# The Inescapable Implication of Uncertainty

In a previous post, we saw that uncertainty is not your friend. In a nutshell, if there is uncertainty, things could be worse than anticipated as well as better.

The previous post encapsulated this in a figure, which I show again below because it serves as a departure point for our next installment of our uncertainty analysis:

The figure shows synthetic distributions of climate sensitivity (if you don’t know what that is, begin by reading the previous post). The means of the four distributions in the above figure are identical but their standard deviations differ considerably, from .49 in the top left to 2.6 in the bottom right. The spread of each distribution characterizes the extent of uncertainty surrounding the mean estimate of 3 degrees.

The consequences of this increase in uncertainty are as obvious as they are inescapable: As uncertainty increases, the probability of a truly catastrophic outcome (defined as climate sensitivity exceeding the highly conservative 5°C mark; indicated by red lines in the figure) increases from a small .07% in the top left panel to a concerning 14% in the bottom right. In other words, increasing the standard deviation of our distribution fivefold, from .5 to 2.5, increases the likelihood of catastrophe by a factor of 200. (The word “catastrophe” has been over-used in the climate arena, but a 5°C increase would be truly catastrophic by most accounts, and most climate scientists are very concerned even about the possibility of considerably lower increases.) See update below.

The first conclusion about the climate system therefore has to be that the greater the uncertainty, the greater the potential for catastrophe.

That said, another aspect of the above figure appears to offer comfort at first glance. As uncertainty increases, from the top left to the bottom right panel, the proportion of the distribution towards the lower end of sensitivity also increases as a trivially necessary consequence of keeping the mean constant. The proportion of low estimates (< 2°C) reaches a whopping 42% when the uncertainty is greatest (bottom-right panel in the figure). This seemingly invites an alternative interpretation: With nearly half of all estimates of sensitivity below the ostensibly-safe “guardrail” of 2°C, perhaps one could legitimately ignore the upper tail, however fat it gets with increasing uncertainty? See update below.

Whether this gamble is advisable might at first glance appear to be a matter of subjective preference—some people may legitimately think that a 40%+ chance of being safe outweighs a 14% likelihood of catastrophe. As it turns out, however, we can push our analysis further and show that the lower end of the climate sensitivity distribution does not offer the comfort it implies at first glance.

## From Sensitivity to Cost

To understand the reasons for this, we must first relate climate sensitivity to the likely damage associated with climate change. Common sense dictates that greater sensitivity translates into greater cost: For example, if sensitivity is low and sea levels rise by only 2 cm, damage will be relatively minimal (i.e., we lose a few meters of beach). If sensitivity turns out to be higher, and sea levels rise by 60 cm, the damage is considerable (i.e., we need to build dams and levees or move people out of harm’s way, all at great cost).

What is even more important than the fact that damage cost increases with climate sensitivity is to ascertain the functional form of that increase: Will a .5°C increase of sensitivity from 2.5°C to 3°C incur the same additional cost as an increase from 3°C to 3.5°C? What about an increase from 4.5°C to 5°C?

It turns out that the precise form of this damage function is subject to debate. However, what does not appear to be subject to debate among economists is the fact that the damage function is convex (e.g., Nordhaus, 2010; Tol, 2011; Weitzman, 2010). “Convex” means that the rate at which damages are increasing with increasing climate sensitivity is itself increasing. This is illustrated in the figure below, using a highly convex (quadratic) function for illustrative purposes.

Consider first the top panel of the figure (Panel A). The panel itself contains three smaller panels, and the largest one in the top-right quadrant displays the cost function used for this example. The horizontal panel at the bottom shows the climate-sensitivity distribution from before, with the mean (3°C) highlighted by a vertical blue line. The left vertical panel shows the distribution of expected damage costs.

The damage-cost distribution was obtained by taking each observation in the sensitivity distribution and “reflecting” it onto the vertical axis using the convex damage-cost function. Units on the damage-cost axis are omitted because the figure is not seeking to convey actual dollar values (although economists believe that they can compute those values for future anticipated warming).

The most important message from the above figure arises from comparison of panels A and B.

The only difference between the two panels is the degree of uncertainty associated with climate sensitivity: The mean sensitivity is identical, but the spread (standard deviation) of the sensitivity distribution is greater in Panel B (standard deviation 2.5) than in Panel A (standard deviation .5).

