Edited by Denis Morel
Thank you very much, Hugh.
It's a great pleasure to be here, and to have a chance just to share with you some very simple ideas about the problems we're facing. Some of these problems are local, some are national, some are global.
They're all tied together. They're tied together by arithmetic, and the arithmetic isn't very difficult. What I hope to do is, I hope to be able to convince you that the greatest shortcoming of the human race is our inability to understand the exponential function.
Well, you say, what's the exponential function?
This is a mathematical function that you'd write down if you're going to describe the size of anything that was growing steadily. If you had something growing 5% per year, you'd write the exponential function to show how large that growing quantity was, year after year. And so we're talking about a situation where the time that's required for the growing quantity to increase by a fixed fraction is a constant: 5% per year, the 5% is a fixed fraction, the “per year” is a fixed length of time. So that's what we want to talk about: its just ordinary steady growth.
Well, if it takes a fixed length of time to grow 5%, it follows it takes a longer fixed length of time to grow 100%. That longer time's called the doubling time and we need to know how you calculate the doubling time. It's easy.
You just take the number 70, divide it by the percent growth per unit time and that gives you the doubling time. So our example of 5% per year, you divide the 5 into 70, you find that growing quantity will double in size every 14 years.
Well, you might ask, where did the 70 come from? The answer is that it's approximately 100 multiplied by the natural logarithm of two. If you wanted the time to triple, you'd use the natural logarithm of three. So it's all very logical. But you don't have to remember where it came from, just remember 70.
I wish we could get every person to make this mental calculation every time we see a percent growth rate of anything in a news story. For example, if you saw a story that said things had been growing 7% per year for several recent years, you wouldn't bat an eyelash. But when you see a headline that says crime has doubled in a decade, you say “My heavens, what's happening?”
OK, what is happening? 7% growth per year: divide the seven into 70, the doubling time is ten years. But notice, if you want to write a headline to get people's attention, you'd never write “Crime Growing 7% Per Year,” nobody would know what it means. Now, do you know what 7% means?
Let's take an example, another example from Colorado. The cost of an all-day lift ticket to ski at Vail has been growing about 7% per year ever since Vail first opened in 1963. At that time you paid $5 for an all-day lift ticket. What's the doubling time for 7% growth? Ten years. So what was the cost ten years later in 1973? (showing slides of rapidly increasing prices) Ten years later in 1983? Ten years later in 1993? What was it last year in 2003, and what do we have to look forward to? (shows "2003: $80; 2013: $160; 2023: $320; audience laughter)
This is what 7% means. Most people don't have a clue. And how is Vail doing? They're pretty much on schedule.
So let's look at a generic graph of something that’s growing steadily. After one doubling time, the growing quantity is up to twice its initial size. Two doubling times, it's up to four times its initial size. Then it goes to 8, 16, 32, 64, 128, 256, 512, in ten doubling times it's a thousand times larger than when it started. You can see if you try to make a graph of that on ordinary graph paper, the graph’s gonna go right through the ceiling.
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