Climate Impacts on Capital Accumulation in the Small Island State of Barbados

: This paper constructs a model of climate-related damage for small island developing states (SIDS). We focus on the loss of private productive capital stocks through extreme climate events. In contrast to most economic analyses of climate impacts, which assume temperature-dependent damage functions, we draw on the engineering literature to allow for a greater or lesser degree of anticipation of climate change when designing capital stocks and balancing current adaptation expenditure against future loss and damage. We apply the model to tropical storm damage in the small island developing state of Barbados and show how anticipatory behavior changes the damage to infrastructure for the same degree of climate change. Thus, in the model, damage depends on behavior as well as climate variables. J.L., and T.L.; funding acquisition, E.K.-B.; investigation, E.K.-B. and C.D.; methodology, E.K.-B. and J.L.; project administration, E.K.-B.; writing—original draft, E.K.-B., J.L., T.L., and C.D.; writing—review and editing, E.K.-B., J.L., T.L., and C.D.


Introduction
Small island developing states (SIDS) are expected to be among the most heavily impacted by climate change [1], including sea level rise, cyclones, rising temperatures, and changing rainfall patterns [2]. While many small island economies perform comparatively well [3], arguably because they must be open to trade due to their narrow resource and export bases [4], their reliance on exports contributes to fluctuations in growth and recurrent high debt levels [5]. The combination of economic and climate vulnerability suggests that understanding climate-economy interactions is particularly important for SIDS. Yet, because of the limited data for most small islands, there have been few studies, particularly those that treat the economy as a whole (the recent study by Moore et al. [6] being a rare exception).
Most economic analyses of climate impact are carried out with global integrated assessment models (IAMs) that combine greenhouse gas emissions from economic activity with a representation of the climate system. Some IAMs, such as DICE [7,8] and FUND [9], as well as the post-Keynesian model developed by Rezai et al. [10], include feedback from the climate system to the economy. Both DICE and FUND implement a cost-benefit analysis and compute optimal emissions trajectories, given their assumptions about social preferences (an optimization mode). Other IAMs, such as GCAM [11,12], can also be run in a simulation mode for exploring alternative non-optimal scenarios. The IAMs that

Model Probability Distributions
In this paper we focus on hydro-meteorological factors, specifically storms. Two approaches for modeling such events are generally used: annual maximum series (AMS) and peaks-over-threshold (POT) analyses [36]. In the AMS approach, a probability distribution is fit to the series of maximum events (e.g., flood or wind speed) in each year of the record. In the POT approach, only the magnitudes and arrivals of events exceeding a threshold are modeled using probability distributions. The POT methods capture the reality that multiple events of interest may occur in a single year, whereas no events of interest may occur in other years. The POT methods, however, typically require more data for calibration. In this paper we apply an AMS model, leaving the more complex POT analysis for future work.
The magnitude-frequency relationship is most often expressed in terms of the quantiles of the probability distribution assumed to approximate the behavior of a particular disaster type, where, F X is the cumulative distribution function (CDF) of random variable X, and x p is the pth quantile of X, where 0 ≤ p ≤ 1 is the non-exceedance probability of magnitude x p over a particular time period, normally taken to be a year. This information is very often communicated in terms of the return period of an event of magnitude x p , where the return period T is defined as the inverse of the exceedance probability, . (2) If the distribution of the disaster magnitude is not changing, the return period can be interpreted in two ways: (1) if F X describes the distribution of maximum observed event in a time period (say the maximum annual flood), then over T periods one expects (in the statistical sense of "expectation") for x p to be exceeded exactly once; (2) if the realizations of X are independent from one time period to the next, then the return period is also the average waiting time to observe an event exceeding x p .

Stationarity and Non-Stationarity
A crucial assumption behind the use of the return period is stationarity, meaning that the probability distribution of events remains unchanged over time. Stationarity has never held exactly in reality. Even in a stationary climate, land use change from human activity (e.g., de-and afforestation, urbanization, agricultural practices, etc.) can affect flood distributions in complex ways [37]. However, even when it is justified, the concept of a return period can be challenging for non-specialists to understand. The case for non-stationarity (e.g., [38]) should not be overstated, and for many analyzes stationarity remains a reasonable assumption. However, non-stationarity is both accelerating and amplifying due to climate change, and as it does, the meaning of the return period becomes problematic even for specialists [39][40][41][42]. Nevertheless, the return period remains a popular means of communicating the frequency-magnitude relationship of extreme events, and it was adopted in the IPCC special report on extreme events [24].
In this paper we allow for forward-looking design in that an engineer is assumed to choose the least-cost design given anticipated changes in the frequency of extreme events.

