Abstract
Uncertainty plays a significant role in evaluating climate policy, and fat-tailed uncertainty may dominate policy advice. Should we make our utmost effort to prevent the arbitrarily large impacts of climate change under deep uncertainty? In order to answer to this question, we propose a new way of investigating the impact of (fat-tailed) uncertainty on optimal climate policy: the curvature of the optimal carbon tax against the uncertainty. We find that the optimal carbon tax increases as the uncertainty about climate sensitivity increases, but it does not accelerate as implied by Weitzman’s Dismal Theorem. We find the same result in a wide variety of sensitivity analyses. These results emphasize the importance of balancing the costs of climate change against its benefits, also under deep uncertainty.
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Notes
There is no consensus on the exact definition of the term ‘fat tail’. However, most climate-change economists use the term as the following: “a PDF has a fat tail when its moment generating function is infinite - that is, the tail probability approaches 0 more slowly than exponentially” (Weitzman 2009a: 2). We follow this definition in this paper.
We do not mean to say that the Dismal Theorem is repugnant. The meaning of the term ‘repugnant’ in this sentence is not ‘disgusting’, but ‘repellent to the senses’. We use such strong language to emphasize that one of the implications that an arbitrarily large carbon tax should be imposed—or that emissions should be driven to zero immediately is repellent to the senses. In this case, there is a risk of starvation to death of a large part of the world population.
As an illustration, Costello et al. (2010) compute the willingness to pay (as a fraction of consumption) equating the expected utility without considering emissions control (see Eq. 3 in their paper) and a discounted utility from reduced consumption (i.e. net of willingness to pay) (see Eq. 4 in their paper). Since the expected utility in the absence of emissions control tends to infinity under fat tails, willingness to pay approaches 100 % of consumption. The results we present in Sect. 2 in the current paper also support this finding.
They also run the optimal policy version of the DICE model, but show the Weitzman effect only in the business as usual case.
Indeed \(\mathop {\text{ max }}\limits _\mathrm{c} \text{ E }_\mathrm{s} U(c;s)\) is generally not equal to \(E_s \mathop {\text{ max }}\limits _\mathrm{c} U(c;s)\), where \(E_{s}\) is the expectation operator over a state of the world s and c is a control variable.
For the effect of learning on climate policy, see Hwang et al. (2013).
Applying Weitzman’s damage function with no emissions control in our model in Sect. 3 does not have a feasible solution, probably because the model is no longer convex. We present here consumption flow in the deterministic model instead.
The consumption constraint (the total consumption in society, \(\text{ C }_\mathrm{t} \ge 20\) trillion US$ (in 2005) in the original DICE model) starts to bind after a certain time-period (e.g. from the year 2245 when climate sensitivity is \(8^\circ \text{ C }\)).
There is a computational limit for running the model. For instance, in our system (with 8GB RAM and Intel\({^\circledR }\) \(\text{ Core }^{\mathrm{TM}}\) i5-3210M CPU @ 2.50GHz) running the model with 2,000 states of the world and the time horizon of 600 years was not possible. Considering the memory constraint and the computation time, we here set the time horizon to 300 years (as opposed to 600 years in DICE). We run the model with many combinations of the number of states of the world and the time horizon within the computational limit, but greater numbers of states of the world and longer time horizons than our choice do not affect the results qualitatively.
The term ‘feedback factor’ refers to the impact of a physical factor such as water vapor and cloud on the total radiative forcing in a way of amplifying the response of climate system (Hansen et al. 1984).
The total feedback factors are assumed to be strictly lower than 1 because an equilibrium cannot be reached if is greater than or equal to 1 (Roe 2009).
Since increasing \(\bar{{f}}\) has a similar effect on the distribution of climate sensitivity, we only look at the effect of increasing \(\sigma _{f}\) .
Applying an MPS to a fat-tailed distribution raises some technical difficulties. First, it is not possible to apply the method to a fat-tailed distribution because, by definition, a fat-tailed distribution does not have a moment generating function, and thus there is no mean (first moment) to preserve. Second, even if we truncate the distribution so that we can calculate the mean and the standard deviation from the simulated PDF, it does not suit for the purpose of our paper. One may think of an iterative way of taking densities from the centre and transferring them into the tails (e.g. Mas-Colell et al. 1995: ch.6), but this may produce several discontinuous jumps on the probability distribution.
