Underestimating Exponential Growth

"The greatest shortcoming of the human race is our inability to understand the exponential function." 
 
-Albert A. Bartlett

Exponential growth is often widely underestimated. Here are a few issues where many people make mistakes because they don’t understand the simple math of exponential growth.

  1. Economic Growth: When there are trade offs between efficiency and equality, many people today think that the taxes, regulation and redistribution are worth a slightly lower growth rate.  However, when this trade off is applied over a long time period, the results can be staggering. If the choice was made in 1870 to have more equality at a cost of 1 percentage point of growth a year, America in 1990 would be no richer than Mexico.
  2. Entitlement Spending and National Debt: As I have pointed out previously, the United States is headed for very high debt levels if entitlement spending is not reformed.  One very simple way to fix this is to index entitlement benefits to inflation and not income. The growth of the economy would make it easy to pay for a safety net at today’s living standards. Unfortunately, this would only work for Social Security and not Medicare as the medical system is structured in a way that leads to health care inflation greater than that of the real economy.  Additionally, there is another problem when the net national debt reaches 100% of GDP. If the market perception of the debt turns negative and nominal interest rates remain higher than nominal GDP growth, then there is no way for the economy to grow itself out of debt.  This is the current situation with Greece, and Japan isn’t doing too much better.
  3. Personal Finance and Pension Plans: If a prudent investor can make 10% real returns in a year, then they can turn 50 thousand dollars into over 1.6 million dollars after 35 years. This simple math explains how many of the rich people today consist of those who have saved and invested prudently. On the other hand, a supposedly fully funded pension fund planning on a world of 8% real returns that finds itself in a world of 4% real returns will find itself underfunded by over 75% 35 years later (In this case, the people making pension return assumptions are underestimating how much they matter, they just know that their books look better if they assume a higher return). Robin Hanson has been proposing that people don't give to the future because they don't care about it, but it may also be that they do not fully understand the impact of exponential growth*.
  4. Overpopulation and increasing Resource Consumption: Overpopulation does not seem to be the exponential problem that we once thought it was. Once become rich enough, their population growth rate slows down. The UK’s Ministry of Defense 2008 Strategic Trends report expects the population to level out at around 9 billion people between 2050 and 2100 (page 25).  While overpopulation is itself not a problem, the exponential economic growth of these emerging economies are coincident with an exponential increase in demand for resources and these limited resources present constraints on growth.

Having established that exponential growth rates are important, here is a handy rule of thumb that will give an intuitive understanding of exponential growth. To calculate the doubling time of an exponentially growing series, take 70 (or 69.3 to be exact) and divide it by the growth rate. This means that a 10% growth rate leads to a doubling every 7 years, a 7% growth rate is a doubling every 10 years and a 3.5% growth rate is a doubling every 20 years.


*It is also possible that someone who both cares about the future and understands exponential growth might think that there were existential problems for the current society that are significant enough to reduce the probability of a far future donation from ever paying off.

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7 responses
Regarding your asterisked comment, there is an obvious reason that those who "care about the future and understand exponential growth" don't donate via trusts that pay off in the long-distant future.

IRRs for many charitable projects, especially in developing countries, have been estimated in the 20% range. Whatever intuitive measure of value one uses, accepting these numbers would eliminate long-lasting trusts from consideration b almost all donors.

— Dikran Karagueuzian
Good point, they only have to believe those rather unrealistic estimates. Of course, giving to the far future also reduces the chance that the funds are spent rationally.
— Jeff Lonsdale
I believe the estimates are realistic, and that such charitable projects will continue to exist for a long time to come. As the Wall Street Journal wrote, donors are regularly faced with the choice of whether to use a single charitable dollar to cure an African suffering from a tropical disease, or buy Yale a pencil.

Most choose to buy Yale the pencil. Or, as Robin Hanson put it "Charity is not about helping."

For those donors for whom charity is about helping, there are many projects that would likely dominate giving to the far-distant future, and there will continue to be such projects as long as the vast majority of donors prefer give to other causes.

— Dikran Karagueuzian
Money can have a much larger impact in increasing someone's quality of life in the developing world if done correctly, but even when helping Africa charity is often more about signaling. I don't know about you, but when one of my friends is working for a charity to help people in Africa, the conversation about whether or not their organization is actually making a lasting and/or cost effective impact at the advertised rates (Save a child for just...) can be rather awkward.

In general, it is highly unlikely that most people who decide to donate money to help in Africa check to make sure that the necessary studies to measure that their method of giving has a real impact. They generally just do an intuitive check to make sure the program makes sense. Even when there is an attempt at a comprehensive study, they might not randomize or use control groups properly (The Millennium Village program seems to fall under this category).

Givewell has looked at a lot charities, and they found that there is a lot of feel good stuff that might not have much of an impact as well as a few proven ways to actually make an impact for a small amount of money, but not the vanishingly small amount claimed by many charities.

— Jeff Lonsdale
Funny how we agree on the facts but not on the conclusion.

I looked at the Givewell recommended charities, of which there are quite a few, and they all strike me as likely to have a tremendous impact, and (although it is not measured) some huge IRR.

My thesis is that the person who cares about the impact of his charitable giving, as opposed to the signaling value, is likely to both do the research and understand exponential growth. For such a person, micronutrient projects or whatever will dominate gifts to the far-distant future.

Thus the failure to give to the far-distant future is because charity is largely for signaling and not because people don't understand exponential growth.

Of course all of this is a side note to your main point, which is that huge mistakes are being made all the time by failing to understand exponential growth. I agree, and would add that huge mistakes are also made by incorrect estimation of future exponential growth.

— Dikran Karagueuzian
Nice facts, It seems like there are quite a few fundamental problems built into our economic system. Until we can turn these exponential growth curves into sustainable systems our efforts will be minimal.
— Andy Blanchard
Nice facts, It seems like there are quite a few fundamental problems built into our economic system. Until we can turn these exponential growth curves into sustainable cycles our efforts will be minimal.
— Andy Blanchard
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