Misinterpreting Growth of Immigrant Populations

I would like to explain an effect that I think is responsible for a lot of misconceptions regarding immigration. If I am right, my explanation shows that some common explanations, while not being entirely false, are mostly irrelevant, or at least of second-order importance. And this also means that conclusions based on such explanations, are off the mark. Explanations that naively extrapolate trends can be especially misleading.

Let me start with an example of what I mean. In 1961, the Federal Republic of Germany and Turkey struck a deal that Turkish “guest workers” (Gastarbeiter) could be hired by German businesses. There were similar treaties also with other countries such as Italy (1955), Greece (1960), or Yugoslavia (1968). 1961 was the year the GDR built the Berlin wall, so rather suddenly the influx of East Germans came to a halt (about 200,000 in 1960). The West German economy was running at full speed with 8.2% annual GDP growth in the 1950s. German businesses were desperately trying to hire additional workers as unemployment fell to below 1%.

So from 1961 to 1971, about 650,000 Turkish guest workers were hired, and until 1976, this grew to about 825,000. What happened in the meantime was that on November 23, 1973, the German government (Social Democrats & Free Democrats) decided to suspend the treaties with Turkey and other countries (except Italy) after a sharp economic downturn and rising unemployment. So no more workers could be hired. In addition, guest workers in Germany were given a choice either to return home or stay permanently. Family reunion was still possible. Only about 135,000 of the 825,000 Turkish guest workers in 1976 were women. Assuming that 135,000 of the male guest workers were married to them and the others would bring in an additional 535,000 women via family reunion, the total would amount to 1.4 million adult immigrants of Turkish descent. Since a return home was encouraged via payments, you would have to subtract a certain number of returnees, though.

And here is what baffles a lot of people: numbers kept going up after 1973. There are now (as per 2013), 2.8 million people of Turkish descent in Germany, roughly a doubling or even more if you assume there was some return migration. Most of the increase came long after 1973, e.g. there were only 2.1 million people with Turkish citizenship in 1998 (that’s when naturalization became rather easy, so up until this point Turkish citizenship usually meant someone was an immigrant or descended from one).

Here are some of the usual suspects to explain the rise: diaspora dynamics, chain migration via family reunion, men marrying women from Turkey and bringing them in, very high birth rates, maybe encouraged by generous welfare benefits, etc. While there is certainly anecdotal evidence for most of this, and also data supporting some of the claims, I think this all is at best a sideshow, and so further conclusions built on such explanations are mostly irrelevant. If I am right, there is nothing baffling about the observation at all, and the explanation is actually rather simple. It is only that a naive analysis tends to overlook the effect I will explain now.

I will work with stylized facts, but the results should not be materially different if you calibrate the model with exact data. Let’s assume that in one year 1 million people immigrate. For the moment let’s also suppose they are 50/50 men and women. To make things more transparent, I assume for the moment that immigrants are 0 years old (of course, that is false, but I will show later how to fix this).

What happens to the initial immigrants over time? Here’s a stylized graph for the percentages over time:

Percentage Alive

So, no one dies until age 51. Then 1% die per year until age 76 (i.e. a quarter), and 3% per year until age 101 (the rest). This is not entirely true as a certain fraction actually die already before 51, and the real graph is certainly not piecewise linear. However, it is not far off, and life expectation comes out slightly above 80 years which is quite close to reality. I also make the simplifying assumption that men and women have the same mortality. This is not true either, but again not too far off.

How about children? Again I make a simplifying assumption which is roughly in line with reality. The following percentages will have a child at the age on the x-axis (lumping men and women together, so per person, not woman):


The peak is at 30 years, positive rates are from ages 16 to 44. The percentages add up to 100%, i.e. everyone (men and women) has exactly one child or two children per woman. Since no one dies in the relevant age groups (by assumption above), this means you have exactly replacement fertility (that’s why I adjusted reality a little to get confusing effects out of the way). To stress the point: there is no growth that I have built into the model. As you will see, no fancy birth rates are needed.

What happens to the one million initial immigrants in year 0? Since I assumed they are 0 years old, they won’t have any children right away, but starting after 15 years they will. And after some more time, their children will have children, etc. So the immigrant population (including descendents) will develop like this (millions):


Don’t be fooled by the slight downward trend at about 100 years. There is some oscillation, but the number the line converges to is about 2.76. So over time, the one million will grow to 2.76 million. Children and grandchildren come on top. However, initial immigrants eventually die out, so great-grandchildren and further generations only make a minor difference, and the whole population settles into a steady-state. All in all you have almost a tripling, and that’s although I assumed replacement fertility, i.e. no growth at all!

