model (starting
from FaIR v2.1.0). A separate manuscript is in preparation describing this
process in detail. We provide an overview relevant to results in this paper
here. We use the procedure and parameter values from v1.0.2 of _fair-
calibrate_ which is available from [23]. This reference also contains the
calibration code which is fully reproducible. The calibration procedure
produced uses the full standalone (rather than reduced, DICE-coupled) FaIR
v2.1.0 model, is run at a one-year rather than three-year time step, and has
internal variability switched on.

###  3.1 Calibration

####  3.1.1 Climate response

The parameters of the three-layer energy balance model ($\kappa_{j}$, $C_{j}$,
$\epsilon$ and $\gamma$, along with the ERF from a quadrupling of CO2,
$F_{4\times\mathrm{CO}_{2}}$) are calibrated to the response of 49
_abrupt-4xCO2_ ESMs participating in the Coupled Model Intercomparison Project
Phase 6 (CMIP6) using the maximum likelihood method of Cummins et al. [19]. We
use the first 150 years from the quadrupled CO2 run for consistency, even for
models where more years are available. This provides us with a set of nine
climate response parameters from 49 ESMs.

####  3.1.2 Carbon cycle feedbacks

The parameters dictating the carbon cycle feedbacks are $r_{0}$, $r_{u}$,
$r_{t}$ and $r_{a}$ defined in eq. 5. The parameters are calibrated from 11
carbon-cycle enabled Earth System models from CMIP6 using _1pctCO2_ and
_1pctCO2-rad_ experiments as described in [9].

####  3.1.3 Non-CO2 forcing

The full FaIR model that is used for the historical spin up also uses CMIP6
model results to calibrate the effective radiative forcing from aerosol-cloud
interactions (ERFaci) to emissions of precursor species in 11 CMIP6 models (a
similar method to [24]), the sensitivities of chemically active species to
ozone formation and destruction in six CMIP6 models (from [25]), and the
sensitivities of chemically active species to methane lifetime in four CMIP6
models (from [26, 27]). There are many more anthropogenic and natural
components that influence non-CO2 forcings, with these sensitivities based on
assessments from the IPCC Sixth Assessment Report (AR6) rather than CMIP6
models [16, 28].

###  3.2 Sampling

We use a prior ensemble size of 1,500,000 parameter sets based on CMIP6 model
calibrations or AR6 distributions, and use these parameter sets to run the
full version of FaIR offline (i.e. uncoupled to DICE) from 1750–2100 under the
historical + SSP2-4.5 scenario. Emissions of 53 greenhouse gas and short-lived
climate forcers under historical + SSP2-4.5 are provided by the Reduced
Complexity Model Intercomparison Project (RCMIP), with data available from
[29]. For short-lived climate forcers (SO2, BC, OC, NH3, CO, NMVOC, NOx),
RCMIP uses the same input emissions sources prepared for running CMIP6 models,
whereas greenhouse gases emissions are based on respected emissions
inventories for CH4 and N2O and estimated from inversions of concentrations
for halogenated greenhouse gases. We do not use the CO2 FFI and CO2 AFOLU
emissions from RCMIP for the historical period up to 2021, instead using data
from Global Carbon Project [30] which is more up-to-date and ensures the
beginning of the DICE projection period uses a more recent estimate of CO2
emissions. To transition to the SSP2-4.5 scenario, the future emissions are
harmonized [31] to the updated GCP historical to ensure a smooth transition
from the updated baseline to the future scenario.

####  3.2.1 Climate response

The nine energy balance model parameters are sampled from a multi-dimensional
kernel density estimate function that is generated from the 49 CMIP6 models
used in the calibration.

####  3.2.2 Aerosol-cloud interactions

There are four parameters that are used to model the ERFaci in FaIR, which are
the sensitivity to SO2, BC, and OC emissions, and a scale factor. The shape
parameters are sampled from correlated kernel density estimates that are
constructed by the fits to 11 CMIP6 models. The scale parameter is selected
such that the resulting ERFaci in each ensemble member for the 2005–2014
period relative to 1750 is uniform in the range $-2$ to 0 W m-2.

####  3.2.3 Aerosol-radiation interactions

A number of species are involved in contributing to the ERF from aerosol-
radiation interactions (ERFari) [32]. Aerosol formation is dominated by
sulfate (primary source SO2 emissions), BC, OC and nitrate (dominant source
NH3 emissions), with minor contributions from NOx, NMVOC, CH4, N2O, and
halogenated greenhouse gases.

