# Humanity does not play DICE

Antoine Ronsain, Alexandre Gondran, Estelle Malavolti

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Antoine Ronsain, Alexandre Gondran, Estelle Malavolti. Humanity does not play DICE. 2025. hal-05037064

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# Humanity does not play DICE

Antoine Ronsain¹,², Alexandre Gondran², Estelle Malavolti²,³

1 : AIRBUS, Toulouse, France

2 : ENAC, Toulouse, France

3 : Toulouse School of Economics, Toulouse, France

April 2025

## Abstract

The DICE model claims that technological progress can prevent global warming at low cost. However, many uncertainties about climate damages and backstop technologies are not taken into account in this model. Our analysis is based on a very large number of scenarios to account for the possible range of the climate damage function and the level of efficiency of future backstop technologies, as well as their financial costs. We show that if DICE's assumptions turn out to be less optimistic, it's a very risky bet to rely on technology alone to tackle climate change, with 82% of our scenarios leading to economic collapse. Then we introduce in the DICE model a social lever called economic sufficiency, defined as the policy possibility to produce less than the maximum production capacity. The optimal DICE projections then choose economic sufficiency to mitigate climate damage or to avoid economic collapse.

## Introduction

A series of numerical models, referred to as Integrated Assessment Models (IAM), have been developed to evaluate the economic impacts of global warming [22, 36]. Among these, the DICE (Dynamic Integrated Climate Economic) model stands out as a foundational IAM in the field of environmental economics. It establishes an optimal policy framework through the implementation of a carbon tax, which facilitates innovation aimed at decarbonizing the global economy. Results from the DICE model indicate that financing backstop technologies that support innovation can enable the pursuit of sustainable economic growth. The significance of technologies in mitigating or potentially reversing climate change is paramount within IAM, particularly in the context of the DICE model. However, DICE is predicated on strong assumptions regarding the effectiveness and economic costs of these backstop technologies. Our study analyzes the sensitivity of DICE outcomes to variations in these assumptions and proposes the integration of additional strategies, such as promoting sufficiency, to enhance the model's results.

With over 400 citations in IPCC reports, the DICE model has become a standard reference in economic and scientific community. However, together the forecasts and policy recommendations
tions are hotly debated because they rely on strongly contested assumptions. <fnref n="1" /> As an example, the DICE model estimates that a +6°C increase leads to a 12.5% loss in global Gross Domestic Product (GDP) [26], while some climate scientists [38] claim that a +5°C increase in temperature would threaten the human as a species. Furthermore, the magnitude of climate damage is highly uncertain, as is the timing of when the situation may become irreversible [4, 38, 21]. Although Nordhaus acknowledges that climate damage and mitigation costs are "very uncertain" [25], the DICE model remains a deterministic model. The DICE model assumes that in 2050 (respectively, in 2120), backstop technologies can reduce global greenhouse gas (GHG) emissions by 80% (resp. 110%) for a mean cost of 98$ (resp. 70$) per tCO2 (3% of GDP in each case). In addition, the literature is rather limited on the subject [14] and our work fills this gap.

The initial phase of our analysis involves testing the DICE model across a broad spectrum of damage functions. Subsequently, since the DICE model demonstrates robustness to variations in damage functions due to the presence of backstop technologies, we enhance our analysis by imposing constraints [13] on the effectiveness of these technologies and examining various cost levels.

By accounting for all these uncertainties, we demonstrate that DICE' conclusions are inverted: in 93% of the 7,007 scenarios analyzed, backstop technologies alone are insufficient to address global warming, and 82% result in climate collapse. Only 7% of the scenarios remain aligned with a technosolutionist perspective. Finally, we show, in a modified DICE model, that if we also consider the lever of sufficiency, then chances of avoiding climate collapse increases to 78%.

# Results

## Anatomy of the DICE model

The first DICE model, based on a cost-benefit analysis, was published in the 1990s and has been regularly updated (5th and last version in 2023 [26]). The anatomy of the DICE model reveals three blocks (see Supplementary Information A). First, an economic block based on the standard neoclassical [31] growth model (Ramsey-Cass-Koopmans model), hereafter referred to as the Ramsey-DICE model. Second, a climate block using the FaIR model [33], an emissions-based climate model that estimates the global temperature increase from world output (i.e. world GDP). In the DICE model, the temperature increase reduces world GDP via a climate damage function (negative feedback loop). These first two blocks, economic and climate, will be referred to as the BAU-DICE (Business-As-Usual DICE) model. The third DICE block formalizes the solution proposed by Nordhaus to solve the climate problem. This solution is based on a carbon tax on GDP that enables the development of backstop technologies. These technologies contribute to reducing carbon emissions and to reducing long-term climate damage. DICE's cost-benefit analysis is a trade-off between the cost to GDP of financing carbon-saving

<footnote n="1">¹For instance the level of the discount rate (too high), the way the carbon tax is modeled, the focus on global GDP only or the low impact of climate change through the damage function [9, 16, 28, 35].</footnote>
technologies and the benefits of avoiding current and future climate damage.