The Perpetual Inventory Model with Climate Damage
In the model developed in this paper, gross domestic output (GDP), which we denote by Y, is given by a capital productivity κ multiplied by the total capital stock, While it would be helpful to distinguish different types of capital and their vulnerability, as in [43], data limitations prevent us from that level of analysis for our case study country of Barbados.
The change in the value of capital stock, K, is given by the value of gross investment, I, net of depreciation, D. The capital stock in period t + 1 is then calculated as The "perpetual inventory" method of accounting for capital stock is a common approach (e.g., it was used for the Penn World Tables [44]). It can be implemented in a straightforward way using data from national accounts, with the initial level of the capital stock as the only free parameter.
Depreciation can be expressed as a rate δ per unit of capital stock multiplied by the value of the capital stock. In practice, depreciation rates vary over time. However, in this paper we simplify the analysis by assuming a constant rate and we provide a justification in Section 2.5. With this assumption, we can write Equation (4) as Our assumption of a constant depreciation rate is consistent with an assumption that climate change affects capital stocks only through extreme events. More gradual changes, such as rising sea levels leading to quicker erosion of sea defenses, are not considered in this model. Extreme climate events lead to loss of capital beyond normal depreciation. Such events are random, and they appear in the model as a series of independent shocks. This should be a reasonable assumption for storms; droughts, in contrast, tend to appear in multi-year groups. To capture periodic changes in global climate, such as the Pacific decadal oscillation (PDO) or El Niño-Southern oscillation (ENSO) the frequency of storm appearance can change over time, while storms in a particular location can be treated as independent of one another.
We express the loss in period t as a fraction δ t C of the existing capital stock (the damage ratio) and assume that at least some of the damage in the previous period is made up for in the current period. The revised expression becomes Equation (6) formulates climate damage as a depreciation rate shock. As noted in the introduction, we work in this paper within a structuralist tradition [5,16,17], in which economic actors face irreducible uncertainty about the future. In the Materials and Methods, we develop a climate damage model where economic actors anticipate future states of the world according to a stochastic wind speed model, but we allow for the possibility that they are mistaken. That is, actors are taken to assume a stochastic wind speed model, and we apply such a model in simulations, but the two models need not agree. Moreover, even when they do agree, it is possible to have an unusual sequence of devastating storms that exceed anticipated damage. In contrast, economic actors in neoclassical models make optimal decisions given probabilistic knowledge of future states of the world as they occur in the model. Real business cycle (RBC) models make a further "new classical" assumption that economic actors respond rapidly to the information available to them. Such models predict that depreciation rate shocks will have very little effect on economic output [45,46]. In "New Keynesian" models [47], where actors may not respond rapidly to new information, or an RBC model in which actors do not take climate shocks into account [19], depreciation rate shocks can produce an effect. In the model developed in this paper, when accurately anticipated, depreciation rate shocks from climate events can be accounted for in the design of physical capital, but as noted above, expected damage may differ from realized damage, either because of the particular sequence of storm events or because the climate is changing in ways that economic actors did not anticipate.
The investment I t that appears in Equation (6) represents gross additions to productive capital stock. However, some capital expenditure is non-productive, including the cost of hardening capital to withstand a particular magnitude of climate event. We use x to denote the magnitude of an event, while x d is the magnitude of the "design event". The engineer's task is to design the physical capital stock such that any event of magnitude less than x d should inflict minimal damage, while allowing for some damage for events above that level. We assume that the total cost of capital, when built to withstand an event of magnitude x d , inclusive of adaptation cost, is a multiple m a (x d ) of the cost of the productive capital. This and subsequent assumptions can be checked and refined using empirical engineering data. Denoting the total cost with I tot , we write a modified Equation (6), If no storms are provided for, so that x d = 0, then there are no adaptation costs, and therefore m a (0) = 1. The adaptation costs rise with the magnitude of the design event, so m a > 0. We assume declining marginal effectiveness of mitigation expenditure, which translates to rising marginal costs at higher design event magnitude, and therefore m a > 0 as well. The final term in Equation (6) is the cost of rebuilding damaged capital (loss and damage). For the purposes of the present paper, we make a simple behavioral assumption, that damage is rebuilt, but at most a fraction of GDP can be devoted to rebuilding in any period. This means that there will ordinarily be a stock of damaged capital waiting to be rebuilt, D, where With that assumption, That is, the entire stock of damaged capital is repaired if funds permit. Otherwise, loss and damage expenditure is limited to the maximum available funds for repairs. This assumption makes loss and damage expenditure endogenous, controlled only by the expenditure fraction . It could be relaxed to allow for different responses, including the choice to migrate rather than rebuild. That is, indeed, a strategy pursued at the household level within US counties [22], but it is significantly more difficult to emigrate from a sovereign small island state than from a county within a large continental economy.

Materials and Methods
In this section we develop a model for climate damage to productive capital stocks. We first discuss the relevant calculations under a stationary climate, then under a non-stationary climate, and then construct a model specifically for Barbados.

Balancing Construction Costs against Climate Damage under Stationarity
For commercial infrastructure, such as we consider in this paper, an explicit cost-benefit calculation is often applied to investment in protective capital. We implement such a calculation in this section. In contrast, public infrastructure is usually built according to a specified return period (e.g., a 50-year event). In that case, the magnitude of the design event can be calculated from the design return period and the probability distribution F X (x) using Equation (2).
For commercial investment, we note that a given level of gross investment I tot includes both the gross increment of productive capital I and adaptation costs. From Equation (7), the relationship is To weigh adaptation cost against the reduction in future damage, we add to the construction cost the discounted potential damage to (depreciated) productive capital. In this case we need the average expected storm damage, which will depend on both the design event and the shape of the distribution.
We write this as δ C (x d ; σ), where σ is a vector of parameters for the distribution. Assuming stationarity, and a discount rate i, the discounted average cost of repairing damage, C d , is equal to In this equation, the discounted value of productive capital declines at the normal depreciation rate, excluding climate damage. We assume that climate damage is fully repaired in the subsequent period, so it adds to the cost with a one-period discount. (The simulation model described later in the paper has a quarterly time step.) This is a more restrictive assumption than in Equation (6), where expenditure on repairs can extend over several time periods. We adopt it both because it greatly simplifies the calculation and because it is meant to represent the calculation of an engineer attempting to find an optimal design threshold. From that vantage point, the engineer would have little basis to guess how long repairs might be delayed due to future cash-flow constraints. The result is an overestimate of the actual discounted repair costs, because the discount applies to the start of the rebuilding period, but it is not applied over the course of rebuilding.
The total cost C can now be expressed as This "engineering" cost contains only internal costs borne by the entity that must build and maintain the capital stock. It excludes actual or imputed external costs, and it does not consider social benefits. Thus, it seeks to represent the costs to which economic actors respond. Alternative assumptions, such as insuring new investment against climate damage, can be implemented by modifying this equation.
Good engineering practice suggests that the design should minimize the total engineering cost [48], which is achieved when x d satisfies This is a general expression that depends on the precise forms for the marginal adaptation cost and damage ratio. For the simulation model we assume specific functional forms, which we introduce below.
We emphasize that the calculation that results in Equation (13) is not a social welfare calculation, so it does not suffer from the problems raised by Pindyck [49] regarding IAMs. Rather, it represents a textbook present-worth analysis of equal-life alternatives for an engineering project (e.g., see [50]). The discount rate in Equation (11) is the one that a firm would choose when comparing between competing investments (a financial discount rate). We, therefore, avoid the contentious debate over the appropriate social discount rate [51]. Unlike social discount rates, for which experts provide a wide range of values [52], discount rates used by firms for investment decisions are comparatively standard and uncontroversial. The alternatives being compared in the present-worth calculation are represented by different design event magnitudes x d . The "equal-life" condition is met through the assumption that any damage will be rebuilt. When the design magnitude satisfies Equation (13), present worth is maximized because the discounted costs are minimized.

