where ECD (t) is the effective carbon dioxide concentration that would give the same radiative forcing as the actual concentration of carbon dioxide, methane, and other greenhouse gases; ECD (1765) is the pre-industrial value; ESO2 (t) is the emission rate of SO2 which is converted to sulfate (SO4) aerosols in the atmosphere; AFSO4 is the sulfate-aerosol radiative forcing in 1990; and g(t) is Gaussian-distributed noise with mean zero and standard deviation Oq (e.g., Hasselmann, 1976) which generates a red-noise-like temperature variability.

We represent our uncertainty about the climate system with a range of plausible values for the climate sensitivity, AT2x, and the sulfate forcing, AFSO4. We choose the range of climate sensitivities, 0.5°C < AT2x < 4.5°C, as in LSB. In practice we can find a best-guess estimate of the climate-sensitivity-dependent sulfate forcing parameter by using the EBC/UDO model to reproduce the instrumental temperature record (Andronova and Schlesinger, 2001). However, there are many sources of error, such as the influence of volcanoes and solar-irradiance variations, that could affect these estimates. Thus, we characterize our uncertainty about the sulfate forcing by the percentage it deviates from the best-estimate value as a function of climate sensitivity. For any pair of values for the sulfate forcing parameter and the climate sensitivity, we find the best estimate of the associated white-noise climate forcing, oq, by regressing our EBC/UDO climate model using each pair against the instrumental temperature record from 1856 to 1995.

We represent our uncertainty about the impacts of climate change using a simple, phenomenological damage function designed to capture, in aggregate, some of the impacts of climate variability and the ability of society to adapt to changes in variability. We write the annual damage in year t, as percentage of gross world product (GWP), as

where AT(t) is the annual global-mean surface temperature change relative to its pre-industrial value, and AT5(t) and AT30(t) are the 5-year and 30-year running averages of AT(t).2 The second term on the right-hand side of Eq. (3.2) represents impacts due to changes in the variability of the climate system that society and ecosystems can adapt to on the timescale of a year or two. The third term represents those impacts that society and ecosystems adapt to more slowly, on the order of a few decades, and thus are sensitive to both the year-to-year variability and the secularly increasing trend in temperature. The first term represents the damages due to a change in the global-mean surface temperature and is similar to the power-law functions used in most simple damage models in the literature, which we can view as representing impacts that society and ecosystems adapt to on century-long timescales. The coefficient ax represents the damages due to a 3°C increase in the global-mean temperature and the coefficient a2 and a3 represent the maximum damages due to climate variability in 1995 at the 90% confidence level.

We use a variety of empirical data to fix the damage-function parameters in Eq. (3.2). Because there are many gaps in the available information, and much of it that exists is heavily debated, we define a wide range of plausible parameter combinations rather than any best estimate. In LSBA we used only the requirement that the model be consistent with past observations to constrain the input parameters, though in general, we could also use future forecasts to generate constraints as well (as in the work on innovation described below). The wide range of plausible parameters supports, rather than hinders, our goals in this study since in the end we show that a simple adaptive-decision strategy can be robust against both very small and very large damages.

We constrain the parameters for the first term in Eq. (3.2) by noting that whatever damages due to climate change have occurred in the last few years and decades, they cannot have been more than a few tenths of a percent of GWP, otherwise we would have observed unambiguous evidence of damages

2 The N-year running average is given by A TN(t) = 2 AT(t).

to date. With AT5 (1995) = 0.5°C, we can write ax < 0.1%-6^, which corresponds to a range for the damage coefficient ax < 20% GWP for cubic damages. We constrained the parameters in the second and third terms using time series data on economic losses from large-scale natural disasters (Munich Re Reinsurance, 1997). This is an imperfect data source because these extremeevent damages (which in 1996, excluding earthquakes, caused $60 billion, 0.2% GWP, in damages) are due at least in part to trends in society's vulnerability to natural disasters rather than to any change in the size or frequency of natural disasters themselves and, conversely, are only one component of the damages due to climate change. Lacking better sources of available information, we used these extreme-event data to define three bounding cases: "Low Variability" with parameters (a2,a3,^2,^3,) =(0.2%,0%,1,na); "High Variability" with parameters (0.4%,0%,2,na); and "Increasing Variability" with parameters (0%,0.33%,na,3). (Note that the "Low" and "High" damages due to variability cases drop the third term in Eq. (3.2), while the "Increasing Variability" case drops the second.) The "High" and "Increasing" cases differ in that damages in the latter grow with increasing concentrations of greenhouse gases while the former do not. Thus damages due to climate variability can be affected by emissions-abatement policy in the latter case but not the former.

