The Safe Carbon Budget

Cumulative emissions drive peak global warming and determine the safe carbon budget compatible with staying below 2oC or 1.5oC. The safe carbon budget is lower if uncertainty about the transient climate response is high and risk tolerance low. Together with energy costs this budget determines the constrained welfare-maximizing carbon price and how quickly fossil fuel is replaced by renewable energy and how much of it is abated. This price is the sum of a gradual damages component familiar from the unconstrained optimal carbon price highlighted in economic studies and a Hotelling component for the additional price needed to ensure that the safe carbon budget is never violated familiar from IAM studies. If policy makers ignore damages, as in the cost-minimizing temperature constraint literature, a more rapidly rising carbon price results. The alternative of adjusting damages upwards to factor in the peak warming constraint leads initially to a higher carbon price which rises less rapidly.


Introduction
Many economic studies derive optimal climate policies from maximizing social welfare subject to the constraints of an integrated assessment model that combines both a model of the economy and a model of the carbon cycle and temperature dynamics (e.g., Nordhaus, 1991Nordhaus, , 2010Nordhaus, , 2014Golosov, et al., 2014;Dietz and Stern, 2015;van den Bijgaart et al., 2016;Rezai and van der Ploeg, 2016). The resulting optimal carbon price is the same throughout the world and is more or less proportional to world GDP if global warming damages are proportional to world GDP.
The ratio depends on ethical considerations such as intergenerational inequality aversion (the lack of willingness to sacrifice consumption today to curb global warming many decades into the future) and impatience or the amount by which welfare of future generations is discounted.
It also depends on detailed aspects of the carbon cycle and heat exchange dynamics (e.g., the fraction of carbon emissions that stays up permanently and the rate at which the remaining parts of the carbon stock returns to the surface of the earth).

The Paris Climate Agreement within the United Nations Framework Convention on Climate
Policy (COP21) has been signed in April 2016 and commits to keep global warming well below 2 o C this century and pursue efforts to limit temperature even further to 1.5 o C. This eschews a welfare-maximizing approach and has the merit of focusing at a clear and easy-to-communicate target for peak global warming. Since climate change is subject to large degrees of uncertainty, it is usual to specify a probability of say 2/3 that this target must be met or equivalently to set a risk tolerance of a 1/3. Since cumulative carbon emissions drive peak global warming, the target for peak global warming determines how much carbon can be emitted in total. This is called the safe carbon budget and depends on three key parameters only: maximum permissible global warming, climate uncertainty, and risk tolerance. The informational requirements for calculating this budget are much less than needed for constrained or unconstrained welfare maximization.
The path-breaking study by Fitzpatrick and Kelly (2017) also investigates the difficult problem of determining the optimal climate policy under uncertainty with the constraint of a probabilistic temperature target. My approach is deliberately much simpler as I exploit that peak global warming is driven by cumulative carbon emissions. The policy problem can therefore be separated into two parts: first, determine the safe carbon budget for cumulative emissions and fossil fuel use, and, then, work out how this budget for fossil fuel use is allocated over time in a way which maximizes global welfare taking due account of production losses resulting from global warming. The resulting recommendations are straightforward to communicate to policy makers, and hopefully by splitting it in two parts it is helps to get countries to agree on the required international climate policy.
My main aim is to show that there is a constrained efficient (cost-minimizing) time path for the price of carbon which ensures that cumulative emissions from now on stay forever within the safe carbon budget. This carbon price and the time paths for mitigation and abatement are derived from an integrated assessment model and, if the safe carbon budget bites, this price exceeds the unconstrained optimal carbon price. I show that it consists of two components: (1) the present discounted value of all future production losses from emitting one ton of carbon today (also called the social cost of carbon) which rises at the same rate as world GDP as in recent studies on simple rules for the optimal carbon price in the absence of temperature constraints (e.g., Golosov et al, 2014;van den Bijgaart et al., 2016;Rezai and van der Ploeg, 2016), and (2) the present discounted shadow costs of staying forever within the safe carbon budget which rises at the real interest rate (called the Hotelling component). One can thus determine how fast fossil fuel is phased out and renewable energies are phased in and how much of fossil fuel is abated. Using the safe carbon budget means that ethically loaded concepts such as how much to discount welfare of future generations and the willingness to sacrifice consumption today to curb global warming play no role in determining the safe budget, but do affect the timing of the energy transition and how much of fossil fuel is abated. Inter alia, I also show that calibrated production damages from global warming are under-estimated if the risk of tipping points is factored in damages by insisting that peak temperature stays below target.
I differ from existing studies on temperature constraints in taking cumulative emissions, peak warming and the safe carbon budget rather than a temperature constraint as drivers of climate policy. This is why the Hotelling component (2) rises at a rate equal to the real interest rate and not at a rate equal to the real interest rate plus the rate of decay of atmospheric carbon as in Nordhaus (1982), Tol (2013) and Bauer et al. (2015). This is also why, in contrast to Lemoine and Rudik (2017), temperature inertia does not lead to an inverse U-shape of the costminimizing path for the carbon price that grows more slowly than exponentially and might temporarily overshoot the safe carbon budget consistent with the temperature target.
My other aim is to put forward these results in the simplest possible integrated assessment framework where cumulative emissions drive peak warming. I make simplifying assumptions such as abstracting from non-CO2 carbon gases for which the transient climate response to cumulative emissions is not valid, other climate uncertainties, detailed marginal abatement costs, endogenous technology and sectoral transformation strategies and more convex damage functions. My aim is not to come up with the best numbers for climate policy as this is better left for detailed integrated assessment models (IAMs), albeit that the time paths for the optimal unconstrained and constrained carbon price are in line with earlier numerical fully fledged IAM studies (e.g., Clarke, et al., 2014). The climate policy/science literature has already addressed the need to tighten climate policy in the light of the 1.5 o C target (e.g., Kriegler et al, 2014;Tahvoni et al., 2015;Rogelj, et al., 2015Rogelj, et al., , 2016, the FEEM Limits Project, the 2016 SSP data base on shared socioeconomic pathways, comparison exercises reported in IPCC studies , and studies that deal with the Hotelling-type carbon taxes needed to deal with peak warming constraints (e.g., Bauer et al., 2015). The analysis in this paper is complementary and more modest in that it builds a bridge between the economics literature based on production damages and the climate policy/science literature on temperature constraints. To put it differently, overshooting a peak warming target bears an unacceptable risk of irreversible tipping points and the implicit (or shadow) costs of meeting this target must be added to the explicit costs based on production damages.

