More on Quadratic Population Growth

[This is part of my series on Thomas Malthus’ “Essay on the Principle of Population,” first published in 1798. You can find an overview of all posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”]

In my previous post, I explained why quadratic population growth makes more sense than exponential population growth as the go-to model for how human populations behave. The underlying logic is this:

If a population has a constant target for its population density, an increase in its size can only be accommodated by geographic expansion. As long as this is possible in two dimensions, the area of settlement will grow quadratically. And since by assumption the size of the population is proportional, it also increases quadratically.

After some time, expansion might only be possible in one dimension, that’s when population growth becomes linear. And eventually, all available land is settled, so population growth has to come to an end. How all this works out depends on the geography and accidental circumstances, eg. whether a continent like Australia or the Americas is discovered.

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One way this can work is via settlement of new lands that are not yet populated. But there is also another way. Suppose the world is already settled at a low population density, eg. by hunter-gatherers. At some point, there is a technological leap: the domestication of animals. This makes much higher population densities possible.

The initial herder population can now take advantage of this and grow into the new opportunity. They find themselves in a world that is practically not populated apart from a few hunter-gatherers. So, an expansion can begin that initially should be with quadratic population growth.

However, it is not necessary that the initial herder population settles the whole world. Other populations of hunter-gatherers might learn herding from those who have already mastered it. The expansion could hence be cultural and not via migration, or a mix of the two. It is immaterial for the aggregate effect how this works.

Note that the theoretical derivation for quadratic population growth does not mean that there is an “inevitable” increase or what is called “population pressure.” There is only growth when there is an opportunity that a population can grow into. If there is not, the population stabilizes at the constant population density they target.

This may sound absurd if you are under the influence of the Malthusian worldview: Populations must always grow, they cannot just stabilize at some level without some brute force! But as I will show in later posts, there is nothing wrong with the idea that human population can and often do stabilize. It has happened many times, actually so often that stable population sizes were the normal situation in the past, not population growth and even at a fast pace.

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Now, so far this was only a theoretical speculation what a plausible model for population growth should be. This proves nothing. As I demonstrated in my last post, though, data for the Amish from 1900 to 2017 are very well explained by quadratic population growth, and this works better than an explanation via exponential population growth. It seems as if the Amish had two regimes: first one with lower growth until about the 1960s, then a transition, and evntually a regime with higher growth from the 1980s on.

The Amish are a rather special example, so I wondered how this works for other populations, preferably at a larger scale. Turns out, quadratic growth is also there a better model than exponential growth. Actually, so far I have not come across any examples where it is the other way around.

Of course, quadratic growth only applies when there is an opportunity that a population can grow into and as long as the expansion is unimpeded. Otherwise there should be stabilization or if opportunities evaporate even shrinkage.

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Here are some examples, all for the time frame from 1820 to 2000 and for European countries where data are rather unreliable. Note that before 1820, growth behavior could be different, eg. slower and also with periods of stagnation or even shrinkage.

Let’s start with Europe as a whole. I always do one analysis with the assumption of underlying quadratic population growth (“Root”) and another with underlying exponential population growth (“Logarithm” or “LN”). That is I transform the populations sizes with the respective functions and then fit a linear function where I supply the equation and the R squared.

Both models yield a good fit. However, quadratic growth is slightly better than exponential growth as an explanation. We also see the effect in the second chart as also with the Amish: there is a slight bulge. The two ends are below the trendline, and in the middle of the function is above. That’s what should happen if the true model is quadratic growth.

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Let’s now look at individual countries, first two where the fit is not as perfect, but still quite good: Germany and the UK. Everything here again supports a case for quadratic over exponential population growth:

Apparently there was a regime change for Germany after World War I, with slower growth afterwards. The situation looks similar for the UK although there seems to be more of a transition between two regimes:

We also see the same phenomenon again as above that there is a slight bulge after a logarithmic transform.

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Next come two examples with a more stable development: Finland and Sweden:

Maybe there is a new regime from about the 1960s on with slightly slower growth, but it is hard to tell. Before that, there are some small fluctuations, but without an apparent break. The situation for Sweden is even more stable:

The fit under the assumption of quadratic growth is very good with only minimal fluctuations.

In sum: for Europe as a whole, Germany, the UK, Finland, and Sweden as well as the Amish quadratic population growth appears to be a better explanation than exponential population growth. It also makes far more sense in theory because exponential population growth leads to absurd conclusions.

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In my last post, I have demonstrated what a huge difference the model can make for extrapolations. On the data set itself, the divergence may be rather modest. But with exponential growth, this is then magnified, in particular for forecasts over long periods of time. Quadratic growth is much slower by comparison and so this can lead to a major gap between extrapolations. The choice of an adequate model is hence much more important than fitting it to the data.

Now, if exponential population growth is already in theory less plausible than quadratic population growth and is also an inferior explanation for the empirical data, there is no reason to view it as the go-to model, especially not for forecasts over the long run. Exponential population growth might just be an idée fixe that Thomas Malthus popularized more than two centuries ago and that has become a part of our culture via the Malthusian worldview.