"Forgotten Fundamentals of the Energy Crisis"

by Albert A. Bartlett

University of Colorado at Boulder

I. INTRODUCTION¹ 

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 pr  oduction 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. 
 

II.  BACKGROUND 

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 sixteenfold (2⁴  =  16); etc.  It is natural then to talk of growth in terms of powers of  2. 
 

III.  THE POWER OF POWERS OF TWO 

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. 

Compound interest on an account in the savings bank causes the account balance to grow exponentially.  One dollar at an interest rate of  5 % / yr  compounded continuously will grow in 500 yr to 72 billion dollars and the interest at the end of the 500th year would be coming in at the magnificent rate of  $114 / s.  If left untouched for another doubling time of 14 yr, the account balance would be 144 billion dollars and the interest would be accumulating at the rate of  $228 / s. 

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. 
 

IV.  EXPONENTIAL GROWTH IN A FINITE ENVIRONMENT  

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?  

Answer: Two more doubling times (minutes)!  See Table III. 

  
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 
 

V.  LENGTH OF LIFE OF A FINITE RESOURCE WHEN THE RATE OF 
     CONSUMPTION IS GROWING EXPONENTIALLY  

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