Let’s fix the one unrealistic assumption. Of course, immigrants do not come at age 0. They come perhaps at age 20 (or 30 if you prefer this). But that’s very easy to fix. Just start at 20 (or 30) in the above graph instead of 0. There may be some children that were born before immigration. Add them in under the heading of family reunion which increases the size of initial immigration some. For immigration at age 20, this means that the factor will be slightly less than 2.76. If you take immigration at age 30 instead, then this will boost initial immigration by about 50%, so the factor is somewhat below 2. Note: 2 was about the factor we had for immigrants from Turkey to Germany (2.8 million people of Turkish descent now versus about 1.4 million initial immigrants, but maybe also some return migration). And I did not need any assumptions about massive further family reunion, brides from Turkey, gigantic birth rates, etc.

Here are some conclusions that are easy to read from the graph. Let me assume that immigrants come at age 20 (i.e. the graph starts at year 20). If you like some other age, adapations do not change the conclusions materially which are stylized anyway:

  • Since there was only immigration in year 0 (i.e. 20 in the above graph), if the government decides to shut down immigration after that, then numbers will keep rising for over 50 years. Restrictionists will be stark staring mad because they are looking for loopholes as an explanation. There should also be calls for even stricter enforcement like a clamp-down on family reunions (although that is not possible by assumption).
  • Over one generation (by construction on average 30 years), numbers will double. So this seems like a population with an extraordinary growth rate of 2.4% per year (for a doubling). If a population in steady-state (!) were to grow at such rate, it would mean a fertility of 4 per woman (by construction it is exactly 2 per woman). In the first years the increase is even steeper, more like 3% which corresponds to a fertility of about 5 in steady-state. So restrictionists will be looking for the huge families and come up with cultural explanations. And that would even apply here, where it is false by assumption. Higher fertility would not be unimportant, but compared to an additional perceived fertility of 2 or even 3 via the effect it will be second-order for a moderate divergence in fertility.
  • After one generation, and even two generations, restrictionists may simply extrapolate the trend and conclude how the immigrant population will grow indefinitely and swamp everybody else. However, again by construction, there is no growth, and things will level off eventually. So the extrapolation is totally unwarranted.
  • After about 50 years, the baffling thing might be that numbers stagnate (as is the case for people of Turkish descent in Germany by now). So there will be amazement over how that could happen.
  • And one final conclusion that restrictionists usually miss (maybe if they did not, they would have to be very dishonest about the other claims): If you let in one million in the beginning, you will have 2.76 million descendents eventually. Not only the initial immigrants come, but their whole part in the population pyramid at home. (There is a twist that I will explore in a further post: this also means that all those people are missing in the source country.) So if restrictionists understood the effect, they could scale everything up by a factor of 2.76 (or whatever it is exactly calibrating with actual data) and make their stories even scarier.

Just as an extra service, here are the headlines in Germany for the next decades:

  • 2015/2016: There are 1 million Syrian refugees to Germany.
  • 2017: We can’t handle so many people and will close the borders for Syrians. (I am optimistic this will not happen wholesale, but then I want to stay true to the above example.)
  • 2037: How could that happen? We let in just 1 million and clamped down on further immigration, now there are 2 million people here of Syrian descent! Are they smuggling more people in? A reporter has noticed a Syrian family with 10 children. It’s probably their culture. Let’s try to clamp down even more on migration from Syria.
  • 2057: Now there are more than 2.5 million people of Syrian descent. Soon everybody will be Syrian here. What went wrong?
  • 2067: Strange, they always had those extremely large families. Are they now migrating back to Syria?

Some further remarks


  • The basic effect is known as “population momentum,” i.e. population growth has a certain inertia. Even if fertility changes (goes up or down), it takes some time before population growth changes to a new level (higher or lower). You can understand the effect in the following way: if you have a population with no growth, but 1% extra children in one year. Then this is equivalent to 1% immigrants at age zero. So you have the above build-up. There will be another 1% after one generation, and another 1% after two generations (minus some initial “immigrants” who have died in the meantime), and then it levels off.
  • The effect that I describe is not just population momentum. My main point is about how population momentum is easily misinterpreted in the context of immigration. If you have an extra 1% children in one year, and it builds up to 2.76% extra population over time, anyone who looked at the development would speak of about 1% population growth per generation, which is not very impressive. The reference point here is the whole population. If the extra 1% population are immigrants, however, one may be inclined to take a different reference point: initial immigrants. So the same development would be described as a growth of about 100% per generation, which is much more dramatic. While the two cases are parallel, the perception is quite different.
  • There is an extra effect that makes the perception even more dramatic. Since immigrants do not immigrate at age 0, but 20, 25, or 30, the first generation appears shortened. It does not take about 30 years for a doubling, but only 5 or 10 years. This adds even more drama. Especially, if someone does a sloppy analysis by comparing data for only two data points (which journalists or the general population are maybe prone to do).

