Catastrophic Climate Change, Population Ethics and Intergenerational Equity

Catastrophic climate change, population ethics and intergenerational equity

Aurélie Méjean∗1, Antonin Pottier2, Stéphane Zuber3 and Marc Fleurbaey4

1CIRED - CNRS, Centre International de Recherche sur l’Environnement et le Développement (CNRS, Agro ParisTech, Ponts ParisTech, EHESS, CIRAD) 2Centre d’Economie de la Sorbonne 3Paris School of Economics - CNRS 4Woodrow Wilson School of Public and International Aﬀairs, Princeton University

Abstract

Climate change raises the issue of intergenerational equity. As climate change threatens irreversible and dangerous impacts, possibly leading to extinction, the most relevant trade-off may not be between present and future consumption, but between present consumption and the mere existence of future generations. To investigate this trade-off, we build an integrated assessment model that explicitly accounts for the risk of extinction of future generations. We compare different climate policies, which change the probability of catastrophic outcomes yielding an early extinction, within the class of number-dampened utilitarian social welfare functions. We analyze the role of inequality aversion and population ethics. A preference for large populations and low inequality aversion favour the most ambitious climate policy, although there are cases where the effect of inequality aversion on the preferred policy is reversed. This is due to two effects: a higher inequality aversion reduces both the welfare gained when postponing climate policy to increase the consumption of present generations and the welfare gained when delaying extinction. We also show that the risk of extinction is the main driver of the preferred policy over climate damages affecting consumption. Keywords: Climate change; Catastrophic risk; Equity; Population; Climate-economy model JEL Classification: D63 ; Q01 ; Q54 ; Q56 ; Q5.

∗Corresponding author: [email protected] - CIRED, 45 bis avenue de la Belle Gabrielle, 94736 Nogent- sur-Marne, France

1 1 Introduction

The risk of abrupt and irreversible changes to the climate is one of the five reasons for concern identified by the IPCC (McCarthy et al., 2001; Field et al., 2014). Such abrupt changes are generally associated with tipping points (Lenton et al., 2008), like the shutoff of the Atlantic thermohaline circulation, the collapse of the West Antarctic ice sheet or the dieback of the Amazon rainforest. They may have indirect impacts, for instance through increased migration and conflicts (Reuveny, 2007; Hsiang et al., 2013). Catastrophic outcomes can also arise from discontinuities in the response of socio-economic systems to climate change.

From the very beginning, taking these catastrophes into account has been a daunting task of the economics of climate change (Dowlatabadi, 1999; Tol, 2003; Azar and Lindgren, 2003). Following the seminal work of Cline(1992) and Nordhaus(1994), the economics literature has mainly considered climate change as an issue of intertemporal consumption trade-oﬀ, where the costs of climate change mitigation lower consumption today, but increase consumption in the future as some damages due to climate change are avoided. In this framework, catastrophes are accounted for in three diﬀerent ways: as the certainty-equivalent loss of risky castastrophic damages (Nordhaus and Boyer, 2000), as a sudden jump in climate damages when the temperature rises above a given temperature threshold (Ambrosi et al., 2003; Pottier et al., 2015), or through a power law damage function with a large exponent (Hope, 2006; Ackerman et al., 2010; Weitzman, 2012). The stochastic nature of catastrophes is also sometimes explicitly accounted for, see for instance Peck and Teisberg(1995) and recent works on climate tipping points (Lontzek et al., 2015; Lemoine and Traeger, 2016).

A smaller strand of literature (Gjerde et al., 1999; Tsur and Zemel, 2009) considers that abrupt climate change introduces an irreversible regime shift in the sense that post-catastrophe welfare is independent from pre-catastrophe actions (usually it is assumed to be zero or constant), in the tradition of modelling environmental catastrophes of Cropper(1976) or Clarke and Reed(1994). The trade-oﬀ is then not only between present and future consumption, but also between present consumption and catastrophic risk reduction (Weitzman, 2009). In this context, Bommier et al. (2015) show that the representation of preferences in terms of risk aversion greatly matters for the appropriate level of mitigation.

The possibility that social welfare may drop to zero after the catastrophe can be interpreted as human extinction. This is indeed an outcome considered in the Stern Review (2007), which

2 uses the possible extinction of humanity to justify pure time discounting (see also Dasgupta and Heal(1979) for an early proposal). Because climate change and climate policy aﬀect the size of Earth’s population in these models, evaluating climate policies then raises the issue of valuing varying population sizes (Broome, 2012). Introducing the risk of human extinction compels to spell out the collective attitudes towards population size, a question that has been largely ignored in the literature and should be treated in an explicit way to inform climate policy analysis (Millner, 2013; Kolstad et al., 2014).

When the risk of climate catastrophe entails a risk of extinction of the human population, population ethics and risk preferences matter. In this paper, we examine the trade-off between consumption and the existence of future generations. To do so, we build an integrated assessment model that explicitly accounts for the risk of extinction of future generations. We evaluate different climate policies within the class of number-dampened utilitarian social welfare functions, and we analyze the role of inequality aversion and population ethics. This paper is thus at the crossroads of the literature on catastrophic climate change and the fledgling litterature on population ethics applied to climate change. If the importance of population growth for future greenhouse gas emissions has long been recognized (Gaffin and O’Neill, 1997; Kelly and Kolstad, 2001), and thus population control has been mentioned as a possible way to mitigate climate change (see Cafaro(2012) for an advocacy), it is only very recently that the role of population ethics for climate policy analysis has started to be discussed. Adler and Treich(2015) propose a prioritarian framework to evaluate scenarios with variable population in the case of climate change. Scovronick et al.(2017) show that putting a larger weight on population size warrants tighter climate policy whereas Lawson and Spears(2018) discuss how the preference for large populations can be overturned when environmental constraints are accounted for.

In this paper, the climate catastrophe is defined as the extinction of the human population. We thus account for the risk of extinction. Note that due to the speculative nature of the risk studied, our analysis should be taken mainly as a thought experiment, pointing to orders of magnitude and qualitative judgements instead of sharp, quantitative statements. This paper differs from previous work as it explicitly defines the population ethics adopted for the purpose of evaluating climate impacts on population size. It explores the impact of inequality aversion and of attitudes towards population size on the preferred climate policy when the possibility of such a catastrophe is accounted for. It adopts number-dampened utilitarianism (Ng, 1986) to evaluate and order climate policies. We use the integrated assessment model RESPONSE

3 (Ambrosi et al., 2003; Pottier et al., 2015), representing the interaction between climate and the economy, to assess the impacts of three scenarios (a business-as-usual case and two climate policies with different levels of stringency) in terms of future consumption paths and associated climate change. Our method differs from the standard approach in climate change economics, which computes the optimal climate policy (often the optimal carbon tax or the social cost of carbon) and examines its sensitivity to various model parameters. Instead, we follow the insight of Michal Kalecki who defines planning as “variant thinking” (Sachs, 1977). Other authors have used a method similar to ours, evaluating a set of exogenous climate policies defined in terms of yearly CO2 emissions, along with a business-as-usual scenario, see for instance Nordhaus and Boyer(1999), Nordhaus(2010), Dietz and Asheim(2012) and more recently Dietz and Matei (2016). The rationale behind our choice is for the analysis to be relevant to on-going policy debates on future emission trajectories.

The paper is structured as follows. Section2 presents the framework used to evaluate the effects of climate policies. The analytical results presented in section3 demonstrate that we cannot predict the impact of changes of ethical parameters (the inequality aversion coefficient and the population ethics coefficient) on policy decisions. This is because the preferred climate policy depends on the relative impact of these ethical parameters on the welfare gained due to a lower hazard rate and the welfare lost due to a lower consumption stream. In the numerical analysis presented in section4, we find cases where increasing inequality aversion favours the most ambitious climate policy. This rather unusual result is explained by the relative effect of inequality aversion on the risk and consumption components of the welfare difference. We also find that the risk of extinction is the main driver of the preferred policy over climate damages. Section5 concludes.