Balancing Construction Costs against Climate Damage under Non-Stationarity
In a changing climate in which storms are expected to become more severe over time, the choice of design period is not straightforward. Designing for the current climate means under-designing, while designing for the expected climate at the end of the design life means over-designing. The minimum cost is achieved somewhere in between [41].
We capture non-stationarity by introducing a time-varying fractional damage cost function into Equation (11), the discounted cost of repairing damage, This is a general expression. It depends on the marginal adaptation cost, the dependence of the damage ratio on the event magnitude, and changes in the parameters of the distribution of storm events. Below, we argue that the mean damage function can be assumed to grow exponentially over time, With this approximation we can explicitly compute the sum in Equation (14) to find Following the same steps as before, we find that the magnitude of the design event should satisfy We return to this expression below, after first constructing a non-stationary statistical model for peak wind speed.

Wind Speed Model
A number of distributions are commonly used to describe the distribution of extremes in hydrology and meteorology [29]. We use the generalized extreme value (GEV) distribution, which encompasses three families of distribution. The extreme value (EV) type I distribution (also called the Gumbel) describes the distribution of the largest observation in samples arising from parent distributions with exponential tails (e.g. the normal and gamma distributions), and it corresponds to a GEV with zero shape parameter. The EV II distribution (also called the Fréchet), corresponding to positive GEV shape parameters, exhibits heavy or fat tails meaning that extreme quantiles can be quite large. The EV III distribution (also called the Weibull), corresponds to negative GEV shape parameters, is bounded by zero and is typically used to model the distribution of the smallest observation in a sample, for instance low-flows in an annual streamflow record. The cumulative density function of the GEV distribution is given by where while the probability density function is In these expressions, x represents the magnitude of a particular event. The distribution has a location parameter µ and scale parameter, σ, each with the same units as x, and a dimensionless shape parameter ξ.
We now turn to the specific case of Barbados. Details of the wind speed model are provided in the Appendix A. We took data on storms from the Caribbean Hurricane Network's StormCARIB website (https://stormcarib.com), which included dates, peak wind speed, storm classification, and name for storms in the Caribbean. Data are available for Barbados specifically, as well as for the Eastern Caribbean as a whole, from the mid-19th Century through 2010. The StormCARIB data are based on "best track" data from the U.S. National Hurricane Center's North Atlantic hurricane database reanalysis project (HURDAT) [53].
The model suggested by the exploratory data analysis, where the probability that a certain peak wind speed will be exceeded in Barbados, is derived from a peak wind speed distribution for the Eastern Caribbean as a whole, which is modeled using a generalized extreme value (GEV) distribution. Therefore, we use the following model for Barbados, where, p s is the strike probability, specifically, the probability that a storm in the Eastern Caribbean passes within 60 nautical miles, or 69 miles of the island. The parameter φ is the average value of the peak wind speed in the Eastern Caribbean divided by the peak wind speed observed in Barbados. As discussed in the Appendix, our estimate for p s is 0.36 and for φ is 1.34. The probability distribution P EC is a cumulative GEV distribution.
To simulate a non-stationary climate, the parameters for the GEV model parameters depend on the global average sea surface temperature (more precisely, the sea surface temperature anomaly relative to the 1961-1990 average). The motivation for this covariate is that tropical storm intensity tends to rise with sea surface temperature [54], although not uniformly, because temperature and pressure in the atmosphere also affect the intensity of storms [55]. Localized temperature extremes can also impact economic performance [18,56], but here our focus is on temperature averaged over large areas as a covariate with storm intensity.
The Appendix details our procedure for estimating the location, scale, and shape parameters for the Eastern Caribbean and gives their values. To obtain estimates for Barbados, we divided the location and scale parameters for the Eastern Caribbean by φ = 1.34, leaving the shape parameter unchanged. Using the central estimates for the parameters, we find µ = 48.9 + 27.2τ mph, σ = 34.2 mph, ξ = −0.37, where, τ is the global average sea surface temperature anomaly. The average anomaly between 1850 and 2010 was τ = −0.13 • C, corresponding to µ = 45.4 mph. Combined with the estimate of 0.36 for p s , that gives the return periods (shown in Table 1) for Barbados for tropical storms and category 1-5 hurricanes according to the Saffir-Simpson scale. Table 1 also shows the observed frequencies from the StormCARIB database. The estimates are in reasonable agreement with observation, given the comparatively small number of observations for hurricanes. We note, that it is possible that the probability of Eastern Caribbean storms to strike Barbados, captured by the parameter p s , may change in the future. Historically, most storms have missed Barbados, but if the hurricane belt migrates southward as the climate changes, then the frequency could increase.

Calculating Average Climate Damage
We expect the damage ratio δ C (x; x d ) (the fraction of productive capital lost in an event that exceeds the design threshold) to rise with the magnitude of the event and fall with the threshold. Below the threshold the loss is zero, while at some magnitude above the threshold, damage will reach 100%. Damage models used in engineering depend on the type of hazard. For storms, structural damage depends on wind speed and the size of the storm [57] and, near the coast, storm surge [58]. Detailed studies consider the vulnerabilities of different structural components [59,60]. In the simplest models, the damage ratio rises as the wind speed to a power [61,62], as shown in Figure 1.
The maximum event magnitude, x max , depends on the distribution. In the specific case of the distribution we use for storm events in Barbados, we write this equation using the rescaled location and scale parameters, as where, the vector of distributional parameters σ = (p s , µ, σ, ξ). This equation gives a general expression for the mean damage ratio when extreme events follow the rescaled GEV distribution that we have estimated for Barbados. We next specify a functional form for the damage ratio. We note, that it is possible that the probability of Eastern Caribbean storms to strike Barbados, captured by the parameter ps, may change in the future. Historically, most storms have missed Barbados, but if the hurricane belt migrates southward as the climate changes, then the frequency could increase.

