Some Thoughts on Exponential Growth

We hear the phrase, "exponential growth" a lot these days, because the phrase describes the manner in which a number of quantities are increasing in time. For example: bank balances; debt; the economy; world population; the extinction of species; the amount of carbon dioxide in the atmosphere; the melting of ice reservoirs are all increasing exponentially with time. The amount of water vapor that the air is capable of holding increases exponentially with temperature. Something grows exponentially when it increases in amount by the same fraction (or percentage) during each successive equal unit of time. Thus a bank account that earns 5% per year interest increases in size each year by 5%, or the fraction 0.05. You will here it said that exponential growth involves increase at a constant rate. This is inaccurate. Rate is the change in amount of something per unit of time. Thus if you travel 10 miles in 10 minutes, you are traveling at a rate of 1 mile per minute, or 1mile/min. Something that grows at a constant rate increases linearly, not exponentially, in time. Exponential growth results in a graph of amount versus time that inexorably curves upward. The increase is much more rapid than linear.

For example, suppose you are growing bacteria in a petri dish. Bacteria grow exponentially. Let's say that at the beginning, you start with 1 bacterium in the dish, and that these particular bacteria divide every minute. After 1 minute, you will have 2; after 2 minutes, you will have 4, and so on:

Time, minutes          Number of bacteria in dish

0                                  1

1                                  2

2                                  4

3                                  8

4                                 16

5                                 32

6                                 64

7                                128

You can see that with each successive minute the number of bacteria doubles from the number in the preceding minute. Here's a graph of number of bacteria on the vertical axis and number of minutes on the horizontal axis. Notice the inexorable upward curving. This curve is sometimes called a "hockey stick."

Generally, if a quantity is growing by a certain fraction, f, in each successive equal unit of time, the size of the quantity after n of these units of time have passed is calculated using the following relationship:

Size after n time units = Initial size * (1 + f)^n.

The symbol ^ means that the n is an exponent. As a concrete example, suppose that you start with $100 in a savings account and that you earn 4% interest compounded annually (yeah, I know--dream on!). How much will you have in the bank after 5 years?

Amount after 5 years = $100*(1 + 0.04)*(1 + 0.04)*(1 + 0.04)*(1 + 0.04)*(1 + 0.04) = $121.66

Yeah, 4% is a lousy interest rate. Note that we have multipled by (1 + 0.04) 5 times , which is what (1 + 0.04)^5 means. After 5 more years, your balance will be $148.02. Note that the amount of increase is bigger than the amount of increase in the first 5 years.

An interesting thing to calculate for a given fraction of increase is the amount of time required for the original quantity to double. In other words, how many time units must pass before the quantity is twice as big as it was initially? This can be estimated pretty easily from the following formula:

Number of time units for doubling = 70/percentage of increase = 0.7/fraction increase = 0.7/f

It will take about 17 years for your balance to double. It will double again in another 17 years, and will then be about 4 times larger than the starting balance.

The key thing about exponential growth is that quantities grow along a path that curves upward in time. Things can get unmanageably large pretty quickly. People don't get too concerned at the idea of an economy that grows at the rate of 5% per year, because most people think in terms of linear growth when they see this statistic. But this meant that 14 years, from now, the economy will be twice as large as it is now. And 14 years after that, it will be 4 times larger than it is now. If the economies of all of the developed countries are growing exponentially (most of them are, particularly those of China and India, which are growing at double digit percentages), doesn't it make sense that the raw materials (metals, wood, fossil fuels, minerals, rock, etc) consumed in this growth will be used up in very short order? What will happen to our growing economies then? Think about it.