A very popular approach to finding the right price of carbon is to assume away all other market failures and imperfections and consider the optimal carbon pricing policy for the global economy. The revenues from the carbon tax (or from selling emission permits) are rebated in lump-sum fashion to the private sector. Hence, important problems such as unemployment, information failures leading to problems of adverse selection or moral hazard and attaining an equitable distribution of incomes within a generation are cast aside. Also, all countries are supposed to cooperate to avoid international free-rider problems or, alternatively, rich countries offer side payments to poorer countries to persuade them to have the same carbon price throughout the globe. The optimal carbon price is then set to the Pigouvian carbon price, also known as the social cost of carbon (SCC). The SCC is defined as the expected present discounted value of all present and future damages to GDP in the world economy resulting from emitting one ton of carbon today, where the discount rate used to evaluate this is called the risk-adjusted or stochastic discount rate (SDR). Since greenhouse mixes perfectly, it does not matter where in the world emissions takes place.
The outline of this section is as follows. Section
2.1 derives tractable expressions for the social discount rate and social cost of carbon when there is uncertainty about the rate of economic growth (i.e. consumption follows a geometric Brownian motion), damages as fraction of aggregate consumption are linear in temperature (cf. Burke et al.,
2015), and temperature is a linear function of cumulative emissions (e.g. Allen et al.,
2009; Matthews et al.,
2009). Section
2.2 then calibrates the model and gives a quantitative assessment of the risk-adjusted social cost of carbon. Section
2.3 extends the model of Sect.
2.1 to allow for the risk of rare macroeconomic disasters (cf. Barro,
2006,
2009). Section
2.4 discusses market-based calibrations based on insights from asset pricing with ethics-based calibration and how these affect the way macroeconomic risks affect the social discount rate and cost of carbon. Section
2.5 discusses the effects of climate tipping points (e.g. melting of the Greenland Icesheet or of the permafrost) and how this affects the optimal carbon price. Finally, Sect.
2.6 explains why in practice the social discount rate might decline with the length of the horizon (i.e. has a declining term structure), and why this is so important for evaluating climate policy as this has consequences many decades or centuries ahead.
2.1 The social cost of carbon under normal macroeconomic uncertainty
To make headway, it is easiest to learn from the asset pricing literature which uses an object called the stochastic discount factor (SDF) to obtain the share price by taking the expected present value of a stream of present and future dividends. In a similar vein, one can evaluate the SCC which corresponds to the expected present discounted value of the stream of present and future marginal damages resulting from emitting one ton of carbon today. We assume recursive utility which distinguishes the coefficient of relative risk aversion,
γ, from the inverse of the elasticity of intertemporal substitution,
η, and specifies a rate of time impatience (or utility discount rate),
ρ. The function describing utility to go from time
t onwards, denoted by
Jt, is then defined recursively using an aggregator function
\(f(C,J)\)(see Epstein and Zin (
1989) for a discrete time and Duffie and Epstein (
1992) for a continuous-time formulation). This recursive definition of utility to go is given by
$$J_{t} = E_{t} \left[ {\int\limits_{t}^{\infty } {f(C_{s} ,J_{s} )ds} } \right]\;{\text{with}}\;f\left( {C,J} \right) = \frac{1}{{1 - \eta }}\frac{{C^{{1 - \eta }} - \rho \left( {\left( {1 - \gamma } \right)J} \right)^{{\frac{{1 - \eta }}{{1 - \gamma }}}} }}{{\left( {\left( {1 - \gamma } \right)J} \right)^{{\frac{{\gamma - \eta }}{{1 - \gamma }}}} }}$$
(1)
Expected utility analysis is a special case corresponding to
\(\gamma \eta = 1\) in which case policy makers maximise at time
t the expected value of the usual discounted welfare loss function
\(\int\limits_{t}^{\infty } {e^{{ - \rho (s - t)}} \left( {\frac{{C_{s} ^{{1 - \eta }} - 1}}{{1 - \eta }}} \right)ds} .\) This special case does not distinguish risk aversion from the inverse of the elasticity of intertemporal substitution and can be used in case there is no uncertainty. If there is uncertainty, this special case implies that policy makers have neither a preference for early nor for late resolution of uncertainty. For these reasons, the more general recursive utility specification is used. The general specification implies a preference for early resolution of uncertainty if risk aversion exceeds the inverse of the elasticity of intertemporal substitution, i.e.
