More Trouble with the Curve

Suppose the earth’s atmosphere were a bathtub, and that human-generated (anthropogenic) carbon dioxide emissions were being dumped into the bathtub through a faucet. An exponential curve included in the last post on this blog shows the growing amount of carbon dioxide in the bathtub starting in 1750, when the industrial revolution began, and extending through the year 2007. The amount of CO₂ in the bathtub grew very slowly for a while but then increased at faster and faster rates. The exponential curve showing the amount of carbon dioxide that has flowed into the bathtub since 1750 is steeper and steeper over time because more carbon dioxide flows through the faucet every year from burning more fossil fuels.

A related graph, below, represents how much carbon dioxide is coming out of the faucet every year from use of fossil fuels, rather than how much has entered the bathtub. The graph shows annual carbon dioxide emissions by year from 1870 to 2010. The shape of this curve is also exponential, like the growing amount that has flowed into the bathtub. The similarity in shape is a special characteristic of exponential curves: a graph of the rate of change of an exponential function is itself an exponential function.

      The figure above comes from p.4 of an International Energy Agency publication called “2012 CO₂ Emissions Overview.”

This graph shows, for example, that between the years 1970 and 2010 annual human-caused carbon dioxide emissions from burning fossil fuel more than doubled, from about 15 gigatons per year to more than 30 gigatons per year. (A gigaton is one billion tons.)

The previous blog post reported that humanity “had not yet begun” to change the exponential growth of carbon dioxide (and thus carbon) in the atmosphere, but that might not be quite accurate. I learned that global emissions “stalled” in 2014, meaning that annual emissions did not grow between 2013 and 2014 but were the same each year. Ignoring the fact that the graph above represents only the years through 2010 (not 2014), stalling means that the graph is flat for the years 2013 and 2014, at a figure of about 32 gigatons per year.

This stalling phenomenon has happened several times over the past 40 years; see, for example, the noticeable dip in the curve above during a recession in the 1980s. Nonetheless, the exponential growth trend has proved to be far more powerful over the years than the few times when annual emissions stalled. Starting in 2015 or soon afterward, the exponential growth in CO₂ emission each year might well increase again, as it has been doing for decades with only brief “blips” in the exponential growth pattern.

However, for sake of argument, suppose that CO₂ annual emissions from burning fossil fuels truly peaked in 2014 and will never grow larger. Nonetheless, in order to avoid more than a 2° C increase in global temperature, scientists say that humanity would need to reduce annual emissions from fossil fuel use to zero by about 2050, i.e. rapidly reduce the flow from our anthropogenic CO₂ “faucet” over the next few decades, and then turn the faucet off entirely by mid-century. The curve in the graph above would need to plummet downward and then reach zero annual emissions by 2050.

The last time annual anthropogenic CO₂ emissions were anywhere near zero was hundreds of years ago, before the industrial revolution, as can be seen above. The exponential growth of annual emissions has been going on so long with such potentially catastrophic results that the World Bank warned in 2012, and continues to warn, that unless current patterns change quickly the earth will probably become 4° C warmer by the year 2100 (equivalent to more than a 7° Fahrenheit increase). Results would be devastating, they say. How many people have understood and accepted this information?

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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.

The 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.