Implicit Population Targets

[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 one of my previous posts, I discussed an observation for South Korea: There seems to be a connection between fertility and population size.

That was my expectation apriori before I even looked at any data because many reasons bring me to the conclusion that human populations effectively target a certain population size. Of course, no population would have an eye on its numbers. That would presuppose rather precise census data that were not available for most of history. And even when they were, perhaps from the 18th century on, many people would not know the results. It also seems weird why a population would care about such an aggregate goal, which also depends on how you aggregate. But then this could work through population density, which is observable also on the individual level and which is proportional to population size. Or it could be the inverse of population density: area per capita.

To achieve a certain target size, a population basically only has to do this: When it is below the target, it has to have fertility above the replacement level, and maybe more so, the farther away it is. And if it is above the target, it has to have fertility below the replacement level, and again perhaps more so the farther away it is. Since there is population momentum (see my post on what that is if you are unsure), it is not easy to hit a target size. Population growth goes on for decades even while fertility goes down. Hence a plausible outcome would be some overshooting that then needs to be corrected downwards with fertility below the replacement level.

As noted, my derivation was completely apriori without regard for actual data. My hunch was that a population would adjust its fertility according to the percentage deviation from the target. The reason I assumed this was that a percentage deviation is rather easy to calculate. And since I think populations do this on an intuitive level, no hard math should be involved. My apriori solution was an adaptive process with some stickiness built in to avoid oscillations when overshooting is corrected that leads to undershooting that would then lead to overshooting and so on.

Up to this point, I had not looked at any data. My model for how fertility adjusts yields a pretty reasonable behavior that is broadly in line with what populations do. Not to be misunderstood: I certainly did not blind myself to any data, I only did not try to fiddle around with them to extract the underlying mechanism. Then I studied the data for South Korea to see whether my idea worked. I chose the country for a reason: There is an ongoing demographic transition after massive population growth. And South Korea has had comparably little emigration and immigration, which I suspected might lead to different results.

To my surprise, my apriori derivation worked astonishing well. But I also realized that my assumption about percentage deviations from the target size appeared to be wrong. Apparently the deviation on a logarithmic scale was behind it. Once I realized this, it made perfect sense: A logarithmic transform leads to symmetrical data. An increase by 200% and a decrease by 50% are of the same size, as they should be because concatenating them means: no change. Percentage deviations are comparably awkward although they are also roughly symmetrical for small values.

Working on a logarithmic scale seems much harder to do than taking percentages, though. But then biological systems are able to do such computations. An example for this is our auditory system, which transforms frequencies with the logarithm. Going to the next octave means doubling the frequency, but we perceive all of them alike although the absolute changes are very different, and we also view an octave up, ie. to 200% the frequency in the same way as an octave down, ie. to 50% the frequency. While I did not expect a logarithmic transform here, my feeling is that it is by no means out of reach for humans.

Once you transform population size with the logarithm, there is also no restriction that would stop humans from transforming fertility in the same way, and also with the same rationale: Fertility twice the replacement level is the opposite of fertility half the replacement level. And when you do this for the South Korean data, you find a very tight relationship, which is linear and hence easy to implement “in hardware” for humans:

ln(population size/target size) = a * ln(actual fertility/replacement level)

with a coefficient a that seems to be around -0.3. You can also write this as:

ln(population size) — ln(target size)
= a * ( ln(actual fertility) — ln(replacement level) )

That does away with the quotients that are perhaps harder to calculate, and all you have to do is take differences instead, which is much easier.

The residuals around the linear function have some further structure that looks like some time series process, which would tie in with my apriori model that this might be adaptive and also with some stickiness. So while it may look like a population just plugs the data into the formula and solves for the actual fertility it should have, what is going on could be a little more complicated. But then this is only under the assumption that what we see in the data is really what I expected to find. The combination of high R² and serially autocorrelated errors might also be a warning sign that this is a an artifact.

