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DEPARTMENT OF ECONOMICS OxCarre Oxford Centre for the Analysis of Resource Rich Economies Manor Road Building, Manor Road, Oxford OX1 3UQ Tel: +44(0)1865 281281 Fax: +44(0)1865 271094 [email protected] www.oxcarre.ox.ac.uk OxCarre Research Paper 123 _ Abandoning Fossil Fuel: How Fast and How Much A third way to climate policy Armon Rezai* Vienna University of Economics & Business & Frederick van der Ploeg OxCarre *OxCarre External Research Associate Direct tel: +44(0) 1865 281281 E-mail: [email protected] ABANDONING FOSSIL FUEL: HOW FAST AND HOW MUCH A third way to climate policy Armon Rezai* and Frederick van der Ploeg**†‡ Abstract Climate change must deal with two market failures: global warming and learning by doing in renewable use. A third way for climate policy is therefore required consisting of an aggressive renewables subsidy in the near term and a gradually rising carbon tax which falls in long run. Such a climate policy ensures that more renewables are used relative to fossil fuel, the carbon-free era is brought forward, more fossil fuel is locked up and global warming is lower. Further, there is an intermediate phase where fossil fuel and renewables are used alongside each other. The climate externality is more severe than the learning-by-doing one. The optimal carbon tax is not a constant proportion of world GDP. Keywords: climate change, integrated assessment, Ramsey growth, carbon tax, renewables subsidy, learning by doing, directed technical change, multiplicative damages, additive damages JEL codes: H21, Q51, Q54 Revised June 2014 * Department of Socioeconomics, Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Vienna, Austria. Also affiliated with IIASA. Email address: [email protected]. ** Department of Economics, Oxford University, Manor Road Building, Manor Road, Oxford OX1 3UQ, U.K. Also affiliated with VU University Amsterdam. Email address: [email protected]. ‡ This version of the paper has benefited from helpful comments from Erik Ansink, Reyer Gerlagh, John Hassler, Per Krusell and seminar participants at Oxford, Tilburg and the Tinbergen Institute and audiences at Annecy, SURED 2014, Ascona, and WCERE 2014, Istanbul. The first author is grateful for support from the OeNB Anniversary Fund grant on Green Growth (no. 14373). The second is grateful for support from the ERC Advanced Grant ‘Political Economy of Green Paradoxes’ (FP7-IDEAS-ERC Grant No. 269788). 1 1. Introduction Climate policy has to deal with two crucial market failures: the failure for markets to price carbon to fully internalize all future damages arising from burning another unit of carbon and the failure of markets to internalize the full benefits of learning by doing in using renewables. To correct for these market failures one must set a carbon tax equal to the social cost of carbon (SCC), which corresponds to the present value of all future marginal global warming damages from burning one extra unit carbon,1 and set a renewable subsidy equal to the social benefit of learning by doing (SBL), i.e., the present value of all future reductions in the cost of renewables from using one unit of renewable energy now. Our objective is to use a fully calibrated integrated assessment framework of Ramsey growth and global warming to gain insights into such a third way to climate policy. Our answer takes account of exhaustible fossil fuel, increasing efficiency of carbon-free alternatives, gradual and abrupt transitions from fossil fuel to renewables, a temporary population boom, and ongoing technical progress and structural change. We show that the third way to climate policy consists, on the one hand, of an aggressive and temporary renewable subsidy, and, on the other hand, a gradually rising carbon tax to price out fossil fuel. This policy mix curbs fossil fuel use and promotes substitution towards renewables, leaves more untapped fossil fuel in the crust of the earth and brings forward the carbon-free era. Initially only fossil fuel is used, then, as a result of endogenous technical progress in renewables, fossil fuel and renewables are used alongside each other, and, finally, the economy only renewables are used. Reductions in climate damages are optimally traded off against welfare losses from less economic activity. Our analysis highlights the following three features. 1 Instead, the optimal price of carbon can be found on an efficient emissions market or can correspond to the shadow price of direct control legislation. 2 First, in contrast to Nordhaus (2008, 2014), Rezai et al. (2012) and Golosov et al. (2014), we analyse not only the optimal carbon tax but also the optimal transition times for introducing renewables alongside fossil fuel and abandoning fossil fuel altogether and the amount of untapped fossil fuel to be left in the crust of the earth.2 Our model is annual instead of decadal as earlier integrated assessment models or semi-decadal as in Nordhaus (2014), because this allows us to pinpoint much more accurately the transition times towards the carbon-free era and such long time intervals can introduce significant numerical biases (Cai et. al, 2012). Building on IEA data, we furthermore suppose that extraction costs rise as remaining reserves falls and less accessible fields are explored. This allows us for the first time to quantitatively assess the important role of untapped fossil fuel and stranded assets in the fight against global warming. We thus offer an estimate of the maximum cumulative carbon emissions (the ‘carbon budget’). In our analysis fossil fuel is exhaustible and extraction costs are stock dependent, hence the price of fossil fuel contains two forward-looking components: the scarcity rent (the present discounted value of all future increases in extraction costs resulting from an extracting an extra unit of fossil fuel) and the SCC. The optimal transition times consequently depend on expectations about future developments such as learning by doing in using renewable energy. In contrast, most integrated assessment models assume that fossil fuel is abundant so its demand does not depend on expectations about the price of future renewables and the optimal distribution of finite resources across time and thus the transition simply occurs when the price of fossil fuel inclusive of the carbon tax reaches the price of the renewable energy source. Second, we suppose that the cost of renewable energy falls as cumulative use increases (Arrow, 1962). Although technological progress, diffusion, and adoption are studied in the economics of climate change, these studies typically 2 We derive our results based on a calibrated and richer version of the theoretical growth and climate model analysed in van der Ploeg and Withagen (2014). 3 do not allow for exhaustible fossil fuel and stock-dependent fossil fuel extraction costs, and do not offer a fully calibrated integrated assessment model.3 If cost reductions from learning-by-doing externalities are not fully internalized, a subsidy is required alongside the carbon tax. We compare “laissez faire” with a global first-best scenario where both the climate and learning-by-doing externalities are fully internalized. Since international climate negotiations have failed miserably and there is room for national renewable energy policies, we also consider a scenario where only learning-bydoing externalities are internalized and the climate externality is not. This leads to faster extraction of fossil fuel and acceleration of global warming, since fossil fuel owners fear that their resources will be worth less in the future. This phenomenon has been coined the Green Paradox by Sinn (2008). Third, we show that with CES production and utility the optimal carbon tax first rises and then declines with world GDP. The result of Golosov et al. (2014) that the optimal carbon tax is proportional to world GDP is thus not robust;4 it depends on logarithmic utility, Cobb-Douglas production, full depreciation of capital in each period, no capital need in fossil fuel extraction, 3 Tiang and Popp (2014) provide recent econometric evidence for significant learning by doing effects in renewable energy generation, suggesting that each new wind power project in China (with 60 GW capacity) leads to a unit cost reduction of 0.25%. De Zwaan et al. (2002) are the first to address optimal climate policy in the face of learning by doing in renewables in integrated assessment models. Popp (2004) studies endogenous technical progress in a fully calibrated IAM; Popp et al. (2010) review the implications of technical innovation and diffusion for the environment; and Goulder and Mathai (2000) study the implications in a stylized model of climate change. Manne and Richels (2004) argue that endogenous technical progress does not alter climate policy recommendations, specifically the transition timing. Jouvet and Schumacher (2012) find the opposite and so does Popp (2004) who studies optimal policies in an adapted version of DICE of Nordhaus (2008). Hübler et al. (2012) present a multi-region IAM with endogenous growth and study region-specific welfare effects. Fisher and Newell (2008) study optimal interaction of policy instruments in a calibrated model of heterogeneous energy producers limited to the US energy sector. None of these studies considers the “laissez faire” decentralized economy. Fossil fuel stocks are assumed abundant and their extraction costless. Thus these studies cannot address how much fossil fuel to lock up in the crust of the earth and the time of phasing in renewables does not depend on expectations about future policies and energy prices. These are crucial features of our analysis. Tsur and Zemel (2005) analyze growth and R&D in a model with scarce resources, but do not offer a calibration and estimates of optimal climate policy. Table F.1 in appendix F provides a detailed comparison of prominent IAMs. 4 This formula is also used in Hassler and Krusell (2012) and Gerlagh and Liski (2013). 