A forcing efficiency for each species is calculated by dividing its
contribution to ERFari as assessed by IPCC AR6 [32] by its change in emissions
(or concentrations for greenhouse gases) between 1750 and 2019. The headline
assessment of ERFari is $-0.3$ W m-2 from 1750 to 2005–2014, which is
reproduced using these forcing efficiencies (the assessed ERFari in 2019 was
$-0.22$ W m-2 owing to reductions in emissions). To sample uncertainties, we
use a uniform distribution for the forcing efficiency of each species that
ranges from zero to a factor of two times the best estimate. The addition of
nine uniform random variables leads to an overall prior distribution of ERFari
that is approximately trapezoidal.

####  3.2.4 Carbon cycle feedbacks

The carbon cycle feedback strengths are dictated by the $r_{0}$, $r_{u}$,
$r_{t}$ and $r_{a}$ parameters in eq. 5. A four-dimensional kernel density
estimate using the 11 model CMIP6 model calibrations in [9] (their table 3)
were used to sample from.

####  3.2.5 Ozone forcing

The starting point for sampling ozone forcing is the IPCC AR6 assessment of
total (tropospheric plus stratospheric) ozone forcing from 1750 to 2019 of
+0.47 W m-2. This is based on the six-model mean historical time series from
CMIP6 models in [25]. A correction for historical warming is applied to this
time series using CMIP6 data of $-0.037$ W m-2 K-1 [27] which is backed out of
the forcing time series; this is because warming produces more water vapour,
which produces more OH radicals, which are a chemical sink for ozone. Ozone is
produced from chemical reactions that are facilitated from CH4, NOx, N2O,
NMVOC and CO, and destroyed by halogenated greenhouse gases. The contribution
of each factor to total ozone (in W m-2) is derived from six CMIP6 models in
[26] for the 1850–2014 period which is then scaled up for each species to
match the overall 1750–2019 assessment from AR6. The uncertainty in the ozone
forcing contribution from each specie is also taken from [26] and scaled.
These values are converted to a forcing efficiency for each specie, as is done
for ERFari. A Gaussian distribution is assumed for the forcing efficiency of
each parameter.

####  3.2.6 Effective radiative forcing uncertainty

IPCC AR6 considered the effective radiative forcing in a few aggregated
categories, and provided uncertainty ranges for each [16] (see table 2). For
many categories we used the same uncertainty ranges as AR6, which are applied
as a scaling factor to the historical and future forcing time series (table
2). For solar forcing, the amplitude of the solar variability around a mean
value is scaled by a factor, and an underlying trend over 1750 to 2019 is
added. For ozone and aerosols, we do not use the AR6 uncertainties as priors
but the priors described in the various preceding sections, which are similar
to or wider than the corresponding AR6 assessments.

####  3.2.7 CO2 initial concentrations

The pre-industrial concentrations of many greenhouse gases are uncertain. As
we use present-day CO2 concentrations as a constraint, we consider the
possibility that the atmospheric concentration of CO2 in 1750 was slightly
higher or lower than the IPCC’s best estimate. Clearly, as we do not have in-
situ observations from this time, the uncertainty in concentrations is greater
than in the present day. We use the AR6 assessment, drawing samples from a
Gaussian distribution with mean 278.3 ppm and 90% range 2.9 ppm [33].
Variation in pre-industrial CO2 values affects both the carbon cycle and the
CO2 forcing calculation (eq. 10).

Table 2: Effective radiative forcing scaling factor distributions used in the prior parameter sampling for FaIR. Symmetric uncertainty ranges are Gaussian, asymmetric ranges are half-Gaussian where two separate distributions are used for values below and above the best estimate. The solar forcing trend is Gaussian around +0.01 W m-2. Some forcers do not take their uncertainties from AR6 and use different prior distribution uncertainties given by the corresponding section number. Forcer | 90% uncertainty around best estimate, or source
---|---
CO2 | $\pm 12\%$
CH4 | $\pm 20\%$
N2O | $\pm 14\%$
Halogenated GHGs | $\pm 19\%$
Stratospheric water vapour | $\pm 100\%$
Ozone | section 3.2.5
Aerosol-radiation interactions | section 3.2.3
Aerosol-cloud interactions | section 3.2.2
Aviation contrails |  $-66\%$ to $+70\%$
Black carbon on snow |  $-100\%$ to $+125\%$
Land use and irrigation | $\pm 50\%$
Solar forcing amplitude | $\pm 50\%$
Solar forcing trend 1750–2019 |  $-0.06$ to $+0.08$ W m-2 (Gaussian)
Volcanic forcing | $\pm 25\%$

