this post is written with Béatrice Cherrier (Research Director, CNRS-ENSAE / CREST)

The first lessons in insurance and financial mathematics address discounting and the value of time, borrowing Christian Gollier’s expression, because insurers must account for this temporal aspect in medium-term annuity calculations. But do these discounting calculations, used for centuries to reflect individual decisions (of policyholders, investors, companies), still make sense when used to guide public policy decisions with long-term consequences, like climate policies?

When Kenneth Arrow joined the IPCC team in 1993, he expressed this concern to the coordinator of certain chapters: discounting in climate economics is as necessary as it is controversial. He wrote: “Your outline is very complete, with one exception. There needs to be discussion of discount rates. To a considerable extent, suggested policies require present costs (reduced carbon consumption) to prevent future disutilities and costs. Clearly, the tradeoff between present and future is very important, controversial though it be” (Cherrier and Duarte 2024).

The history of this transfer of a mathematical tool from the individual to the collective dimension since the 1930s, summarized here, is rich with lessons.

• The Mathematics of Individual Behavior: A Brief History of Exponential Discounting

To recap, the theory of discounting stems from a relatively simple hypothesis: the rate of change in the value of a good must be proportional to its current value, that is y'=r\cdot y or \frac{dy}{dt}=r\cdot yThis differential equation has a fairly simple solution, of the form y(t)=y(0)\cdot\exp[rt]. In fact, this is the very definition of the exponential function: the only function that is proportional to its derivative. This is the “Cauchy problem,” proposed by Augustin-Louis, Baron Cauchy, born in 1789. This solution has the advantage of offering a form of temporal consistency, since: y(s+t)=y(s)\cdot\exp[rt]=y(t)\cdot\exp[rs]This formulation in “continuous time” is simply expressed in “discrete time” as: y(t+1)-y(t)=r\cdot y(t)or y(t+1)=(1+r)\cdot y(t)which generalizes to y(t+h)=(1+r)^h\cdot y(t)This is the famous discount formula, providing the value today of an amount in  h years. Proposing other approaches to discounting the future, as Richard Thaler did, leads to temporal inconsistencies.

Numerous experiments in economics and neuroscience (e.g., McClure et al. (2004)) show that decisions about future gains or losses seem better explained by hyperbolic rather than exponential discounting.

While the mathematical formalism (in continuous time) of discounting dates back to the French Revolution, discounting calculations go back at least to medieval Italy, with the work of mathematician Leonardo of Pisa (known as Fibonacci) in his Liber Abaci, as Charpentier (2016) recalled, at the turn of the 13th century. According to William N. Goetzmann, the problem posed in Liber Abaci “is the earliest known case in which the present value of two cash flow streams are compared. Although I have searched earlier surviving works in mathematics, I can find no direct precedent for it in mathematical history. The method of ‘net present value’ is the most important tool in modern finance.” Subsequently, this formula for discounting future cash flows y(t)=\frac{y(t+h)}{(1+r)^h}was found in writings by ecclesiastics in the late 16th century and then by merchants in the 17th century, as shown by William Deringer, and it was omnipresent in 20th-century commercial practices (Doganova 2024).

Formally, Irving Fisher formalized this approach by emphasizing that the interest rate is the link between the value of capital and the value of income. As he writes, it is the operator that allows one to “capitalize income,” meaning to derive the value of capital from the income it generates. To take one of his examples, if you hold a property right that entitles you to receive $104 in one year, the present value of this right, assuming a 4% interest rate, will be $100. If the property is the right to $100 in one year, its present value is “evidently” 1/1.04 or $96.2.

However, the price of temporal consistency is a loss of intuition. In 2001, Martin Weitzman made the following observation: suppose you consult two experts on the interest rate to choose, and they offer two different opinions, namely r_1​ and r_2​, with r_1<r_2​. Which value should you use? It might seem reasonable to choose the average value, namely (r_1+r_2)/2. But as Weitzman points out, this is incorrect. The rate must satisfy: \frac{1}{t}\log\left(\frac{1}{2}\exp[r_1t]+\frac{1}{2}\exp[r_2t]\right)(which is not constant). In the short term, the approximation r\approx(r_1+r_2)/2 is relevant, but in the long term, r\approx\min\{r_1,r_2\}=r_1<r_2. In other words, it is the lowest rate that should prevail. This counterintuitive rule is crucial when facing uncertainty about a long-term future.

