Thursday, October 5, 2006

The Exponential Function

“Anyone who believes we can have exponential growth in a finite world is either a mad man – or an economist” - Aldus Huxley


Many a mathematician, physicist, and geologist in the country is familiar with, and most do not dispute, the validity of the argument that we are faced with an unprecedented problem in the form of a permanent energy crisis (just Google "Hirsch Report Peak Oil" and read what the U.S. Department of Energy published in February, 2005). Yet the mainstream press and the public are either uniformed or in complete denial.

The mathematical necessity of the matter can be better illuminated if we become familiar with the concept of the exponential function. Dr. Albert Bartlett, professor emeritus of physics at the University of Colorado (Boulder), has been speaking about the subject for the past several decades and has written an excellent book, which I heartily recommend, titled “The Essential Exponential Function for the Future of our Planet”. For those of you disinclined to read this tome, this paper might suffice.

We have all heard the word “exponential”, but I think a definition is in order. The following is from a lecture given by Dr. Bartlett at the University of Colorado to describe the exponential function.

“This is a mathematical function that you'd write down if you're going to describe the size of anything that was growing steadily. If you had something that was growing at 5% per year, you'd write the exponential function to show how large that growing quantity was year after year. And so we are talking about a situation where the requirements required for the growing quantity to increase by a fixed fraction is a constant 5% per year. The 5% is a fixed fraction, the three years a fixed length of time. So that's what we want to talk about. Its just ordinary steady growth.

Well if it takes a fixed length of time to grow 5%, it follows it takes a longer fixed length of time to grow 100%. That longer time's called the doubling time and we need to know how you calculate that doubling time. It's easy. You just take the number 70, divide it by the percent growth per unit time and that gives you the doubling time. So our example of 5% per year, you divide that into 70, you find that growing quantity will double in size every 14 years.
Well, you might ask, where did that seventy come from, well, the answer is that it's approximately one hundred multiplied by the natural logarithm of two. If you wanted the time to triple you would use the natural log of three. So it's all very logical. But you don't have to remember where it came from, just remember 70.”
Please don’t let your eyes glaze over, I promise not to mention logarithm again; for our purposes here, as Dr. Bartlett says, just remember 70, and doubling time (“T2”).

So what is so important about the rule of 70 and doubling time? Just this, if something is growing at 7% per year, it will double every 10 years (70/7). Something growing at 5% doubles every 14 years (70/5). More importantly, the total amount of the unit measured at the end of the doubling period will be greater than the total of ALL of the preceding doubling periods, and at the end of 10 T2’s is over 1000 times the size of the original amount (1,2,4,8,16,32,64,128,256,512,1024 – 1024 is 10th doubling time). For example, total world Oil consumption grew at 7% per year during the 1950’s, 60’s and most of the 70’s. Using the rule of 70 that means that the world consumed more Oil in the 1960’s than it had in all of human history prior to January of 1960 (look at the above numerical progression - 16 is greater than the total of all the numbers before it (1+2+4+8=15), as are each subsequent number). During the decade of the‘70’s the world consumed more oil than it had from 1859 (Colonel Drake drill’s his well) to December 31, 1969. After the Oil shocks of the 70’s, the period 1980 to present saw much lower exponential growth in Oil consumption, 100% by geological necessity. Had the world continued to double its consumption of Oil each decade from 1980 to the present, the world’s entire endowment of oil would have been consumed, and you and I would be cooking over a dung fire tonight.

To date, the world has consumed just over 1,000,045,000,000 (1 trillion, 45 billion) barrels of oil out of a likely endowment of 2.06 trillion barrels (Dr. Ken Deffeys, professor of geology, Princeton University, and author of "Beyond Oil", another book I strongly recommend). We are currently consuming roughly 30,000,000,000 (30 billion) per year. Well, if there were no steady increase in the use of oil (exponential growth) we would be completely out of oil in 33.3 years (If you want to add a slush number, go right ahead). But this is not the case because A: Demand is increasing at roughly 1.8 % per year and (however) B: Once an oil field has produced more than half of its total endowment of recoverable oil, its production declines each and every year (this is called terminal decline). The last barrels do not come out of the ground at the same speed as the barrels coming out of the ground today. Think about it: what finite resource continues to increase in production until the last unit is consumed? Not a one. The production of gold, oil, iron, coal, etc… from any one field follows a bell curve production cycle; slow but rising in the beginning, the left side of the curve, peaking in the middle, and declining on the right side, the down slope of the bell curve.

So what does this mean to you? Desire to consume oil (notice I did not use the word “demand”) will continue to increase exponentially, but the supply available to satisfy that desire will not be able to grow once the world’s oil supply enters terminal decline. Exponential growth collides head on with terminal decline. No one disputes this. No government, no research university, no oil company executive. Oil is a finite resource and by mathematical necessity its production must peak at some point, and then enter terminal decline. The only debate is when. If the world’s reserve estimates are correct, and I believe they are hopelessly exaggerated by oil producing countries for political gain (future posts will coer this issues), and future discoveries total 160 to 200 billion barrels, the peak is RIGHT NOW, + - 5 years.

We have been receiving price signals from the Oil and Natural Gas markets (some would argue those signals have broken down with the recent decline in price, but volatility is one of the signals), political signals from national governments in the form of resource wars (Iraq), political blackmail (Russia turning off Europe’s Natural Gas Pipeline last January), and the 9/11 attacks on the World Trade Center (Radical Islam is much more aware of this crisis the average American). et al.

Once you understand the exponential function, (its mirror image, the logarithmic function, is the subject of a future post), and how its tenets impact the concept of sustained growth of anything – the economy, population, compound interest, inflation… - you will never view anything that is growing steadily the same again.

mentatt (at) yahoo (dot) com

1 comment:

KrediK said...

This formula isn't quite correct. If you have for example 70% growth per year, divide it into 70, you find that growing quantity will double in size every 1 year. But wait... We have 70% growth per year AND 100% growth for 1 year. Well, 70 isn't the same as 100, I think.