Optimal Carbon Pricing in General Equilibrium: Temperature Caps and Stranded Assets in an Extended Annual DSGE Model

The general equilibrium model developed by Golosov et al. (2014), GHKT for short, is modified to allow for additional negative impacts of global warming on utility and productivity growth, mean reversion in the ratio of climate damages to production, labour-augmenting technical progress, and population growth. We also replace the GHKT assumption of full depreciation of capital each decade by annual logarithmic depreciation. Furthermore, we allow the government to use a lower discount rate than the private sector. We derive a tractable rule for the optimal carbon price for each of these extensions. We then simplify the GHKT model by modelling temperature as cumulative emissions and calibrating it to Burke et al. (2015) damages. Finally, we consider how the rule for the optimal carbon price must be modified to allow for a temperature cap, and what this implies for stranded oil and gas reserves. We illustrate our analytical results with a range of optimal policy simulations.


Introduction
Carbon emissions are at the root of the most important global externality (Stern, 2007).
The first-best response is to price carbon, either via a carbon tax or an emissions market, at a uniform price throughout the globe and rebate the revenue as lump-sum rebates. The tractable and influential general equilibrium model of growth and climate change developed by Golosov et al. (2014), denoted GHKT from hereon, presents a simple and intuitive rule for the optimal pricing of carbon in general equilibrium: the price of emissions should be proportional to aggregate output and thus grow in line with trend growth. The proportionality factor depends on the discount rate, the severity of damages, and the dynamics of atmospheric carbon. The GHKT model has become very popular due to its analytical versatility. Its rule for the optimal carbon price is intuitive and easy to calculate but can be criticized for the strong assumptions necessary to derive it. Our aim is therefore to offer various extensions of the GHKT model, some more important than others, which still yield an analytically tractable rule for the optimal carbon price. We also offer a simplification of the model where temperature is driven by cumulative emissions and damages are calibrated to Burke et al. (2015) rather than to Nordhaus (2016). We thus take stock of the merits, uses, and limitations of the GHKT model, and believe that the extended GHKT model is more realistic and can be used in teaching.
We first extend the GHKT model to allow for logarithmic depreciation. Using theoretical and empirical arguments, Anderson and Brock (2021) demonstrate that logarithmic depreciation is better able to fit the aggregate data than geometric depreciation and they suggests its tractability lends itself for tractable expressions for optimal policy. Full depreciation each decade, as is in GHKT, is a special case of logarithmic depreciation.
We adopt their depreciation model and find a tractable expression for the optimal carbon price. This allows us to use finer time resolutions (yearly instead of decadal), opening the model to the inclusion of business-cycle interactions. 1 Global warming damages in the GHKT model are proportional to aggregate production.
We also allow for disutility from global warming and climate damages to the growth of total factor productivity. These extensions also lead to a tractable expression for the 2 optimal carbon price, allowing any combination of damages to consumption and utility.
Using recent calibration studies, we show that the optimal carbon price in 2019 rises from $64 to $456 per ton of carbon if damages affect productivity growth permanently.
Exogenous growth in population (cf. Kelly and Kolstad, 2001) and productivity largely determine the severity of climate change. We, therefore, include positive rates of population growth and labour-augmenting technical progress in the GHKT model. This has important implications for the rates of growth and interest, while still allowing us to obtain a generalized tractable expression for the optimal carbon price. A one percent growth rate in population increases the optimal carbon price per ton of carbon from $64 to $164 per ton of carbon (and to $2,039 per ton of carbon if damages affect productivity growth). We also allow for mean reversion in the process of total factor productivity growth and show how this affects the rule for the optimal carbon price.
Following Belfiori (2016) and Barrage (2018) we allow the government to have a lower discount rate than the private sector. This implies that the carbon pricing policy must be complemented with a subsidy for natural and man-made capital to offset the relative myopia of the private sector. In this case, the social cost of carbon increases as the government places greater weight on future generations. Given that a handout to the owners of capital and, more importantly, oil reserves is politically unlikely, we show by how much the carbon price would be increased to offset the missing subsidy. This increase is a second-best way to compensate future generations for the lower material wealth left to them in the form of capital stocks and fossil fuel reserves. We derive the tractable rules for each of these policy instruments in general equilibrium and illustrate why such an equilibrium is time consistent. In our numerical simulations the second-best carbon price is up to 10% above its first-best level.
We also apply two simplifications of the GHKT model based on recent developments in atmospheric science and economics, which lead to a general-equilibrium extension of the framework set out in van der . First, recent atmospheric science insights suggest that temperature is driven by cumulative emissions (Allen et al., 2009;Matthews et al., 2009;Anderson et al., 2014;Dietz and Venmans, 2018;Dietz et al., 2021). While the carbon cycle of GHKT is fairly accurate on a decadal scale, this is not the case for shorter time periods (Dietz et al., 2021). Second, the GHKT model 3 models the ratio of damages to output as a negative exponential function of the atmospheric carbon stock and calibrates the function to the DICE-13 model of Nordhaus (2008). We have calibrated our damages to the GHKT model. Since these damages are very modest compared to those found econometrically in Burke et al. (2015), we also calibrated damages to these much higher estimates. Both simplifications increase the realism of the GHKT further and bring it up to date with the most recent atmospheric insights and empirical evidence on global warming damages. They yield a simpler expression for the optimal carbon price and a much higher optimal carbon price of $1,507 instead of $64 per ton of carbon.
Finally, we add realism to our welfare analysis by adding a ceiling on temperature or equivalent a cap on cumulative emissions. This captures the 2015 Paris Climate Accord where countries agreed to limit global warming to 2°C while aiming to keep warming at or below 1.5 degrees Celsius from pre-industrial levels. We show that, if the temperature cap bites, this requires adding a Hotelling term that rises at a rate equal to the market rate of interest to the welfare-maximizing carbon price. The size of this term increases for tighter temperature caps with the initial carbon prices $281 and $1,152 for the 2°C and 1.5°C targets. We also the effects on stranded oil and gas reserves and the fossil fuel mix.
Section 2 presents the original GHKT model and our extensions. Section 3 derives the social optimum and decentralized equilibrium. Sections 4 discusses the case when the government discounts the future at a lower rate than private agents. Section 5 simplifies the GHKT model by letting temperature be driven by cumulative emissions and damages be calibrated on Burke et al. (2015). Section 6 explores how our results are modified by a temperature cap. Section 7 calibrates our annual version of the GHKT model and discusses the baseline optimal policy and business as usual results. Section 8 demonstrates the numerical sensitivity of the optimal policy simulation with respect to each of our extensions of the GHKT model. Section 9 takes stock and discusses the merits and versatility, uses, and limitations of the GHKT model. Section 10 concludes.

