[Note: For convenience of the reader, MFS has separated the paper into two sections.]

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 Procedings 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)
  

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)  III.  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) VI.  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) IX.  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.
 

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?

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 approximately  1 %  of the annual consumption of gasoline by vehicles in the U.S.  So one would have to multiply corn production by a factor of about  100  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.

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!

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.
 

The energy crisis has been brought into focus by President Carter's message to the American people on April 18 and by his message to the Congress on April 20, 1977.  Although the President spoke of the gravity of the energy situation when he said that it was "unprecedented in our history," his messages have triggered an avalanche of critical responses from national political and business leaders.  A very common criticism of the President's message is that he failed to give sufficient emphasis to increased fuel production as a way of easing the crisis.  The President proposed an escalating tax on gasoline and a tax on the large gas guzzling cars in order to reduce gasoline consumption.  These taxes have been attacked by politicians, by labor leaders, and by the manufacturers of the "gas guzzlers" who convey the impression that one of the options that is open to us is to go ahead using gasoline as we have used it in the past.

We have the vague feeling that Arctic oil from Alaska will greatly reduce our dependence on foreign oil.  We have recently heard political leaders speaking of energy self-sufficiency for the U.S. and of "Project Independence."  The divergent discussion of the energy problem creates confusion rather than clarity, and from the confusion many Americans draw the conclusion that the energy shortage is mainly a matter of manipulation or of interpretation.  It then follows in the minds of many that the shortage can be "solved" by congressional action in the manner in which we "solve" social and political problems.

Many people seem comfortably confident that the problem is being dealt with by experts who understand it.  However, when one sees the great hardships that people suffered in the Northeastern U.S. in January 1977 because of the shortage of fossil fuels, one may begin to wonder about the long-range wisdom of the way that our society has developed.

What are the fundamentals of the energy crisis?

Rather than travel into the sticky abyss of statistics it is better to rely on a few data and on the pristine simplicity of elementary mathematics.  With these it is possible to gain a clear understanding of the origins, scope, and implications of the energy crisis.
 

When a quantity such as the rate of consumption of a resource (measured in tons per year or in barrels per year) is growing at a fixed percent per year, the growth is said to be exponential.  The important property of the growth is that the time required for the growing quantity to increase its size by a fixed fraction is constant.  For example, a growth of  5 %  (a fixed fraction) per year (a constant time interval) is exponential.  It follows that a constant time will be required for the growing quantity to double its size (increase by 100 %).  This time is called the doubling time  T₂ , and it is related to P, the percent growth per unit time by a very simple relation that should be a central part of the educational repertoire of every American.

As an example, a growth rate of  5 % / yr  will result in the doubling of the size of the growing quantity in a time  T₂  =  70 / 5  =  14 yr.  In two doubling times (28 yr) the growing quantity will double twice (quadruple) in size.  In three doubling times its size will increase eightfold (2³  =  8); in four doubling times it will increase sixteen fold (2⁴  =  16); etc.  It is natural then to talk of growth in terms of powers of  2.
 

Legend has it that the game of chess was invented by a mathematician who worked for an ancient king.  As a reward for the invention the mathematician asked for the amount of wheat that would be determined by the following process: He asked the king to place 1 grain of wheat on the first square of the chess board, double this and put 2 grains on the second square, and continue this way, putting on each square twice the number of grains that were on the preceding square.  The filling of the chessboard is shown in Table I.  We see that on the last square one will place  2⁶³  grains and the total number of grains on the board will then be one grain less than  2⁶⁴.

How much wheat is  2⁶⁴  grains?  Simple arithmetic shows that it is approximately 500 times the 1976 annual worldwide harvest of wheat?  This amount is probably larger than all the wheat that has been harvested by humans in the history of the earth!  How did we get to this enormous number?  It is simple; we started with 1 grain of wheat and we doubled it a mere 63 times!

Exponential growth is characterized by doubling, and
a few doublings can lead quickly to enormous numbers.

The example of the chessboard (Table I) shows us another important aspect of exponential growth; the increase in any doubling is approximately equal to the sum of all the preceding growth!  Note that when 8 grains are placed on the 4th square, the 8 is greater than the total of  7 grains that were already on the board.  The 32 grains placed on the 6th square are more than the total of  31 grains that were already on the board.  Covering any square requires one grain more than the total number of grains that are already on the board.
 

On April 18, 1977 President Carter told the American people, "And in each of these decades (the 1950s and 1960s), more oil was consumed than in all of man's previous history combined."

