Hurricane damage distributions and trends
We depict the distribution of the stream of annual hurricane damages over the whole time period t = 0, ... , 100 for the four climate models under the 20th century and RCP 8.5 scenarios in Fig. 1 as quantile-quantile plots.Footnote 9 As can be seen, for all four climate models most damages are zero or fairly small, generally less than 50 billion USD. As a matter of fact, zero damage years under current (future) climate constitute 52 (48), 58 (41), 50 (29), and 36 (29) per cent for the CCSM4, IPSL5, MIROC5, and MRI5 models, respectively. If one considers the mean average annual damages due to hurricanes then the CCSM4, IPSL5, MIROC5, and MRI5 models predict these to be 0.6 (0.7), 0.9 (1.7), 1.2 (3.0), and 2.0 (3.0) for current (future) climate, respectively. Thus, relative to the total exposed assets in the Caribbean (2,257 billion USD), expected percentage damaged per year is small, ranging from 0.03 and 0.13%. Considering the largest damaging years in our synthetic set, which by construction have a return period of 10,000 years, damages under these climate models would, respectively, be 110 (147), 136 (248), 172 (401), and 303 (452) billion USD for current (future) climate. This constitutes a range of between 5 and 20% of total assets in the Caribbean being destroyed for such events.
Comparing the two different climate outcomes, one finds that for smaller damages the 20th Century climate hurricane annual damage distributions all lie to the right of the reference line, although not completely for the CCSM4 model. This suggests that smaller damages are relatively more likely than under future climate. Comparing the quantiles for larger damages, it becomes apparent that these are fairly infrequent regardless of climate model or climate scenario, but still of non-negligible probability. For three of the models—MIROC5, IPSL5, and CCSM4—these large, but low probability, annual damage years are much more common under future climate. The exception in this regard are the damages generated from hurricanes under the MRI5 model, where damages above 80 USD Billion constitute a larger proportion of the total distribution under current climate.
The depiction of damages in Fig. 1 of course ignores their time dimension. We thus in Fig. 2 show the evolution of mean cumulative damages, including the 95% range across simulations, over time for each of our four climate models for the 20th climate and RCP 8.5 contexts, where for the latter we assume that the climate signal emerges linearly with time as outlined in Section 2.3. Under all four climate models, mean average cumulative damages increases more over time under future climate. This is most prominent for the MIROC5 model, while it is substantially less pronounced for the CCSM4 model series. In this regard, average (across simulations) cumulative damages over a 100 years for the former would be in the region of 90 billion USD higher under climate change, but only be 7 billion lower for the latter. Importantly, however, for all four models, the 95% range of estimates of cumulative damages is fairly wide. For example, for the MIR5 model, cumulative damages after a 100 years under climate change could range between 149 and 459 billion USD. Even for the CCSM4, which shows the lowest variability, the range for future (current) climate damages falls between 32 (26) and 115 (99) billion USD. Nevertheless these numbers are much larger than those found by Moore et al. (2017), who discovered a range between 350 and 520 million USD. This is likely due to the fact they did not actually use a set of hypothetical storms under different climate scenarios, and consequently did not explicitly model damages. Rather they used an estimated historical relationship between GDP and a power index of storms, as well predictions in this power index from global temperature changes under climate change, to infer future impacts.
In order to get a better picture of the distribution of hurricane damages over time we also depict the cumulative distribution functions (CDF) of these for current and future climate for our four climate models at snapshots t = 50 and t = 100 in Figs. 3, and 4, respectively, where the blue (red) line refers to the cumulative probability distribution of current (future) climate. Additionally, at the bottom of each graph we also include a green line indicating up to which point on the damage range current climate damages lie to the left of the future climate damages.
Examining the CDFs at 50 years (Fig. 3), it is apparent that for all models, except MIROC5, the distributions cross at some stage. This suggests that at t = 50 under the current climate conditions hurricane damages are relatively more characterized by high damage low probability events than the distribution of damages under future climate for most models. In contrast, at t = 100 (Fig. 4) the picture is reversed, where for all models except MRI5 the CDF of future climate now lies to the left of that of 20th climate. Taken together, the two time snapshots of relative CDFs suggest that over time future climate becomes more characterized by high damage low probability events rather than high probability low damaging storms than under current climate conditions.
Time-stochastic order dominance tests
Time-stochastic first- and second-order dominance
The results of the time-stochastic dominance tests for three terminal periods (50 and 100 years) and our four climate models are given for a linear signal and where exposure grows at the same rate as income in the upper panel of Table 1. Accordingly, for all models, for all terminal periods, pure time-stochastic first-order (TSFD) and second-order (TSSD) dominance fail. This may not be surprising given the results regarding the cumulative distributions F1 and F2 at various points in time as just described.
