- , “Critical issues for the calculation of the social cost of CO2: why the estimates from PAGE09 are higher than those from PAGE2002,†Climatic Change, 2013, 117 (3), 531–543.
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- , “Estimates of the social cost of carbon: concepts and results from the DICE-2013R model and alternative approaches,†Journal of the Association of Environmental and Resource Economists, 2014, 1 (1/2), 273–312.
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- , “Revisiting the social cost of carbon,†Proceedings of the National Academy of Sciences, 2017, 114 (7), 1518–1523.
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- , “The social cost of CO2 from the PAGE09 model,†Economics Discussion Papers, 2011, (201139) .
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- (2013). The background concentration of CO2 in the atmosphere is initialised on the observed 2010 level, i.e. 389ppm or 829GtC.21 We assume a pre-industrial atmospheric CO2 concentration of 275.8ppm, resulting in an excess concentration of 113.2ppm in 2010.
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- A method of apportioning the differences between DICE 2016 and DICE-FAIR-Geoffroy to factors (a) to (c) is to plot the percentage difference in carbon prices and temperatures – always relative to DICE-FAIR-Geoffroy – in DICE 2016, DICE-Geoffroy (i.e. combining the DICE 2016 carbon cycle with the Geoffroy et al. (2013) temperature dynamics model), DICE-Joos (i.e. combining the Joos et al. (2013) carbon cycle with the DICE 2016 temperature dynamics model) and DICE-Joos-Geoffroy. Figure 6 does this.27 The way to intuit this figure is that whichever model is closest to DICE 2016 explains most of the difference between it and DICE-FAIR-Geoffroy. Hence the main contributing factor to the difference in optimal carbon prices between DICE 2016 and DICE-FAIR-Geoffroy is (b) insufficient removal of atmospheric CO2 in DICE 2016 (top left panel).
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- • For Golosov et al. (2014), we assume that 51.4% of the excess emissions in 2010 are in the permanent box and 48.6% are in the slow-decaying box. These numbers are obtained by using the authors’ initial values in 2000 and running their model on historical emissions between 2000 and 2010.
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- • For Lemoine and Rudik (2017), we can directly impute the initial atmospheric CO2 concentration and temperature.
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- • For the PAGE and FUND models, it is most convenient to adjust the timing of the emission impulse so that the background CO2 concentration is 389ppm – 2008 in FUND, 2009 in PAGE.
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- • Gerlagh and Liski (2018) do not explicitly model temperature. CO2 emissions map on to atmospheric concentrations and these in turn map directly on to damages. They define a common adjustment speed of temperature and damages in a one-box model. This gives Tt+1 = Tt − ε(ECS × log2(Mt/M1850) − T).
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- Aengenheyster, Matthias, Qing Yi Feng, Frederick Van Der Ploeg, and Henk A Dijkstra, “The point of no return for climate action,†Earth System Dynamics, 2018, 9 (3).
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- All the models are fed with emissions from the IPCC RCP scenarios, including both CO2 and other greenhouse gases and forcing agents. The CMIP5 multi-model mean response is obtained from Stocker et al. (2013). The CMIP5 response is quasi-linear. By contrast, most of the economic models produce a convex response, with warming increasing more than proportionately as a function of cumulative CO2 emissions, except for the high emissions RCP8.5 scenario and except for the Golosov et al. (2014) model. FAIR is a reasonably close approximation of the complex CMIP5 models.
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Allen, Myles R, “Drivers of peak warming in a consumption-maximizing world,†Nature Climate Change, 2016, 6, 684–686.
- Arrow, Kenneth, Maureen Cropper, Christian Gollier, Ben Groom, Geoffrey Heal, Richard Newell, William Nordhaus, Robert Pindyck, William Pizer, Paul Portney et al., “Determining benefits and costs for future generations,†Science, 2013, 341 (6144), 349– 350.
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- b = È = (0.2173, 0.2240, 0.2824, 0.2763)0 and d = (1, 1, 1, 1)0 on an annual basis. The mean lags for the temporary boxes are 277, 25 and 3 years. Aengenheyster et al. (2018) also estimate a 4-box model in continuous time.