4 and multiplicative, exponential climate damages in production. The proportional carbon tax performs poorly if policy needs to address multiple market failures and intergenerational inequality aversion differs from unity. The proportional tax suggested by Golosov et al. (2014) rise too slowly relative to the first best and is, therefore, insufficient in pricing fossil fuels out of the market. Like the renewable subsidy in isolation, it causes more fossil fuel to be extracted in the near term and leads to a Green Paradox effect. Section 2 discusses a simple two-stock model of carbon accumulation in the atmosphere and global mean temperature due to Golosov et al. (2014) and discusses our benchmark specification of climate damages which are bigger at higher temperatures than Nordhaus (2008, 2014) following recent suggestions by Stern (2013) and Dietz and Stern (2014). We do not adopt the approximation to damages used in Golosov et al. (2014), since this leads to unrealistic low damages at higher temperatures. Section 3 formulates our general equilibrium model of climate change and Ramsey growth. Section 4 uses our calibrated integrated assessment model to highlight the different outcomes for the optimal carbon tax, the renewables subsidy, untapped fossil fuel, the time it takes to phase in renewable energy and to reach the carbon-free era, and welfare under the optimal, the constrained optimal and the “laissez faire” scenarios. It also discusses the performance of the simple formula for the carbon tax put forward by Golosov et al. (2014). Section 5 discusses the sensitivity of climate policy. Section 6 concludes. 2. The carbon cycle, temperature and global warming damages We use an annual version of the decadal model of the carbon cycle due to Golosov et al. (2014) and based on Archer (2005) and Archer et al. (2009): (1) EtP1 EtP L Ft , L 0.2, (2) EtT1 (1 )EtT 0 (1 L )Ft , 0.002304, 0 0.393, E0T 699 GtC, E0P 103 GtC, 5 where EtP is the part of the stock of carbon (GtC) that stays thousands of years in the atmosphere, EtT the remaining part of the stock of atmospheric carbon (GtC) that decays at rate , and Ft the rate of fossil fuel use (GtC/decade).5 About 20% of carbon emissions stay up ‘forever’ and the remainder has a mean life of about 300 years, so 1 (1 0.0228)1/10 0.002304, where 0.0228 is the parameter proposed for the decadal model in Golosov et al. (2014). The parameter 0 0.393, is calibrated so that about half the carbon impulse is removed after 30 years. The equilibrium climate sensitivity (ECS) is the rise in global mean temperature following a doubling of the total carbon stock in the atmosphere, Et EtP EtT . A typical estimate for the ECS is 3 (IPCC, 2007). To facilitate comparison with Golosov et al. (2014), we ignore lags between the stock of atmospheric carbon and global mean temperature:6 (3) Tt ln Et / 596.4 / ln(2), 3, Et EtP EtT , where 596.4 GtC (280 ppmv CO2) is the IPCC figure for the pre-industrial stock of carbon.7 The evolution of fossil fuel reserves follows from the depletion equation: (4) St 1 St Ft , S0 4000 GtC, where St denotes the stock of reserves at the start of period t. Nordhaus (2008) has combined detailed micro estimates of the costs of global warming to get aggregate macro costs of global warming of 1.7% of world 5 The three reservoirs used by Nordhaus (2008) and the linear representation used in Gerlagh and Liski (2013) highlight the exchange of carbon with the deep oceans arising from the acidification of the oceans limiting the capacity to absorb carbon. Our carbon cycle ignores time-varying coefficients as in Bolin and Erikkson (1958). It also abstracts from diffusive rather than advective transfers of heat to the oceans (Allen et al., 2009) which leads to longer and greater warming (Bronselaer et al., 2013; Baldwin, 2014). Finally, it does not consider positive feedback effects which are more severe at higher temperatures. 6 This lag lowers the SCC (see section 5.3 and appendix C). 7 To convert to GtCO2, we multiply by 44/12. 6 GDP when global warming is 2.5o C. This figure is used to calibrate the fraction of production output that is left after damages from global warming: Z (Tt ) (5) 1 1 1Tt 2 3Tt 4 , so Z ( Et ) Z ln Et / 596.4 / ln(2) , with ζ1 = 0.00284, ζ2 = 2, and ζ3= ζ4 = 0 and plotted as the solid line in fig. 1.8 Golosov et al. (2014) approximate this with the exponential net output function Z ( Et ) exp ( Et 581) with 2.379 105 , which gives rise to the dotted line in fig. 1. The fit to the Nordhaus (2008) damages is good for small degrees of global warming from the present, but at higher degrees the fit is much worse and too optimistic about the size of global warming damages.9 Net output after daamge (% of gross) Figure 1: What is left of output after damages from global warming 100% 90% 80% 70% 60% 50% 0 1 2 3 4 5 6 Temperature change above pre-industrial (°C) DICE-07 DICE-2013R Golosov et al. (2014) Nordhaus-Weitzman Weitzman (2010) and Dietz and Stern (2014) argue that damages rise more rapidly at higher levels of temperature than suggested by (5), but empirical studies on the costs of global warming at higher temperatures are not available. With the heroic assumptions that damages are 50% of world GDP at 6o C and 8 The damage function resulting from the DICE-07 model is almost distinguishable (up to temperatures of 7 Celsius) from that of the DICE-2013R model (see http://www.econ.yale.edu/~nordhaus/homepage/Web-DICE-2013-April.htm ) 9 The fit deviates by less than 1%-point only between 2°C and 4°C warming. Z(Et) = 1.0435 0.0001 Et is a much better fit, also at higher temperature and carbon stocks (R 2 = 0.9994). 7 99% at 12.5o C, Ackerman and Stanton (2012) recalibrate (5) with ζ1 = 0.00245, ζ2 = 2, ζ3 = 5.021 x 10-6, and ζ4 = 6.76. The extra term in the denominator captures potentially catastrophic losses at high temperatures.10 The dashed line in fig. 1 plots this recalibrated net output function. Although the calibration of Nordhaus (2008) can be empirically justified for moderate degrees of global warming, the revised calibration is more appropriate for higher temperatures and matches it very closely with deviations of less than 0.5%-point up to 3°C of warming. 3. An integrated assessment model of Ramsey growth and climate change The social planner's objective is to maximize utilitarian social welfare (Ct / Lt )11/ 1 (1 ) LtU t (Ct / Lt ) (1 ) Lt . 11 / t 0 t 0 (6) t t Here Lt is the size of the exogenous world population, Ct aggregate consumption, U the instantaneous CES utility function, > 0 the rate of pure time preference and > 1 the elasticity of intertemporal substitution. The ethics of climate policy depend on how much weight is given to future generations and on how small intergenerational inequality aversion is or how easy it is to substitute current for future consumption per head. The most ambitious climate policies result if society has a low rate of time preference and little intergenerational inequality aversion (low , high ). Output at time t is produced with three inputs: manmade capital Kt, labour, Lt, and energy. Types of energy are renewables Rt, (e.g., solar or wind energy), and fossil fuels (oil, natural gas and coal), Ft. The production function H(.) has constant returns to scale, is concave and satisfies the Inada conditions. Renewables are subject to learning, so their unit production cost b(Bt) falls with 10 We capture catastrophic losses at high levels of atmospheric carbon, but abstract from positive feedback and uncertain climate catastrophes that occur once temperature exceeds certain thresholds (e.g., Lemoine and Traeger, 2014; van der Ploeg and de Zeeuw, 2013). 