Discounting and Climate Policy

The social rate of discount is a crucial driver of the social cost of carbon ( SCC ), i.e. the expected present discounted value of marginal damages resulting from emitting one ton of carbon today. Policy makers should set carbon prices to the SCC using a carbon tax or a competitive permits market. The social discount rate is lower and the SCC higher if policy makers are more patient and if future generations are less affluent and policy makers care about intergenerational inequality. Uncertainty about the future rate of growth of the economy and emissions and the risk of macroeconomic disasters (tail risks) also depress the social discount rate and boost the SCC provided intergenerational inequality aversion is high. Various reasons (e.g. autocorrelation in the economic growth rate or the idea that a decreasing certainty-equivalent discount rate results from a discount rate with a distribution that is constant over time) are discussed for why the social discount rate is likely to decline over time. A declining social discount rate also emerges if account is taken from the relative price effects resulting from different growth rates for ecosystem services and of labour in efficiency units. The market-based asset pricing approach to carbon pricing is contrasted with a more ethical approach to policy making. Some suggestions for further research are offered. different measures for both intra-regional and intergenerational inequality aversion. They show that Thomas


Introduction
Climate policy is about making sacrifices now to curb global warming in the distant future. To assess climate policy, it is thus important to assess what the future benefits in terms of less global warming damages are worth today so that they can be compared with the sacrifices that must made today to curb emissions. Instead of looking at what the benefit of curbing emissions today, one could look at the cost of emitting one ton of carbon today. Both require one to know what social discount rates to use to assess what future benefits are worth today. For this purpose, it helps to define the social cost of carbon or SCC for short. The SCC is defined as the expected present discounted value of all current and present damages to aggregate consumption of emitting one ton of carbon today. It corresponds to a Pigouvian tax. 2 It is of immense practical interest to know what the appropriate social discount rate or SDR to use for evaluating climate policy should be both from a theoretical and an empirical point of view. This discount rate depends on ethical and economic considerations and has been subject to huge debate with on one end of the spectrum the descriptive market-based approach of Nordhaus (2017) with a relatively high discount rate and a low carbon price, and on the other hand of the spectrum the prescriptive ethical approach of the economist Stern (2008) and the philosopher  with a low discount rate and a high carbon price (e.g. Beckerman and Hepburn, 2007;Dietz et al., 2009;Rendall, 2019). It also depends on the uncertainties that affect future economic growth and emissions growth. In particular, the discount rate depends on the volatility and skewness of these growth rates. Most important of all, a strong case can be made for the social discount rate to decline with the length of the horizon of climate policy.
The outline of this survey is as follows. Section 2 discusses the celebrated Keynes-Ramsey rule for the SDR and a simple relationship between temperature and cumulative carbon emissions.
These are then used to derive the SCC and interpret its various ethical and economic drivers.
Section 3 gives numerical estimates of the SDR and the SCC. Section 4 derives a related expression for the SCC from underlying models of the dynamics of atmospheric carbon and of temperature. Section 5 extends the basic model of the SDR and the SCC to allow for stochastic shocks to the rate of economic growth and emissions. This allows one to identify the effects of prudence and self-insurance as well as of impatience, affluence, and growing damages on the SCC. Section 6 discusses the large upward effects of tail risk in the climate sensitivity and damages, risks of tipping of the climate system and long-run risk about future economic growth on the SCC. Section 7 discusses the effects of correlation between damage risks, climatic risks, and economic risks on the SDR and the SCC. Sections 8 and 9 discuss the effects of macroeconomic disasters on the SDR and SCC and highlights the connections and differences between the theory of the SCC under uncertainty and asset pricing theory. Section 10 discusses arguments for a declining expected value of the SDR and its effect on the SCC if the SDR is itself uncertain. Section 11 examines the survey and empirical evidence on the appropriate, certainty-equivalent SDR to use and by how much it declines. Section 12 reviews evidence from experts on impatience and intra-and intergenerational inequality aversion. Section 13 highlights dual discounting and the endowment approach, and their impact on the SCC. Section 14 concludes with a summary of the key insights. Section 15 offers some nuancing comments related to the differences between the social and the private discount rates and intergenerational aspects of climate policy and suggestions for further research.

