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Robust management of climate risk damages

Risk Management volume 25, Article number: 15 (2023) Cite this article

Abstract

We consider the case of a risk manager or policymaker who does not know the true climate and economic parameters of the Dynamic Integrated Climate Economy (DICE) model and who, because of political or social constraints, cannot act optimally. We find that the impact of parameter uncertainty on economic outcomes is much more pronounced away from optimality than along an optimal path. We also find that for this non-omniscient and politically constrained actor the most desirable of the feasible courses of actions depends strongly on which model is most uncertain. When we consider uncertainty in the growth rate of the economy or in the cost of abatement, a gradual ramp-up is preferred to a steep (‘Stern-like’) abatement schedule. This result is extremely robust to the choice of a number of non-expected-utility-maximization decisional criteria that do not make use of probabilities: minimax, maximax and maximin all give the same recommendation. Ambiguity aversion does not change these results. However, when even a small uncertainty in the damage function is considered, a steeper abatement schedule becomes a strong contender, and is preferred by some decisional criteria. Furthermore, the ‘cross-over point’ for the damage exponent (the point, that is, above which an aggressive abatement schedule becomes preferred) is very close to the DICE value. This suggests that researching this aspect of climate modelling would have the greatest policy relevance. Finally, we note that a gradual (‘Nordhaus-like’) ramp-up of the abatement efforts abatement schedule is always preferred to a slower (‘business-as-usual’) schedule of abatement even in the case of much stronger future economic growth or much milder climate damage than the central estimates of the DICE model.

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Notes

  1. IPCC Report (2014), Chapter 4, Strengthening and Implementing the Global Response, Sect. 4.2.1.1, p. 321.

  2. A description of earlier versions of the model, and the changes to the model and its parameters, can be found in Nordhaus and Sztorc (2013). The structure of the model and its relationship to similar IAMs can be found in Nordhaus and Moffat (2017) and Nordhaus (2007b). The links between the DICE model and DSGE models are discussed in detail in Hassler et al. (2016).

  3. Dietz et al. (2007) claim that a non-aggressive ramp-up can only by justified by those who deny the anthropogenic origin of climate change; by those who stress the adaptability of the human species; and by ‘those who accept the science of climate change and the likelihood that it will inflict heavy costs, but simply do not care much for what happens in the future beyond the next few decades’. Several other authors take objection to this characterization: see, e.g. Dasgupta (2008, 2020); Nordhaus and Moffat (2017); Nordhaus (2007c).

  4. It is extremely difficult if not impossible to meaningfully estimate discount rates for future costs and benefits [...] Thus standard cost–benefit analyses become difficult to justify [...] and are not used as an assessment tool in this report’. IPPC (2014), Chapter 1, p. 76.

  5. Constrained policies are also considered in Nordhaus (2007b); however, these still explore the optimal policies consistent with the achievement of exogenous targets, such as a maximum acceptable increase in the temperature anomaly by a certain date. As discussed in Sect. 2, Tol (2020) also considers policy actions away from optimality, but from the perspective of a ‘selfish bureaucrat’ is the spirit of Niskanen (1960).

  6. ‘A possible explanation of the carbon pricing puzzle is based on the existence of political constraints related to the social acceptability of climate policies...’, Gollier (2020), p. 5, emphasis added.

  7. Parameter uncertainty in IAM models is usually dealt with by calculating the optimal paths associated with parameter sets drawn from a posited parameter distribution. Needless to say, this falls well short of reaching a decision on the basis of a procedure that feeds the uncertainty into a convex utility function.

  8. Here we use the term ‘risk’ to denote collectively what Knight (1957/2006) describes as ‘a priori probability’ and ‘statistical probability’, and we use the term ‘uncertainty’ to denote what he call ‘estimates’. See p. 225.

  9. The precise definition of the Gradual, Aggressive and Slow schedules is given in Sect. 5.

  10. The decarbonization rate and the abatement costs are in the DICE model a deterministic function of time. We endogenize them in Sect. 5, but they become a function of cumulative investment efforts, not of economic growth.