2 Evaluating the eﬀects of climate policies

Evaluating climate policies involve two steps: ﬁrst, each climate policy is translated into a social outcome by a climate-economy model; second, social outcomes are evaluated using a social welfare function and policies are ordered in terms of social welfare. We present the social welfare function in section 2.1 and the climate-economy model in section 2.2.

4 2.1 Number-dampened utilitarianism with a risk of extinction

Here we introduce the social welfare function used when social outcomes are certain, before extending it to risky situations.

Consider a social outcome x, characterized by a population stream n and a consumption stream c, indexed by t ∈ 0,T . Time horizon is T , which depends on social outcome x. Generation J K t is of size nt and enjoys a consumption per capita ct. We do not consider intra-generational inequalities, and we consider that consumption is equally distributed among people of the same generation, so that we can refer either to a generation or to a (representative) individual of this generation.

How to value such an outcome? We follow most of the existing literature and assume that aggregate welfare V from outcome x has the following generalized utilitarian form1.

Deﬁnition 1 (Number-dampened utilitarian social welfare functions) A social welfare function is a number-dampened utilitarian social welfare function (NDU SWF) if there exist real numbers β ∈ [0, 1], c ∈ R++ and η ∈ R+ such that:

T " 1 η 1 η # β X nt c − c − V (x) = N t − , (1) T N − η − η t=0 T 1 1

PT where NT = t=0 nt is the total (cumulated) population that comes into existence in social 1−η 1−η ct c outcome x. The utility enjoyed by an individual of generation t is 1 η − 1 η . The utility − − averaged across individuals and generations is:

T " 1 η 1 η # X nt c − c − AU = t − , T N − η − η t=0 T 1 1

β so that the social welfare (1) is the product of average utility and a population weight NT .

The NDU SWF is a variant of number-dampened utilitarianism proposed by Ng(1986), where welfare is the product of average utility and a concave function of total population size. The slight difference is that Ng’s concave function has a finite limit at infinity to avoid Parfit’s Repugnant conclusion, whereas ours has not. Related SWF average over periods, not over the

1There exists a limited literature in population ethics proposing alternatives to this generalized utilitarian formula, for instance equally-distributed equivalent criteria (Fleurbaey and Zuber, 2015), egalitarian criteria (Blackorby et al., 1996) or rank-dependent criteria (Asheim and Zuber, 2014, 2016).

5 entire history, and compute the weight accordingly, see Boucekkine et al.(2014) or Lawson and Spears(2018). The NDU SWF embodies the preferences of an evaluator through three important ethical parameters.

1. Parameter η is the (intergenerational) inequality aversion coeﬃcient. It determines the marginal utility of individual consumption. A high value of η implies that the utility function of individuals is strongly concave in consumption, so that marginal utility at high consumption levels is much lower than at low consumption levels. A more inequality averse social welfare function means that we are willing to sacriﬁce more to equalize consumption levels across generations.

2. Parameter β is the population ethics coeﬃcient and determines how valuable large populations are. Indeed, Eq. (1) embeds well-known views of utilitarianism when population size varies. Total or classical utilitarianism values the total sum of utilities, which happens when β = 1. Average utilitarianism, on the contrary, values the average utility, which happens when β = 0. Varying β between 0 and 1 spans cases between the total and the average utilitarian views.

3. Parameter c is a consumption per capita threshold. In the case of total utilitarianism (β = 1), it is the critical level of consumption, i.e. the level of consumption such that, if enjoyed by an additional individual, total welfare is left unchanged when that individual is added to the population. This parameter plays a key role in critical-level generalized utilitarianism (Broome, 2004; Blackorby et al., 2005). When β =6 1 however, the critical level is not constant and depends on population size and average utility. Parameter c still inﬂuences aggregate welfare, the value of changing population size, and thus of the risk of extinction.

Consider now a risky social outcome y, characterized as the previous one by a population stream n and a consumption stream c, but indexed by t ∈ N. The main diﬀerence is that, due to the risk of extinction, the time horizon is unknown. The risk of extinction is speciﬁed as a stream of hazard rate p. The hazard rate pt is the probability of extinction at the end of period t, conditional on existence at the beginning of period t. The survival probability at t Qt 1 is P t = τ− (1 − pτ ) (probability that the time horizon is greater than or equal to t). The ≥ =0 probability that humanity becomes extinct after generation t is Pt = ptP t, which we name the ≥ horizon probability, i.e. the probability that the time horizon is t. The social outcome y is fully

6 characterized by the triplet (n, c, p).

To extend the NDU SWF to social outcomes with unknown time horizons, we can view social outcome y as a collection of social outcomes of known ﬁnite horizons, each associated with a probability of realization (the horizon probability). We then simply assume that aggregate welfare W is the expected value of the NDU welfare of each possible outcome.

Deﬁnition 2 (Expected number-dampened utilitarian social welfare functions) A social welfare function is an expected number-dampened utilitarian social welfare function (ENDU

SWF) if there exist real numbers β ∈ [0, 1], c ∈ R++ and η ∈ R+ such that:

T " 1 η 1 η #! X∞ β X nt ct − c − W (y) = PT N − (2) T N 1 − η 1 − η T =0 t=0 T

X∞ β = PT NT AUT . (3) T =0

The NDU SWF does not exhibit pure time discounting. Contrary to the standard approach, it treats generations in a fair (ie. symmetric) way2. Yet in the ENDU SWF, we can recover an endogenous time discounting, which is similar to Tsur and Zemel(2009), where the endogenous discounting comes in addition to pure time discounting. Indeed, one can write Eq. (2) as follows: ! " 1 η 1 η # X∞ X∞ β 1 ct − c − W (y) = PT N − nt − . (4) T 1 − η 1 − η t=0 T =t | {z } θt

In Eq. (4), the scalar θt is tantamount to a time discount factor on the utility of generation t. This discount factor is not pure time discounting but stems from the uncertainty about the horizon. It depends on the stream of hazard rate and the attitudes towards population size as embodied in the population ethics coeﬃcient β.

Table1 summarizes the main notations of our evaluation framework.

2.2 Modelling the social outcomes of climate policies

We present the climate-economy model (section 2.2.1) and the way the risk of extinction is modelled and calibrated (section 2.2.2).

2See Ramsey(1928); Stern(2007) for arguments on why we should treat generations in a fair way.

7 nt size of generation t Nt total population up to date t pt hazard rate Pt horizon probability (there exists exactly t generations) ct consumption per capita at date t c consumption threshold parameter η inequality aversion coeﬃcient β population ethics coeﬃcient AUt average utility at date t W welfare

Table 1: Main notations

2.2.1 The climate-economy model

To translate climate policies into social outcomes to be evaluated by the ENDU SWF, we use the Integrated Assessment Model Response. Response (Ambrosi et al., 2003) belongs to the tradition of compact integrated assessment models (Nordhaus, 1994; Hope et al., 1993; Tol, 1997). It combines a simple representation of the economy and a climate module. The economic module is a Solow-like growth model with capital accumulation and exogenous population. It includes climate mitigation costs and a climate damage function, that is chosen here to be equivalent to that of DICE. A key parameter is exogenous technical progress (TFP growth), which decreases over time3

The climate module describes the evolution of the global temperature increase and radiative forcing, taking the emissions, that depend on the abatement level, as an input. The temperature increase feedbacks on the economy through the damage function. A thorough description of the model and its equations can be found in Dumas et al.(2012) and Pottier et al.(2015).

A policy (or scenario) i is speciﬁed as the decision variables of the model, that is a saving rate stream and an abatement stream. The model computes outputs from these decision variables. Model outputs include the stream of consumption per capita ci. Because population (conditional on existence) is exogenous, the generation size stream n does not depend on policy i (it will thus be common to all social outcomes). To complete the description of the social outcome yi = (ci, n, pi) of policy i, we only need to specify how the hazard rate is computed.

3In this paper, we made this hypothesis due to the very long time horizon considered. Indeed, assuming constant labour productivity growth would bring unrealistically high levels of consumption per capita at the time scales considered. This issue is usually not discussed in the literature due to the shorter time horizon used in models (typically a few hundred years).