Paris COP21 target for peak global warming and the safe carbon budget
The key driver of peak global warming measured as deviation from pre-industrial temperature, , PGW is cumulative carbon emissions, E (e.g., Allen et al., 2009a,b;Matthews et al. 2009;Gilllett, et al. 2013;IPCC, 2013;Allen, 2016), which are measured here from 2015 onwards and thus do not contain historical emissions. Cumulative emissions ignore the slow removal of part of atmospheric carbon to oceans and surface of the earth and thus under-estimate peak global warming, but only by a small amount (see Appendix A1  Allen (2016), which allows for uncertainty in the TCRE and abstracts from additive uncertainty in PGW. Uncertainty in the TCRE may follow from a more complicated stochastic process with dynamics and non-normal features such as skewedness and fat tails or result from a number of underlying shocks to the climate system, but (1) keeps it simple. Paris COP21 has agreed to keep PGW below 2 o C (and to aim for 1.5 o C). I assume that this target has to be met with probability 01   : (2) o prob 2 C . PGW     IPCC typically sets  to 2/3. The safe carbon budget compatible with (2) is deduced from (1) and denoted by E . Cumulative emissions at any time t cannot exceed the safe carbon budget: F  is the cumulative normal density function with mean  and variance 2  .
Equation ( (2015) suggest that reserves and probable reserves (resources) are 3 to 10-11 times higher than the carbon budget compatible with peak temperatures of 2 o C. They calculate that 80% of global coal reserves, half of global gas reserves and a third of global oil reserves must be left unburnt. In practice, much more needs to be abandoned as many oil and gas reserves are owned by states instead of private companies. Not only carbon assets will be stranded but also energyintensive irreversible investments in electricity generation such as coal-fired stations. A more ambitious PGW target of 1.5 o C as stated in the Paris COP21 agreement requires tightening the safe carbon budget to 411 GtCO2 if uncertainty is ignored. At current global yearly uses of oil, coal and gas this implies the end of the fossil fuel era in one decade instead of four decades.
Equation (3) Table 1 reports the safe carbon budget for these two calibrations, for peak global warming targets of both 2 o C and 1.5 o C, and for a range of risk tolerance values. The qualitative results are the same for the two calibrations of the , TCRE but the calibration based on Matthews et al. (2009) and Gillett et al. (2013) yield higher safe carbon budgets due to the lower mean value of the TCRE (despite the slighted higher standard deviation). Below I focus on the results using the calibration based on Allen et al. (2009).
Focusing at a PGW target of 2 o C, Table 1 indicates that a risk tolerance of 1/3 (in line with the value reported by the IPCC) gives a safe carbon budget from 2015 onwards of 1,228 GtCO2.
Tightening up risk tolerance to 10% and 1% curbs the safe carbon budget to 994 GtCO2 and 766 GtCO2, respectively. Less risk tolerance thus implies that less carbon can be burnt in total.
If PGW has to be kept below 1.5 o C, the safe carbon budget drops dramatically from 1,228 GtCO2 to 381 GtCO2 if risk tolerance is a third and from 766 GtCO2 to a mere 238 GtCO2 if risk tolerance is 1%. For future reference, we choose a risk tolerance of 1%.

