Exponential Growth

Posted by jns on 11 May 2007

Here’s a quick question with a pedagogical purpose. Would you buy a battery from this man?

“The energy capacity of batteries is increasing 5 percent to 8 percent annually, but demand is increasing exponentially,” Mr. Cooper[, vice president for business development of PolyFuel Inc., a company working on battery technology,] said.

[Damon Darlin and Barnaby J. Feder, "Need for Battery Power Runs Into Basic Hurdles of Science", New York Times, 16 August 2006.]

Forget basic hurdles of science, the basic hurdle here would seem to be an executive in a technical industry who doesn’t understand what exponential growth is.

In short: growth of something that is proportional to the current size of that thing is exponential growth. Thus, demand for batteries that grows 5% to 8% annually — i.e., 0.05 to 0.08 times current demand — is exponential growth.

The constant that governs how fast something grows exponentially is the “growth rate”. Small growth rate = slow growth; large growth rate = fast growth. In symbols, an exponential function of time, t, is

f(t) = A × e^st

where A is a constant amplitude and s is the growth rate. If s is relatively large, f(t) changes values rapidly; is s is very small, f(t) changes values slowly. If s happens to be a negative number, f(t) disappears over time, quickly or slowly depending on the size of s. The letter ‘e’ represents the base of natural logarithms. Why it shows up in the exponential function takes some explanation; for now, just think of it as a constant number nearly equal to 2.17 and don’t lose any sleep over it.^*

Many people think “exponential growth” means “grows really, really quickly”, but this is a misconception. It is true that power-law growth is generally faster than algebraic growth (for instance, multiplying a number over and over again by some number, say, 47) all other things being equal, but any particular exponential function will grow slowly or quickly depending on its growth rate. Think of a $0.15 deposit in a bank account that pays compound interest; the account grows exponentially but it’s going to be awhile before you’re a millionaire.

So please, please can we stop saying things like “Wow! That growth is so exponential! It’s huge!”

And if I were you, I don’t think I’d buy a battery from Mr. Cooper, either.
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^* In fact, ‘e’ is irrational (not expressible as the fraction of two integers, or whole numbers) and transcendental (not the solution to an algebraic equation, which is to say a polynomial with rational coefficients and integer powers). But that’s a lot of other story that we needn’t go into right now.