8 cumulated past production Bt and thus b < 0. Fossil fuel extraction cost is G(St )Ft , with St remaining reserves. Extraction is harder as less accessible fields have to be explored, so G ' 0. What is left of production after covering the cost of resource use is allocated to consumption Ct , investments Kt 1 Kt , and depreciation Kt with a constant depreciation rate : (7) Kt 1 (1 ) Kt Z ( Et ) H ( Kt , Lt , Ft Rt ) G(St ) Ft b( Bt ) Rt Ct , where damages follow from (5). The initial capital stock K0 is given. The development of atmospheric carbon follows from (1) and (2), that of fossil fuel stock reserves from (4), and that of knowledge for renewable use from (8) Bt 1 Bt Rt , B0 0. Current technological options favour fossil energy; complete decarbonisation requires substantial reductions in the cost of renewables versus that of fossil fuel. Apart from carbon taxes, technological progress is an important factor in determining the optimal combination of fossil and renewable energy sources (Acemoglu et al., 2012; Mattauch et al., 2012). We thus capture learning and lock-in effects by making the cost of renewables a decreasing function of past cumulated renewable energy production, b ' 0 with Bt Rs . 11 s 0 t Proposition 1: The social optimum maximizes (6) and satisfies (1)-(8), the Euler equation for consumption growth (9) Ct 1 / Lt 1 1 rt 1 , rt 1 Z t 1 H Kt 1 , Ct / Lt 1 and the efficiency conditions for energy use (10a) Z ( Et ) H Ft Rt ( Kt , Lt , Ft Rt ) G( St ) tS tE , Ft 0, c.s., 11 We prefer learning-by-doing over other specifications of endogenous technical change, such as investment in R&D in Bovenberg and Smulders (1996) and Acemoglu et al. (2012), due to the availability of empirically validated learning curves. 9 (10b) Z ( Et ) H Ft Rt ( Kt , Lt , Ft Rt ) b( Bt ) tB , Rt 0, c.s. , where the scarcity rent , the SCC and the social benefit of learning by doing in final goods units (the SBL) are, respectively, given by (11) tS G '(St 1s )Ft 1s t s , s 0 (12) tB b '(Bt 1s )Rt 1s t s and s 0 (13) tE s0 L 0 (1 L )(1 )s t s Z '( Et 1s ) H ( Kt 1s , Lt 1s , Ft 1s Rt 1s ), with the compound discount factors given by t s (1 rt 1s ' )1, s 0. s '0 s Proof: see appendix B. The Euler equation (9) indicates that the growth in consumption per capita rises with the social return on capital (rt+1) and falls with the rate of time preference, especially if intergenerational inequality aversion (1/) is small. Equation (10a) implies that, if fossil fuel is used, its marginal product should equal total marginal cost, which is the sum of current extraction cost G(St) the scarcity rent tS and the SCC tE . If fossil fuel is not used, its marginal product is below marginal cost. Equation (10b) states that, if the renewable is used, its marginal product must equal its current cost b(Bt) minus the SBL tB . Equation (11) states that the scarcity rent of keeping an extra unit of fossil fuel unexploited is the present discounted value of all future reductions in fossil fuel extraction costs. It follows from the Hotelling rule which states that the return on extracting an extra unit of fossil, selling it and getting a return on it, the rate of interest rt 1tS minus the increase in future extraction cost G '(St 1 )Ft 1 , must equal the expected capital gain from keeping an extra unit of fossil fuel in the earth tS1 tS . Equation (12) indicates that the SBL equals the present discounted value of all future learning-by-doing reductions in the cost of renewable energy. 10 Equation (13) states that the SCC is given by the present discounted value of all future marginal global warming damages from burning an additional unit of carbon, taking due account of that part stays in the atmosphere for ever and the rest gradually decays at a rate corresponding to roughly 1/300 per year. Proposition 2: If the utility function is logarithmic, the production function is Cobb-Douglas, global warming damages are Z ( Et ) exp ( Et 581) , depreciation of physical capital is 100% every period and energy production does not require capital input, the SCC (13) per Giga ton of carbon becomes (13) 1 1 tE , Golosov c.s. L 0 (1 L ) Z ( Et ) H ( K t , Lt , Ft Rt ). Proof: see Golosov et al. (2014). This result that the optimal SCC is proportional to world GDP requires bold assumptions which are much less appropriate in an annual than in a decadal model. The factor of proportionality is independent of intergenerational inequality aversion and the factor shares in production; it increases if the social rate of discount is smaller, the permanent fraction of the atmospheric stock of carbon L is larger, and the lifetime of the transient component of the atmospheric stock of carbon 1/ is larger. Decentralized market outcome In a decentralized market economy firms choose inputs to maximize profits, Z (Tt ) H ( Kt , Lt , Ft Rt ) wt Lt (rt 1 ) Kt ( pt t )Ft qt t Rt , under perfect competition taking the wage rate wt, the market interest rate rt+1, the market price for fossil fuel pt, the specific tax t on carbon emissions, the market price for renewable energy qt, the specific subsidy t on the use of renewable energy, and temperature Tt as given. Capital accumulation follows from Kt 1 (1 ) Kt I t , where It is the investment rate at time t. Fossil fuel owners also operate under perfect competition and maximize the present value of their 11 profits, p F G(S ) F t t t t t t 0 with t (1 r1s )1, t 0, subject to the s 0 t depletion equation (4), taking the market price of fossil fuel pt as given and internalizing the effect of depletion on future extraction costs. Producers of renewable energy also operate under perfect competition and maximize the present value of their profits, q b( B ) R , taking the market price of t t t t t 0 renewable energy qt and the stock of accumulated knowledge about using renewable energy Bt as given. Households maximize utility (6) subject to their budget constraint AtH1 (1 rt 1 ) AtH wt Lt Tt Ct , where AtH denotes household assets and Tt lump-sum transfers from the government at time t. The government budget constraint is t Ft t Bt Tt , where we abstract from government consumption and investment. Without loss of generality, the government balances its books and hands back revenue from carbon taxes minus cost of renewable subsidies as lump-sum transfers. Asset market equilibrium requires that AtH Kt and the final goods market equilibrium condition is Z (Tt ) H ( Kt , Lt , Ft Rt ) Ct It G(St ) Ft b( Bt ) Rt , so the market must produce enough final goods to satisfy demand for consumption and investment goods and cover fossil fuel extraction costs and production costs of renewables. Proposition 3: The decentralized market outcome satisfies (1)-(8), the Euler equation (9), the energy producers’ optimality conditions (10a) pt G(St ) tS , Ft 0, c.s., (10b) qt b( Bt ), Rt 0, c.s. , where the scarcity rent tS is (11), and the final good firms’ optimality conditions wt Z (Tt ) H Lt , rt 1 Z (Tt ) H Kt , pt t qt t Z (Tt ) H Ft Rt . 12 Proof: Follows directly from the optimality conditions for firms, fossil fuel producers and households. Proposition 4: The social optimum is replicated in the decentralized market economy if t tE and t tB , t 0, where these follow from (12) and (13). Proof: Comparing conditions of proposition 2 with those of proposition 3. The first best is thus sustained in the market economy if the specific carbon tax is set to the SCC, the specific subsidy on renewable subsidy is set to the SBL, and net revenue is handed back in lump-sum fashion to households. Definition 1: A decentralized market economy has the following outcomes. I. First best or social optimum with t tE and t tB , t 0. II. Constrained first best with t 0 and t tB , t 0. III. Constrained first best with t tE and t 0, t 0. IV. “Laissez faire” or business as usual with t t 0, t 0. Policy scenario I leads to the social optimum and sets the specific carbon tax t to the SCC (13) and the renewable subsidy t to the SBL (12). Scenario II internalizes the benefits of learning to use the renewable but not the SCC, so that the optimal renewable subsidy is given by t tB and the carbon tax t is set to zero (i.e., (13) is irrelevant). No international climate agreement can be reached but national governments move ahead using subsidies (e.g. feed-in tariffs) to stimulate cost reduction in the production of renewable energy. Scenario III internalizes the SCC but not learning to use the renewable, so that t tE and the optimal renewable subsidy is zero (i.e., (12) is irrelevant). Scenario IV corresponds to “laissez faire” and corrects neither for climate nor for learning by doing, so t = t = 0. Given that initially renewable energy sources are not competitive, the market economy starts with an initial phase of only fossil fuel use where fossil fuel demand follows from setting its marginal product, Z (Tt ) H Ft , to the sum of 13 extraction cost, scarcity rent and carbon tax, G ( St ) tS t . After some time, the relative cost of renewable energy has fallen (due to the rising carbon tax and technical progress) and extraction costs have risen sufficiently (as less accessible fields have to be extracted) so that it becomes attractive to phase in clean energy. During this intermediate phase of simultaneous use, demand follows from Z (Tt ) H Ft Rt G ( St ) tS t b( Bt ) t . After some more time, fossil fuel is phased out and the final carbon-free phase starts. Since fossil fuel extraction costs become infinitely large as reserves are depleted, reserves will not be fully exhausted and thus fossil fuel will be left untapped in the crust of the earth at the end of the intermediate phase. Since fossil fuel and renewable energy are perfect substitutes, simultaneous use is