###  3.3 Constraining

From the 1,500,000 member prior ensemble we seek to produce a smaller
constrained ensemble of parameters that fit both observational constraints and
IPCC assessments. The prior ensemble produces a very wide range of climate
projections, most of which are implausible and not useful for using for future
projections (fig. 3(a)).

####  3.3.1 First step: root-mean-square comparison to historical warming

The first constraining step uses a root-mean-square error (RMSE) comparison
between the time series of historical temperature observations from the IPCC
assessment [33] and each FaIR ensemble member. Ensemble members that differ
from the historical time series with a RMSE of more than 0.16 K are rejected.
This rules out 92% of the original ensemble, and produces projections that are
much closer to historical observations (fig. 3(b)).

####  3.3.2 Second step: Distribution fitting

The second constraining step takes the ensemble members that pass the RMSE
test and attempts to simultaneously fill distributions of observations and
climate system metrics that were assessed in the IPCC AR6 WG1 [28]. The
distributions are given in table 3. A three-parameter skew-normal distribution
is used to fit a distribution to the percentiles given in table 3 (“Target”
column), which reduces to a Gaussian where the upper and lower ranges are
symmetric around the best estimate. Distribution fitting further narrows the
range of climate projections (fig. 3(c)), particularly for future warming.

(a) Prior

(b) After RMSE constraint

(c) Final reweighed ensemble

Figure 3: SSP2-4.5 emissions run in FaIR for the prior, RMSE-only constraint
and final, reweighted ensemble. In each subplot, the full minimum to maximum
range is in light grey, the 5-95% range mid-grey, 16-84% range dark grey and
ensemble median black. Historical temperature observations are shown in blue.

The size of the final reweighted posterior is user-definable, but is limited
by the number of runs passing the RMSE step, which itself depends on the size
of the prior. We choose 1001 ensemble members for the final ensemble. This
number is sufficiently large to fill the uncertainty space (consistent with
constraints) and small enough to be computationally lightweight in the FaIR-
DICE coupled system. Typically, ensemble sizes of a few hundred or a few
thousand are used in reduced-complexity climate model applications [34]. The
final 1001-member constrained, reweighted posterior distribution is, in most
cases, close to the target (table 3 “Ensemble” column).

Table 3: Target distributions, and percentiles from the reweighted constrained posterior, for climate system assessments used to constrain FaIR. All distributions are from the IPCC AR6 Working Group 1. | Target | Ensemble
---|---|---
Variable | 5% | 50% | 95% | 5% | 50% | 95%
ECS [K] | 2.0 | 3.0 | 5.0 | 1.97 | 3.04 | 5.20
TCR [K] | 1.2 | 1.8 | 2.4 | 1.34 | 1.84 | 2.48
Historical warming 1850–1900 to 1995–2014 [K] | 0.67 | 0.85 | 0.98 | 0.71 | 0.85 | 0.99
ERFari 1750 to 2005–2014 [W m-2] | $-0.6$ | $-0.3$ | 0.0 | $-0.60$ | $-0.30$ | +0.02
ERFaci 1750 to 2005–2014 [W m-2] | $-1.7$ | $-1.0$ | $-0.3$ | $-1.75$ | $-1.04$ | $-0.34$
Total aerosol forcing 1750 to 2005–2014 [W m-2] | $-2.0$ | $-1.3$ | $-0.6$ | $-2.06$ | $-1.37$ | $-0.59$
CO2 concentration in 2014 [ppm] | 396.95 | 397.55 | 398.15 | 396.87 | 397.56 | 398.24
Ocean heat content change 1971 to 2018 [ZJ] | 286 | 396 | 506 | 275 | 401 | 514
SSP2-4.5 warming 1995–2014 to 2081–2100 [K] | 1.24 | 1.81 | 2.59 | 1.22 | 1.83 | 2.70
Figure 4: Constraining the FaIR posterior to nine IPCC assessments of
observable and emergent climate metrics. In each distribution, the green curve
shows the prior distribution derived from a 1,500,000 member ensemble in
section 3.2. The purple curves shows the distributions after the RMSE
filtering step (section 3.3.1). The red curves show the final reweighted
posterior distributions (section 3.3.2). The aim is to match the red curves as
closely as possible to the target distributions, constructed from table 3, in
black curves.