But Weitzman writes in a context entirely different from Fisher’s. He is not trying to capture individual behaviors but rather to understand how we, collectively, should account for the risk of extreme climate events. Indeed, between the 1930s and the 2000s, discounting found its place in public decision-making models, from cost-benefit analysis to climate models.

• The Rise of Collective Discounting in Public Decision-Making: The Social Discount Rate and Ramsey’s Rule

Since the 1930s, discounting has been increasingly used in public decision-making models emerging among American engineers who wanted to determine water management projects (dikes, canals): cost-benefit analysis, in which future values are discounted. While this practice mirrored how, as we have recalled above, individual behaviors are mathematically modeled, debates arose about the relevance of this alignment as cost-benefit analysis was applied to all areas of state intervention. Should the discount rate reflect the government, the citizens, or a “collective”?

The concept of the “social discount rate” appeared in the 1950s and 1960s (Deringer 2024). But theorist Kenneth Arrow proposed a methodology that, by its rigor and clarity, gained widespread acceptance. He argued that market interest rates cannot be used as discount rates in public calculations because markets are imperfect. Instead, the discount rate should be mathematically derived from a model that rigorously describes the optimization of welfare over time. This is precisely what growth theorists proposed in the 1960s. Using a mathematical framework proposed in 1928 by British mathematician Frank Ramsey, a polymath who died at only 26, Arrow derived a formula that describes the social discount rate if the government makes optimal decisions:r=\rho+\gamma gwhere:

• \rho is the pure time preference rate or pure impatience rate: it reflects individuals’ intrinsic preference for immediate well-being over the future. A high \rho indicates a strong preference for consumption today rather than tomorrow. This parameter, in effect, reflects the value a society places on future generations. The higher it is, the less future generations are considered in decisions.
• \gamma is the coefficient of aversion to inequality or risk: it measures sensitivity to future well-being and aversion to inequalities between generations. A high \gamma indicates greater importance placed on redistribution and intergenerational equity.
• g is the expected growth rate of income or consumption in the future: it reflects the expected level of future prosperity. If future generations are expected to be wealthier due to sustained growth, a higher discount rate is justified (as future generations will need less of that wealth). Conversely, low or uncertain growth implies a lower discount rate, making future resources or benefits more valuable.

Arrow gradually convinced economists working on energy and climate issues to adopt his method. Over time, this equation would be called the Ramsey formula or rule. This is paradoxical and reflects the ambiguities associated with discounting in public decision-making models.

For Ramsey himself had not derived this formula in his 1928 article, merely using a discount rate in the last pages. He considered this practice “ethically indefensible… [it] arises merely from the weakness of imagination.” The choice of these parameters, particularly the preference for the future, therefore involves an ethical component. This is why many philosophers debated economists’ discounting choices starting in the 1970s. John Rawls, for example, extensively discussed intergenerational solidarity in A Theory of Justice, perhaps the most famous political philosophy treatise of the 20th century. He proposed the difference principle, where inequalities should benefit the least advantaged. However, he clearly indicated that such a principle could only apply within a single generation: “future generations have no means to improve the situation of the least fortunate first generation.” But then, how should the problem of justice between generations be treated? His idea was that “each generation must not only preserve the achievements of culture and civilization and maintain intact the just institutions that have been established, but also set aside an appropriate amount of real capital accumulation each period.”

• The Disputes on Discounting in Energy and Climate

Paradoxically, as the use of the Ramsey formula became more widespread, particularly in climate economics from the 1990s onward, disagreements about how to use the equation intensified.

The first issue concerns the time horizon. While growth economists, the first to use this formula, thought in terms of what might happen over the next 50 years, economists of the 1970s were abruptly compelled to consider horizons of 200, 500, or even 1,000 years: the long-term horizon. Following the energy crisis, governments tasked economists with producing models to guide decisions on the optimal energy mix, particularly regarding the role of nuclear power (Cherrier and Duarte 2024). However, nuclear waste management and its environmental impacts extend over hundreds of years. The application of the Ramsey formula to assess the links between the economy and the climate, beginning in the 1980s, confirmed the long-term stakes.

