Exponential Functions On A Finite Planet

September 27, 2008 – A lightweight, lighthearted and hopefully thought provoking examination of why so many problems seem to be coming to a head today.

By The Cerebral Aesthetic Vagabond

With all the problems becoming evident in our world today – financial collapse, food shortages, peak oil – it seems opportune to examine the root cause of all these problems. The fundamental cause of all of them is our foolish quest for exponential growth on a finite planet. Here is what an exponential function looks like:

Classical exponential function
Classical exponential function, y = ex (x -> 0.5 ... 10.0)

Does that graph look familiar? Doesn’t it look like so many graphs you’ve seen? Of population, pollution levels, debt? If we ever observe any growth pattern here on finite Earth that looks like that graph, we ought to be scared to death! Exponential growth – or any form of boundless growth – on a finite planet is obviously impossible. Sooner or later such a growth pattern will slam right into the finite confines of our planet. In fact, with respect to debt, food production, oil supply and population growth, I think those sounds you hear of crumpling metal and tinkling glass means it just did.

Finance-Based Economy

Our entire finance-based economy depends on exponential growth in prices, simply because of interest. When debt is used to purchase assets, those assets must rise in price sufficiently to repay the principal plus the interest, not even taking into consideration one’s net profit. So P1 = P0 * I, where P0 is the price at time 0 (zero), and I is the interest (one plus the interest expressed as a decimal fraction). To realize that price appreciation, the owner of the asset must sell it, and if the new buyer also buys the asset with debt, that debt is now exponentially larger because it includes the previous principal plus the previous interest. The asset must then appreciate again in order for the new buyer to pay back the new debt and the new interest, so P2 = P1 * I, which can be reduced to P2 = ( P0 * I ) * I. After a third iteration, P3 = ( ( P0 * I ) * I ) * I. Notice how I is getting propagated with each iteration? This is an exponential function that will repeat endlessly if permitted to, and can be expressed more concisely as Pn = P0 * In.

The problem is that such an exponential function simply cannot repeat endlessly. What would happen if house prices, which increased by 10% or more each year between say 2001 and 2006, kept on increasing at that rate for another 50 years? By 2056 they’d be 117 times the price they were in 2006 (1.10 raised to the 50th power). Assuming that the median house price in 2006 was $200,000, then the median price in 2056 would be $23,478,000. It matters not whether we’re talking about houses or stocks or derivatives, the problem is the same for all of them.

Consider the U.S. GDP, shown in the graph below.

U.S. GDP, 1929-2007 [1]
U.S. GDP, 1929-2007 [1]

Doesn’t that graph look eerily similar to the classical exponential function that appears at the top of this post? That’s because the government seeks to maintain economic “growth” in the neighborhood of 3% per year. That rate is considered “healthy”; less is considered anemic; more is considered overheated. But any figure above zero is an exponential function, as can be clearly seen in the graph above, and is ultimately unsustainable.

Population Growth

Consider the U.S. population, shown in the graph below.

U.S. Population, 1790-2008 [2]
U.S. Population, 1790-2008 [2]

Again, it looks a lot like all the preceding exponential functions. Fortunately, the rate of population growth is slowing, as can be seen in the table below. Nevertheless, even an annual growth rate as low as 1.22% is still an exponential function which cannot continue forever.

U.S. Population Growth Rate
Years Annual Growth Rate (%)
1790-1850 3.00
1850-1900 2.41
1900-1950 1.38
1950-2008 1.22
Source: U.S. Census Bureau [2]

What’s interesting when comparing population growth to GDP growth is that while the rate of population growth has been slowing, the rate of GDP growth has apparently been increasing! From 1900-1950 the population grew at 1.38% annually, while from 1929-1950 the GDP grew at 5.09% annually, outpacing population growth. Yet from 1950-2008 population growth slowed to 1.22% annually, while from 1950-2007 GDP growth increased to 6.99% annually! The fact that GDP growth has continually outpaced population growth probably reflects growth of the money supply (i.e. monetary inflation) more than it does real growth. In a perfect world, GDP should rise in lockstep with population, or perhaps a little faster owing to productivity increases. However, productivity cannot rise forever any more than anything else can; eventually we will reach the limit of what a single human being can produce.

Crop Land Use

Not surprisingly, the amount of land used for growing crops has increased along with the human population, in other words, exponentially.

Global Crop Land, 1700-1992 [3]
Global Crop Land, 1700-1992 [3]

It might be difficult to understand how much land that is in the chart above. So consider these facts:

Global Crop Land Use Facts
Total land area of the Earth [4] 150,000,000 square kilometers
Total crop land used in 1992 [3] 1,761,561,000 hectares (a hectare is 1/100 of a square kilometer)
Crop land growth rate from 1700-1992 0.6% per year

Dividing 150,000,000 square kilometers of total land on Earth by 17,615,610 square kilometers used for crop land in 1992 gives us 8.515. In other words, we are presently using (or were using in 1992) 1/8.5 of all available land on earth to grow crops. I’m stunned that it is such a high proportion! So the question is how long, at a growth rate of 0.6% per year, will it be until we are using 100% of all available land on Earth for growing crops? The answer is approximately 358 years (1.006 raised to the 358th power is approximately 8.5). Clearly, we cannot keep increasing the amount of land used for crops for more than another 358 years at the pace of the last 300 years, because by then we will have devoted 100% of all the land on Earth – including all the mountains, forests, deserts and cities – to growing crops!