The distribution matching procedure to produce a constrained ensemble used
here is similar and updated from that that was developed as part of the IPCC’s
Sixth Assessment Report (AR6) WG1, supported by the Reduced Complexity Model
Intercomparison Project (RCMIP) [35, 34, 16, 28]. This evaluated the ability
of simple climate models to reproduce historically observed climate change and
expert assessments of emergent climate variables. Models that were fit for
purpose were carried forward to provide climate projections from PB-IAM
emissions scenarios for the IPCC AR6 WG3 report [36, 37]. FaIR v1.6.2 was one
simple climate model that was deemed to be fit for purpose [16]. It was not
the only one, but of the three that passed the test it is (a) structurally
simple and (b) open source (at time of AR6 publication), lending it to be
easily used inside the optimization code of a CB-IAM [38, 39].

##  4 Historical spin-up and non-CO2 forcing

In the reduced version of FaIR in DICE, the carbon pools $R_{i}$, temperatures
in each energy balance model layer $T_{j}$ and non-CO2 effective radiative
forcing $F_{\mathrm{ext}}$ need initial values to start from in 2023.
Furthermore, $F_{\mathrm{ext}}(t)$ must be provided as an exogenous time
series in each ensemble member for 2023–2500.

To do this, we run the 1001-member final calibrated, constrained ensemble from
section 3.3.2 in FaIR, with all forcings (but no internal variability) and
uncoupled to DICE, for 1750 to 2500 in 3-year time steps. We run the SSP1-1.9,
SSP1-2.6 and SSP2-4.5 emissions scenarios, which provide the non-CO2 forcing
time series for the 1.5°C overshoot, well-below 2°C and “optimal” scenarios
that we run in DICE. These non-CO2 forcing time series are saved out for 2023
to 2500 for each ensemble member and scenario. The values of $T_{j}$ and
$R_{i}$ are saved out for 2023 for each ensemble member and scenario, and used
as initial conditions.

As described in section 2.2 in the main paper the historical all-forcing
implementation of FaIR uses a different relationship for calculating ERF from
CO2 than the CO2-only DICE-coupled implementation. We also save out the value
of $F_{2\times\mathrm{CO}_{2}}$ used in the computation of future CO2 forcing
in FaIR-DICE (eq. 10).

##  5 Climate uncertainty is not just ECS

Consideration of climate uncertainty in CB-IAM analyses may be limited to
cases where the eqilibrium climate sensitivity (ECS) is varied but other
climate system parameters are held at their best estimate or default values
[40, 41, 42, 43]. Notably, there is a strong anti-correlation between ECS and
aerosol radiative forcing in historically-consistent climate simulations [10,
44]. A high ECS could also be masked by a slow response timescale of the deep
ocean or a sea-surface-temperature mediated pattern effect [45].

We briefly demonstrate why full climate uncertainty needs to be taken into
account, and varying the ECS alone is not sufficient. In fig. 5, we run FaIR
over the historical period (1750–2023, at three year time steps) with median
parameter values from the 1001-member calibrated, constrained FaIR ensemble,
except for the climate feedback parameter $-\kappa_{1}$ which is inversely
proportional to ECS. We choose $-\kappa_{1}$ to produce ECS=2°C, 3°C and 5°C
(5th, 50th and 95th percentiles of the IPCC AR6 distribution). While ECS=3°C
(by design) reproduces historical temperatures well, ECS=2°C is too cool, and
ECS=5°C is too warm (substantially so). Different parameter combinations would
permit historically consistent climates with ECS=2°C (or lower) and ECS=5°C
(or higher) but with very different future warming outcomes. Therefore,
varying only ECS and not the full spectrum of climate uncertainty can result
in historically inconsistent climates which reduces confidence in projections
made using climate-economic models.

Figure 5: Historical climate projected with FaIR with the climate feedback
parameter varied to produce ECS=2°C, 3°C and 5°C and all other climate
parameters at their ensemble median values from the 1001-member calibrated and
constrained ensemble. Observed global mean surface temperature in black.