I know that my analysis is far too weak to establish that it must be so. There are many dangerous steps here if this were meant as statistical inference. But from my point of view, it was not supposed to be that. All I wanted to see was (1) whether my apriori derivation would be borne out by real data, (2) what the probable relationship between population size and fertility might be, and (3) maybe even to pin some coefficients down.

All in all, I was surprised how good it turned out: (1) was a clear “check,” as for (2), I have to revise my assumptions, but in a very sensible way, from percentanges to differences on a logarithmic scale, and I could even get a ballpark for a coefficent as for (3). No proof, but no refutation either. Any argument here would have to be stronger than a high R² for a limited data series anyway.

— — —

My assumption in the ensuing analysis is that I broadly get this right, ie. that the connection is not an artifact. If this assumption is wrong, any conclusions are also wrong. I might be telling a story for what is essentially just noise. Please bear this in mind and do not get carried away with the conclusions, a strong temptation for me. I view this as exploratory and tentative. At best, I am trying to derive a reasonable hypothesis that would need much better confirmation.

If we can make this assumption (!), we can also turn the analysis around and back out “implicit population targets” from the data for population size and fertility for a country. I would like to report how that works out for some examples. I set the coefficient to -0.3 although my estimation yielded something slightly higher. I do not think that the estimate is precise anyway even if it is warranted, I only want to have a ballpark.

Backing out the implicit population target is simple. All you have to do is sort the terms differently as:

ln(target size)
= ln(population size) — a * ( ln(actual fertility) — ln(replacement level) )

And then you have to exponentiate the righthand side to get the implicit population target.

Here is what I calculate for South Korea:

The blue line is the actual population size, and the red line is the implicit population target. Note that it has to be something that is almost constant by construction. I fit a line to the data on a double-logarithmic scale. Then I subtract a linear function with the same slope. That must yield almost a constant. When I transform this back, it should hence also be almost a constant. What we have here is only a different view on the same result. But then I think it gives you a better idea of where the target level is.

Here we see that the implicit target size rises from the 1920s to the 1970s. Note that my explanation in the background is not that a population always targets the same size, but that that can change with conditions. Things got much better for the South Koreans over this time period. However, from about 1970 on, the target size has remained more or less flat. I would not interpret the small wiggles here, which may be artifacts because the figure for fertility can be distorted in many ways, though only to some extent. If there is an underlying adaptive process that would also lead to oscillations, which should not be interpreted literally. And finally, I would also expect an impact from conditions, especially economic crises, though a small one at this level of development.

All in all, South Korea seems to have had a stable target size for more than four decades of perhaps 42 million or so. Since the point in my previous post was to push back against predictions that South Korea will die out, I would interpret the development as some overshooting after a fast ramp-up. Over the longer run, I would expect the population size to converge to the target size, which is definitely not zero, and also much higher than where South Korea started out a century ago. As noted, I had a lower target size in my initial analysis, which was pretty sloppy. But I would now increase this somewhat, and I also amend my offer for a bet to: South Korea will never have less than 35 million inhabitants until 2100, barring extreme events like an all-out nuclear war.

As you can see, the target size was reached some time in the 1980s where the overshooting begins. Apparently, population growth due to momentum might go on into the 2020s when the population size should peak. If the way down to the target size takes as long as to the peak, about 40 years, then South Korea would reach it around 2065. There might then be some slight undershooting and a correction upwards, but perhaps only a small fluctuation around the target size. Of course, that all depends not only on my estimate, but also on whether my general take is correct, which is unproven.

All in all, I would see no reason for panic from my perspective. A decrease by, let’s say twelve million over 40 years from a peak is 300,000 fewer people on average a year, sometimes less, sometimes more. That is like 0.6 to 0.7 percent of actual population, hardly a sudden collapse as many imagine it.

— — —

How critical is setting the coefficient to -0.3?