\(\gamma \eta > 1,\) which is in line with empirical evidence. As we will see in our calibration, the more general specification of recursive preferences allows a market-based calibration (e.g. Barro,
2009; Cai & Lontzek,
2019; Cai et al.,
2016; Lontzek et al.,
2015) which attempts to explain the equity premium puzzle whilst the simpler expected utility specification with
\(\gamma \eta = 1\) allows for an ethics-based calibration (e.g. Arrow et al.,
2014; Gollier,
2002,
2011).
We need three further assumptions about the endowment, damages, and temperature.
First, world consumption follows a geometric Brownian motion
\(dC_{t} = \mu C_{t} dt + \sigma C_{t} dW_{t} ,\) where
μ denotes the drift of this stochastic process and
σ > 0 the volatility and
Wt is a unit Wiener process. This gives the solution
\(C_{s} = C_{t} \exp \left( {\left( {\mu - \frac{1}{2}\sigma ^{2} } \right)(s - t) + \sigma W_{s} } \right),\) so mean consumption is
\(E\left[ {C_{s} } \right] = \exp \left( {\mu (s - t)} \right)C_{t}\) and expected consumption growth is
\(E\left[ {\ln \left( {C_{s} /C_{t} } \right)} \right] = g(s - t)\) with
\(g \equiv \mu - \frac{1}{2}\sigma ^{2} .\) It is important to account for uncertainty about the rate of economic growth, because this negatively affects the social discount rate and increases the social cost of carbon via the prudence effect in Eq. (
3) discussed below. Furthermore, it positively affects the social discount rate and curbs the social cost of carbon via the insurance effect in Eq. (
3) discussed below.
Second, damages resulting from a marginal increase in temperature are proportional to world consumption. The coefficient of proportionality is called the marginal damage ratio (MDR). We thus do not adopt the convex (quadratic) specification used for the DICE model by Nordhaus (
2017), but the linear specification found in the detailed empirical evidence provided by Burke et al. (
2015). We assume that these damages are known with certainty, but we will discuss the effects of damage uncertainty briefly at the end of this sub-section.
Third, temperature is a linear function of cumulative carbon emissions, where the marginal effect of cumulative emissions on temperature is the transient climate response to cumulative emissions or the TCRE. Recent insights in atmospheric science suggest that this is a good approximation to detailed models of the dynamics of carbon in the atmosphere and in the oceans and of the dynamics of temperature of the atmosphere and the oceans (e.g. Allen et al.,
2009; Matthews et al.,
2009).
Armed with these assumptions, the social cost of carbon is then defined by
$$SCC_{t} = E_{t} \left[ {\int_{t}^{\infty } {\frac{{H_{s} }}{{H_{t} }} \times MDR \times TCRE \times C_{s} ds} } \right]\;with\;H_{s} = \exp \left( {\int_{0}^{s} {f_{J} (C_{u} ,J_{u} )du} } \right)f_{C} (C_{s} ,J_{s} ),$$
(2)
where
Hs denotes the stochastic discount factor to discount from time
s to the present for preferences (1). It can be shown that Eq. (
2) gives the optimal carbon price
Pt or SCC at time
t as
$$P_{t} = SCC_{t} = \frac{{MDR \times TCRE \times C_{t} }}{{r - g}}{\text{ with }}r = \rho + \eta ^{{ - 1}} g - \frac{1}{2}(\eta ^{{ - 1}} - 1)\gamma \sigma ^{2} ,$$
(3)
where the growth-adjusted SDR is given by
\(r - g = \rho + (\eta ^{{ - 1}} - 1)(g - \frac{1}{2}\gamma \sigma ^{2} ).\) The SCC is proportional to the marginal damage ratio, the transient climate response to cumulative emissions, and current world consumption, and inversely proportional to the difference between the risk-adjusted discount rate and the growth rate. We correct the discount rate for the economic growth rate since global warming damages rise in line with world economic activity. Hence, this
growing-damages effect pushes up the SCC and the carbon price.