Five Extensions of the GHKT Model
We build on the familiar Brock-Mirman (1972) and Golosov et al. (2014) assumptions: logarithmic utility, Cobb-Douglas production function, full depreciation of capital within 4 a decade, exponential climate damages in production, fossil fuel extraction and production of renewable energy requiring no capital, and a two-box carbon cycle with a part of carbon staying up permanently in the atmosphere and another part that gradually decays and is returned to the surface of the earth and oceans. Energy is derived from coal, oil/gas, and renewable sources.
Replacing the assumption of manmade capital depreciating fully in every decade, we allow for logarithmic depreciation as in Anderson and Brock (2021). Here, the case of full depreciation corresponds to a special case but importantly we can vary temporal resolution to yearly or even quarterly. We extend the GHKT model in four other directions by allowing for a direct negative effect of climate change on household utility and on productivity growth including allowing for mean reversion in total factor productivity, population growth, and growth in labour-augmenting technical progress.
We assume that the public and private discount rates are the same but in section 4 we will investigate differential discount rates.
The social planner maximizes utilitarian social welfare, which consists of utility derived from per capita consumption, ln( / ). tt CN Climate change lowers output via production damage and, in a first extension of the GHKT model, via instantaneous per capita welfare loss due to climate change, Subscripts denote periods of time t = 0, 1, 2, …. The time impatience factor is constant and denoted by 0 <  < 1. Population at time t is Nt and, in a second minor but quantitatively important extension to the GHKT model, its constant gross growth factor equals Nt+1/Nt = . 2 The stock of atmospheric carbon is Et.
Production in the GHKT framework occurs in energy and final goods sectors and is given by a nested production function. Final goods output, Yt, is produced by combining labour, Lt, capital, Kt, and energy from a Cobb-Douglas production function with a unit elasticity of substitution, while energy output is described by a CES production function and produced from (i) a finite stock of fossil fuel St which can be extracted without cost, but 2 We assume that the discount rate corrected for population growth is positive, i.e., 1.

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subject to a Hotelling rent, (ii) an infinite stock of fossil fuel (i.e. coal), and (iii) renewable energy sources, with the latter two requiring only labour input, L2t and L3t, but no capital.
A1t and A2t are the corresponding exogenous labour productivities. Production is thus given by fossil fuel use in period t. The elasticity of substitution between different energy types is constant and given by 1/ (1 ).

 −
The share parameters in the energy sub-production function are positive and satisfy 1 2 3 1.
The formulation in (2) captures GHKT model as a special case, since with 1  == it boils down to full depreciation, 1 .
The dynamics of fossil fuel depletion are In our numerical simulations, we annualize the decadal calibration of (4) in GHKT or replace it by a climate model solely based on cumulative emissions.
Total factor productivity in final goods production, At, falls as the stock of atmospheric carbon increases. The instantaneous damage of the stock of atmospheric carbon to total factor productivity is denoted by  > 0. In our fifth extension of the GHKT model we allow for mean reversion around an exogenous trend growth path for total factor productivity, .
t A With mean reversion in the log of total factor productivity denoted by 0 1 1,   −  the development of total factor productivity is given by The GHKT model corresponds to δ = 0 in which case climate change (i.e. atmospheric carbon concentrations in excess of pre-industrial level ) E lowers the level of total factor productivity in that period. However, if δ = 1, climate change affects growth of total factor productivity permanently. Intermediate values of  allow for transitory effects, i.e. mean reversion in the effects of climate change on total factor productivity. Since Harrodneutral and Hicks-neutral technical progress are equivalent for Cobb-Douglas production 3 The carbon cycle can be extended to include arbitrary many boxes. The model of Gerlagh and Liski (2017a) has 3 boxes. Joos et al. (2013) and Aengenheyster et al. (2018)  Rezai and van der Ploeg (2016) introduce a lagged response of temperature to emissions and derive a tractable expression for the optimal SCC which is akin to the one given in (8) of Proposition 1. In the GHKT notation ε = 1 − φ. 7 functions, we capture the former by  > 0 and abstract from the latter and thus assume t A is constant. In addition, we allow for labour-augmenting (Harrod-neutral) technical progress in the energy sector.

The Social Optimum
The social optimum maximizes utilitarian social welfare, eqn. (1), subject to equations (1 ) The saving and consumption functions (6) follow from the Euler equation and the capital accumulation equation (2) These are modified versions of those presented in Brock and Mirman (1972) and Anderson and Brock (2021) to allow for population growth and logarithmic depreciation. They indicate that a higher capital share, more patience, higher population growth, and lower depreciation boost incentives for aggregate investment and, hence, curb the propensity to consume. If capital depreciates fully and population growth is absent (i.e. 1  == and  = 0), the consumption share equals (1 )  − and the equations in (6) reduce to that in the GHKT model. As in the GHKT model, the propensity to consume, c, and the marginal propensity to save are constant and independent of assumptions on the climate and its interactions with the economy.
Proposition 2: Demand for the three energy types follow from the efficiency conditions c.s., c.s., c.s., Equations (7) indicate that an energy good is not used in the production of final goods if its marginal product is less than its social marginal cost. For the scarce fossil fuel type (oil and gas), this cost consists of the scarcity rent plus the SCC. If fossil fuel use is used in production, its marginal product exactly equals its social marginal cost. As fossil fuel reserves are fully depleted (asymptotically), the marginal product of fossil fuel must rise indefinitely. Similarly, the abundant fossil fuel type (coal) is only used if its marginal product equals unit labour cost (i.e., the wage divided by sector-specific labour productivity) plus the SCC. The marginal social cost of renewable energy consists of the wage divided by (potentially) endogenous labour productivity. If the marginal product of renewable energy is less than its marginal social cost, it is not used in production.
Equation (8) is the Hotelling rule. It states that the growth rate of the scarcity rent must equal the social rate of interest as only then will society be indifferent between depleting an extra unit of oil or gas and getting a return equal to the social rate of interest and leaving this unit in the ground and getting the social capital gains. To see this, define growth of output and consumption as 11 ( ) / ,  states that the optimal growth rate in per-capita consumption equals the difference between the social rate of interest and the rate of time impatience, . tt g n r  − = − Proposition 3: The first-best optimal SCC is Equation (9)  However, if we calibrate these damages in dollars, this component of the SCC increases if the utility discount factor is small (i.e. society has more patience and population grows more rapidly, higher ) and if carbon resides in the atmosphere for a longer period (higher L  , lower ).

