When thinking about converting values from the distant past or the distant future to the present, the actual values themselves are often considerably less important than what is called the “discount rate”–that is, the annual percentage rate at which you do the conversion from past or future to the present. An example of converting future values into “what it would be worth today” is the debate over the costs that climate change may impose decades or a century into the future. An example of converting a past value into “what it would be worth today” is the effort to put a value on the costs imposed on American slaves before the US Civil War.

In general, the problem that arises is applying a percentage growth rate over long periods of time can lead to counterintuitive conclusions. As one example, consider the old story that the Dutch purchased Manhattan from the local Native American tribe for $24. Here, let’s set aside the details about the actual transaction, and whether it was in the form of guilders or trade goods. (For details, this useful article argues that the purchase price was equal to about 3-4 months wages for a skilled artisan in Holland, or what would have been paid for 30 beaver skins.)

For my purpose, the key question here is: “What was $24 (or whatever the relevant amount is) that was paid 400 years worth in today’s currency?” For simplicity and clarity, let’s say that the Dutch paid an amount that we will just set equal to 100.

Well, if we apply an interest rate of 0% over the intervening 400 years, that amount of 100 paid for Manhattan would still be 100 today.

If we apply an interest rate of 1% per year over the intervening 400 years, that amount of 100 paid for Manhattan would be (about) 5,300 today.

If we apply an interest rate of 3% per year over the 400 years, that amount of 100 paid for Manhattan would be (about) 13.6 million today.

And if we apply an interest rate of 6% per year over 400 years, that amount of 100 paid for Manhattan would be about 1.3 quadrillion today.

This example should illustrate that in thinking about converting the amount paid for the island of Manhattan 400 years ago to a modern value, the actual amount paid is almost irrelevant to the conversation. The original amount paid 400 years ago could be one-tenth as much or 10 times as much, but what really makes the difference in the conversion to a modern-day value is the annual interest rate you choose to apply. In addition, the example emphasizes that relatively small differences in interest rates, even just a percent or two per year, can lead to really dramatic differences in value when compounded over long periods of time.

Consider a more hot-button example: What is the value of the wages not paid to American Slaves from 1776 to 1860, converted to a modern value? Thomas Cramer carried out a set of calculations under various scenarios. In one of this scenarios, as a commenter notes, projecting his estimate of historical costs forward to 2009 at an annual rate of 3% leads to a total value of $14.2 trillion; however, projecting the same costs forward to 2009 at an annual rate of 6% leads to a total value of $7 quadrillion.

For perspective, US GDP is now about $26 trillion. A proposal to pay slavery reparations of $14 trillion is thus at least at the edge of conceivable, if spread out over time. However, a cost of $7 quadrillion for slavery reparations would require paying the equivalent of all of current US GDP for the next 260 years, which is not conceivable in practical terms.

Again, the choice of how to translate a value from the distant past into present-value terms is not primarily about the historical cost, but about the interest rate that one chooses to adjust the past value to the present. As Cramer wrote in his 2015 article: “Debt estimates over long periods of time are extremely sensitive to the choice of interest rate … Thus, settling on an interest rate would likely represent the most important topic of political negotiations in any reparations debate.”

The example of translating a future cost into a present cost which comes up most often these days relates to climate change issues. Back in 2006, the very eminent Nicholas Stern published what became known as the “Stern review” on The Economics of Climate Change. Soon after, the also very eminent William Nordhaus published a review of the book. Stern and Nordhaus are both prominent economists arguing that the risks of climate change are large and real and need policy action. However, they differ in the discount rate they would apply to future harms: Stern argued for a discount rate of 1.4%, while Nordhaus argued for a rate of around 4.5%.

The percentage difference may seem small, but remember that these are annual percentage rates, so when projected out over long periods of time, they make a huge difference.

If you are calculating how much to pay in the present to avoid a cost of $1 that will be incurred 100 years from now, and you apply an annual rate of 1.4%, the answer is about 25 cents. That is, $0.25 x (1 + .014)¹⁰⁰≈ $1.

However, if you apply an annual rate of 4.5%, then the answer of how much to pay in the present to avoid a cost of $1 that will be incurred 100 years from now is about 1.2 cents. That is, $0.012 x (1 + .045)¹⁰⁰≈ $1.

The Trump administration proposed applying a discount rate of 7%. With that discount rate, then the answer of how much to pay in the present to avoid a cost of $1 that will be incurred 100 years from now is about one-tenth of a cent. That is, $0.001 x (1 + .07)¹⁰⁰≈ $1.