  11. See Chapter 1 in particular.

  12. See p. 158 and passim.

  13. The value of the ‘utility impatience’ parameter of \(\beta\) of 0.0010 chosen by Stern does not even allow for a very modest reduction of the degree of concern for the welfare of infinitely distant generations: it is only meant to cater for the probability that humanity may be wiped out by an asteroid-like event. If this stance is taken, arguably the possibility of a nuclear war or of a global pandemic—which currently appear vastly more likely than the impact of an asteroid—should raise the discount rate above 0.0010.

  14. We follow and further the cost–benefit analyses of climate change in the line of Nordhaus (2008), Stern (2007), and Hope (2006, 2013) and related IAMs. We are aware of objections such as those in Pindyck (2013), who have criticized these cost–benefit analyses for their unrealistic assumptions, but we consider the addressing of these concerns beyond the scope of this work.

  15. A vibrant strand of current research tries to link the DICE model with non-separable utility functions. These invariably make use of some variant of the Bellman equation. See, e.g. Cai et al. (2015b), Ackerman et al. (2013), Daniel et al. (2018). Due to the computational challenges, a full application of the DICE model to a recursive-utility setting is still very incomplete, but it appears that static risk aversion plays a much smaller role than aversion to intergenerational inequality of consumption, which the elasticity of marginal utility chosen in the DICE model (1.45) tries to capture.

  16. Tol (2020) also considers the effect of policy heterogeneity.

  17. Joughin et al. (2014) argue that marine west Antarctic ice sheet collapse is already under way, while Lenton et al. (2008) evaluate that the Greenland ice sheet melting could lead to a global sea level rise of up to one metre per century. The risk of multiple interacting tipping points is also discussed by Cai et al. (2016).

  18. In this paper, we do not consider the effect of uncertainty in the utility discount rate—if of ‘uncertainty’ one can speak in this case—but we do consider uncertainty in the growth rate of the economy, which is one component of the Ramsey-equation discount rate. For a general and in-depth discussion of uncertainty in climate modelling see Brock and Hansen (2018). For a discussion of the effect of uncertainty in the discount rate, see Weitzman (1998). The attending debate about gamma discounting with Gollier can be found in Gollier (2002, 2004), and the resolution of its ‘paradox’ in Gollier (2010).

  19. The paper by Woodward and Bishop (1997) was based on one of the first versions of the DICE model. The model output have changed both because of different parameter choices and improved modelling.

  20. The output available for non-abatement investment and consumption, y is given by \(y= y_{gross}\left( 1-damfrac-abatefrac\right)\), with abatefrac equal to the fraction of output devoted to abatement efforts. See Sect. 4 for a detailed discussion.

  21. To facilitate cross-referencing with the DICE model, we have used the same symbols for the variables used in this paper as the variable names in the GAMS code of the DICE model made public by Prof Nordhaus.

  22. Apart from the need to break the non-sensical link between high carbon intensity and high negative emissions for values of \(\mu\) above 1, Barreto and Kypreos (1999) confirm that endogenizing technological progress produces significantly different outcomes when it comes to modelling energy systems.

  23. 2014 is the closest date to the release date of the version of the DICE model (2016) that we use.

  24. We say ‘implicit’ because a social discount rate is obtained by the first-order Euler condition at the optimum: for ‘small’ (marginal) investments. The DICE approach obtains, and does not start from, optimality, and does not require the investment to be ‘small’.

  25. In different contexts, the utility discount factor is often referred to as the ‘impatience’ parameter. This interpretation is unwarranted when intergenerational utilities are aggregated. Therefore we do not use the term impatience.

  26. Available at https://www.pwc.co.uk/services/sustainability-climate-change/insights/net-zero-economy-index.html.

  27. This limit is discussed in Sect. 4.

  28. The far right tail of the Slow abatement schedule is numerically unreliable: given the high discount rate, the discount factor for horizons over two centuries becomes extremely small (the utility discount factor for the final horizon of 500 year is \(2.5 ^{-8}\)), and therefore large changes in distant consumptions have a negligible effect on the target function. Indeed, for the Slow optimal schedule the default optimization routines offered by two popular software packages—the MatLab function fminunc which uses the quasi-Newton algorithm, and the Python function minimize of the Jones et al. (2001) package SciPy which uses the SLSQP method—generate significantly different behaviours for the abatement fraction at the very long end, but produce a virtually identical total utilities.

