What do compound interest and epidemics have in common? The extremes of exponential processes.

Much of Nassim Taleb’s provocative book Black Swan is devoted to the theme of mediocristan versus extremistan. These are two profoundly different aspects of our world that exist side-by-side. Our cognitive wiring equips to handle the first, but not the second. What are these?

Taleb uses a sports stadium to illustrate. Let’s say that National’s Park (home stadium for U.S. Major-League baseball club Washington Nationals) is filled to capacity. It reportedly holds 41,339 spectators. Now imagine that the ten heaviest people — in terms of body weight — leave the stadium. The change to the total human body mass in the stadium is negligible. Human body mass exists within the realm of mediocristan.

Now suppose that the ten people with the highest net worth leave the stadium. In this case, the aggregate net worth of all people present might change significantly. If a certain Mr. J. Bezos were taking in a Nats game while inspecting his new HQ2 location, the total net worth after he leaves might be significantly less than half its original. This is what Taleb describes as extremistan.

Extremistan tends to result from exponential processes. Compound interest is an example of an exponential process that most are familiar with. The graphic to the left illustrates its affect (J. Reddin, "What Is Compound Interest? Definition And Examples", thecalculatorsite.com, November 2, 2018). Saving under compound interest leads to exponential growth. Compounding of interest in debt has the inverse affect for the borrower.

Extremistan grows out of exponential processes.

My first exposure to exponential processes came in a lecture by a University of Colorado professor. Albert A. Bartlett published a paper “Forgotten fundamentals of the energy crisis” in the American Journal of Physics. Bartlett was an early promoter of the “Peak Oil” theory. He observed that between 1870 and 1970 the total global consumption of petroleum grew at a rate of 8.27% per year.

Bartlett framed energy consumption as an exponential process. He specifically focused on doubling periods. This is the time required for an exponentially-growing process to double. At a compound annual growth rate (CAGR) of 8.27%, petroleum consumption doubles every 8.72 years. Hypothetically, were the global consumption in 1980, say, 65.5 million barrels per day, the consumption in 2020 would be 1,560 million barrels per day. This trend clearly was not sustainable.

What happened? First, energy consumption leveled off. Short-term estimates by the U.S. Energy Information Administration show linearly-increasing consumption over recent years, even pre-COVID-19. The 2019 installment of BP Statistical Review of World Energy shows production growing at an apparently linear rate. This represents a very small CAGR.

Second, new extraction technologies — like hydraulic fracturing (aka “fracking”) — and deep-sea drilling have expanded the available reserves. Oil shale, first seriously explored in the Rocky Mountain region during the 1970s, became economical. So did tar sands in Canada. This incidentally doesn’t mean that reserves are infinite.

Moore’s Law gives us another famous example of an exponential process. Computer-chip magnate Gordon Moore estimated a 40% CAGR for transistor densities on semiconductor chips. This corresponds to a doubling period of approximately two years.

Incidentally, the exact opposite happened with Moore’s law to what happened with Bartlett’s analysis. The Institute for Electrical and Electronic Engineers (IEEE) periodically publishes in its Proceedings articles on the fundamental limits of various technologies. A 1975 article “Physical limits in digital electronics” anticipated limits to semi-conductor manufacturing. The author of a 2001 update “Fundamental limits of silicon technology” was much more timid.

Exponential growth is the “holy grail” in business. Blogger Oleh Kombaiev, gives us the example to the left in “Facebook: Undervalued By 37%” (Seeking Alpha, September 3, 2019). Enjoying a CAGR of 13.4% over a seven-year period, it became one of the most-powerful companies in the world. This corresponds, incidentally, to about a 5?-year doubling time.

Referring back to N. Taleb’s extremistan concept, Moore’s law and Facebook fit within this realm. Exponential growth led to a world-changing industries and firms. It also led to pronounced shifts in socioeconomic balances.

Epidemics are exponential processes.

The COVID-19 pandemic gives us a very painful example of another exponential process. Financial Times produced the image to the right (S. Bernard, et al, “Coronavirus tracked: the latest figures as the pandemic spreads”, as of about 19:00 GMT, Tuesday, March 24, 2020). These are mortality trends in major countries for which data are available.

The FT graphic uses a distinct mechanism for the plotting of this particular exponential process. Specifically, the vertical axis is logarithmic (See K. Change, “A Different Way to Chart the Spread of Coronavirus”, NY Times, March 20, 2020). This stretches things out at the bottom of the axis and compresses them at the top.

Notice also the emphasis on doubling times in the FT graphic. As of the evening of Tuesday, March 24, 667 U.S. citizens had succumbed to COVID-19. Most-importantly, this fits a trend line such that the total number of deaths appears to double about every three days. Exponential processes grow very quickly!

At this trend, only about seven doubling periods are required to grow the mortality count from 667 to 100,000. That is, the U.S. could see its 100,000th COVID-19 death in about 7×3=21 days or about three weeks. From there about another 3? doublings, or less than another two weeks, gets us to the millionth COVID-19 casualty. In logarithmic arithmetic —the expression of exponential processes — 3? doublings translates into an increase by a factor of ten.

Exponential processes and the hazards of extrapolation.

Extrapolation of current trends too far into the future is very difficult. It assumes that circumstances remain unchanged. This is hard enough for non-exponential processes. For exponential ones, it's particularly challenging.

Because of their explosive growth, exponential processes invariably change the circumstances in which they play out. Epidemiology is a perfect illustration. The figure to the right comes from a simulation of multiple viruses (J. Fox, et al, "How epidemics like covid-19 end", Washington Post, February 19, 2020). Eventually, the infectious organism — the pathogen — infects all accessible patients. Like a forest fire consuming all available trees, eventually the epidemic burns itself out.

The figure to the left is from Johns-Hopkins Medicine (L. L. Maragakis, M.D. M.P.H, "Coronavirus, Social Distancing and Self-Quarantine"). We see the number of active infections ultimately reaching a peak. This happens as the size of the unexposed population becomes less than the exposed.

Now, usually the same thing happens with most exponential processes. Price pressures, environmental concerns, and other factors dampened the exponential growth in petroleum consumption that occurred from the late-nineteenth to late-twentieth century. Also, Bartlett's purported 8.27% CAGR really looks like it had dampened out by about 1930. His forecast did not, as a result, play out.

Exponential growth in firms is also not sustainable indefinitely. It leads to extreme concentration in economic (and political) power. London School of Economics professor Paul De Grauwe describes longer-term macro-economic and -societal consequences (P. De Grauwe, The Limits of the Market, Oxford, 2016).

Can Facebok's revenue grow to $195.5 billion in by 2028 as Kombaiev suggests? Google's 2019 revenue was only $160.7 billion. The scope of their businesses — online marketing — overlaps substantially. Whether the economy can support two such behemoths remains to be seen.

Moore's Law is the exponential-process exception that proves the rule. Semiconductor manufacturing maintained exponential growth over a period of nearly half a century. Clever engineering overcame a series of technical obstacles. Some set of fundamental limits presumably exists. The IEEE declined to describe them 20 years ago.