Optimal energy transition given the safe carbon budget
What are the optimal timing of fossil fuel use and carbon emissions, the mitigation and abatement rates, and end of the fossil fuel era? These depend crucially on the costs of fossil fuel versus that of renewable energy, the cost of abatement, and the various rates of technical progress. It is thus not surprising that the IPCC and climate scientists stress a tight target for PGW with reference to geo-physical conditions and risk. I augment a very simple integrated assessment model put forward in van der Ploeg and Rezai (2016) with the constraint on the safe carbon budget (3). This model has constant trend growth in world GDP, g, and constant rates of technological progress in fossil fuel extraction, mitigation of energy (which lead to a gradually rising share of renewable energy) and abatement. It has a two-box carbon cycle (Golosov et al., 2014) and a lag between temperature and increases in atmospheric carbon concentration (Appendix A1). The calibration is based on DICE-2013R (Nordhaus, 2010(Nordhaus, , 2014. Maximizing global welfare subject to the constraint that income net of damages must equal spending on consumption, energy generation, mitigation and abatement yields the unconstrained optimal carbon price. Calculation of this price requires additional parameters for the carbon cycle, i.e., the fraction of carbon emissions staying up in the atmosphere forever, 0 ,  the rate of return of remaining emissions to the surface of the earth and oceans, 1 ,  and the mean lag between the temperature rise following an increase in atmospheric carbon, , Tlag and for the ethical considerations, i.e., the rate at which welfare of future generations is discounted, RTI, and the coefficient of relative intergenerational inequality aversion, IIA. The unconstrained optimal carbon price is proportional to world GDP (see Appendix A2): 3 is the growthcorrected social discount rate, and d > 0 is the flow damage coefficient defined as the fraction loss of world GDP measured in trillion US dollars per trillion ton of carbon in the atmosphere.
The flow damage coefficient is adjusted to allow for the delayed impact of the carbon stock on global mean temperature (see Appendix A2). The unconstrained optimal carbon price is thus high and climate policy ambitious if a large part of emissions stay up forever (high  0 ), the absorption rate of the oceans is low (low  1 ), the temperature lag is small, welfare of future generations is discounted less heavily (low RTI), and there is less willingness to sacrifice consumption to curb future global warming (low IIA). With higher economic growth (high g) future generations are richer so current generations are less prepared to curb global warming (especially if IIA is high), but growth in damages from global warming is also higher and thus a higher carbon price is warranted. The net effect of economic growth on the carbon is negative if intergenerational inequality aversion exceeds 1.
Maximizing welfare given the damages from global warming but subject to the additional constraint that cumulative carbon emissions cannot exceed the safe carbon budget yields the 3 Our formulation of damages is equivalent to that of Golosov et al. (2014) except for adding a temperature lag. The optimal carbon price is independent of the carbon stock. With more convex damages (e.g., Dietz and Stern, 2015) d in (4) needs to be replaced by the reduced-form damage function and E is the delayed carbon stock. Hence, the optimal carbon price increases with global warming as well as world GDP. Convex damages capture the risk of tipping points but this risk is already captured by having an explicit additional temperature constraint. This justifies our specification with flat marginal damages.
8 constrained optimal climate policy. If the safe carbon budget constraint (3) bites, the constrained optimal carbon price, , t P is given by (see (A17b) in Appendix A2): The optimal constrained carbon price (4) that ensures that the safe carbon budget is never violated consists of two components: (i) one akin to the one found in the literature on simple rules for the optimal unconstrained carbon price which grows at the same rate as world GDP (cf. Golosov et al., 2014;van den Bijgaart et al., 2016;Rezai and van der Ploeg, 2016); and (ii) a faster rising Hotelling component that grows at the rate of the real interest rate, i.e., SDR + g = 0. RTI IIA g    If policy makers ignore production damages from global warming as is usual in the literature on temperature constraints (e.g., Nordhaus, 1982;Tol, 2013;Bauer, et al. 2015;Lemoine and Rudik, 2017), the constrained optimal carbon price boils down too: is the real interest rate and *  ensures that the safe carbon budget is never violated. The constrained carbon price is simply the Hotelling component. Matters can become more complicated if there is also a temperature lag, since then the carbon tax has an inverse U-shape and might overshoot (Lemoine and Rudik, 2017). This does not occur if the peak temperature constraint is formulated in terms of cumulative emissions. This is also why the Hotelling component of the carbon price in (4) rises at the real interest rate and not at the real interest rate plus the rate of decay of atmospheric carbon.
Cost minimization given the carbon price (4) (A21)). I assume cost conditions are such that fossil fuel is fully mitigated before it is fully abated.