This damage model has important shortcomings. Among the most important, the white-noise forcing, the driver of the variability in our model, is fitted to the instrumental temperature record and does not change as we run our simulations into the future. Thus the damage distribution due the variability terms change over time only due to increases in the rate of change in the average temperature, though we expect that in actuality changes in variability are more important than changes in the mean (Katz and Brown, 1992; Mendelsohn et al., 1994). Nonetheless, the crude phenomenological damage function in Eq. (3.2) provides a sufficient foundation to support initial explorations of alternative abatement strategies and the impacts of variability on near-term policy choices.

We can now use this model to examine the potential impacts of climate variability on near-term policy choices. Figure 3.3 shows the difficulties an adaptive-decision strategy might have in making observations of the damages due to climate change. The thin lines show the damages due to climate change generated by Eq. (3.2) for two distinct cases. In the first there is a large trend but low variability. In the second there is a large variability, but no trend. The thick line in each case shows the trend a decision-maker (that is, a decision-making process that includes the making and reporting of scientific observations) might reasonably infer from the respective damage time series, calculated using a linear Bayesian estimator that rapidly detects any statistically

2000

2010

2020

2030

2040

Figure 3.3 Actual (thin lines with markers) and estimated (thick lines) damage time series for two plausible climate impact futures. The lower pair of time series has high damages due to variability and no trend (a = 0%). The upper pair has low damages due to variability and significant trend (aj = 3.5%). Both are calculated using the

1990

2000

2010

2020

2030

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Figure 3.3 Actual (thin lines with markers) and estimated (thick lines) damage time series for two plausible climate impact futures. The lower pair of time series has high damages due to variability and no trend (a = 0%). The upper pair has low damages due to variability and significant trend (aj = 3.5%). Both are calculated using the climate parameters (AT2x,AFS04,ctq) :

significant trend in the damage time series. The estimator does a reasonably good job of tracking the damage time series in both cases, but because of the high variability, the estimates for the trend and no-trend cases do not diverge until about 2020. Thus, an adaptive-decision strategy attempting to distinguish between these two cases based on observations of the damage time series would have to wait at least two decades before being able to act. While there are many cases that the estimator can distinguish more quickly, the question nonetheless remains, how can we design an adaptive-decision strategy that can perform successfully when the observations have the potential for such ambiguity?

In order to address this question, we posit a simple, two-period, threshold-based adaptive-decision strategy, similar to the one used in LSB. The strategy can respond to policy-makers' estimate of any trend in damages based on annual observations of the noisy time series D(t) or to any observed changes, neglecting noise, in abatement costs, represented here by the incremental cost of non-emitting capital. As shown in Figure 3.4, our adaptive strategies begin with a pre-determined abatement rate 1/Rj and can switch to a second-period abatement rate 1/R2 in the year, ttrig, when either the damages exceed, or the abatement costs drop below, some specified target values. The logistic half-life R represents the years needed to reduce emissions to one-half the basecase. The damage target (in % GWP) is given by Dest (t) > Dthres (Dest (t) is the decisionmakers' estimate shown in Figure 3.3) and the abatement-cost target (in $/ton carbon abated) is given by K(t) < Kthres. The second-period rate depends on the year ttrig. If ttrig < Tnear, then the second-period abatement is given by R2 = R2near.

Figure 3.4 Flow chart describing a set of simple adaptive-decision strategies designed to address climate variability.

Figure 3.4 Flow chart describing a set of simple adaptive-decision strategies designed to address climate variability.

If Tnear £ ttrig < Tfar, the second-period abatement is R2 = R2mid. If neither condition is met by the year Tfar, the strategy switches to a second-period abatement R2 = R2far. We express the decision facing policy-makers as a choice among the eight parameters defining these adaptive-decision strategies.

We now compare the performance of a large number of alternative adaptive-decision strategies. Figure 3.5a compares the expected regret of alternative strategies to the expected regret of the static DAL and ES strategies as a function of our expectations about the future. The horizontal axis gives the likelihood we ascribe to the possibility that DAL, as opposed to ES, is the better response to the climate-change problem. On the left-hand-side of the graph we are sure that DAL is better. Not surprisingly, the expected regret of the DAL strategy at this point is small ($0.2 billion/ year) while the regret of the ES strategy is high ($91 billion/ year). On the right-hand-side we are sure that ES is better, and not surprisingly the expected regret of ES is small ($0.9 billion/year) here and the expected DAL regret is large ($230 billion/year). In the middle of the graph, we ascribe even odds that DAL or ES is the best strategy.3

This figure is similar to the cube in Figure 3.1, except that we have collapsed our expectations about the future - the independent variables - into a single

3 Formally, we divide the states of the world into exclusive sets depending on whether DAL, ES, or drastic reductions (DR) is the better policy. (Drastic reductions eliminates anthropogenic greenhouse gas emissions over the first half of the 21st century.) We assign a probabiliy (1 — pES)(1 — pDR) to the states where DAL is better, ^ES(1 — pDR) to the states where ES is better, andpDR to the states where DR is better. The horizontal axis in Figure 5a spans 0% £pES £ 100%, with pDR = 0%. The static DAL and ES policies do not necessarily have zero expected regret atpES = 0% and 100%, respectively, because one of the adaptive-decision strategies may perform better across the DAL or ES states of the world.