<figure><img src="image_2.png" /><figcaption>Figure 1: Forecasts of world production (GPD) per capita according to the different DICE blocks taken into account. Ramsey-DICE model with only the economic block of DICE taken into account. BAU-DICE model (Business-As-Usual) uses only the economic and climatic blocks. Full-DICE model with economics, climatic and backstop technologies blocks. Increasing temperature is indicated by a change in the line color to red (no temperature in Ramsey-DICE curve).</figcaption></figure>

In Figure 1, the Ramsey-DICE model shows the reference forecast of long-term world growth without taking global warming into account. The BAU-DICE model modifies the forecast to account for climate impacts : GDP per capita indeed reaches an economic ceiling around year 2300 due to an atmospheric temperature increase of +8°C, followed by a slight recession until year 2500 when the temperature rises to +12°C. Hence, in the BAU-DICE model, no climate collapse is seen, but long-term economic growth is no longer sustainable. Finally, the full DICE model shows that GDP per capita returns almost to the Ramsey level when a carbon tax is introduced: climate change is resolved by the development of backstop technologies on a global scale. As a consequence, temperature increase is kept below 2.9°C. The cost-benefit analysis is highlighted in Figure 1: the area between the Ramsey-DICE and the Full-DICE lines corresponds to the costs of the optimal carbon tax policy, while the area between the BAU-DICE and the Full-DICE lines corresponds to its benefits. In a nutshell, the Full-DICE model relies on technological fixes which need low level of investments (average investment < 1% GDP).

DICE sensitivity to climate damage function

The estimation of the climate damage function for temperatures below +2°C is made possible
by considering the local disasters associated with global warming (floods, droughts, megafires,
heat waves, hurricanes, population displacements, etc.). However, for situations above +2°C,
it is unfortunately impossible the system enters in an out of equilibrium state. Dietz et al. [9]
consequently modified DICE quadratic damage function for an S-curve based on Weitzman's
work [37] for low temperature increase (< +2°C) and rely on qualitative studies [21, 34] for higher increase of the temperature. We, instead, decide to create 169 different S-curves so that it covers a wide range of climate damage functions (see Method), from low impact to high impact and with different tipping points considered.

<figure><img src="image_3.png" /><figcaption>Figure 2: GDP per capita (1) and increase in temperature (2) forecasts, depending on the climate damage functions generated.</figcaption></figure>

Results on BAU-DICE model are as follows: GDP per capita (GDPc) falls between 2060 and 2130 for all damage functions tested. It corresponds in the figure Figure 2.1) to the lines in red. At the same time, we demonstrate that temperature rises by more than +4°C in 2150, as shown by the red lines in Figure 2.2). The collapses of the economies are caused by climate damage.

Notice that the BAU-DICE model is very sensitive to the damage function considered. In fact, using a stronger damage function has the effect of shortening time and accelerating the economic effect of climate change.

When mitigation policies (carbon tax and backstop technologies) are introduced with Full-DICE, the 169 scenarios systematically return to growth (blue lines in Fig. 2.1) close to that of Ramsey-DICE (black line). Moreover, the maximum investment (carbon tax) over this period of time is between 2% and 4% of GDP (see Fig. SI-2) compared to 2% in Nordhaus version. This shows that Full-DICE forecasts are robust to changes in the damage function, unlike the BAU-DICE model. In all scenarios, Full-DICE predicts growth very close to that without climate change. Therefore, the main conclusion from this first analysis is that the use of mitigation technologies in DICE is both very effective and inexpensive. Whatever the damage function considered, these technologies alone can mitigate and invert the temperature increase.

## Criticality of effectiveness and price levels of backstop technologies

The Full-DICE model allows sustainable GDP growth thanks both to the high efficiency and low cost of future backstop technologies. However, innovation is a costly and uncertain process. Hence, a scenario-based approach is developed to explore different possible futures. The scenario is based on two parameters that define the technology: the efficiency, $\lambda_\mu$, and the price of backstop technologies, $\lambda_B$.
<figure><img src="image_4.png" /><figcaption>Figure 3: For an average climate damage function (see Method), the evolution of the GDPc (Fig. 3.1 and 3.3) and temperature (Fig. 3.2 and 3.4) for different levels of efficiency $\lambda_\mu$ (Fig. 3.1 and 3.2) and different levels of price $\lambda_B$ (Fig. 3.3 and 3.4) of backstop technologies.</figcaption></figure>

Figures 3.1 and 3.2: The 100 lines refer to 100 values for $\lambda_\mu$: 1, 0.99,...0.01. Each line in blue corresponds to a rather high technological efficiency, if $\lambda_\mu$ is close to 1, while lines in red, when $\lambda_\mu$ is close to 0, stand for low efficiency technologies. More precisely, $\lambda_\mu = 1$ corresponds to Full-DICE assumption ; $\lambda_\mu = 0$ to BAU-DICE assumption<fnref n="2" /> ; $\lambda_\mu = 0.7$ means that the efficiency of the backstop technologies is at 70% of the Full-DICE model.

Figures 3.3 and 3.4: The 64 lines refer to 64 values of $\lambda_B$: 1, 2,...64. Each blue line corresponds to low price to implement the backstop technology while each red line stands for a higher price to implement the backstop technology. Having $\lambda_B = 1$ corresponds to the Full-DICE price for the technology. When $\lambda_B = 10$, this means that the cost is 10 times higher than the Full-DICE price.

Results presented in Fig. 3 are obtained considering an average damage function (see Method), but similar findings are found with our other damage functions. An economic downturn occurs when the maximum efficiency is less than 70% of Full-DICE assumption ($\lambda_\mu \le 0.7$, Figure 3.1). This downturn reveals to be critical for scenarios in which backstop technologies are only half as efficient as the Full-DICE model ($\lambda_\mu \le 0.5$).

The model also proves to be highly sensitive to changes in technology pricing assumptions. The effect of costs on temperature control is not linear (Fig. 3.4), leading to significant drops

<footnote n="2">²There is no investment in backstop technologies because they have no effect on reducing GHG emissions.</footnote>
<table><caption>Table 1: Classification of the 7007 scenarios according to GDPc in DICE</caption><thead><tr><th>Category</th><th>Sustainable growth</th><th>Negative outcome</th><th>Cimate collapse</th></tr></thead><tbody><tr><td># scenarios</td><td>465/7007 (7%)</td><td>799/7007 (11%)</td><td>5743/7007 (82%)</td></tr></tbody></table>

in GDP. (Fig. 3.3).