Notice that even if there is no difference at all in the estimates of the future costs of climate change a century from now, the choice of discount rates leads to dramatic differences in how much we should be willing to spend in the present to prevent those costs. The 1.4% rate implies spending 20 times as much as the 4.5% rate. The 7% rate suggest spending close to nothing.

The Environmental Protection Agency has recently been updating its estimates of the “social cost of carbon”–that is, the cost of emitting a ton of carbon into the atmosphere when all future environmental costs are taken into account. As I wrote in an earlier post: “The current EPA estimate of the social cost of carbon is $190 per metric ton, using a 2% discount rate. But to give a sense of how much the discount rate matters, the cost estimate would be $120/ton with all the same underlying estimates and a discount rate of 2.5%, but $340/ton with all the same estimates and a discount rate of 1.5%.” Notice how much difference moving the discount rate by just half-a-percent makes!

To repeat the earlier lessons, the choice of how to translate a value from the distant future into present-value terms is not primarily about the estimates of future cost, but about the interest rate that one chooses to adjust the future values back to the present. Even small differences in the interest rate that is used will lead to dramatic differences in the policy prescriptions.

So what discount rate is the correct one to choose? This question is both of central importance and brutally hard. As a useful starting point, The most recent Annual Review of Financial Economics has a four-paper symposium, along with an introduction, on the issues of choosing an appropriate discount rate. The papers are:

• “Introduction to the ARFE Theme on the Social Discount Rate,” by Deborah Lucas (pp. 115–125)
• “The Social Discount Rate: Legal and Philosophical Underpinnings,” W. Kip Viscusi (pp. 127–145)
• “Fixing Our Public Discounting Systems,” by Frédéric Cherbonnier and Christian Gollier (pp. 147–164)
• “Missing Participants, Missing Markets, and the Social Discount Rate: Borrowing Constraints, Intergenerational Transfers, Altruism, and the Desire for Legacy,” by Andrew Caplin and John Leahy (pp. 165–184)
• “Reflections on What Financial Economics Can and Cannot Teach Us About the Social Discount Rate,” by Deborah Lucas (pp. 185–195)

I can’t hope to review the papers here. Instead, I’ll just offer a list of themes and questions that seek to highlight some of the issues involved.

1) The choice of an appropriate social discount rate is a bear-trap of complexities. It will depend on interest rates, technological change, judgements about risk, choices about intergenerational fairness, and other factors. The choice might reasonably differ across specific issues.

2) Those who are focused on costs imposed in the past often prefer to apply a high discount rate–so that when projecting costs from the past up to the present, these past costs will look larger. Conversely, those who are concerned about taking steps in the present to avoid costs that may arise in the future tend to prefer a low discount rate–so that the costs from the future will look larger in the present, as well. Taking these together, a common implication that the present generation should bear both high costs for long-ago past injustices and also high costs for long-distant future benefits.

3) Thinking about questions of fairness in the present–what do those who have more money and wealth owe to those with less money and wealth–is difficult. Questions of fairness and equity become even harder when they reach across multiple generations. Almost none of the people who would be affected by climate change in 100 years are even born yet. Those who suffered deep and grave injustices a century or more ago are dead and gone.

4) A common belief is that social costs should be imposed more heavily, where possible, on the better-off. Thus, an issue with fairness choices across multiple generations is that, for the last couple of centuries, incomes have been rising in high-income countries. Given the long-term real growth rate of 2% for the US economy, the average person from the the present has about 7 times as much income as someone from 100 years ago–and the present also has a much longer life expectancy and access to a much wider array of technologies. It’s likely that people who are living 100 years from now will have much higher incomes, life expectancies, and technologies as well. Thus, imposing costs on the present generation for the benefit of those living 100 years from now will, in all likelihood, be a transfer from the relatively poor to the relatively rich.

5) Basing the discount rate on the interest rates might work fairly well when costs are, say, a decade or two in the past or a decade or two in the future–but then not work so well when the past or future events are many decades or centuries in the future. Lucas writes in her paper in the symposium:

For policies with long-term impacts, intergenerational concerns become paramount, projections of cash flows and discount rates become highly uncertain, and present value calculations are an intrinsically unreliable measure of value. No approach to discount rate selection can overcome those problems; alternative decision criteria need to be established. However, most government investments involve much shorter horizons, and the adoption of standard approaches to risk adjustment could significantly improve social welfare.