Almost ideal stochastic first-order dominance
The restriction placed on the utility function, i.e., 𝜖1T, in order to achieve almost time first-order stochastic dominance (ATFSD) are shown in the 7th column of the first panel in Table 1 for a linear (LIN) climate signal. If one chooses terminal time T = 50, 𝜖1 is found to be relatively large—ranging from 0.06 to 0.46—across all models, although still below the 0.5 threshold. At time T = 100, however, all models experience a fall in the parameter, which is just the share of the violations at the terminal period. As a matter of fact, for the MIROC5 model, this parameter reduces to zero.
In the 8th column of Table 1, we depict the restriction on the product of the utility and discount function marginals. These differ across models, with the lowest restrictions for the MIROC5 model, and the highest for the CCSM4 model, regardless of the choice of terminal period T. For all four models, as one increases the terminal period, the size of γ1 falls, thus increasing the set of admissible utility and discount functions.
We depict the spaces of feasible sets of elasticities of the marginal utility of consumption (η) and discount rates (ρ) under a CRRA utility function under ATFSD at T = 100 in the spirit of Pottier (2015) with a linear climate signal as solid lines in Fig. 5. One should note in this regard that the feasible set includes all possible pairs beneath the given lines. Accordingly, for all models only extremely small discount rates are feasible in order for hurricane damages under current climate to be preferable to a setting under climate change. This set is largest for the MIROC5 (up to 0.033%) and lowest (up to 0.009%) under the CCSM4 model. In terms of η and assuming a zero discount rate, the highest admissible elasticity of the marginal utility of consumption ranges from 1.2% (CCSM4) to 3.0% (MIROC5).
Almost ideal stochastic second-order dominance
Even if ATFSD cannot be achieved, under less restrictive criteria one might still be able to obtain preference ranking by invoking ATSSD. In the final three columns of the first panel in Table 1, we provide the estimated parameter restrictions for ATSSD. Considering the space of decreasing pure discount functions and non-decreasing, weakly concave utility functions, the restriction (γ1,2) for all four models falls as one increases the time of the terminal period. As a matter of fact, for the MIROC5 and CCSM4 model, this restriction is not binding by the time 100 years have passed. Examining, 𝜖2T, one finds nevertheless that the set of admissible u will be bounded, but again, as one considers a further period in the future this condition becomes less restrictive. Regardless of the time horizon, it is least bounding for the MIROC5 and most restrictive for the CCSM4 model. Finally, the remaining restriction on the first-order condition of the discount function, i.e., γ1b, is not binding for the IPSL5, MIROC5, and MRI5 models, regardless of the the time horizon. It is only restrictive for the CCSM4 model, but with some variability according to the choice of T.
Cubic climate signal
We repeated all stochastic dominance tests for the setting where we allow the climate signal to emerge in a cubic manner with time (CUB). The results of these are shown in the second panel of Table 1. Accordingly, as before strict time-stochastic first- or second-order stochastic dominance are not achieved under any of the climate models. In terms of their almost counterparts, generally the estimated parameters under a cubic climate signal imply a less restrictive set of utility functions, discount functions, and the product of the marginals of the utility and discount functions for both ATFSD and ATSSD. Examining the set of admissible implied values of the elasticity of marginal consumption and the time discount rate at T = 100 for a constant relative utility function with exponential discounting, Figure 5 shows a similar pattern in terms of which models provide a greater set of elasticity and discount rate pairs for the cubic climate signal model (depicted as dashed lines). The actual sets are substantially larger though, generally about twice in value.
The results of allowing positive (+) and negative (-) adaptation by letting exposure increase at lower and higher rates than income, respectively, are presented in the last two panels of Table 1. As for the case of zero (0) adaptation, under positive adaptation, neither TFSD nor TSSD is achieved. Additionally the degree of violation in 𝜖1T and in the marginals of the product of the utility and discount function to rise substantially under positive adaptation. As a matter of fact, except for the MIROC5 model and the IPSL5 model at T = 50, all estimated values of this parameter are above the permissible threshold of 0.5, eliminating the possibility of ATFSD. Examining the restrictions on ATSSD shows that except for the MIROC5 model there also exists no admissible set of utility functions, regardless of model or terminal period. Even for the MIROC5 model the violations on the discount rate and the marginals of the utility and discount functions appear high enough to suggest that the space for agreement will be limited.
The findings from the assumption of exposure increasing at a greater rate shown in the last panel of Table 1 suggest that there is no case in which there is TFSD. Nevertheless for all models the estimated violation parameters are small enough to indicate a larger set of admissible utility and discount functions to achieve ATFSD than under zero adaptation. As a matter of fact, using Pottier’s (2015) framework suggests the plane of admissible degrees of risk aversion and discount rates to be contained within ranges between 3.5 and 10.3 and 0.06 and 0.16, respectively. In contrast, TSSD is achieved for all models except for MRI5 at T = 100. However, even for the latter the only restriction needed to achieve ATSSD is in terms of the utility function and the value of the estimate on 𝜖2T indicates that there is a large set of non-decreasing utility functions displaying weak risk aversion that will be able to do so.