Hansjörg is a mathematician by training with a doctorate from the University of Bonn, Germany. After a year at Stanford University as a guest scientist, he went on to work in the financial sector and managed corporate bond funds. Currently, he is building his publishing company Libera Media.

See our blog post introducing Hansjörg, or all blog posts by Hansjörg.

16 thoughts on “Misinterpreting Growth of Immigrant Populations”

  1. It’s interesting that most of the Internet discussion of population momentum in connection with immigration is on restrictionist websites, e.g.:

    http://www.susps.org/overview/birthrates.html and the next page http://www.susps.org/overview/immigration.html doesn’t directly link population momentum and immigration but talks of both of them as reasons to be pessimistic about the US’ ability to keep overpopulation under control

    However, this point is generally brought up only to increase estimates of populations from existing or future immigration rather than as an alternative explanation for things people might otherwise attribute to family reunification and diaspora dynamics.

    I’m not really sure of the extent to which proponents of diaspora dynamics are making the error of ignoring the population momentum effect. However, at a conceptual level the two aren’t really that different: instead of reunifying with family from the home country they create new family in the destination country, that would counterfactually have been created in the home country.

  2. One problem I see with the disaspora dynamics explanation is that you could only observe it with open borders. With mostly closed borders 90% or so of all people who would like to immigrate are not allowed to do so. Immigration restrictions are mostly structured as a quota. So you would expect the quota to fill anyway because there are so many applicants. Since a quota is usually rather sticky, even if people find it easier to emigrate to a country with many compatriots, their numbers will not rise. Only if you assume that the quota were responsive and increased to accomodate more interest, would you be able to see diaspora dynamics. But that’s like saying: Germany took in half a million Syrians this year, so more Syrians would like to be around, and that’s why Germany will make it much easier to immigrate next year. Hardly so.

    One could test the relative relevance within countries, though. Momentum would predict similar growth after a founder effect that is driven by other things (maybe diaspora dynamics at a very low level which makes more sense to me, random effects). Diaspora dynamics would predict that people move to centers of their respective group from other parts of the country. So there should be concentration and much more growth for certain centers). As for people of Turkish descent in Germany: they are pretty much spread out over the West of Germany and with a gradient towards the South. There is literally no German city with a majority of people of Turkish descent (I think the most is something like 20%) although that has been a bogeyman for a long time, how immigrants take over one town after the other (don’t be fooled by reports that seem to indicate otherwise: the trick usually is to take a quarter in a town like Duisburg-Marxloh with less than 20K inhabitants of whom only 45% have foreign citizenship).

  3. “Any culture that will not defend itself against displacement through mass immigration faces extinction. That includes both time-tested and successful cultures. Embracing diversity results in cultural suicide. America’s multicultural path guarantees its destruction via cultural clashes and conflict with Islam, Mexican and African cultures that diametrically oppose American culture. The more diverse a country, the more destructive and broken-down its future. The more people, the more it destroys its quality of life and standard of living. The more it adds immigrants, the more destruction to its environment. The more it imports refugees, the faster America, Canada, Europe and Australia lose their own ability to function and worse, their identities. Exponential growth of any civilization leads to ultimate collapse. You see it in Africa, India and China today. You will see it in Europe, Canada, Australia and America in the coming years, “IF” Western countries don’t stop all forms of immigration.“ Frosty Wooldridge, 6 continent world bicycle traveler, witness to what’s coming to Western countries as to endless refugee immigration.

  4. If possible I would like to work on formulating your entire argument into an abstract or perhaps a couple of lines?

    Essentially that restrictionists, journalists, and the general population misperceive the initial size of the immigrant population and do not take into the lag between arriving (without children) and then having children, and grandchildren and their eventual death. At which point the immigrant population ceases to “grow” and levels off to replacement levels.

    Is that an accurate summation?

    With regards to the levelling off, is this something that happens to all immigrant populations, or only ones to western societies? Are there any statistical examples/studies of this?

    1. I think you get the main point right: young immigrants are only the lower part of a population pyramid. Over time this develops into the whole population pyramid. Immigrants bring their descendents with them although you cannot see them when they immigrate, only later when they are born. After the whole population pyramid has built up and you are in steady-state (roughly after two generations), the respective population grows according to fertility (by construction this was “no growth” here).

