Optional Topics
Exponential Growth
Albert A. Bartlett
University of Colorado, Boulder
If something is growing at a constant rate, such as P = 5% per year, we refer to the growth as steady growth. It is also called exponential growth because the size, N, of the growing quantity at some time t in the future is related to its present size, N0 (at time t = 0), by the exponential function
(E.1)
where e = 2.718 . . . is the base of the natural logarithms and k is the annual percent growth rate P divided by 100:
K = (E.2)
Note that if k is positive, N increases exponentially with time (exponential growth). If k is negative, N decreases exponentially with time (exponential decay). Some examples of exponential decay are radioactive decay (Section 44.5) and the decay of the quantity of charge on the plates of a capacitor as the capacitor is discharging through a resistor (Section 28.4). Figures 1 and 2 are representative graphs of exponential growth.
The condition of steady growth is represented by the simple differential equation
(2)
In words, this equation says that the rate of change of the quantity N is proportional to N. We can rearrange Equation 3 to give
In this form we see that the fractional change of N per unit time is constant. An example would be the case where the quantity N is growing 6% per year. In this case P = 6% per year and k = 0.06% per year.
The solution of Equation 3 is Equation 1. If we can show that a quantity N changes with time according to Equation 3, it follows automatically that N will obey Equation 1.
For example, the fundamental concept of compound interest on a savings account is that the interest 
N added to the number N of dollars in the account in the time interval t is proportional to the number of dollars in the account. The constant of proportionality is the interest rate. If N = $250, P = 8% per year, and t = 1 year, we have simple compounding once a year. The interest in that year is N = 0.08 $250 = $20 so that the value of N at the end of the year would be $250 + $20 = $270. In the next year N = 0.08 $270 = $21.60, and at the end of the second year, N = $270 + $21.60 = $291.60. Suppose the interest were 8% per year, compounded semiannually. In the first half year, N = (0.08/2) $250 = $10; and at the end of the first half-year, N = $260. In the second half year, N = 0.04 $260 = $10.40; and at the end of the first full year, N = $270.40. At the end of the first year, compounding annually gave N = $270. Compounding semiannually gave N = $270.40. This suggests a very fundamental fact. The more frequently we compound the interest, the more rapid is the increase in the size of N. Equation 3 represents the limiting case where t approaches zero and the interest is compounded continuously. In this case at the end of one year we use Equation 1:
Many banks calculate interest by compounding continuously. This leads to newspaper ads such as one that says "9.54% annual yield" which corresponds to a "9.11% annual rate." What this means is that the rate of 9.11% compounded continuously for one year gives the same result (yield) as a rate of 9.54% compounded once. In other words
(E.4)
It has been shown that the number of miles of highway in the United States obeys Equation 3 so that the number of miles of highway will grow exponentially according to Equation 1. In steady growth, it takes a fixed length of time for a quantity to grow by a fixed fraction such as 50%. From this, it follows that it takes a fixed longer length of time for that quantity to grow by 100%. Let us calculate the time required for the quantity N to double in value, which is called the doubling time, T2. We can obtain an expression for T2 by writing Equation 1 as N/N0 = ekt and taking the natural logarithm of each side:
If we set N = 2N0 (that is, we double N0), then T2 (which is the time t when N = 2N0) is
Since k = P/100, this becomes
(E.5)
Likewise, if we want the time needed for N to triple in size, we use the natural logarithm of 3:
EXAMPLE 1 E.1 Compound Interest÷The Eighth Wonder
Suppose you put $15 in a savings account at 9% annual interest to be compounded continuously. How much is in the account at the end of 200 years?
Solution N0 = $15, k = 9/100 = 0.90 per year, and t = 200 years. Therefore, from Equation 1 we have
Now you can see why a famous financier once said that he could not name the seven wonders of the ancient world but surely the eighth wonder would have to be compound interest!
EXAMPLE 2 The Consequences of Inflation
Let us use the doubling time instead of the quantity e to estimate the consequences of an annual inflation rate of 14% that continued for 50 years.
Solution First, we calculate T2 from Equation 5
This inflation rate causes prices to double every five years!
In the next step, we calculate the number of doubling times in 50 years:
Finally, we count up the consequences of each doubling by making use of a table, which shows us that, in 10 doubling times, prices increase by a factor of 1024, which is approximately 1000. Thus, in 50 years of 14% annual inflation, the cost of a $4 ticket to the movies increases to roughly $4000! (See Fig. 3)
It is very convenient to remember that 10 doublings give an increase by a factor of approximately 103, that 20 doublings give an increase of a factor of approximately 106, that 30 doublings give an increase by a factor of approximately 109, and so on.
EXAMPLE 3 The Increasing Rate of Energy Consumption
For many years before 1975, consumption of electrical energy in the United States grew steadily at a rate of about 7% per year. By what factor would consumption increase if this growth rate continued for 40 years?
Solution In this case, P = 7, and so the doubling time from Equation 5 is
and the number of doublings in 40 years is
Therefore, in 40 years the amount of power consumed is 24 = 16 times the amount used today. That is, 40 years from now we will need 16 times as many electric generating plants as we have at the present. Furthermore, if those additional plants are similar to today's, then each day they will consume 16 times as much fuel as our present plants use, and there will be 16 times as much pollution and waste heat to contend with!