Calibration of carbon stock dynamics, damages and the economy
The top panel of Table 2 gives the benchmark estimates of the variance of the lognormally distributed shock to the , TCRE the target for , PGW and risk tolerance. The middle panel gives the parameters needed for finding the optimal energy mix and transition to the carbon-free era from cost minimization given the carbon price, and the bottom panel the additional parameters needed for calculating the welfare-maximizing carbon prices (cf. van der Ploeg and Rezai, 2017). Global energy use measured in GtC is 0.14 percent of world GDP, which matches current energy use of 10 GtC and initial world GDP of 73 trillion dollars. We focus at using mitigation and abatement, so set exogenous technical progress in energy needs to zero. Initial fossil fuel and renewable energy costs are calibrated to give current energy cost shares of 7% of GDP and an additional cost of 5.6% of GDP for full de-carbonization. The cost of fossil fuel is set to 515 $/tC and rises at the rate of 0.1 percent per year to capture resource scarcity.
Technical change leading to a reduction in the costs of mitigation and abatement is 1.25% per year, which matches the cost of 1.6% of GDP for full de-carbonization in 100 years. The cost of full abatement is calibrated to an initial value of 20% of GDP, which then falls at the rate of non-carbon technologies and decreases to 5.7% of GDP in 100 years.
Turning to the bottom panel, the rate of time impatience or the utility discount rate is set to 1.5 percent per year and shows how impatient policy makers are. The coefficient of relative intergenerational inequality aversion is set to 1.45 and indicates how little policy makers are prepared to sacrifice utility of current generations for the benefit of future generations. Given a trend growth rate in world GDP of 2 percent per year, this implies a long-run real interest rate of 4.4 percent per year. Global warming damages in any year are $1.9 per ton of effective carbon in the atmosphere (0.5 percent of world GDP at current levels of atmospheric carbon). These damages rise at the same rate of growth as world GDP and the discount rate to be used is thus the growth-corrected long-run run real interest rate: 2.4 percent per year.
Effective carbon in the atmosphere takes account of the delay between a rise in the stock of carbon and mean global temperature often ten years. A fifth of carbon stays to all intents and purposes permanently up in the atmosphere; the remainder slowly returns to the oceans and the surface of the earth at a rate of 0.23 percent per year. These geo-physical details are not necessary for the calculation of the Hotelling component of the constrained carbon price.

Constrained optimal climate policy simulations with a safe carbon budget
Using this calibration, not pricing carbon at all leads to zero mitigation and zero abatement, cumulative emissions of 6,519 GtCO2, 118 years for the end of the fossil fuel era to occur, and PGW of 4.6 o C, which is much too high. The globally best unconstrained climate policy is portrayed by the purple solid lines in Figure 1. It has an initial price of carbon of $12/tCO2 (or $44/tC), and grows at 2% per annum from then on. The mitigation rate is driven by technological progress and the rising price of carbon, and rises from 20% to 100% in 78 years at which date the carbon-free era starts. The abatement rate rises from a mere 1.5% to 19% at the end of the fossil fuel era. In total 2,328 GtCO2 is burnt, which implies PGW of 2.6 o C. The unconstrained climate policy thus overshoots the 2 o C target agreed at the Paris COP21 conference by 0.6 o C. The safe carbon budget from 2015 onwards corresponding to a risk tolerance of 10% and a peak warming target of 2 o C is 994 GtCO2 (see Table 1). 4 Figure 1 portrays three policies to ensure that cumulative emissions stay within this budget: (1) constrained welfare-maximizing carbon price (4) with d calibrated to estimated production damages (black dashed lines); (2) constrained cost-minimizing carbon price (4") ignoring these damages and thus with d = 0 (black dotted lines); and (3) the welfare-maximizing carbon price with damages adjusted upwards to stay within budget (red dashed-dotted lines).