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Corollary 2: The GHKT model has a constant population, a negative effect of global warming on the level of total factor productivity but not on utility (γ = 1,  =  = 0). Its optimal SCC is 0 (1 ) 11 . If global warming affects growth of total factor productivity (γ =  = 1,  = 0), 0 (1 ) . 1 1 1 If the GHKT model is only extended to allow for population growth and mean reversion in effects of global warming on total factor production, the optimal SCC is

This expression boils down to the GHKT expression if global
warming affects only current total factor productivity,  = 0. If global warming also affects future total factor productivity, 0 <  < 1, the optimal SCC is higher. Note that, if utility damages are absent, the optimal SCC does not depend on the depreciation scheme.
With growth rather than level damages the effect on the optimal SCC is bigger. The elasticity of the SCC with respect to the productivity damage parameter  is which increases in β, γ, and . If the growth in population is higher (larger γ), equation (8) indicates that the SCC increases for production damages but falls for utility damages. The former is intuitive, the latter is the result of two opposing effects: higher population growth increases the attractiveness of capital accumulation due to higher consumption possibilities in the future but also increases damage created by carbon. The utility cost of damages to utility is constant in our formulation. This leaves the effect on the attractiveness of capital accumulation and, hence, the SCC decreases as population growth increases. All other parameters affect the SCC as in the GHKT model.
As discussed above, the SCC is higher under a lower discount rate and a longer residence time of carbon emission in the atmosphere, either because of a larger fraction staying permanently or a lower dissipation rate of the transient fraction (higher β, 4 Population growth is absent from the expression for the optimal SCC in the GHKT model but can be accommodated for by simply replacing  by the discount factor adjusted for population growth .

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Note that the coefficients of the production functions for final goods and for energy outputs and the coefficients driving the process of logarithmic depreciation (2) do not affect the expression for the optimal SCC (9) directly. They do affect energy use and world GDP, and thus only affect the expression for the optimal SCC indirectly.
Under the assumption of the model, all components of the efficiency conditions in equation (7) scale to output and, thus, the energy system decouples from the rest of the economy. 5 To see this, note that Cobb-Douglas technology in the final goods sector implies proportionality of the marginal products of energy and labour to GDP. Further, the combination of logarithmic utility and logarithmic depreciation ensures that the inverse of marginal utility equals GDP times the constant consumption share. Given the equilibrium outcomes of the energy sector, the evolution of climate change is fully determined. Aggregate sector variables, such as total factor productivity, output, and capital stock, follow from those. This sequence of equilibria depends on the fact that the energy system is solely using labour as an input. We summarize this general finding of the model in the following corollary.

Different Discount Rates for the Private and Public Sector
There has been a debate on what the appropriate choice of discount rate for designing climate change policies is. On the one hand, Nordhaus (2008) adopts a utility discount rate of 1.5% per annum ( = 0.985). With trend growth of 2% per annum and an elasticity of intertemporal substitution (EIS) of 1/1.45, this gives a consumption discount rate of 4.4% per annum. This approach calibrates the consumption discount rate to match market returns on assets. Stern (2007) takes the stance that it is unethical to discount the welfare of future generations and therefore chooses EIS = 1 and very small utility discount rate of 0.1% per annum (corresponding to the risk of a meteorite ending the earth as we know it). This reflects ethical preferences, which lead to a much higher SCC. Rather than trying to reconcile the "descriptive" and the "normative" approaches, it seems more realistic to use a high discount rate for the private sector and a low one for the government. Lower public discount rates are equivalent to greater Pareto weights on future generations (Farhi and Werning, 2007;Belfiori, 2018).
Under differential discounting, the first-best solution for the optimal carbon prices requires a capital income subsidy alongside as an additional instrument (von Below, 2012;Belfiori, 2017;Barrage, 2018). 6 The reason is that lower social discount rates lead to socially insufficient saving by private agents with relatively high private discount rates, even in the absence of climate change, and can warrant action to overcome the excessive consumption bias as an additional policy goal. We thus assume that the government is more patient than the private sector who has discount factor 0, P   hence we assume . P   In Proposition 4 we extend the findings of von Below (2012) and Belfiori (2018) and state the optimal capital income subsidy, denoted by , and carbon tax for our model.

Proposition 4:
The social optimum is replicated in the decentralized market economy with a government that is more patient than private agents, , is given by equation (9), where the superscript FB denotes first-best policies. The first-best optimal capital income 6 Von Below (2012) and Belfiori (2017Belfiori ( , 2018 allow for scarce fossil fuel reserves and Hotelling price dynamics, while Barrage (2018) does not.