8 2.2.2 The risk of extinction

Before describing how the hazard rate is modelled, let us motivate the introduction of a risk of extinction due to climate change. In our model (which follows standard speciﬁcations), climate damages on consumption are not suﬃcient per se to trigger the extinction of the human population. We consider that the risk of extinction does not stem from dwindling consumption, but from a societal regime shift induced by climate change. Such a regime shift can be iniated by extreme climate events, climate tipping points, or for instance an epidemic outbreak due to global warming. Such initiating events would provoke a chain reaction, e.g. through inap- propriate policy responses, large migration waves, general warfare, which would challenge the physical, political and social infrastructures of global society, leading to its collapse. This is the type of event described under the label “risk of extinction”: a catastrophe that destroys society and its institutions.

We model extinction as an abrupt event that occurs over the course of one period (a decade). This is of course a harsh simpliﬁcation of actual processes, which would be gradual and involve feedbacks between institutions, consumption, mortality and fertility. In reality, the ultimate consequences of such a catastrophe would probably take decades to unfold. Following the work of Cropper(1976) and Clarke and Reed(1994) in modelling environmental catastrophes, we consider that what is crucial for our analysis is that post-catastrophe variables do not depend on pre-catastrophe variables, so that the transfer of wealth (or population, which is more important for population ethics) cannot occur through the catastrophe, i.e. that the course of post- catastrophe events is inevitably set and cannot be acted upon. Since post-catastrophe cannot be changed, we abstract from the details of the extinction process and consider the simplifying assumption that what triggers the catastrophe also represents the end of time.

Whether such a catastrophic chain of events would actually lead to the extinction of all human beings is controversial and may be considered as highly unlikely. Given the large uncertainty surrounding the probability of such events, we attempt at covering a wide range of possible values, as described below. Due to the speculative nature of the risk studied, our analysis should be taken as a thought experiment, pointing to orders of magnitude and qualitative judgements instead of sharp, quantitative statements.

To further simplify this ﬁrst inquiry into the role of the risk of extinction, we assume that the hazard rate p depends on the temperature increase T only, and not, for example, on wealth or on a proxy of adaptative capacity. We therefore disregard any factor, social or natural, that

9 may mitigate the eﬀect of climate change on the risk of extinction. More precisely, we model the hazard rate p as a linear function of temperature increase T , above a temperature threshold

T0. The hazard rate at date t, pt, is truly p(Tt), a function of temperature increase Tt at date t. We assume that this function is valid in a range of temperature increase spanning from 1◦C to 10◦C, which includes the values of temperature increase presented in the numerical analysis (cf. section4). Figure1 illustrates the chosen functional form for the hazard rate.

  p0, if T ≤ T0   p(T ) = 1 p0 (5) p0 + b · (T − T0), if T0 ≤ T ≤ T0 + −b    1 p0 1, if T0 + −b ≤ T

p(T ) hazard rate as a function of temperature increase T −3 p0 minimum hazard rate, set at 10 per annum T temperature increase compared to pre-industrial levels (◦C) ◦ T0 temperature increase above which the hazard rate starts rising with temperature, set at 1 C ◦ b marginal hazard rate per C above T0

Given the very nature of the risk considered, there is no available data about its realization from which the parameters of the hazard rate function could be inferred. The calibration can thus only be illustrative and temptative. The calibration does not rely on the frequentist approach to probability but on the Bayesian or subjectivist approach, where the probability reﬂects informed though subjective beliefs about the links between events.

We build on the Stern Review (2007) which treats generations in a symmetric way, yet uses a 3 non-zero time discount rate, set at 10− to account for the possible extinction of humanity. We

p(T ) 1

b p0

1°C T

Figure 1: The hazard rate function

10 3 follow the argument and set p0 at 10− .

The marginal hazard rate b, i.e. the additional hazard rate per ◦C above T0, should be chosen in 3 conjunction with p0. To grasp its effects, consider that, at the minimum hazard rate p0 = 10− , the probability of survival after a hundred years is 90%. On top of that, if we assume that the temperature increase T is constant at 2 ◦C (1 ◦C above the threshold) for a hundred years, the 4 probability of survival after a hundred years would become 89% for b = 10− per ◦C, 82% for 3 2 b = 10− per ◦C and 33% for b = 10− per ◦C. If a b much smaller than p0 has an insignificant effect on the survival probability, a large b can have dramatic effects, provided the temperature increase exceeds the threshold. As b cannot be readily calibrated, we need to investigate how the choice of climate policy is sensitive to its value. We therefore explore a range of several orders of magnitude for b.

Although the calibration of the marginal hazard rate (b) is difficult (if not impossible) due to the inherent lack of data, we attempt at providing a reasonable range for that parameter. In order to define that range, we consider all the values of b that cannot reasonably be excluded. The maximum value of b can be derived from the upper bound of the temperature increase above T0 at which there is no certain extinction due to climate change. We cannot affirm with certainty that life on Earth would be impossible for the human species if the temperature increase T 1 reached 10◦C above T0. Therefore, we know that b should be lower than 10− per ◦C. We thus 1 set the maxmum value for b, bmax, at 10− per ◦C. In order to define a minimum value for b, we can argue that the additionnal hazard rate above p0 due to an increase of temperature of 1◦C above T0 would be at the minimum a few orders of magnitude below p0. We retain 4 7 bmin = p0 · 10− = 10− per ◦C. With this minimum value, the survival probability after a hundred years at a temperature increase of 10◦C becomes 0.90471, instead of 0.90479 for b = 0.

This shows that bmin would have little impact on the hazard rate and is therefore appropriately small.

3 Analytical results for marginal and non-marginal policy change

Climate policies aﬀect social outcomes, as they alter both the consumption path and the probability of catastrophic outcomes yielding an early extinction. For example, a stringent mitigation policy that reduces greenhouse gas emissions compared to a business-as-usual case will reduce consumption now and in the short run, but increase consumption in the long run by avoiding

11 climate damages; the hazard rate in all future periods will be also reduced, as temperature increase is lower.

We want to better understand how the preferred policy varies with the ethical parameters of the social welfare function, namely the inequality aversion parameter η and the population ethics parameter β. In the following, we analyse welfare variations in the case of a marginal policy change (section 3.1) and in the case of a non-marginal policy change (section 3.2).

3.1 Marginal policy change

We introduce two quantities that allow us to decompose welfare change in the case of a marginal change in climate policy. A ﬁrst key quantity to analyze climate policy is the well-known social discount rate.

Deﬁnition 3 (Social discount rate) The social discount rate from generation 0 to generation t, denoted ρt, is:

1 1 η ∂W ! t P β 1 ! t c t P N − ρ ∂c0 t T∞=0 T T . 1 + t = ∂W = β 1 (6) c0 P ∂ct T∞=t PT NT −

The formula of the social discount rate indicates that increasing η (when ct ≥ c0) increases the discounting of future beneﬁts and may thus reduce the value of the policy. Increasing β decreases the social discount rate, because future generations become more valuable as they increase total population size (see proof in Appendix A.1).

We have previously shown that the ENDU SWF exhibits endogenous time discounting, with a P β 1 discount factor θt = T∞=t PT NT − . From that we can recover a Ramsey-type formula.

t Let gt be the growth rate of consumption between 0 and t: ct = c0 · (1 + gt) , and δt be the t endogenous time preference rate that corresponds to the discount factor: θt = θ0/(1 + δt) . We can then write the social discount rate as a Ramsey formula in discrete time4.

η 1 + ρt = (1 + δt)(1 + gt) (7)

Contrary to the standard approach where only consumption discount is endogenous and time discounting stems purely from the normative view adopted, here both consumption and time

4 The Ramsey formula is mostly known in continuous time: ρt = δ + η.gt. See Dasgupta(2008) for discounting in discrete time in the context of climate policy.

12 discounts are endogenous. In our evaluation framework, the risk of extinction translates into an endogenous time preference rate.

A second important quantity describes how much the evaluator is willing a generation to pay to reduce extinction before the next period.