Constrained welfare-maximizing carbon price with calibrated damages
The constrained welfare-maximizing carbon price manages to keep cumulative emissions to 994 GtCO2 and has two components: the unconstrained carbon price and the Hotelling component (the difference between the dashed black and the purple solid line). The first component rises at the rate of growth of world GDP (2% per year) and the second component rises at the real interest rate (4.4% per year). The initial Hotelling component of the carbon price necessary to ensure that the safe carbon budget is not violated is $10/tCO2, so that the initial carbon price has to increase from $12 to $22/tCO2. The carbon era now ends in 49 instead of 78 years.
During this period the mitigation rate rises from 28% to 100% and the abatement rate rises from 2.8% to 34%. Note that a peak warming target of 1.5 o C implies that only 308 GtCO2 can be burnt. It necessitates a much higher path for the constrained optimal carbon price that starts at $58/tCO2 and rises in a mere 28 years to $179/tCO2 at the end of the carbon era (not shown).

Constrained cost-minimizing carbon price ignoring calibrated damages
The cost-minimizing time path for the carbon price that ignores damages from global warming also ensures that cumulative emissions do not exceed 994 GtCO2 and consists of only the Hotelling component. It rises more rapidly than the path that does take account of damages. It starts somewhat lower than at $16 instead of $22/tCO2 and rises in 47 years to a final carbon price of $128 instead of $119/tCO2. As a result, mitigation starts somewhat more modestly (at 24%) too. Abatement is more modest and rises from 2.0% to 29% at the end of the carbon era.

Welfare-maximizing carbon prices with damages adjusted upwards
Since welfare maximization with calibrated damages lead to overshooting of the peak warming target, this suggests that calibrated damages are an under-estimate of the true risk of global warming in that they ignore the risks of tipping points and climate disasters which are captured by the safe carbon budget constraint. Adjusting the damage coefficient upwards by a factor 2.8 (i.e., from 1.9% to 5.4% of world GDP per TtC) ensures that cumulative emissions never exceed the safe carbon budget when welfare is maximized. The end of the fossil fuel era then occurs more than two decades earlier than with the unconstrained optimal carbon price (after 56 instead of 78 years, but more slowly than with the constrained welfare-maximizing carbon price (49 years). The initial carbon price almost triples from $12 to $34/tCO2, and then rises at 2% per annum in line with the rate of economic growth. 5 As a result of this more ambitious climate 5 The average adjusted carbon price over 2015-2100 is $89/tCO2 for a safe carbon budget of 994 GtC02. The initial and average adjusted carbon price for a budget of 1,327 GtCO2 (i.e., ignoring uncertainty; see Table 1) are $25 and $65/tCO2, respectively. The latter is only a little higher than the range of $50-60/tCO2 for the average carbon price to keep temperature below 2 o C during 2015-2100 reported in Clarke et al. (2014) for a real interest rate of 5% per year, perhaps due to our real interest rate being a little lower. policy, the path for the mitigation rate is higher and starts at 36% and rises to 100% during the fossil fuel era. The abatement rate is also higher; it starts at 4.2% and rises to 21% towards the end of the fossil fuel era.