Calculating Average Climate Damage
We expect the damage ratio δ C (x; xd) (the fraction of productive capital lost in an event that exceeds the design threshold) to rise with the magnitude of the event and fall with the threshold. Below the threshold the loss is zero, while at some magnitude above the threshold, damage will reach 100%. Damage models used in engineering depend on the type of hazard. For storms, structural damage depends on wind speed and the size of the storm [57] and, near the coast, storm surge [58]. Detailed studies consider the vulnerabilities of different structural components [59,60]. In the simplest models, the damage ratio rises as the wind speed to a power [61,62], as shown in Figure 1. Statistics for Barbados do not include casualty losses to commercial capital stocks. To find an estimate for the historical value for the average damage ratio, we took the estimate of GDP losses in Barbados due to storms from Acevedo [33]. For storms passing within 60 nautical miles, he found average annual losses of 0.2% of GDP. Not all of that will be associated with damage to productive capital. Assuming (somewhat arbitrarily) that half of the loss of GDP is due to loss of productive capital, we multiply 0.1% per year of GDP by the long-run average capital-output ratio of 4.2 years (estimated from data from the Penn World Table v. 9.1 [44]), to find an estimated capital loss of 0.42% per year.
Past studies have found that climate damage can be assumed to increase with wind speed to a power. Nordhaus [61], in a widely-cited study, found damage in the US to rise as the 9th power of the maximum wind speed, while Bouwer and Wouter Botzen [62] found damages to rise as the 8th power. Both estimates are well above conventional models based on physical processes that suggest a power of two or three. In a study that accounted for the size of the storm, as well as peak wind speed, Zhai and Jiang [57] found an exponent on wind speed of 5. In a US context, Murphy and Strobl [63] and Strobl [22] adopted values of 3 and 3.17. In a study of Latin American and the Caribbean, Strobl [20] estimated a value of 3.8, while for the Caribbean alone, Acevedo [33] found the power to be 3. We adopt the Acevedo's estimate as the power in our model, and apply it to wind speeds above the design threshold. Thus, we assume The parameter A is a scale factor to be found through calibration. To make A dimensionless, we divide it by the initial value of x d , x d0 , to the power n. Substituting into Equation (23), we find an expression for the mean damage ratio. Computing the resulting integral numerically for x d = x d0 = 65 mph and the estimates of µ, σ, and ξ reported earlier gives and setting the result equal to 0.42% per year (our estimate for the initial mean damage ratio) gives a value for A of 0.12.
We constructed a numerical estimate of the mean damage ratio using these parameters and a range of possible values for x d and the sea-surface temperature anomaly τ. To a good approximation, we found that ln δ From this expression we can identify the parameter a in Equation (15) as 1.54 r τ , where, r τ is the rate of increase in sea surface temperature in • C per year. An automated search for breakpoints in a piecewise linear fit to the Hadley temperature anomaly series identified breakpoints in 1876, 1913, 1939, and 1973. From 1973 to the end of the series, the temperature has been rising at 0.013 • C per year, giving an estimate for a of 0.022/year. However, as we discuss below when describing the scenarios, it is likely to rise more rapidly in the future.
Next, we specify the form of the adaptation cost function. In principle, and in actual engineering practice, this can be calculated from cost data for structures that can withstand events of different magnitudes (for an example, see [64]). However, aggregate data are not readily available, so for this paper, we assume a one-parameter function. As discussed earlier, when taking the extreme case in which capital stocks are not hardened at all (x d = 0), there are no adaptation costs and m a (0) = 1. Under an assumption of decreasing returns to adaptation expenditure, we adopt an exponential cost function, Using Equation (13) and the approximate function in Equation (25), Barbados' depreciation rate has been falling over time, and particularly sharply since 1982. From the Penn World Table v. 9.1 [44], the 1960-2017 average was 7.7% per year. Assuming a discount rate of 7.0% per year (a typical value for engineering projects), this gives an estimate for θ of 0.0015/mph.
Using these parameter values, and Equation (25) as an approximation for the average damage ratio, we computed x d using (17), to find: We added a superscript "e" on the strike probability p s and the rate of increase in the sea surface temperature anomaly r τ because they represent (possibly incorrect) expectations of future climate. The sea surface temperature anomaly at the time of construction has a subscript "accept" to capture the possibility that the accepted value may not reflect current conditions. We use this expression in the simulations.

Linking to a Macroeconomic Model
In this section we develop a model of capital accumulation for Barbados to illustrate the operation of the climate damage model. Three of the authors (EKB, CD, and TL) previously developed a macroeconomic model for Caribbean SIDS that includes export dependence and external debt [5]. In that model, capital accumulation is endogenous, depending on anticipated demand and capital utilization. It could be extended to respond to losses, as well; Miethe [65] has shown that financial activity in small islands declines after a hurricane, except for offshore financial centers (OFC). (The volume of international investment flowing to Barbados is not sufficiently large relative to its GDP for it to be classified as an OFC [66].) In this paper, we focus on climate impacts and anticipatory behavior, and specify capital accumulation exogenously leaving the combination of the models for future work.
Capital stocks with different design thresholds x d are affected to different extents by climate damage. Therefore, we construct a vintage model, with vintages v = 65, 66, . . . , 150, corresponding to ranges for the design thresholds x d of (65, 66), (66, 67), . . . , (149, 150) mph. The design threshold at a given time is selected using Equation (28) given expectations for future climate and observations of current conditions. Capital accumulation follows Equation (6) for each vintage, Climate damage is calculated using the actual, not anticipated, climate, while the vintage corresponding to the design threshold is determined based on the anticipated climate.
GDP, Y, is given by a capital productivity κ multiplied by the total capital stock, For the loss and damage calculation, we maintain a stock of damaged capital for each vintage, constrain total loss and damage expenditure to lie below a fixed share of GDP, (set to 20% in model runs) and allocate it to different vintages based on their representation in the pool. Specifically, We base our economic parameters on the Penn World Table v. 9.1 [44]. For the purposes of this paper we make simplifying assumptions in order to focus on climate damage.
First, we assume that investment grows at a steady rate g, which we anchor to the historical growth rate of the capital stock. Barbados' capital stock growth has been very slow since the Great Financial Crisis. Assuming a recovery to pre-crisis patterns, but not to the extraordinarily high growth rates of the 1960s and 1970s, we assume g to equal the 1980-2007 average rate of 2.7% per year. All investment at time t flows to the vintage corresponding to the design threshold at time t as calculated from Equation (28), Second, we link capital stock to GDP using a constant capital productivity, and we assume a constant depreciation rate. Neither of these assumptions is strictly true (see Figure 2). However, for both parameters, a fit to the historical data is consistent with a gradual approach toward asymptotic values. Extrapolating those trends and taking the average over the scenario period (2017 to 2050) gives an average capital productivity of 0.17 per year and an average depreciation rate of 3.8% per year. .
For the loss and damage calculation, we maintain a stock of damaged capital for each vintage, constrain total loss and damage expenditure to lie below a fixed share of GDP, ℓ (set to 20% in model runs) and allocate it to different vintages based on their representation in the pool. Specifically, We base our economic parameters on the Penn World Table v. 9.1 [44]. For the purposes of this paper we make simplifying assumptions in order to focus on climate damage.
First, we assume that investment grows at a steady rate g, which we anchor to the historical growth rate of the capital stock. Barbados' capital stock growth has been very slow since the Great Financial Crisis. Assuming a recovery to pre-crisis patterns, but not to the extraordinarily high growth rates of the 1960s and 1970s, we assume g to equal the 1980-2007 average rate of 2.7% per year. All investment at time t flows to the vintage corresponding to the design threshold at time t as calculated from equation (28), Second, we link capital stock to GDP using a constant capital productivity, and we assume a constant depreciation rate. Neither of these assumptions is strictly true (see Figure 2). However, for both parameters, a fit to the historical data is consistent with a gradual approach toward asymptotic values. Extrapolating those trends and taking the average over the scenario period (2017 to 2050) gives an average capital productivity of 0.17 per year and an average depreciation rate of 3.8% per year. We initialize GDP to the 2017 value of Bds$9.35 billion (from the World Bank World Development Indicators database). Over the five-year period 2014-2018, the mean sea surface temperature anomaly was τ = 0.53 • C, corresponding to µ = 63.5 mph. We adopt that as the starting value.