The intuition of the drivers of the SDR and their effects on the SCC follow from rewriting the SDR as
\(r = \rho + \eta ^{{ - 1}} g - \frac{1}{2}(1 + \eta ^{{ - 1}} )\gamma \sigma ^{2} + \gamma \sigma ^{2} .\) The term
ρ is the
impatience effect: more impatient policy makers use a higher discount rate and thus price carbon less vigorously. The term
\(\eta ^{{ - 1}} g\) is the
affluence effect: richer future generations and more intergenerational inequality aversion (or a lower elasticity of intertemporal substitution) boost the discount rate and depress the carbon price. The term
\(- \frac{1}{2}(1 + \eta ^{{ - 1}} )\gamma \sigma ^{2}\) is the
prudence effect: the more risk-averse policy makers and the bigger their coefficient of relative prudence
\((1 + \eta ^{{ - 1}} )\) and the volatility of economic growth and emissions, the lower the social discount rate and the higher the carbon price. This term reflects precautionary saving in response to income uncertainty (cf. Kimball,
1990), but is typically small. These first three terms can in case of exponential utility (i.e.
\(\gamma \eta = 1\)) be rewritten as
\(\rho + IIA \times \mu - \frac{1}{2}IIA^{2} \times \sigma ^{2} ,\) where
\(IIA = 1/\eta\) is the coefficient of relative intergenerational inequality aversion or the elasticity of intertemporal substitution (cf. Gollier,
2011, Eq. (10)). The final term in the expression for the risk-adjusted discount rate,
\(\gamma \sigma ^{2} ,\) captures the
insurance effect: in future states of nature where economic growth is high, damages are high too as damages are proportional to world economic activity. Since abatement is a procyclical investment with higher yields in good times, policy makers can take less climate action, which is reflected in a higher risk-adjusted discount rate and a lower SCC and carbon price, especially when risk aversion (
γ) is large.
The growth-corrected social discount rate
\(r - g = \rho + (\eta ^{{ - 1}} - 1)(g - \frac{1}{2}\gamma \sigma ^{2} )\) simplifies to
\(r - g = \rho\) if the elasticity of intertemporal substitution
η equals one in which case uncertainty has no impact whatsoever. If the elasticity of intertemporal substitution
η exceeds (is less than one), uncertainty increases (depresses) the growth-corrected discount rate. This point is well known in the literature on asset pricing (cf. Smith,
1996a,
1996b; Svensson,
1989; Weil,
1990). It implies that uncertainty curbs (boosts) the social cost of carbon.
Before we quantify the risk-adjusted social cost of carbon, we briefly discuss how Eq. (
3) should be adjusted to allow for uncertainty in the damage ratio. Damages as fraction of consumption increase in temperature but also depend on potentially skewed shocks. Van den Bremer and van der Ploeg (
2021) show that the expression of Eq. (
3) is unaffected if the damage ratio uncertainty is not skewed, but that expression (
3) needs to have a positive multiplicative markup if the distribution of damage ratio shocks is right skewed. The markup is then bigger if damage ratio shocks display a higher volatility and more mean reversion. More analysis of the effects of damage ratio uncertainty on the optimal social cost of carbon is given by Crost and Traeger (
2014) who allow for uncertainty in the power coefficient on temperature in the damage function. Rudik (
2020) assumes that the damage ratio increases in temperature to the power of some coefficient and investigates the effect of uncertainty in this coefficient on the social cost of carbon. He also studies the effects of learning and damage misspecification.
2.2 A quantification of the risk-adjusted carbon price
A numerical example gives further insights. Let us first ignore the econometric estimates of asset pricing and follow Ramsey (
1928) and Stern (
2007) who argue that discounting the welfare of future generations is ethically indefensible and therefore take a zero or very small rate of time impatience (say,
ρ = 0) and let us also follow Gollier (
2011) who assumes IIA = 2 (i.e.
η = 0.5). In contrast, in asset pricing the rate of time impatience is empirically much higher and it is assumed that volatility curbs share prices and therefore the elasticity of intertemporal substitution
η should exceed 1. We take risk aversion to be
γ = 5, which implies a preference for early resolution of uncertainty. We assume that world economic activity in 2017 is about 80 trillion US dollars and trend growth of world consumption is
g = 2% per year. Following Kocherlakota,
1996), we set the annual volatility to
σ = 3.6%. We thus find that the risk-adjusted discount rate equals
\(r = 2 \times 2 - 0.324 = 3.676\%\) per year. This adjustment for risk to the so-called Keynes–Ramsey rule for the discount rate seems modest (cf. Arrow et al.,
2014; Gollier,
2002). When we split up the discount rate into its time impatience (zero), affluence (4% per year), prudence (− 0.972% per year), and insurance (0.648% per year) components, we see that the last two terms offset each other and that the net effect on the discount rate is quite small.