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subsidy depends on the gap between the public and private discount factors. The firstbest optimal social cost of carbon increases in the public patience.
Proof: see Appendix B.
The interpretation of Proposition 4 is as follows. A constant capital income subsidy curbs the interest rate to its socially optimal level. The optimal capital income subsidy is independent of the carbon tax, and of the capital income tax when measured in utility units. Since households own all assets in the economy, i.e. capital and fossil fuel reserves, this amounts to an identical subsidy on capital and fossil fuel reserves. 7 Proposition 5: If the capital subsidy cannot be positive, revenue of the taxes on carbon and capital income is rebated as lump-sum transfers. This policy is time consistent.
Proof: see Appendix B.
The constrained-optimal or second-best carbon tax is relevant when the capital subsidy is fixed at a too low level. It decreases as the capital subsidy is raised to the socially optimal level. Society wants to compensate future generations, who are relatively worse off due to the suboptimal capital subsidy, by improving the climate and lowering damages from global warming. If the capital subsidy is set too high, the planner would set the carbon tax below the Pigouvian rate to take some of the pressure off current generations. Hence, we can determine second-best policies if governments are constrained in the choice of their instruments.
Our rule for the second-best optimal carbon tax is unaffected by re-optimization and is therefore time consistent. If each period capital fully depreciates as in the GHKT model ( 1  == ), the equilibrium consumption share simplifies to and the second-best consumption share as fraction of the first-best consumption share is We thus see that, if the subsidy for capital and fossil fuel reserves cannot be given, the second-best carbon tax has to be set 14 higher than the first-best carbon tax to compensate. This holds also for the case of partial depreciation. If the government were to re-optimize, it does not have an incentive to deviate from its second-best optimal carbon tax path. The second-best policy is thus time consistent. This is quite a special result, which is due to the assumptions of the GHKT model muting all strategic intertemporal considerations.
There are various other ways of implementing policies with different discount rates for the government and private sector. One is to subsidize the fossil fuel owners and the final goods producers instead of subsidizing the households. Another one is if to have a carbon tax that rises faster than the rate of growth of GDP as this would also force the economy to deplete less quickly (von Below, 2012;Belfiori, 2018).

Two Simplifications of the GHKT Model: Global Warming and Damages
Here we give two science-based simplifications of the GHKT model. First, we use recent atmospheric science insights that suggest that temperature is driven by cumulative emissions (Allen et al., 2009;Matthews et al., 2009). This has been already applied in various economic applications (e.g. Anderson et al., 2019;Dietz and Venmans, 2018) and captures the climate dynamics better than most economic models, especially at grids that are finer than a decadal time grid (Dietz et al., 2021). Hence, we replace the two difference equations for the temporary and permanent component of temperature (4) by an equation linking temperature to cumulative emission and one difference equation for the stock of cumulative emissions:  Both extensions simplify the GHKT model and make it more realistic and up to date with the most recent atmospheric insights and empirical evidence on global warming damages.
Proposition 6: With temperature given by (4) and total factor productivity by (5), the expression for the optimal carbon price is Proof: analogous to proof of Proposition 3. See also Appendix C.
The optimal SCC thus still grows in line with the level of aggregate output. Without utility damages from global warming and no mean reversion in total factor productivity (i.e.

Implications of a cap on temperature or cumulative emissions
Using the finding that temperature is driven by cumulative emissions and given that climate policy is often articulated in terms of temperature targets, we now suppose that there is a cap on cumulative emissions, , Proposition 7: With temperature given by (4) and total factor productivity by (5), the expression for the optimal carbon price under a cap on cumulative emissions is given by Equation (10) shows that the optimal carbon price consists of two terms. The first term is the usual term corresponding to the optimum without a (biting) temperature cap as in equation (9)  The first two conditions of (7) give efficient use for coal and gas, The price of the oil-gas aggregate is currently less than that of coal, so that a higher carbon tax curbs at least initially emissions from coal relative to that of oil and gas. As time increases, the scarcity rent on oil and gas increases exponentially and thus eventually a carbon tax boosts emissions from coal relative to that of oil and gas.
As time goes to infinity, the carbon tax and wage rise in line with aggregate output which is at a smaller rate than the growth rate of the scarcity rent on oil and gas which equals the interest rate. It follows that as time goes to infinity and with asymptotic depletion of oil and gas, the ratio of emissions of coal to that of fossil fuel goes to infinity. However, as time goes to infinity, coal use and oil/gas use tend to zero, for else the cap on cumulative emissions will be violated, while renewable energy grows to a positive asymptote and grows forever if the economy grows. It is not immediately clear yet how the cap on cumulative emissions,