Discounting and the Social Cost of Carbon: Deterministic Case
What is the cost today of one dollar of damages at some future time t? Let this cost be X. To is the Keynes-Ramsey rule and can be interpreted as follows. First, the more impatient policy makers are (the higher RTI) the higher the social discount rate. Second, the richer future generations (the higher the future rate of economic growth GR) and the bigger the aversion to intergenerational inequality (the higher IIA) the higher the social discount rate and thus current generations are less willing to make sacrifices today to avoid future global warming damages.
This expression for the SDR can be used to evaluate the SCC. To do this, two additional assumptions are required. First, the ratio of damages to aggregate production Y is a simple linear function of temperature and denote the marginal effect of temperature on the damage ratio by MDR. Second, temperature is a linear function of cumulative emissions as suggested by insights from climate science (e.g. Matthews et al., 2009;Allen et al., 2009) and has been applied to derive climate policy (e.g. van der Dietz and Venmans, 2019a;Dietz et al., 2020 the SCC is also unaffected by IIA and the rate of growth of the economy.
Variants of these formulae for the SCC have been used to advise climate policy. In the absence of any other distortions or market imperfections the first-best optimal response for policy makers is to ensure that carbon emissions are priced at a price equal to the SCC. One way of 6 doing this is to have a carbon tax and rebate the revenues in lump-sum manner. Another way is to have a competitive global market for emission permits.  Ramsey (1928) argued that discounting the welfare of future generations is ethically indefensible and arises from "the weakness of imagination". Broome (2012) takes a normative stance and arrives at a similar position. It is thus reasonable to set the rate of time impatience to zero, RTI = 0. This is close to the 0.1% per year used by the Stern Review to reflect the remote possibility of a meteorite ending the world as we know it (Stern, 2007). The trend rate of growth of the economy is taken to be 2% per year, GR = 0.02. Finally, following Gollier (2011 a coefficient of relative intergenerational inequality aversion of 2 is taken, so that IIA = 2. Armed with these assumptions, it follows that SDR = 0 + 2 x 0.02 = 4% per

Estimates of the Social Cost of Carbon
year. The growth-corrected discounted rate is thus 2% per year. Equation (2) becomes Since one ton of carbon is 12/44 tons of CO2, the SCC starts at 18.5 $/tCO2 and grows from then on at a yearly rate of 2%. Each year the SCC must also be adjusted for inflation. 4 Burke et al. (2015) show that overall economic productivity is non-linear in temperature for all countries with productivity peaking at an annual average temperature of 13  C and declining strongly at higher temperatures. They find that expected global losses are approximately linear in global mean temperature, with median losses 2.5-100 times larger than prior estimates for 2  C. It is thus assumed that the damage ratio is linear in temperature.
It is of interest to examine the sensitivity of the SCC with respect to various assumptions.
Nordhaus (2017)  This yields for the base case with SDR -GR = 2% per year a SCC of 53.4 $/tC or 14.6 $/tCO2. This is a bit less than the 68 $/tC or 18.5 $/tCO2 in our base scenario (see equation (2) above).
The bigger the proportions of carbon that stay a short time in the atmosphere, the lower the SCC. Analogous expressions can be obtained to evaluate the SCC for the linear 2-box model of atmospheric dynamics of Golosov et al. (2014) and the linear 3-box model of Gerlagh and Liski (2018). 7 Most of the models of the carbon cycle and temperature dynamics in the integrated assessment models used by economists have too much temperature inertia and give misleading insights into the magnitude of the SCC (Dietz et al., 2020). A more reliable approach is to model temperature as linear function of cumulative emissions as will be done from now on.

Discounting and the Social Cost of Carbon: Stochastic Case
Returning to the base model that we started with and where cumulative emissions are the driver of temperature, now consider how the SDR and the SCC are affected if future economic growth 9 and emissions are stochastic. The SCC at time t is the expected present discounted value of all current and future damages from time t onwards of emitting one of carbon at time t. Hence, Using these results in (4), the social cost of carbon for a power utility function becomes Power utility functions do not separate intergenerational inequality aversion or IIA (the inverse of the elasticity of intertemporal substitution) from risk aversion or RA. If recursive preferences are used as put forward by Epstein and Zin (1989) in discrete time and Duffie and Epstein (1992) in continuous time, van den Bremer and van der Ploeg (2019) show that (5) becomes Note that the social discount rate in (6) can be rewritten as The first term RTI is the impatience effect: more impatient policy makers use a higher SDR and lower SCC, so price carbon less vigorously. The second term IIA GR  is the affluence effect: richer future generations and more intergenerational inequality aversion push up the SDR and lower the SCC. The third term is the prudence effect: the more riskaverse policy makers and the bigger their intergenerational inequality aversion and the higher the volatility of economic growth and emissions, the lower the SDR and the higher the SCC.
This term arises if utility has a positive third derivative in which case there is precautionary saving in response to income uncertainty in the consumption-saving problem for the household (cf. Leland, 1968;Kimball, 1990). Also, climate policy makers behave in a prudent fashion by pursuing a more vigorous climate policy if future economic growth is uncertain. The coefficient of relative prudence is 1 + IIA for this class of preferences, so the prudence effect is stronger if relative prudence is stronger. The first three terms in (7) (2)).
The fourth term in (7), 2 , RA   is the self-insurance effect: in future states of nature where economic growth is high, damages are high too as damages are proportional to world GDP.
Therefore, abatement is a procyclical investment with higher yields in good times. Hence, policy makers can take less climate action, which is reflected in a higher SDR and a lower SCC.
This term is also known as the risk premium. In fact, if the elasticity of damages with respect to GDP is , the risk premium generalises to are uncorrelated with the future state of the economy, but positive if they are positively (negatively) correlated with the future state of the economy.
Finally, there is the growth-correction to the SDR in the denominator of (6), the term -GR, which reflects the growing damages effect and calls for a lower SDR and a higher SCC.
The SDR derived from the Keynes-Ramsey rule must be adjusted downwards to allow for the precautionary effect, but this adjustment is small (e.g. Gollier, 2002;Arrow et al., 2014). To see this, set the annual volatility to 3.6% (Kocherlakota, 1996), IIA = 2, RTI = 0 and GR = 2% per year as before. However, we take the coefficient of relative risk aversion to be RA = 5 > IA = 2 and allow for the self-insurance effect in (7). The prudence effect in (7) is then 0.74% per year (a bit higher than in Arrow et al. (2014) due to the higher RA) and the self-insurance effect in (7) is 0.49% per year. These two effects thus depress the SDR from 4 to 3.75% year, which boosts the SCC from 18.5 to 21.1 $/tCO2. For higher (lower) values of IIA, the downward adjustment of the SDR and upward adjustment of the SCC is bigger (smaller).

