Recently I was looking at some data about world food production on the excellent Our World in Data site, and I discovered something very simple, but very surprising about the world’s population. We often hear (and I used to teach) about the threat of an exponentially growing population and the pressure it is supposed to be putting on our food supply and the natural resources that sustain it (land, water, nutrients, etc). But I found that the global population isn’t growing exponentially, and hasn’t been for at least half a century.

It has actually been growing in a simpler way than exponentially—in a straight line.

### What exponential growth is

Exponential growth (sometimes also called geometric or compound-interest growth) can be described by an equation in which time is raised to a power, i.e. has an exponent—hence the name. But it also can be described in simpler terms: the growth rate of the population, as a fraction of the population’s size, is a constant. Thus, if a population has a growth rate of 2%, and it remains 2% as the population gets bigger, it’s growing exponentially. And there’s nothing magic about the 2; it’s growing exponentially whether that growth rate is 2% or 10% or 0.5% or 0.01%.

Another way to put it is that the doubling time of the population—the number of years it takes to grow to twice its initial size—is also a constant. So, if the population will double in the next 36 years, and double again in the following 36 years, and so on, then it’s growing exponentially. There’s even a simple rule-of-thumb relationship between doubling time and the percentage growth rate: Doubling Time = 72/(Percentage Growth Rate). So that population with a 36 year doubling time, is growing at a rate of 2% per year.

But probably the simplest way to describe exponential growth is with a graph, so here’s how it looks:

This graphic not only shows the classic upward-curving shape of the exponential growth curve, but also how it contrasts with growth that is linear, i.e. in a straight line. Additionally, it demonstrates a simple mathematical result: if one quantity is growing exponentially and a second quantity is growing linearly, the first quantity will eventually become larger than the second, no matter what their specific starting points or rates of growth.

This isn’t just abstract math; it also illustrates the most famous use of exponential growth in political debate. It was put forward by the English parson Robert Malthus over two centuries ago. He argued that the human population grows exponentially while food production can only grow linearly. Thus, it follows inevitably that the population will eventually outgrow the food supply, resulting in mass starvation. This is the case even if the food supply is initially abundant and growing rapidly (but linearly). The upward-bending-curve of an exponentially-growing population will always overtake it sooner or later, resulting in catastrophe.

### Looking at real data

Critics ever since Malthus’ time have pointed out that his assumption that food production grows in a straight line is just that—an assumption, with little basis in theory. So I wasn’t surprised to see that the OWID data showed faster-than-linear (upward-curving) growth in global food production over the past half-century. What did surprise me was that the growth of the world’s *population* over that time period has actually been very close to a straight line.

Here’s that graph:

The graph looks very much like a straight line rather than the upward-curving exponential, but is that really the case? We can test this by calculating the value of what statisticians call the R^{2} (or “coefficient of determination”) for this curve. The closer it is to a straight line, the higher R^{2} will be, and if the data fits a straight line perfectly then R^{2} will be exactly 1.0.

So, what’s the actual value for this data? It’s 0.9992. I.e. the fit to a straight line isn’t quite perfect, but it’s very, very close.

### Is this some sort of artifact?

I was actually quite surprised at how well the data fit a straight line—so much so that I wondered if this was just an artifact of the method I used, rather than a real result. So I applied the same method—plot the data, fit a straight line to it, and calculate the value of R^{2}—to the data for some of the world’s largest countries and regions, rather than the world as a whole.

For several of these, the lines looked very straight and the value of R^{2} was almost as high as in the graph for the world as a whole, or even slightly higher, e.g.:

Country or Region |
R^{2} for the linear equation of Population vs. Time |

Brazil | .9977 |

India | .9954 |

Indonesia | .9995 |

Latin America and the Caribbean | .9994 |

North America | .9966 |

Pacific island small states | .9991 |

But for others it was considerably lower (e.g. .9777 for China, .9668 for the European Union) and two graphs proved clearly that the excellent fit to a straight line is a real result, not an artifact. These were the ones for Sub-Saharan Africa and Russia:

The point about the Sub-Saharan African graph is not simply that it has a lower value of R^{2} (0.964), but that its data deviates from the straight line in the way that an exponential curve should: higher than the straight line at the lowest and the highest time values, and lower than the straight line at the intermediate ones. It does fit an exponential curve quite well, thus showing that the method can pick out an exponential curve if the data do follow one. But this is the only region or large country for which that’s actually true.

The Russia graph doesn’t fit an exponential curve well at all—it actually curves downward overall, rather than upward as it should if it were an exponential—but it does show that the value of R^{2} can be much lower than 1.0 for real data. For Russia it’s 0.632. So as with the Sub-Saharan Africa case, it proves that the high value of R^{2} for the world as a whole is not an artifact caused by the method. It reflects the reality of the past 55 years.

Finally, since I and many of my readers are from the United States, here’s that graph:

For the US as for the whole world, population growth over the past half-century has been quite close to a straight line; the R^{2} is 0.9956.

### A direct test of whether growth is exponential

These graphs and R^{2} values seem to indicate that linear growth is the best model for the world population over the past 55 years, but there’s another way to show that it’s not exponential. As I said above, exponential growth occurs when the __percenta__g__e__ growth rate remains __constant__ as the population gets bigger. So a simple test is to graph the percentage growth rate over time, and see whether it’s a constant—i.e., a horizontal line. So here’s that graph:

This result, like the others, is quite clear. The percentage growth rate is not a constant, as it should be if the population were growing exponentially. Rather, it has been dropping steadily over the past half-century, from over 2.0% in the early sixties to below 1.2% now.

### What exponential growth is __Not__

So, we should stop saying that the world’s population is growing exponentially. That hasn’t been the case for at least 50 years. Exponential growth clearly doesn’t describe the global reality of the twenty-first century.

But there’s actually a second reason to stop saying that the global population is growing exponentially, and that’s because the term is so commonly misused and misunderstood. Note the next few times that you hear someone use the word, and I think you’ll find that it’s not being used in the sense of “constant percentage-growth-rate” or “constant doubling-time” or even just “an upward-bending curve.” Rather, it’s being used—often with an emphatic stress on the “-nen-” syllable and an implicit exclamation mark at the end of the phrase—to mean “rapidly” or “quickly” or “fast” or “big.”

That way of speaking is common, but it’s also just plain wrong. Remember the example that I started with: the exponential growth rate can be high (e.g. 10%) or low (e.g. 0.01%) or intermediate (e.g. 2%). In every case it’s exponential growth, but it’s very fast exponential growth if the growth rate is 10% and very slow exponential growth if it’s 0.01%.

I’m not that sanguine about getting people to go back to using “exponential” in its correct sense, but I think it’s at least worth a try. After all, we already have several other good words for that other, incorrect meaning—e.g. “fast” or “big.”

### Implications

The results don’t just imply that we should talk about population growth differently, but also that we need to re-think how it relates to food production. There is good news in these data, because they show that hunger and environmental catastrophe is not at all inevitable. Malthus’ argument just doesn’t fit reality.

While linear growth has its challenges, it’s far easier to deal with than exponential growth. The distinction between growing exponentially and growing in a straight line does matter. On that point, at least, Malthus got it right.

Posted in: Food and Agriculture, Science Communication
Tags: agriculture, demography, exponential growth, Hunger, linear growth, Malthus, Population

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