Indeed, while the climate policy ability to control temperature seems linear with a change in efficiency (Fig. 3.2), it is not the case when the cost of technology varies (Fig. 3.4). The marginal impact of technology cost on temperature control decreases when cost increases.

Our scenario-based analysis shows that if the efficiency of backstop technologies is not as high as expected in Full-DICE model, or if their prices are higher than expected, then DICE model conclusions can be completely opposite: technology can't fix global warming.

## Accumulation of uncertainties challenges DICE model recommendations

We have analyzed the sensitivity of the DICE model to three parameters studied separately one-at-a-time: climate damages, technology efficiency and technology costs. We now construct 7007 scenarios combining these three parameters (see Method). The DICE model projections for these scenarios are classified into three categories according to their economic outcomes<fnref n="3" />. The first category, called *sustainable growth*, corresponds to the case where GDPc continues to grow over the horizon time 2450. The initial version of DICE model is to be included in this category. In this category, the technology alone fix climate change. Conversely, if GDPc falls below half of its 2020 level (during the horizon period, 2450), the scenarios are labeled *climate collapse*. Between these two extreme situations, the scenarios referred to as *negative outcome* correspond to a GDPc that experiences stagnation or recession before the end of the period studied.

The table 1 shows the number of scenarios which outcome is classified in each of the three categories. Most of the GDPc projections lead to *climate collapse* (82% of the scenarios). Two thirds of the remaining scenarios are classified as *negative outcome*. This means that in all these scenarios the climate crisis leads to at least a temporary recession. Only in a small part of the scenarios, the model is able or willing to maintain growth.

When the damage exceeds the predictions made by the DICE model and the technological assumptions are not fulfilled, the model identifies optimal solutions that result in a lower GDP per capita. The model struggles to sustain growth because the efficiency of backstop technologies—the only tool available in DICE to address climate change—is inadequate. When faced with the challenge of maintaining GDP growth, the decision-maker may respond in one of two ways: either they invest immediately and to the fullest extent in backstop technologies to postpone a collapse (where $\mu[t] = \mu_{max}[t]$), as long as the costs of these technologies are manageable, or they choose to delay or reduce their investment in backstop technologies when the costs become prohibitive.

<footnote n="3">³We have noted a huge disparity in the level of the Social Cost of Carbon (SCC) between our scenarios; it varies from 308 $/tCo2 to 11.095 $/tCo2 in 2100. This is why the classification according to economic outcomes seems more relevant.</footnote>
<table><caption>Table 2: Classification of the 7007 scenarios according to GDPs in sufficiency-DICE</caption><thead><tr><th>Category</th><th>Sustainable growth</th><th>Negative outcome</th><th>Climate collapse</th></tr></thead><tbody><tr><th># scenarios</th><td>2240/7007 (32%)</td><td>3195/7007 (46%)</td><td>1572/7007 (22%)</td></tr></tbody></table>

The combination of these hypotheses helps us understand why decision-makers may not prioritize addressing climate change. The objective of the DICE model is to maximize short-term consumption. As a result, investing in costly measures to reduce climate damage and enhance long-term consumption is often seen as less appealing than taking no action at all.

## Sufficiency, a lever for addressing the problem of technological uncertainty

We have modified the DICE model to incorporate a social action lever, as we previously observed that relying solely on technological solutions is unlikely to effectively address the climate crisis when more realistic technological assumptions are applied in DICE. The policymaker, now represented as a decision variable in the model, has the ability to reduce CO2-emitting production (see the method for details). The GDP, assessed using a Cobb-Douglas production function, is now considered the maximum production capacity. The policymaker must then decide whether it is optimal to maximize production or if it is more beneficial to decrease carbon-emitting output to mitigate global warming and prevent future collapse.

Furthermore, we have made a further modification to the DICE model to adopt a long-term perspective that aligns with the physical realities of climate change. This adjustment in time preference places greater value on future generations by lowering the *rate of time preference* from 3.2% (the initial rate in DICE) to 1% in the utility function (see method for detailed information). We refer to this model as *sufficiency-DICE*.

If we apply the previous test to sufficiency-DICE, the classification of the results changes dramatically (see Table 2). The sufficiency lever is utilized in all scenarios. The share of production relative to capacity ranges from 18% to 100%, depending on the scenario and the time period. With sufficiency-DICE, the percentage of scenarios classified as "climate collapse" decreases from 82% to 22%, while those classified as sustainable growth rise from 7% to 32%. In addition, the number of scenarios categorized as "negative" also increases. The nature of these negative outcomes differs significantly from previous ones. In DICE, the decline of GDP occurs in the long term due to climate damage, whereas in sufficiency-DICE, most GDP declines occur in the short term as economic activity is intentionally reduced to better manage emissions.

Note that the Social Cost of Carbon (SCC) increases between the DICE and sufficiency-DICE versions. The range of SCC numbers at 2100 in our 7007 scenarios goes from 308$/tCo2 to 11,095$/tCo2 for the DICE model and from 594$/tCo2 to 14,939$/tCo2 for sufficiency-DICE.

Although policymakers previously focused solely on the technological aspects of mitigating climate change, they can now also influence emissions by adjusting production levels to what they deem optimal. Changes in pure-time preferences enhance incentives for long-term strategies. Policymakers are willing to invest in technologies, even if they are costly, and may reduce economic activity if these technologies are not sufficient to manage temperature control. These adjustments lead to significantly more optimistic projections and help avert potential
catastrophes.