Effects of Tail Risks, Tipping Risks and Long-Run Risks on the SCC
Although uncertainty about future economic activity has a modest effect on the SDR and the SCC, skewed uncertainty about the climate sensitivity has a substantial upward effect on the SCC especially if the damage ratio is a convex function of temperature. More specifically, if shocks to the climate sensitivity are more persistent, more volatile, and more skewed, this pushes up the SCC by more (van den Bremer and van der Ploeg, 2019). Uncertainty about shocks to the ratio of damages pushes up the SCC but only if the distribution of these shocks is skewed. 8 The effects of these two types of uncertainty on the optimal carbon price can be substantially higher than that of growth uncertainty.
It has also been shown that the risk of climatic tipping (e.g. melting of Ice Sheet or reversal of Gulf Stream) leads to substantial increases in the SCC because global warming increases the risk of tipping (e.g. Traeger, 2014, Lontzek et al., 2015;van der Ploeg and de Zeeuw, 2018;Cai and Lontzek, 2019). Furthermore, if one tip raises the likelihood of another tipping point, this domino-effect boosts the SCC even more and thus even more vigorous climate action must be undertaken (Cai et al., 2016;Lemoine and Traeger, 2016). Pindyck (2011) gives an excellent survey of the effects of fat-tailed and thin-tailed uncertainty on climate policy and warns that cost-benefit analysis of climate policy is difficult as we do not even know the probability distribution of future temperature impacts. Fat-tailed distributions when combined with power utility functions give rise to the "dismal" theorem, which states that the SCC is unbounded and thus that society is prepared to sacrifice all of GDP to curb emissions (Weitzman, 2009(Weitzman, , 2011. For utility functions with bounded marginal utility, however, this so-called "dismal" theorem no longer holds. Still, skewed distributions for the climate sensitivity and damage ratio and tipping points call for more stringent climate policies.

Correlated Shocks to the Climate, the Damage Ratio, and the Economy
Shocks to the economy, to damages from global warming, and to the climate have often taken to be independent, but these different types of shocks may be correlated. To illustrate this, consider the case RA = IIA = 1 discussed by Golosov et al. (2014) For  = 1, this boils down to SDR = RTI with no effects of uncertainty (i.e. ) at all. In general, there are two additional effects of a "beta" smaller than one on the SDR: (i) as marginal damages grow at a less rapid rate than world GDP, the present discounted value of marginal damages is smaller and this increases the SDR (see second term in (8)) and thus lowers the SCC; (ii) in future states of nature shocks to future damages are now less than perfect correlated with future world GDP, so self-insurance is less and the risk premium and the SDR is pushed down (see third term in (8)) and the SCC must increase. 9 The relative magnitudes of these opposing effects can be seen from With a growth rate of around 2% per annum and an annual volatility of about 3.6% (Kocherlakota, 1996), this expression is negative for all values of  between -1 and 1 so effect (i) dominates effect (ii). Dietz et al. (2018) argue on basis of theory and integrated assessment modelling with uncertainty about both the growth rate of emissions and about the damage ratio and the climate sensitivity that this climate "beta" is close to unity for maturities up to one hundred years.
Effectively, the positive effect on this beta of uncertainty about exogenous, emissions-neutral technical change swamps the negative effect on this beta of uncertainty about the climate sensitivity and the damage ratio. They conclude that mitigating climate change increases aggregate consumption risk, which justifies a higher discount rate on the expected benefits of emission reductions. However, the stream of undiscounted expected benefits also increases in this beta and this dominates the effect on the discount rate, so that on balance the present value of emissions abatement (the SCC) increases in this beta (as also indicated by our effect (ii)).