Deﬁnition 4 (Social value of catastrophic risk reduction) The social value of catastrophic risk reduction in period t, denoted ξt, is:

∂W P ∂PT β ∂p T∞=0 ∂p NT AUT ξ = − t = − t . (8) t ∂W η P β 1 ∂ct ct− T∞=t PT NT −

The social value of catastrophic reduction ξ was introduced in Bommier et al.(2015), and is related to ‘the value of statistical civilization’ as introduced by Weitzman(2009). The social value of catastrophic risk reduction bears resemblance to the value of a statistical life that plays a key role in analyzing individual attitudes towards mortality risk. It is formally very close as it measures a risk-consumption trade-oﬀ. However, the social value of catastrophic risk reduction has to do with the willingness to add people to a population rather than extending the life of existing individuals. The impact of parameters η and β on ξ is unclear.

With P t the survival probability at t, we have (see Appendix A.2): ≥

P β β T∞=t PT NT AUT − P tNt AUt ξ = ≥ . (9) t η P β 1 (1 − pt)ct− T∞=t PT NT −

The numerator is the diﬀerence between the expected welfare if the world follows its course and the welfare if the world last no longer than t. Thus, ξt will be positive if living longer than for just t generations yields an expected welfare gain. The denominator involves the chance of survival at t, the marginal social value of consumption at t and the discount factor θt.

Let us show that the social discount rate and the social value of catastrophic risk reduction can shed light on the welfare effects of climate policy. Consider a change in climate policy that does not affect the size of generation, but may alter consumption per capita and the risk profile. This change is small in the sense that it induces a marginal variation in social outcome dn = 0, dc, dp. We will take this marginal variation in social outcome as the starting point of our analysis

The welfare variation is: X∞ ∂W ∂W dW = dc + dp , (10) T ∂c T ∂p T =0 T T

13 which can be written:

1 1 ∂W ∂W !− ∂W !− ! ! ∂W X∞ ∂c0 ∂c0 ∂pT dW = ∂W dcT − ∂W − ∂W dpT (11) ∂c0 T =0 ∂cT ∂cT ∂cT ! ∂W X∞ 1 = (dcT − ξT dpT ) (12) ∂c (1 + ρ )T 0 T =0 T ∂W X∞ dcT ∂W X∞ −ξT dpT = + (13) ∂c (1 + ρ )T ∂c (1 + ρ )T 0 T =0 T 0 T =0 T | {z } | {z } dWc dWp

Equation 13 decomposes the welfare variation by separating the impact on welfare from a variation of consumption (dWc) and from of variation in risk (dWp). This way, we can attribute the welfare change either to a change in consumption or to a change in the risk proﬁle. Consider dW > 0: if dWc > 0 but dWp ≤ 0, the welfare gain is due to higher consumption. On the contrary, if dWp > 0 but dWc ≤ 0, the welfare gain is due to a lower risk. If both dWc > 0 and dWp > 0, the welfare gain is due to both higher consumption and a lower risk.

To see how this decomposition can be used to understand the eﬀects at play, consider a marginal increase in climate change mitigation (i.e. a marginally higher rate of abatement). The consumption proﬁle will then be marginally lower in the short term while it will be higher in the long-term due to avoided climate damages; the hazard rate will be also reduced.

The ﬁrst term (dWc) of equation (13) captures the intertemporal consumption trade-oﬀ. It may or may not play in favour of more stringent mitigation. The second term of equation

(13)(dWp) plays in favour of more mitigation (provided that all ξt are positive). The trade-oﬀ between consumption and the probability of future generations existing occurs if those terms have opposite signs (in our example, if the intertemporal consumption trade-oﬀ plays against more stringent mitigation).

At least two effects diminish the welfare gains that stringent climate policies may bring. First, as future generations are assumed to be richer (the model uses a declining but positive rate of growth of total factor productivity), a high inequality aversion favours present over future consumption, tilting the intertemporal consumption trade-off towards less mitigation. This is captured by the impact of inequality aversion η on the social discount rate ρt in the first term of equation (13)(dWc). Second, a high risk of extinction can be associated with large welfare gains from mitigation if climate policy happens to significantly reduce the hazard rate.

14 However, a high risk of extinction also shortens the time horizon and favours the short term in the intertemporal consumption trade-off. This effect is captured by the influence of the horizon probability Pt on the social discount rate ρt.

3.2 Non-marginal policy change

The above decomposition holds only for marginal variations in social outcomes. We now compare non marginal climate policy scenarios resulting in discrete variations in social outcomes. The change from policy i to policy j results in a move from social outcome yi = (n, ci, pi) to yj = (n, cj, pj), with a common generation size n.

The welfare diﬀerence between these two social outcomes5 is ∆W = W (cj, pj) − W (ci, pi). When ∆W is positive, social outcome yj (and policy j) is preferred; when it is negative, social outcome yi (and policy i) is preferred.

We want to understand how ∆W depends on η and β. It is impossible to obtain general results due to the complexity of the possible variations of social outcomes. It is however possible to make general statements when only the consumption streams or only the hazard rate streams vary, and can be univoquely ranked. We explore these two situations (3.2.1 and 3.2.2) before proposing a decomposition of welfare variation in the general case (3.2.3).

3.2.1 Welfare variations due to higher consumption

j j i i j i Suppose we have two social outcomes y = (n, c , p) and y = (n, c , p), such that ct ≥ ct. That is, social outcomes share the same characteristics, except that consumption is always greater in social outcome yj. We therefore have:

X β j i ∆W = Nt Pt AUt(c ) − AUt(c ) . t

This welfare variation is induced by a variation in consumption only and will be noted ∆cW by analogy with the decomposition of marginal welfare change (it will prove consistent with the general decomposition to be introduced later).

j i j i Given that ct ≥ ct, AUt(c ) ≥ AUt(c ) for any time horizon t. Hence ∆cW is positive and β increases in β, as Nt increases in β (as long as Nt > 1).

5With a slight abuse of notation, W (c, p) := W (y) = W ((n, c, p)).

15 Furthermore: t " j 1 η i 1 η # X nτ c − c − AU (cj) − AU (ci) = τ − τ . t t N − η − η τ=0 t 1 1

1−η 1−η j i j i cτ cτ However, when cτ ≥ cτ , 1 η − 1 η is decreasing in η, provided consumption per capita is − − 6 larger than c, which is itself larger than 1. So that ∆cW decreases in η: as inequality aversion increases, the welfare gain associated with higher consumption is reduced. These results are summed up in the following proposition.

Proposition 1 Assume that social outcomes yj = (n, cj, p) and yi = (n, ci, p) are such that:

i j 1. ct and ct are larger than c

j i 2. ct ≥ ct for all t

then ∆cW (= ∆W ) is positive, increasing with β and decreasing with η.

3.2.2 Welfare variations due to lower risk

j j i i j i Now consider two social outcomes considered y = (n, c, p ) and y = (n, c, p ), such that pt ≤ pt. That is, social outcomes share the same characteristics, except that the hazard rate is always lower in social outcome yj. Assume furthermore than the consumption stream is increasing and above the consumption threshold c.

We therefore have:

X β j i ∆W = Nt · (Pt − Pt ) · AUt(c) t .

This welfare variation only stems from a difference in risk profile and will therefore be noted j j Q j ∆pW . As Pt = pt . τ t 1(1 − pτ ), the contribution of a lower hazard rate at time t has two ≤ − opposite effects: there are fewer states of the world with exactly t generations, which reduces total welfare, but there are more states of the world with strictly more than t generations, which increases total welfare, since teh consumption of these generations is above the threshold c.

6 a1−η −b1−η Indeed, let a, b two positive real numbers such that a ≥ b, and let f the function f(η) = 1−η . We 0 (− ln(a)a1−η +ln(b)b1−η )(1−η)+a1−η −b1−η (1−(1−η) ln(a))a1−η −(1−(1−η) ln(b))b1−η have f (η) = (1−η)2 = (1−η)2 . But the function g(x) = (1 − (1 − η) ln(x))x1−η is decreasing in x provided x > 1, so that f 0(η) < 0.

16 Reducing the hazard rate at t comes down to swapping extinction at period t for extinction at later periods (where the welfare of the state of the world where there are exactly t generations, β Nt · AUt(c), is higher). If AUt(c) is above the subsistence level c and if AUt(c) is an increasing j i sequence, a lower hazard rate at time t increases social welfare. Then, when pt ≤ pt for all periods t, ∆pW is positive.

