The depletion of finite natural resources like coal and oil also follows the exponential pattern described by Professor Bartlett. The IRS recognizes depletion of natural resources as analogous to the depreciation of plant and equipment and therefore allows certain related tax deductions. That natural resources are finite is undeniable. There may be some limited new discoveries in the future but in the end the recoverable world and U.S. reserves are limited and will ultimately be fully depleted or used up.
Many scientists believe fossil fuels are the primary cause of environmental pollution and climate change. This is a second good reason for seeking alternative fuels and energy sources while pursuing a vigorous conservation program and a stable population. One question raised by the recent election is whether we should develop all potential alternate energy sources simultaneously as a bridge to the future, or instead, focus our efforts solely on the development of clean energy sources? Because the State’s environmental problems and population seem to be increasing rapidly accompanied by perennial disastrous forest fires, mud slides, and fuchtbares wetter (generally foul weather), California Governor Arnold Schwarzenegger has initiated a very ambitious alternative energy and conservation program. He admits that to achieve his goals will probably require re-thinking California’s current ban on nuclear power plant construction and a significant state investment in solar conversions for those who cannot afford them.
Professor Bartlett wrote about the exponential depletion of oil and coal, relying in part on the work of Dr. M. King Hubbert, a geophysicist now retired from the United States Geological Survey, a world authority on the estimation of energy resources and on the prediction of their patterns of discovery and depletion. Bartlett observed that “we have the vague feeling that oil from Alaska and from beneath the Arctic ice cap will greatly reduce our dependence on foreign oil. During the recent presidential election campaign, we heard politicians speaking at length of the need for energy self-sufficiency in the U.S….” Republicans have urged drilling in all the offshore areas where there are proven or prospective oil deposits, not to perpetuate our output of carbon pollutants, but to assure that we have adequate bridging energy sources and the petrochemicals needed to restore our economy and keep it moving forward. Democrats prefer to focus almost entirely on the development of renewable energy sources. While commendable, this approach places our economy in some degree of jeopardy financially if we have to continue to buy petroleum on the world market and the price returns to the peaks of a few months ago. Few believe alternative energy and chemical sources can be developed on a timely basis to meet all our needs. How many airplanes will be able to fly on wind or solar energy? How much chemical fertilizer will come from the air and the sun? In the minds of many 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 and the rolling blackouts in California, one may begin to wonder about the long-range wisdom of the way that our society has developed.
But I digress. How much coal and how much petroleum do we really have and how long will it last in the face of the potential for exponentially increasing demands? According to Dr. Bartlett, this can be best expressed in terms of the exponential expiration time (EET) or Te which is equal to ( 1 / k ) ln ( k R / r(0) + 1 ) where: r(0) is the current rate of consumption (i.e. at time t = 0), e, which has a value of 2.71828, is the base of natural logarithms (ln), k is the fractional growth per year, and R is the size of the resource in tons. 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, driven in part by population growth, is the central theme of our economy. It is regarded as disastrous when actual rates of growth fall below the planned or expected 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) or Te of the resource.
Obviously, the rate of consumption of a resource will not grow at a constant rate until the last resource is consumed as assumed above. This suggests that the exponential solution provides some sort of a lower bound for the EET that we all should keep in mind. Although the gradual tapering off of the rate of consumption of a resource will cause it to last somewhat longer than the EET, that will not be of much help to most consumers. Therefore, we should pay some attention to the exponential solution after all.
The growing quantity will increase to twice its initial size in the doubling time T2 where: T2 (in years) = (ln 2) / k » 70 / k. This represents a rule of thumb that can be used to estimate how long it will take for a quantity to double at growth rate of k. For example, $100 invested at 5% would double to $200 in 70/5 or 14 years; at 10% it would take half that long.
If we 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 is a measure of the total consumption C of the resource in the time interval. We can find the time Te 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.
In the table below are exponential expiration times Te for domestic petroleum, based on an initial 1970 rate of consumption r0 of 3.29 billion barrels per year, various annual growth rates in consumption k, and three different assumptions about the size of our domestic petroleum resource, 93.4, 103.4, and 206.8 billion barrels respectively. The middle column adds 10 billion barrels for Alaskan crude and the last column adds another 103.4 billion barrels for potential production from oil shale. With the most optimistic assumption about the size of the domestic resource and zero growth in the annual consumption rate, the expiration time is 62.8 years. Based on a starting year of 1972, this would mean the year 2035 is the first year in which shortages of domestic petroleum should become apparent. To put it another way, during that period it will be necessary to increase our oil imports or offset the decline in domestic production through conservation and the development of alternative energy sources. Sixteen years is precious little time to get our act together with hybrid or electric vehicles, wind, solar, and nuclear power. Offshore oil drilling will not produce immediate results but is an important bridge to begin the process of weaning ourselves from foreign oil. Eventually, drilling could also help continue the downward pressure on oil prices.
k Expiration Times
0% 28.4 31.4 62.8
1% 25.0 27.3 48.8
2% 22.5 24.4 40.7
3% 20.5 22.1 35.3
4% 19.0 20.4 31.4
5% 17.7 18.9 28.4
6% 16.6 17.7 26.0
7% 15.6 16.6 24.1
8% 14.8 15.7 22.4
9% 14.1 14.9 21.1
10% 13.4 14.2 19.9