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# Einstein on Wall Street, a Time-Money Continuum: Mark Buchanan

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July 28 (Bloomberg) -- What is the value of time? This question was once a matter for philosophers such as Plato or Aristotle. Today economists claim to know the answer. The future, they say, is “discounted” because the value of having something or some amount of cash is greater than the value of having that same thing or amount of money a year from now.

How much so? Well, if you can put \$96 today into an investment earning 4 percent interest, it will be worth almost \$100 in a year. So \$100 a year from now is worth only \$96 today.

This way of thinking about future assets can make a big difference if we look at things more complicated than cash, especially if we try to assess their value many, many years into the future.

Pretend, for example, that we want to see if it is worth creating a costly marine sanctuary that could take many years to establish and even longer to effectively protect a population of endangered salmon or cod. The costs of the sanctuary come mainly in the first years -- the expense of setting it up and policing it, as well as perhaps millions of dollars a year in lost fishing revenue. The most important benefits, on the other hand, may come in the distant future. Even if the fish population might increase 100-fold, creating a sustainable fishing industry 500 years from now that is far more valuable than today’s, economists would discount the advantages coming in those distant years almost to nothing.

Discounting 4 percent for 500 years in a row means dividing the future value by a number close to 500 million. So those future benefits contribute virtually zero in the calculation to determine if eventual benefits justify the upfront costs.

## Calculating a Payoff

This so-called exponential discounting -- reducing the value of something by a fixed percentage for each unit of time -- is standard practice in economics. It comes into play whenever people consider investing for long-term payoff, whether by building railroads for high-speed trains or reining in carbon emissions to preserve the climate. And it discounts the distant future especially drastically. This is why economists and others often squabble over the right annual percentage to use -- should it be 5 percent, 7 percent, 1 percent? Change this a little, and values change a lot.

Some economists and philosophers have even argued that in cases of environmental preservation 0 percent is the only ethically defensible number. That is, no discounting at all. But this debate over percentages may actually be a distraction from a more serious problem with the formulas used in discounting.

That’s the message of a recent paper written jointly by John Geanakoplos, an economist at Yale University, and Doyne Farmer, a physicist at the Santa Fe Institute, in New Mexico. They argue that economic discounting as currently practiced is logically incorrect and, as a result, the cost/benefit analyses done by such authorities as the International Panel on Climate Change and the U.S. Environmental Protection Agency may drastically undervalue the future.

Economists like exponential discounting because it seems “rational”; in particular, it discounts equal periods of time equally. The standard analysis also assumes that the discount rate remains constant. That assumption is rather peculiar, Geanakoplos and Farmer point out, given that interest rates bounce up and down all the time -- and the interest rate at any moment should be closely linked to the discount rate, to reflect how cash investments gain value through time.

Revising the assumption of a never-changing discount rate leads to results totally at odds with current economic practice, Geanakoplos and Farmer have shown. To understand their argument, consider the next half-century. Year by year, the true discount rate (which no one knows precisely) will probably fluctuate in some complicated way, following one of many possible up-and-down paths. Since we don’t know the future, to determine the effective discount rate for the 50-year period, we should average over all possibilities.

## Counting the Paths

That’s simple enough, but here is where things get interesting: In calculating this average, some paths turn out to contribute far more than others. In particular, paths that descend into relatively low rates and stay there for many years have a disproportionate effect -- a path at 1 percent for 50 years, for instance, counts 20 times as much as a path running along at 7 percent. Change 50 to 500 years, and the difference becomes 10 trillion times.

This demonstrates how simple thinking about the future can lead to terrific mistakes. When something fluctuates, we often suppose we can use the average rate over time. And sometimes this works. The amount of food you will eat over 20 years, for example, will be roughly equal to 20 times what you ate last year, because your appetite doesn’t fluctuate that much. But averaging to get a true effective discount rate isn’t so easy. Some of the paths of fluctuation -- the lower paths -- carry extraordinary weight, and hence dominate the outcome.

Not surprisingly, Geanakoplos and Farmer find that the correct formulas for discounting over long periods don’t follow the textbook exponential form. The math is tricky (I’ve put some discussion of the technical stuff on my blog. But the consequences are not. Using a standard model from finance for interest rate movements (with an average rate of 4 percent), the authors show that, for the first 100 years or so, their correct form of discounting gives results that are similar to those that come from traditional calculations. But at 500 years the standard exponential discounts the future not just a little too strongly, but a million times too strongly. And it gets worse after that.

Going back to the example of the marine sanctuary, and using the Geanakoplos-Farmer formula, you find that the present value of benefits 500 years from now gets multiplied millions of times compared with the standard analysis. A thriving marine ecosystem in the future, linked with a much larger fishing industry, might well be worth investing in today.

In effect, today’s standard economic methods make the distant future count for almost nothing. And those who always thought this seemed hopelessly naïve turn out to be right.

The important point, Geanakoplos and Farmer emphasize, is that everything about discounting depends on what assumptions you make about how variables fluctuate with time. A fixed interest rate doesn’t work. And neither does the strong exponential discounting to which it leads -- even though most economists continue to use it.

This is a prime example of what John Maynard Keynes meant when he said the ideas of economists “are more powerful than is commonly understood.” Economists calculating the value of the future to perform cost-benefit analyses that influence how we take care of our world have been making some astronomically large mistakes.

(Mark Buchanan, a theoretical physicist and the author of “The Social Atom,” is a Bloomberg View columnist. The opinions expressed are his own.)