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

In my previous posts in this series (you can work backwards from here), I have tried to make the point that exponential growth should not be the go-to model for how human populations behave. Yet, a negative result is perhaps unsatisfactory. This raises the question what an appropriate model should be.

I would have developed this part in the context of a non-Malthusian explanation for population dynamics anyway, but then I came across something that I can use very well. It is in a post by Lyman Stone on the Amish. His question is: How Long Until We’re All Amish? He there supplies the data for the Amish population since 1900 for which I am very grateful. Spoiler alert from my side: It is not going to happen.

So, what if not the exponential function? The answer is a bit complicated, but I think quite intuitive. I will start with an example:

Suppose a population, Adam and Eve, start at some point on a plane. They have children who set up shop close-by when they are old enough. Next those have children of their own who do the same, and so forth. If they try to have a fixed population density or equivalently for the inverse: fixed space per person, the size of the population is proportional to the area settled.

Let’s assume the settled area lies within a circle of a certain radius (you could also take a square or whatever). If the area expands at a constant speed, its size grows quadratically, and so does also the size of the population because it is proportional. Hence quadratic growth of population is a plausible model with geographical expansion.

— — —

As a short aside:

The argument actually also shows why exponential growth is impossible (!!!) even under ideal conditions. Suppose a population lives on an infinite plane with space no end and settles again within a circle whose radius can now go to infinity. The exponential function grows faster asymptotically, ie. as time goes to infinity, than any polynomial. With a constant speed of expansion, the area grows quadratically, ie. as the square of time, which is a polynomial.

The quotient of the square divided by an exponential function goes to zero, which just means that the exponential functions grows faster asymptotically. If the former is the area, the latter population size, then this means that area per capita goes to zero or equivalently that population density goes to infinity. But humans need at least some minimum space, and so that is not possible.

Now this was under the assumption that the circles expand with a constant speed. How about relaxing that to make more room for the next generation? That is, of course, possible because we are anyway in a fantasy world. But this leads to another contradiction. With a fixed population density, the radius of the circles has to grow proportionally to the square root of an exponential function (the area is proportional to population size, and that is proportional to the square of the radius).

But root(exp(x)) = exp(x/2), which is again an exponential function. However, then also the speed at which the circles expand is an exponential function because the derivative is. And so the speed at which the area of settlement expands has to go to infinity to accomodate a population that grows exponentially. At some point humans just have to throw the towel in. And so this is not possible either.

— — —

The reason why we have quadratic growth in the above example of a population that lives on an area within a circle is that we have two dimensions for expansion. So if the radius doubles, this yields twice as much area twice, which leads to the square. But then there are also situations where expansion in two directions is not possible.

Here is an example: Suppose the geography is bounded in the x-direction, but not in the y-direction. The population lives now on an infinite band with a certain width. Assume in addition that all the space below a line for some value of y is already settled and not above it. In this case, it is only possible to expand in one direction, the y-direction. So, we now have one dimension, and hence linear growth. A concrete case is when a population expands along a sea coast or the shore of a river, but does not move inland beyond some distance. Then there will be only linear growth, which is slower than quadratic growth with the same speed of expansion.

But it can be even worse. We do not live on an infinite plane or an infinite band, but basically on a sphere with a finite surface. So growth cannot go to infinity anyway. At some point, the remaining area dwindles, and any expansion will have to come to a halt. That is then growth to the zero-th power, ie. a constant, while linear growth is to the first, and quadratic growth to the second power.

We can now answer the question what a natural model should be for population growth, at least if geographic expansion is the dominant theme: Some function between the zero-th and the second power. If there is no obvious constraint on the number of dimensions, quadratic growth is the go-to model, not exponential growth.

It is not as easy as a knee-jerk reply: I don’t have to look because it must be the exponential function! The exact behavior depends on the geography: Sometimes it can be more like quadratic growth, at other times also linear growth, or a mix of the two, and then it will have to stall at some point in time. There may also be a certain stop and go: For example, it might take some time before people find the way to Australia or the Americas. Before that population growth comes to a halt, and then it can get going again.

— — —

Actually, even Malthus and his followers would have to draw this conclusion. Once population has grown to the max in a region, it can only expand geographically, and then we obtain the same results. But curiously I am not aware of a single example over more than two hundred years where a Malthusian has even understood the point. Neither for the problem that exponential growth is impossible already in theory. I am afraid there is no explanation here that is flattering for Malthusians.