Results
We built the model described above as a system dynamics model in Vensim. (Code is available from the authors upon request; the model requires Vensim DSS.) We ran three scenarios: stationary (a stationary climate); non-stationary no anticipation (non-stationary climate but a design threshold that does not anticipate any climate change); and non-stationary with anticipation (non-stationary climate with accurate anticipation of future climate). We note that a further (and arguably far more likely) scenario is a non-stationary climate in which climate change is anticipated, but inaccurately. However, the scenarios we have chosen are sufficient for the purpose of this paper, which is both to demonstrate the model and to explore whether anticipatory behavior (or lack of it) can substantially affect both adaptation costs and loss and damage.
Each scenario was run from 2017 to 2050 in Monte Carlo mode, with storm parameters drawn from a GEV distribution. For the temperature anomaly we used trends from MAGICC/SCENGEN 5.3 [67] (which reports global average temperature rather than sea surface temperature) with the "no policy" P50 scenario. We adopted a piecewise linear rate of increase with breaks in 2030 and 2040, from 0.53 • C in 2017, to 0.85 • C in 2030, 1.17 • C in 2040, and 1.52 • C in 2050. Because storms represent extreme events, it takes a very large number of runs to generate a representative distribution. However, a smaller number of runs is sufficient to give an idea of trends. We ran each scenario 10,000 times, using the same pseudo-random number sequence for each scenario. The results are shown in the figures below. In each figure, the bands correspond, under stationary conditions, to 5-year return events (80%), 20-year events (95%), 100-year events (99%), and 500-year events (99.8%). In addition, the outer boundary for all events (100%) is shown. The mean value is shown as a yellow line. Figure 3 shows adaptation expenditure as a share of GDP in the three scenarios. In the stationary scenario, even at the 99.8% level, adaptation expenditure remains below 5% of GDP. It rises through anticipatory behavior in the non-stationary with anticipation scenario. In the non-stationary no anticipation scenario, delays in rebuilding lead to a fall in GDP in the more extreme scenarios, so although costs are the same as in the stationary case, they rise as a share of GDP. Figure 4 shows loss and damage expenditure as a share of GDP in the three scenarios. Following the model assumptions, loss and damage expenditure in any given year is capped at 20% of GDP, so repairs may take several years to complete. (The backlog of damaged capital starts at zero, giving the discontinuity in the graph in the first years.) Under stationary conditions, mean loss and damage is around 3% of GDP, and in 80% of cases loss and damage expenditure is below 10% of GDP, but due to the accumulation of a backlog, there is a good chance that expenditure can be higher. In at least 1% of cases (that is, 100-99%), it reaches the maximum level. Note that total damage will be even higher than is shown in the graph because the model does not take into account the duration of the storm and only tracks damage to productive capital stocks. Damage to houses, crops, municipal buildings, public infrastructure, and so on is not accounted for.
the outer boundary for all events (100%) is shown. The mean value is shown as a yellow line. Figure 3 shows adaptation expenditure as a share of GDP in the three scenarios. In the stationary scenario, even at the 99.8% level, adaptation expenditure remains below 5% of GDP. It rises through anticipatory behavior in the non-stationary with anticipation scenario. In the non-stationary no anticipation scenario, delays in rebuilding lead to a fall in GDP in the more extreme scenarios, so although costs are the same as in the stationary case, they rise as a share of GDP.  Figure 4 shows loss and damage expenditure as a share of GDP in the three scenarios. Following the model assumptions, loss and damage expenditure in any given year is capped at 20% of GDP, so repairs may take several years to complete. (The backlog of damaged capital starts at zero, giving the discontinuity in the graph in the first years.) Under stationary conditions, mean loss and damage is around 3% of GDP, and in 80% of cases loss and damage expenditure is below 10% of GDP, but due to the accumulation of a backlog, there is a good chance that expenditure can be higher. In at least 1%  of cases (that is, 100-99%), it reaches the maximum level. Note that total damage will be even higher than is shown in the graph because the model does not take into account the duration of the storm and only tracks damage to productive capital stocks. Damage to houses, crops, municipal buildings, public infrastructure, and so on is not accounted for.
The difference between anticipation of climate change and building in line with historical climate change can be clearly seen in the non-stationary scenarios in Figure 4. The optimal design threshold trades off anticipated damage against mitigation cost, so damages rise in both nonstationary scenarios, with damage at the 95% (or 20-year) level overwhelming the capacity to rebuild in both scenarios. Yet, mean loss and damage expenditure rises only slightly in the non-stationary with anticipation scenario, while it rises substantially in the non-stationary no anticipation scenario. IAMs report damage as a GDP loss. For purposes of comparison, we calculated the loss relative to a baseline in which GDP grows at a steady rate of 2.7% per year. The results are shown in Figure  5. The mean values are comparatively small in the stationary and non-stationary with anticipation scenarios (less than 1.0% of GDP) because damaged capital is rebuilt, and the investment expenditure is part of GDP. These values are higher than the 0.2% per year estimated by Acevedo [33] and the study of Moore et al. [6], which found output losses for Barbados between 0.20% and 0.25% of GDP in 2050. This is because the model tracks a backlog of unrepaired damaged capital stocks, and cumulative damage increases subsequent GDP losses (see the discussion in [19]). Without anticipation, mean losses are higher still, on the order of 4% by 2050. Moreover, due to cumulative The difference between anticipation of climate change and building in line with historical climate change can be clearly seen in the non-stationary scenarios in Figure 4. The optimal design threshold trades off anticipated damage against mitigation cost, so damages rise in both non-stationary scenarios, with damage at the 95% (or 20-year) level overwhelming the capacity to rebuild in both scenarios. Yet, mean loss and damage expenditure rises only slightly in the non-stationary with anticipation scenario, while it rises substantially in the non-stationary no anticipation scenario.
IAMs report damage as a GDP loss. For purposes of comparison, we calculated the loss relative to a baseline in which GDP grows at a steady rate of 2.7% per year. The results are shown in Figure 5. The mean values are comparatively small in the stationary and non-stationary with anticipation scenarios (less than 1.0% of GDP) because damaged capital is rebuilt, and the investment expenditure is part of GDP. These values are higher than the 0.2% per year estimated by Acevedo [33] and the study of Moore et al. [6], which found output losses for Barbados between 0.20% and 0.25% of GDP in 2050. This is because the model tracks a backlog of unrepaired damaged capital stocks, and cumulative damage increases subsequent GDP losses (see the discussion in [19]). Without anticipation, mean losses are higher still, on the order of 4% by 2050. Moreover, due to cumulative damage and the rebuilding backlog, even with anticipation there is a significant probability of greater losses, as shown by the rise in the 80% and 95% confidence intervals. Consumption expenditures, and, therefore, living standards, are reduced because of expenditure for mitigation and loss & damage as well as from lower output due to damaged capital stocks. This is shown in Figure 6. In a stationary climate, even the most severe storms reduce consumption by no more than 0.5 percentage points of GDP. In a non-stationary climate, consumption losses can be substantial even when climate change is anticipated. We emphasize again that while our model includes forward-looking actors, they are not the social planners of the IAMs. Rather, they are engineers who attempt a minimum-cost design given their expectations of climate change at the time of construction. IAMs seek to maximize social welfare, which is normally taken to be an increasing function of household consumption. Thus, an IAM might produce a scenario with lower consumption losses and higher adaptation costs compared to the non-stationary with anticipation scenario that we present in this paper. Consumption expenditures, and, therefore, living standards, are reduced because of expenditure for mitigation and loss & damage as well as from lower output due to damaged capital stocks. This is shown in Figure 6. In a stationary climate, even the most severe storms reduce consumption by no more than 0.5 percentage points of GDP. In a non-stationary climate, consumption losses can be substantial even when climate change is anticipated. We emphasize again that while our model includes forward-looking actors, they are not the social planners of the IAMs. Rather, they are engineers who attempt a minimum-cost design given their expectations of climate change at the time of construction. IAMs seek to maximize social welfare, which is normally taken to be an increasing function of household consumption. Thus, an IAM might produce a scenario with lower consumption losses and higher adaptation costs compared to the non-stationary with anticipation scenario that we present in this paper.