In addition, the Roe and Baker’s distribution is more scientifically founded than the lognormal distribution (note that the Roe and Baker’s distribution is suggested by physical scientists, whereas the lognormal distribution we used is due to economists, namely Ackerman et. al.). Furthermore, applying the lognormal distribution produces an abnormal decreasing carbon-tax pattern in uncertainty as we see in Fig. 5. Thus we use the Roe and Baker’s distribution as our reference PDF.
When we apply an upper bound of 1,000, the variance and the mean of the CS distribution changes, so does the point at which we stop increasing the variance.
If we use the theoretical variance in Fig. 5, then the concave property of carbon tax function is more transparent.
As we can see from the left panel in Fig. 4, during an MPS, both tails become fatter. Fattening only the right tail may be possible (by holding the left tail and adjusting the right tail), but it would produce a discontinuous jump around the mean (or the mode). The problem is that such a discontinuous PDF is not realistic for the CS distribution.
One of the alternative ways to avoid decreasing carbon tax is to use a much bigger upper bound. If we use a higher upper bound, the density added to the right tail during the MPS is more considered, and thus the effect of the imbalance of the newly added densities between the left and the right tail becomes smaller.
Calibrating to the estimated damage costs gives a similar result to calibrating the exponent, of which default value is 2 in the DICE model, to the estimated damage costs because both calibrations should give the same damage costs at the calibrated data points.
A lognormal or a gamma distribution may be used instead of this truncated normal distribution. For our purpose, however, those distributions did not perform better than the truncated normal distribution used here.
Because of memory constraints, we reduced the number of states of the world on climate sensitivity to 100. In this case, climate sensitivity increases by \(0.25^\circ \text{ C }\) from 0 to \(25^\circ \text{ C }\). The number of states on damage costs is 25, of which value increases by 0.4 % from \(-\)4.2 to 5.0 %. Thus, the total number of states is 2,500.
Models surveyed in Tavoni and Tol (2010) have three alternative stabilization targets (450, 550, 650 \(\text{ CO }_{2}\)eq ppm in 2100). Since the DICE model does not have such corresponding targets, we used the results of the models, which have the similar characteristics with the DICE optimal run.
The number of states of the world is 100 for climate sensitivity and 25 for abatement costs, which increases by 0.025 % from \(-\)0.15 to 0.45 %.
Recall the Ramsey formula: \(r=\rho +\alpha \times g\), where \(r, g, \rho \) are the discount rate, the growth rate of consumption per capita, the pure rate of time preferences, respectively.
As we have seen in Sect. 2, if we do not consider the option of emissions control—or in the business as usual scenario - the social cost of carbon can be increasing and convex in uncertainty. Notice that the social cost of carbon is not equal to the optimal carbon tax in the presence of emissions control (Tol 2012).
Note that conventional values for the coefficient of constant relative risk aversion are \(\alpha \approx 2\pm 1\) (Weitzman 2007).
Note that the equilibrium climate sensitivity of \(1,000^\circ \text{ C }/2\times \text{ CO }_{2}\) means that the average atmospheric temperature increases \(1{,}000^\circ \text{ C }\) when carbon concentration is doubled.
The historical data used for this calibration are as follows: atmospheric temperature (Hadley center, CRUTEM3), ocean temperature (NOAA, global anomalies and index data), radiative forcing (Hansen et al. 2007).
Air temperature may decrease over time if \(\text{ CS } <1.2^\circ \text{ C }\) (Baker and Roe 2009), but in this experiment the sign of the slope changes at \(\text{ CS }=1.5^\circ \text{ C }\). This may be caused by the sign of radiative forcing in the slope equation or observational-errors of the data used (Slope \(=\) \(\beta _1 [T_{AT} (t)-T_{AT} (t-1)]/F(t-1)\), where \(\beta _1 >0\) is a constant).
