Quadratic Population Growth and Cowen’s Second Law

[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 my posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”]

What baffles me is this: I have been writing about the question of whether quadratic or exponential population growth is the go-to model for human populations (see here, here, and here). My data are for the Amish from 1920 until 2017, Europe as a whole, Germany, the UK, Finland, and Sweden from 1920 until 2000, and England and Wales from 1815 until 1869 (overlap here with the data for the UK).

My empirical finding is this: Over the longer run, an explanation via quadratic population growth always beats one via exponential population growth although you see some deviations from it, too. Over shorter time periods (the Amish in two growth regimes from 1920 to 1960 and from 1990 to 2017 as well as England and Wales from 1815 until 1869), an explanation via quadratic population growth is almost too perfect to believe.

Now, I may be hunting for confirmation. But that is not so. I first developed a theoretical prediction without looking at any data. As I will explain in a moment, I expected quadratic population growth to be a good approximation, but not such a good one. Only then did I begin to look at concrete data sets. It began by pure accident when I found a post about the Amish by Lyman Stone who kindly also supplied his data. Next, I picked some other examples at random. So this is a sample. Perhaps I just happened on a few extraordinary cases, but that seems unlikely.

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Here are the reasons for my theoretical prediction in a nutshell:

• Exponential growth is basically the assumption that population growth is independent from population size. Humans always have the same fertility when they have a chance to grow their numbers. Malthus thinks that this is even maximum fertility. If you also have fixed mortality (Malthus’ implicit assumption: a minimum baseline), then there will be a fixed percentage of new people per time, which is just exponential growth on a continuous time-scale.
• Quadratic growth is basically the assumption that a population grows from a lower population density to a higher population density. At a location, this might work via growth that should probably be somewhat like logistic growth (not exactly, I will explain that in further posts). However, if population growth first occurs at a center and then spreads out from there at a fixed speed, the main effect comes from geographical expansion: growth is proportional to area, which grows quadratically in this case. That’s only as long as it can work this way. Later, growth will slow down, become linear, and eventually fizzle out. Population growth here is not independent from population size because we live in a world with a two-dimensional geography and also on a sphere with only so much area. (For beings that can grow in three dimensions, eg. in water or in the soil, you would expect cubic growth.)

What contradicts a model with exponential population growth from the start is that it is not even theoretically possible over the longer run for beings that live essentially in two dimensions even if there were no constraints for the food supply or anything else. The reason is that exponential growth implies that the relevant area has to expand at a speed that goes to infinity or that population density has to go to infinity or both. Quadratic growth could go on forever on an infinite two-dimensional plane with no other constraints.

My expectation was that quadratic growth (and at times also slower linear growth or fizzling out) should have played the main role in the early expansions of humankind that were clearly linked to geography, but rather slow: the settlement of the world — first Africa, then the other continents — the domestication of animals and later the domestication of plants that made higher population densities possible.

It is not as clear for population growth in modernity. If the expansion were very fast and growth mostly in parallel, you would see the local behavior with somewhat logistical growth also on the aggregate level. That was what I expected. But the data have convinced me that also here geographical expansion played the dominant role.

All in all, already on a theoretical level, quadratic population growth seems to win over exponential population growth. Note that even Malthusians would have to come to this conclusion if geographical expansion is the main driver. They must also have a shift from a lower population density to a higher one with technological advances. Their explanation would only be different from mine in one inessential regard: the higher level should be determined by a hard constraint for the food supply, while I think the higher population density may be related to the food supply, but not in this way, and even that is not necessary in all cases.

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Now, here is what baffles me and what I allude to in my title. Tyler Cowen’s Second Law posits:

There is a literature on everything.

This is not meant as a serious law of nature, but is more tongue-in-cheek. If you think about something, you can expect that, with many smart people around, there were already others before you who have thought about it as well. And if it is of any scientific interest, there should have been even enough people who have written about it, and so there must be a whole literature on it. That’s certainly a good rule of thumb.

Of course, this cannot be true in general. There are real advances where someone has an idea for the first time. If Newton had tried to locate a literature on his theories, Einstein one on general relativity, etc., they would have failed. Otherwise, this would lead to the absurd conclusion that everything was already written up as soon as there were literatures at all.

My theoretical argument for quadratic population growth is very elementary once you think about it. For me, it was a revelation once I got the idea. But then it is not a very difficult one. Throwing the names of Newton or Einstein around does not mean I think of myself as a genius. Quadratic population growth is more of a no-brainer. If it turned out that I am really the first one to think about it, I would not be proud, but appalled by how stupid everybody else has been before me: even more stupid than I am.

It would be even more baffling if my preliminary observation turned out to be correct that quadratic population growth is all over the place. You only have to look at almost any data set for a population that grows and it just stares you in the face. Hard to see how someone could not find this purely by playing around with the data and then start to think about why that is so. The leap does not seem very far from growth to the second power to humans living essentially in two dimensions and hence quadratic growth for settling area at a fixed speed.

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However, here is the thing: I have not been able to find the literature on quadratic population growth. That may be my fault. What I have done so far is this: I have skimmed through all Wikipedia entries that could be broadly related. There are oozes of them for exponential population growth as a model. There is even a page on quadratic growth in general. However, it only lists a few examples from mathematics and computer science. You are then immediately linked back to exponential growth.

Next, I looked through different blogs, eg. Marginal Revolution run by an infovore like Tyler Cowen. A search for “quadratic growth” turns up no results. Extensive searches on Google did not yield anything, and more appropriately also on Google Scholar. I found a few things where people discussed fitting different functions to data in a statistical setting. One even did that for population data and found an extremely good fit with a quadratic function, but then concluded that that could not be so and it must be exponential growth.

Maybe my problem is that I search for the wrong thing. Perhaps quadratic population growth is known by a different name although I find the term almost inevitable. Sometimes such theories go by the name of a person, and then it would be “X’s Law of Population Growth.” Since I have no idea who went before me, I also have no idea who X might be.

So, my question to anyone who reads this is simple: Could you please point me to the literature on quadratic population growth? Discussions about population growth have been around for centuries. That actually started long before Malthus who was anything but original. There are lots of people who have worked on demographics. Many should have looked at least at the data and noticed that quadratic growth is a very good approximation. Some of them must also have thought about why that might be so. It does not seem hard to find the explanation that I have given above.

Can it really be so that I am the first one here? I can’t believe this. Please help me to get over the horror that all the many people over the past centuries who worked on demographics could have been too stupid to even consider quadratic population growth and find the rather obvious explanation. I am very willing to cede my claim to precedence if it means I can think of the whole genre better. And maybe I then also find the next ideas in the literature that I have so far missed myself.