Trouble with the Curve (The Exponential Curve, That Is)

Most human beings find it challenging to understand how exponential functions work in the real world. For example, I was shocked to read in The Collapse of Western Civilization that more than half of anthropogenic carbon dioxide has been emitted since the mid-1970s. I shouldn’t have been surprised, because I do understand the mathematics, but I was. My shock caused me to look for graphs of greenhouse gas emissions over time. The one below comes from the U.S. Department of Energy.

cumulative_global_1751_2007The shape of this curve is characteristic of all exponential functions: it is almost flat at the outset (at left), with rapid and accelerating growth over time (i.e., moving right). Change happens faster and faster as time goes by. In this case, world population growth and increasing global wealth per-capita after the industrial revolution largely explain the exponential growth in CO2.

Although exponential functions are taught in high school math and science classes all over the globe, people are not very good at understanding exponential growth in the real world.

A classic teaching example of an exponential function is based on an ancient Persian or Indian story about the king who was so pleased with one of his wise men, who had invented the game of chess, that he offered the man a reward of his own choosing. The wise man (too clever for his own good, it turned out) asked for very little: just one grain of rice on the first square of the chess board, two grains on the second, four on the third, eight on the fourth, and so on, with the number of rice grains doubling on each successive square. The king agreed that the request was modest. He was wrong. When his courtiers tried to complete the task they found that it would require more rice than existed in the entire kingdom, indeed in the entire world. Instead of a reward, the so-called wise man lost his head.

The psychological difficulty faced in understanding exponentials stems from the fact that the left side of an exponential function is so flat. Adding a few grains of rice at a time, square after square, is an achievable task. Some people refer to the rapid acceleration characteristic of the right side of the exponential curve as “the second half of the chessboard.” That is where intuition and daily experience fail, and where, for many people, the rapid acceleration of change is unexpected, even shocking. A few doublings of the rice in the story is not hard for the king’s courtiers but 63 doublings is impossible.

Here is another example that is easier to try yourself. Take a sheet of newspaper and fold it in half. Now fold it in half again. How many times do you think you can do that, doubling the thickness of the folded paper each time? Naturally you aren’t going to guess you can fold it 100 times—and that’s a good thing, because calculations show that folding the paper in half only 100 times (one source says 103) the thickness of the stack of paper, doubled in size every time, would be about the size of the known universe!

Poor understanding of exponential functions has implications not only for policymakers but for ordinary people, such as in managing their household finances. In a study published in 2009, researchers present evidence that decisions related to borrowing, savings, portfolio choices, and wealth are all affected by what they call “exponential growth bias,” or the tendency of many people to think of exponential functions (like the compounding of interest in tax-deferred retirement accounts) as linear. But looking at a graph shows how different an exponential function is compared to a straight line function.

For greenhouse gas emissions, the world must change the graph shown above into an S shape; that is, the rapidly increasing curve shown in the graph needs to bend to the right over time until it levels off (meaning humanity stops adding excess greenhouse gases to the atmosphere). It is not intuitive for policymakers how much of a difference timing makes for exponential functions. If we had begun to reduce emissions in 1975 we could have leveled off at a much, much lower cumulative total of emissions than is possible now (and the world hasn’t yet begun).

The “wise man” in the story of the chessboard lost his head. If the world’s policymakers fail to act very, very quickly, consequences will not be about fictional people and will occur worldwide.


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