References
Ackerman F, Stanton EA, Bueno RN (2010) Fat tails, exponents, extreme uncertainty: simulating catastrophe in DICE. Ecol Econ 69:1657–1665
Annan JD, Hargreaves JC (2011) On the generation and interpretation of probabilistic estimation of climate sensitivity. Clim Chang 104:423–436
Anthoff D, Tol RSJ (2010) Climate Policy under fat-tailed risk: an application of FUND. Economic and Social Research Institute (ESRI) Working Paper 348
Anthoff D, Tol RSJ (2011) Schelling’s conjecture on climate and development: a test. Economic and Social Research Institute (ESRI) Working Paper 390
Baker MB, Roe GH (2009) The shape of things to come: why is climate change so predictable? J Clim 22(17):4574–4589
Clarke L, Edmonds J, Krey V, Richels R, Rose S, Tavoni M (2009) International climate policy architectures: overview of the EMF 22 International Scenarios. Energy Econ 31:S64–S81
Costello CJ et al (2010) Bounded uncertainty and climate change economics. Proc Natl Acad Sci 107(18):8108–8110
Dietz S (2011) High impact, low probability? An empirical analysis of risk in the economics of climate change. Climatic Change 108:519–541
Forest CE et al (2002) Quantifying uncertainties in climate system properties with the use of recent climate observations. Science 295(5552):113–117
Hansen J et al (1984) Climate sensitivity: analysis of feedback mechanisms. Geographysical Monograph Series 29:130–163
Hansen J et al (2007) Dangerous human-made interference with climate: a GISS modelE study. Atmospheric Chem Phys 7(9):2312
Hegerl GC et al (2006) Climate sensitivity constrained by temperature reconstructions over the past seven centuries. Nature 440:1029–1032
Hennlock M (2009) Robust control in global warming management: an analytical dynamic integrated assessment. RFF Discussion Paper No. 09–19 University of Gothenburg
Horowitz J, Lange A (2008) What is wrong with infinity-a note on Weitzman’s dismal theorem. University of Maryland Working Paper
Hwang IC, Reynès F, and Tol RSJ (2013) The effect of learning on climate policy under fat-tailed uncertainty. Unpublished manuscript (submitted for publication)
Ikefuji M et al. (2010) Expected utility and catastrophic risk in a stochastic economy-climate model. CentER Discussion Paper Series No. 2010–122 Tilburg University
Karp LS (2009) Sacrifice, discounting and climate policy: five questions. CESifo Working Paper Series No. 2761 Center for Economic Studies & Ifo Institute for economic research
Marten A (2011) Transient temperature response modeling in IAMs: the effects of over simplification on the SCC. Econ Open-Access Open-Assessment E-J 5:2011–2018
Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford university press, New York
Newbold SC, Daigneault A (2009) Climate response uncertainty and the benefits of greenhouse gas emissions reductions. Environ Resour Econ 44:351–377
Nordhaus WD (2008) A question of balance: weighing the options on global warming policies. Yale University Press, New Haven
Nordhaus WD (2011) The economics of tail events with an application to climate change. Rev Environ Econ Policy 5:240–257
Phillips AW (1957) Stabilisation policy and the time-forms of lagged responses. Econ J 67:265–277
Pindyck RS (2011) Fat tails, thin tails, and climate change policy. Rev Environ Econ Policy 5:258–274
Pycroft J et al (2011) A tale of tails: uncertainty and the social cost of carbon dioxide. Econ Open-Access Open-Assessment E-J 5:2011–2022
Roe GH, Baker MB (2007) Why is climate sensitivity so unpredictable? Science 318(5850):629–632
Roe GH (2009) Feedbacks, timescales, and seeing red. Annu Rev Earth Planet Sci 37:93–115
Rothschild M, Stiglitz JE (1970) Increasing risk: I. A definition. J Econ Theory 2:225–243
Salmon M (1982) Error correction mechanisms. Econ J 92:615–629
Schneider SH, Thompson SL (1981) Atmospheric \(\text{ CO }_{2}\) and climate: importance of the transient response. J Geophys Res Oceans (1978–2012), 86(C4):3135–3147
Solomon S et al (2007) Climate change 2007: the physical science basis. Contribution of working group I to the fourth assessment report of the Intergovernmental panel on climate change. Cambridge University Press, New York
Sterner T, Persson UM (2008) An even sterner review: introducing relative prices into the discounting debate. Rev Environ Econ Policy 2:61–76
Tavoni M, Tol RSJ (2010) Counting only the hits? The risk of underestimating the costs of stringent climate policy. Climatic Change 100:769–778
Tol RSJ (2012) Targets for global climate policy: an overview. Economics Department Working Paper Series No. 37–2012 University of Sussex
Tol RSJ (2003) Is the uncertainty about climate change too large for expected cost-benefit analysis. Climatic Change 56:265–289
Tol RSJ, Yohe GW (2007) Infinite uncertainty, forgotten feedbacks, and cost-benefit analysis of climate policy. Climatic Change 83:429–442
Tol RSJ (2009) The economic effects of climate change. J Econ Perspect 23:29–51
Weitzman ML (2007) Subjective expectations and asset-return puzzles. Am Econ Rev 1102–1130
Weitzman ML (2009a) On modeling and interpreting the economics of catastrophic climate change. Rev Econ Stat 91:1–19
Weitzman ML (2009b) Additive damages, fat-tailed climate dynamics, and uncertain discounting. Econ Open-Access Open Assess E-J 3:29–39
Weitzman ML (2010) What is the damages function for global warming-and what difference might it make? Climate Change Econ 1:57–69
Weitzman ML (2011) Fat-tailed uncertainty in the economics of catastrophic climate change. Rev Environ Econ Policy 5:275–292
Weitzman ML (2012) GHG targets as insurance against catastrophic climate damages. J Public Econ Theory 14(2):221–244
Yohe GW (2003) More trouble for cost-benefit analysis. Climatic Change 56:235–244
Yohe GW, Tol RSJ (2010) Precaution and a dismal theorem: implications for climate policy and climate research. In: Geman H (ed) Risk management in community markets. Wiley, New York
Acknowledgments
Funding by the CEC-DG RTD FP7 project ClimateCost is gratefully acknowledged. William Nordhaus deserves praise for making his model code freely available. The authors of this paper are also very grateful to three anonymous reviewers and David Popp, the associate editor of the journal Environmental and Resource Economics, for valuable comments and suggestions.