infeasible without learningby-doing or without renewable subsidy or carbon tax (except possibly for a single period of time). Learning by doing introduces convexity in the production cost of the renewable so an intermediate phase with simultaneous use emerges. From that moment on the in-situ stock of fossil fuel remains unchanged, but the carbon in the atmosphere gradually decays leaving ultimately only the permanent component of the carbon stock. During the final phase Z (Tt ) H Rt b( Bt ) t , which shows that renewable use increases in capital, the stock of renewable knowledge and the renewable subsidy but decreases in global mean temperature. We are interested in how the optimal timing of transitions from introducing the renewable alongside fossil fuel and from phasing out fossil fuel altogether differs across scenarios I-IV; for example, we wish to know by how much carbon taxes and renewable subsidies bring forward the carbon-free era. These transition times follow from the conditions that energy prices at these transition times should not jump. The stock of untapped fossil fuel at the end of the intermediate phase follows from the condition of indifference between using fossil fuel and renewable energy and a vanishing scarcity rent: 14 (14) G(St ) t b( Bt ) t for 0 t tCF , G(St ) t b( Bt ) t , St StCF , t tCF , where tCF is the time at which the economy for the first time relies on using only renewable energy. From (14) we see that the stock of untapped fossil fuel clearly increases in the renewable subsidy and carbon tax. 4. Policy simulation and optimization In our simulations time runs from 2010 till 2600 and is measured in years. The functional forms and calibration of the carbon cycle, temperature module and global warming damages have been discussed in section 2. The functional forms and benchmark parameter values for the economic part of our model put forward in section 3 are discussed in appendix D. Our benchmark parameters assume low extraction costs and an initially high cost for renewable energy generation which biases our outcomes toward fossil fuel use. The reported simulations use the CES production function (with elasticity of substitution between energy and the capital-labour aggregate equal to = 0.5). Apart from I-IV table 1 gives two further scenarios in which the carbon tax (13) is implemented by either (V) choosing the renewable subsidy optimally or (VI) ignoring learning by doing in renewable use. The results for these scenarios are reported in fig. 2 and tables 2 and 3. Table 1: Scenarios Carbon tax, τt 0 Proportional to GDP (13) (I) first-best (II) only subsidy (V) proportional carbon tax and optimal renewable subsidy (III) only carbon tax (IV) “laissez faire“ (VI) only proportional carbon tax Renewable subsidy, t from (13) from (12) 0 15 Figure 2: Benchmark policy simulations Consumption, Ct Output after Damage, Yt 300 $trillions / yr $trillions / yr 200 150 100 50 250 200 150 100 50 0 0 2010 2060 2110 2160 2210 2260 2310 2010 2060 2110 2160 2210 2260 2310 Mean Global Temperature, Tt 6 °C (above pre-industrial) $trillions (2010) Capital Stock, Kt 600 400 200 0 5 4 3 2 1 0 2010 2060 2110 2160 2210 2260 2310 2010 2060 2110 2160 2210 2260 2310 Renewable Energy Use, Rt 25 40 20 30 GtC / yr GtC / yr Fossil Fuel Use, Ft 15 10 20 10 5 0 0 2010 2060 2110 2160 2210 2010 800 500 600 400 400 200 2110 2160 2210 Renewable Energy Subsidy, νt $ / tC $ / tC Social Cost of Carbon, τt 2060 300 200 100 0 0 2010 2060 2110 2160 2210 2260 2310 2010 2060 2110 2160 2210 2260 2310 Key: laissez faire ( ), carbon tax only ( ), proportional carbon tax only ( ), renewable subsidy only ( ), first best ( ), prop. tax & optimal subsidy ( ) 16 Figure 2: Benchmark policy simulations (continued) Hotelling Rent, ϴst Cumulative Emissions 3000 400 2500 300 $ / tC GtC 2000 1500 1000 200 100 500 0 0 2010 2060 2110 2160 2210 2010 2060 2110 2160 2210 2260 2310 Key: laissez faire ( ), carbon tax only ( ), proportional carbon tax only ( ), renewable subsidy only ( ), first best ( ), prop. tax & optimal subsidy ( ) 4.1. How to quickly de-carbonize and leave more fossil fuel untapped Under the first-best scenario I (blue, solid) consumption, GDP and the capital stock are monotonically increasing. The transition to renewable energy takes place smoothly as soon as 2035 and fossil energy is phased out completely by 2040. Over this period 320 GtC are burnt, so most of the 4000 GtC of fossil fuel reserves are abandoned. This leads to a maximum increase in temperature of 2.1°C or a maximum atmospheric carbon stock of 970 GtC (from (3)) which is close to the maximum of a trillion tons of carbon used in Allen et al. (2009). This rapid and unambiguous first-best transformation towards a carbon-free economy is achieved through the implementation of a carbon tax and a renewable subsidy policy. Both follow an inverted U-shaped time profile. The global carbon tax is set to the SCC which starts at 109 $/tC or 30 $/tCO2 and reaches a maximum of 409 $/tC or 112 $/tCO2 in 2240, long after the transition to renewable energy has occurred. The renewable subsidy starts at 175 $/tC or 47$/tCO2 and quickly rises to 387 $/tC or 105 $/tCO2 in 2038 and rapidly falls to zero as all learning has occurred by the end of this century. The optimal policy mix, therefore, combines a quick and aggressive subsidy to phase in renewable energy quite early and a carbon tax which gradually rises (and falls) to price fossil energy out of the market once renewable energy sources are competitive. 17 Table 2: Transition times and carbon budget I II III IV Social optimum Carbon tax only Renewable subsidy only Laissez faire Only fossil fuel Simultaneous use Renewable Only Carbon used 2010-2035 2035-2040 2040 – 320 GtC 2010-2080 x 2080 – 770 GtC 2010-2075 2075-2085 2085 – 1080 GtC 2010-2175 x 2175 – 2500 GtC In the “laissez faire” scenario IV (orange, dashed) both externalities remain uncorrected. As a result the economy uses much more fossil fuel, 2500 GtC in total, so much less fossil fuel is left in the crust of the earth. Global mean temperature increases by a maximum of 5.1°C matching recent IPCC and IEA estimates for business as usual. The transition to renewable energy occurs much later, in 2175, and abruptly. The reason is that climate benefits of renewable energy and learning by doing are not taken into account; hence the potential cost reductions are not recognized and take longer to materialize. Table 3 indicates that these inefficiencies (see section 5.2 for more detail) cause a substantial welfare loss relative to the first best of almost 6 times today’s world GDP. 12 Table 3: Social costs of carbon, renewable subsidies, and welfare losses I II III IV Social optimum Carbon tax only Renewable subsidy only Laissez faire Welfare Loss (% of GDP) 0% -48% -95% -598% Maximum carbon tax τ ($/tC) 410 $/GtC 690 $ / GtC Maximum renewable subsidy ($/tC) 390 $/GtC 460 $/GtC max T (°C) 2.1 °C 3.0 °C 3.5 °C 5.1 °C Failing to reach an international climate agreement on carbon taxation and implementing only a renewable subsidy to encourage learning by doing up to a rather higher level of 460 $/GtC or 125 $/tCO2 is insufficient to bring forward the transition to renewable energy and to avoid severe climate change. This 12 Stern (2007) expresses cost of inaction in annuity terms; Nordhaus (2008) in terms of today’s consumption. 18 “renewable subsidy only” case III performs better than “laissez faire” in terms of welfare and climate outcomes but temperature rises by as much as 3.5°C above pre-industrial levels and 1080 GtC are used in total (a bit less than half of that under “laissez faire” case but still 3 to 4 times the optimal carbon budget). Subsidizing renewable energy induces simultaneous use of renewable along with fossil energy sources as late as 2075 with a complete transition by 2085; this indicates the power of the carbon tax in quickening the uptake of renewables. Not internalizing the climate externality, however, still lowers welfare by 95% compared to the first best. There are Green Paradox effects (Sinn, 2008) as fossil fuel use is, compared with under “laissez faire”, increased in anticipation of fossil fuel being made less competitive relative to the renewable energy.13 Fossil fuel deposits are depleted more quickly to avoid them becoming obsolete. Establishing carbon markets, taxing emissions, or direct control but ignoring the learning externality (see the orange, solid lines) is politically perhaps less likely, but it is more effective in avoiding global warming and increasing welfare than a subsidy for renewable energy. In scenario III the carbon tax rises to a much higher level of 690 $/tC or 188 $/tCO2 than the first-best carbon tax. A carbon tax ends fossil fuel use not until 2080 but does this abruptly as the learning externality is still not recognized. This policy limits global warming to 3.0°C and carbon use to 770 GtC. The inefficiency of the learning externality seems to matter less in that not internalizing it only lowers welfare by 48%. The comparison of welfare indicators across partial policy scenarios gives further credence to the assessment of Stern (2007) that climate change is the biggest externality our planet faces and that policy makers should prioritize climate negotiations over renewable policy. If such negotiations fail due to insurmountable free-riding problems with international treaties, (national) 13 Under Leontief production technology, there are no Green Paradox effects as the energy demand is a fixed proportion of output. 