Macroeconomic Disasters and the SCC: Insights from Asset Pricing Theory
The SCC can be viewed as an asset with a negative price, since it is the expected present discounted value of all future marginal damages caused by emitting one ton of carbon today.
In the asset pricing theory put forward by Lucas (1978), the idea is that trees grow fruits each year and that the growth rate in the harvest of fruits is stochastic. The objective is to put a price on these trees. This tree metaphor is used to analyse the idea of an asset, which is like a tree in that it generates a stream of unknown future dividends. The price-dividend ratio, say V, is the expected present discounted value of a tree where the dividend is the unleveraged claim on consumption. Asset pricing theory extends the stochastic process for growth of consumption and output for the incidence of rare macroeconomic disasters (wars, great recessions, virus outbreaks, etc.) as well as geometric Brownian motion (Barro, 2016). If disaster shocks occur with instantaneously probability  and destroy ln(1-b)Yt of the endowment, the expected endowment growth is where EIS (cf. 1/IIA) denotes the elasticity of intertemporal substitution which can differ from 1/RA for Epstein-Zin preferences (e.g. Barro, 2009). If one accepts that preferences from policy makers correspond to those in financial markets, one could replace 1/EIS in (9) by the coefficient of intergenerational inequality aversion IIA so instead of (6) the SCC becomes corresponds to the expected return on unleveraged equity and 1/V is akin to SDR − GR in equation (6). If f r denotes the risk-free return, the equity premium is The asset markets calibration of Barro (2009) aims to explain both the equity-premium puzzle which from (11) requires a high value of the RA and the idea that uncertainty depresses the 15 price-dividend ratio which from (9)   temperature-induced tail risks is large. Bansal et al. (2017Bansal et al. ( , 2019 conclude that temperature is a source of long-run economic risks and shows that it is important to allow for forward-looking capital markets for understanding the cost and impact of climate change.
9. What can one learn from the asset pricing interpretation of climate policy?
First, whereas the dividend-price ratio increases and thus the SCC decreases in uncertainty for the market-based calibration, the ethics-based growth-corrected social discount rate decreases and consequently the carbon price increases in uncertainty. The reason is that Barro (2009) has EIS = 2 > 1 whilst Gollier (2011) has IIA = 2 which corresponds to EIS = 0.5 < 1. Growth rate uncertainty thus demands more vigorous climate action if IIA > 1 as in Nordhaus (2007), Gollier (2011 and van den Bremer and van der Ploeg (2019) but requires less strong carbon pricing if EIS > 1 or equivalently IIA < 1.
Second, allowing for macroeconomic disasters substantially depresses the dividend-price ratio and thus the growth-corrected social discount rate by much more than normal growth uncertainty (i.e. geometric Brownian motion). This means that disaster uncertainty has substantially larger positive or negative effects on the SCC than normal growth uncertainty depending on whether the IIA is above or below one.
Third, as pointed out by Epstein et al. (2014), Epstein-Zin preferences are too restrictive as they only have two parameters to capture three aspects of preferences, i.e. aversion to risk, aversion to intergenerational inequality and preference for early resolution of uncertainty. What is needed is general enough preferences, so one can separate calibration of all three aspects.
Fourth, some argue that it is important to use the market rate of interest to discount returns from social investments (or damages from global warming) thus respecting private preferences, Caplin and Leahy (2014) argue that this is only justified if preferences over all choices 17 including past ones are time invariant. Under reasonable conditions policy makers should be more patient than private citizens, whose choice define the most short-sighted Pareto optimum.
Finally, future work needs to face the challenge of private versus social preferences head-on. Barrage (2018) and Belfiori (2017) show in a deterministic setting with more patient policy makers than private agents that the optimal policy is to have a carbon price equal to the SCC and a capital subsidy to correct for the excessive impatience of private agents. If the capital subsidy is not implemented, climate policy will be time inconsistent. Future research needs to combine the ethical calibration of climate policy put in place by relatively patient policy makers with IIA > 1 and combine this with market-based consumers and investors with EIS > 1.

Discounting the Far Future with Uncertain Discount Rates
So far, the analysis has assumed a constant SDR, irrespective of the horizon of the intended policy. The rate to discount a project in year 101 to year 100 is thus the same as the rate used to discount a project in year 11 to year 10. If more distant discount rates are smaller (larger), there is a downward-(upward-sloping) term structure for the discount rate. In fact, it has been argued that a declining term structure for the SDR is more appropriate (e.g. Arrow et al., 2014).
The reason for this is that, if shocks to interest rates and thus to consumption growth rates are persistent, the resulting schedule of efficient discount rates must decline for longer horizons.