Matching the GHKT Model: Logarithmic Depreciation on an Annual Time Scale
Our model is an extension of the original version presented in GHKT. We mean by this that if capital depreciates fully, population and TFP growth are absent, damages only impact production rather than utility or productivity growth, private and public discount rates are the same, and time is on a decadal scale, our extension reproduces the findings of GHKT exactly. However, since our analytical solution allows for logarithmic depreciation, we can abandon the restrictive assumption of a decadal time scale. In this section, we show that our model reproduces the numerical findings of GHKT even on an annual scale with partial logarithmic depreciation. We then illustrate how the equilibrium trajectories under business-as-usual and optimal policy change if we allow for the various extensions of our model and estimates for depreciation in Anderson and Brock (2021). In 18 our baseline numerical simulations, we adopt the following parameter values from the calibration of GHKT given in Table 1.
This implies a capital share of 30% and an energy share of 4% of value added and a rate of time impatience of 1.5% per annum. The carbon cycle is calibrated to the following points: 20% of carbon emissions stay up forever in the atmosphere, of the remainder 60% is absorbed by oceans and the surface of the earth within a year and the rest has a mean life of 300 years. Half of a carbon emissions impulse is removed from the atmosphere after thirty years. Production damages from global warming are 2.35% of global GDP for each trillion ton of excess carbon in the atmosphere. Initial output is calibrated to match $700 trillion per decade. 8 Using these parameters and a decadal time scale, setting population and TFP constant, depreciation of capital to 100% (ι = κ = 1), and with only production damages (γ = 1, ψ = δ = 0), the optimal carbon price directly follows from expression (9) and equals $56.5 per ton of carbon (tC) or $15.4 per ton of CO2 in 2010.  Population is constant and normalized to unity, Nt = 1. GHKT set the elasticity of substitution between energy types to 0.945 < 1 (i.e. ρ = −0.058). This implies that all energy factors are essential to production and can never be phased out completely.
Climate policy, therefore, solely aims at depressing fossil energy use rather than a transition to a carbon-free economy where emissions are zero. Relative prices and demand of different energy types (in GtC) and extraction costs of coal are used to calibrate the energy share parameter κ1 = 0.5008, κ2 = 0.08916 and κ3 = 0.41004, and the initial labourefficiency parameters for coal, A2,0 = 7683 and renewables, A3,0 = 1311. The efficiency of labour in coal and renewables is assumed to grow at 2% per annum (i.e. A2,t+1/A2,t = A3,t+1/A3,t = 1.02 10 ). The finite stock of oil is set to 300 Gt of oil which converts to 253 GtC. It is assumed that there is no productivity increase in the aggregate goods sector,  = 1. Together with the initial conditions for the atmospheric stocks of carbon, 0 684, p E = 0 t E = 118, the equilibrium trajectories of energy use and climate change can be computed. 9 Using energy inputs, total factor productivity is calibrated to reproduce initial output Y0 with K0 = $128,922 billion from Barrage (2014). This gives A0 = 18,298.
The equations of Propositions 1-3 can readily be solved numerically using standard routines. 10 Figure 1 reproduces the findings of GHKT for the optimal policy (orange) and business-as-usual (blue) cases. The decadal time scale is visible by the stepwise increments. We compare these with annual version of our model (smooth solid lines).
Adjustment of all time-dependent parameters to the annual scale is reported in Table 1 (see also Appendix D). We change depreciation from 100% in each decade to annual partial (logarithmic) depreciation with ι = 1.26 and κ = 0.1 to match the output and capital dynamics of the decadal GHKT model with 100% depreciation.
We base our calibration of income-per-capita damages on the detailed empirical estimates in Burke et al. (2015). The blue line in figure 5(d) of this study is the "differentiated response, lung-run effect", i.e. a middle-range estimate, and suggests that for every increase in temperature by 1  C, the global warming damage in terms of lost GDP per capita increases by 12.5% of global economic activity. 11 This is much higher than the damages in the DICE models (e.g. 2.35% of global GDP for each trillion ton of excess 9 Note that in the supplementary material to GHKT Barrage (2014) reports initial values of 699GtC for permanent and 103 GtC for transitory atmospheric carbon and κ1 = 0.5429 and κ2 = 0.1015. 10 The source code containing our solution routines is available upon request. 11 The empirical results of Burke et al. (2015) also suggest that the change in GDP per capita is smallest at an annual temperature of about 13 degrees Celsius; it drops off rapidly if temperature is either lower or higher than that (see their figure 2). The top two panels of Figure 1 show the effects of pricing carbon on global equilibrium temperature and coal use. As described in detail in GHKT, pricing carbon effectively avoids the worst effects of climate change, virtually exclusively through a reduction in coal use as this is the most carbon-intensive fossil fuel. Temperature increases beyond 2°C at the end of this century and 3°C at the end of next century even under carbon pricing. Since oil can be used without cost, its stock of reserves will always be fully depleted (asymptotically) in this model. Policy intervention hardly affects the time profiles of oil use and renewable energy output, but coal use falls significantly albeit it takes a century or so for this to occur.
Capital stock and output (net of damages) are reported in the bottom panels of Figure 1.
Both increase rapidly within the first decades but due to the absence of any growth engines, both converge to their steady state level around 2050. This steady-state level is falling over time due to the continued emissions of carbon and increases in damages.
Given that carbon-based energy inputs are essential in production, this is an unavoidable

Pricing Carbon in our Extended GHKT Model
We use our annualized version of GHKT model as a baseline to compare how extensions regarding long-run growth in population and productivity and damages to productivity and utility change the model's predictions. Table 2 presents key environmental variables for these scenarios. The reported social cost of carbon are updated to 2019's GDP level of $85 trillion in constant 2010 US$ (World Bank, 2020). This increases the baseline carbon tax from $56 to $64 and increases to grow at the rate of (real) GDP. The sensitivity of the optimal carbon tax to differences in public and private discounting, to different climate and damage formulations, and caps of temperature is presented in Table 3.

Population growth and technological progress
The introduction of population growth lowers the population-adjusted discount rate and increases the saving rate and the social cost of carbon as the current generations are more 23 willing to provide for a larger population in the future and to make sacrifices to curb future global warming. With annual population growth of 1% (γ = 1.01), the optimal carbon price more than doubles to $164/tC in 2019 in Table 2 and temperature at the end of the century falls by half a degree to 1.9°C. Peak warming falls from 3.7 to 2.7°C. In contrast, TFP growth of 1% per annum does not affect the initial carbon price. However, over time the economy grows at a faster pace and thus the optimal price of carbon grows at a faster pace too. As a result, energy output increases slightly and temperature by 0.1°C. Bohn and Stuart (2015) consider endogenous population size, calculate the externality from an extra birth on climate change and find it to be large. This requires policy to be less pronatalist and may require a sizeable Pigouvian tax on having children. We abstract from these issues here. (1), (2), (3), and (4) $232 1.8 2.5