19 renewable subsidies are yet a relatively efficient way to avert the most severe aspects of global warming. 4.2. Time paths for the market price of fossil fuel and the renewable Market prices for both types of fossil and renewable energy are depicted in fig. 3. Regular lines depict prices of fuels in use, faint lines prices of fuels not in use. The price of fossil energy consists of the sum of marginal extraction cost and the Hotelling rent plus any carbon tax (see equation (10a)). The market price of renewable energy is set to its production cost minus any learning subsidy (see equation (10b)). Initially prices are rising in all scenarios and only on these rising sections are fossil fuels used. A renewable subsidy enables a period of simultaneous use (indicated by boxed sections) and smoothes the transition to the carbon-free era. Figure 3: The market price of fossil and renewable energy ($/tC) 1000 1000 800 800 $ / tC 1200 $ / tC 1200 600 600 400 400 200 200 0 2010 2060 2110 2160 2210 2260 2310 Key left: laissez faire ( ), carbon tax only ( right: renewable subsidy only ( 0 2010 2060 2110 2160 2210 2260 2310 ) ), first best ( ). Boxes indicate simultaneous use. First, consider the “laissez faire” scenario IV (orange, dashed, left panel) where the carbon tax and the subsidy are set to zero. The market cost of renewable energy is above the market price of fossil energy and constant in the absence of any production and/or subsidy. Fossil fuel is in use initially and its price rises due to increasing extraction costs and the increasing Hotelling rents. As extraction costs approach the back-stop price, the fossil fuel era draws to an end and the Hotelling rent falls. This fall in the Hotelling rent mitigates rising 20 extraction costs and prolongs the era of fossil fuel use. In its final period, the scarcity value reaches zero and extraction costs equal renewable unit costs. After this switch point, the cost of fossil energy, consisting now only of the extraction cost component, remains constant (coloured in faint orange as not in use). The cost of producing renewable energy falls quickly and approaches its lower floor due to learning by doing. This scenario is clearly sub-optimal, since renewable energy is too expensive for much too long relative to fossil fuel. Next, consider the case where only the climate externality is internalized using a carbon tax, scenario II (orange, solid, left panel). Adding the SCC to the price of fossil energy increases its initial price and its growth rate significantly. It surpasses the market price of renewable energy much earlier but the transition to the carbon-free era is still abrupt with a price spike in 2080 as the learningby-doing externality is still not recognized. If instead of the carbon tax, a subsidy for renewable energy is introduced, scenario III (dark blue, dashed, right panel), the market price of energy falls below its “laissez faire” level since fossil fuel owners anticipate that their resources will be worth less in the future. Lower market prices stimulate higher fossil fuel use (of up to 20%), faster extraction of fossil fuel, and acceleration of global warming (the Green Paradox). The subsidy lowers the market price of renewable energy to allow for simultaneous use (indicated by boxed sections). Once renewable energy is brought into use, its price rises with that of fossil energy. The market price of renewable energy starts declining only as the transition to carbon-free energy is complete. To implement the first best, scenario I (dark blue, solid, right panel), a carbon tax needs to be added to the renewable subsidy.14 The carbon tax increases the 14 If the SCC is taken account of, the price of fossil energy continues to rise even as no fossil energy is produced (see the faint solid orange and dark blue lines). The SCC rises initially, since decay is limited and consumption is increasing. This yields a smaller marginal utility of consumption and thus a higher SCC (see (13)). However, after some point of time decay of atmospheric carbon dominates the decrease in marginal utility of consumption and the SCC – and with it the market price of fossil energy – start to fall. If the fall is sufficiently large, fossil 21 price if fossil energy beyond its “laissez faire” level. This lowers the required subsidy for renewable energy, brings forward the switch points to simultaneous use and to full renewable energy use, and precludes the Green Paradox effect. 4.3. “Laissez faire” overinvests in dirty and underinvests in clean capital How does “laissez faire”, scenario IV, differ from the first best? Initially, output, consumption and capital accumulation take place at very similar levels. However, the impacts of the climate and learning externalities are large enough to drastically change accumulation paths as temperatures rise. This is also reflected in the substantial welfare loss of about 6 times initial GDP. “Laissez faire” leads to inefficient allocation of resources, since the economy overvalues the returns to conventional capital accumulation and undervalues investments in green energy sources. Failure to cooperate induces excessive fossil fuel extraction and capital accumulation leading to high global warming damages. This inefficient use of resources lowers welfare because it keeps consumption low in early periods of the program to allow for capital accumulation and consumption low in future periods due to high global warming damages. Damages under “laissez faire” are large enough to lower factor returns sufficiently to induce decumulation of capital and a fall in consumption. From 2140-2190 the capital stock falls by 25% from a peak of $410 trillion to a trough $310 trillion, consumption drops by 17% from a peak of $107 trillion to a trough of $88 trillion. This climate crisis is ended as extraction costs rise above the cost of renewable energy. At this point all of energy use is sourced from renewables and the climate and economy recover from previous excessive use of fossil fuels. The inefficiency of this scenario is also reflected in the high share of GDP expended on energy which rises from around 7% in 2010 to 10% in 2110, compared to 6.5% and 6%, respectively, under first best. Once the economy switches to the renewable and stocks of atmospheric carbon recede, energy becomes competitive again. As fig. 3 indicates, the time horizon that we consider is too short and the learning-based reduction in renewable costs too large to make such re-switching optimal. A carbon tax proportional to GDP or consumption does not allow for re-switching. 22 the return to capital and the interest rate increase. This leads to a resumption of global growth. As only a fraction of carbon dissipates, welfare remains below the social optimum even in steady state. Close inspection demonstrates that consumption in the social optimum is the lowest for some initial periods. This implies that internalization of the climate externality has to be phased in more slowly which is the case with higher intergenerational inequality aversion (lower ).15 4.4. The optimal carbon tax versus aggregate consumption or world GDP To examine whether the linear formula for the optimal carbon tax (13) put forward by Golosov et al. (2014) holds up in our more general and detailed integrated assessment model of Ramsey growth and climate change, fig. 4 plots the ratio of the optimal carbon tax to both world GDP and aggregate consumption. We immediately see that the optimal carbon tax (dark blue line) is not well described by a constant proportion of world GDP or aggregate consumption. The general pattern is that during the initial phases of fossil fuel use the social cost of carbon rises as a proportion of world GDP as more carbon emissions raise marginal damages of global warming and during the carbon-free phases the SCC falls as a proportion of world GDP as a significant part of the stock of carbon in the atmosphere is gradually returned to the surface of the oceans and the earth. The required rise and fall of the SCC is substantial and non-linear due to rapid rises in productivity and population. The dotted lines in fig. 2 provide further details on the time profiles of key variables under scenarios V and VI which set climate policy according to (13). This proportional tax internalizes part of the SCC and improves welfare relative to no policy. A carbon tax proportional to GDP increases the price of 15 Alternatively, the social optimum might have to be abandoned and instead one has to devise second-best intergenerational compensation schemes for policy to increase consumption in all periods. Karp and Rezai (2014) and Karp (2013) discuss the identification of generations in Ramsey models, intergenerational discounting, and the various effects of environmental policy in an OLG setting. 