Bounded uncertainty and climate change economics

Abstract

It has been argued recently that the combination of risk aversion and an uncertainty distribution of future temperature change with a heavy upper tail invalidates mainstream economic analyses of climate change policy. A simple model is used to explore the effect of imposing an upper bound on future temperature change. The analysis shows that imposing even a high bound reverses the earlier argument and that the optimal policy, as measured by the willingness to pay to avoid climate change, is relatively insensitive to this bound over a wide range.

The central question in the economic analysis of climate change policy concerns the degree to which current consumption should be reduced to mitigate future consequences of climate change. In answering this question, two issues must be addressed: how to discount losses incurred far in the future and how to deal with substantial uncertainty about these losses. Much of the discussion stimulated by the recent Stern Review on the economics of climate change (1) revolved around the first of these issues (2, 3). This paper focuses on the second.

Although there remains little if any uncertainty about the reality of anthropogenic climate change, substantial uncertainty surrounds key aspects of it. For example, under one emissions scenario from the most recent assessment by the Intergovernmental Panel on Climate Change (IPCC) (4), there is a two-thirds probability that mean global surface temperature will increase by 2.4–6.4 °C by 2100 with a remaining one-third probability that the increase will fall outside this range. The corresponding range of welfare effects and society's willingness to pay to avoid them is wide, and there is a need to balance the risks of under- and overreacting.

The standard economic calculus for decision-making under uncertainty rests on consideration of the expected discounted utility. Under this calculus, in the face of risk aversion, a premium accrues to policies that avoid the risk of high-cost outcomes even when they are highly improbable. In an influential paper, Weitzman (5) pointed out that analyses that do not explicitly incorporate uncertainty about future climate change fail to account for this risk premium and thereby understate the benefits of aggressive policies that avoid future risk. Further, using a previous theoretical result (6), Weitzman showed that the combination of a heavy-tailed probability distribution for temperature change and a common model of risk aversion implies that the risk premium for avoiding climate change is infinite. Although the practical policy implications of this result are unclear, it clearly calls into question analyses pointing to a moderate policy response.

A necessary (but not sufficient) condition for Weitzman's result is that the upper tail of the distribution of temperature change is unbounded. This paper explores the effect on the optimal policy of placing an upper bound on temperature change. Weitzman argued against this kind of truncation, claiming that the truncation point perforce must be arbitrary and that the results therefore would be highly sensitive to this arbitrary choice. As discussed later, we disagree with the first of these points, and we show that, whether the choice of truncation point is arbitrary or not, the second point is not correct.

In related work, Newbold and Daigneault (7) evaluated the risk premium of responding to climate change under different scenarios. To avoid the pathological result of Weitzman, they truncated the distribution of temperature sensitivity, defined as the equilibrium temperature response to a doubling of atmospheric CO₂, using 100 °C as a base case. The focus of the present work is on the sensitivity of the optimal policy to this truncation point.

Results

We begin by sketching out the model under which the analysis will proceed. This model is stylized. Stylized models are used in economics, atmospheric science, and other fields to develop and sharpen qualitative insights. For example, a stylized model of climate feedback was used by Roe and Baker (8) to explain the difficulty of reducing uncertainty about temperature sensitivity. The model outlined here mirrors the general set-up of Weitzman and others.

Let denote the temperature increase at time s in the future. We assume that

Under this model, temperature increases linearly until time , at which point temperature levels off at a final increase of . We will assume that is known and focus on uncertainty in . If we adopt a Bayesian perspective, this uncertainty can be expressed through a probability density function , as discussed later. Let be consumption at time s, normalized so that current consumption is 1. In the absence of temperature change, consumption increases at rate g. However, increasing temperature adversely affects consumption, for example, through effects on human health, natural hazards, or ecosystem services. We assume that the proportion of consumption retained at time s is so that consumption at time s is given by

Economists measure the welfare that society gains by consumption through a utility function. Among other things, the shape of the utility function determines the degree of risk aversion held by society. Risk aversion implies a willingness to pay to reduce uncertainty over future climate change impacts beyond their expected value. Here, we adopt the standard constant relative risk aversion utility function . Under this model, the percentage change in utility associated with a fixed percentage change in consumption is independent of the level of consumption. When , utility takes the limiting form log C.

The uncertainty in propagates to utility. Let be certain utility at time s for fixed . The expected discounted utility is given by

where is the pure rate of time discounting (9). The inner integral represents discounted utility for fixed , and the outer integral averages this discounted utility over the distribution of . The absent bounds on the outer integral correspond to the range of over which is positive.

A central question concerns the willingness of society to pay to avoid the expected loss in utility associated with climate change. Following Weitzman (5), one way to measure this willingness is by the fraction of consumption that society would be willing to forego in perpetuity to avoid this loss in utility. Let

be the certain discounted utility if consumption is reduced by a factor at each time and climate change is avoided (i.e., ). What we refer to, for convenience, as the optimal value of is found by equating EU and . That is, represents the maximum fraction of consumption that society would be willing to forego to avoid the loss of consumption caused by climate change.

The model outlined above includes an economic component and a climate component. On the economic side, we take g = 0.015 and and, as a base case, . The specification of and is discussed by Dasgupta (9) in the context of climate change policy. Turning to , the IPCC (4) reported a range for the fractional loss in consumption of 1–5% for a warming of 4°. As a base case, we take , giving a 3% consumption loss for a 4° warming. The Stern Review (1) referred to the possibility of economic losses of up to 20% of GDP. For this value of , this level of loss would occur with a warming of around 11°.