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- Bréon, J.A. Church, U. Cubasch, S. Emori, P. Forster, P. Friedlingstein, N. Gillett, J.M. Gregory, D.L. Hartmann, E. Jansen, B. Kirtman, R. Knutti, K. Krishna Kumar, P. Lemke, J. Marotzke, V. Masson-Delmotte, G.A. Meehl, I.I. Mokhov, S. Piao, V. Ramaswamy, D. Randall, M. Rhein, M. Rojas, C. Sabine, D. Shindell, L.D. Talley, D.G. Vaughan, and S.-P. Xie, “Technical Summary,†in T.F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, and P.M. Midgley, eds., Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press, 2013.
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- C Further results DICE 2016 compared with DICE-FAIR-Geoffroy Here we compare standard DICE 2016 (row 5) with DICE-FAIR-Geoffroy (row 1). This comparison is affected by differences in: (a) temperature dynamics between the DICE 2016 and Geoffroy et al. (2013) models; (b) removal of atmospheric CO2 between the DICE 2016 and Joos et al. (2013) carbon cycles, and; (c) assumptions about (non-)diminishing marginal removal of atmospheric CO2 between DICE 2016/Joos et al. (2013) and FAIR. Therefore this comparison is of the combined effect of all the modifications to DICE that we have identified, which would bring it fully into line with the climate science models we have assembled.
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- Clarke, L., K. Jiang, K. Akimoto et al., “Asessing transformation pathways,†in O. Edenhofer, R. Pichs-Madruga, Y. Sokona et al., eds., Climate Change 2014: Mitigation of Climate Change.
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- Collins, M., R. Knutti, J. Arblaster, J.-L. Dufresne, T. Fichefet, P. Friedlingstein, X. Gao, W.J. Gutowski, T. Johns, G. Krinner, M. Shongwe, C. Tebaldi, Weaver A.J., and M. Wehner, “Long-term climate change: projections, commitments and irreversibility,†in T.F. Stocker, C. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, and P.M. Midgeley, eds., Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge, UK and New York, NY, USA: Cambridge University Press, 2013.
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- Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge, UK and New York, NY, USA: Cambridge University Press, 2014.
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- Default FAIR uses a climate sensitivity of 2.75◦ C. We now turn to the economic models included in Figure 1. We take each of these models “off the shelfâ€Â, except that, in order to be consistently compared following the experimental protocol of Joos et al. (2013), we ensure all the models are initialised on the same atmospheric carbon stock and atmospheric temperature: • In DICE 2016, the carbon stocks are initialised on the year 2015, when the atmospheric CO2 concentration is assumed to be 399.4ppm. Hence we reduce the excess carbon content of the three carbon boxes in DICE 2016 by 9.2% to obtain comparable 2010 initial conditions.24 We do not change the initial deep ocean temperature in DICE 2016.
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- DICE 2016 DICE 2016 is formulated in discrete time with a time unit of 5 years and, like the model of Geoffroy et al. (2013), has two heat boxes, one for the temperature of the atmosphere, land and upper oceans, and one for the temperature of the deep oceans: Tt = Tt−1 + 1 CUP
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Dietz, Simon and Frank Venmans, “Cumulative carbon emissions and economic policy: in search of general principles,†Journal of Environmental Economics and Management, 2019, 96, 108–129.
- Es, where the term in square brackets shows how much of an emission impulse at time s is left in the atmosphere at time t. Roughly a fifth of carbon stays up in the atmosphere “foreverâ€Â, half of an emission impulse is removed after 30 years, and the remaining carbon in the atmosphere has a mean life of 300 years. This yields θL = 0.2, θ0 = 0.393 and Õ = 0.0228. It follows that the half-life equals ln(0.5)/ ln(0.9772) = 30 decades. The initial values for 2010 are S0(1) = 684 GtC and S0(2) = 118GtC. Our starting date is 2015, so we update these and use S0(1) = 712GtC and S0(2) = 159GtC instead.
