Once In A Centillion Years

The 9.0 earthquake that shook Japan in March 2011 was six times larger than the most powerful Japanese earthquake ever measured, making it a statistical curiosity. If the probability distribution of earthquakes looked anything like that for people’s height, this quake would have been the equivalent of a 20-foot tall woman.

But certain probabilities, including those of earthquakes, stock returns, and war, deviate from the bell or normal distribution (with lots of minor events clustered near the center of the bell and just a few toward the edges) that we are all familiar with. These events are said to have “fat tails”: once in a while something very dramatic happens that doesn’t fit into our “normal” expectations.

How much should we spend to guard against these catastrophic events that probably won’t happen? A recent paper by William Nordhaus, “The Economics of Tail Events with an Application to Climate Change,” examines the policy implications of disastrous “tail events.” Answering this question accurately, Nordhaus concludes, depends on our risk preferences, the probability of an event, and society’s ability to apply new knowledge to the event’s prevention. But even then, coming up with a good response is challenging because conventional cost-benefit analysis is difficult to apply in the case of events with low-probability, high-loss consequences.

Nordhaus’ paper is motivated by Martin Weitzman’s Dismal Theorem, which explains this failure of cost-benefit analysis. Economist Paul Krugman, citing Weitzman’s work in his analysis of climate change for the New York Times Magazine, interpreted the Dismal Theorem to imply that “the non-negligible probability of utter disaster…should dominate our policy analysis.” For Krugman, the slightest chance of a future disaster due to climate change is sufficient cause for taking action now to curb emissions. The theorem suggests that the large uncertainties surrounding climate change make the case for reducing emissions today stronger, not weaker.

But Nordhaus warns that the conditions under which the Dismal Theorem holds are restrictive, and we should be skeptical of its relevance for actual policymaking. There are, he argues, good reasons to question its assumptions about how we value catastrophes. For instance, applying the Dismal Theorem might lead us to spend infinite amounts of money in order to avoid a catastrophe, no matter how small its probability. This, Nordhaus writes, is unrealistic:

For example, assume that there is a very, very tiny probability that a killer asteroid might hit Earth and that we can deflect that asteroid for a huge expenditure. The CRRA utility function [a common assumption about risk aversion] implies that we would spend virtually all of world income no matter how small the probability. That is, even if the probability were 10^-10, 10^-20, or even 10^-1,000,000, we would still spend most of our income to avoid these infinitesimally low probability outcomes (short of going extinct to prevent extinction).

Nordhaus points out that time and time again society has revealed its willingness to overlook such minuscule probabilities of catastrophe. Physicists run experiments in large particle colliders that, they report, create a tiny chance of Earth being reduced instantly to a black hole.  Yet, the experiments continue. Further, an asteroid of the size that extinguished the dinosaurs will hit Earth each year with a probability of 1 in 100 million, yet the U.S. government spends a paltry four million dollars each year tracking the threat.

Nordhaus also thinks that the Dismal Theorem doesn’t allow for enough societal learning. Some of the potentially catastrophic events associated with climate change, for example, may have fat-tailed distributions. He is sanguine, however, that as we learn more about humankind’s impact on the climate, we will have time to update policy and avoid the worst outcomes.

Nordhaus is not ready to dismiss Weitzman’s Dismal Theorem outright. Instead, he sees it as a warning to policymakers performing cost-benefit analyses: not so fast. Catastrophic events may not be the usual fodder for cost-benefit analyses. But when they are applied to extreme events like climate change they should be employed with care.