We show in Appendix A.3 that ∆pW is increasing with β and decreasing in η. The intuition behind this result follows. As η increases, the concavity of the utility of consumption increases, bringing the utility from the consumption of an individual closer to the utility of the threshold consumption c. Said diﬀerently, when inequality aversion increases, the welfare gain of consumption well above the critical level decreases, and therefore a population of individuals with large consumption levels has less value. Therefore, as η increases, the value in terms of welfare of an additional generation is reduced, i.e. the welfare gained due to a lower risk proﬁle decreases. The following proposition summarizes our results.

Proposition 2 Assume that social outcomes yj = (n, c, pj) and yi = (n, c, pi) are such that:

1. ct is larger than c and increasing

j i 2. pt ≤ pt for all t then ∆pW (= ∆W ) is positive, increasing with β and decreasing with η.

3.2.3 Trading oﬀ consumption and risk

When both risk and consumption vary, we can decompose the welfare diﬀerence ∆W = W (cj, pj)− W (ci, pi) as7:

∆W = W (cj, pj) − W (ci, pi)

= W (cj, pj) − W (cj, pi) + W (cj, pi) − W (ci, pi)

= ∆pW + ∆cW (14)

j j j i The first term ∆pW = W (c , p )−W (c , p ) is the part of the welfare difference that is explained j i i i by the variation of risk. The second term ∆cW = W (c , p )−W (c , p ) is the part of the welfare difference that is explained by the variation of consumption.

7We could also have written ∆W = (W (cj , pj )−W (ci, pj ))+(W (ci, pj )−W (ci, pi)). Using this decomposition give similar results than the one presented in4.

17 The difference in welfare ∆W can either be explained by a difference in welfare due to consumption (∆cW ), a difference in welfare due to risk (∆pW ), or both. Table2 sums up the possible cases as a function of the sign of the quantity at the top of each column.

product of welfare diﬀerences diagnostic

∆cW ∆W ∆pW ∆W · · + + social outcome preferred due to higher consumption and lower risk + - social outcome preferred due to higher consumption - + social outcome preferred due to lower risk

Table 2: Which of risk or consumption explains the diﬀerence in welfare?

If a social outcome yj exhibits a lower risk and higher consumption8 than social outcome yi, it is obvious that once the welfare diﬀerence is decomposed, ∆W will be positive, increasing with β and decreasing with η.

Consider now a diﬀerent situation where social outcome j exhibits a lower risk and higher consumption than social outcome i. In a sense, we have traded less consumption against less risk. What is the result of this consumption-risk trade-oﬀ? If we decompose ∆W in ∆pW and

∆cW , we see that Proposition2 (resp. Prop1) can be applied to ∆pW (resp. −∆cW ). The welfare diﬀerence ∆W is thus the diﬀerence of two positive functions, both increasing in β and decreasing in η.

Lower risk and higher consumption have opposite impacts on social welfare. As a consequence, the total impact is ambiguous, so is the dependence in ethical parameters β and η, even in this simplified case where the intertemporal consumption trade-off does not interfere with the consumption-risk trade-off. We thus need to use numerical examples to find out how the parameters may affect the ranking of social outcomes of contrasted climate policies.

4 Numerical evaluation of three climate policies

A numerical analysis is necessary to go beyond the analytical results presented in previous section. We set out three climate policies (section 4.1) and rank them using the ENDU SWF. We examine speciﬁcally the role of the risk of extinction (section 4.2) and the role of ethical parameters (section 4.3) in determining the preferred climate policy among the set of policies considered. 8Precisely “higher consumption” refers to the conditions of proposition1 and “lower risk” to those of propo- sition2.

18 4.1 The numerical experiment: Climate policies and their social outcomes

We introduce the climate policies and parameter ranges used in the numerical experiment. We consider three climate policies: one business-as-usual scenario, i.e. continuing past trends of emissions, calibrated on the middle of the range of business-as-usual scenarios from the database used for the ﬁfth assessment report of the IPCC, see (IIASA, 2014); and two abatement scenarios which are expected to limit the increase of the global temperature above the pre-idustrial level to 3 ◦C and 2 ◦C.

These policies are deﬁned in terms of saving rate and abatement compared to the baseline. The saving rate is constant for all policies, like in Golosov et al.(2014). Following Dennig et al.(2015), we choose the value of 25.8%, which is consistent with the observed world average gross saving rate (The World Bank, 2017). Figure2 illustrates the policies considered in terms of emissions and temperature. Figure3 illustrates the associated damages and consumption streams (for a speciﬁc set of technical parameters).

100 5 bau bau 3 ◦C 3 ◦C 80 ◦ ) 4 ◦ 2 C C 2 C ◦ per year)

2 60 3 GtCO 40 2

20 temperature increase ( 1 emissions (

0 0 2010 2050 2100 2150 2010 2050 2100 2150 time time (a) Emissions (b) Temperature increase

Figure 2: Emissions and temperature increase over time (bau, 3 ◦C, 2 ◦C)

Figure4 shows the evolution of the hazard rate p for the three policies considered, for two 4 diﬀerent values of the marginal hazard rate b. For b = 10− per ◦C (which is three orders of magnitude higher than the minimum value of b considered in the study), the hazard rate shows very little change over time.

In terms of ethical parameters used for the evaluation, we choose 0 ≤ β ≤ 1, which means there is at least a weak preference for large populations. The case β = 0 corresponds to average

19 8 800 bau bau 3 ◦C 3 ◦C ◦ ◦ 6 2 C 600 2 C of GDP)

% 4 400

damages ( 2 200 consumption per capita

0 0 2010 2050 2100 2150 2010 2050 2100 2150 time time (a) Damages (b) Consumption per capita

Figure 3: Damages and consumption over time (bau, 3 ◦C, 2 ◦C)

·10−3 ·10−3 6 6 bau bau 3 ◦C 3 ◦C 2 ◦C 2 ◦C 4 4

2 2 hazard rate (per annum) hazard rate (per annum)

0 0 2010 2050 2100 2150 2010 2050 2100 2150 time time (a) b = 10−4 per ◦C (b) b = 10−3 per ◦C

Figure 4: Hazard rate (bau, 3 ◦C, 2 ◦C) utilitarianism, while the case β = 1 corresponds to total utilitarianism. We test a wide range of values for the inequality aversion coeﬃcient η, spanning from 0.5 to 5.

Table3 shows a summary of the parameter values considered.

20 parameter description value η inequality aversion coeﬃcient 0.5 to 5.0 β population ethics coeﬃcient 0 to 1 b marginal hazard rate 10−7 to 10−1 per ◦C

Table 3: Value ranges of scenario parameters

21 4.2 Results: The role of the risk of extinction

This section examines the effect of the risk of extinction on the preferred policy between the 9 business-as-usual scenario and a 3 ◦C scenario . In a first approach, we do not account for climate damages on consumption, in order to primarily focus on the trade-off between consumption and the risk of extinction. In doing so, we do not consider the intertemporal consumption trade-off determined by the balance between abatement costs and climate damages. Abatement reduces both the stream of consumption and the stream of hazard rate over the whole period10. The most ambitious climate policy would therefore be favoured because of a reduction in the risk of extinction, while the least ambitious policy would be favoured because of an increase in consumption at early periods. We later examine the impact of climate change damages on the preferred policy.

4.2.1 Consumption vs. risk of extinction

We examine the preferred policy option between BAU and 3 ◦C as a function of the probability of extinction for a given degree of inequality aversion (η = 2.0) and for a given value of the population ethics coeﬃcient (β = 1.0, i.e. the case of total utilitarianism)11. The results presented in table4 show that only a zero marginal hazard rate b (i.e. the case of a purely exogenous hazard rate) leads to favour the BAU scenario (i.e. no abatement)With a marginal hazard rate equal to zero, mitigation has no impact on extinction and only reduces consumption. So climate policy necessarily reduces welfare given that we exclude climate damages for now. With a marginal hazard rate superior to zero, there is a chance that climate action may avoid extinction, and the 3 ◦C scenario is favoured over the BAU.