However, despite social leverage and positive outlooks, one-fifth of the scenarios still indi-
cate the risk of climate collapse. This is mainly due to the insufficient adoption of affordable
and efficient technologies. Even with a reduced emphasis on technological criticality through
sufficiency-DICE, policymakers cannot effectively tackle the climate crisis without technological
advancements.

Discussion

Long-term prospective models like DICE offer a framework for public policy discussions. The DICE model was designed to address the following question: Can backstop technologies ef- fectively mitigate global warming? If so, what level of GDP investment is required? The deterministic version of DICE, developed by Nordhaus, answers the first question affirmatively and suggests allocating a small percentage of GDP (less than 2%) for the second.

Our study indicates that these results are significantly influenced by the assumptions regard-
ing the future development of backstop technologies and their associated costs. By employing
a scenario-based approach, we are better able to capture the variability and uncertainties sur-
rounding future technological advancements.

Our scenario-based analysis yields a conclusion that contrasts with previous findings on the first question: there are only rare instances—just 8% of scenarios—where backstop technologies effectively address global warming. Relying solely on technological solutions is a risky gamble. This is why we have introduced and tested an additional mitigation strategy in the DICE model: sufficiency, which is defined as the capacity to produce less than the maximum potential output of a given period. In our scenario-based approach, sufficiency proves to be a far more effective strategy to mitigate global warming. Without sufficiency, 82% of scenarios lead to climate collapse, whereas this figure drops to only 22% when sufficiency is considered. Our study recommends acting on both levers: technological and social [5, 15].

In this study, Nordhaus damage function has been challenged and modified based on Dietz
and Stern study [10] : an increase in temperature leads to a loss of GDP. However, damages
can be applied on many others parameters of DICE model such as the productivity [10, 23],
the economic growth rate [2, 3, 18], the capital stock [10], the population [1, 30]... The impact
on results is diverse: For example, applying damage on productivity or on the economic growth
rate generates more inertia. The DICE damage function is reversible and does not take into
account the irreversibility of climate tipping points [4]. Reaching a tipping point leads the
system into a new state and no return is possible, resulting in permanent damage [17].

Defining the impact level of a damage function is complex because, by definition, we only
have data on the impacts at low levels, which we then extrapolate to higher level [27]. This
uncertainty surrounding the climate damage function in a model can render the model invalid.
Therefore, when using a model like DICE, it is essential to adopt a scenario-based approach;
otherwise, the results may lack legitimacy.

The categorization of scenarios proposed here is only based on GDPc. Notions of regional
and social distribution are not treated at all. This comment does not apply only to our study
but also to the whole neoclassical theory. A. Krueger, a former World Bank chief economist, resumes the neoclassical idea as fixing poverty reduction is by "making the cake bigger (more GDP), not by trying to cut it up in a different way"[19]. DICE model is not meant to represent the rise or decrease of inequalities. Yet there is increasing empirical evidence of a correlation between rising economic inequality and global warming [11]. These distribution and allocation issues are widely studied, in the field of ecological economics [7, 8], for example through demand-side approaches [5, 15].

Much of the success and popularity of the DICE model stems from its simplicity, which can be encapsulated in just a dozen equations. However, this simplicity also presents a significant drawback: it leads to a rough aggregation of concepts on a global scale, such as GDP, population, capital, technology, and temperature. DICE only takes into account the environmental impact of global warming. Other planetary boundaries, such as biodiversity loss, are not modeled in DICE or in our study and left for future work.

Nonetheless, the straightforward nature of a predictive model is crucial for facilitating discussions within the scientific community regarding the relevance of the model's assumptions and interactions. Without this clarity, a model risks becoming a black box—an oracle whose validity is challenging to substantiate. Our study demonstrates that the conclusions drawn from a model as simple as DICE can change dramatically with slight modifications to certain initial parameters.

# Method

## Scenarios for climate damage function

The climate damage function is one of the key points of the model and at the same time one of the main sources of criticism and uncertainty [35]. It links the FAiR climate model to the economic growth model. An increase in temperature leads to more damage ($\Omega \in [0; 1]$), which affects economic production ($Y$):

$$Y[t] = (1 - \Omega[T[t]])Y^R[t] \quad \text{with } Y^R[t] \text{ production if no climate damage.} \tag{1}$$

Calibrated with US data on temperature and economic activity [16], Nordhaus draws a quadratic damage function in term of atmospheric temperature rise, $T$:

$$\Omega[T] = 0.003467 \times T^2 \tag{2}$$

He finds that a rise of +3°C would result in about 3.12% damage to GDP, while a rise of +6°C would result in 12.5% damage to GDP [26]. His justification for such low damage is that most economic activity is indoors and should not be much affected by global warming [16].

To calibrate damages, economists use cross-sectional but mostly panel data [2, 3]. However, the methodology of calibration is quite complex [23] and the results are fraught with uncertainty [24]. The uncertainties surrounding the estimation of the impacts of climate change make the DICE cost-benefit analysis highly critical. To get rid of this problem, cost-effectiveness
analyses have been proposed, with the aim of stabilising GHG emissions at a level deemed satisfactory. There is no longer any climate-economy coupling, just constraints on emission levels.

Some economists [9, 37] uses a S-curve damage function. There are many methods to define an S-curve [12, 29]. The one we have chosen follows a polynomial form (eq. 3) similar to that of Dietz and Stern [9].

$$ \Omega[T] = 1 - \frac{1}{1 + a_2 \times T^2 + a_n T^n} \qquad (3) $$

The quadratic coefficient $a_2 = 0.00284$ is calibrated to define damage for a small increase in temperature below +2°C [9]. For these values, the damage is similar to that of Nordhaus (see figure 4).