A typical figure for the TCRE is 1.8 °C per trillion tons of cumulative emissions (e.g. Allen et al.,
2009; Matthews et al.,
2009; van der Ploeg,
2018). To get an estimate of the marginal damage ratio, we deduce from (Burke et al.,
2015, Fig. 5, panel (d)) an approximate figure of 12.5% damages to world economic activity for every increase in temperature by 1 °C so that we set MDR = 0.125. The carbon price then follows from Eq. (
2) as $1074 per ton of carbon or $293 per ton emitted CO
2. This is very much higher than the carbon prices that follow from DICE. There are two reasons for this. First, Nordhaus uses a higher rate of time impatience,
ρ = 1.5% per year, in which case the growth- and risk-corrected discount rate is not 1.676% but the higher value of 3.176% per year. This depresses the carbon price from $293 to $155 per ton of CO
2. Second, in contrast to Burke et al. (
2015), the DICE specification for the ratio of damages to output is quadratic in temperature, and therefore, marginal damages are not constant but rise linearly with temperature. Evaluating the DICE marginal damages at reasonable temperature yields much lower marginal damage estimates than the ones of Burke et al. (
2015), a factor 2.5–100 times smaller at 2 °C. Nordhaus (
2017) calibrates damages as 0.236% loss in global income per degree Celsius squared, so that marginal damages are 2.1% and 8.5% of world GDP at, respectively, 3 and 6 °C. The marginal damage ratio thus equals 0.472% loss of global income per degree Celsius. At 2 °C, this gives for DICE a MDR of 0.944% instead of for Burke et al. (
2015) a constant 12.5% of global economic activity. It follows that the SCC shrinks to $12 per ton of CO
2. DICE also has a higher elasticity of intertemporal substitution of
η = 2/3 and ignores growth uncertainty which implies that the SDR is 2.5% per year and the carbon price finally becomes $16 per ton of CO
2. We summarise this quantitative assessment of the social cost of carbon in Table
1, and we will compare it in Sect.
2.3 when we also allow for the risk of small macroeconomic disasters.
Table 1
Quantitative assessment of the social cost of carbon
Calibration |
: impatience ρ = 0, elasticity of intertemporal substitution η = 0.5, IIA = 2, relative risk aversion γ = 5, C0 = $80 trillion, g = 2%/year, annual volatility σ = 3.6%, transient climate response to cumulative emissions (TCRE) 1.8 °C/TtC, MDR = 12.5% |
Social cost of carbon with regular macroeconomic uncertainty: |
$293/tCO2 for baseline calibration, $16/tCO2 if with ρ = 1.5%/year, η = 2/3, and MDR = 0.944% (Nordhaus at 2 °C) |
Social cost of carbon with also risk of small macroeconomic disasters: |
$102/tCO2 if ρ = 5.2%/year, η = 2, γ = 4, drift μ = 2.5%/year, annual volatility σ = 2%, g = 2%/year, risk of disasters π = 1.7%/year, mean size of disaster 0.29%, V = 20.7 |
In the remainder, we use the (Burke et al.
2015) estimate of the marginal damage ratio since their empirical work suggests that it is roughly constant and they obtained these estimates from detailed scientific estimates.
1 Although this estimate of the MDR is very large compared to the DICE estimate, it still does not include other economic damages associated with global warming such as tropical cyclones or sea-level rises which are typically included in other damage estimates such those of DICE. From panel (d) of Fig. 5 in Burke et al. (
2015), we see that this estimate of the MDR also has a wide range from 0.06 to about 0.18, which includes our estimate of 0.125. Furthermore, the TCRE has a 5–95% probability range of 1.4–2.5 °C (Allen et al.,
2009) or 1.0–2.1 °C per trillion tons of carbon (Matthews et al.,
2018). Depending on the outcomes of the MDR and the TCRE, there is thus a wide range of the optimal carbon price varying from $78 to $586 per ton of CO
2 which includes our figure of $293 per ton of CO
2.