The results show that the difference in the hazard rate always explains the preference for the 3 ◦C scenario, while the difference in consumption counteracts (in grey). Conversely, the difference in consumption explains the preference for the BAU scenario when the hazard rate is purely exogenous (i.e. for a marginal hazard rate b = 0.0 per ◦C), as hazard rate streams are identical in the 3 ◦C scenario and the BAU scenario. This means that when the 3 ◦C scenario is preferred

9We do not include the 2 ◦C scenario in this section for the sake of clarity 10Again, the stream of consumption is lower over the whole period due to the fact that we do not account for climate damages. This would not be the case if climate damages were accounted for, cf. Figure 3b. 11 Welfare is calculated by adding welfare over states of the world, not over time periods. In practice, Eq. (2) is truncated at a chosen time horizon. We ﬁnd that the choice of the time horizon of the model is crucial to correctly interpret the results. As the hazard rate p depends on the emissions path, the time horizon should be chosen so that the probability of survival at the end of the period is close to zero for all the emission paths considered, in order to ensure that long term welfare is not overlooked when calculating aggregate welfare.

22 b (per ◦C)

0 10−7 10−6 10−5 10−4 10−3 10−2 10−1 BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C

∆ct causes ∆W , ∆pt plays no role

∆pt causes ∆W , ∆ct counteracts

Note that the preferred policy is written inside each cell, while the color of the cell indicates what determines the welfare diﬀerence (see section 3.2.3)

Table 4: BAU vs. 3 ◦C (β = 1.0, η = 2.0)

to the BAU, this is due to its eﬀect on reducing the hazard rate.

4.2.2 The role of damages

So far, the analysis has not accounted for the intertemporal consumption trade-oﬀ between abatement costs and climate damages. We now test the sensitivity of the results to the inclusion of climate damages in the model. Climate damages occur due to temperature increase, and are subtracted from production, thus reducing consumption. As before, they can be mitigated thanks to abatement, which comes at a cost. The results show that the preferred policies are unchanged whether or not climate damages are accounted for, with the exception of b = 0 per

◦C , i.e. for a purely exogenous hazard rate. This result means that the risk of extinction due to climate change is the main driver of the policy choice over the eﬀect of climate damages on consumption.

b (per ◦C)

0 10−7 10−6 10−5 10−4 10−3 10−2 10−1 w/o damages BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C

w damages 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C

∆ct causes ∆W , ∆pt counteracts or plays no role

∆pt causes ∆W , ∆ct counteracts or plays no role

∆ct and ∆pt cause ∆W

Table 5: BAU vs. 3 ◦C with and without damages (β = 1.0, η = 2.0)

The table shows that the 3 ◦C scenario is preferred due to both the risk and the consumption eﬀects (again with the exception of b = 0 per ◦C). This contrasts with the case without

23 climate damages, where the 3 ◦C scenario was preferred due to the risk effect alone, while the consumption effect played in favour of the BAU. The effect on consumption is indeed expected to favour the 3 ◦C scenario, as the reduced consumption of future generations due to climate damages can impact total welfare in a significant way. The results also show that when damages are accounted for and when there is no risk of extinction (b = 0 per ◦C), the 3 ◦C scenario is preferred to the BAU scenario. This result is in accordance with the results of the Stern Review 3 (2007), which, assuming a pure time preference of 10− , concludes that the avoided damages of a 3 ◦C scenario outweigh its abatement costs, and that such a policy should be pursued. It is 3 interesting to note that, with damages, ∆ct no longer causes ∆W when b increases from 10− 2 to 10− per ◦C. Indeed, the 3 ◦C scenario is then preferred due to the risk of extinction only, as the intertemporal consumption trade-off no longer plays in favour of the 3 ◦C scenario for higher values of b. This is due to the fact that with a higher marginal hazard rate, the benefits of abatement on future consumption through avoided climate damages have less weight in total welfare, as future generations are less likely to exist. This is an example of how the hazard rate and climate damages interact in the intertemporal consumption trade-off.

4.3 Results: The role of ethical parameters

We now examine the role of ethical parameters in determining the preferred policy in a case with climate damages in addition to the catastrophic risk of extinction due to climate change. We consider in turn the role of the population ethics coeﬃcient (β), and the role of the inequality aversion (η).

4.3.1 The role of population ethics

12 In table6, we examine preferred policy options (between the BAU, 3 ◦C and 2 ◦C scenarios) as a function of the marginal hazard rate b for various population ethics coefficients (various β) and for given values of the inequality aversion coefficient (η = 1.5 and η = 2.5). The contributions of the difference in hazard rate and of the difference in consumption to the difference in welfare is calculated when comparing the preferred policy and the next best policy (see section 3.2.3).

For a given marginal hazard rate b, a higher population ethics coeﬃcient favours the most ambitious climate policy. This result is consistent with intuition, as ambitious climate policies

12See Appendix B.2 for the comparison between the 3 ◦C and BAU scenarios, and between the 3 ◦C and 2 ◦C scenarios.

24 allow for larger cumulative population. Above a certain value of the marginal hazard rate, the value of β plays no role on the preferred policy. We show that the difference in hazard rate always explains the preference for the most ambitious climate scenario, while the difference in consumption counteracts. Conversely, the difference in consumption always explains the preference for the least ambitious climate scenario while the difference in hazard rate counteracts, except when the hazard rate is purely exogenous (i.e. for a marginal hazard rate b = 0).

β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−5; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [4.10−6; 9.10−6]

3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [2.10−6; 3.10−6]

3°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−6

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

(a) η = 1.5

β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [3.10−6; 10−5]

3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−6; 2.10−6]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

(b) η = 2.5

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

Table 6: BAU vs. 3 ◦C vs. 2 ◦C as a function of β and b (with climate damages)

25 4.3.2 The role of inequality aversion

We examine preferred policy options as a function of the marginal hazard rate b for various degrees of inequality aversion (η) and for given values of the population ethics coeﬃcient (β = 13 0, 0.1, 1). We examine the preferred policy between the BAU, 3 ◦C and 2 ◦C scenarios in table 7.

In the case of average utilitarianism (β = 0, table 7a), at a given b, a higher inequality aversion favours the least ambitious climate policy. As the marginal hazard rate b decreases, the minimum level of inequality aversion that justiﬁes the least ambitious scenario is reduced: richer generations are added, which enhances inequalities between generations. At high values of the marginal hazard rate b, the most ambitious climate scenario is favoured for all values of inequality aversion.

As climate damages are accounted for, the 2 ◦C scenario is preferred due to both the differences in hazard rate and consumption streams for relatively low values of inequality aversion (η ≤ 1.0) in the case of average utilitariansim (i.e. β = 0), see table 7a. This contrasts with the case without climate damages, where the most ambitious scenario is preferred due to the difference in hazard rate alone, while the consumption effect played in favour of the BAU (cf. Appendix C.1). The difference in consumption streams is indeed expected to favour the most ambitious climate policy for low inequality aversion, as climate damages reduce the consumption of future generations more than that of present ones. As future generations are richer than present ones in the baseline scenario due to technical change, this effect only occurs for low inequality aversion.

Note that, as before, the contributions of the difference in hazard rate and of the difference in consumption to the difference in welfare is calculated when comparing the preferred policy and the next best policy. When the 3 ◦C policy is preferred, the next best policy is either the BAU or the 2 ◦C. When the 3 ◦C scenario is preferred (see for instance table 7a), when the next best policy is the 2 ◦C, the corresponding cell is white (i.e. the 3 ◦C scenario is preferred to the 2 ◦C due to the difference in consumption, while the difference in hazard rate counteracts); when the next best policy is the BAU, the conrresponding cell is grey (i.e. the 3 ◦C scenario is preferred to the BAU due to the difference in hazard rate, while the difference in consumption counteracts).

The pattern is diﬀerent for β = 0.1 (table 7b). Increasing the inequality aversion η still favours the least ambitious climate policy for low values of η, but the eﬀect is reversed for η > 2.0. As

13see Appendix C.2 for the comparison between the 3 ◦C and BAU scenarios, and between the 3 ◦C and 2 ◦C scenarios.