<figure><img src="image_5.png" /><figcaption>Figure 4: S-curve damage functions controlled by two parameters: $T_{50\%}$ and $T_{99\%}$; $T_{50\%} = 4.5^{\circ}C$, $T_{99\%} = 8^{\circ}C$ define the black S-curve. The Nordhaus damage function is the blue line. Red intervals indicate the range values of $T_{50\%}$ and $T_{99\%}$.</figcaption></figure>

In our study, we define two thresholds $T_{50\%}$ and $T_{99\%}$ to calibrate $\Omega[T]$ (eq. (3)) where $T_{50\%}$ is the temperature that leads to 50% of the damage on GDP (ie. $\Omega[T_{50\%}] = 0.5$) and $T_{99\%}$ the one that leads to 99% (ie. $\Omega[T_{99\%}] = 0.99$).

$T_{50\%}$ corresponds to a tipping point. For Lenton [21], a tipping point<fnref n="4" /> is a critical threshold near which a tiny perturbation significantly changes the state of the system, here our economic activity. Other climate experts add the notion of irreversibility to the tipping point definition [4]. Keller [17] introduces irreversible points in the damage function. Not so with DICE, where all damage is reversible. Climate experts have estimated many tipping points, such as the collapse of the AMOC<fnref n="5" /> or the melting of GIS<fnref n="6" />. According to the same studies, it is difficult to quantify either the temperature that triggers the tipping point, the magnitude of the damage, or the

<footnote n="4">⁴Here, tipping points are only considered through the damage function. But they can also be introduced into the climate model. For example, they can affect carbon uptake by the atmosphere or climate sensitivity to radiative forcing [20].</footnote>

<footnote n="5">⁵AMOC: Atlantic meridional overturning circulation.</footnote>

<footnote n="6">⁶GIS: Greenland ice sheet.</footnote>
time of occurrence (not necessarily instantaneous). Nevertheless, it seems that most tipping points occur between +3°C and +6°C [21] which corresponds to the range chosen for $T_{50\%}$.

Concerning $T_{99\%}$, there is few literature about a temperature that threatens the entire human species. Xu and Ramanathan [38] consider a species-level risk after a +5°C temperature rise. Based on their work, +5.5°C is considered the lower limit for $T_{99\%}$. In the absence of information on the upper bound, it is arbitrarily set at +10°C. We define 169 damage functions by fixed $T_{50\%}$, $T_{99\%}$ values, following :

$$3^{\circ}\mathrm{C} \leq T_{50\%} \leq 6^{\circ}\mathrm{C} \\ T_{50\%} + 2.5^{\circ}\mathrm{C} \leq T_{99\%} \leq 10^{\circ}\mathrm{C}$$

and a temperature step of 0.25°C. Some combinations of thresholds are not considered because they are not realistic: if $T_{50\%}$ is higher than $T_{99\%}$ or if the jump from one threshold to the other is too sharp : $T_{99\%} \geq T_{50\%} + 2.5C$ (see Fig. SI-5).

## Scenarios on maximum efficiency of the backstop technologies

The first parameters of the DICE model that play a key role in the technology is, $\mu_{max}[t]$, the maximum ability of backstop technologies to reduce global GHG emissions (level of effectiveness). In the DICE model, this corresponds to the upper bound of the decision variable, that is the emission control rate : $0 \leq \mu[t] \leq \mu_{max}[t]$. If $\mu[t] = 1$, this means that the global economy produces no GHG emissions at year $t$; this maximal abatement increases almost linearly from a few percent in 2025 ($\mu_{max}[2025] = 0.05$) to 100% in 2070 ($\mu_{max}[2070] = 1$) and even reaches 110% from 2120 onwards ($\mu_{max}[2120] = 1.1)^7$. This assumption is optimistic and so we introduce 100 scenarios that reduce this maximum effectiveness linearly by a scalar, $\lambda_{\mu}$, between 1 to 0 with a step of 0.01 : $\mu_{max}^{new}[t] := \lambda_{\mu} \times \mu_{max}^{old}[t]$ (see Method and Fig. SI-4).

## Scenarios on the price of backstop technologies

The second parameter of the DICE model that play a key role in the technology is $pback[t]$, the financial cost of the backstop technologies per ton CO$_2$ to decarbonize the whole production. Nordhaus fixes the $pback[2050] = 515$ means, for example, that the cost of decarbonized the half of the global emission can not exceed 1% of GPD in 2050 (see Method and Fig. SI-3). These assumptions seem very optimistic and there are many uncertainties about the prices of future backstop technologies. Then, we add 64 scenarios where the cost $pback[t]$ is multiplied by a scalar, $\lambda_B$, between 1 and 64 ($pback^{new}[t] := \lambda_B \times pback^{old}[t]$) [32].

## The introduction of uncertainty

There are many ways to introduce uncertainty into an optimisation problem such as the DICE model. It mostly depends on what needs to be tested: a policy, a model... The aim of this study

<footnote>$^7$With $\mu[t] > 1$, this means that the more GDP increases, the more GHG is removed from the atmosphere: there is a positive coupling between GDP growth and cooling of atmospheric temperature.</footnote>
is to assess the robustness of Nordhaus' results in the face of a change in model assumptions. We want to test the extent to which the model is able to maintain growth when some assumptions are questioned. We therefore performed a scenario-based analysis by running many separate optimisations on different parameter scenarios. Each optimisation, called a scenario, is defined by given specific assumptions that change from those of Nordhaus.

