Exponential Growth - Bartlett's Law

Professor Bartlett's lesson remains valid


Mathematics of Exponential Growth

Exponential growth is a statistical regularity of great compass. You will easily be able to adduce many examples of it. It is the most important of predictive tools, especially in economics. A sound theoretical basis for the exponential growth is sometimes not apparent. If the rate of growth of a quantity is proportional to the quantity itself, then the growth of the quantity will be exponential. That is, if dq/dt = q/T, then q = q0 exp (t/T), where q is the growing quantity, t is the time, T the e-folding time, and q0 is the amount of the quantity at t = 0. The exp() function is the power t/T of the mysterious number e = 2.718..., the base of natural logarithms, which appears in the solution of the differential equation. The doubling time, Td = T ln 2 or about 0.693 T, is more often used than the e-folding time. If the quantity is decreasing (negative T), Td is the familiar half-life as used for radioactivity.

["Linear If we plot q against t in a straightforward linear plot, what appears is the familiar rising curve of exponential growth. The slope of the curve at any point is the ratio of the growing q to the constant T, as shown in the figure, and which is the meaning of the differential equation we began with. A powerful analytical tool is the semilog plot, in which the logarithm of q (to any base) is plotted against t. The curve now becomes a straight line that is easy to draw and analyze. The figure shows how to get the doubling time directly from a semilog plot. The intercept on the log axis is q0, the value of q when t = 0. The time axis can begin anywhere, not just zero, since the curve is a straight line. Semilog graph paper is available, so logarithms do not actually have to be used. The linearity of the plot proves the exponential nature of the growth. Should the curve bend upwards as time increases, the growth is superexponential, and if it bends downwards, the growth is subexponential. It is significant and important to recognize both these cases.

Beginnings and Endings

Whenever exponential growth is encountered, the question of beginnings and endings is very interesting. The mere detection of exponential growth, and the estimation of a doubling time, are not as fundamental. Exponential growth has been observed in bacterial colonies, the population of humans with technology and weapons available, the number of scientific periodical titles, the production of crude oil, and the number of PhD's granted yearly in the United States [see D. Goodstein, Amer. J. Phys., 67, 183-186 (1999)]. Each of these examples has its own dynamics, but each has produced very nice linear semilog plots.

Considering PhD's in the United States, if we extrapolate the line to earlier dates, we find that in 1850 there would be 0.1 PhD granted. This shows that the curve cannot be extrapolated backwards into a region where the statistics are invalid due to small sample size. Beginnings, therefore, are not subject to statistical regularities. In the case of scientific journal titles, exponential growth was not seen until there were about ten titles.

Whereas beginnings are of interest to historians, endings are of vital importance in economics. Professor A. A. Bartlett of the University of Colorado made a vigorous effort to awaken the general population to the fact that all exponential growth must end. If this proposition has not already received a name, it may well be called Bartlett's Law, as in this note. Most economists, politicians, and promoters blindly extend the line to the right, with conclusions as ridiculous as those obtained by extending it to the left. Of the examples quoted above, only human populations have not seen the end of exponential growth. Scientific journal titles were supposed to reach 1,000,000 by 2000; they leveled off at 40,000 shortly after exponential growth was boasted in 1950. There should be 20,000 Physics PhD's in 2000; there will actually be about 1000 or fewer, having abruptly leveled off at this number in 1970. United States crude oil production peaked in 1970 after exponential growth since the beginning of the century, and is now in exponential decline. Bacterial colonies grow exponentially until food is exhausted; then they die and release spores that drift around until more food is located, so the pointless process can be repeated. Humans appear to do the same, except for the spores.

Endings

Three kinds of endings can be recognized. The first is the abrupt catastrophe, when some necessity becomes exhausted. The second is an abrupt leveling-off, when increasing mortality balances the growth. The third is the Hubbert logistic decline, which eventually becomes an exponential decline. All three types are well-documented. They may apply to complete systems made up of many subsystems so that statistical regularities become important, or to subsystems large enough for similar regularities to be valid. Individual cases are unpredictable by statistics. For example, individual oil provinces have been observed to obey Hubbert's Law quite well, but world-wide production may well suffer a catastrophe, due to the nature of the market.

Exponential growth is characterized as growth without either restriction or forced encouragement. Primitive humans endured local cycles of growth and catastrophe, while the overall population remained stable, the rate of reproduction balancing the mortality. Where weather or civilization lowered the rate of loss, the population increased gradually. The encouragement of reproduction was regarded as very desirable, and has always been present in human society, usually in weird and unpleasant ways.

The first recognizable example of Bartlett's Law in human history will probably be the exhaustion of cheap, flexible energy in the forms of liquid and gaseous petroleum. The reserves of this definitely finite resource can now be estimated relatively accurately, the accuracy of Hubbert's Law has been established, and the workings of the market (which is totally ignorant of Bartlett's Law) are all fitting together nicely to ensure that low prices are maintained up to the moment when the supply first falters, and then prices will be driven up vertically when pricing becomes the means of allocation of an insufficient supply. The actual catastrophe will be economic, not material, and there still well be a good deal of petroleum remaining at this epoch. After this, energy will still be available; it simply will no longer be cheap and convenient.

Predictions

Thomas Malthus predicted disaster from increasing population when he observed that population increased exponentially (using figures from the early United States census), while arable land could be brought into production only arithmetically (at a constant rate). When the curves crossed, famine would ensue. These observations were made before 1800. It is easy to see why they were not borne out, except locally and temporarily, but this does not invalidate his analysis. Now we have exponential growth versus a fixed resource, not an arithmetically increasing one.

Petroleum is not simply a source of gasoline. It is essential to the low- cost production of food, in the form of Diesel fuel and as a chemical feedstock for the manufacture of fertilizers. The rise in price of oil will deny it to the poorest of the human population, depriving it of the food that has been forced from the ground by the use of cheap energy and chemicals. This is only one aspect of the problem of the exhaustion of resources that will become worse and worse as the world population increases. Human population cannot increase forever at an exponential rate; it will level off -- but the Earth cannot support indefinitely even its current population. The pain does not come gradually, but all at once, in these cases.

It is exceedingly difficult for humans, especially politicians and priests, to realize the scale of the universe in time and space, and that seemingly substantial and everlasting things are merely temporary and insignificant bright flashes in eternity.


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Composed by J. B. Calvert 1999
Last revised 10 May 1999