Our most important decisions—about slowing climate change, funding childhood education, eating ice cream—often require weighing present costs and benefits against future ones. Crucially, we are usually pretty uncertain about how much, if at all, to discount the future in favor of the present. A federal Interagency Working Group (IWG) is about to pick a single number to try to answer this complicated question, a "discount rate." Whatever number IWG settles on, the U.S. government will use it to decide if the future benefits of green energy investments are worth the present costs.
The point of this blog post is to show, using a bit of math, that picking a discount rate while ignoring our uncertainty about the number we've picked makes us systematically undervalue future costs and benefits. Ignoring uncertainty makes us short-sighted.
Ask an economist about weighing present and future benefits, and they will give you an answer like this: if you received $100 now, you could invest it at some rate of return "r" (for example r might be 0.03 or 3 percent) and in a year you'd have (1 + r)*$100 (in our example, $103). But if instead you just received $100 a year from now, well, you'd just have $100. With some hand-waving, the economist concludes that a dollar benefit is worth 1 + r times more if we get it now, instead of getting it a year from now. Or turning things around, that same future benefit is worth 1/(1 + r) times less if we get it a year from now, instead of now; and if we get it t years from now, the value is further reduced to 1/(1+r)t. Here, r is the "discount rate": the bigger it is, the more we discount future benefits when considering them in the present; and the further into the future the benefits will occur, the more we discount them.
Source: IWG, 2021
Now, even if we buy this story (and the sneaky shift from "benefit" in general to "dollars" in particular), there's a catch: we don't really know what rate of return our investments will actually deliver. At best, we might have a range of plausible values for r, some values perhaps more probable than others.
For example, suppose we believe there are three equally probable values for r: 0.01, 0.03, and 0.05—this is roughly the range that IWG is considering for discounting future climate damages. If you are a policymaker considering a cost of climate change (e.g. $1 billion of storm damage) that will occur in 50 years, you need some way to turn $1 billion in 50 years into a what it's worth today, the expected present value of that climate cost. How should you do it?
It turns out there are two ways to calculate an expected present value: 1) the easy way, and 2) the easy and correct way. Both are just a matter of taking an average.
The first way is to calculate the average of the plausible values of r, then plug that average into our present value formula. In our example, the average value of r is 0.03, so we get ($1 billion)/(1.03)50 = $230 million. This method is simple, but wrong: it's equivalent to pretending we know r will be 0.03, ignoring our uncertainty about that prediction. It's not clear whether IWG will end up taking an average, but they do seem to be trying to pick a single number to use in any given analysis.
The second way, the correct way, is to plug each plausible value of r into the present value formula, one at a time, then take the average of the resulting present values. In our example, this is ($1 billion)*(1/1.0150 + 1/1.0350 + 1/1.0550)/3 = $310 million. That's 80 million dollars higher, or 35 percent higher, than the mistaken estimate.
The difference between these two estimates comes from uncertainty: the first ignores uncertainty about the discount rate, and the second correctly accounts for it.
This is just an example; a more realistic calculation would use more than three possible discount rates, giving greater weight to the more likely discount rates. But the phenomenon we've illustrated — that ignoring uncertainty necessarily leads to an underestimate of present values —is extremely general. It comes from a theorem of probability called Jensen's Inequality, which applies regardless of the form the uncertainty takes.
Underestimating the present value of future costs and benefits could be catastrophic. Such myopia may lead to the erroneous conclusion that the long-run benefits of green infrastructure investments are not worth their present costs, when they are; or that present inaction on climate change is cheap, when it is not. Such myopia endangers long-run prosperity and survival. To avoid the dangers of short-sightedness, the first step is to admit to uncertainty regarding how much to value things that will happen in the future.