23 fossil fuel and encourages a transition to renewable energy earlier on. However, the proportional tax starts at a too low level and rises too slowly (and even falls in scenario VI) compared to first best. Still, in this sub-optimal scenario a Green Paradox effect emerges around 2080 arising from the anticipation of the imminent phasing in of renewables. The welfare loss of such an inefficient policy is equivalent to 155% of current GDP. Once the proportional carbon tax is supplemented with an optimal renewable subsidy, compensating the non-optimal climate policy, the social optimum is matched quite closely and the welfare loss is less than 1% of initial GDP. Figure 4: The SCC as fraction of aggregate world GDP and consumption (i) Social Cost of Carbon / Output (ii) Social Cost of Carbon / Consumption 4 5 1/$ 1/$ 3 2 4 3 2 1 2010 2060 2110 2160 2210 2260 2310 Key: social optimum ( ), carbon tax only ( proportional tax & optimal subsidy ( 1 2010 2060 2110 2160 2210 2260 2310 ), ), proportional carbon tax only ( ) 5. Robustness of the social cost of carbon Fig. 5 shows the sensitivity of the social optimum to some other key parameters and to the proportional tax rule proposed by Golosov et al. (2014) and table 4 give the corresponding transition times and carbon budgets. Our general finding of an inverted-U time profile for the social cost of carbon is robust. The exact timing and magnitude depends on specific parameters. Introducing a time lag in the temperature response to carbon increases as in Gerlagh and Liski (2013), lowers the SCC and slows the transition to carbonfree sources of energy. 24 Table 4: Transition times and carbon budget (extended) Scenario Fossil fuel Simultaneous use Renewable Only Carbon used Baseline Temperature lag 2010–2035 2010–2055 2035–2040 2055–2060 2040 – 2060 – 320 GtC o 600 GtC o Nordhaus damage 2010–2105 2105–2110 2110 – 845 GtC o = 0 (Leontief) 2010–2035 2035–2040 2040 – 325 GtC o 2010–2040 2040–2045 2045 – 340 GtC o 2010–2105 x 2105 – 1,340 GtC o proportional tax & optimal subsidy (V) prop. tax only (VI) Climate policy is also less ambitious if damages are lower and less convex as in Nordhaus (2008), which implies that more carbon is burnt and much less fossil fuel is left abandoned in the crust of the earth. Lowering the substitutability between the capital-labour aggregate and energy in the CES production function ( = 0) leads to higher (fossil and renewable) energy use, which causes higher stocks of atmospheric carbon and a higher SCC as the destructive and costly energy production cannot be substituted. Climate policy is less ambitious with a proportional carbon tax set according to (13) where the presence of an optimal subsidy reduces the amount of carbon burnt to mimic the first-best closely. Figure 5: Sensitivity analysis for the time paths of the optimal SCC $/tC Social Cost of Carbon, τt Renewable Energy Subsidy, νt 500 500 400 400 300 300 200 200 100 100 0 2010 2210 CES = 0 $ / t C Key: 2110 0 2010 2310 Baseline 2060 Nordhaus Damage Social Cost of Carbon, τt 2110 2160 2210 Temperature lag 500 400 300 200 100 0 The subsidy for renewable energy use and the optimal carbon tax are 2010 2060 2110 2160 2210 2260 2310 complements. The subsidy policy increases in cases in which the carbon tax 25 (and with it the market price of fossil energy) falls: lower damages and less responsive temperature. We also conducted further robustness checks: A much lower social rate of discount ( = 0) leads to a much more ambitious climate policy with a much higher social cost of carbon, earlier phasing in of renewables and more fossil fuel left in situ. A higher elasticity of intertemporal substitution implies less intergenerational inequality aversion. For example, with zero intergenerational inequality aversion ( instead of = ½), this implies in a growing economy that the carbon tax hurts earlier generations much more than later generations. The social planner is relatively more concerned with fighting global warming than with avoiding big differences in consumption of different generations. Climate policy is also more aggressive with a higher equilibrium climate sensitivity ( = 6). Starting with half the initial capital stock (K0 = 100) or increasing population and productivity growth (A(∞) = 6 and L(∞) = 10.6) hardly affect the SCC. The increase in the SCC under lower discounting and intergenerational inequality aversion and a more sensitive climate allows for less aggressive renewable subsidies because of instrument complementarity. 6. Conclusion Our main findings based on an integrated assessment model of climate change and Ramsey growth are that it is important to not only internalize the climate externality but also learning by doing in using renewables. Pricing carbon and subsidizing renewable use curbs fossil fuel use and promotes substitution away from fossil fuel towards renewables, increases untapped fossil fuel, and brings forward the carbon-free era. Our benchmark results give rise to a global carbon tax which rises from about 110 $/tC (or 30 $/tCO2) initially to 190 $/tC (52 $/tCO2) in 2050 and a renewable subsidy which rises from 175 $/tC (48 $/tCO2) initially to 380 $/tC (104 $/tCO2) in 2040 and then falls quickly to zero. The optimal policy mix, therefore, consists of an aggressive subsidy 26 making renewable energy competitive early on and a gradually rising carbon tax pricing fossil energy out of the market. The total amount of carbon burnt is 320 GtC which is much less than the 2,500 GtC under “laissez faire”. Consequently, the social optimum manages to limit the maximum temperature to 2.1 °C instead of 5.1 °C under “laissez faire”. The welfare loss without policy is almost 6 times today’s world GDP. Climate policy becomes less ambitious as the social cost of carbon is higher, fossil fuel is abandoned less quickly and more carbon is used in total if the discount rate is higher, intergenerational inequality aversion is weaker, and the equilibrium climate sensitivity is lower. Less substitutability between energy and the capital-labour aggregate leads to less energy use especially in capital-scarce economies and less global warming. Hence, the required global carbon tax is lower. If international agreements on cooperation in climate change mitigation cannot be reached, governments can move forward unilaterally by introducing subsidies to renewable energy. Such policy internalizes the learning externality and accelerates fossil fuel extraction and global warming as fossil fuel owners fear that their resources will be worth less in the future. The level and the duration of the subsidy policy increases compared to the social optimum to compensate for the missing carbon tax. This limits global warming to 3.5 °C and the loss in welfare to 95% relative to the social optimum. If a carbon tax is introduced instead of the subsidy, the welfare loss reduces to 50% of today’s world GDP. The comparison of welfare indicators across partial policy scenarios gives further credence to the assessment of Stern (2007) that climate change is the biggest externality our planet faces and leads us to conclude that policy makers should prioritize climate negotiations over renewable policy. If such negotiations fail, due to insurmountable free-riding problems associated with international treaties, (national) renewable subsidies are yet a relatively efficient instrument to avert the most severe aspects of global warming. Renewable subsidies in isolation, however, are insufficient to combat climate change. Making total factor and energy productivities also endogenous (using 27 the empirical estimates of the determinants of growth rates given in Hassler et al., 2011) allows further substitution possibilities between energy and the capital-labour aggregate in the longer run and justifies a more ambitious climate policy. R&D subsidies should be set such that the price of renewable energy follows the (rising) price of fossil energy and need to be complemented by a carbon tax in order to avoid excessive extraction associated with the Green Paradox. A recent interagency working group (2010) suggests that US institutions use a social cost of carbon of initially $80/tC (22 $/tCO2) rising to 165 $/tC (45 $/tCO2) in 2050 in project appraisal based on a discount rate of 3% per annum. A discount rate of 2.5% per annum would imply an initial social cost of carbon of 129 $/tC (35 $/tCO2) rising to 238 $/tC (65 $/tCO2) in 2050, which in line with our estimates.16 These figures are typically based on existing integrated assessment models which are often very elaborate and in which consumers rarely maximize utility as in the Ramsey model. Hence, it is not always clear what the underlying assumptions and the crucial parameters deriving the results are. Golosov et al. (2014) offer a tractable fully consistent general equilibrium model of climate change and Ramsey growth, but employ unrealistically low damages at higher temperatures and need to make some very bold assumptions to ensure that both aggregate consumption and the carbon tax are a fixed proportion of world GDP. Much too much carbon is burnt and the welfare losses are substantial with this proportional carbon tax, especially if there is no policy in place for encouraging learning by doing in renewable production. In the absence of an optimal renewable subsidy, the proportional tax starts at a too low level and rises too slowly and causes extraction rates to rise, as predicted by the Green Paradox. 