A Short Explanation of Population Momentum

[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 experience, demographics is a playground for charlatans. I don’t mean the scientific discipline here, but the treatment of respective topics by writers, journalists, but often also scientists from other fields. The problem in my view is that demographics is at base not very complicated, and hence many people feel called upon to “add their mustard” as one would say in German (ihren Senf dazu geben). Anyone can do some calculations in Excel and then tell you that world population will be one zillion in 4000 AD or Japan will die out in five hundred years or whatever.

Note that I am not a “certified” demographer either, just a plain mathematician by training. My point is not to defend the discipline against encroachments. I think anyone can do such calculations and set up models. What counts is whether the arguments and the results are correct, not whether someone has a degree. However, it does not hurt to know about the basics first. And unfortunately, very often it is clear that someone who poses as an expert lacks an elementary understanding of what they are writing about.

I would like to explain two concepts in this and another post that anyone commenting on demographics should be aware of. That’s not some pedantic requirement. If you don’t understand these concepts, you can get so much wrong that your conclusions may be totally worthless. And this is not an exceptional problem. Such errors have shaped public debates for decades. It is not only journalists or writers, though, who fail here. I could point you to many papers even in the scientific literature where people don’t get it.

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There are some warning signs for demographic charlatans in my view. A really good one is when people argue with birth and death rates instead of fertility and mortality, even better when they are unaware that these are different concepts and call “birth rates” fertility and “death rates” mortality.

I have already explained the fundamental problem in another post of mine (see here), and might explore it further in the future to have a reference. Basically it is that birth and death rates are relative to a reference population and can and often do change because that varies, and not because the propensity to have children or to die changes. If you miss this, you seriously misinterpret the data.

Not to be misunderstood: It is not verboten to work with birth and death rates. Often that is all you have. But there are some pitfalls here you have to avoid. So, if someone not even bothers to address the potential problems, is confused about concepts, and so forth, a plausible assumption is either stupidity or, more benevolently, ignorance.

But that’s not my main point here. There are two more advanced, though still very basic concepts that everyone who makes claims about demographics should have understood. Those are more subtle and admittedly easy to miss. Hence ignorance is the first assumption. Yet, that does not shield you from making major errors, quite the opposite. And although the points are subtle, the consequences are not. You can get a lot wrong here.

The two basic concepts I mean are: (1) Population Momentum, and (2) the Tempo Effect. Unfortunately, it is very common in my experience that people with strong opinions on demographics have never heard of either. However, if you have not understood these points, there is a good chance that your conclusions are faulty in some way. Since this is very important and I often appeal to these concepts, I would like to write dedicated posts as a reference. Don’t be scared, this is not very complicated. And it is a valuable indicator to spot charlatans.

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In this post, I will address “population momentum.” The phenomenon is easy to understand on an intuitive level: Population growth (or shrinkage) is sluggish and can go on for decades even after a population has changed course. The horizon here is between half and a full century. Hence when you observe population growth (or shrinkage), this may mostly not be an effect of decisions now, but an echo of events decades ago.

Here is a stark example: the population size for South Korea (you can find the data here, kinks in the curve on the left are spurious because there aren’t data for all years, and I have German settings, so dots and commas are the other way around):

As you can see, the population of South Korea has grown steadily for almost a century. It also looks like it is set to grow even further. Yet, if you look at fertility (“TFR” = the total fertility rate in the data), ie. roughly how many children a woman will have over her life, then that went below 2.1 in 1983 and has remained there ever since. Now, in modern times a value of 2.1 is the “replacement level.” If women have 2.1 children on average, the next generation will be just as large as theirs. You need slightly more than two children because there is still some mortality of a few percent until adulthood and also sex ratios are not perfectly 50/50 at birth.

Fertility in South Korea not only went below the replacement level in 1983, it kept falling afterwards, and reached a value of 1.05 in 2017, ie. half the replacement level. That means the next generation will be half as large in an approximate analysis. Beware: the “tempo effect” may be at work here and so this interpretation is not entirely correct.

Many observers then conclude from these data that the population of South Korea will soon collapse, in a particularly naive analysis: by 50% over the next thirty years. However, note that despite fertility below the replacement level since 1983, the population of South Korea has grown by about a quarter from 40 to more than 50 million. How can that be?

The answer is: population momentum, which leads to further population growth even after decades although fertility may indicate otherwise. South Korea had high fertility well into the 1970s, and that led to population growth for decades down the line. It will not go on forever, though. Over the longer run low fertility will lead to shrinkage, but that is really over the longer run, and before that the population will grow somewhat further. All in all, current fertility may not tell you anything about population growth at the moment or even over the medium run. That may depend on developments decades ago.

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This was only a qualitative description what population momentum feels like and what to watch out for. The situation is hard to disentangle for a concrete example like South Korea because so many things are going on at the same time. It is not possible to do all this in your head, at some point only calculations can get this right. However, it is possible to understand where population momentum comes from and how it works. I hope I can elucidate this below. And I would say that it is also possible to handle it to some extent even on an intuitive level.