We can now see that this astounding observation is a simple consequence of a growth rate whose doubling time is  T₂  =  10 yr (one decade).  The growth rate which has this doubling time is   P   =  70 / 10  =  7 % / yr.

When we read that the demand for electrical power in the U.S. is expected to double in the next 10-12 yr we should recognize that this means that the quantity of electrical energy that will be used in these 10-12 yr will be approximately equal to the total of all of the electrical energy that has been used in the entire history of the electrical industry in this country!  Many people find it hard to believe that when the rate of consumption is growing a mere 7 % / yr, the consumption in one decade exceeds the total of all of the previous consumption.

Populations tend to grow exponentially.  The world population in 1975 was estimated to be 4 billion people and it was growing at the rate of  1.9 % / yr.  It is easy to calculate that at this low rate of growth the world population would double in 36 yr, the population would grow to a density of 1 person / m2 on the dry land surface of the earth (excluding Antarctica) in 550 yr, and the mass of people would equal the mass of the earth in a mere 1,620 yr!  Tiny growth rates can yield incredible numbers in modest periods of time!  Since it is obvious that people could never live at the density of 1 person / m² over the land area of the earth, it is obvious that the earth will experience zero population growth.  The present high birth rate and / or the present low death rate will change until they have the same numerical value, and this will probably happen in a time much shorter than 550 years.

A recent report suggested that the rate of growth of world population had dropped from 1.9 % / yr to 1.64 % / yr.²   Such a drop would certainly qualify as the best news the human race has ever had!  The report seemed to suggest that the drop in this growth rate was evidence that the population crisis had passed, but it is easy to see that this is not the case.  The arithmetic shows that an annual growth rate of  1.64 %  will do anything that an annual rate of  1.9 %  will do; it just takes a little longer.  For example, the world population would increase by one billion people in 13.6 yr instead of in 11.7 years.

It is very useful to remember that steady exponential growth of  n % / yr  for a period of  70 yr  (100 ln2) will produce growth by an overall factor of 2ⁿ.  Thus where the city of Boulder, Colorado, today has one overloaded sewer treatment plant, a steady population growth at the rate of  5 % / yr  would make it necessary in 70 yr (one human lifetime) to have 2⁵  =  32 overloaded sewer treatment plants!

Steady inflation causes prices to rise exponentially.  An inflation rate of  6 % / yr will, in 70 yr, cause prices to increase by a factor of 64!  If the inflation continues at this rate, the $0.40 loaf of bread we feed our toddlers today will cost $25.60 when the toddlers are retired and living on their pensions!

It has even been proven that the number of miles of highway in the country tends to grow exponentially.^1(e),3 

The reader can suspect that the world's most important arithmetic is the arithmetic of the exponential function.  One can see that our long national history of population growth and of growth in our per-capita consumption of resources lie at the heart of our energy problem.
 

Bacteria grow by division so that 1 bacterium becomes 2, the 2 divide to give 4, the 4 divide to give 8, etc.  Consider a hypothetical strain of bacteria for which this division time is 1 minute.  The number of bacteria thus grows exponentially with a doubling time of 1 minute.  One bacterium is put in a bottle at 11:00 a.m. and it is observed that the bottle is full of bacteria at 12:00 noon.  Here is a simple example of exponential growth in a finite environment.  This is mathematically identical to the case of the exponentially growing consumption of our finite resources of fossil fuels.  Keep this in mind as you ponder three questions about the bacteria:

(1) When was the bottle half-full?  Answer: 11:59 a.m.!

(2) If you were an average bacterium in the bottle, at what time would you first realize that you were running out of space?  Answer: There is no unique answer to this question, so let's ask, "At 11:55 a.m., when the bottle is only 3 %  filled (1 / 32) and is 97 % open space (just yearning for development) would you perceive that there was a problem?"  Some years ago someone wrote a letter to a Boulder newspaper to say that there was no problem with population growth in Boulder Valley.  The reason given was that there was 15 times as much open space as had already been developed.  When one thinks of the bacteria in the bottle one sees that the time in Boulder Valley was 4 min before noon!  See Table II.
 

Suppose that at 11:58 a.m. some farsighted bacteria realize that they are running out of space and consequently, with a great expenditure of effort and funds, they launch a search for new bottles.  They look offshore on the outer continental shelf and in the Arctic, and at 11:59 a.m. they discover three new empty bottles.  Great sighs of relief come from all the worried bacteria, because this magnificent discovery is three times the number of bottles that had hitherto been known.  The discovery quadruples the total space resource known to the bacteria.  Surely this will solve the problem so that the bacteria can be self-sufficient in space.  The bacterial "Project Independence" must now have achieved its goal.