<table><caption>Table 3: Scenario-based analysis</caption><thead><tr><th>Sensitivity analysis</th><th>Type</th><th>Parameters Range/Step</th><th>Nb scenarios</th><th>Results</th></tr></thead><tbody><tr><td>Damage climate function</td><td>One-at-a-time (OAT)</td><td>T<sub>50%</sub> ∈ [3<sup>o</sup>C; 6<sup>o</sup>C] step = 0.25<sup>o</sup>C<br>T<sub>99%</sub> ∈ [T<sub>50%</sub> + 2.5<sup>o</sup>C; 10<sup>o</sup>C]</td><td>169</td><td>Fig. 2</td></tr><tr><td>Maximum effectiveness of backstop technologies</td><td>OAT</td><td>λ<sub>μ</sub> ∈ [0; 1] step = 0.01<br>T<sub>50%</sub> = 4.5<sup>o</sup>C and T<sub>99%</sub> = 8.5<sup>o</sup>C</td><td>101</td><td>Fig. 3.1</td></tr><tr><td>Cost of backstop technologies</td><td>OAT</td><td>λ<sub>B</sub> ∈ {1, 2, ...64}<br>T<sub>50%</sub> = 4.5<sup>o</sup>C, T<sub>99%</sub> = 8.5<sup>o</sup>C</td><td>64</td><td>Fig. 3.3</td></tr><tr><td>Technologies in DICE</td><td>Three-at-a-time</td><td>T<sub>50%</sub>, T<sub>99%</sub> (step = 0.5<sup>o</sup>C)<br>λ<sub>μ</sub> ∈ {0, 0.1...1}, λ<sub>B</sub> ∈ {1, 5, 11...61}</td><td>7007 = 49 × 11 × 13</td><td>Table 1</td></tr><tr><td>Technologies of Sufficiency-DICE</td><td>Three-at-a-time</td><td>T<sub>50%</sub>, T<sub>99%</sub> (step = 0.5<sup>o</sup>C)<br>λ<sub>μ</sub> ∈ {0, 0.1...1}, λ<sub>B</sub> ∈ {1, 5, 11...61}</td><td>7007</td><td>Table 2</td></tr></tbody></table>

## From DICE to sufficiency-DICE

The modification of DICE we propose is to allow the policy maker to reduce its production in order to reduce global carbon emissions. The production function defined in DICE, by a Cobb-Douglas function, Eq. (4), is now considered as the maximum production capacity and not the level of gross production :

$$Y_{gross}[t] \leq Y_{capa}[t] := a[t] \times l[t]^{1-\gamma} \times K[t]^{\gamma} \quad (4)$$

$$Y_{gross}[t] = (1 - \eta[t]) \times Y_{capa}[t] \quad (5)$$

The policy maker is able to define the quantity of goods and services to be produced each year. This choice is modeled by the parameter of realization, $\eta[t]$, which is an additional decision variable to the system, Eq. (5). It corresponds to the percentage of voluntary production reduction ($0 \leq \eta[t] \leq 1$). If $\eta[t] = 0$, then we find the Full-DICE model. The decision-maker now has 2 levers for reducing GHG emissions, one technological, $\mu[t]$, and one social, $\eta[t]$.

The control rate, $\mu[t]$, is a value between 0 and 1.1. Nordhaus links this parameter to the adoption of backstop technologies. If this parameter is between 0 and 1, it is a classical emission reduction, while if it is between 1 and 1.1, it implies negative emissions, with carbon capture and storage (CCS) technologies for example. Nordhaus does not provide further details and is deliberately rather vague about the technologies behind his model definition. Thanks to this control rate, total industrial GHG emissions, $\sigma[t] \times Y_{capa}[t]$, are reduced by $\mu[t] \times \sigma[t] \times Y_{capa}[t]$.

The social lever, $\eta[t]$, allows to reduce only the production that emits GHGs, therefore, in
sufficiency-DICE, industrial emissions are:

$$E_{IND}[t] = (1 - \mu[t] - \eta[t]) \times \sigma[t] \times Y_{capa}[t] \quad (6)$$

$$with \quad 0 \le \eta[t] \le \max(0, 1 - \mu[t]) \quad (7)$$

Indeed, it makes no sense, in this study, to voluntarily reduce carbon-free production. Then we have $\eta[t] = 0$ if $\mu[t] \ge 1$ and $\eta[t] \le 1 - \mu[t]$ if $\mu[t] \le 1$. In this way, the policy maker can choose the optimum balance between technological and social reduction. We don't go into detail about the political measures that can be taken to achieve sufficiency, but the latest chapter 5 of IPCC AR6 WGIII report [6] provides a few pointers, as do post-growth measures [15].

An additional modification is made to obtain sufficiency-DICE model, we reduce the pure time of preferences from 3.5% (in the original DICE version) to 1% to take more account of future generations. Finally, to overcome the side effects associated with a low pure time of preferences, we set the savings rate to its optimal value (see supplementary information).

## Acknowledge

Estelle Malavolti acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future ("Investissements d'Avenir") program, grant ANR-17-EURE-0010.

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# Supplementary Information

## A DICE model structure

The core of the DICE model is a neoclassical Ramsey-Cass-Koopmans (RCK) long-run growth model. It uses the total factor of productivity (TFP) as technological progress. The Finite Amplitude Impulse Response (FAIR) simple climate model is integrated into the RCK model to form a business-as-usual model (BAU-DICE). CO₂ emissions in the FAIR input are calculated by multiplying output (GDP) by a carbon intensity factor (emissions model). The temperature calculated by the FAIR model reduces the production of the RCK model by a quadratic damage function (climate damage model). To complete the DICE model, a carbon abatement sub-model is added, which defines the financial cost of each tonne of CO₂ emitted that is abated. This cost or tax simultaneously reduces the GDP allocated to consumption or savings, but also promotes backstop technologies that reduce CO₂ emissions and mitigate climate damage.