##  6 Comparison of results with DICE-2023 and DICE-2016

We now turn to the beta version of DICE-2023R which incorporates the climate
and carbon cycle modules from FaIR v1.0 [3]. Alongside the climate module
update, DICE-2023R includes more recent updates to the economic and population
assumptions over DICE-2016R (as does our version), but the economic part of
the IAM is structrually unchanged. For completeness, we also compare our
results to DICE-2016R. We compare “optimal” scenarios across the three
versions of DICE. DICE-2023R and DICE-2016R both produced 2°C and 1.5°C style
scenarios but under different definitions to ours and the comparisons are less
meaningful. The GAMS code for DICE-2023R and DICE-2016R was obtained from the
online appendix to [3] (accessed 16 June 2023), and included as part of the
software repository that accompanies this paper.

The comparison of the three models’ “optimal” scenarios is shown in fig. 6.
The improvement in carbon cycle dynamics between DICE-2016R (orange) and
DICE-2023R (green) is evident in the trajectory of atmospheric CO2
concentrations (fig. 6b) over the next 100 years. The DICE-2016R
concentrations are substantially higher than both our FaIR-DICE (blue) and
DICE-2023R despite lower total CO2 emissions (fig. 6a). Nevertheless, the
carbon cycle parameterisation in DICE-2023 is more sensitive than in our study
given higher CO2 concentrations over the next 100 years despite lower
emissions before 2100. As our CO2 concentrations in the present-day are
specifically constrained for in the posterior ensemble (main paper fig. 1b;
supplement fig. 4) and the carbon cycle models in FaIR-DICE and DICE-2023R are
similar, the differences are likely due to a slightly outdated set of carbon
cycle parameters used in DICE-2023R.

Figure 6: Climate projections under the “optimal” scenario for the FaIR-DICE
implementation used in this study (blue lines ensemble medians; shaded regions
5–95% ranges) compared to DICE-2023R (green) and DICE-2016R (orange).

Temperature (fig. 6c) responds to effective radiative forcing (fig. 6d). The
ERF is similar between the three versions, particularly our version compared
to DICE-2023R. This leads to fairly similar temperature projections until
2075, at which point DICE-2023R warms slower than our version. DICE-2023R uses
a two-layer impulse response model for climate which is similar to our three-
layer model, and has the same ECS and TCR as our ensemble median. The
differences may arise from the timescales of climate response; DICE-2023R uses
FaIR v1.0 default values which are a CMIP5 multi-model mean, whereas we use a
constrained probabilistic set drawn from CMIP6 model priors. The temperature
response in DICE-2023R is clearly a large improvement over DICE-2016R and well
within our uncertainty range.

In summary, DICE-2023R is capturing many of the key behaviours of our version
of FaIR-DICE and could be improved further with a re-calibration of the carbon
cycle and possibly the climate response. The models differ in other ways
including their population and economic assumptions, time step, time horizon,
and the fact that DICE-2023R considers non-CO2 greenhouse gases as a separate
forcing category. A full comparison of the two models is beyond the scope of
this study.

##  7 Comparison to Rennert et al. (2022)

One additional case we run is attempting to recreate the National Academies of
Science, Engineering and Medicine (NASEM) update to the SCC from Rennert et
al. [4]. We do this in order to attempt to benchmark our results with a more
commonly used discount rate of 2%.

Rennert et al. use a different method to calculate SCC, where 10,000
projections of population (which we adopt, fig. 1) and emissions of CO2, CH4
and N2O are generated following expert elicitation, and SCC is calculated as
the marginal discounted damages from an additional pulse of CO2 into the
future. They also use a different integrated assessment model (GIVE) and a
different climate damage function to calculate SCC. Climate uncertainty is
considered in the overall system and is provided by the IPCC AR6 calibration
of FaIR v1.6.2 [10, 28].

The discount rate $r$, equivalent to a risk-free interest rate, is given by
the Ramsey equation

$r=\rho+\eta g$ (13)

where $\rho$ is the pure rate of time preference, $\eta$ is the marginal
utility of consumption and $g$ is the per-capita annual growth in consumption
[46]. In order to achieve a near-term discount rate of 2%, Rennert et al.
calibrated the $\rho$ and $\eta$ parameters to fit future expected evolution
in interest rates from the economic literature that matched the desired value
of $r$ in the near term. They obtain $\rho=0.2\%$, $\eta=1.24$, implying
$g=1.45\%$.