The inertia of the physical and biological systems governing the Earth’s climate is such that the consequences of policies adopted today will not be felt for another 50 years. The real question is what happens after that, as illustrated by the report on the economic consequences of global warming led by British economist Nicholas Stern, as well as the successive IPCC reports published every five years since 1990. These reports describe alternative scenarios spanning 50 to 200 years.

The publication of the Stern Report in 2006 sparked a controversy (known as the “Stern-Nordhaus controversy”) that highlighted a second point of contention regarding the Ramsey formula. The central figures of this controversy—the second being economist William Nordhaus, who received the 2018 Nobel Prize for his work integrating economic and CO₂ emissions models—both use the Ramsey formula. But the issue was no longer, as it was for Arrow, purely theoretical. It was about parameterizing the equation, deciding which value to use for this discount rate, as increasingly recommended by government reports (Quinet 2014).

Nordhaus and Stern disagreed, as Nordhaus had also disagreed, as early as the 1990s, with other climate economists. When he published his DICE model (Dynamic Integrated Climate-Economy model) in 1992, Nordhaus adopted a discount rate of 6–7% defined by the Ramsey formula. In contrast, William Cline, who simultaneously published one of the first books on climate economics, advocated for a much lower rate (1–2%), supporting more rapid and significant CO₂ emissions mitigation policies than Nordhaus recommended.

Nordhaus clarified these differences in the application of the Ramsey rule in subsequent years: “While this [Cline’s] approach is philosophically satisfying, it is inconsistent with actual societal decisions on saving and investment (…) it is essential that the discount rate be based on actual behavior and returns on assets rather than on a hypothetical view of how societies should behave or an idealized philosophy about treatment of future generations.”

As debates on parameterizing the Ramsey equation proliferated, another economist working on these issues, Alan Manne, asked Arrow to elucidate these disagreements in his chapter on discounting in the second IPCC report. The chapter highlighted an opposition between prescriptivists (like Arrow and Cline), who believed that selecting certain parameters for the formula should result from an ethical decision, and descriptivists (like Nordhaus and Manne), who argued that the discount rate should be based on observing economic agents’ behavior (e.g., market rates, surveys, etc.).

• Recent Research: Between Individual and Social Discounting

These debates on discounting in climate models reveal that the link between individual and collective discounting has not disappeared and is resurfacing. These hesitations are evident in a letter economist Tom Schelling sent to several colleagues, including Cline, Nordhaus, and Manne, during discussions about IPCC models (Cherrier and Duarte 2024).

What temporal preferences are we trying to model, he asked? Should we consider the temporal preferences of a single agent (often referred to as “representative”) or policymakers? In the case of a “representative agent,” why consider a single, constant rate, given that there is no theoretical or empirical justification (in psychology or physiology) to establish a solid basis for such reasoning? More complicated still, does it even make sense to consider such distant time horizons? No economic agent can discern between what might happen in 150 years versus 250 years.

While these reflections on the coherence of individual and collective discounting are theoretical and philosophical for Schelling, they are also fueled by a growing body of empirical research. Richard Thaler began testing individuals’ preferences for the future in laboratory settings in 1980, and by the mid-1980s, Maureen Cropper and David Laibson, among others, concluded that empirical discount rates decrease over time (a phenomenon known as hyperbolic discounting).

At the same time, Weitzman, Gollier, and others expressed dissatisfaction with how the Ramsey formula reflects individual and collective attitudes toward risk. Other economists sought to rethink the normative foundations of discounting (for example, by incorporating extinction risks), aligning more closely with philosophers who rejected the Ramsey formula or even the very idea that discounting is an appropriate way to consider the future.

Climate change forces us to think long-term while acknowledging that the future will be turbulent. The goal is not to predict the future but to prepare for it. The climate expected for 2100 is now forecast for next year! When the future accelerates, understanding how to account for it becomes essential to making timely, necessary decisions.

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