Discussion
We drew on standard methods in engineering risk assessment to develop a model for damage to commercial productive capital in the small island state of Barbados. Detailed impact models like ours are often used for local studies; such studies inform the damage functions used in the FUND model [9]. However, to our knowledge there is no bottom-up macroeconomic model that includes damage to productive capital. The closest we have found are the papers of Fankhauser and Tol [26], which introduced a temperature-dependent depreciation rate into different types of neoclassical growth models; and of Rezai et al. [10], which applied a greenhouse gas concentration-dependent depreciation rate in a post-Keynesian model.
The behaviorally-based damage model presented in this paper allows for greater flexibility than do temperature-dependent damage functions. Anticipating a changing climate or designing for current conditions produce different effective depreciation rates for the same change in global temperature, in contrast to Fankhauser and Tol [26], Moore et al. [6], or Rezai et al. [10]. Firms may rebuild damaged capital (as assumed in this paper) or suffer an extended loss of output. They may target a particular rate of growth, while damage costs are made up through lower consumption (as in this paper and FUND) or damages may be reflected in lower output and correspondingly lower saving and investment (as in DICE and Fankhauser and Tol [26]).
The simulation results presented in this paper provide a counterpoint to the climate damage models used in integrated assessment models. When running IAMs in an optimizing mode, the results are highly dependent on an assumed social discount rate [68][69][70][71][72][73]. The analysis in IAMs is normative: the investment trajectory maximizes social utility, taken to be an increasing function of consumption. In contrast, in this paper the analysis is descriptive. The discount rate is one that might be used by a private firm deciding between different investments. The simulation results produced by the model can be reviewed and critically assessed, and policy instruments chosen to make a socially desirable outcome more likely. For example, we assumed that capital of a given vintage would be rebuilt to its original specifications. Instead, it could be built to specifications of new investment, thus following the recommendation to "build back better" [74]. Alternatively, sufficiently