Utility damages and damages to productivity growth
To allow for utility damages we follow the calibration of Barrage (2018) and assign 74% of damages at 2.5°C to production and 26% of damages to utility (χ = 1.806  10 -5 and 24  = 7.376  10 -6 but δ = 0). 12 The optimal carbon price falls only slightly to $63/tC, which causes peak temperature to edge up by 0.1°C in Table 2.
In the GHKT formulation damages only affect the current level of TFP but not its growth rate. Dell et al. (2012) find that a temperature increase of 1°C lowers per capita income growth of an economy on a balanced growth path by 1.171%-points in poor and 0.152%points in rich countries (although the latter result is not statistically significant) which requires setting δ = 0.367 in our model. The resulting optimal carbon price increases significantly from $64 to $100/tC. Pricing emissions at this level is still insufficient to be consistent with the ambitions of the Paris Agreement to keep the temperature increase by the end of the century well below 2°C, because temperature increases at the end of the century to 2.2°C and peak temperature is 3.1°C. Table 2 also reports combinations of the effects discussed so far. Most notably the effects of population growth (1) and damages to productivity growth (4) compound to an initial carbon price of $259/tC and temperature of 1.8°C in 2100 and a peak temperature of 2.4°C. Adding the effects of TFP growth (2) to the effects of damages to productivity growth (4) does not change the optimal carbon price: it stays at $100/tC.
Allowing for damages to utility (3) does not alter scenarios (1) or (2) much, except when combined with damages affecting economic growth (4). Here it lessens the lasting effect of productivity damages as these only affect the damage component affecting production.
When utility damages (3) are added to damages to the growth rate (4), the optimal SCC falls from $100/tC to $90/tC. Finally, adding effects (1), (2), (3) and (4) leads to an optimal SCC of $232/tC and a temperature increase of 1.8°C at the end of the century.

Different discount rates for the private and public sector
If the public sector applies a lower discount rate, the policymakers can reproduce the social optimum if they price carbon and simultaneously subsidize saving. If the government discounts the future at 0.1% per year while the private sector maintains the baseline rate of 1.5% per year, the SCC increases to $601/tC in Table 3 and, following proposition 2, the required capital income subsidy is 1.4%. If the government cannot subsidize capital income, the second-best policy defined in proposition 3 is to increase the carbon tax by 9% to $650/tC to compensate the future for the inefficiently low savings rate. Note that the missing capital subsidy also makes saving in the form of fossil fuel less attractive. Oil use is brought forward. Temperature at the end of the century increases by 0.2°C as a result, while peak temperature remains unchanged. If the government uses a discount rate of 1%, the optimal carbon tax is $93/tC, the capital subsidy 0.5%, and the second-best carbon tax $96/tC.

Temperature driven by cumulative emissions and damages of Burke et al. (2015)
We can simplify the climate model of GHKT by assuming that cumulative emissions drive temperature, as in section 5. This is equivalent to assuming that 1.
L  = If carbon remains in the atmosphere permanently, the social cost of emitting increases to $135/tC in Table 3 (see also the discussion of equation (9)).

Cap on temperature or on cumulative emissions and stranded oil and gas reserves
The finding that temperature is driven by cumulative emissions has given rise to temperature targets being expressed in carbon budgets. Using budgets presented in Table   2.2 of the IPCC's Fifth Assessment Report, we can add temperature targets as discussed in section 6 and quantify the 1.5°C and 2°C as a cap on cumulative emissions of 150 GtC and 355 GtC, respectively. This is consistent with our calibration on cumulative emissions for the 2°C target. These constraints on cumulative emissions are based on the baseline GHKT rather than the model of cumulative emissions to make the figures reported below consistent with those displayed in Figures 1 and 2.
The introduction of emission caps increases the optimal carbon tax to $281 and $1,152 for the more stringent target but also augments the carbon tax in that a fraction grows at the interest rate, and therefore each year 1.5% (using ( ) This permits a low temperature peak around 2100 after which temperature falls. Panel (a) in Figure 3 plots the time profiles of the carbon taxes for temperature caps and compares them to the baseline case of no cap. Since in the baseline calibration, all exogenous growth engines are turned off, the optimal carbon tax is fairly flat in the case of no cap. Following equation (9), however, the carbon taxes increase exponentially if the cap is binding (note the log-scale in the plot). Under the budgets used, 50% of the models used by the IPCC remain within the respective temperature target. The IPCC also reports carbon budgets for the 33% and 66% mark, which are captured by the shaded areas in Figure 3.
Coal use is most affected by carbon taxation. As shown in Figure 1, the use of renewable sources and oil and gas only decrease by 0.3% and 6%, while coal use nearly halves from 4 GtC to 2 GtC per annum initially. Given that oil and gas are available without cost (save the intertemporal scarcity rent), all reserves are generally used up eventually, slashing coal use is the primary function of the carbon tax. Similar results hold for carbon taxation under a 2°C cap. The lower panel in Figure 3(a) shows that renewable use and oil and gas extraction barely change, while initial coal use drops to 1 GtC. The more forceful pricing of carbon under the 1.5°C target has a similar effect, again reducing coal to 10% of its business-as-usual level and less than half its use under the 2°C target. While renewable use falls only marginally, pricing emissions at $1,152 per tC also hits oil and gas, cutting emissions from its use in half.

Figure 3: Caps on cumulative emissions and temperature (a) Time paths for 1.5° and 2°C targets (b) Initial values in 2010 across cumulative emissions
The qualitative difference in oil and gas use between the 1.5°C and 2°C caps is illustrated in Figure 2.b, which plots initial carbon taxes and scarcity rents and long-run stranded oil and gas assets as a function of the carbon budget. In the baseline case of no cap, where carbon is only taxed to avoid marginal damages, cumulative emissions amount to roughly 2,000 GtC. As the budget becomes more smaller, climate policy must become more stringent and the carbon tax gradually deviates from the baseline tax of $56/tC (gray dashed). As the budget approaches the level of 300 GtC the initial carbon tax rises rapidly to above $400/tC, after which it continues to increase at a lower rate. The behaviour of the carbon tax is mirrored by the scarcity rent on oil and gas reserves. Since a lower carbon budget implies less carbon-intensive use, the Hotelling rent declines as the carbon budget increases. Below 500 GtC of cumulative emissions, the decline accelerates before reaching zero at 300 GtC. This is the budget of cumulative emissions, which directly 28 equates supply and use of oil and gas reserves. Lower levels of cumulative emissions keep total use below supply and some oil and gas assets will be abandoned underground (see green slope in Figure 2(b)). The (natural resource) wealth effects of temperature caps can be measured in terms of write-off due to depressed prices or in the amount of unburnable fossil fuel. At the introduction of a 2°C cap the value of oil and gas reserves drops from $105bn to $62bn. This is $43bn reduction is 128 times the $334mn drop when carbon prices based on baseline damages are introduced. The cap for 1.5°C makes reserves worthless as half (or 128 GtC) will never be used and the scarcity rent is, therefore, zero.
The lower panel of Figure 2(b) plots energy use across the carbon budget. Oil and gas use is flat as long as the scarcity rent is positive. From equation (7) this implies that the carbon tax is set to offset the decrease in the scarcity rent and only coal use is affected at carbon budgets above 300 GtC, which is consistent with the Herfindahl rule. At lower levels, coal and oil and gas use also drop linearly, although reduction in cumulative use are proportionally larger for oil and gas (due the greater slope) which contradicts the Herfindahl rule and stems from the fact that coal and oil and gas use are regulated by a uniform tax. Carbon taxation also lowers renewable energy use, since the GHKT calibration assumes that all energy types are cooperative factors of production. These reductions are small, reaching at most 10% at very high levels of carbon taxation.