(3) How long can the bacterial growth continue if the total space resources are quadrupled?

James Schlesinger, Secretary of Energy in President Carter's Cabinet recently noted that in the energy crisis "we have a classic case of exponential growth against a finite source."⁴
 

Physicists would tend to agree that the world's mineral resources are finite.  The extent of the resources is only incompletely known, although knowledge about the extent of the remaining resources is growing very rapidly.  The consumption of resources is generally growing exponentially, and we would like to have an idea of how long resources will last.  Let us plot a graph of the rate of consumption  r(t)  of a resource (in units such as tons / yr) as a function of time measured in years.  The area under the curve in the interval between times  t  =  0 (the present, where the rate of consumption is  r₀ ) and   t  =  T  will be a measure of the total consumption  C  in tons of the resource in the time interval.  We can find the time  T_e  at which the total consumption  C  is equal to the size  R  of the resource and this time will be an estimate of the expiration time of the resource.

Imagine that the rate of consumption of a resource grows at a constant rate until the last of the resource is consumed, whereupon the rate of consumption falls abruptly to zero.  It is appropriate to examine this model because this constant exponential growth is an accurate reflection of the goals and aspirations of our economic system.  Unending growth of our rates of production and consumption and of our Gross National Product is the central theme of our economy and it is regarded as disastrous when actual rates of growth fall below the planned rates.  Thus it is relevant to calculate the life expectancy of a resource under conditions of constant rates of growth.  Under these conditions the period of time necessary to consume the known reserves of a resource may be called the exponential expiration time (EET) of the resource.  The EET is a function of the known size  R  of the resource, of the current rate of use  r₀  of the resource, and of the fractional growth per unit time  k  of the rate of consumption of the resource.  The expression for the EET is derived in the Appendix where it appears as Eq. (6).  This equation is known to scholars who deal in resource problems⁵  but there is little evidence that it is known or understood by the political, industrial, business, or labor leaders who deal in energy resources, who speak and write on the energy crisis and who take pains to emphasize how essential it is to our society to have continued uninterrupted growth in all parts of our economy.  The equation for the EET has been called the best-kept scientific secret of the century.⁶

The question of how long our resources will last is perhaps the most important question that can be asked in a modern industrial society.  Dr. M. King Hubbert, a geophysicist now retired from the United States Geological Survey, is a world authority on the estimation of energy resources and on the prediction of their patterns of discovery and depletion.  Many of the data used here come from Hubbert's papers.^{7 - 10}  Several of the figures in this paper are redrawn from figures in his papers.  These papers are required reading for anyone who wishes to understand the fundamentals and many of the details of the problem.
 

Let us examine the situation in regard to production of domestic crude oil in the U.S. Table IV gives the relevant data.  Note that since one-half of our domestic petroleum has already been consumed, the "petroleum time" in the U.S. is 1 minute before noon!  Figure 1 shows the historical trend in domestic production (consumption) of crude oil.  Note that from 1870 to about 1930 the rate of production of domestic crude oil increased exponentially at a rate of  8.27 % / yr  with a doubling time of 8.4 yr.  If the growth in the rate of production stopped and the rate of production was held constant at the 1970 rate, the remaining U.S. oil would last only  (190 - 96.6) / 3.29  =  28 yr!  We are currently importing one-half of the petroleum we use.  If these imports were completely cut off and if there was no growth in the rate of domestic consumption above the 1970 rate, our domestic petroleum reserves would last only 14 yr!  The vast shale oil deposits of Colorado and Wyoming represent an enormous resource.  Hubbert reports that the oil recoverable under 1965 techniques is  80 x 10⁹  barrels, and he quotes other higher estimates.  In the preparation of Table V, the figure  103.4 x 10⁹  barrels was used as the estimate of U.S. shale oil so that the reserves used in the calculation of column 4 would be twice those that were used in the calculation of column 3.  This table makes it clear that when consumption is rising exponentially, a doubling of the remaining resource results in only a small increase in the life expectancy of the resource.
 

A reporter from CBS News, speaking about oil shale on a three-hour television special feature on energy (August 31, 1977) said, "Most experts estimate that oil shale deposits like these near Rifle, Colorado, could provide more than a 100-yr supply."  This statement should be compared with the figures given in column 4 of Table V.  This comparison will serve to introduce the reader to the disturbing divergence between reassuring statements by authoritative sources and the results of simple calculations.  Anyone who wishes to talk about energy self-sufficiency for the United States (Project Independence) must understand Table V and the simple exponential calculations upon which it is based.