Now consider the consequences of increasing uncertainty on damage costs: Although mean sensitivity is the same across both panels, the average expected damage increases with uncertainty—the mean damage in Panel A is lower than in Panel B. The comparison is made easy by comparing the lengths of the two vertical double-headed arrows, which point to the mean damage in each panel. It is quite clear that the arrow is longer—representing greater expected cost—in Panel B than in Panel A.

In a nutshell, if we expect X°C warming, the expected damage cost associated with that degree of warming is a function not (only) of X but also of the uncertainty associated with our estimate of X—and the greater that uncertainty, the greater the expected damage.

Not only is uncertainty not your friend, but greater uncertainty translates into greater expected loss.

## And There is More

And this is just the beginning, because there are a few more points worth making about the above figure: First, not only does increasing uncertainty about climate sensitivity increase the mean expected damage cost (i.e., best mean prediction), but it also increases the uncertainty around that expected damage cost—and that increase in uncertainty is particularly dramatic. Of course, uncertainty surrounding expected damages is highly relevant because it must be taken into account when assessing the total expected risk from climate change. To illustrate, whereas expected (mean) damage increases by “only” around 50% between the two panels, the associated standard deviation (uncertainty) of the damage increases 10-fold.

Another point worth making about the figure is that greater values of climate sensitivity likely translate into quicker evolution of the climate, all other things being equal (e.g., Bahn et al., 2011, Fig. 2). In other words, greater uncertainty about sensitivity (Panel B) not only translates into greater expected damage, but that damage is also likely to arrive sooner rather than later because the rate of temperature increase is greater with greater sensitivity. (This must not be confused with the fact that greater sensitivity may entail a longer asymptotic response time of the climate system; e.g. Hansen et al., 1985. That is, with greater sensitivity the climate system warms more quickly, but because it ultimately reaches a higher temperature, getting to that asymptote may take longer than with lesser sensitivity.)

This is an important point to bear in mind because if the greater damage were delayed, rather than accelerated, economists could claim that its absolute value should be temporally discounted (as all economic quantities typically are; see Anthoff et al., 2009). But if greater damage arrives sooner, then any discounting would only further exacerbate the basic message of the above figure: Greater uncertainty means greater real cost.

To sum up, uncertainty is no one’s friend. Greater uncertainty means things can be worse than you think. And greater uncertainty means you’ll pay more for the damages arising from climate change than if there were less uncertainty. In fact, you may end up paying much more than anticipated.

Uncertainty is no one’s friend.

The next post in this series we will examine how uncertainty affects the likely cost of mitigation.

Update 28/3/12: It has been drawn to my attention that the 5°C cutoff for an outcome to be labeled “catastrophic” was too conservative; that is, temperature increases considerably less than that would be associated with outcomes that most people would consider catastrophic. Conversely, limiting temperature increases to 2°C may not be “safe.”

I do not necessarily disagree, but those issues are not central to the point made here: Wherever one places a cutoff above or below the mean, the implications of the fat tails are identical, and it does not matter where exactly the “catastrophe” lurks. The crucial fact is that greater uncertainty translates into greater likelihood of catastrophic (or bad or terrible) outcomes, all other things being equal.

## References

Anthoff, D.; Tol, R. S. J. & Yohe, G. W. Discounting for Climate Change Economics: The Open Access Open Assessment E-Journal, 2009, 3.

Bahn, O.; Edwards, N. R.; Knutti, R. & Stocker, T. F. Energy policies avoiding a tipping point in the climate system Energy Policy, 2011, 39, 334-348.

Hansen, J.; Russell, G.; Lacis, A.; Fung, I.; Rind, D. & Stone, P. Climate Response Times: Dependence on Climate Sensitivity and Ocean Mixing Science, 1985, 229, 857-859.

Nordhaus, W. D. Economic aspects of global warming in a post-Copenhagen environment Proceedings of the National Academy of Science, 2010, 107, 11721-11726.

Weitzman, M. L. What is the “Damages Function” for Global Warming – and What Difference Might it Make? Climate Change Economics, 2010, 1, 57-69.

Tol, R. S. J. The Social Cost of Carbon Annual Review of Resource Economics, 2011, 3, 419-443.