<figure><img src="image_6.png" /><figcaption>Figure SI-1: Structure of the DICE model. Black boxes represent main models while yellow boxes are intermediary model (or transition model) that link main ones. Ramasay-DICE and BAU-DICE models have only one decision variable, the saving rate (in blue); Full-DICE model adds a second decision variable, the emissions control rate (in blue).</figcaption></figure>
Full-DICE model : main equations

• Exogenous data

<table>
  <tr>
    <td>Time step</td>
    <td>T = {1, 2..., 101} and Δt = 5</td>
  </tr>
  <tr>
    <td>Years</td>
    <td>y[t] = 2015 + Δt × t; Y = {2020, 2025, 2030..., 2520}</td>
  </tr>
  <tr>
    <td>Population</td>
    <td>l[t] = l[t - 1] (10825 / (l[t - 1]))<sup>0.145</sup> with l[0]=7752.9</td>
  </tr>
  <tr>
    <td>TFP</td>
    <td>a[t] = a[t - 1] / (1 - g<sub>a</sub>[t]) with a[0] = 5.84164<br/>and g<sub>a</sub>[t] = 0.082 × e<sup>-0.0072 × Δt × (t - 1)</sup></td>
  </tr>
  <tr>
    <td>Carbon intensity factor</td>
    <td>σ[t + 1] = σ[t] × e<sup>Δt × g<sub>σ</sub>[t + 1]</sup> with σ[0] = 0.291<br/>and g<sub>σ</sub>[t] = min(-0.015 × 0.96<sup>t</sup>, -0.005)</td>
  </tr>
  <tr>
    <td>Backstop price per ton CO<sub>2</sub></td>
    <td>pback[t] = pback<sub>2050</sub> × e<sup>-Δt × b[t] × (t - 7)</sup><br/>with pback<sub>2050</sub> = 515 and<br/>b[t] = 0.01 if t &lt; 8, b[t] = 0.001 otherwise</td>
  </tr>
  <tr>
    <td>Maximum abatement rate</td>
    <td>μ<sub>max</sub>[t]</td>
  </tr>
  <tr>
    <td>Discount rate</td>
    <td>β = 0.0344</td>
  </tr>
</table>

• Decision Variables:

<table>
  <tr>
    <td>Saving rate</td>
    <td>0 ≤ S[t] ≤ 1, ∀t ∈ T</td>
  </tr>
  <tr>
    <td>Emissions control rate</td>
    <td>0 ≤ μ[t] ≤ μ<sub>max</sub>[t], ∀t ∈ T</td>
  </tr>
</table>

• Objective function: $\max \sum_{t \in T} (1 - \beta)^{\Delta t \times t} \times l[t] \frac{(C[t]/l[t])^{1-\phi}}{1-\phi}$ with $\phi = 0.9$

• Constrains:

<table>
  <tr>
    <td>Production gross</td>
    <td>Y<sub>gross</sub>[t] = a[t] × l[t]<sup>1-γ</sup> × K[t]<sup>γ</sup><br>with γ = 0.3 (capital elasticity)</td>
  </tr>
  <tr>
    <td>Production net</td>
    <td>Y[t] = (1 - Ω[t]) × (1 - Λ[t]) × Y<sub>gross</sub>[t]</td>
  </tr>
  <tr>
    <td>Capital</td>
    <td>K[t + 1] = K[t](1 - δ<sub>K</sub>) + S[t] × Y[t]<br>with δ<sub>K</sub> = 0.1 and K[0] = 302</td>
  </tr>
  <tr>
    <td>Consumption</td>
    <td>C[t] = (1 - S[t]) × Y[t]</td>
  </tr>
  <tr>
    <td>Industrial emission</td>
    <td>E<sub>Ind</sub>[t] = (1 - μ[t]) × σ[t] × Y<sub>gross</sub>[t]</td>
  </tr>
  <tr>
    <td>Atmospheric temperature rise</td>
    <td>T<sub>ATM</sub>[t] = FAIR_model(E<sub>Ind</sub>)</td>
  </tr>
  <tr>
    <td>Damage in % of GDP</td>
    <td>Ω[t] = 0.003467 × T<sub>ATM</sub>[t]<sup>2</sup></td>
  </tr>
  <tr>
    <td>Abatement cost in % of GDP</td>
    <td>Λ[t] = pback[t] × σ[t] × μ[t]<sup>θ<sub>2</sub></sup>/1000θ<sub>2</sub> with θ<sub>2</sub> = 2.6</td>
  </tr>
</table>

B Abatement cost in GDP rate

In the DICE model, backstop technologies allow a switch to less GHG emission pathways.
To introduce these technologies, a carbon tax is levied on GDP. This extra-cost of use those
backstop technologies, called abatement cost, is evaluated, each year, in GPD rate by the
following equation :

$$
\Lambda[t] = pback[t] \times \sigma[t] \times \mu[t]^{\theta_2} / 1000\theta_2
$$

with pback[t], σ[t] and θ₂ exogenous data. Only 0 ≤ μ[t] ≤ μmax ≤ 1.1 is a decision variable,
called emission control rate. The black line in Fig. SI-3 gives the maximum possible value of the
<figure><img src="image_7.png" /><figcaption>Figure SI-2: Share of GDP allocated to climate policy depending of the 169 damage functions.</figcaption></figure>

abatement cost ($pback[t] \times \sigma[t] \times \mu_{max}[t]^{\theta_2}/1000\theta_2$). The decision maker can control emissions with the decision variable $\mu[t]$. The cost of decarbonizing the world economy increases to 4% of GDP until 2070, then falls. $\mu[t]$ may vary under the black curve. In the ordinate, the cost is indicated and, using the five colored curves, the GHG emission rate is evaluated (green curve: 80% less emissions).

$$0 \le \Lambda[t] \le pback[t] \times \sigma[t] \times \mu_{max}[t]^{\theta_2}/1000\theta_2 \le 0.04$$

In the Nordhaus study, in the baseline DICE scenario, the control of emissions does not remain at its maximum value. Therefore, the maximum share does not exceed 2.5% of GDP in any period.