26 shown in Prop.1, there are two opposite effects of increasing inequality aversion: it decreases the value of reducing the risk of extinction but it also decreases the cost of implementing the policy by making the first (poor) generations pay. It turns out that the decrease in the value of risk reduction is first faster and then slower that the decrease in the cost of the policy (see Appendix C.3).

The choice of β signiﬁcantly changes the preferred policy option for low values of the marginal hazard rate (b), but plays no role for higher values of b (top lines of the tables). In the case of average utilitarianism (β = 0.0), inequality aversion plays a role on the preferred policy option for a wider range of b values than in the case of total utilitarianism (β = 1.0). The choice of β does not inﬂuence the preferred policy when the hazard rate is exogenous. η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1]

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−3

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−4

2°C 2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−5

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [8.10−6; 9.10−6]

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [4.10−6; 7.10−6]

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3.10−6

2°C 2°C 3°C 3°C 3°C 3°C 3°C BAU BAU BAU 2.10−6

2°C 2°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 10−6

2°C 2°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 10−7

2°C 2°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

(a) β = 0 η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1]

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [4.10−6; 10−3]

2°C 2°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 3.10−6

2°C 2°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2.10−6

2°C 2°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 10−6

2°C 2°C 3°C 3°C 3°C 3 ◦C 3 ◦C 3°C 2°C 2°C 10−7

2°C 2°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

(b) β = 0.1 η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1] 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−6; 10−3] 2°C 2°C 3°C 3°C 2°C 2°C 3°C 2°C 2°C 2°C 10−7 2°C 2°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

∆pt and ∆ct cause ∆W

Table 7: BAU vs. 3 ◦C vs. 2 ◦C as a function of η and b (with climate damages)

28 5 Conclusion

With a probability of extinction that depends on temperature increase compared to pre-industrial levels, two eﬀects are competing. On the one hand, future generations are assumed to be richer, and a high inequality aversion thus gives preference to present consumption. This plays in favour of the least ambitious climate policy which preserves the consumption of the present, poorer generation. On the other hand, emission reductions can prevent extinction, which can favour ambitious climate policies. The main results are summarised below.

• We cannot predict a priori the impact of changes in the preference for large populations (parameter β) and inequality aversion (parameter η) on the preferred policy, even in the case without climate damages. The preferred policy depends on the relative eﬀect of η and β on the welfare gained due to a lower hazard rate and the welfare lost due to a lower consumption stream. A high inequality aversion reduces the welfare gained due to higher consumption stream. It also reduces the value of postponing extinction. A preference for large populations (high β) increases both the welfare lost due to a lower consumption stream, and the welfare gained as the size of the cumulative population increases due to a lower hazard rate.

• In the numerical analysis, a preference for large populations (high β) always favours the most ambitious climate scenario. This result is consistent with our ﬁrst intuition: a higher β gives as a larger weight to the welfare of future generations. Low inequality aversion (η) also usually favours the most ambitious scenario, as future generations are assumed to be richer. This is consistent with results from the literature. More precisely, the results conﬁrm that total utilitarians who are little inequality averse would tend to favour ambitious climate policies.

• However, we find cases where a higher inequality aversion coefficient favours the most ambitious climate policy. For instance, the 2°C policy is preferred to the 3°C policy as inequality aversion increases for the combination of parameters (η ≥ 1.5; β = 0.1, b ≤ 4 5.10− ). This pattern is also observed for higher values of β. This result is explained by the relative effect of inequality aversion on the risk and consumption components of the welfare difference. This analysis may thus help identifying new spaces of compromise between various ethical stances to set the ambition of climate policies. Indeed, a coalition could be formed between opposite sides of the ethical spectrum regarding inequality aversion.

29 4 • Finally, except for very low hazard rates (b ≤ 3.10− , i.e. a probability of survival in 2100 of 97% if the temperature increase compared to the industrial level reaches 2°C), the risk of extinction is the main driver of the preferred policy over climate damages, as the preferred policy is unchanged whether or not climate damages are accounted for.

Due to the speculative nature of the risk studied, our analysis should be taken as a thought experiment, pointing to orders of magnitude and qualitative judgements instead of sharp, quantitative statements. Also, the effect of ethical parameters on the preferred policy should be further explored. Indeed, the values of the ethical parameters may play a significant role on the optimal climate policy, which was not the focus of this exercise. The next step would thus be to examine the influence of the inequality aversion coefficient and of the population ethics coefficient on the optimal policy. A further step would be to consider more general welfare functions that disentangle risk aversion and inequality aversion, to explore whether one of these parameters has more influence on the results. Finally, it would be interesting to include the possibility of partial extinction, i.e. to allow for a variable population size as a function of the severity of climate change instead of the ‘all-or-nothing’ set-up used in this paper.

30 Acknowledgements

This research has been supported by the Chair on Welfare Economics and Social Justice at the Institute for Global Studies (FMSH - Paris), the Franco-Swedish Program on Economics and Philosophy (FMSH - Risksbankens Jubileumsfond), the ANR project EquiRisk, Grant # ANR-12-INEG-0006-01, the ANR project FairClimpop, Grant # ANR-16-CE03-0001-01 and Princeton University’s Program in Science, Technology and Environmental Policy (STEP). We are grateful for comments from Mark B. Budolfson, Francis Dennig, Maddalena Ferranna, Noah Scovronick, Dean Spears, Robert H. Socolow, Nicolas Treich and Céline Guivarch. We thank an anonymous referee from the French Association of Environmental and Resource Economists (FAERE).

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36 Appendices

A Proofs

A.1 The social discount rate

The social discount rate is

1 η P β 1 ! t ct t ∞ PT N − ρ = T =0 T − 1. t c P β 1 0 T∞=t PT NT −

P∞ P N β−1 Pt−1 P N β−1 T =0 T T T =0 T T Let us show that h(β) = ∞ = 1 + ∞ is decreasing in β. Indeed, P P N β−1 P P N β−1 T =t T T T =t T T

Pt 1 β 1 P β 1 Pt 1 β 1 P β 1 T−=0 PT ln(NT )NT − T∞=t PT NT − − T−=0 PT NT − T∞=t PT ln(NT )NT − h0(β) = P β 12 T∞=t PT NT − (A-1)

But given that NT is the total population of the T ﬁrst generations (hence increasing in T ), we have NT ≤ Nt 1 for all T ≤ t − 1, NT ≥ Nt for all T ≥ t and Nt > Nt 1. Hence: − −

Pt 1 β 1 P β 1 Pt 1 β 1 P β 1 T−=0 PT ln(NT )NT − T∞=t PT NT − − T−=0 PT NT − T∞=t PT ln(NT )NT − h0(β) = P β 12 T∞=t PT NT − Pt 1 β 1 P β 1 Pt 1 β 1 P β 1 ln(Nt 1) T− PT NT − T∞=t PT NT − − ln(Nt) T− PT NT − T∞=t PT NT − ≤ − =0 =0 P β 12 T∞=t PT NT − Pt 1 β 1 P β 1 ln(Nt 1) − ln(Nt) T−=0 PT NT − T∞=t PT NT − = − < 0. P β 12 T∞=t PT NT −

Hence, h0(β) < 0 and the social discount rate is decreasing in β.

37 A.2 The social value of catastrophic risk reduction

The social value of catastrophic risk reduction is

P ∂PT β T∞=0 ∂p NT AUT ξ = − t . t η P β 1 ct− T∞=t PT NT −

∂PT ∂Pt Here we explain how to derive Eq. (9). Note that = 0 if T < t, = Pt/pt = P t and ∂pt ∂pt ≥ ∂PT −P / − p T > t ∂pt = T (1 t) if . Hence,

X∞ ∂PT β β X∞ PT β NT AUT = P tNt AUt − NT AUT ∂pt ≥ (1 − pt) T =0 T =t+1   1 β X∞ β = P t(1 − pt)Nt AUt − PT NT AUT  (1 − pt) ≥ T =t+1 ! 1 β X∞ β = P tNt AUt − PT NT AUT , (1 − pt) ≥ T =t and P β β T∞=t PT NT AUT − P tNt AWt(c) ξ = ≥ . (A-2) t η P β 1 (1 − pt)ct− T∞=t PT NT −

A.3 Evolution of welfare diﬀerences

We show here in that ∆pW is increasing with β and increasing in η when social outcome j Q i has less risk. Recall that the horizon probability is Pt = pt. τ

P P Lemma 1 Let (ut)t N be a non null sequence such that τ t uτ ≥ 0 for all t and τ 0 uτ = 0. ∈ ≥ ≥ P If a is an increasing (resp. decreasing) sequence then τ 0 aτ uτ is positive (resp. negative). ≥

Proof. The proof relies on the transformation of the sequence (ut)t N. If we introduce Ut = ∈ P τ t uτ , the conditions on (ut)t N become that Ut is non-negative. We have ut = Ut − Ut+1. ≥ ∈ P P P So τ 0 aτ uτ = τ 0 aτ (Uτ − Uτ+1) = a0.U0 + τ 1 Uτ .(aτ − aτ 1). ≥ ≥ ≥ − j i The sequence Pt − Pt veriﬁes the condition of sequence u in the lemma.