On the climate side, we calibrate the base case by referring to the A1F1 scenario of the IPCC (4). Under this scenario, which is the most pessimistic considered by the IPCC, the “best estimate” is a warming of 4 °C between the baseline period 1980–1999 and 2090–2099 with a likely range of 2.4–6.4 °C. To capture this scenario in a rough way, we take to be 100 years in the future and assume that has a Cauchy distribution, shifted to have a mode at 4.4 °C and truncated on the left at 0 °C. The Cauchy distribution, which corresponds to the Student t distribution with 1 degree of freedom, is the canonical heavy-tailed distribution. For this distribution, the density of which is shown in Fig. 1, the probability that lies between 2.4 °C and 6.4 °C is ≈0.75, which is slightly larger than the value of 0.67 given by IPCC (4). In the sequel, we will truncate this distribution at high levels of . This truncation has minimal effect on this probability.

Fig. 1.

A modified Cauchy distribution of temperature increase (τ) with mode at 4.4 °C and truncated on the left at 0 °C. This distribution is truncated on the right at values of between 20 °C and 50 °C and renormalized to give the probability density function used in Eq. 3.

As guaranteed by the results in ref. 4, when is unbounded on the right, essentially implying that . However, the picture changes dramatically when is truncated, even at a very high level. This scenario is shown in Fig. 2, where is plotted against the truncation point for in the range 20–50 °C. The optimal value of increases slowly from around 0.01 at to around 0.03 at . Although interest here centers on the sensitivity of to and not on its actual value, the range of values of in Fig. 2 is broadly consistent with most mainstream economic analyses, lending a further degree of support to the base case calibration of the model.

Fig. 2.

The maximum fraction of consumption that society would be willing to pay to avoid the loss of consumption resulting from climate change as a function of the maximum possible temperature increase with g = 0.015, for and (solid line) and 0.0032 (dashed line).

The relative insensitivity of to extends to other combinations of and . For example, Fig. 2 also shows results for , which is essentially the value used in the Stern Review, and , corresponding to a 5% consumption loss at a 4° warming and a 20% loss at a warming of 8.4°. Only in the case does begin to increase relatively rapidly with , and then only when exceeds 40°.

It is notable that, for the higher value of , eventually increases rapidly with only for the higher value of . In general, controls the utility tradeoff between consumption by the rich and by the poor. The higher is, the more willing is society to divert consumption from the rich to the poor. The results of the Stern Review, which did not allow for highly improbable but highly costly climate change, depended on the adoption of a low value of to induce the relatively poor present to reduce consumption for the benefit of the relatively rich future. Here, however, as increases, highly improbable but highly costly climate change becomes possible, so that the future actually may be poorer than the present. In this case, the logic of the Stern Review is reversed, and a high value of is needed to induce a larger diversion of consumption from the present to the future.

The results presented here suggest that, under a reasonably calibrated model, the optimal level of response to climate change, as measured by the factor , is relatively stable over a wide range of upper bounds on the uncertainty about future warming. This finding is important for climate policy, because it implies that there is no need to establish a precise upper bound. To be sure, if the consumption loss caused by warming increases much more rapidly with temperature, this result could be overturned. In this regard, although the model of consumption loss used here is in broad accord with the literature, its extrapolation to very large temperature changes may be problematic. Along the same lines, for fixed, the distribution for used here has the heaviest tail within the truncated Student t family. Other things being equal, the adoption of a different member of this family would reduce the value of for all values of .

Discussion

In ref. 5, the heavy tail of the distribution of temperature sensitivity (defined as the equilibrium response of mean global surface temperature to an increase in radiative forcing equivalent to doubling of atmospheric CO₂) arises from the adoption of the standard reference prior distribution with improper heavy-tailed density . The posterior distribution resulting from the combination via Bayes's Theorem of this prior with any likelihood satisfying very mild conditions inherits this heavy tail. In other words, no amount of learning that supports a low value of can eliminate this heavy tail and avoid an infinite risk premium in avoiding climate change. Although the use of reference priors can be appealing on the grounds of objectivity, it can also give rise to pathologies such as this one (10). In contrast, as suggested above, if this problem is eliminated by truncation, then a premium attaches to continued learning about future climate change and its impacts on society.

Technical statistical issues aside, the question remains whether it is scientifically reasonable to bound uncertainty about future climate change. A convincing argument can be made that it is. Temperature sensitivity can be constrained by empirical studies of the actual temperature response to changes in radiative forcing in both recent (11–13) and historical (14–16) periods (see ref. 17 for a review). With the exception of one pair of studies (18, 19) that explicitly ruled out greater than ∼8 °C, these studies have not addressed directly an upper bound. However, it is fair to say that their results consistently rule out values of as high as those considered here.

Finally, we have focused here on bounding uncertainty about future climate change on the grounds of scientific reasonableness. As noted, however, this approach is not necessary to reverse the result in ref. 5: The result also is reversed if the upper tail of the distribution of future temperature change declines at a rate faster than polynomial. Thus, in qualitative terms, the results presented here do not require attaching zero probability to the event that future temperature change exceeds a fixed threshold.