Not much. Here are two charts where I play around with it somewhat. The first is for a coefficient of -0.25:

In this case, you have a slight drift upwards. But the basic conclusion would be almost the same, up to a few million. And with a coefficient of 0.35, you obtain this:

That makes no real difference either, perhaps by one or two million over the longer run.

— — —

As I have remarked above, I deliberately took South Korea as an example because there was little emigration and immigration. Another example of the same type would be Japan, so I did the analysis also for that country, and again with a coefficient of -0.3.

Note that I simply use the coefficient that I derived for South Korea. My reasoning here is that if I am right, the underlying mechanism should be a part of human nature and would not vary a lot between countries and even different times and under different circumstances. Of course, this need not be so. The coefficient could also be responsive to many things, ie. how strongly a population reacts to a crisis. If I get plausible results with even this strong assumption, though, it might be an indication that we are dealing with something like a human constant. However, to be fair: if it is not so, results could also be spurious.

Here is what I get for the implicit population target in Japan (thousands on the vertical scale). Note that it does not have to be so by construction as in the case of South Korea.

There are some differences from the start. Japan was already more populated and richer than South Korea after World War II. The country was not in the club of rich countries as late as the 1960s, though. Japan only had a headstart by about a decade versus South Korea. All in all, this would make it plausible that the ramp up should come somewhat earlier, but also be more modest and slower. The same would apply for the overshooting because there was less leeway to expand the population and also less population momentum that keeps running even after the population lowers its fertility.

That is borne out by the data: Not only is the overshooting more modest, by about 15% versus 30% for South Korea, it also takes less time to reach a peak after hitting the target level: roughly 30 versus 40 years. Together with an earlier start by a decade, this means that maximum population size was reached around 2005, while it might take until the late 2020s for South Korea.

But then the situation for the implicit population target seems quite similar. There is a ramp-up after World War II, especially when Japan becomes one of the rich countries. As for the high targets early on, you have to keep in mind that I work with a replacement level of 2.1 in the background. That is correct for modern industrial countries now. But it depends on the level of mortality. My hunch here is that mortaltiy was still higher after World War II, and hence also the replacement level. That means target sizes appear as too large in my analysis. The same should also apply for the early data in the case of South Korea and also of other countries. A better analysis would have to work with the actual replacement level, and not a fixed value of 2.1.

Apart from these quibbles, the target size for Japan seems to have been pretty stable for about forty years, and at a level of 113 million or so with some slight fluctuations. Note that the scale starts at 60 million, which enlarges these changes visually. Since I have seen an article only a few days back that Japan will soon see its population shrink by 50 million or so, here is my offer: I will bet that the population of Japan will not go below 100 million until the end of the century. More information below.

Actually, Japan also illustrates a point that I made for South Korea, but only as a forecast that I would be willing to bet on: When the population starts to shrink, fertility will begin to rise. That has happened in Japan almost to a tee. The peak for population came around 2005, and that was exactly when fertility in the country started to turn around. For that reason, doomsayers have probably also moved on to South Korea because Japan looks like you could be proven wrong in no time. Germany fell out of fashion even earlier. While my explanation is eerily good, I would not expect a turnaround exactly at the peak, that could perhaps also take a few years for other reasons.

— — —

Finally, I would like to discuss another example with a different, though perhaps somewhat comparable setting: Russia. Immigration and emigration played no role until the 1990s. But then that was for the Soviet Union. I assume there was internal migration although that was heavily regulated, some of which would have been from, but some of it also to what is now the Russian Federation. I have no idea about the net flows. If those were non-negligible, which seems plausible to me, then the country is not exactly comparable because my conclusion is that migration changes the equation.

There is also another difference. South Korea and Japan suffered during World War II, South Korea into the 1950s from the Korean War. Arguably it was even worse for Russia. So all three countries started out at depressed levels after 1945, but maybe relatively at different ones. Then the Soviet Union did not have explosive growth as had Japan and South Korea. Things could only get better after World War II with a normalization to the low levels before. But afterwards any improvements should have been slow and drawn out.