  29. ‘Forcing’ is the balance of energy in and energy out for the Earth system, i.e. the difference between solar irradiance absorbed by the Earth, and the energy radiated (not reflected) back into space via black-body radiation. Reflection is accounted for, via albedo, as an effective reduced irradiance.

  30. The concentrations for the RCPs are CO2 equivalent gases, aggregating gases under the Kyoto Protocols. The CO2 concentration fo the DICE model is what is referred to in the documentation as ‘concentration in the upper strata’.

  31. The technical meaning of the term ‘likely’ in the IPCC reports is ‘with likelihood higher than 66%’.

  32. This is true by construction for the ‘objective’ schedules, but happens to be approximately true as well for the subjective schedules—confirming again that the subjective schedules have been reasonably chosen.

  33. The values for \(K_1\) and \(K_2\) are 0.03025 and \(-10,993.704\), respectively.

  34. Several other non-probabilistic criteria have been proposed, such as the \(\alpha\)-maximin criterion, defined by \(\max _{a \in A} [\alpha \min _{s \in S} a(s) + (1-\alpha ) \max _{s \in S} a(s)]\) by Arrow and Hurwicz (2010). However, they often depend on an arbitrary ‘mixing’ parameter \(\alpha\). A detailed analysis over a range of possible values for \(\alpha\) would make the present work, already very rich in permutations, too heavy.

  35. Actually, we simply impose the milder requirement that the social planner only knows the first two moments of the distribution of the parameter values. Given this level of knowledge, a Guassian distribution is the corresponding maximum-entropy distribution. The planner therefore uses the discrete probabilities \(q_i\) associated with a Gaussian density with these two moments. For the optimization over five values the discrete probabilities were chosen to be \(q_{\pm 40\%}=0.13\), \(q_{\pm 25\%}=0.20\) and \(q_{0}=0.34\).

  36. For ease of cross-reference with the DICE code we use in this section the symbols and variable names used in the DICE code.

  37. “...An important example that has been used in policy discussions is an approach which assumes a sharp threshold at a temperature increase of 2 C; implicitly this implies a very sharp kink in the curve near that threshold...’, p. 16.

  38. “...By far the most important uncertain variable for climatic outcomes is the growth in total factor productivity. This is the main driver of economic growth in the long run, and output trends tend to dominate emissions and therefore climate change...” Nordhaus (2007b), p. 109.

  39. emphasis added.

  40. This value refers to an earlier version of the DICE model, and the expected growth per annum cannot therefore be directly compared. The standard deviation, however, should be more transportable. See the discussion on p. 226 of Nordhaus (2007b).

  41. We note again that there are no reasonable values of the damage exponent for a cross-over between the Gradual and Slow schedules to occur. We also note that, for damage exponents below thresholds in the range of values between 1.75 and 2.45 the Slow schedules can have a greater discounted utility than the Aggressive schedules.

  42. A case has been made [see, e.g. Lovejoy (2019)] that not only the weather, but also the climate may be governed by chaotic-dynamics behaviour, with fat-tailed (power-law) outcomes. This would clearly require far deeper changes of the DICE equations than altering its parameters, and is beyond the scope of our study.

  43. See, in particular, Chapter 20, Must a concern for the environment be centred on human beings?.

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  1. EDHEC Risk Climate Impact Institute, 10, Fleet Place, London, EC4M 7RB, UK

    Riccardo Rebonato & Lionel Melin

  2. Universita’ Bocconi, Milan, Italy

    Riccardo Ronzani

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  1. Riccardo Rebonato
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  2. Riccardo Ronzani
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  3. Lionel Melin
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Rebonato, R., Ronzani, R. & Melin, L. Robust management of climate risk damages. Risk Manag 25, 15 (2023). https://doi.org/10.1057/s41283-023-00119-z

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