Farm fields landscape

Arithmetic, Population, and Energy

Dr. Albert Bartlett

The Greatest Shortcoming of the Human Race is
our Inability to Understand the Exponential Function!


This talk examines the arithmetic of steady growth, such as 5% per year, the doubling time for such growth, and the large numbers one gets when steady growth continues over modest periods of time. The examination then turns to what happens when one has steady growth in a finite environment. These concepts are applied to populations and to fossil fuels such as petroleum and coal. A series of recommendations is given for dealing with the problems that are revealed by the very simple arithmetic.

A copy of the original (1978) paper is also included. [Ed note - published separately on this site - CS]

Reflections in 1998 on the Twentieth Anniversary of the Paper, "Forgotten Fundamentals of the Energy Crisis"


Around 1969, college and university students developed a major interest in the environment and, stimulated by this, I began to realize that neither I nor the students had a good understanding of the implications of steady growth, and in particular, of the enormous numbers that could be produced by steady growth in modest periods of time. On September 19, 1969 I spoke to the students of the pre-medical honor society on "The Arithmetic of Population Growth." Fortunately I kept my notes for the talk, because I was invited to speak to other groups, and I gave the same talk, appropriately revised and enlarged. By the end of 1975 I had given the talk 30 times using different titles, and I was becoming more interested in the exponential arithmetic of steady growth. I started writing short numbered pieces, "The Exponential Function," which were published in The Physics Teacher. Then the first energy crisis gave a new sense of urgency to the need to help people to gain a better understanding of the arithmetic of steady growth, and in particular of the shortening of the life expectancy of a non-renewable resource if one had steady growth in the rate of consumption of such a resource until the last of the resource was used.

When I first calculated the Exponential Expiration Time (EET) of U.S. coal for a particular rate of growth of consumption, using Eq. 6, I used my new hand-held electronic calculator, and the result was 44 years. This was so short that I suspected I had made an error in entering the problem. I repeated the calculation a couple of more times, and got the same 44 years. This convinced me that my new calculator was flawed, so I got out tables of logarithms and used pencil and paper to calculate the result, which was 44 years. Only then did I begin to realize the degree to which the lifetime of a non-renewable resource was shortened by having steady growth in the rate of consumption of the resource, and how misleading it is for leaders in business and industry to be advocating growth of rates of consumption and telling people how long the resource will last "at present rates of consumption."

This led to the first version of this paper which was presented at an energy conference at the University of Missouri at Rolla in October 1976, where it appears in the Proceedings of the Conference. In reading other papers in the Proceedings I came to realize that prominent people in the energy business would sometimes make statements that struck me as being unrealistic and even outrageous. Many of these statements were quoted in the version of the paper that is reprinted here, and this alerted me to the need to watch the public press for more such statements. Fortunately ( or unfortunately ) the press and prominent people have provided a steady stream of statements that are illuminating because they reflect an inability to do arithmetic and / or to understand the energy situation.

As this is written, I have given my talk on "Arithmetic, Population, and Energy" over 1260 times in 48 of the 50 States in the 28 years since 1969. I wish to acknowledge many constructive and helpful conversations on these topics I have had throughout the 20 years with my colleagues in the Department of Physics, and in particular with Professors Robert Ristinen and Jack Kraushaar, who have written a successful textbook on energy. (Energy and Problems of a Technical Society, John Wiley & Sons, New York City, 2nd Ed. 1993)

Reflections on the "Fundamentals" Paper Twenty Years Later

As I read the 1978 paper in 1998, I am pleased to note that the arithmetic that is the core of the paper remains unchanged, and I feel that there are only a few points that need correction or updating.

1) When I derived my Eq. 6 in the Appendix, I was unaware that this equation for the Exponential Expiration Time (EET) had been published earlier by R. T. Robiscoe (his Eq. 4) in an article, "The Effect of Growth Rate on Conservation of a Resource." American Journal of Physics, Vol. 41, May 1973, p. 719-720. I apologize for not having been aware of this earlier derivation and presentation of this equation.

2) The world population was reported in 1975 to be 4 billion people growing at approximately 1.9% per year. In 1998 it is now a little under 6 billion people and the growth rate is reported to be around 1.5% per year. The decline in the rate of growth is certainly good news, but the population growth won't stop until the growth rate has dropped to zero.