Discussion
We drew on standard methods in engineering risk assessment to develop a model for damage to commercial productive capital in the small island state of Barbados. Detailed impact models like ours are often used for local studies; such studies inform the damage functions used in the FUND model [9]. However, to our knowledge there is no bottom-up macroeconomic model that includes damage to productive capital. The closest we have found are the papers of Fankhauser and Tol [26], which introduced a temperature-dependent depreciation rate into different types of neoclassical growth models; and of Rezai et al. [10], which applied a greenhouse gas concentration-dependent depreciation rate in a post-Keynesian model.
The behaviorally-based damage model presented in this paper allows for greater flexibility than do temperature-dependent damage functions. Anticipating a changing climate or designing for current conditions produce different effective depreciation rates for the same change in global temperature, in contrast to Fankhauser and Tol [26], Moore et al. [6], or Rezai et al. [10]. Firms may rebuild damaged capital (as assumed in this paper) or suffer an extended loss of output. They may target a particular rate of growth, while damage costs are made up through lower consumption (as in this paper and FUND) or damages may be reflected in lower output and correspondingly lower saving and investment (as in DICE and Fankhauser and Tol [26]).
The simulation results presented in this paper provide a counterpoint to the climate damage models used in integrated assessment models. When running IAMs in an optimizing mode, the results are highly dependent on an assumed social discount rate [68][69][70][71][72][73]. The analysis in IAMs is normative: the investment trajectory maximizes social utility, taken to be an increasing function of consumption. In contrast, in this paper the analysis is descriptive. The discount rate is one that might be used by a private firm deciding between different investments. The simulation results produced by the model can be reviewed and critically assessed, and policy instruments chosen to make a socially desirable outcome more likely. For example, we assumed that capital of a given vintage would be rebuilt to its original specifications. Instead, it could be built to specifications of new investment, thus following the recommendation to "build back better" [74]. Alternatively, sufficiently extensive damage could lead to abandonment and migration [75]. Additional behavioral extensions could include insurance against climate damage and publicly-funded recovery efforts in the calculation of total costs in Equation (12).
A further contrast is between the use of specific moments of distributions (such as the mean) and a full distribution, as shown above in the model outputs. The importance of looking at the full distribution in climate damage studies was urged by Weitzman [69,76]. IAMs, when run in an optimizing mode, assume that agents choose an optimal expected future path over which expectations are calculated as the mean across different possible future states. The results in this paper make clear how misleading mean values can be when distributions are asymmetrical and have broad tails. In Figure 5, when climate damage is anticipated, mean GDP losses rise only slightly over the current value. However, losses at the 95% level (corresponding in the stationary case to a 20-year event) roughly double by 2050, suggesting a substantial probability of hardship arising from storm damage.

Conclusions
The dominant approach to computing climate damages in economic models is to use a temperature-dependent damage function. This has some advantages at global level, where damage estimates are aggregates over highly heterogeneous local impacts. However, they are less appropriate for local studies, where a wide variety of modelled climate variables may be available, such as the frequencies of extreme climate events, and it is possible and relevant to explore alternative behavioral assumptions.
Local studies are particularly needed for small island developing states (SIDS), which must contend with the compound uncertainties of heavy reliance on export markets and potentially rising climate damage. SIDS rely on capital-intensive export industries, and both the capital stocks and transport costs can be affected by tropical cyclones.
In this paper we drew upon the literature on engineering design and risk assessment to develop a model for damage to commercial productive capital and applied it to the small island state of Barbados. The model features behavioral variables that are not captured by temperature-dependent damage functions, such as anticipation of future climate change. We found that anticipatory behavior can substantially affect climate impacts on the economy.

Acknowledgments:
The authors would like to thank John Agard and Adrian Cashman for their guidance on the GoLoCarSce and Water-aCCIS projects, which provided the foundation for the work presented here, and for their encouragement to build on that foundation in the writing of this paper.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A. Estimating Parameters for the Wind Speed Model
Data on storms for the Eastern Caribbean extends from 1851 to 2010, while that for Barbados extends from 1855 to 2010. We note that a bias in historical data identified by Landsea [77] appears to have been corrected in the HURDAT database [78]. Thus, in contrast to Acevedo [33], we do not adjust historical wind speeds. For three out of 58 storms that reached Barbados, the peak wind speed recorded in Barbados exceeded that recorded for the Eastern Caribbean. Since Barbados is part of the Eastern Caribbean, we considered those to be recording errors and set the ratios equal to one. Otherwise, we used the ratio of the recorded peak wind speed in the Eastern Caribbean to that in Barbados. The average ratio, which is our parameter φ, we found to be 1.34, with a standard deviation of 0.41. This parameter is comparatively stable over time, despite an apparent rising trend. A one-sided, two-sample, Wilcoxon rank sum test for whether the mean before 1931 (the median year for storms in Barbados) is lower than after 1931 could not reject the null hypothesis of no difference between the means (p = 0.41).
We used count data of storms per year to estimate a Poisson model for storms with a peak wind speed of at least 40 mph (using the R fitdistrplus package ver. 1.0-11) for both the Eastern Caribbean and Barbados. Graphical representations of the fit for the Eastern Caribbean are shown in Figure A1 and for Barbados in Figure A2. The parameter estimate for the Eastern Caribbean is λ = 1.56 ± 0.20 (to two standard deviations), corresponding to a return period of 1.24 years, and for Barbados λ = 0.37 ± 0.10, corresponding to a return period of 3.22 years. To estimate the strike probability in Equation (21), we computed a Poission fit for the Eastern Caribbean for storms with wind speed w t = φ × 40.0 mph = 53.6 mph and compared the exceedance probability to that of a 40-mph storm in Barbados. The result is p s = 0.36.