2
2.3 The role of the risk of macroeconomic disasters
In asset pricing (e.g. Lucas,
1978), trees grow fruits each year and the growth rate of the harvest is stochastic. The objective is to put a price on these trees. This metaphor is used to price assets, where the fruits correspond to a stream of unknown future dividends. But the climate is also an asset because the social cost of carbon corresponds to the expected present discounted value of all future marginal damages caused by emitting one ton of carbon today. Barro (
2006,
2009) extends the stochastic process for consumption growth for the risk of macroeconomic disasters (e.g. virus outbreaks, recessions, wars, natural disasters). If the instantaneous probability of a disaster is
π and the disaster destroys ln(1–
b)
Ct of the endowment, expected consumption growth is
\(g = \mu + \frac{1}{2}\sigma ^{2} - \pi E\left[ b \right] < \mu + \frac{1}{2}\sigma ^{2} .\) The price-dividend ratio,
V, is the expected present discounted value of a tree with the dividend an unleveraged claim on consumption. It follows that the dividend-price ratio is given by
$$\frac{1}{V} = \rho + (\eta ^{{ - 1}} - 1)\left( {g - \frac{1}{2}\gamma \sigma ^{2} } \right) - \pi \frac{{\eta ^{{ - 1}} - 1}}{{\gamma - 1}}\left( {E\left[ {(1 - b)^{{1 - \gamma }} } \right] - 1 - (\gamma - 1)E\left[ b \right]} \right)$$
(4)
(Barro,
2009). We thus generalise (3) and obtain the social cost of carbon or carbon price as
$$P_{t} = SCC_{t} = V \times MDR \times TCRE \times C_{t} {\text{ with }}V = 1/(r^{e} - g),$$
(5)
where
\(r^{e} = 1/V + g\) is the expected return on unleveraged equity. The equity premium is
$$r^{e} - r^{f} = \gamma \sigma ^{2} + \pi \left( {E\left[ {(1 - b)^{{ - \gamma }} } \right] - E\left[ {(1 - b)^{{1 - \gamma }} } \right] - E\left[ b \right]} \right),$$
(6)
where
\(r^{f}\) denotes the risk-free return. To explain the equity premium puzzle, one needs a high degree of risk aversion
γ and a high risk of disasters. Uncertainty is seen to depress the price-dividend ratio so one needs
η > 1 (see (4)). Barro (
2009) sets the risk of disasters to
π = 1.7%/year and the average shock size to E[
b] = 0.29%. This gives
\(E\left[ {(1 - b)^{{ - 4}} } \right] = 7.69\) and
\(E\left[ {(1 - b)^{{ - 3}} } \right] = 4.05.\) Furthermore, the annual volatility and drift of consumption growth are set to
σ = 2% and
μ = 2.5%/year, respectively. Preferences are calibrated as
γ = 4 >
η = 2, and
ρ = 5.2%/year. Note that
η > 1 whilst most ethics-based calibrations have IIA = 1/
η > 1 (e.g. Gollier,
2011). This gives trend growth of
g = 2%/year, a price-dividend ratio of
V = 20.7, a safe interest rate of
rf = 1%/year, a return on risky assets of
re = 6.9%/year, and a risk premium of 5.9% per year. Note that disaster risk has a much bigger impact than normal macroeconomic uncertainty: e.g. to compensate for a rise of the volatility
σ by 10%, one needs an increase of only 0.38% in consumption for all years but to compensate for a 10% rise in disaster risk
π this figure is 2.6%.
To see what Barro’s market-based calibration with disaster risks implies for the optimal risk-adjusted carbon price, we use 1/
V = 4.83% per year for the risk- and growth-corrected interest rate (instead of 1.676% per year) and find with our marginal damage estimate from Burke et al. (
2015) that the carbon price drops from $293–$102 per ton of emitted CO
2 (See final row of Table
1.) Hence, a price of about $100 per ton of CO
2 can be justified based on the scientific evidence about the production damages presented in Burke et al. (
2015) and the asset pricing with disaster risk evidence of Barro (
2009). Despite a market-based calibration, it is a high carbon price. It is about four times higher than the current market price for European emission allowances.
Bansal and Yaron (
2004) offer an alternative for explaining the equity premium puzzle to the macroeconomic disaster risk approach of Barro (
2006,
2009). They use time-varying variance of shocks to the long-run economic growth rate which gives rise to long-run risks. Cai and Lontzek (
2019) in their integrated assessment of climate policy under a wide range of economic and climate uncertainties and tipping points use Epstein–Zin preferences in their market-based calibration with
η = 1.5 > 1,
γ = 10 > 1/
η and long-run risks as in Bansal and Yaron (
2004). Their numerical policy simulations confirm our analytical findings that macroeconomic growth uncertainty depresses the optimal risk-adjusted carbon price (consistent with
η > 1). This important paper also deals with climate tipping points (e.g. melting of Greenland or of the permafrost, or the reversal of the Gulf Stream) which differ from the risk of rare macroeconomic disasters discussed by Barro (
2006,
2009) and are discussed in Sect.