France and the United Kingdom indeed employ declining SDRs.
A declining SDR occurs if shocks to the consumption growth rate are not independently and identically distributed, but positive correlated over time. The SDR is then no longer constant and the downward adjustment to allow for the uncertain growth rate is not necessarily small anymore. Furthermore, the downward adjustment of the SDR becomes more substantial for long horizons as discussed in the excellent survey of . The point is that, due to positive correlation, positive shocks to consumption make future consumption riskier, which magnifies the prudence term in equation (7) for distant horizons. Gollier (2008) shows that, if the growth in consumption, 1 ln( / ) t t t C C x −  follows an AR(1) process, the SDR is lower for longer horizons provided the autocorrelation coefficient of the process, , is between zero and one. In fact, the prudence effect is multiplied by 2 (1 ) 1  − − as the horizon tends to infinity.
Hence, the more autocorrelation in shocks to consumption growth (higher ), the bigger the amplification of the prudence effect at very long horizons.
Vasicek (1977) had shown much earlier that, if the spot interest rate follows an Ornstein-Uhlenbeck process with positive serial correlation as proposed by Merton (1971) and the initial interest rate is high enough, the conditional expectation of the interest rate declines over time towards its long-term mean, the term structure slopes downwards. If the initial interest rate is low enough, the term structure slopes upwards. For intermediate values, it is a humped curve.
It turns out that estimated models with autocorrelation in consumption growth only imply modest declines in the SDR. However, Weitzman (2007) and Gollier (2008) show that subjective uncertainty about the trend and volatility of the growth rate of aggregate consumption can lead to a declining SDR too. Gollier (2008) gives an example where the mean growth rate of aggregate consumption takes the values 1% and 3% per year with equal probability and shows that this implies that with IIA = RRA = 2 the SDR excluding the selfinsurance term falls from 3.5% today to 2% per year in three centuries. 10 This is not that different from the 4% per year for the first three decades and 2% per year thereafter that the French government uses for project appraisal. Arrow et al. (2014) argue that it need not make sense to have a higher RRA then IIA, since recursive preferences have been used to explain saving decisions in financial markets. Hence, it is more appropriate to use (5) than (6). 10 With IIA = RRA = 2 and RTI = 0, the SDR takes on 6% and 2% with equal probability so the certainty-equivalent value is The mean instantaneous rate is 0.5 x 0.06 + 0.5 x 0.02 or 4% but for a horizon t of 300 years it is 2.23% and as the horizon tends to infinity it tapers off to 2% per year. Weitzman (1998Weitzman ( , 2001Weitzman ( , 2007    Key: Columns 2-5 give present value today in millions of 1 billion in the future 11 The astute reader may ask why the SCC in (5) or (6) uses a constant SDR in (7). The reason is that the growing variance of consumption growth demands a rising SDR, which offsets the declining SDR highlighted here.

20
Groom and Hepburn (2009) examine the result of Gollier (2004) that as the evaluation date moves further into the future, the discount rate at a given point in time will increase. They show, however, that for a given evaluation date the schedule of discount rates will decline in line with the seminal paper of Weitzman (2001). Gollier and Weitzman (2010) attempt to reconcile the positively correlation consumption growth and the expected net present value approaches for a declining SDR by showing that the latter approach is equivalent to utility maximisation with a logarithmic utility function. Groom and Hepburn (2019) survey the various approaches to the SDR and the mechanisms for why it might decline with the horizon.
Differences in opinions do not necessarily reflect uncertainty in the statistical sense but may reflect differences in ethical judgements. In fact, Gollier and Zeckhauser (2005) offer theoretical arguments why heterogeneity in rates of time impatience can lead to a declining utility discount rate and possibly also to a social discount rate that declines for longer horizons. 12 They show that this requires that all individual agents have a constant discount rate and display decreasing absolute risk aversion. They also discuss the possibility that the discount rate decreases with GDP per capita. Millner and Heal (2018) consider a dynamic social choice problem where a sequence of committees decides on how to consume a public asset and each committee takes account of the behaviour of future committees. Furthermore, each committee member has a different view of the pure rate of time preference. They show that deciding by majoritarian vote in each period is superior to aggregating preferences in utilitarian manner, since the latter leads to time inconsistent and inefficient decision making.

Survey and Empirical Evidence on Declining Discount Rates
Using responses from a couple of thousand professional economists about the constant discount rate giving a mean of 4% and a standard deviation of 3%, Weitzman (2001) modelled uncertainty about this rate with a gamma distribution and found that the immediate future (1-5 years), near future (6-25 years), medium future (26-75 years), distant future (76-300 years) and far-distant future (more than 300 years) future should be discounted at 4%, 3%, 2%, 1% and 0% per year, respectively. Freeman and Groom (2015a) warn against this interpretation of survey evidence. If the variation in opinions about discount rates is due to irreducible differences in ethical judgements, then as Weitzman (2001) has shown the term structure of the SDR declines rapidly. However, if this variation is due to respondents forecasting future rates under uncertainty, Freeman and Groom (2015a) show that the term structure of the SDR is much flatter as opinions from additional experts provide new information and can be used to cut forecasting errors, especially if forecasts by experts are not much correlated. This leads to a much lower SCC than the one suggested by a rapidly declining term structure. When interpreting survey evidence, it is thus important to distinguish heterogeneity and uncertainty. Newell and Pizer (2003), , Hepburn et al. (2009) andGroom (2015b) have estimated certainty-equivalent interest rates from historical data series. For example, Newell and Pizer (2003) find that the certainty-equivalent value of the social discount rate falls from 4% today to 2% in a century if a random walk for interest rates is assumed, but falls from 4% to 2% in more than three centuries if a mean-reverting auto-regressive model for interest rates is used.  find that these declining patterns also occur for autoregressive models with conditional heteroscedasticity and for regime-switching and statespace models, especially if returns to capital are uncertain and persistent. Hepburn et al. (2009) also find that the regime-switching model is a better model of past interest rates and that this implies a faster decline in certainty-equivalent discount rates than Newell and Pizer (2003). 22 Freeman and Groom (2015b) obtain a declining SDR by using real rather than nominal interest rates and estimating a co-integration model of inflation and nominal interest rates.

