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.

Aging and Saving

The Consumer Expenditure Survey by the BLS just released an interesting article highlighting their 30 years as a continuous survey.  I decided that it could be interesting to see what their data combined with Census demographic projections imply about the future trend in savings. One way to do this is to break down the savings rate by age and project it forward.

Years: 2004-2008

All consumer units

Under 25 years

25-34 years

35-44 years

45-54 years

55-64 years

65-74 years

75 years and older

Average Savings Rate









As the savings rate obviously wasn’t 17.5% over the past few years, it should be noted that the CES implied measure of savings is rather different than the personal savings rate data calculated by the BEA.

In 2004, the CES changed the way it calculated income for incomplete survey responders, so the savings bias is calculated as an average of 2004 through 2008, which is about 15%.

The analysis assumes that the savings rate of each group will remain constant, and combines CES data with the US Census projections with constant net immigration to predict the future path of savings rate. The census cohorts are adjusted by the relative amount of consumer units because they suffer the same discrepancies as household data, with fewer young consumer units per capita.  This savings rate is adjusted downward by about 15% to be comparable to the BEA’s personal savings rate data.

The results are interesting in their mildness. The savings rate is only expected to drop a little over 1% due to demographic factors.  Of course, this study is biased for many reasons.

  1. The accuracy of the using the aggregate consumer expenditure data without adjustment is questionable.
  2. Social security and other retirement benefits are assumed to exist at levels comparable to the past few years. This biases the savings rate of the old cohorts higher compared to a likely scenario in which government and pensions have solvency issues due to the increasing dependency ratio. 
  3. The educational and racial composition of the projections is held constant when the savings behavior of these cohorts is actually rather variable.
  4. Wealth effects are completely ignored.

Introductory Post

The goal of this blog is to focus on long term problems while avoiding the pessimistic bias by recognizing the positive impact of human ingenuity compounding over long time periods

I’ve free ridden off of some of the very insightful analysis of the blogosphere for some time. This blog is my attempt to get some of my ideas out there and free ride off of the insightful commenters.

Below are a few of the themes that I think are important:

1. The time preference of the average person and average policy maker

There is a problem with the excessive focus on the short term. The existence of this focus is well documented in behavioral economic studies of hyperbolic discounting.  Policies that have tradeoffs with benefits in the short term vs. large expenses in the long term are often favored.  To take one example, the current state and local pension problems occurring all around the United States exist in part because the time preferences of politicians are far too short term.  Further more, demographic pressures mean that Keynes’s famous quote “The long run is a misleading guide to current affairs. In the long run we are all dead.” is changing to “In the medium run we are all dead” for more and more voters as the population of the developed world ages.

2. Globalization and its impact on the middle class of the developed world

Globalization is unequivocally increasing the wealth of this world. Basic comparative advantage proves that when trade is opened up, the pie grows. However, the inefficient producers of goods produced by new trading partners do often lose out. In this day and age, those inefficient producers tend to consist largely of the middle class of the developed world. The upper class has capital that is benefiting from globalization while the lower class has service jobs with manual labor that can’t be replicated by someone working for much cheaper when they are halfway across the globe.  The interesting question is whether the benefits of cheaper goods outweigh the costs of slightly worse jobs for the middle class.  Even with globalization, total compensation has been trending upwards for labor, people just don’t notice it because a lot of compensation goes towards labor costs and many simplistic commentators like showing the inflation adjusted household median income chart that leaves out noncash forms of compensation.  

Source: US Census and BLS

The other question is whether there is any policy that could have prevented these harms to the middle class short of attempts to keep the developing world undeveloped.  Even if the answer to that is no, this does not mean that there will not be a counter productive political attempts to reverse recent trends.

3. Resource constraints

Peak oil is the main resource constraint that comes to mind, but there are many long term issues with resources that are not being adequately addressed. Peak oil generally raises the cost of substitutable goods, making other forms of energy more expensive. This bleeds into other assets such as raising the cost aluminum, which is produced in an energy intensive fashion.  Water constraints are another important resource limitation on growth.  Other resources like rare earth elements or industrial metals may be more expensive due to short term supply and demand imbalances driven in part by the developing world. What makes peak oil more interesting than other basic resource problems is that the daily oil supply is running up against geologic constraints while other resource shortages generally last only as long as it takes for new mines to be built and brought online.

4. The entitlement mindset

A lot of people feel like they are owed a lot of things by society. They also believe that the average person shouldn’t be paying for these things. This is a trend that cannot continue.  Also relevant to this theme is the long term performance of countries with lots of foreign aid compared to countries that are less dependent on foreign aid.

5. Status driven trends

As the developed world gets richer, more and more money is spent on status and signaling related behavior. This has many unfortunate side effects.

6. Demographic issues

Source: UN Medium Variant Projections: The 2008 Revision

Charts like the one above are very common when discussing upcoming demographic problems.  The developed world is aging, and long term demographic trends are important beyond the basic “How are we going to pay for all of these old people?” questions.