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- FAIR calculates α as a function of the integrated CO2 impulse response function (iIRF) over the first 100 years of the model horizon. The assumed relationship between α and iIRF100 in FAIR has no analytical solution, but can be well approximated by fitting an exponential function, which results in the following solution: α = 0.0107 exp (0.0866iIRF100) . (13) 23 DICE assumes a climate sensitivity of 3.1◦ C. The mean climate sensitivity in Geoffroy et al. (2013) is between 3.05◦ C and 3.25◦ C, according to how models are aggregated (λ̄ × TCO2x2 = 3.05◦ C while λ × TCO2x2 = 3.25◦ C).
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- FAIR is identical to the model of Joos et al. (2013), except that the residence time of CO2 in each of the four atmospheric boxes is modified by a parameter α representing carbon cycle feedbacks.
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- For each of the 16 carbon cycle models that formed part of the CMIP5 study, the four-box carbon cycle model of Joos et al. (2013) is used as a reduced-form representation. Joos et al. (2013) document the fitting procedure and resulting parameter values. The initial excess atmospheric CO2 concentration of 113.2ppm relative to pre-industrial needs to be distributed among the four boxes of the Joos et al. model. The same need arises for the FAIR model, which shares the same four-box structure. Moreover, as the Joos et al. model was not designed to reproduce historical removal of CO2 from the atmosphere (Millar et al., 2017), it is the FAIR model that we use to initialise the boxes in all of these models. To do this, we feed historical emissions into FAIR from 1890 to 2010.22 This results in the following distribution of the initial excess concentration between the four boxes: 52.9% in box 1; 34.3% in box 2; 11.1% in box 3; 1.6% in box 4.
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- Geoffroy, Olivier, David Saint-Martin, Dirk JL Olivié, Aurore Voldoire, Gilles Bellon, and Sophie Tytéca, “Transient climate response in a two-layer energy-balance model. Part I: Analytical solution and parameter calibration using CMIP5 AOGCM experiments,†Journal of Climate, 2013, 26 (6), 1841–1857.
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- Gerlagh and Liski (2018) Gerlagh and Liski (2018) have a simple lag with partial adjustment of 0.183 per decade (or 2% per year), so they have Tt = Tt−1 + 0.183 (0.842Ft − Tt−1).
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Gerlagh, Reyer and Matti Liski, “Consistent climate policies,†Journal of the European Economic Association, 2018, 16 (1), 1–44.
- Gillett, Nathan P, Vivek K Arora, Damon Matthews, and Myles R Allen, “Constraining the ratio of global warming to cumulative CO2 emissions using CMIP5 simulations,†Journal of Climate, 2013, 26 (18), 6844–6858.
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Gollier, Christian, Pricing the Planet’s Future: The Economics of Discounting in an Uncertain World, Princeton University Press, 2012.
- Golosov et al. (2014) Golosov et al. (2014) have no temperature lag, so they have Tt = 0.842Ft.
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- Golosov, Mikhail, John Hassler, Per Krusell, and Aleh Tsyvinski, “Optimal taxes on fossil fuel in general equilibrium,†Econometrica, 2014, 82 (1), 41–88.
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- Goodwin, Philip, Richard G Williams, and Andy Ridgwell, “Sensitivity of climate to cumulative carbon emissions due to compensation of ocean heat and carbon uptake,†Nature Geoscience, 2015, 8 (1), 29–34.
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- Hope, Chris, “The marginal impact of CO2 from PAGE2002: An integrated assessment model incorporating the IPCC’s five reasons for concern,†Integrated Assessment, 2006, 6 (1).
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- Interagency Working Group on Social Cost of Carbon, “Technical update on the social cost of carbon for regulatory impact analysis under executive order 12866,†Technical Report, Interagency Working Group on Social Cost of Carbon, United States Government 2013.
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- IPCC, Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press, 2013.
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- Joos, Fortunat, Raphael Roth, JS Fuglestvedt, GP Peters, IG Enting, W von Bloh, V Brovkin, EJ Burke, M Eby, NR Edwards et al., “Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: a multi-model analysis,†Atmospheric Chemistry and Physics, 2013, 13 (5), 2793–2825.