The other major difference is that Japan and South Korea basically only had one direction: always up. Not so for Russia. Maybe the reforms in the 1980s led to some enthusiasm that the country would finally also have its chance to become a rich country. But then the developments in the 1990s were a rude shock that should have curbed such exuberance. With rising oil prices, the country recovered after 2000 somewhat. But then a repressive regime as well as rampant corruption and mismanagement put the brakes on growth.

For all these reasons, Russia should have a less stable development than Japan or South Korea regarding its implicit population target. As noted above, I do not assume that a population stubbornly pursues a constant level independent of circumstances. Target sizes can go up when there are improvements, but only in line with their extent, and they can also go down when there is a deterioration. My reasoning here is that humans do this for a very good reason: To take advantage of opportunities, but also to balance this with staying away from trouble. Note that you can draw similar conclusions also without assuming my explanation. Hence, it is no proof for my contention that I can tell a consistent story for what happened.

Here is what I get for population size and implicit population targets in the case of Russia, again with the South Korean coefficient of -0.3:

Also Russia had a “baby boom,” which is very understandable, when things got better after the horrible times during World War II, but also the famines and in the 1930s and 1940s. The target size levels off in the 1960s, but then slowly begins to rise over the 1970s. The 1980s seem to have been a time of optimism that things would now get really good. The target size rises massively. However, then the actual developments turn out far worse than expected, and the target is yanked down by about 20%. Only in the late 2000s does the target size begin to rise again, perhaps to levels as in the 1970s and even beyond.

But I am bit skeptical about the reliability of the figures here. It is easy to understand how population can be overcounted. Germany had this experience a few years back when an exhaustive census was replaced with a sampling method. That led to the discovery that population figures had been exaggerated perhaps for decades and by as much as one or two million. Some cities, like Berlin, lost quite a bit of virtual population over night.

There were several reasons for this: Foreigners have to register with an agency. But many had moved back home, maybe as early as the 1970s and had not told the government about it. So they remained on the record as what are known as “Karteileichen” in German, literally: file corpses. Another bias came from an effort by cities to inflate their population because this is important to get a larger share of taxes. As far as I can see, there was no deliberate falsification. That’s not how it works here. But there were many incentives to always err on the high side.

I even know this from direct experience. I moved away from the city of Bonn some 20 years ago, but failed to tell them about it. Technically, I could be fined a small sum, but that did not happen because they never found out about it in the first place. So a few years ago, I got a letter in the mail that I should declare my actual residence in Frankfurt as a second residence, which would make it possible to collect a tax. Silly as it is, the city of Bonn assumed I was still living in my old apartment although also someone else had informed them they live there, or maybe even several people over time. I just smiled and threw the letter into the green garbage can for paper. Ordnung muß sein! (There has to be order.)

Now, it is easy to see how cities might count people who have actually moved away or the country people who do the same. But it is a little hard to understand how the opposite can happen. That should perhaps be “Karteibabys” (file babies) then: people who continually live there, but always remain under the radar of the government. I gather that the Russian government has other things to do than run the country to the benefit of its citizens, but that all stretches my belief. A few years back, some three million people were discovered!

My hunch here is that there is extreme pressure to find extra people, eg. count those who have moved away or just falsify the figures. Since data for fertility go into the calculation that are even more malleable than for population, I would not be surprised if there were some bias upwards there, too. A shrinking population is a massive PR problem for the regime, and so I assume all the incentives are lined up to inflate figures always in one direction. That said, and with appropriate adjustments downwards, I would say that Russia is another country that could tone its demographic panic somewhat down.

— — —

Granted, all these interpretations make only sense if my underlying assumptions are sound. Suppose the coefficient is not stable or the same for all human populations, then the implicit population targets may be meaningless or at least way off. If the dynamics are not what I assume they are, then everything here can be wrong. I have by no means made my case for all that. It is also easy to tell “just so”-stories about how the Russians reacted to events and moved the implicit target size around. It should be suspicious if I can explain everything with something as untested as this. At best, I expect to get a grip on very long-run developments.

