How economists model the future, Season 1, Episode 1
The uncertain origins of the Ramsey Formula for discounting
This post is part of a series on the history of how economists model the future with the Ramsey formula, based on joint work with Pedro Garcia Duarte. Full Paper here
Part of economists’ role is to model the aggregate consequences of individual choices that extend into the future. To do so, some of them use a mathematical calculation: they discount future flows by dividing them by an interest rate, effectively making present and future values comparable by assuming that tomorrow’s streams of money are worth less than today’s. This practice, as historians show, has centuries-old roots: exponential discounting, which has the nice property of being time consistent, has been around among late 16th century clergymen, 17th century merchants, 19th century foresters, and has been ubiquitous in 20thcentury business practices.
After World War II, discounting became intertwined with cost-benefit analysis in the public sector, one shaping decisions on which investment to pursue. Academic economists increasingly joined the fray, engaging in longstanding controversies over how to choose the crucial discounting rate. The stakes grew higher as economists were tasked with modeling even more distant future. To illustrate: 1000$ cost or benefit in 20 years is worth $360 today at a 5% discount rate, $149 at 10%. Extend this a 100 years, and your great-grandkids receiving or paying $1000 is worth a mere $8 today at 5%, and peanuts at 10% (a term I took from Solow). By the 1990s, academic economists involved in policy work had settled on a specific formula for calculating the discount rate.
This formula (see picture below) posits that the discount rate equals the sum of two components. First, the "pure rate of time preference," representing impatience or how much the present generation is favored over future ones. Second, the product of the per capita consumption growth rate and the elasticity of the marginal utility of consumption, which measures the additional utility derived from each extra unit of consumption (sometimes linked to risk or inequality aversion). The second term often accounts for the assumption that future generations will be wealthier.
The formula has been widely used in the past decades. It featured prominently in the 2nd IPCC report, published 1995, in which economists participated in numbers unseen since. Crowning the chapters introducing several models to evaluate the costs of mitigating greenhouses gas emissions were methodological chapters on cost-benefit analysis, social justice, and a dedicated chapter on discounting which opened by pointing to economists’ consensus in using the Ramsey formula.
A decade later, the formula became the centerpiece of the famed Stern-Nordhaus controversy. At its core was the question: should the 'pure rate of time preference' be derived from economic parameters or ethical considerations? There’s a ton of literature analyzing these disagreements. Less attention has been paid to Stern and Nordhaus’s agreement to disagree within the formula’s framework.
The formula was also extensively discussed in the revised US Office in Management Budget’s (OMB) cost-benefit analysis guidelines proposed in the Spring of 2023. The draft garnered thousands of comments, with some suggesting that the formula “is a better guide to prescriptive discounting rather than descriptive discounting” since it’s “inherently linked to the growth rate.” Consequently, the final guidelines, published November 2023, largely omitted discussions of the “model-based approaches that endogenize the discount rate,” hinting at growing challenges to the formula.
Despite its prominence, the formula's exact origins and path to widespread adoption in economic work remain unclear. Since it’s named after a genius British mathematician who tragically died at 26, Frank Ramsey, many economists believe that it “has been in the literature since 1928.” That is, that we took the formula from the 1928 article in which Ramsey models how much a nation should save.
But as a happy few know, the Ramsey formula doesn't appear in Ramsey's paper. And the why highlights the nature of economists’ enduring struggles with discussing discounting. Ramsey wrote down a model of a stationary economy (without growth), with an objective function that maximized an intertemporal sum of utilities over the indefinite future – a key reason for the paper’s fame. Should the future utilities be discounted was the next question. No, Ramsey answered. For if individuals are known to discount the future (because of myopia, selfishness, or else), societies at large should not consider that the “unborn” count for less: “it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination,” he famously wrote.
However, Ramsey the mathematician recognized a problem: without discounting, the objective function, an infinite sum, might not converge. The no-discounting raised tractability issues. He addressed this by assuming a maximum obtained utility, which he called “bliss,” and rather sought to minimize the distance between bliss and current utility. Later in the paper, he turned to considering a single individual, and reintroduced discounting. A foundational ambiguity, if not ambivalence.
While Ramsey provided the toolbox from which the discounting formula could be derived, he didn't formulate it himself. When did economists start doing so, and how did it become a framework for discussing the choice of a discount rate in water resources, public investment, development, energy and eventually, climate models? These are the question Pedro Garcia Duarte and I have spent some time in the archives answering. We found that Ramsey’s struggles with theoretical consistency, ethical considerations and tractability constraints form the backbone of the next 70 years of discounting debates. Against a backdrop of evolving economic and social crises, economists continually debated which of these 3 aspects to prioritize. As it gained traction, the status of the Ramsey formula evolved from an optimality condition derived in a growth model to a definition and a framework for empirically parametrizing models.
The following series of 10 posts, in which I’ll unpack what we found, is a writing exercise, aimed at (1) weeding out the details and clarify the core narrative in our companion 16 000-word paper; (2) addressing my longstanding struggles with writing; (3) testing whether history of economics is more digestible in shorter formats and (4) keeping me sane and focused throughout the chaotic French election round. I’m sole responsible for the mistakes, approximations and interpretations in these posts.
Next (S1E2). “The number game”: discounting struggles in the 1950s and 1960s”
This will be fun, Beatrice. I remember working through the Ramsay paper some years ago -- not even knowing enough to realize how little I understood of it -- but still being struck by the no-discounting stipulation not making sense to me (as no-growthers' arguments don't make sense to me today) when the whole paper seemed to scream discounting. I couldn't tell whether his moral objections to it were serious or just a Cambridge-humor way of bracketing an issue until a framework was in place. It also seemed to me that later economists referred to "Ramsey model" only out the belief that the Arrow-Debreu-McKenzie findings justified their taking the discounted part of the paper as the whole show. A "representative agent," under the A-D-Mck restrictions, could stand for the whole economy.
Now you're going to make me read this paper again! Sigh. :-)
Ms. Cherrier, I congratulate you on this valuable and highly enjoyable series on the Ramsey Formula for discounting. I am reading each post in this series with great interest, and learning a lot -- thank you! Some of the ancillary material you have included is also quite delightful. The letter from Koopmans, Nordhaus and Litan (as part of their CONAES effort) to Ken Arrow is just mind blowing (at least to me, it is!). I always stood in awe of the great Arrow, but now I think of him as a god!
Just one small quibble about this post. You write -- "To illustrate: $1,000 cost or benefit in 20 years is worth $360 today at a 5% discount rate, $149 at 10%." When I perform the appropriate math, I get $377 at a 5% discount rate; you are right, at a 10% rate, I do get $149 as well. Am I missing something, or did you just make a boo-boo with the $360?
Please keep up the good work!