Incidentally also linear growth can happen, which is also implied by Malthus’ claim that the food supply grows linearly. As I have shown in a previous post, he has no argument for this and so also no argument for the conclusion. And even if Malthus had one, it would have to be completely different. So this is only a coincidence and not some hidden insight.

Note that also faster growth than quadratic growth is conceivable, at least for some time. But that implies either a speedup for the expansion or an increasing population density or equivalently a shrinking area per capita. Or it might also be that a population starts new colonies far away whose growth does not interfere with the growth of the original area of settlement. However, all this can last only so long.

— — —

After these general discussions, we can now tackle the case of the Amish. It is rather unimportant here that they have very high fertility. I will look into this in another post. What interests me is only what the growth behavior is, ie. the functional form.

Here is how the size of the Amish population has grown from 1900 to 2017 (note that I have German country settings in the background, which means that dots and commas are the other way around):

Lyman Stone starts from sparser data and fills the gaps in with linear interpolation. I would say that is okayish. Linear growth is at least a conceivable candidate. The best way here would be some quadratic interpolation. I will explain how to do it below. What is not a plausible interpolation, though, is exponential interpolation. But for many historical data sets, people naively assume that only exponential growth is possible, and do this. That can then lead to the circular effect that others who work with the data sets find exponential growth, which does not surprise them at all because they “know” this from the start. I will discuss such examples in another post.

— — —

Now, you can look at the above graph and exclaim: Wow, that is an exponential function! But be careful. That you can know this so fast means that you use your intuition. And it works like this: You notice that the function grows faster than a linear function, its slope increases over time. And then you jump to the conclusion: It must be the exponential function! However, that is a fallacy. The premise indeed holds for the exponential function, but there are many other functions where it holds, too, eg. the square. So the conclusion is not warranted. It is like any bell-shaped function in a statistical context will evoke an outcry: The normal distribution again!

Since the Malthusian worldview is part of our culture and one of its central tenets is that it can only always be the exponential function, you have to make this point explicitly because noone bothers to question whether something is really an exponential function. Look closely here: In the first part, the function grows rather slowly, then it has kind of a smooth kink from the 1960s to the 1980s after which it grows faster. There are two regimes here, and that is not what an exponential function does.

— — —

As noted above: the go-to model should be quadratic growth. I will compare it to the assumption that we have underlying exponential growth.

It is not easy to see what is going on with the absolute numbers. That is so because the scale is determined by the last and by far largest data point, and that means that anything to the left looks rather flat. To see whether quadratic growth may be the case, we have to transform the data with the inverse function: the square root. This turns quadratic behavior into linear behavior, which is rather easy to spot visually. Here is the result:

As already alluded to above, we apparently have two different regimes: one before the 1960s and another after the 1980s. In between, there seems to be a transition where things speed up. Actually, there is also another transition at the start from no growth. But that may be an artifact. I don’t know. Apart from the transitions, the transformed function looks pretty linear, ie. the untransformed function is quadratic. That is confirmed if we zoom in on the two regimes (no dates, only a running number for years, German country settings again with commas instead of dots):

I have added a linear trendline, which comes extremely close, especially for the more recent regime. So it looks good that we indeed have quadratic growth here, though with different regimes and transitions between them.

— — —

How about an assumption that we are dealing with exponential growth instead? We can do the same analysis, now only with the inverse of the exponential function: the natural logarithm. Here is what we get for the whole time frame:

The result looks similar. If something is originally exponential, it should transform to a linear function. There are again two regimes with a transition, also one at the start. And the behavior for the regimes is also rather linear. But if you look closely, there are small bulges here. The slope starts out a little steeper and then becomes less steep. That is what you should get for an underlying quadratic function. It grows faster than an exponential function at first, but then cannot keep up.

Here is what we get if we zoom in on the two regimes:

I have again added linear trendlines. If we had exponential growth, the fit should be very good. It is good as you can see from the R squared, but slightly worse than for the assumption that we deal with quadratic growth. In additon, the deviation is also quite systematic. The linear function is above the transform at both ends, and below in the middle.

So, it looks like quadratic growth is the better model, not exponential growth.

— — —

Why is there a transition? If it were exponential growth, all we could say is: because it is so. If it is quadratic growth because of an underlying geographic expansion, we can name candidates where this is not exhaustive: the speed of expansion might change, the population density might go up, which means that the same speed can accomodate a larger increase, or both. My hunch is that the Amish improved their productivity and that meant they could settle at a higher population density.