My goal is very modest and so the purpose here is only exploratory and extremely tentative: Under the assumption that it works like this, what could be learned from concrete examples? All in all, they do not yield a refutation. Especially, Japan and the turnaround for fertility right when population peaks looks almost too good for my expectations.

One thing I have alluded to is that migration might change the equation. In a further post, I will, therefore, look into examples where that plays a role: France, Germany, and the US. The description in my previous post now seems wrong how migration interacts with population targets after I have looked more closely at it. It is actually the other way around from what I thought: Initially, the resident population keeps its population target, which means it lowers its fertility to make up for the extra people. However, after a generation, about thirty years, the population target starts to shift up to account for immigration.

In a way, that has to be so: The US has not seen a phenomenon where the population tries to shrink back to somewhat less than 4 million as in 1790. Population targets evidently have to shift upwards in such a case. As I will show in another post, the move from pre-industrial to modern conditions quite regularly leads to an increase in population by a factor of about 12 with some leeway around it, but very little if you break it down to average growth rates per year or average fertility versus the replacement level. However, countries of immigration have sometimes seen increases by a factor of more than hundred since 1500, as in the case of Brazil, the US, and Canada.

Still, I find the “pure” examples without immigration and major upheaval, as in Russia, impressive. It does not really refute my contention that these populations quite consistently target a population size. If you think I am totally wrong, here is my offer to bet:

• The population of South Korea will not go below something like 35 million until 2100 (that’s actually more ambitious than in my previous post).
• Fertility in South Korea will be higher than 1.5 in 25 years (I had: twenty years, but I guess it might take a few years longer to reach the peak than I first thought).
• The population of Japan will not go below 100 million until 2100 (new bet!).
• I would have to think through what my bet for fertility in Japan twenty-five years on is, but I would guess that it will be pretty close to the replacement level, a huge surprise for doomsayers.

Those are very courageous bets indeed. Read any forecasts for those countries and you will see that I should resoundingly lose all four bets. It is funny that someone on Twitter, a well-known blogger on demographics, claimed that my explanation is unfalsifiable. It very much is! The only problem is that I will not live until 2100, so those bets are somewhat awkward to operationalize. I have a fair chance to live for another quarter century. So that might be the way to go. Of course, I have to exclude extraordinary events like atomic wars, pandemics that lead to mass infertility, and what have you. And I will not bet anyone to be honest, preferably someone with a reputation on the line. So, who wants to become the next Paul Ehrlich?

But then I think it is fun to predict the end of the world if deep in your heart you don’t believe it either. And there are also ominous portents: a lot of doomsayers lie along the way who have already been proven wrong. Most of them relied on exponential extrapolations with a positive rate. Noone would be so stupid these days, and hence it is more fashionable to extrapolate with a negative rate. But who says you cannot be wrong with that, too?

— — —

I will address the case with immigration in another post. The data have sent me back to the drawing-board. For me this is very interesting because it shows something about how target sizes are implemented on the micro-level. My current hunch is that the lag of about 30 years before immigration begins to kick in has to do with how the next generation learns about the population target that their parents were pursuing.

For the parents, immigrants are just another type of extra population that should be kept in check with lower fertility. But for their children, the immigrants are taken into account. My idea is that they infer a higher population target for their parents than was actually the case. I have not yet thought this through well enough and may have to revise this initial hypothesis later. But I have a tentative interpretation that can not only account for this phenomenon, but also others, eg. with internal migration from the countryside to the cities. Or why the resident population reacts in this way while the migrants more or less do not, only from the second generation on.

Work in progress, should only write about it when I am ready. But then this is blogging and not writing the last word on the matter. You have to keep that in mind and be extra careful with my claims.