16 Such models of climate change yield estimates of the social cost of carbon starting from 5 to 35 $ per ton of carbon in 2010 and rising to $16 to $50 per ton in 2050 (e.g., the DICE, PAGE and FUND models of Nordhaus (2008), Alberth and Hope (2007) and Tol (2002), respectively), but the Stern Review obtains much higher estimates of the social cost of carbon with a much lower discount rate (Stern, 2007). 28 Our model of Ramsey growth and climate change has more realistic damages and finds that the optimal carbon tax is a hump-shaped function of world GDP. Our analysis also pays careful attention to how fast and how much fossil fuel should be abandoned and how quickly and how much renewables should be phased in. 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(2014) use a decadal Cobb-Douglas production function with negative exponential damages: (A1) Yt Z ( Et ) AK t Ft exp (2.13 Et 581) AK t Ft , 2.379 10 5 , where Kt is the capital stock at the start of period t, A is the calibrated total factor productivity (including the contribution of fixed factors such as labour and land), is the share of capital in value added, and is the share of energy in value added. Apart from (1), (2) and (A1), Golosov et al. (2014) assume logarithmic utility, 100% depreciation of manmade capital each period, zero fossil fuel extraction costs and thus full exhaustion of initial fossil fuel reserves. With these bold assumptions one can show that the social cost of carbon or the optimal carbon tax is a constant fraction of world GDP: tGolosov c.s. 1000 L 1 (1 ) 1 0.004758 (1 L )0 0.00748 Y Y, 1 t 1 1 t 1 (1 )(1 ) 1 0.9772(1 ) 1 (1 ) where > 0 is the rate of time preference (see (13)). Decadal world GDP is 630 US$ trillion in 2010, so that using a discount rate of 10% per decade (or 0.96% per year), = 0.1, we get for 2010 a SCC of 75 US$/tC (or 20 $/tCO2). This estimate is lower if society is less patient. The beauty of this expression is that no detailed integrated assessment model of growth and climate change is needed. The carbon tax follows directly from the rate of time preference , world GDP and some technical damage and carbon cycle parameters. The formula breaks down if intergenerational inequality aversion or factor substitution differs from unity, extraction costs are non-zero (especially if they are stock dependent) and the bad fit of the exponential approximation of (5) plays up. II Appendix B: Proof of proposition 1 The adjoined Lagrangian for our model reads: L t 0 t 0 t 0 S B (1 )t LU t t (Ct / Lt ) t (St 1 St Ft ) t (Bt 1 Bt Rt ) (1 )t tPE ( EtP1 EtP L Ft ) tTE EtT1 (1 ) EtT 0 (1 L ) Ft (1 )t t Kt 1 (1 ) Kt Z ( EtP EtT ) H ( Kt , Lt , Ft Rt ) G(St ) Ft b( Bt ) Rt Ct , where tS denotes the shadow value of in-situ fossil fuel, tB the shadow value of learning by doing, tPE and tTE the shadow disvalue of the permanent and transient stocks of atmospheric carbon, and t the shadow value of manmade capital. Necessary conditions for a social optimum are: (A2a) U '(Ct / Lt ) (Ct / Lt )1/ t , (A2b) Zt H Ft Rt G(St ) tS L tPE 0 (1 L )tTE / t , Ft 0, c.s., (A2c) Zt H Ft Rt b( Bt ) tB / t , Rt 0, c.s., (A2d) (1 Zt 1HK )t 1 (1 )t , t 1 (A2e) tS1 (1 )tS G '(St 1 )Ft 1t 1, (A2f) tB1 (1 )tB b '(Bt 1 )Rt 1t 1, PE (A2g) tPE Z '(EtP1 EtT1 )Ht 1t 1, 1 (1 )t TE (A2h) (1 )tTE Z '( EtP1 EtT1 ) Ht 1t 1. 1 (1 )t Equations (A2a) and (A2d) yield the Euler equation (9). The Kuhn-Tucker conditions (A2b) and (A2c) can be rewritten as (10a) and (10b) after defining E PE TE tS tS / t , tB tB / t and t L t 0 (1 L )t / t in final good units. Equations (A2e) and (A2d) yield the Hotelling rule (A3) tS1 (1 rt 1 )tS G '(St 1 )Ft 1 . Forward summation over time of (A3) gives (11). (A2f) and (A2d) yield III tB1 (1 rt 1 )tB b '(Bt 1 )Rt 1 . (A4) Forward summation over time of (A4) yields (12). Defining tPE tPE / t and tTE tTE / t in final good units we use (A2g), (A2h) and (A2a) to get: PE (A5a) tPE Z '(EtP1 EtT1 )Ht 1, 1 (1 rt 1 )t TE (A5b) (1 )tTE Z '(EtP1 EtT1 )Ht 1. 1 (1 rt 1 )t Solving (A5a) and (A5b) it can be shown that tE LtPE 0 (1 L )tTE is given by (13). Appendix C: A temperature lag Suppose a lag between atmospheric stock of carbon and temperature: Tt 1 (1 T )Tt T ln ( EtP EtT ) / 581 , (3) where T is the speed at which a higher stock of atmospheric carbon gets translated into a higher global mean temperature. Since it takes on average about 70 years, we set T = 1.70. With this additional equation the Lagrangian function becomes: (1 ) L U (C / L ) (S S F ) (B B R ) (1 ) T (1 )T ln ( E E ) / 581 (1 ) ( E E F ) E (1 ) E (1 ) F (1 ) K (1 ) K Z (T ) H ( K , L , F R ) G ( S ) F b( B ) R C . L t t t 0 t T t t 0 t t 0 t t t t t 1 PE t t 0 t t 1 P t 1 T P t S t t 1 t T t B t t P t TE t L t t t t 1 t t T t T t 1 t t T t t t 0 t L t t t t t where tT is the marginal disvalue of global warming. Necessary conditions for a social optimum are (11a)-(11f) as before and: PE (11g) tPE T ( EtP1 EtT1 )1 tT1, 1 (1 )t TE (11h) (1 )tTE T ( EtP1 EtT1 )1 tT1, and 1 (1 )t IV (11i) (1 T )tT1 (1 )tT Z '(Tt 1 )Ht 1t 1. We now get as before (12)-(15). The dynamics of the permanent and transient components of the SCC become: (16) PE tPE ( EtP1 EtT1 ) 1tT1 , 1 (1 rt 1 ) t TE (1 )tTE ( EtP1 EtT1 ) 1tT1 , 1 (1 rt 1 ) t where tT tT / t . Solving (16) yields the SCC as the present discounted value of all future marginal damages from global warming: (A6) tC s 0 L 0 (1 L )(1 ) s t s ( EtP s 1 EtT s 1 ) 1tT s 1 . Solving (11i) together with (11d) yields: (A7) (1 T )tT1 (1 rt 1 )tT Z '(Tt 1 ) Ht 1. This equation yields the marginal cost of global warming: (A8) tT s0 (1 T )s t s Z '(Tt s 1 )Ht s 1 . Upon substituting (A8) into (A6) we get the SCC. Relationship to Golosov et al. (2014) Golosov et al. (2014) assume that there is no temperature lag. Following Gerlagh and Liski (2013) we do allow for such a lag. Under this set of assumptions, (A8) shows that the marginal cost of global warming at the social optimum is proportional to global GDP (cf. equation (6)): T 5 1 (A8) t 2.379 10 T Z (Tt ) H ( Kt , Lt , Ft Rt ). Upon substitution of (A8) into (A6), we get: 1 s P T 1 tC 2.379 105 s 0 L 0 (1 L )(1 ) t s ( Et s 1 Et s 1 ) Z (Tt s 1 ) H t s 1 . T This expression does not simplify to a simple expression depending only on V current global GDP. However, if we use a dynamic reduced-form temperature module with a distributed lag between carbon stock and damages, Zt Z (Et ), (A9) Et 1 (1 T )Et T (EtP1 EtT1 ), we have the Lagrangian function (1 ) L U (C / L ) (S S F ) ( B B R ) (1 ) E (1 ) E ( E E ) (1 ) ( E E F ) E (1 ) E (1 ) F (1 ) K (1 ) K Z ( E ) H ( K , L , F R ) G ( S ) F b( B ) R C . L t t t 0 t E t t 0 t t 0 t t T P t 1 P t t 1 t S t t t 1 PE t t 0 t t t 1 t 1 t t T t 1 TE t t B t t P t 1 T L t t t T t 1 t t T t t t 0 t L t t t t t and thus we get: PE (11g) tPE T tE ), 1 (1 )(t TE (11h) (1 )tTE T tE ), 1 (1 )(t (11i) (1 T )tE1 (1 )tE Z '(Et 1 )Ht 1t 1. It thus follows that (A10) tC s 0 L 0 (1 L )(1 ) s t stE s 1 , (A11) tE (1 T )s t s Z '(Et s1 )Ht s1 . s 0 Under the assumptions made by Golosov et al. (2014), we get (A11) tE 2.379 105 1 Z ( Et ) H t T and thus the following simple expression for the SCC: (A10) 1 tC 2.379 10 5 T 1 1 L 0 (1 L ) Z ( Et ) H ( K t , Lt , Ft Rt ). VI A lag between atmospheric carbon and damages (T > 0) thus pushes down the SCC so our estimates of the optimal global carbon tax will be biased upwards. Appendix D: Functional forms and calibration Preferences As stated in (8), we suppose a CES utility function. We set the elasticity of intertemporal substitution to = ½ and thus intergenerational inequality aversion to 2. The rate of pure time preference is 1% per year. Cost of energy We employ an extraction technology of the form G(S ) 1 (S0 / S ) , where 1 and 2 2 are positive constants. This specification implies that reserves will not be fully be extracted; some fossil fuel remains untapped in the crust of the earth. Extraction costs are calibrated to give an initial share of energy in GDP between 6%-7% depending on the policy scenario. This translates to fossil production costs of $350/tC ($35/barrel of oil), where we take one barrel of oil to be equivalent to 1/10 ton of carbon, giving G( S0 ) 1 0.35. The IEA (2008) long-term cost curve for oil extraction gives a doubling to quadrupling of the extraction cost of oil if another 1000 GtC are extracted. Since we are considering all carbon-based energy sources (not only oil) which are more abundant and cheaper to extract, we assume only a doubling of production costs if a total 2000 GtC is extracted. With S0 4000 GtC,1 this gives 2 1. 2 This implies that we assume very low extraction costs and a high initial stock of reserves which biases our findings toward using more fossil fuel longer. 