Expert and Other Evidence on Preferences of Policy Makers
Since market data cannot be relied on to estimate preferences needed to formulate climate policy, scholars have turned to other methods. Grijalva et al. (2014) use a laboratory experiment to elicit discount rates over a 20-year horizon when government bonds can be used for payment.
Using exponential discounting, they find an implied average discount rate of 4.9% per year, much lower than in previous experimental studies that used horizons of days or months. They also find strong support for discount rates that decline with longer horizons (falling to 0.5% in a century). There is also evidence that more optimistic people with more optimistic views about technological progress have higher discount rates, which is in line with the Keynes-Ramsey rule (1). However, Drupp et al. (2018) surveyed 200 experts to disentangle the various effects on the SDR but found that most experts when recommending which SDR to use did not follow the Keynes-Ramsey rule. Despite disagreements, they found that three-quarters of the experts found a median risk-free social discount rate of 2% per year acceptable. This is much lower than the figure of 4.9% per year found in Grijalva et al. (2004).
Pindyck (2019)  It is also important to have solid experimental or survey evidence on what intergenerational inequality aversion might be.  gives various arguments based on surveys and introspection for why IIA of 2 is plausible. Weitzman (2007) and Nordhaus (2016) also use IIA = 2, Arrow (2007) uses an IIA of 2 to 3, and the Intergovernmental Panel on Climate Change uses a range of 1.5 to 2 for the IIA. Stern (2007) uses IIA = 1, but Dasgupta (2007Dasgupta ( , 2008 criticises this for being much too low implying too much indifference between welfare of current and future generations. Dasgupta (2007) finds it absurd that the current generation literally must starve itself so that future generations can enjoy ever-increasing consumption levels and argues that a value for IIA of 2 to 4 is much more reasonable. The point is that none of these studies recommend a value of IIA less than one (corresponding to EIS > 1) and thus take a radically different view than taken in the asset pricing literature.
To assess the appropriate SDR, it is thus not only necessary to form an opinion or have evidence on the value of the rate of time impatience but also to have survey or other empirical evidence on intergenerational inequality aversion. Inequality aversion can be estimated across individuals from progressive taxation or across countries from development aid to be 0.7 (Tol, 2010), but this offers no guidance on inequality aversion across generations. Evans (2005) uses evidence on the structure of personal income taxes for OECD countries that the average elasticity of marginal utility is 1.4, but this is more like an estimate of intra-generational than intergenerational aversion. Groom and Maddison (2019) use the same method (the equalsacrifice income tax method) and three other methods (the Euler equation approach, the Frisch additive preference approach, and risk aversion in insurance markets) to come up with an estimate of the elasticity of marginal utility of 1.5. The same value then captures intragenerational and intergenerational inequality aversion and risk aversion, but that is a big leap. Dennig et al. (2015) put forward an integrated assessment model with different measures for both intra-regional and intergenerational inequality aversion. They show that Thomas

24
Schelling's conjecture that the inequality parameter can have the opposite effect on the intensity of climate policy to what is suggested by the Keynes-Ramsey rule (Budolfson, et al., 2017). Hence, despite that higher IIA implies a higher SDR and thus a lower SCC, when inequalities are properly accounted for, it is possible that climate policy should become more ambitious under higher (intergenerational and intra-regional) inequality aversion.
Others have tried to estimate RTI and IIA from stated preferences, but this suffers from both methodological and conceptual issues as people have different values for these parameters. Yet another alternative is to derive RTI, IIA and RA from financial market data, but as discussed in section 8 it is not clear that this has much to do with preferences of climate policy makers. This explains why some researchers have turned to survey evidence from experts.