## C Maximum abatement rate

Nordhaus introduces backstop technologies that allow carbon emissions to be reduced. The rate of abatement is determined by a tax taken from production, thereby marginally limiting consumption and global welfare. Nordhaus introduces a hard constraint on the ability to reduce emissions: the maximum rate of abatement. The purpose of this maximum rate is to make the behaviour of the model more realistic. The author does not want to allow a strong and unrealistic climate policy. This maximum abatement rate increases almost linearly from 2020, where it is 5%, to 2070, where it is 100% (see figure SI-4). In 2120, the abatement rate reaches 110%.

It should be noted that when the abatement rate is above 100%, the higher the production, the higher the negative emissions. In other words, more than a simple decoupling between economic activity and carbon emissions, the model also allows for a negative correlation between these two parameters.
<figure><img src="image_8.png" /><figcaption>Figure SI-3: Abatement cost in GDP rate is $\Lambda[t] = pback[t] \times \sigma[t] \times \mu[t]^{\theta_2}/1000\theta_2$ with $pback[t]$, $\sigma[t]$ and $\theta_2$ exogenous data. Only $\mu[t]$ is a decision variable. The curves correspond to the value of CO$_2$ abatement in GDP rates, $\Lambda[t]$, over time for various $\mu[t]$ constant values. Reading example: the green curve corresponds to the percentage of GDP that needs to be invested to eliminate 80% of GHG emissions each years ($\mu[t] = 0.8$). The black line corresponds to abatement cost if $\mu[t] = \mu_{max}[t]$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure SI-4: Maximum abatement depending time, $\mu_{max}[t]$. A hundred scenarios are defined, in which the maximal efficiency of backstop technologies to decarbonate atmosphere are reduced by weighting the value by $\lambda_{\mu} \in [0, 1]: \mu_{max}[t] := \lambda_{\mu}\mu_{max}[t]$. If $\lambda_{\mu} = 0.7$, it means that the efficiency is reduced of 70% from the Nordhaus parameters.</figcaption></figure>

## D Justification for fixing the saving rate

In the case of sufficiency-DICE, two main changes has been done. First, we decreased the pure time of preference fixed at 3.5% in standard DICE to 1% and then we decided to fix the saving rater to its optimal long-term value based on Nordhaus computation (see eq 8)

$$s_{\infty}^{*} = \frac{\delta_{k} + .004}{(\delta_{k} + .004 \times \epsilon_{mu} + prstp) \times \gamma} \qquad (8)$$
<figure><img src="image_10.png" /><figcaption>Figure SI-5: Set of the 169 configurations of $T_{50\%}$ and $T_{99\%}$ in order to define the 169 damage functions. $3^{\circ}\text{C} \le T_{50\%} \le 6^{\circ}\text{C}$ and $T_{50\%} + 2.5^{\circ}\text{C} \le T_{99\%} \le 10^{\circ}\text{C}$ with a temperature step of $0.25^{\circ}\text{C}$.</figcaption></figure>

Where $\delta_K$ is the discount factor for capital, $\gamma$ is the elasticity of substitution for production, $\epsilon_{mu}$ is a factor that plays a role in how consumption increases welfare, and $prstp$ is the pure time of preferences. The reason we have fixed the saving rate is because of the change in the pure time of preferences. The lower this parameter is, the greater the side effect in the optimization calculation. In other words, the fact that there is a limit to the time horizon for the optimization has a strong impact on the optimal solution (see table 4).

When the rate of preferences is high (=3.5%), fixing or not fixing the saving rate is of little importance; but when the rate of preferences is low (=1%), it has a big impact on the result distribution. The side effect only affects the last time steps of the optimization if the rate of preferences is high, but when it is low, it affects almost all the time steps if the rate is low. Here, it leads the saving to 0, which is not realist). The social planner cannot solve climate change (technology is not effective enough), therefore he stops production to keep the temperature (and climate damage) low until the last periods. Then, it produce a lot as the total factor of productivity is very high.

A much more distant horizon should be defined to avoid this side effect. Fixing saving also avoids this problem. The same methods were used by Dietz and Stern [9] when they decided to reduce the pure time of preferences.

## E Comparison of economic forecasts in the different models
<table><caption>Table 4: Classification of scenarios according to GDPc tends in DICE depending on saving rate and pure time of preferences</caption><thead><tr><th></th><th>Saving rate</th><th>Sustainable growth</th><th>Negative outcome</th><th>Cimate collapse</th></tr></thead><tbody><tr><td rowspan="2">DICE-3.5%</td><td>fixed</td><td>6%</td><td>9%</td><td>85%</td></tr><tr><td>not fixed</td><td>7%</td><td>11%</td><td>82%</td></tr><tr><td rowspan="2">DICE-1%</td><td>fixed</td><td>9%</td><td>21%</td><td>70%</td></tr><tr><td>not fixed</td><td>7%</td><td>53%</td><td>40%</td></tr><tr><td rowspan="2">sob-DICE-3.5%</td><td>fixed</td><td>7%</td><td>18%</td><td>75%</td></tr><tr><td>not fixed</td><td>7%</td><td>16%</td><td>77%</td></tr><tr><td rowspan="2">sob-DICE-1%</td><td>fixed</td><td>32%</td><td>46%</td><td>22%</td></tr><tr><td>not fixed</td><td>11%</td><td>60%</td><td>29%</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>Figure SI-6: GDP per capita in different scenarios and in the different models. Scenarios are characterized by the following parameters: $T_{50\%}$, $T_{99\%}$, $\lambda_{\mu}$, $\lambda_{B}$.</figcaption></figure>