3) In 1978 I reported that "We are currently importing one-half of the petroleum we use." The data now indicate that, except for brief periods, this could not have been true in 1978. The basis for my statement was a newspaper clipping that said that the U.S. had experienced, in 1976, the first month in its history in which more oil was imported than was produced domestically. However, the imported fraction of the oil consumed in the U.S. has risen, and in early 1995 the news said that the calendar year 1994 was the first year in our nation's history when we had to import more oil than we were able to get from our ground ourselves. (Colorado Daily, February 24, 1995)

4) The paper reported that by 1973 nuclear reactors (fission) supplied approximately 4.6% of our national electrical power. By 1998 this had climbed to approximately 20% of our electrical power, but no new nuclear power plants have been installed in the U.S. since the 1970s.

5) A table that I wish I had included in the original paper is one that would give answers to questions such as, "If a non-renewable resource would last, say 50 years at present rates of consumption, how long would it last if consumption were to grow say 4% per year?" This involves using the formula for the EET in which the quotient ( R / r0 ) is the number of years the quantity R of the resource would last
at the present rate of consumption, r0. The results of this simple calculation are shown in Table I.

Forgotten Fundamentals of the Energy Crisis - Table I

Example 1. If a resource would last 300 years at present rates of consumption, then it would last 49 years if the rate of consumption grew 6% per year.

Example 2. If a resource would last 18 years at 5% annual growth in the rate of consumption, then it would last 30 years at present rates of consumption (0% growth).

Example 3. If a resource would last 55 years at 8% annual growth in the rate of consumption, then it would last 115 years at 3% annual growth rate.

6) In the end of Section VIII of the 1978 paper I quoted Hubbert as writing in 1956 that "the peak of production of petroleum" in the U.S. would be reached between 1966 and 1971. The peak occurred in 1970. Hubbert predicted that "On a world scale [oil production] will probably pass its climax within the order of half a century...[2006]" My more recent analysis suggests the year 2004, while Campbell and Laherrère predict that the world peak will be reached before 2010 (Scientific American, March 1998, pp. 78-83). Studies by other geologists predict the peak within the first decade of the next century. Hubbert's analysis appears thus far to be remarkably good.

7) The "Fundamentals" paper was followed by a paper titled, "Sustained Availability: A Management Program for Non-Renewable Resources." American Journal of Physics, Vol. 54, May 1986, pp. 398-402. This paper makes use of the fact that the integral from zero to infinity of a declining exponential curve is finite. Thus, if one puts production of a non-renewable resource on a declining exponential curve, one can always find a rate of decline such that the resource will last forever. This is called "Sustained Availability," which is somewhat analogous to "sustained yield" in agriculture. This paper explores the mathematics of the options that this plan of action can give to a resource-rich nation that wants to divide its production of a resource between domestic use and exports.

8) Many economists reject this sort of analysis which is based on the assumption that resources are finite. A colleague in economics read the paper and later told me that "It is all wrong." When I asked him to point out the specific errors in the paper, he shook his head, saying, "It is all wrong."

9) The original paper dealt more with resources than with population. I feel that it is now clear that population growth is the world's most serious problem, and that the world's most serious population problem is right here in the U.S. The reason for this is that the average American has something like 30 to 50 times the impact on world resources as does a person in an underdeveloped country. (A.A. Bartlett, Wild Earth, Vol. 7, Fall 1997, pp. 88-90)

We have the jurisdiction and the responsibility needed to permit us to address our U.S. population problem, yet many prefer to focus their attention on the population problems in other countries. Before we can tell people in other countries that they must stop their population growth, we must accept the responsibility for working to stop population growth in the United States, where about half of our population growth is the excess of births over deaths and the other half is immigration, legal plus illegal. This leads me to offer the following challenge:

Can you think of any problem, on any scale, from microscopic to global, whose long-term solution is in any demonstrable way, aided, assisted, or advanced by having larger populations at the local level, the state level, the national level, or globally?

Horror Stories

Here are more recent horror stories to add to those that were recounted in the original paper.

1) The Rocky Mountain News of October 6, 1993 reported that: Shell Oil Co. said "... it planned to spend $1.2 billion to develop the largest oil discovery in the Gulf of Mexico in the past 20 years. The discovery ... has an estimated ultimate recovery in excess of 700 million barrels of oil and gas." The 700 million barrels of oil sounds like a lot -- until you note that at that time the U.S. consumption was 16.6 million barrels / day, so that this "largest oil discovery in the Gulf of Mexico in the past 20 years" would supply the needs of the U.S. for only 42 days!