Dual Discount Rates, Relative Scarcity and The Endowment Effect
A classic paper highlights that over time the relative benefits of preservation increase as there may be limited substitution with economic development, technical progress in economic development, and demand for environmental quality might rise more than proportionally with wealth (Fisher and Krutilla, 1975). This leads to dual discount rates, where the discount rate for future benefits from preservation is smaller than that for future benefits and costs of developing a project. This makes environmental policy more ambitious. More generally, if the consumption of environmental quality is a driver of household satisfaction, its relative price is likely to change over time. To value an environmental project, future environmental gains are thus converted at current relative prices and an "environmental" discount rate equal to the SDR minus the rate at which the relative price of environmental goods in terms of consumption goods is used. An alternative way to value such a project is to convert future environmental gains into consumption units using the future relative price and use the consumption discount rate. When thinking about discounting it is thus important to focus at the relative scarcity of the natural environment in the provision of ecosystem services (e.g. Hoel and Sterner, 2007;Sterner and Persson, 2008;Gollier, 2010;Traeger, 2011;Zhu et al., 2019). Because the growth rates of the economy and ecosystem services differ, relative prices change over time and this affects the SDR. Empirical work suggests that the discount rate for ecosystem services is about 1%-point smaller than that for consumption (e.g. Drupp, 2018). economic growth. This then implies that the consumption discount rate declines towards a low value that is given by the standard Keynes-Ramsey rule with a low growth rate. If ecosystem services can be easily substituted, this effect does not occur. The effect on the discount rate of limited substitutability of ecosystem services in production is much stronger than the relative price effect that results from limited substitutability of ecosystem services in utility. Hoel and Sterner (2007), Traeger (2011) and Zhu et al. (2019) also investigate the effects of ecosystem services in the utility function on the time pattern of the discount rate. They conclude that even for a given growth of consumption the discount rate is not constant due to the timevarying value share of ecosystem services. If the elasticity of intra-temporal substitution between consumption and ecosystem services exceeds one but is less than the elasticity of intertemporal substitution, the discount rate declines with time. This also occurs if the elasticity 26 of intra-temporal substitution is less than one but greater than the elasticity of intertemporal substitution (see Figure 7 in Zhu et al., 2019). This effect appears to be relatively small. Dietz and Venmans (2019b) also find a declining discount rate, but here the mechanism is due to habit persistence or reference dependence and loss aversion. They show that loss aversion affects the discount rate via an instantaneous endowment effect and via a reference-updating effect, which reduces the incentive to smooth consumption. On a path of rising material consumption, this reduced incentive tends to lower the discount rate, but on a declining path of environmental quality it leads to a higher discount rate.
Venmans and Groom (2019)  Finally, in a very impressive extension of Nordhaus's DICE model to allow for the relative scarcity of non-market goods, Drupp and Hänsel (2020) show that for their core calibration accounting for relative prices is equivalent to a decreasing pure rate of time preference by 0.6%points and leads to a more than 50% higher SCC.

Conclusion
The appropriate choice and term structure of the SDR to be used for climate policy matters enormously and is hotly debated. One billion dollars of damages a century from now is worth today only 7.6 million dollars if the discount rate is 5% but 370 million dollars if the discount rate is 1% per annum. In the latter case, the SCC is much higher. The stochastic discount factors that arise naturally in asset pricing with Epstein-Zin preferences and long-run risks 13 have been used to derive the optimal carbon price, where the latter corresponds to the expected value in terms of less future global warming damages of reducing emissions by one ton of carbon today. Since the asset pricing literature assumes that uncertainty depresses the price-dividend ratio, it assumes that the elasticity of intertemporal substitution exceeds one (and calibrates risk aversion to explain the equity premium puzzle). However, if this is applied to climate policy, this implies that normal growth uncertainty and the risk of macroeconomic disasters lead to lower carbon prices. Although the insights from asset pricing regarding discount rates are invaluable, it is important to realise that impatience and attitudes to risk and intergenerational inequality of policy makers cannot be deduced from savings and investment decisions in financial markets. Ethics-based preferences of policy makers typically have much lower rates of impatience and higher degrees of intergenerational inequality aversion. Such preferences may allow risk aversion to exceed intergenerational inequality aversion as this would also allow for a preference for early resolution of uncertainty.