We re-run our 1001 member ensemble of FaIR-DICE using $\rho=0.2\%$,
$\eta=1.24$, retaining SSP2-4.5 as the non-CO2 time series and include this
scenario as an additional projection in fig. 7. We find this leads us to a
median near-term discount rate of 2.5% and a median SCC of $84 (t CO2)-1
(table 4). Using the NASEM discount parameters leads to an SCC that is less
than half of that reported for the 2% discount case of $185 (t CO2)-1 in [4].
Rennert et al. also show results from a 2.5% discount rate, where SCC is $118
(t CO2)-1, 40% higher than our value with the same discount rate.

Figure 7: Emissions, climate and economic projections for four scenarios using
FaIR-DICE, with the additional scenario included that uses the discount
parameters from Rennert et al.. (a) CO2 emissions from energy and industrial
processes for the “optimal” (dark blue), Rennert et al. (light blue), 2°C
(pink) and 1.5°C (yellow) scenarios. (b) Temperature projections. (c) Total
effective radiative forcing projections. (d) Histogram of year-2023 SCC (in
2020$) on a log-log scale. In (a-c), light shading shows 5-95% range and solid
lines show ensemble medians. Table 4: Key results from the four scenarios
including our implementation of the Rennert et al. discounting parameters. All
correlations are significant at the 1% level.

Variable | “optimal” | Rennert et al. | Well below 2°C | 1.5°C overshoot
---|---|---|---|---
CO2 emissions 2050 (Gt CO2 yr-1) | 45 (39–49) | 33 (24–40) | 15 (2–24) | 2 (−14 to +12)
CO2 emissions 2100 (Gt CO2 yr-1) | 25 (5–41) | −6 (−21 to +12) | −19 (−23 to −5) | −23 (−23 to −13)
Net zero CO2 year | 2129 (2105–2152) | 2096 (2080–2112) | 2077 (2053–2094) | 2054 (2040–2079)
Social cost of carbon 2023 (2020$ (t CO2)-1) | 26 (15–44) | 84 (49–142) | 439 (237–934) | 1759 (821–4434)
Peak warming (°C relative to 1850–1900) | 3.1 (2.7–3.7) | 2.5 (2.1–2.9) | 1.8 (1.5–2.2) | 1.6 (1.3–2.1)
Warming 2100 (°C relative to 1850–1900) | 2.9 (2.4–3.6) | 2.5 (2.1–2.9) | 1.7 (1.5–2.0) | 1.4 (1.2–1.7)
Effective radiative forcing 2100 (W m-2) | 5.2 (4.4–5.9) | 4.2 (3.3–4.9) | 2.7 (1.9–3.3) | 1.9 (1.4–2.6)
ECS/SCC correlation coefficient | .51 | .66 | .74 | .74
ECS/2050 CO2 emissions correlation coefficient | −.48 | −.61 | −.72 | −.76
2014 aerosol forcing/SCC correlation coefficient | −.64 | −.63 | −.60 | −.59
2014 aerosol forcing/2050 CO2 emissions correlation coefficient | .61 | .61 | .59 | .56
Near-term discount rate (%) | 3.1 (3.1–3.2) | 2.5 (2.4–2.6) | 1.4 (1.2–1.6) | 0.6 (0.2–0.8)

The higher $r$ value that we obtain with the same discounting parameters
implies a higher near-term growth rate in our model compared to [4]. Indeed,
our first to second period per capita growth is 1.84% per year.

DICE and GIVE are different integrated assessment models, and should not
necessarily be expected to give the same results even if the climate system
components are the same in both models (which they are not). Rennert et al. in
their Table 1 show that switching from DICE to GIVE, and then updating the
damage function from DICE-2016R to theirs increases the SCC by 80%. Therefore,
it is expected that our SCC estimate would be lower than that provided in [4]
for a 2.5% discount rate. It should also be noted that achieving an ensemble
median 2% discount rate in DICE would require an iterative approach to
adjusting the $\rho$ and $\eta$ parameters, as unlike in GIVE, per-capita
growth rates in DICE are dynamic and solved as part of the overall system.

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