Merits and versatility of the GHKT model
The GHKT model has very bold assumptions but nevertheless has become a reliable workhorse for many applications in climate economics. This literature has shown a multitude of interesting applications of the GHKT model that permit a tractable and convenient analysis. Li et al. (2016), Anderson et al. (2017) and Gerlagh and Liski (2017a) have used the model to allow for climate model uncertainty, Engström and Gars (2016) for abrupt climate change and tipping points, Gerlagh and Liski (2017b) and Iverson and Karp (2020) for hyperbolic discounting, and Hassler and Krusell (2011) for multi-country analysis. 29 We have extended the model to set of relevant policy questions by allowing for population growth and labour-augmenting technical progress in the final goods and energy sectors and for an annual instead of a decadal time scale by having logarithmic depreciation, while still obtaining an exact tractable expression for the optimal SCC. This is also the case if global warning negatively impacts utility or the growth rate of total factor productivity, changes in global warming only affect total factor productivity gradually, and the government uses a lower discount rate than the private sector. Tractable expressions of the optimal SCC can only be obtained if a more realistic model of temperature depending on cumulative emissions is used and damages are a linear function of cumulative emissions. Finally, we have shown that these rules for the optimal SCC can be easily modified to allow for temperature caps.

Extensions of the GHKT model with approximate rules for the optimal SCC
There are some important modifications of the GHKT model that break the exact simple rule for the optimal SCC but nevertheless yield approximate expressions for the optimal SCC that lead to outcomes that are close to the numerical optimum and yield negligible welfare losses (e.g. van den Bijgaart et al., 2016;Rezai and van der Ploeg, 2016).  (2016)), where gt, n and  denote the long-run growth rate of the economy, the population growth rate, and the rate of time impatience , respectively. This leads to a good approximate rule for the optimal SCC. Provided EIS < 1, higher growth and higher intergenerational inequality aversion (i.e. lower EIS) increase the discount factor that must be applied to calculate the optimal SCC, since current generations are then less willing to make sacrifices to lower temperature for future generations. Consequently, the optimal SCC is lower than in the GHKT model with a unit EIS. 30 Second, if parameters or functional forms are changed that only affect the optimal carbon price via GDP, our optimal policy rules still give good outcomes that are not too far from the true optimum. This is so if, for example, the Cobb-Douglas production function for final goods is changed to a CES function or depreciation is not quite 100% per period.
This is also so if the exhaustible energy type (oil) is replaced by an abundant energy type or if this energy type has strictly positive extraction costs. 13 All these changes will affect GDP but will have only a (minor) effect on the rule for the optimal SCC.
Third, if damages to total factor productivity are not proportional to aggregate output but additive, replacing current GDP ( t Y ) by initial GDP ( 0 Y ) and replacing  (2016)). The path of the optimal SCC is now flat rather than rising at the rate of economic growth and the initial level is lower than with multiplicative damages as the growth-corrected discount factor is bigger (Rezai et al., 2020).
Fourth, perturbation methods yield tractable rules for good approximations to the optimal SCC in dynamic stochastic general equilibrium models with uncertain shocks to the economy, temperature response, and damages, allowing for correlated shocks and for the coefficient of relative risk aversion to differ from the inverse of the elasticity of intertemporal substitution (van den Bremer and van der Ploeg, 2021). The optimal SCC then depends on precautionary, insurance, and hedging motives. Note that with the logarithmic utility of the GHKT model all the effects of uncertainty about the rate of economic growth and all hedging effects cancel out, albeit uncertainty about the temperature response and damages does affect the optimal SCC.

Limitations of the GHKT and the extended GHKT model
The model runs, however, into limits and fails to generate tractable closed-form rules for the optimal SCC when reasonable features are considered. For example, the GHKT model does not allow the cost of fossil fuel extraction to increase as reserves are depleted. This would allow for partial exhaustion of reserves, so that policy can affect how fast to deplete 13 One can get a correct tractable expression for the optimal SCC if oil uses only labour input and no capital.

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fossil fuel reserves and how much of the reserves to abandon, introducing considerations of economic obsolescence to the discussion of stranded assets (Rezai and van der Ploeg, 2015). Moreover, even under stringent climate policy, scarcity rents could remain positive as depleting reserves is costly. Furthermore, as Golosov et al. (2014) point out, their model can be extended for learning by doing in production of renewable energy. The carbon price must then be complemented with a renewable energy subsidy, which is set equal to the present discounted value of all present and future benefits of using one unit of renewable energy for short or the social benefit of learning for short (van der Zwaan et al., 2002;Popp, 2004;Rezai and van der Ploeg, 2017). This leads to a spike in renewable energy subsidies and a gradual ramp in carbon prices (cf. Acemoglu et al. (2012) who show this in a context with directed technical change). 14 Unfortunately, the GHKT model also cannot speak to relative price effects as the atmospheric carbon stock (or cumulative emissions) is separable from consumption of final goods in utility. It is infeasible to obtain tractable expressions for the optimal SCC if the utility of non-market environmental goods is non-separable from the utility of consumption of market goods. If that were the case and environmental goods grow more slowly than consumption goods, relative prices increase over time and damages becomes more costly over time (Hoel and Sterner, 2007;Sterner and Persson, 2008;Zhu et al., 2019;Drupp and Hansel, 2021;Bastien-Olvera and Moore, 2021). This leads to higher mitigation rates and a declining term structure for the discount rate.
Finally, a major limitation of the GHKT model is that it is effectively static, i.e. income and intertemporal substitution effects in the consumption-saving decisions cancel out.
Hence, issues of time inconsistency only play a limited role and that the GHKT model is of limited interest for the analysis of second-best policy issues. Similarly, strategic interactions between countries are severed as the pre-commitment and subgame-perfect outcomes coincide (e.g. Hambel and Kraft, 2020). More realistic analysis of strategic issues thus requires a model which allows for more interesting dynamic interactions.