2) The headline in the Wall Street Journal for July 18, 1986 proclaimed that "U.S. Oil Output Tumbled in First Half as Alaska's Production Fell Nearly 8%." In the body of the story we read that the chief economist for Chevron Corporation observes that, "The question we can't answer yet is whether this is a new trend or a quirk." The answer to his question is that it is neither; it is an old trend! It is exactly what one expects as one goes down the right side of the Hubbert Curve.

3) Another headline on the front page of the Wall Street Journal (April 1, 1997) said: "Four Decades Later, Oil Field Off Canada is Ready to Produce. Politics, Money and Nature Put Vast Deposit on Ice; Now It Will Last 50 Years: Shot in the Arm for U.S." In the body of the story we read that:

The Hibernia field, one of the largest oil discoveries in North American in decades, should deliver its first oil by year end. At least 20 more fields may follow, offering well over one billion barrels of high-quality crude and promising that a steady flow of oil will be just a quick tanker-run away from the energy-thirsty East Coast.

Total U.S. oil consumption in 1996 was about 18 million barrels a day. Do the long division and one sees that the estimated "one billion barrels of high-quality crude" will supply the needs of the U.S. for just 56 days! This should be compared with the "50 Years" in the headline.

4) In the Prime Time Monthly Magazine (San Francisco, September 1995) we find an article, "Horses Need Corn" by the famous radio news broadcaster Paul Harvey. He emphasizes the opportunity we have to make ethanol from corn grown in the U.S. and then to use the ethanol as a fuel for our cars and trucks: "Today, ethanol production displaces over 43.5 million barrels of imported oil annually, reducing the U.S. trade balance by $645 million. . For as far ahead as we can see, the only inexhaustible feed for our high horsepower vehicles is corn."

There are two problems with this:

A) The 43.5 million barrels must be compared with the annual consumption of motor gasoline in the U.S. In 1994 we consumed 4.17 billion barrels of motor vehicle gasoline. (Annual Energy Review, 1994, DOE / EIA 0384(94), p. 159) The ethanol production is seen to be approximately1% of the annual consumption of gasoline by vehicles in the U.S. So one would have to multiply corn production by a factor of about100 just to make the numbers match. An increase of this magnitude in the farm acreage devoted to the production of corn for ethanol would have profound negative dietary consequences. Editor's note: new technologies allow ethanol to be made from "hemicellulosic" materials currently wasted, such as corn stover, rice stalks, waste paper, and yard and wood wastes. Although corn is still the primary feedstock used today, it is not the only (or even preferred) feedstock envisioned for the future.

B) It takes energy (generally diesel fuel) to plow the ground, to fertilize the ground, to plant the corn, to take care of the corn, to harvest the corn, and then more energy is needed to distill the corn to get ethanol. So it turns out that in the conventional production of ethanol, the finished gallon of ethanol contains less energy than was used to produce it! It's an energy loser! The net energy of this "energy source" is negative!

Editor's note: although the net energy balance of ethanol production from corn was negative in the 1970s and early 1980s, modern ethanol production is significantly more efficient, and now has a positive net energy balance. Ethanol production from other feedstocks, such as molasses or hemicellulosic materials, has an even greater positive net energy balance. As the author has said in numerous presentations, "don't let others do your thinking for you." If this issue is of interest to you, look into it further.

5) The Clinton administration, in a "Draft Comprehensive National Energy Strategy" (February 1998) talks about America's oil as being "abundant," (pg. 4) and it advocates "promoting increased domestic oil ... production" (pg. 2) to reverse this downward trend in U.S. oil production. The peak of the Hubbert Curve of oil production in the U.S. was reached in 1970 and we are now well down the right side of the Curve. The Draft Strategy calls for "stabilization of domestic oil production" (pg. 12) which is explained in "Strategy 1" (pg. 12) "By 2005, first stop and then reverse the decline in domestic oil production." The Hubbert Curve rises and falls in a manner like that of a Gaussian Error Curve, and once one is over the peak, one can put bumps on the downhill side, but except for such "noise," the trend after the peak is always downhill. A large national effort might reverse the decline in U.S. oil production for a year or two, but it hardly plausible to propose to "stabilize" domestic oil production for any extended period of time. It almost seems as though the U.S. Department of Energy has not studied the works of Hubbert, Campbell & Laherrère, Ivanhoe, Edwards, Masters and other prominent petroleum geologists.

Source [Above portion, as well as rest.]