EXAMPLE 4 Annual Increase in World Population
Populations tend to grow steadily. In July 1987 we saw reports that the population of the earth had reached 5 109. The world birth rate was estimated to be 28 per 1000 each year, while the annual death rate was estimated to be 11 per 1000. Thus, for every 1000 people, the population increase each year is 28 ö 11 = 17. For this growth rate we find
P = 100 k = 1.7% per year
This growth rate seems so small that many people regard it as trivial and inconsequential. A proper perspective of this rate appears only when we calculate the doubling time:
This simple calculation indicates that it is most likely that the world population will double within the life expectancy of today's students! At the most elemental level, this means that we have approximately 41 years to double world food production.
What is the annual increase in the earth's population? Since for one year N << N, we can get a good answer from Equation E.3:
This annual increase in the world population is roughly one third of the population of the United States.
Some illuminating calculations can be made based on the assumption that this rate of growth has been constant and will remain constant. These calculations demonstrate that the growth rate has not been constant at this value in the past and cannot remain this high for very long.
EXAMPLE 5 When Did Adam and Eve Live?
When we use Equation 1, setting N = 5 109, N0 = 2 (Adam and Eve) and k = 0.017 (from Example 4), we have
This gives t = 1273 years ago, or about 714 A.D., for the time when A and E were around. This result proves that through essentially all of human history the population growth rate was very much smaller than it is today. In fact, it must have been near zero through most of human history.
EXAMPLE 6 Growth of Population Density
The land area of the continents (excluding Antarctica) is 1.24 1014 m2. If this modest annual growth rate of 1.7% were to continue steadily in the future, how long will it take for the population to reach a density of one person per square meter on the continents?
Solving, we find t is slightly less than 600 years.
EXAMPLE 7 Growth of the Mass of People
If this very low rate of growth continues, how long will it take for the mass of people to equal the mass of the earth (5.98 1024 kg)? (Assume that the mass of an average person is 65 kg.)
This gives a value of t of about 1800 years! We have assumed that the mass of a person is 65 kg.
The last two examples prove that the growth rate of world population cannot stay as high as it presently is for any extended period of time. Although world agricultural production has been just barely keeping pace with world population growth, millions are malnourished and many people are starving. However, we will not have to double food production in 41 years if we can lower the worldwide birth rate. If we fail to double world food production in 41 years, then the death rate will rise. Dramatic increases in world food production in recent decades are due almost exclusively to the rapid growth of the use of petroleum for powering machinery and for manufacturing fertilizers and insecticides. Indeed, it has been noted that "modern agriculture is the use of land to convert petroleum into food." The student must wonder how much longer we can continue the long history of approximately steady population growth when our food supplies are tied so closely to dwindling supplies of petroleum.
This brief introduction to the arithmetic of steady growth enables us to understand that, in all biological systems, the normal condition is the steady-state condition, where the birth rate is equal to the death rate. Growth is a short-term transient phenomenon that can never continue for more than a short period of time. Yet in the United States, business and government leaders at all levels, from local communities to Washington, D.C., would have us believe that steady growth forever is a goal we can achieve. They would have us believe that we should continue our population growth (the U.S. population increases by about 2 million people per year) and the growth in our rates of consumption of natural resources. We now hear about "sustainable growth" as though the addition of the adjective "sustainable" would render inoperable the laws of nature.
In contrast to all this optimism, please remember that someone once noted that "The greatest shortcoming of the human race is our inability to understand the exponential function."
Suggested Readings
Bartlett, A. A., Civil Engineering, December 1969, pp. 71 - 72.
Bartlett, A. A., "The Exponential Function," The Physics Teacher, October 1976 to January 1979.
Bartlett, A. A., "The Forgotten Fundamentals of the Energy Crisis," Am. J. Physics 46:876, 1978.
Kerr, R. A., "Another Oil Resource Warning," Science, January 27, 1984, p. 382.
Problems
- In 1626, Manhattan Island was purchased for $24. Assuming a continually compounded interest rate of 4.4%, calculate the current land valuation of the island.
- The following "mystery" was taken from Deborah Hughes-Hallett: Elementary Functions, W. W. Norton, 1980, p. 264.
The police were baffled by what seemed to be the perfect murder of a girl who had been found, apparently suffocated, in her kitchen. Finally, Sherlock Holmes was called in. With the aid of Dr. Watson's knowledge of botany, the mystery was solved and the following story told. The girl had been making bread in her kitchen, whose dimensions were 6 ft by 10 ft by 10 ft. She had formed the dough into a ball of volume 1/6 cubic feet and turned away to wash some dishes. At that moment Holmes' enemy, Professor Moriarty, had added a particularly virulent strain of yeast to the bread. As a result, the bread immediately started to rise, tripling in volume every 4 minutes. Before long, the dough filled the room, stopping the clock at 3:48 and squashing the girl to death against the wall. By the time Inspector Lestrade of Scotland Yard reached the scene the next day, the yeast had worked itself out and the dough returned to its original size.
At what time did Professor Moriarty add the yeast?
Figure Captions
Figure 1 If you try to draw an ordinary graph of the size versus time of anything that is growing steadily, the graph will go right through the ceiling.
Figure 2 Exponential growth curve N/N0= ekt. Note, N/N0 has the value 1 at t = 0.
Figure 3 Price of a $4 ticket with 14% annual inflation. The cost at t = 0 is N0 = $4.
|
No. of Doublings |
Price Increase Factor |
|
1 |
2 = 21 |
|
2 |
4 = 22 |
|
3 |
8 = 23 |
|
4 |
16 = 24 |
|
5 |
32 = 25 |
|
6 |
64 = 26 |
|
7 |
128 = 27 |
|
8 |
256 = 28 |
|
9 |
512 = 29 |
|
10 |
1024 = 210 |
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