1 Stocks of carbon-based energy sources are notoriously hard to estimate. IPCC (2007) assumes in its A2- scenario that 7000 GtCO2 (with 3.66 tCO2 per tC this equals 1912 GtC) will be burnt with a rising trend this century alone. We roughly double this number to get our estimate of 4000 GtC for initial fossil fuel reserves. Nordhaus (2008) assumes an upper limit for carbonbased fuel of 6000 GtC in the DICE-07. 2 Since G(2000) / G(4000) (4000 / 2000) 2 2 2 and 2 2 2. VII Initial capital stock and depreciation rate The initial capital stock is set to 200 (US$ trillion), which is taken from Rezai et al. (2012). We set the depreciation rate to be 0.1 per year. Global production and global warming damages Output before damages is H t (1 ) AKt ( AtL Lt )1 0 1. This 11/ ( Ft Rt 1 11/ 11/ ) , 0, 0 1 and is a constant-returns-to scale CES production function in energy and a capital-labour composite with the elasticity of substitution and the share the parameter for energy. The capital-labour composite is defined by a constant-returns-to-scale Cobb-Douglas function with the share of capital, A total factor productivity and AtL the efficiency of labour. The two types of energy are perfect substitutes in production. Damages are calibrated so that they give the same level of global warming damages for the initial levels of output and mean temperature. It is convenient to rewrite production before 1 11/ 11/ 11/ AKt ( AtL Lt )1 Ft Rt . We set the damages as H t H 0 (1 ) H H 0 0 share of capital to = 0.35, the energy share parameter to = 0.06, and the elasticity of factor substitution 0.5 which we will refer to as the CES run. Initial world GDP in 2010 is $63 trillion. Given A1L 1 , we calibrate A = 3.78 to yield initial output under “laissez faire”. The energy intensity of output is calibrated to an initial energy use of 9 GtC under “laissez faire”, 0.15. Population growth and labour-augmenting technical progress Population in 2010 (L1) is 6.5 billion people. Following Nordhaus (2008) and UN projections population growth is given by Lt 8.6 2.1e0.35t . Population growth starts at 1% per year and falls below 0.1% percent within six decades VIII and flattens out at 8.6 billion people. In the sensitivity analysis in section 4.3 we assume faster growth and a higher plateau to reflect more recent forecasts. Without loss of generality the efficiency of labour AtL 3 2e0.2t starts out with A1L 1 and an initial Harrod-neutral rate of technical progress of 2% per year. The efficiency of labour stabilizes at 3 times its current level. Cost of the renewable and learning by doing We model learning by doing with initial cost reductions and a lower limit for the cost of the renewable, i.e., b( Bt ) 1 2e 3Bt , 1 , 2 , 3 0. This formulation differs from the usual power law definition of learning curves (Manne and Richels, 2004) but allows us to better calibrate initial learning rates (which can reach infinity for power law) and specify a lower limit for unit cost. We calibrate unit cost of renewable energy to the percentage of GDP necessary to generate all energy demand from renewables. Under a Leontief technology, with 0 , energy demand is Zt Ht . The cost of generating all energy carbon free is Zt Ht b / Zt Ht bt . Nordhaus (2008) states that it costs 5.6% of GDP to decarbonise today’s economy in a model of back-stop mitigation. We increase this cost significantly to 7-8% of GDP. To calibrate the production cost of renewable energy, this cost estimate needs to be combined with the cost of producing conventional energy which ranges between 6%-7%. This gives b1 0.14 or with 0.15 we get b1 b(0) 1 2 0.94 or $940/tC. Through learning by doing this cost can be reduced by 60% to a lower limit of 9% of GDP, so that b() 1 0.563 and thus 2 0.375. In our simulations with 0, this lower limit falls to about 6% of GDP due to substitution of energy through capital. We assume that cumulative renewable production lowers unit cost at a falling rate and the parameter 3 measures this speed of learning. We calibrate learning such that costs would decrease slowly. We suppose a mere 1.25% reduction in cost if all of world energy use would be supplied by renewable sources in the initial IX period, so that 3 = 0.00375. These assumptions imply at most a 14% reduction for a doubling of cumulative production at the peak of learning-by-doing and are very conservative: Manne and Richels (2004) assume a 20% cost reduction (and consequently need to impose unrealistic constraints on renewable use due to the strong curvature of the power law learning curve). Alberth and Hope (2007) assume a cost reduction of 5%-25% for a doubling of cumulative experience. Our calibration implies a shallow learning curve and, together with the assumption with the calibration of abundant fossil fuel, biases the model toward fossil fuel use. Appendix E: Computational implementation (for online publication only) In our simulations we solve the model for finite time and use the turnpike property to approximate the infinite-horizon problem. All equilibrium paths approach the steady state quickly such that the turnpike property renders terminal conditions essentially unimportant. We allow for continuation stocks to reduce the impact of the terminal condition on the transitions paths in the early periods of the program. We use the computer program GAMS and its optimization solver CONOPT3 to solve the model numerically. The social planner solution, scenario I, in which the externality is taken into account fit the program structure readily. To solve the “laissez-faire” equilibrium paths, we adopt the iterative approach discussed in detail in Rezai (2011). Conceptually, the aggregate economy is fragmented into N dynasties in the externality scenarios. Each dynasty has 1/Nth of the initial endowments and chooses consumption, investment and energy use in order to maximize the discounted total utility of per capita consumption. (All production and cost functions are homogeneous of degree 1 and invariant to N). The dynasties understand the contribution of their own emissions to the climate change and learning in renewable energy, but take carbon emissions and knowledge generation of others as given. The climate and knowledge dynamics are X affected by the decisions of all dynasties. This constitutes the market failures. Under “laissez faire” all dynasties essentially play a dynamic non-cooperative game, which leads to a Nash equilibrium in which each agent forecasts the paths of emissions and renewable generation correctly and all agents take the same decisions. With identical dynasties, equilibrium requires RtExog ( N 1) / N Rt . Ft Exog ( N 1) / N Ft and If N 1 the externalities are internalized and we obtain the social optimum. As N , we obtain the “laissez faire” outcome characterized in section 2. For this outcome, the transition equations (cf. equations (1), (2) and (8)) become EtP1 EtP L Ft Exog , EtT1 (1 ) EtT 0 (1 L ) Ft Exog , Bt 1 Bt RtExog . The numerical routine starts by setting the time path of emissions exogenous to the planner’s optimization, 0 Ft Exog and 0 RtExog , at an informed guess. (The left superscript denotes the index of iteration.) GAMS solves for the welfaremaximizing investment, consumption, and energy use choices conditional on this level of exogenous emissions. These decisions define the time profile of exogenous fossil and renewable energy use in the next iteration, and i 1 RtExog i Rt . i 1 Ft Exog i Ft The routine is repeated and Ft Exog and RtExog are updated until the difference in the time profiles between iterations meets a pre-defined stopping criterion. In the reported results iterations stop if the deviation at each time period is at most 0.001%. XI Appendix F: Comparison of prominent IAMs (for online publication only) Table F.1 compares the key properties of various IAMs: the DICE-2013R model as discussed in the decadal version of Nordhaus (2008) and the semidecadal version of Nordhaus (2014), the FUND model as presented in Tol (2002), the GHKT model as put forward in Golosov et al. (2014), the GL model as developed by Gerlagh and Liski (2013), and the AABH model as set out in Acemoglu et al. (2012), and finally the RP model as put forward in the present paper. The recent Dietz and Stern (2014) extends the DICE model to make the case for stricter climate policy by allowing for the endogeneity of growth, the convexity of damages and climate risk, but still has decadal time resolution and abundant fossil fuel and can thus not discuss issues to do with how much fossil fuel to lock in the crust of the earth. Table F.1: Comparing Integrated Assessment Models of Climate Change Unit of time (in yrs) DICE FUND GHKT GL AABH RP 10 or 5 1 10 10 5 1 Scarce fossil resources Stranded reserves Explicit energy market Endogenous technical progress in renewables Population growth Productivity growth Optimal growth N. A. C.-D. C.-D. Aggregate production function (incl. energy) Leontief CES (not logarithmic) utility N. A. 100% depreciation Climate damages (at high temperature) low N. A. very low very low Prod. Health Prod. Prod. & Utility Type of damage CES CES high Utility Prod.