Conclusion
We have shown that the GHKT model still offers a tractable rule for the optimal SCC and the price of carbon emissions when we relax the underlying restrictive assumptions. First, we use logarithmic depreciation (Anderson and Brock, 2021) to allow for a finer time resolution with an annual time scale. Second, we allow global warming to negatively affect utility and the rate of growth of total factor productivity (as estimated in Dell et al.
(2012) and studied in Dietz and Stern (2015)), and also allow more generally for mean reversion in total factor productivity. Third, we follow Von Below (2012), Belfiori (2017Belfiori ( , 2018 and Barrage (2018Barrage ( , 2019 and allow policymakers to be more patient than the private sector. This requires the carbon tax to be complemented with a capital subsidy.
We also consider the second-best optimal carbon tax in case a capital subsidy is infeasible and discuss its time consistency. We also show that the rule for the optimal SCC can easily be adapted to allow for positive long-run growth by introducing population growth and labour-augmenting technical progress.
We have also adopted the GHKT model to make it more realistic by adopting recent atmospheric insights that temperature is a linear function of cumulative emissions. If damages are then a function of cumulative emissions instead of the stock of atmospheric carbon, we get a tractable modified rule for the optimal SCC. We calibrated this to both the detailed econometric damage estimates of Burke et al. (2015) and of the GHKT model. We then use this modified model to derive a tractable expression for the optimal carbon price under a temperature cap. If the cap bites, this requires adding a term to the unconstrained optimal carbon price that rises at a rate equal to the rate of interest. Such a policy leads to stranded oil and gas reserves. We illustrate all our results with numerical simulations which can be done with a simple programme (available upon request).
If policymakers internalize global warming damages to total factor productivity, the optimal SCC will grow at the same rate as GDP. If damages are unrelated to aggregate output, the optimal SCC will be stationary. The initial SCC increases in the degree of patience of policymakers, the effect of global warming on utility, the effect of global warming on total factor productivity and the persistence of this effect, the sensitivity of temperature to emissions, and the rate of population growth. In case temperature is determined by the dynamics of atmospheric carbon as in the GHKT model, it decreases 33 in the degree to which atmospheric carbon decays. However, if policymakers implement a temperature cap, the optimal SCC will grow at a rate equal to the rate of interest which is typically higher. The initial carbon price then increases in the level at which temperature is capped, and the final cost of decarbonizing the economy and decreases in the rate of growth of the carbon price (i.e. the interest rate).
Although we might have come to the limits of this popular and tractable general equilibrium model, we believe that the lasting contribution of the extended GHKT model is to offer analytical insights in key drivers of optimal carbon pricing that cannot be obtained through numerical optimal policy simulations, serve as a useful work horse for future applications, and be invaluable in teaching. We also think that these type of tractable rules for the optimal SCC might serve as good rules of thumb for the optimal trajectory of carbon pricing in more complicated, numerical large-scale integrated assessment models. Such rules have the added advantage that they are easier to communicate and to commit to by policymakers.
where t, t, and ϕt denote the shadow values of capital and fossil fuel reserves and the shadow value of the log of total factor productivity at time t, respectively, and the t t  and p t  denote the shadow disvalues of the transitory and permanent components of atmospheric carbon, respectively. In choosing consumption, the marginal benefit of using output for consumption today has to equal the benefit of transferring output into the future, the shadow price of capital. Under geometric depreciation this term is constant, but here depreciation depends on the level of investment and, therefore, and thus the growth rate of per-capita consumption at time t must equal In case 1, The Euler equation (A2) and (2)

   + − 
The stable manifold is given by , which gives equation (6) in Proposition 1. Using this and (6)   Hence, using (6) and 0 (1 ) , we finally get equation (9) of Proposition 3. The first-order optimality conditions for fossil fuel and renewables give rise to the Kuhn-Tucker conditions stated in (7)    increase of 1°C leads to a statistically significant long-run reduction in the annual income per capita growth rate of 1.171 percentage points. We use this value under the assumption that most long-run growth in the future will result in today's poor countries. To convert from p.c. income into a growth rate of TFP, we multiply by the labour share which equals 1 -α -ν = 0.66. This gives a calibrated value of δ = 0.367. Our framework assumes a mean-reverting process, implying that TFP growth is higher, the further TFP is from its trend (the case 0 <  < 1). This allows for catching-up processes and some forms of adaptation to climate change. The frameworks used in Dietz and Stern (2015) and Diaz and Moore (2015) trade the conventional assumption in the DICE model and in Golosov et al. (2014) of "level" effects (i.e. global warming negatively affects the level of TFP) which are one-off effects for "growth" effects (global warming negatively affect the growth rate of TFP) in which case losses can never be caught up. The meanreverting process used in our paper captures these cases with δ = 0 (level effects) and δ = 1 (growth effects). Given that our calibrated case uses the value δ = 0.367, the effects of climate change on TFP growth are lower (relative to GHKT's level effect specification) than those in other frameworks. This can be seen from Table 2, which indicates that if global warming affects the growth rate rather than the level of TFP the optimal carbon tax increase from $64 to $100/tC while they rise from $44 to $118/tC in Dietz and Stern (2015). In general, the higher the value of , the more persistent the effect of global warming on TFP and thus the higher the optimal carbon tax.