To isolate the effect, I look at an ideal situation where I can eliminate all other influences that distract from the main point. What I assume is a stable population which roughly behaves like populations in developed countries now, and then I look at what happens when the population has more children in just one year than are necessary to stabilize the population.

Let me first describe my assumptions for the population, which are somewhat idealized, but close to reality. I exclude all migration whether in or out. This means that there are only two ways how the size of the population can change: either new children are born or people die. To get a grip on this I have to model fertility and mortality, and I start with the latter.

Mortality is the probability to die for someone of a certain age. It is lower when you are young and rises as you get older. One way to look at it is to plot what happens with a cohort (those born in a year) as they age. With each year, their numbers shrink with mortality. Here is how that might look:

The horizontal axis is for age in years. The vertical axis shows the percentage of the 100% initially who are still alive at that age . My assumption here is that there is no mortality until age fifty, which is not exactly true in reality, but also not too far away these days. After age fifty, there is some mortality, and later, after age eighty, that picks up. By age hundred, all people in the cohort are dead. As noted, this is only meant as a caricature of reality. My point does not depend on the specifics, only the exact magnitude of the effect depends on that.

For a stable population, the shape of this curve is also the shape for the age structure of the population, ie. the percentages in cohorts for different ages. All you have to do is normalize the sum of all values to 100%. Why that should be so is perhaps not obvious. To understand it, think of it in this way: A new cohort is born at age zero each year. Since the population is stable, it must always be of the same size.

Now, there is only one way for a cohort to shrink, namely by mortality (I exclude migration here). All cohorts are the result of this in a stable population, and so you get the form above as the cohorts shrink to zero. Note that it is different for a population that grows or shrinks. Older cohorts then started out smaller or larger at an earlier time because the whole population was smaller or larger.

You have probably seen “population pyramids” for the age distribution. I have implicitly assumed here that men and women have the same mortality, which is only approximately true in reality. If you want to keep it separate you can plot the age distributions for both of them separately and on two sides of a vertical, not a horizontal axis, which then may look like a pyramid (it practically never does, so this is somewhat of a misnomer). What I have here is half a population pyramid that lies on the ground.

You can already get a feel for the momentum effect from this vantage point: Suppose twenty years ago, there was a cohort that was larger than usual, then you would see it here as a larger cohort at age 20. But that also means that there are more people who could have children now, and that then leads to population growth even if those of age 20 individually have rather few children. With a larger cohort, it might still be more than usual. In this way, larger and smaller cohorts can ripple through the population structure, and that’s one way to understand population momentum.

But let me pursue a different approach, which makes the effect easier to track. Apart from mortality, I also have to model fertility for the population. Here is my stylized assumption:

What you see is the share of children that are born to parents in a certain age group. The sum is 100% over a whole life. I implicitly assume that there is no difference between men and women, which is not true. But adding more realism is only confusing and does not help with understanding population momentum.

Most children are born around 30 years of age for their parents, rather few for teenagers or after age 40. Although I simplify here, the shape for actual populations in developed countries is quite similar and also has such a triangular shape with a peak at about 30 years. In earlier times, the peak came more to the left, perhaps at age 25, and so the shape was less symmetrical.

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All this was only a preparation for the main point. By assumption, we have a stable population here that has always the same number of children each year. Now, as an exception, one million children are born on top of that in just one year. This means the population grows immediately, but that is not all. To see this, let’s track the extra population of the one million children separately, the rest is still a stable population and nothing changes there.

Here is what happens (the horizontal scale is time after the extra children are born, and the vertical axis is for millions):

Initially, there are one million extra children by assumption. Nothing changes until they get to fertile age (recall that I set mortality to zero until age 50). However, then the children have children of their own, some earlier, some later, but with a mean age of about 30 years as per my assumption for fertility. If the extra children behave as the rest of the population, which is stable and has replacement fertility, there are now one million children in the second generation. Since fertility works out over about thirty years, they do not appear all at once, and hence the curve rises smoothly to two million after about 45 years. That is the first and the second generation combined.

The first children in the second generation were born already after 15 years, and some may then have children of their own after another 15 years, ie. already after 30 years: a third generation. However, initially there are only very few grandchildren. Over time that picks up, though. Since the second generation is spread out and fertility in the second generation spreads that out even further, the increase with the third generation is even smoother. It now comes on top of the first two generations, mostly after fifty years, and that goes on until about eighty years.

You might expect that this goes on and on with ever new generations. But there is also mortality. So after fifty years, some from the first generation begin to die, and that picks up after eighty years. From some point on, mortality cancels more grandchildren out, and the total population stabilizes. The third generation is also one million strong, but the increase is only by about 700,000 because of mortality, mostly for the first and a little for the second generation. After about eighty years, all this stops and not a lot happens any more. There are only some slight oscillations around a level of about 2.7 million, if fertility and mortality don’t change, in principle until the end of time. That’s when population momentum is over.

Think about this: Although we start with only one million extra children, there will be about 2.7 million people on top of the stable population eventually. Population growth not only occurs right now, but there are echos about thirty years down the line and until eighty years afterwards. Actually, most of the population growth comes later: 1.7 million out of 2.7 million. That’s baffling for people who don’t understand population momentum. They will search for some development at the time, but cannot find it: fertility is at the replacement level. The real reason is that you have an effect that can come from events that occurred decades before.