38 Recall that: X∞ β j i ∆pW = Nt · (Pt − Pt ) · AUt(c), t=0

1−η 1−η AU c Pt nτ (cτ ) − c where t( ) = τ=0 Nt 1 η 1 η . − − β The derivative of Nt with respect to β is positive and increasing in t (because Nt is increasing); assuming that per capita consumption is increasing, AUt(c is increasing. The product of the two is increasing, and we can apply lemma1 to that the derivative of ∆pW with respect to β is positive. So that ∆pW is increasing in β.

Suppose the derivative of AUt(c) with respect to η is negative and decreasing in t. Since Nt is increasing, the product of the two would be decreasing (and negative). We could then apply lemma1 to show that the derivative of ∆pW with respect to η is negative. So that ∆pW would be decreasing in η.

Therefore, we only need to show that the derivative of AUt(c) with respect to η is negative and decreasing in t

Assuming that cτ > c, we know by the same reasoning as in the main text that the derivative 1−η 1−η (cτ ) c with respect to η of each term 1 η − 1 η is negative and decreasing in τ (since cτ is increas- − − ing). The derivative with respect to η of AU(t) is a weighted average of these derivative, so taht it is also negative and decreasing in t. This concludes our proof of the claim.

B The role of population ethics

B.1 Without climate damages

B.1.1 BAU vs. 3 ◦C

39 β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [3.10−6; 10−1]

BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−6; 2.10−6]

BAU BAU BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−7

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(a) η = 2.0 β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [4.10−6; 10−1]

BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−6; 3.10−6]

BAU BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−7

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(b) η = 2.5

Table A: Preferred policy and contributions as a function of β and b without damages (BAU vs. 3 ◦C)

40 B.1.2 3 ◦C vs. 2 ◦C

β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [5.10−6; 10−5]

3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [3.10−6; 4.10−6]

3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2.10−6

3°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−6

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(a) η = 2.0

β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [3.10−6; 10−5]

3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2.10−6

3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−6

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(b) η = 2.5

Table B: Preferred policy and contributions as a function of β and b without damages (3 ◦C vs. 2 ◦C)

41 B.2 With climate damages

B.2.1 BAU vs. 3 ◦C

β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−2; 10−1]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−3

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 10−4]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

∆ct and ∆pt cause ∆W

(a) η = 2.5 β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [2.10−6; 10−1]

BAU 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 10−6]

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(b) η = 3.0

Table C: Preferred policy and contributions as a function of β and b with damages (BAU vs. 3 ◦C)

B.2.2 3 ◦C vs. 2 ◦C

42 β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [3.10−6; 10−5]

3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−6; 2.10−6]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

∆ct and ∆pt cause ∆W

(a) η = 2.5 β b (per ◦C)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−6; 10−5]

3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

(b) η = 3.0

Table D: Preferred policy and contributions as a function of β and b with damages (3 ◦C vs. 2 ◦C)

43 C The role of inequality aversion

C.1 Without climate damages

C.1.1 BAU vs. 3 ◦C

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [4.10−6; 10−1]

3°C 3°C 3°C 3°C BAU BAU BAU BAU BAU BAU 3.10−6

3°C 3°C 3°C BAU BAU BAU BAU BAU BAU BAU 2.10−6

3°C 3°C BAU BAU BAU BAU BAU BAU BAU BAU [10−7; 10−6]

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(a) β = 0.0 η b (% per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−6; 10−1]

3°C 3°C BAU BAU BAU 3°C 3°C 3°C 3°C 3°C 10−7

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(b) β = 0.1 η b (% per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 10−1]

BAU BAU BAU BAU BAU BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

Table E: Preferred policy and contributions as a function of η and b without damages (BAU vs. 3 ◦C)

44 C.1.2 3 ◦C vs. 2 ◦C

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−4; 10−1]

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−6; 10−5]

2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(a) β = 0.0

η b (% per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [6.10−6; 10−1]

2°C 2°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 5.10−6

2°C 2°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C [3.10−6; 4.10−6]

2°C 2°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2.10−6

2°C 2°C 3°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 10−6

2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

(b) β = 0.1

η b (% per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−7; 10−1]

2°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 10−7

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

Table F: Preferred policy and contributions as a function of η and b without damages (3 ◦C vs. 2 ◦C)

45 C.2 With climate damages

C.2.1 BAU vs. 3 ◦C η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−2; 10−1]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−3

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [3.10−6; 10−4]

3°C 3°C 3°C 3°C 3°C 3°C 3°C BAU BAU BAU 2.10−6

3°C 3°C 3°C 3°C 3°C BAU BAU BAU BAU BAU [10−7; 10−6]

3°C 3°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

(a) β = 0

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−2; 10−1]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−3

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [3.10−6; 10−4]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2.10−6

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 10−6]

3°C 3°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

(b) β = 0.1

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−2; 10−1]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−3

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [3.10−6; 10−4]

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 2.10−6

3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 10−6]

3°C 3°C 3°C 3°C 3°C BAU BAU BAU BAU BAU 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

∆ct and ∆pt cause ∆W

Table G: Preferred policy and contributions as a function of η and b with damages (BAU vs. 3 ◦C)

47 C.2.2 3 ◦C vs. 2 ◦C

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1]

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−3

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 10−4

2°C 2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 10−5

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C [10−7; 4.10−6]

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

(a) β = 0

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1]

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [9.10−6; 10−3]

2°C 2°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 3.10−6

2°C 2°C 3°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C [10−6; 2.10−6]

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 2°C 2°C 10−7

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

(b) β = 0.1

η b (per ◦C)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−2; 10−1]

2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C 2°C [10−6; 10−3]

2°C 2°C 3°C 3°C 2°C 2°C 2°C 2°C 2°C 2°C 10−7

2°C 2°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 3°C 0

∆ct causes ∆W , ∆pt counteracts (or plays no role)

∆pt causes ∆W , ∆ct counteracts (or plays no role)

∆ct and ∆pt cause ∆W

Table H: Preferred policy and contributions as a function of η and b with damages (3 ◦C vs. 2 ◦C) C.3 Evolution of the welfare diﬀerence with inequality aversion

We examine the evolution of the terms of the welfare difference with η. FigureA shows the 6 behaviour of ∆cW and ∆pW with η in the case (β = 0.1, b = 10− per ◦C). This figure should be compared to the third row from the bottom of table Fb. The figure clearly shows that both terms decrease with η, but at a different pace.

The terms of the welfare difference cross twice. This coincides with the result that the preferred policy switches twice, first from the 2°C scenario to the 3°C scenario at low η, then from the 3°C 6 scenario to the 2°C scenario at high η (for b = 10− per ◦C). These results are consistent with the analysis presented in section3 which showed that anything coud be expected regarding the evolution of the preferred policy with respect to ethical parameters, as the welfare difference ∆W is a difference between two positive quantities that behave in the same way with respect to each ethical parameter.

2 10− of probability change of consumption change

3 10−

4 10−

5 10− diﬀerences in welfare 6 10−

7 10−

1.0 1.5 2.0 2.5 3.0 3.5 aversion to inequality η

Figure A: Contributions to the welfare diﬀerence between 2°C and 3°C as functions of η (β = 0.1, b = 10−6 per °C)