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Appendices
Appendix
1.1 Appendix A: Calibration of the Atmospheric Temperature Equation
The DICE model represents the climate system by a multi-layered system consisting of the atmosphere, the upper oceans, and the lower oceans. It adopts a box-diffusion model (Schneider and Thompson, 1981). This representation is a simple way of incorporating climate system into IAMs but has some problems. Apart from the criticism that it fails to capture the real mechanism of the climate system (Marten 2011), one of the practical problems encountered during simulations is that the model does not produce a feasible solution when the value of climate sensitivity is lower than around \(0.5^\circ \text{ C }\). We find that this problem is induced by the fact the original specification creates a fast cyclical adjustment when we change only the parameter on climate sensitivity. To see this, notice that the air-temperature evolution equation can be rearranged with simple algebra into Eq. (17), which is an error-correction model (Phillips 1957; Salmon 1982) with an adjustment speed \(\alpha _1 \) and target \(\alpha _2 F(t)-\alpha _3 \left[ {T_{AT} (t-1)-T_{LO} (t-1)} \right] \):
Where \(T_{AT} (t)\) is the global mean surface air temperature increases, \(T_{LO} (t)\) is the global mean lower ocean temperature increases, \(F(t)\) is the total radiative forcing increases, \(\alpha _1 ,\alpha _2 \), and \(\alpha _3 \) are calibrated parameters. Since \(\alpha _2 \) is defined as climate sensitivity (CS) divided by the constant (the estimated forcing of equilibrium \(\text{ CO }_{2}\) doubling), decreasing climate sensitivity artificially increases the adjustment speed, \(\alpha _1 =\xi _1 /\alpha _2 \). With the default value of the DICE model (\(\text{ CS }=3^\circ \text{ C }\)), the adjustment speed \(\alpha _1 \) is 0.22. For CS lower than 0.8, the adjustment speed becomes higher than one. This leads to a cyclical adjustment to the equilibrium temperature, which does not make much sense scientifically speaking. Assuming \(\text{ CS } = 0.5^\circ \text{ C }, \alpha _1 =1.7\) implies important and irrelevant jumps up and down in the temperature every period and leads to an infeasible solution.
To avoid this problem, we recalibrate the parameter values in the air-temperature evolution equation so as to ensure a coherent adjustment process. We calculate atmospheric temperature \(T_{AT} (t)\) according to Eq. 17 using various values of adjustment-speed \(\alpha _1 \) and climate sensitivity CS. Then we fit \(T_{AT} (t)\) against the historical observation data.Footnote 30 Through the experiment we find that the adjustment-speed is linearly related to the inverse of climate sensitivity and the slope of the function changes around \(\text{ CS }=1.5^\circ \text{ C }\) and \(\text{ CS }=3^\circ \text{ C }\).Footnote 31 Thus, we obtain three different functional forms according to the range of CS as follows.
Appendix B: The Behavior of Carbon Tax in Uncertainty
See Fig. 11.
The behavior of carbon tax in uncertainty
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Hwang, I.C., Reynès, F. & Tol, R.S.J. Climate Policy Under Fat-Tailed Risk: An Application of Dice. Environ Resource Econ 56, 415–436 (2013). https://doi.org/10.1007/s10640-013-9654-y
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DOI: https://doi.org/10.1007/s10640-013-9654-y