Conclusion
Climate uncertainty, a higher transient climate response to cumulative emissions and a tighter risk tolerance imply a lower safe carbon budget and that less fossil fuel can be burnt in total, thus requiring a more ambitious climate policy. The safe carbon budget is easy to negotiate and communicate, and does not depend on ethical considerations regarding welfare of current and future generations. Furthermore, the informational requirements are less than needed for welfare-optimizing climate policies. The relatively modest identified damages from global warming in integrated assessment models imply that the unconstrained welfare-maximizing carbon price leads to overshooting of the peak warming target and thus that the safe carbon budget constraint bites. There are three options of staying within the safe carbon budget.
The first option is the welfare-maximizing carbon price that ensures that the safe carbon budget constraint is never violated. It consists of two components: a gradual component corresponding to the social cost of carbon based on calibrated damages which rises at a rate equal to the growth rate of world GDP and a Hotelling component that rises at a faster rate equal to the real interest rate. The second option is relevant if policy makers ignore damages, as in the cost-minimizing temperature constraint literature. This leads to a more rapidly rising carbon price which is not recommended. The third option of adjusting damages to factor in the peak warming constraint leads to a less rapidly rising carbon price than the first option. It has a smoother time path for the carbon price and has some merit if calibrated damages indeed under-estimate true damages.
More generally, carbon prices are affected by a wide range of other climate and economic uncertainties. For example, uncertainty about future growth of aggregate consumption depresses the social discount rate used by prudent policy makers and pushes up the unconstrained optimal price of carbon even more (e.g., Gollier, 2012). Other types of uncertainty about the damage flows resulting from atmospheric carbon, the climate sensitivity, and sudden release of greenhouse gases into the atmosphere boost the risk-adjusted unconstrained optimal carbon price even more and take account of hedging risks (e.g., Dietz et al., 2017;Hambel et al., 2017;van den Bremer and van der Ploeg, 2017). Mitigating the risks of interacting, multiple tipping points can push up the carbon price by a further factor of 2 to 8 (Lemoine and Traeger, 2016;Cai, et al., 2016). As uncertainty about the climate sensitivity has the biggest effect on carbon prices 6 , it may not be bad to start with the risk-adjusted safe carbon budget. Still, for future research it is important to extend the literature on risk-adjusted carbon prices to allow for peak warming constraints.
It has been argued that an approach based on probabilistic stabilization targets is ad hoc and incurs welfare costs of 5% as the targets are inflexible and do not respond to changes in climatic conditions, the resulting policies tend to overreact to transient shocks, and the temperature ceiling is lower than the unconstrained optimal temperature under certainty (Fitzpatrick and Kelley, 2017). 7 The relatively small welfare costs may be a price worth paying if an easy-tocommunicate temperature target prompts policy makers into action.
In fact, the IPCC approach of focusing attention at cumulative emissions and the safe carbon budget focuses at what matters most for global warming. The role of economics is to show how these cumulative budgets translate in the most cost-efficient manner to time paths of fossil fuel use, renewable use, and abatement. This paper has extended the IPCC approach to allow for various forms of climate uncertainty, since these curb the safe carbon budget significantly. This is related to the point-of-no return approach (van Zalinge et al., 2017), which prompts the question what to do once the climate has moved outside the viable region and can no longer be moved with traditional carbon pricing policies into the viable region. Negative carbon emissions and therefore unconventional policies such as geo-engineering are then called for (e.g., Keith, 2000;Crutzen, 2006;McCracken, 2006;Bala et al., 2008;Lenton and Vaughan, 2009;Barrett et al., 2014;Moreno-Cruz and Smulders, 2016) and some argue that they are already called for to keep global warming below 2 o C (e.g., Gassler et al., 2015). Such policies act as insurance and are needed before the climate moves outside the viable set and reaches the point of no return.
More work is needed on the reversible and irreversible uncertainties driving the climate (both the stock of carbon in the atmosphere and temperature) and what they imply for the safe carbon budget, climate mitigation and adaptation policies, and the need for negative-emissions policies.
Rogelj, J., G. Luderer, R.C. Pietzcker, E. Kriegler, M. Schaeffer, V. Krey and K. Riahi (2015). Energy system transformations for limiting end-of-century warming to below 1. Maximizing global welfare subject to the resource constraint that income available after damages has to equal spending on consumption, energy generation, mitigation and abatement and the carbon cycle discussed in Appendix 1 yields the unconstrained optimal climate policy. Maximizing welfare subject to the additional constraint that cumulative carbon emissions cannot exceed the safe carbon budget yields the constrained optimal climate policy. 8 Global welfare is  Note that damages to economic production are proportional to the delayed stock of carbon in the atmosphere. This is a reduced-form relationship, since temperature is a concave (typically logarithmic) function of past stocks of atmospheric carbon and damages a convex function of temperature, so this formulation supposes that the convexity and concavity wipe each other roughly out as argued in Golosov et al. (2014). Strictly speaking, the uncertainty in the climate sensitivity and transient climate response affects the damage coefficient d but we will ignore this for simplicity. Allowing for this would boost the unconstrained optimal price of carbon, mitigation rate and abatement rate somewhat, but will not affect the constrained optimal climate policies. Equation (A5) is the cumulative emissions constraint and follows from the identity 00 0 (1 )(1 ) . (