With climate change, doing the math matters

Mark Buchanan

sometimes say that there’s no need to rush in addressing climate change — that our constantly increasing wealth will eventually make the problem relatively inexpensive to solve. Thanks to some careful mathematical research, that argument is looking increasingly wrong.
Placing a value on something that may happen 50 or 100 years from now isn’t easy. Economists typically rely on market interest rates, taking a historical average and projecting it into the future. If, for example, $10 would compound to $100 over the next century, then they’ll estimate that avoiding $100 of climate-related damage in 2116 is worth just $10 today.
Unfortunately, that approach can lead to some big errors. To understand why, imagine a simple bet. You put up $10, which I’ll then multiply by one of two numbers, 0.6 or 1.5, selected randomly for 30 rounds. If the resulting amount is more than $10, you win — and you get to keep the money.
This might look like a good deal for you. Each of the two numbers, 0.6 and 1.5, should come up a roughly equal number of times, averaging to 1.05. So on average, you would get whatever you put in times 1.05 — a 5 percent return on each round. Over 30 rounds, $10 would grow to about $43.
Except that’s not how it works. In reality, you never experience that 5 percent average gain, only either a 40 percent loss or a 50 percent gain on each occasion. Say you win the first round, leaving you with $15 ($10 times 1.5). If you lose the next, you’ll find yourself with only $9 ($15 times 0.6) — a loss of 10 percent on your initial investment. Over time, as a recent paper by two physicists demonstrates, you should actually anticipate losing about 5 percent per round. After 30 rounds, you’ll be lucky to have $2 left.
It’s the multiplication of factors through time that leads to this counterintuitive result. Of course, this is precisely also how the growth of value through compound interest works — factors over shorter times multiplying together to give the result over longer times. This insight has some big implications.
Consider the long-running squabble between American economist William Nordhaus and British economist Nicholas Stern over the value of actions to mitigate climate change. Nordhaus, using conventional market methods, estimates that the future benefits should be discounted at an annual rate of 4 percent or more, making $100 of savings in 2116 worth $2 or less today. Stern, employing an approach that focuses more on the well-being of future generations, argues that the proper discount rate is just 1.4 percent, making the present value of action at least 10 times greater.
Various academics, including Princeton’s Angus Deaton, have weighed in over the years. More recently, a group of economists and physicists applied the math from our betting game to the problem. Looking at data on interest rates from 14 nations over a century — and, crucially, accounting for the way rates fluctuate from year to year, and even turn negative at times — they found the proper discount rate to be much lower than the Nordhaus estimate, and much more in line with Stern’s.
Economists have had great influence in convincing people to think about climate change in terms of discount factors and cost-benefit analysis. A deeper understanding of the math, though, suggests that we should be wary of such naive optimization exercises — and recognize the need to act.

Mark Buchanan, a physicist and science writer, is the author of the book “Forecast: What Physics, Meteorology and the Natural Sciences Can Teach Us About Economics.”