2.5.
Here and in Sect.
2.1, we highlighted the role of risks of rare macroeconomic disasters and Epstein–Zin preferences, respectively, in explaining the equity premium puzzle. Long-run risks also help to explain this puzzle. In fact, there are various other methods that have been proposed such as habit persistence (Campbell & Cochrane,
1999), idiosyncratic risk (Constantinides & Duffie,
1996), and debt, balance sheets and institutional finance (Brunnermeier,
2009). Cochrane (
2017) critically surveys all these methods and shows their similarities and discusses their merits and their problems.
2.4 Ethics-based versus market-based calibration
As we have seen, an ethics-based calibration typically sets the coefficient of intergenerational inequality aversion bigger than one (e.g. IIA = 2 and thus
η = ½ < 1 as in Gollier (
2011) and many other studies of optimal climate policy) in which case uncertainty
boosts the optimal carbon price. A market-based calibration, in contrast, assumes that uncertainty depresses equity prices and thus sets the elasticity of intertemporal substitution
η greater than 1 in which case macroeconomic uncertainty
reduces the optimal carbon price. A market-based calibration gives a big role for the risk of macroeconomic disasters, which cuts the dividend-price ratio and thus the SDR much more than normal growth uncertainty. Hence, the risk of macroeconomic disasters has much bigger negative effects on the carbon price if
η > 1. However, Epstein et al. (
2014) argue that Epstein–Zin preferences are too restrictive because they cannot capture all three aspects of preferences, namely risk aversion, intertemporal substitution or intergenerational inequality aversion, and preference for early resolution of uncertainty. For example, one might calibrate
γ and
η to market data, but this may lead to an unrealistic preference for early resolution of uncertainty.
So, what should one use: an ethics-based or a market-based calibration? The choice seems to matter as the comparative statics results with respect to uncertainty depend on which one is chosen. The answer might well be that one needs to use both in the sense that the best approach may be to have a model where policy makers adopt ethics-based preferences with a near-zero impatience
ρ (and possibly also an IIA > 1 or
η < 1) and the private sector which is much less patient (and has an
η > 1). It can be shown that, if stochastic shocks are ignored, more patient policy makers than private agents implies that the optimal policy consists of two components: a carbon price equal to the social cost of carbon and a capital subsidy to correct for the excessive impatience of private agents (Barrage,
2018). In such a context, if policy makers do not want to implement the required capital subsidy, carbon pricing is time inconsistent.
2.5 Effects of climate tipping points and tail risks on the optimal carbon price
Just like a small risk of rare macroeconomic disasters has a big effect on the SDR and thus on the carbon price (in contrast to the modest effect of conventional macroeconomic uncertainty), small risks of climatic tipping and abrupt shifts in the climatic regime (e.g. caused by melting of the Greenland or the Antarctic Ice Sheet, the melting of the permafrost, or the reversal of the Gulf Stream) generate big increases in the optimal risk-adjusted carbon price because a hotter planet increases the risk of tipping (e.g. Cai & Lontzek,
2019; Lemoine & Traeger,
2014; Lontzek et al.,
2015; van der Ploeg & de Zeeuw,
2018). Policy makers must thus not only internalise the global warming externalities arising from damages to aggregate output but also internalise the adverse effect of temperature on the risk of climate disasters. The main insight is that the risk of climate tipping leads to very big increases in the optimal carbon price. To make things worse, if the setting off of one climate tipping point raises the likelihood of another tipping point being set off, the ensuing risk of cascading climate tipping points can boost the carbon price even more (Cai et al.,
2016; Lemoine & Traeger,
2016).
Furthermore, it has been argued that the combination of fat-tailed distributions and power utility functions implies that the optimal carbon price is infinite in which case the world is willing to give up all national income to cut emissions (Weitzman,
2009,
2011). However, if there is a bound on marginal utility, the “dismal” theorem no longer holds. Pindyck (
2011) surveys the effects of fat-tailed and thin-tailed uncertainty on climate policy. He argues that cost–benefit analysis of carbon pricing is fraught with difficulties as not even the probability distribution of future temperature impacts is known. Still, skewed uncertainties boost the carbon price a lot.