x lu

x lu

Probability of ES Future

100%

Probability of ES Future

100%

Probability of Drastic Future

Figure 3.5 (a) Performance of alternative adaptive-decision strategies as a function of the probability ascribed to climate-change futures that require an "Emission-Stabilization" as opposed to "Do-a-Little" response. Strategies are labeled by a triplet representing rate of near-term abatement (in years until emissions are reduced by half), the estimated damages that will trigger a change of strategy (in % of gross world product), and the observed marginal cost of abatement that will trigger a change in strategy (in $/ton-carbon). (b) The most robust adaptive-decision-strategy as a function of expectations about the probability of a drastic future and of high and increasing damages due to variability.

Probability of Drastic Future

Figure 3.5 (a) Performance of alternative adaptive-decision strategies as a function of the probability ascribed to climate-change futures that require an "Emission-Stabilization" as opposed to "Do-a-Little" response. Strategies are labeled by a triplet representing rate of near-term abatement (in years until emissions are reduced by half), the estimated damages that will trigger a change of strategy (in % of gross world product), and the observed marginal cost of abatement that will trigger a change in strategy (in $/ton-carbon). (b) The most robust adaptive-decision-strategy as a function of expectations about the probability of a drastic future and of high and increasing damages due to variability.

dimension so that we can compare the relative cost of the strategies. There is another major difference. In Figure 3.1 we considered only one adaptive-decision strategy; here we examine the performance of thousands (5120 to be precise). Note that we have found adaptive-decision strategies that dominate the DAL and ES strategies no matter what our expectations about the future. That is, this chart shows that even with the potential of potentially ambiguous observations, adaptive-decision strategies are still the best response to climate change.

Of course, not all possible adaptive-decision strategies are better than ES and DAL. In fact, Figure 3.5a shows only a very small number of dominant strategies, those that perform best compared to the others as a function of expectations about the future. For each of many points across the horizontal axis we find the strategy with the lowest expected regret and label each of these dominant strategies with the three parameters - the rate of near-term emissions reductions, the damage threshold, and the innovation threshold -(Rj,.Dthres, Ehres), that are most relevant to policy-makers' near-term decisions. As our expectations increase that ES, as opposed to DAL, is the better strategy, we pass through five different adaptive-decision strategies, with Rx ranging from infinity to 40 years.

Two interesting patterns emerge as one examines this set of best adaptive strategies as a function of expectations about the future. First, there is a tradeoff between the rate of near-term emissions reductions and the confidence one should require in observations of damage trends before acting on them. That is, in the face of variability in the climate system, policy-makers can choose a response threshold for observed damages that can compensate, to a greater or lesser extent, for any choice of near-term emissions-reduction target. Second, the rate of near-term emissions reductions depends most sensitively on expectations about the future, while the aggressiveness with which society ought to respond to innovations are the least sensitive. That is, the policy choice at the focal point of the current negotiations may be the most controversial component of a robust strategy, in part because stakeholders with different expectations will have the most divergent views as to the proper target, while the least controversial components may be at the periphery of the negotiations.

Figure 3.5a also shows the most robust strategy across the range of expectations, which we find by searching for the strategy with the least squares regret across the points on the horizontal axis. This most robust strategy, given by (60 yrs, 1.2%, $65), begins with moderate emissions reductions, is relatively insensitive to observations of climate damages, and is very sensitive to observations of decreasing abatement costs. Its annualized, present-value expected regret is $22 billion/year, independent of our expectations about the ES future. While this low number does not mean climate change is costless (the regret measures the difference between the performance of a given strategy and that of the best strategy for that state of the world), it does suggest that even faced with deep uncertainty, we can find adaptive-decision strategies that perform nearly as well as the strategies we would choose if we knew the future with clarity.

The adaptive-decision strategies shown in Figure 3.5a still contend with a rather narrow range of uncertainty. For instance, in making this figure, we have assumed equal likelihoods of each of our variability cases - low, high, and increasing - that sulfate emissions are low, as in the scenarios of the Special Report on Emissions Scenarios (SRES; Nakicenovic et al., 2000), and zero probability of climate impacts so severe as to require immediate and rapid phase-out of anthropogenic greenhouse gas emissions. In Figure 3.5b, we examine the sensitivity of the most robust strategy to these assumptions. To make this figure, we find the most robust strategy as a function of: (i) our expectations that immediate and draconian emissions reductions (we call this the Drastic Reductions strategy) are the best response to the threat of climate change and: (ii) the probability that the damages due to climate variability are high or increasing. Our finding that the most robust adaptive-decision strategy has moderate near-term emissions and is more sensitive to observations of abatement costs than to observations of damages, is true in the shaded region on the left of the figure where the likelihood of drastic future ranges from about 0% to 12%, depending on the likelihood of low climate variability.

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