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Lemoine, Derek and Ivan Rudik, “Steering the climate system: using inertia to lower the cost of policy,†American Economic Review, 2017, 107 (10), 2947–57.
- M0 ∆Tt ∆F for the Geoffroy et al. (2013) model and ∆Tt ∆S = F2×CO2 ln 2 1 M1 ∆Tt ∆F for the discrete-time models such as DICE. Note that the response to a step increase in atmospheric carbon decreases in the values of the atmospheric carbon stock. To calculate these convoluted step responses, we suppose that the concentration of atmospheric carbon stays constant at its initial value. Hence, we set Ms to 3038 GtCO2 or 389 ppmv for all s. For the DICE model we thus get ∆Tt ∆S = F2×CO2 ln 2 1 M1 ∆Tt ∆F = 0.0012585 as t → ∞. For a small step change in atmospheric carbon of 100 GtC, the steady-state increase in temperature would then equal 0.0012585 × 100 × 44/12 = 0.46 K, which is consistent with the plot in Figure 4.
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- MacDougall, Andrew H, “The transient response to cumulative CO2 emissions: a review,†Current Climate Change Reports, 2016, 2 (1), 39–47.
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Manoussi, Vassiliki, Anastasios Xepapadeas, and Johannes Emmerling, “Climate engineering under deep uncertainty,†Journal of Economic Dynamics and Control, 2018, 94, 207–224. Mattauch, Linus, H Damon Matthews, Richard Millar, Armon Rezai, Susan Solomon, and Frank Venmans, “Steering the climate system: Comment,†American Economic Review, forthcoming.
Matthews, H Damon, Nathan P Gillett, Peter A Stott, and Kirsten Zickfeld, “The proportionality of global warming to cumulative carbon emissions,†Nature, 2009, 459 (7248), 829–832.
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- National Academies of Sciences, Engineering, and Medicine, Valuing Climate Damages: Updating Estimation of the Social Cost of Carbon Dioxide, National Academies Press, 2017.
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- The 16 CMIP5 carbon cycle models emulated by Joos et al. (2013) are then combined with 16 CMIP5 temperature models (i.e. atmosphere-ocean general circulation models), which are represented in reduced form using the model of Geoffroy et al. (2013), as described in their paper. We set the climate sensitivity equal to 3.1◦C in all models.23 This allows us to focus on temperature inertia in the climate models. For all models, we use 0.85◦C as initial atmospheric warming relative to pre-industrial in 2010. The initial lower ocean temperature is 0.22◦C above pre-industrial, obtained by running FAIR on historical emissions.
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- The Delay 56 and Delay 112 carbon cycle models use the same values as Joos et al. (2013) for È, but multiply the mean lags by five and ten respectively. In other words, any point on the impulse response function will be a point on the Delay 56 (112) impact response function five (ten) years later. 26 In continuous time, their model is ṁ = bE − (A − I) m.
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- The DICE 2016 carbon cycle model: 3 boxes The DICE 2016 carbon cycle of Nordhaus (2017) has three boxes: (1) the atmosphere, (2) the upper oceans and biosphere, and (3) the lower/deep oceans. The diffusion matrix is A =       0.88 0.196 0 0.12 0.797 0.001465 0 0.007 0.998535        and b = d = (1, 0, 0)0. No carbon leaves the system, so the elements of the columns of A sum to 1.
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- The FUND model The FUND carbon cycle model, which is based on Maier-Reimer and Hasselmann (1987), has 5 boxes with shares of emissions flowing into each of them equal to b = (0.13, 0.20, 0.32, 0.25, 0.1)0, d = (1, 1, 1, 1, 1)0 and A has diagonal elements equal to exp(-1/lifetime), where the lifetimes for the 5 boxes are ∞, 363, 74, 17 and 2 years respectively. These correspond to half-lives of ∞, 252, 51, 12 and 1.4 years respectively.