Further Research
There are many promising avenues for further research into how to use discounting and asset pricing in the formulation of climate policy. First, the decisions to undertaken abatement projects require long-run climate investments and a careful application of cost-benefit analysis.
For example, Gollier (2020b) stresses that policy makers should use project-specific riskadjusted discount rates. In practice, due to financial illiteracy, the dogma that the government in contrast to the private sector can pool all risks 14 , or the misguide use of using a single discount rate corresponding the weighted average cost of capital 15 , many public (and private) decision makers use an all-purpose discount rate that does not depend on the risk profile of their investment projects. The welfare loss of using a single discount rate rather than different discount rates for different projects with different risk profiles leads to a welfare loss corresponding reduction in permanent consumption of 15 to 45%, depending on which discount rate is used. Policy makers should reform the way to discount and evaluate investment projects to abate emissions in line with the well-known principles of asset pricing theory. This also has crucial implications for the SDR to be used for different climate investment projects. Projects with a negative correlation with the future state of the economy (e.g. dikes) should be evaluated using a lower SDR than projects with a positive correlation.
14 This goes back to Arrow and Lind (1976), who argued that the pooling argument implies that all public investment projects should be invested at the risk-free rate. But from the consumption capital asset pricing model, it is well known that this proposition is only true for projects with a zero beta. Using a single discount rate thus means that projects with a positive beta (e.g. railroads, highways) get implemented more easily. 15 This implies there is too much investment in risky and not enough in safe projects (Krueger et al., 2015).
Second, asset pricing can also be used to better understand how best to implement caps on temperature as required by the Paris Agreement on climate policy. A temperature cap implies a cap on cumulative emissions as temperature is a simple function of cumulative emissions.
Intertemporal efficient arbitrage then requires that the carbon price must grow at a rate equal to the rate of interest, because only then is society indifferent between emitting one ton less today (thus saving the carbon tax which can yield a rate of return equal to the interest rate) or emitting one ton less next year (thus facing an expected increase in the carbon price). Gollier (2020a) uses an intertemporal asset pricing approach with both conventional and uncertainty and disaster risks about the rate of economic growth and with uncertainty about future abatement technologies to show that the appropriate risk-adjusted interest to use is about 3.75% per year, which is higher than the risk-free rate implying a positive carbon risk premium (provided marginal abatement costs and aggregate consumption are positively correlated) but a lot lower than the return on risky assets. The 7% per year or even higher rate of growth of the carbon price typically used in integrated assessment models thus leads to intertemporally inefficient outcomes, grossly under-estimating the efficient carbon price that is needed today.
Third, more thinking and sound empirical work is needed on the appropriate term structure. A recent macro-finance study uses tools from the fixed-income literature on government yields to specify and estimate a Bayesian time-series model to show that the equilibrium real interest rate is a crucial anchor for the term structure of discount rates and that empirically this anchor has fallen by about 1%-point per year and has thus roughly doubled the estimated present value of the economic loss from climate change, the SCC, since the 1990s (Bauer and Rudebusch, 2020). An exciting other recent study estimates a downward-sloping term structure for real estate with an average return of 6% and a discount rate for a century ahead of about 2.6% per year (Giglio et al., 2020). It also shows that real estate performs badly during consumption disasters and is thus a risky asset and that real estate is exposed to climate change risk (proxied 30 by rising sea levels, hurricanes). It combines these findings with asset pricing insights to come up with a social discount rate for climate policy appraisal which has similar horizons as real estate but a different risk profile. Their key assumptions are that disasters are more likely when growth is high, and that economic growth picks up temporarily after a disaster. These assumptions imply that the appropriate discount rate to use for climate appraisal starts off very low and then rises, i.e. an upward-sloping, not downward-sloping term structure of the social discount rate. The point is that hedging against the adverse effect of disasters on short-term cash flows is more valuable than hedging long-term cash flows as these are affected less due to adaptation. The discount rate is below the risk-free rate of 1-2% per year at all horizons.
Fourth, as this survey's discussion of the application of asset pricing to climate policy has highlighted, it is crucial to allow for integrated assessments of the economy and the climate that allow for different preferences for policy makers and private agents. If the government uses a lower (ethically based) social discount rate for climate change than for investments in other domains, the government policy is not Pareto-efficient. Higher welfare can be obtained by transferring part of the investments for climate with a low internal rate of return to the other sectors with higher hurdle rate. This requires an analysis of second-best optimal climate policies. Barrage (2018) shows that, if patient policy makers that are more patient (farsighted) than the private sector, a capital subsidy as well as a carbon price is needed. In the absence of such a subsidy, climate policy is time inconsistent. Others use the hyperbolic discounting framework put forward by Laibson (1977) and Krusell and Smith (2003) to analyse climate policy (e.g. Karp, 2005;Karp and Tsur, 2011;Iverson and Karp, 2017;Gerlagh and Liski, 2018). For example, Iverson and Karp (2017) study the Markov-perfect equilibrium to a dynamic game where private agents choose savings and policy makers decide on climate policy. They show that with hyperbolic discounting, convex damages give rise to significant strategic interactions across generations of planners. Being able to commit for over a century 31 significantly boosts welfare of the first generation but being able to commit for only a few decades has hardly any benefit. Gerlagh and Liski (2018) show that with a declining discount rate the delay and persistence of climate impacts act as a commitment device to policy makers.
They also focus at the Markov-perfect equilibrium and find that the returns on capital and climate investments are no longer leading, which implies a substantial boost to the carbon price.
The commitment value increases the carbon price by a factor of 20.
Finally, future research should deal with the problem of different generations in a realistic fashion. For example, Kotlikoff et al. (2019) use an overlapping generations model to design Pareto-improving green tax reforms by taxing future generations to give transfers to current generations in a way that makes all generations better off. The challenge is thus to study the drivers of the social discount rate and climate policy in second-best frameworks where private agents and governments have diverging preferences, commitment matters, and both the economic and climate system are subject to both normal uncertainties and tipping risks.