Appendix: Estimating parameters for the wind speed model
Data on storms for the Eastern Caribbean extends from 1851 to 2010, while that for Barbados extends from 1855 to 2010. We note that a bias in historical data identified by Landsea [77] appears to have been corrected in the HURDAT database [78]. Thus, in contrast to Acevedo [33], we do not adjust historical wind speeds. For three out of 58 storms that reached Barbados, the peak wind speed recorded in Barbados exceeded that recorded for the Eastern Caribbean. Since Barbados is part of the Eastern Caribbean, we considered those to be recording errors and set the ratios equal to one. Otherwise, we used the ratio of the recorded peak wind speed in the Eastern Caribbean to that in Barbados. The average ratio, which is our parameter ϕ, we found to be 1.34, with a standard deviation of 0.41. This parameter is comparatively stable over time, despite an apparent rising trend. A onesided, two-sample, Wilcoxon rank sum test for whether the mean before 1931 (the median year for storms in Barbados) is lower than after 1931 could not reject the null hypothesis of no difference between the means (p = 0.41).
We used count data of storms per year to estimate a Poisson model for storms with a peak wind speed of at least 40 mph (using the R fitdistrplus package ver. 1.0-11) for both the Eastern Caribbean and Barbados. Graphical representations of the fit for the Eastern Caribbean are shown in Figure 7 and for Barbados in Figure 8. The parameter estimate for the Eastern Caribbean is λ = 1.56 ± 0.20 (to two standard deviations), corresponding to a return period of 1.24 years, and for Barbados λ = 0.37 ± 0.10, corresponding to a return period of 3.22 years. To estimate the strike probability in equation (21), we computed a Poission fit for the Eastern Caribbean for storms with wind speed wt = ϕ × 40.0 mph = 53.6 mph and compared the exceedance probability to that of a 40-mph storm in Barbados. The result is ps = 0.36. Next, we selected the peak wind speed in each year for the Eastern Caribbean. Wind speeds are not recorded for years without storms. Thus, there is no data for years when peak wind speed data fell below 40 mph, the minimum for a tropical storm. In those years we set the peak wind speed to a common value-the mean below the threshold-and fit a stationary GEV distribution (using the R package ismev ver. 1.42) that is truncated to the left at x = 0. We found the mean value below the threshold in an iterative procedure in which we: initialized the mean below threshold to one-half the threshold, fit the GEV, calculated the mean below the threshold with the fitted parameters, used that value for the next iteration, and iterated until the estimated mean below threshold converged to a tolerance of 10 -7 mph. This procedure found an estimated mean (non-storm) peak wind speed below threshold of 3.74 mph. Next, we selected the peak wind speed in each year for the Eastern Caribbean. Wind speeds are not recorded for years without storms. Thus, there is no data for years when peak wind speed data fell below 40 mph, the minimum for a tropical storm. In those years we set the peak wind speed to a common value-the mean below the threshold-and fit a stationary GEV distribution (using the R package ismev ver. 1.42) that is truncated to the left at x = 0. We found the mean value below the threshold in an iterative procedure in which we: initialized the mean below threshold to one-half the threshold, fit the GEV, calculated the mean below the threshold with the fitted parameters, used that value for the next iteration, and iterated until the estimated mean below threshold converged to a tolerance of 10 −7 mph. This procedure found an estimated mean (non-storm) peak wind speed below threshold of 3.74 mph. Diagnostic plots for the fitted GEV distribution are shown in Figure 9. As shown in the figure, the full GEV parameter set is needed to capture the behavior at high wind speeds (i.e., a Gumbel distribution would be inappropriate). The estimated parameters for the Eastern Caribbean are μEC = 60.1 ± 8.1 mph, σEC = 46.0 ± 5.9 mph, ξEC = -0.34 ± 0.11 (again to two standard deviations). This corresponds to a return period of 1.29 years for storms with the threshold peak wind speed (40 mph), in reasonable agreement with the Poisson estimate of 1.24 years. For the non-stationary GEV model we used as a covariate the global average sea surface temperature anomaly, τ, relative to the 1961-1990 average from the Hadley Climate Research Unit (https://www.metoffice.gov.uk/hadobs/). The justification is provided in the main text. Sea surface temperature does not drive non-storm peak wind speeds, so we expect peak wind speeds in the upper Diagnostic plots for the fitted GEV distribution are shown in Figure A3. As shown in the figure, the full GEV parameter set is needed to capture the behavior at high wind speeds (i.e., a Gumbel distribution would be inappropriate). The estimated parameters for the Eastern Caribbean are µ EC = 60.1 ± 8.1 mph, σ EC = 46.0 ± 5.9 mph, ξ EC = -0.34 ± 0.11 (again to two standard deviations). This corresponds to a return period of 1.29 years for storms with the threshold peak wind speed (40 mph), in reasonable agreement with the Poisson estimate of 1.24 years. Diagnostic plots for the fitted GEV distribution are shown in Figure 9. As shown in the figure, the full GEV parameter set is needed to capture the behavior at high wind speeds (i.e., a Gumbel distribution would be inappropriate). The estimated parameters for the Eastern Caribbean are μEC = 60.1 ± 8.1 mph, σEC = 46.0 ± 5.9 mph, ξEC = -0.34 ± 0.11 (again to two standard deviations). This corresponds to a return period of 1.29 years for storms with the threshold peak wind speed (40 mph), in reasonable agreement with the Poisson estimate of 1.24 years. For the non-stationary GEV model we used as a covariate the global average sea surface temperature anomaly, τ, relative to the 1961-1990 average from the Hadley Climate Research Unit (https://www.metoffice.gov.uk/hadobs/). The justification is provided in the main text. Sea surface temperature does not drive non-storm peak wind speeds, so we expect peak wind speeds in the upper For the non-stationary GEV model we used as a covariate the global average sea surface temperature anomaly, τ, relative to the 1961-1990 average from the Hadley Climate Research Unit (https://www.metoffice.gov.uk/hadobs/). The justification is provided in the main text. Sea surface temperature does not drive non-storm peak wind speeds, so we expect peak wind speeds in the upper part of the distribution to be more sensitive to sea surface temperature than in the lower part. Consistent with this assumption, Elsner et al. [79] found the upper quintiles of peak tropical storm wind speed to rise over time and with changing sea surface temperature, but not the lower quintiles. Accordingly, for this fit we again set the peak wind speed for years with no data to 3.74 mph, the value that we estimated for the stationary distribution.
Only the location parameter had a statistically significant correlation to sea surface temperature at the 5% level. Setting up a model with a temperature-dependent location parameter, the residual probability and quantile plots are shown in Figure A4. The estimated parameters are µ EC = (65.6 + 36.7 τ) ± (9.1 + 29.7 τ) mph, σ EC = 45.9 ± 6.1 mph, ξ EC = -0.37 ± 0.12. part of the distribution to be more sensitive to sea surface temperature than in the lower part. Consistent with this assumption, Elsner et al. [79] found the upper quintiles of peak tropical storm wind speed to rise over time and with changing sea surface temperature, but not the lower quintiles. Accordingly, for this fit we again set the peak wind speed for years with no data to 3.74 mph, the value that we estimated for the stationary distribution. Only the location parameter had a statistically significant correlation to sea surface temperature at the 5% level. Setting up a model with a temperature-dependent location parameter, the residual probability and quantile plots are shown in Figure 10. The estimated parameters are μEC = (65.6 + 36.7 τ) ± (9.1 + 29.7 τ) mph, σEC = 45.9 ± 6.1 mph, ξEC = -0.37 ± 0.12.