7) In thinking ab0ut how to project future costs back to what we should be willing to spend in the present, it seems important to remember that climate change is not the only issue here. Public resources are limited, and a standard guideline is to spend where the social rate of return, broadly understood, is highest. For example, how much should society be willing to spend now on young children to improve their changes of being healthy and productive adults some decades in the future? How much should we be willing to spend now to reduce the risks of future pandemics (naturally occurring or human-created, or nuclear wars, that might arise in future decades? How much should society be willing to spend now to reduce the solar storms, a supervolcano, or risk of the Earth being hit by a wayward comet some decades in the future?

8) I have occasionally seen interest groups who support aggressive spending against climate change argue that the idea of a discount rate is more-or-less just an excuse for inaction, and the appropriate discount rate for the future is zero percent. Frankly, I’m not sure these folks have thought through the implications of their position. A discount rate of zero implies that all future costs should be weighted as if they are occurring in the present. But is it really worth spending an equal amount to save the life of an existing person as it is to spend that amount to save the projected life of someone living 50 or 100 or 1000 years from now? We can have reasonable arguments about what discount rate makes sense in what context, but a zero discount rate that wipes out all distinctions between present, near future, and distant future is not sensible.

7) For a sense of the difficulties that arise if you wipe out the distinctions between present and future costs, consider the court case back in 2004 about storage of nuclear waste. Government regulators argued that they had projected 10,000 years ahead, and the risks of storing nuclear waste over that time interval were acceptable. The court ruled that looking “only” 10,000 years ahead wasn’t enough. Even a tiny positive discount rate will compound, over 10,000 years or more, to a point where the present value of taking action will be essentially zero. But the court was, in effect, arguing for a discount rate of zero percent. (Viscusi discusses this issue in the symposium.)

8) The expected path of future technology should affect one’s willingness to spend in the present as well. Imagine that we knew there was a high chance that new technology would provide batteries for electric cars in the next 5-10 year that used many fewer scarce resources, as well as being longer-lasting and easier to recharge. In that scenario, pushing everyone to buy today’s technology would be wasteful. Indeed, as Viscusi points out, if one assumes a discount rate of zero for the future, it can, perhaps counterintuitively, make sense to defer immediate spending on future problems, and instead use the better technology that will emerge.

9) For the record, none of this discussion is intended to express an opinion on appropriate current policy for climate change. However, I will say that, in my own mind, some of the strongest arguments for taking action to reduce fossil fuel emissions have to do with immediate and near-term costs of air pollution on human health. As the environmental economists say, there could be a “double dividend” of reducing the use fossil fuels, in both immediate and long-term ains.

10) In thinking about how to project past values to the present, it also seems important remember that the current topic under discussion is not the only event. If one thinks about past American injustices, for example, certainly slavery would loom large. But treatment of, say, Native Americans would also loom large, as would discriminatory events and practices against other ethnic groups and against women. One might then need to add other economic, social, and legal, and even foreign policy injustices. What the present owes to the past is a wide-ranging topic. It’s fine to argue that one might apply different interest rates to different examples. But

10) My own bias, for what it’s worth, is that achieving some understanding the past is enormously important. But I’m mostly not a fan of taking long-ago past events and turning them into present day conflicts. There are a lot of places around the world where some groups of people continue to bear deep and even violent grudges against other groups of people for injustices that occurred between these groups literally centuries ago. My preference is for current generations to be given a clean slate. In effect, I would argue for a very low or perhaps even a negative interest rate to be applied to the past.

11) For the record, my unwillingness to project costs of the distant past into the present doesn’t mean that one should ignore problems of the present! For example, some prominent proposals that the US should pay black Americans some form of reparations for slavery have shifted away from basing their argument on costs from the past, projected up to the present. Instead, their argument for reparations is based on the current black-white wealth gap, or the current socioeconomic gap between families of black and white children. This approach has the advantage of avoiding the social discount rate altogether. It also shifts the discussion, in a subtle way, from being directly about “reparations for slavery” to focus instead on current inequalities.  

12) There is a way for current generations to spend money, and then have future generations repay the loans. As a thought experiment, imagine a form of deficit spending in which no repayments are made for, say, 20 or 50 or 100 years. (The realistic version of this borrowing is that we just keep rolling over debts and interest as they grow, decade after decade.) Will the generations that eventually face repaying these debts be glad that we borrowed this money, or not? If the borrowed funds are spent on infrastructure, human capital, and long-term environmental preservation, and future generations are better-off than we are today, then such borrowing will look like a wise choice. But from the standpoint of 20 or 50 or 100 years into future, our announced intentions with regard to the future will not matter much. They will be able to judge our decisions by the actual consequences.

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