Before we move on, we want to point out that mitigating climate change via pricing carbon is important but in practice adaptation policies are essential too. Adaptation is especially important after a climate tipping point when large parts of the economy may be destroyed. The most important adaptation strategies are investment to protect against disasters and for people to move and migrate after a disaster has hit. Not taking account of this will lead to overestimates of the costs of global warming. For example, Desmet et al. (
2021) show that ignoring adaptation leads to costs of 4.5% of GDP in 2200 compared to 0.11% of real GDP if there is adaptation. In case adaptation is infeasible after a macroeconomic disaster prudence and mitigation is even more important, and thus, the cost of carbon needs to be higher.
2.6 Effects of a falling term structure for the social discount rate
Equations (
3) and (
4) imply a flat term structure for the SDR or the dividend-price ratio. In other words, the same rate is used to discount marginal damages from year 51 to 50 as from year 1 to 0. But many economists argue that the term structure of the SDR slopes downwards and that the one-year discount rate used in the future is lower than that used today (e.g. Arrow et al.,
2014). If more distant discount rates are smaller, there is a downward-sloping term structure for the SDR. There are various reasons for a falling term structure.
The first reason is that, if shocks to consumption growth rates and the interest rate are positively correlated over time and thus persistent, the schedule of efficient discount rates may decline as the horizon lengthens (Vacisek,
1977). It then follows that the downward adjustment of the SDR due to uncertainty becomes bigger for longer horizons provided the autocorrelation coefficient of the process,
φ, is between zero and one (e.g. Gollier,
2012). The positive correlation in consumption growth shocks makes future consumption riskier and thus makes the prudence effect in the SDR (see Sect.
2.1) bigger for more distant horizons. The prudence effect is multiplied by
\((1 - \varphi )^{{ - 2}} > 1\) as the horizon tends to infinity. Hence, the more autocorrelation in shocks to consumption growth (higher
φ), the bigger the amplification of the prudence effect at very long horizons.
3 However, estimated models with autocorrelation in consumption growth imply only modest declines in the SDR. But if subjective uncertainty about the trend and volatility of consumption growth is introduced, this can lead to more rapid declines in the SDR (e.g. Gollier,
2008; Weitzman,
2007). For example, if mean consumption growth is 1% or 3% per year with 50–50 chance (with zero impatience,
ρ = 0 and risk aversion and intergenerational inequality aversion equal to two,
γ = 1/
η = 2), the SDR (excluding the insurance term, see Sect.
2.1) falls from 3.5% today to 2% per year in three centuries (Gollier,
2008).
The second reason for a declining term structure of the SDR argues that evaluating the expected net present value of a stream of uncertain marginal damages using a constant SDR is equivalent to calculating the net present value with a certain, but decreasing certainty-equivalent value of the SDR, i.e.
\(- \ln \left( {E\left[ {e^{{ - SDR \times t}} } \right]} \right)/t\) if the horizon is
t (Weitzman,
1998,
2001,
2007). Jensen’s inequality gives
\(E\left[ {e^{{ - SDR \times t}} } \right] > e^{{ - E\left[ {SDR} \right]t}} ,\) so that the certainty-equivalent value of the SDR is less than
\(E\left[ {SDR} \right]\) and this difference increases as the horizon lengthens. Table
2 gives the certainty-equivalent value of the SDR for different horizons when the SDR is 2%, 4% or 6%, each with a probability of one third. This value equals the mean value of the SDR (i.e. 4%) at infinitesimally small horizons and tends to the minimum value of the
SDR (i.e. 2%) as the horizon becomes infinite. At very long horizons, the payoff with the lowest SDR dominates the payoffs under the other values of the SDR. Uncertainty about future discount rates thus calls for a decreasing term structure of the certainty-equivalent value of the
SDR. This boosts the SCC and the carbon price.
Table 2
The certainty-equivalent value of the SDR declines with the horizon
1 | 980 | 961 | 942 | 961 | 3.99 |
50 | 368 | 135 | 50 | 184 | 3.34 |
100 | 135 | 18 | 2 | 52 | 2.96 |
200 | 18 | 0.33 | 0.006 | 6.2 | 2.54 |