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- The Gerlagh and Liski (2018) carbon cycle model: 3 boxes Gerlagh and Liski (2018) have boxes for (i) the atmosphere and the upper oceans, (ii) the biosphere and (iii) the lower oceans. Since within a decade (their unit of time) carbon mixes perfectly between the atmosphere and the upper oceans, these are combined into box one. The stock of atmospheric carbon St is a constant share of the contents of box one, i.e. d = (0.914, 0, 0)0. They have A =       0.6975 0.2131 0.029 0.1961 0.7869 0 0.1063 0 0.9706        and b = (0.8809, 0.0744, 0.0447)0. The eigenvalues of A are 0.5286, 0.9264 and 1, and we calculate the corresponding Èi to be 44.5%, 18.2% and 16.2%. The eigenvalues imply that the half-lives for the two temporary boxes are 90 and 11 years.
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- The Golosov et al. (2014) carbon cycle model: 2 boxes Golosov et al. (2014) have A =   1 0 0 1 − Õ   , b =   θL θ0(1 − θL)    and d =   1 1    , where 0 < θL < 1 and 1 − θL are the proportions of emissions that flow into the boxes holding the permanent and transitory components of atmospheric carbon respectively, 0 < θ0 < 1 is the proportion of atmospheric carbon in the transitory box that decays within the span of a unit of time (i.e. within a decade), and Õ > 0 denotes the speed of decay of carbon in the transitory box. Hence Eq. (3) becomes Mt = m0(1) + (1 − Õ)t m0(2) + t X s=1 h θL + θ0(1 − θL)(1 − Õ)t−s i
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- The Joos et al. (2013) carbon cycle model: 4 boxes Joos et al. (2013) use a continuous-time model with one permanent and three transitory boxes to fit impulse response functions to an ensemble of Earth System model simulations.26 They get A =          1 0 0 0 0 0.9975 0 0 0 0 0.9730 0 0 0 0 0.7927           ,
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- The rate of uptake by the biosphere and oceans is independent of the amount of carbon stored in each box, so positive feedback between atmospheric CO2 and CO2 uptake is ruled out. There is no direct interchange of carbon between the atmosphere and the lower/deep oceans. The lower/deep oceans can store a large amount of carbon, but the rate of diffusion into the lower/deep oceans is only 0.007. The eigenvalues of A are (0.6796, 0.9959, 1) and V =       0.6991 0.5075 0.3173 −0.7148 0.3002 0.1942 0.0157 −0.8077 0.9282        , so b̄ = (0.5283, 0.8085, 0.6946)0, d̄ = (0.6991, 0.5075, 0.3173)0 and thus the Èi are 37%, 41% and 22%. Since no carbon leaves the boxes, one of the eigenvalues equals 1. The smallest eigenvalue corresponds to a half-life of 9 years (5 x ln(0.5)/ ln(0.6796)) and the middle one corresponds to a half-life of 851 years.
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- The resulting background scenario is compared to a scenario with the same emissions path, but with an impulse of 100GtC added to the atmosphere at time zero (the year 2010). The 100GtC is added to each carbon box in proportion Èi.
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- This is a feature shared by DICE 2016 and DICE-Geoffroy, but not by the other models, which incorporate the four-box carbon cycle of Joos et al. (2013). However, when it comes to the 2◦C carbon price, or temperature on either path, the main contributing factor to the difference between DICE 2016 and DICE-FAIR-Geoffroy is (a) temperature dynamics. Excessive delay, offset by excessive long-term warming, is a feature shared by the DICE 2016 and DICE-Joos variants. Excessive delay and excessive long-term warming are responsible for the temperature trajectories in DICE 2016 that start below DICE-FAIR-Geoffroy but end up higher, significantly so on the optimal path. Excessive long-term warming also explains the high 2◦C carbon price, because it significantly limits the 2◦C carbon budget. 27 For this comparison we omit emissions, because when emissions approach or reach zero the differences between the models can explode or be undefined respectively.
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van der Ploeg, Frederick, “The safe carbon budget,†Climatic Change, 2018, 147 (1-2), 47–59.