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We can now play around with the assumptions. If there are not more children in one year, but fewer than for a stable population, you have the same thing, but times minus one. You get the same development, but now for the missing children and their missing descendants over two further generations. There is immediate shrinkage, but further shrinkage comes over the next eighty years or so, and that is even the main part.

Then the assumption is strange that there are extra children only in one year. This was only to isolate the effect and make it transparent. In reality, higher than replacement fertility might go on for many years. However, that only means that you have an overlay of the above curve with shifted and scaled versions of itself. The main dynamic is the same. Apart from the immediate effect, there will be additional population growth for another eighty years even if the population goes down to the replacement fertility. Note that it is false to extrapolate this to further growth. Population momentum runs its course, and then it is over.

Since most of the growth comes later, it is perhaps not surprising why it can overcome low fertility anyway. South Korea had high fertility of more than four into the 1970s. That then led to more growth 20 to 40 years on, ie. from 1990 to 2010. And there will be even more growth until eighty years later, ie. until the 2050s. That could swamp lower growth from low fertility from the 1980s on, and so the population of South Korea kept growing anyway.

Of course, population momentum interacts with fertility. In my example above, the extra children and their descendants have replacement fertility. If they have less than that, the second generation is less than one million strong, and the third generation even less. Yet, both generations come on top, and that’s why even with below replacement fertility, you still have population momentum and further growth for eighty years. Only when earlier generations start to die and are replaced by generations which are smaller, shrinkage begins to work out. But that comes from a higher level via population momentum. For decades the outcome is still population growth.

It is tempting to look at fertility of 1.05 and conclude that there is a decrease of 50% from one generation to the next, which is true, but misleading for population dynamics where the population structure plays a role. It is also false to expect that the population will suddenly collapse. All these effects in both directions come on top of each other, which smoothes things out. Developments are hence slow and not sudden.

The population of South Korea will peak perhaps over the next decade and then start to shrink, but only marginally at first. That will speed up then, but it will still take a long time before it goes back to the levels as in the 1980s with 40 million people. And that is under the assumption that fertility will remain well below the replacement level for decades and even at as a low level as now. But then this is an assumption. I would think it is plausible that that will not happen, and fertility will rise again towards the replacement level and maybe even above it. In that case, there will be some shrinkage from a higher peak, but not by much, and that will take decades and might run out or even turn around. Hardly a scenario to panic. (I develop a tentative argument in my post here).

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I hope it has become clear why you can go seriously wrong if you are not aware of population momentum. Noone is able to think this through in their heads and account for all the many things that are going on in parallel in a real population. That’s where you just have to do the calculations. But you can get a rough idea why despite low fertility in South Korea, there is still a lot of population momentum left that will play out until the 2050s. If you see people who argue with current fertility as if that were the same thing as current population growth or growth over the medium run, then you are put on notice that they might be charlatans.

Here is another example where population momentum leads to a lot of confusion: immigration. Suppose the one million extra were not children, but immigrants. They will probably not come at birth, but only later. Still the same thing happens, you only start around age 25 perhaps in the above graph, which is a typical average for immigration.

The first generation is already grown up then, and they even have a few children that they bring along. Hence the reference point is something like 1.2 or 1.5 million. But then you have further population momentum for about half a century. You had only 1.2 or 1.5 million actual immigrants, but together with the second and third generation born in the country, the population grows to 2.7 million or about double that over the next half century.

Now, if someone has not understood population momentum and observes this, they regularly get the wrong idea. You know 1.2 or 1.5 million immigrated, but now the population keeps growing although there was no further immigration. Common fallacies here are to search for immigrants who sneaked into the country in some way. There is always some family reunification, and then that was perhaps it.

Or even more common, the conclusion is that the immigrants have extraordinary fertility, and that’s why the population doubles over half a century. However, by assumption you have replacement fertility in the above example. Even if the immigrants have somewhat higher fertility, that has less of an effect than population momentum for decades. So this whole line of argument — and I assure you that you can find that a lot — is completely mistaken.

Another false conclusion is also to extrapolate such population growth. Population momentum for immigrants will go on for half a century, and then it is over. It is wrong to treat this as if it could go on forever because it is a temporary effect. Yet, there are plenty of people who get this wrong anyway. If you are interested in further details and examples, feel free to read my article here.

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Unfortunately, I would say that there are a lot of people around who pose as experts on demographics or write dramatic stories as journalists about how countries like South Korea will soon die out, and who have no idea what population momentum is. Stupid as it is, they have not even understood the basics of what they want to lecture others about. And you have the same nonsense also in the context of immigration.

I hope I have convinced you that population momentum is not some trick question in an exam without any import. It is very important to get this right. But as I wrote at the start: Demographics is a playground for charlatans. Just watch out when someone is not aware of population momentum, and you have another good indicator to catch them. I will address the “tempo effect” in another post. It is not as devastating to get this wrong, but then I would say even fewer “experts” have understood it.