Conclusion

I could go on and on, citing examples of one exponential function after another, but I hope the foregoing handful of examples are sufficient to persuade the reader that we cannot continue striving for a stable growth rate, as we do now with inflation, the GDP, population and so forth. Stable growth is not stable at all – it’s an exponential function! The only truly stable and sustainable condition is stasis: constant population, GDP, standard of living, land use, energy and food production. We should be striving for zero growth in all these areas. Of course, as I suggested at the beginning of this post, our finance-based economy is partly to blame for our omnipresent need to pursue perpetual growth. The reason we’ve become a finance-based economy is because it’s a heck of a lot easier to sit on one’s butt and collect interest from someone else than it is to toil away in the field or the factory. The only way to end the attraction of finance as a means to “earn” a living is to eliminate usury, popularly known as interest. The Bible – the U.S. is a Christian nation, right? – prohibits usury in a couple of dozen different places. It doesn’t stipulate how much interest constitutes usury:

Leviticus, 25:37. Thou shalt not give him thy money upon usury: nor exact of him any increase of fruits.

Notice the word “any”? No amount of interest is permissible. A similar prohibition can be found in the Koran as well:

They who swallow down usury, shall arise in the resurrection only as he ariseth whom Satan hath infected by his touch. ... O believers! fear God and abandon your remaining usury, if ye are indeed believers.

Perhaps the authors of these ancient, revered books wisely understood that usury leads societies down the path of ruin. I’ll close with this delightful chart showing the inflation rate in Zimbabwe over the last few years. Let it serve as a warning of what happens to a society that pursues unlimited growth.

Zimbabwe Inflation Rate, 2005-2008 [5]
Zimbabwe Inflation Rate, 2005-2008 [5]

References

1. U.S. Department of Commerce. Bureau of Economic Analysis.

2. U.S. Census Bureau.

3. University of Wisconsin, Madison; Center for Sustainability and the Global Environment.

4. Wikipedia: Earth; Area of Earth's Land Surface.

5. I informally collected these numbers myself by doing a search for “Zimbabwe inflation rate” and recording the date and estimated inflation rate from each of the eight articles I examined.

Update – October 4, 2008

After writing this post I ran across a marvelous paper from 1974 by the esteemed geologist, Dr. M. King Hubbert, titled M. King Hubbert on the Nature of Growth (PDF). While Dr. Hubbert is best known for his peak oil theory, this paper discusses exponential growth in general, making it a nice addition to this post. I did not cite energy production in my post as an example of exponential growth even though intuitively it ought to be. But as Dr. Hubbert persuasively shows, energy production is most definitely an example of exponential growth, at least during limited time intervals.

A couple of points in Dr. Hubbert’s paper that stand out are:

1) Human population growth, which was more or less zero (“steady state” in Hubbert’s terminology) until the discovery of fossil fuels, has since grown at an exponential rate that parallels the exponential growth in the production of fossil fuels.

2) By the year 2032 we will have consumed 90% (the first 10% plus the middle 80%) of the Earth’s entire estimated oil reserves.

Combining these two points yields a frightening future for us humans. Dr. Hubbert stated in this paper that the heyday of oil production is the period between 1968 and 2032, with the peak of oil production occurring in the year 2000 (remember, this paper was written in 1974). According to recent estimates by other geologists, the peak of oil production occurred around the year 2005, so Hubbert’s prediction was very close, just as was his prediction about the peak of U.S. oil production. The bottom line is that we are smack in the middle of the 80% area that Hubbert talks about, and according to him there are no easy alternatives to energy from oil. So what happens to the human population when 90% of the oil is gone? Human population growth has paralleled energy production on the way up. It stands to reason that human population growth will parallel energy production on the way down as well.

I’ll conclude with this quote from Dr. Hubbert’s paper:

Without further elaboration, It is demonstrable that the exponential phase of the industrial growth which has dominated human activities during the last couple of centuries is drawing to a close. ...

Yet, during the last two centuries of unbroken industrial growth we have evolved what amounts to an exponential-growth culture. Our institutions, our legal system, our financial system, and our most cherished folkways and beliefs are all based upon the premise of continuing growth. Since physical and biological constraints make it impossible to continue such rates of growth indefinitely, it is inevitable that with the slowing down in the rates of physical growth cultural adjustments must be made.

One example of such a cultural difficulty is afforded by the fundamental difference between the properties of money and those of matter and energy upon which the operation of the physical world depends. Money, being a system of accounting, is, in effect, paper and so is not constrained by the laws within which material and energy systems must operate. In fact money grows exponentially by the rule of compound interest. If M0 be a national monetary stock at an initial time, and i the mean value of the interest rate, then at a later time t the sum of money M0 will have grown exponentially to a larger sum M given by the equation

M=M0eit. (6)

Wow! If I may pat myself on the back, these quotes succinctly summarize my entire post. There is much more in Dr. Hubbert’s paper. It’s highly readable and I highly recommend reading it.

Update – October 13, 2008

I was talking with a friend about financial derivatives and told him that I thought the explosive growth of derivatives had followed an exponential function. I also said that the explosive use of financial derivatives was due to the omnipresent need to generate more and more returns, thanks to the compounding of interest, which is an exponential function. So I expended some effort to find out if my suspicion about the growth function of financial derivatives was merited. Looking at the graph below, it certainly looks like a classic exponential curve (source: http://www.bis.org/statistics/otcder/dt1920a.csv, http://www.bis.org/statistics/derstats.htm; the final data point for 12/08 was added by me, based on estimates of $1-1.25 quadrillion being thrown around).

Global Derivatives (notional amounts outstanding), 1998-2008
Global Derivatives (notional amounts outstanding), 1998-2008

The End