As a bonus: I can now also explain how to get a better interpolation than linear interpolation for the sparse data points if quadratic growth is the appropriate model. All you have to do is transform the data with the square root, then use linear interpolation, and transform this back with the square. The same would also work for exponential interpolation, only then with a logarithmic transform, linear interpolation, and an exponential transform back.

— — —

All this may seem like nitpicking. Both a quadratic and an exponential function grow fast, both go to infinity. So what?

Here is why it matters: Lyman Stone wants to draw conclusions about the future, even two centuries ahead. If exponential growth is the correct model, then he should use that to extrapolate, otherwise quadratic growth. Let me now do this for both types of extrapolation. Since I am only after a qualitative result I forecast for only one century with both models (German settings again):

The red line is under the assumption of exponential growth, the blue line for quadratic growth. I calculate the values in this way: I transform the data and then add the difference between 1990 and 2017 to the value 27 years before. Then I transform this back with the square and the exponential function, respectively. This approach creates slight kinks for exponential extrapolation. But then I do not bother to improve on it because the exact values are not my point, only the general behavior.

What seems like a minor quibble makes a huge difference now. The reason is that the exponential function speeds up more and more and so does also the acceleration, the second derivative, while the acceleration remains fixed for the quadratic function. The difference on the data set may seem minimal, but it leads to a massive divergence.

With the assumption of underlying exponential growth, we obtain 13 million Amish in 2117, with the assumption of underlying quadratic growth only about 1.9 million. Most of the explanation comes from the model we use. In the first case, it is plausible to wonder whether the Amish will overrun everything, in the second case not, at least not for a very long time.

— — —

There is also another reason to doubt that the Amish will swamp the US. They have a preindustrial lifestyle in many ways, which means they can only live on the countryside. There are about 50 million people all in all who live in rural areas in the US. Even if the Amish managed to replace them completely that would be a far cry from a majority.

And then due to their lifestyle it seems as if they cannot even achieve population densities as other rural Americans. You can see this in the data in Lyman Stone’s post. It seems as if the Amish can only go to about 15% of the resident population. If they could increase their population density more, they would simply swamp the respective countries and would expand only afterwards. But apparently they branch out much earlier.

Now, if the Amish can only get to about 15% of the population density of rural Americans in general, then this puts an even lower bar on how large their population can grow. That would be something like 15% of 50 million, or 7.5 million. If a quadratic extrapolation is reasonable, it looks like even this is still centuries away. I am skeptical whether it will happen, but at least it seems like a plausible upper estimate for the Amish.

One consequence of this is also that fertility for the Amish will have to go down in the future, eventually to the replacement level that may be similar to that for the general population with about 2.1 children or somewhat above if the Amish have higher child mortality. That’s also why it is pointless to try and learn anything from the Amish regarding faster population growth. They cannot do it either, only as long as they can expand geographically.

If all Americans adopted the Amish lifestyle that would only mean that population density and the size of the population would have to go down massively. Anything you learn here goes totally in the other direction from what Lyman Stone is after. My estimate above may then be even too generous because the Amish can live like they do only because they are embedded in a society and economy that is not Amish. If that were no longer the case, their productivity would probably suffer.

— — —

There is also another observation that is relevant to my discussion of the Malthusian argument. The Amish are obviously not Malthusians. They do not remain put and procreate until enough of their children die of starvation and related diseases to keep the population at maximum size. Instead, they grow their population only to a point and expand long before they reach the brink of starvation. Their high fertility may seem like validation for a Malthusian view. But that is not true either as I will discuss in a further post.

There is also a somewhat speculative, but in my view plausible explanation for why Lyman Stone finds lower fertility for those with telephones. They are perhaps the Amish who are in already settled areas in the interior where there is less space to expand further. Maybe they can have population growth mostly only from increasing population density, which is only possible via higher productivity and hence through the adoption of new technologies. Telephones would then be a proxy for this feature. However, that can only go so far, and so those Amish have to cut down on their fertility, and they do.

Although I think practically all the conclusions are wrong that Lyman Stone wants to draw, I am very grateful for the data he has aggregated. Up until I found his post, I had derived the result about quadratic growth only in theory. I had not checked it on empirical data. And here it is! Seems like I am on to something.

— — —

PS

Check out also my next post with further examples: “More on Quadratic Population Growth.”