      I am not sure what you mean by “western societies.” Iran has sub-replacement fertility, but I assume that you mean fertility above replacement. A rough estimate would go like this: if fertility were 2.5 and not 2 in the model, then each immigrant has 1.25 children and 1.5625 = (1.25 * 1.25) grandchildren. So you’d have an additional 0.25 + 0.5626 = 0.8125 descendants over the first two generations which accelerates growth. However, the momentum effect accounts for 1.76 or more than twice that.

      After two generations, fertility becomes the only driver, and the population will keep growing because now it is above replacement fertility by assumption. Note, though: you have to make the assumption that fertility is very persistent even into the third or fourth generation. It would be reasonable to describe the situation after two generations as due to fertility. But no one ever does this, no one discusses growth of an immigrant population more than half a century after the fact (maybe some scientists, but no one in the media, policitians, the general populace). They always try to explain the build-up in the initial phase in this way. However, there it is only a second-order contribution, the main driver is momentum.

      In a way, this is not an empirical question: if you make these assumptions, the conclusions follow necessarily. It would be fair to ask whether the assumptions are in accordance with what you have in reality. You might want to calibrate the model to actual data, etc. As a check on how close to reality my explanation is I discussed the data for people of Turkish descent in Germany, where it looks like there was about a doubling over half a century and after a clamp-down on immigration (even some return migration probably). It looks like momentum can explain this mostly. I think fertility is or was slightly higher, but not a lot. Another example would be migration from East to West Germany after 1989. The total for net migration was about 1.7 million people, but there was a decrease of about 3.5 million people in the East German states. Actually, I would make the prediction that the East German states will lose another million or so because this was only the first half of the momentum effect.

    2. Maybe another remark to clarify my point here: This is the case for immigration in one year and no immigration afterwards. The examples that I give (immigrants from Turkey to Germany, or in my other comment: East Germans to West Germany) are rougly of this type: a peak over a short timeframe and then only rather low further migration.

      It is also interesting to look at the case where you have a fixed number of immigrants each year, so you get an overlay of many such simple cases with shifts. The basic result is that also in this case the number of first-generation immigrants (and also for all further generations) will level off (assuming replacement fertility) although it takes longer. Since there is always new immigration, the total population will keep growing. I will explore this in further posts. Just wanted to keep things simple for didactic reasons because the momentum effect is perhaps not intuitive at first.

  5. If you’re counting “people of Turkish descent in Germany”, then surely that includes people with only one Turkish parent, or only one Turkish grandparent. If you take into account a certain level of intermarriage, then presumably you can expect that eventually nearly everyone in Germany will be of Turkish descent, even if most Germans will have only one or two Turkish ancestors many generations back, and little or no connection to Turkish culture.

    1. That’s a very good point. As far as I understand it, the definition for “Turkish descent” in the statistics is that someone or at least one of their parents immigrated from Turkey (which does not have to mean they are ethnically Turkish, many are also Kurds), I am unsure how people are counted who are third-generation. Probably they are not counted as “of Turkish descent” in any event, cf. https://en.wikipedia.org/wiki/Turks_in_Germany#Demographics

      I find your argument plausible that over time the descendents of immigrants will probably merge into the general population, and you would only notice a Turkish name (like you have French names from the Huguenots who immigrated in 1685, or Polish and Italian names where immigration occured a century ago or more). However, as for Turkish immigrants, in the short run, rates of intermarriage are pretty low. For first-generation immigrants they are in the lower single digits. This rises somewhat for the second generation, but only to 12% for men who marry out. Here is an article on the topic: http://www.migrationpolicy.org/article/wedding-bells-are-ringing-increasing-rates-intermarriage-germany

      I would also agree that “blood accounting” is essentially silly and meaningless. In German, it also reminds one of Nazi practice to classify people as “quarter-Jews,” etc. My point here is not that I think it is relevant in itself (maybe partly as a proxy for other things, e.g. a common heritage could be the basis for political organization, etc.), but that restrictionists often treat immigrants and descendents like an alien, homogeneous and rather unchangeable block in a country. Even if a group stayed separate and close-knit over generations, I am not convinced this has to be a problem in and of itself. But even if someone thinks it is a problem, they tend to make some assumptions that are not warranted.

      For example, the consequence of this post is that immigrants bring their own assimilation program with them: their children will be closer to the native culture, and their grandchildren even more. And they outnumber the initial immigrants after some time, so the impact of unassimilated immigrants (if there was one) would automatically be cushioned by their descendents. It is almost impossible that first-generation immigrants become a majority which pretty much goes against the dramatic visions of some restrictionists that natives will become a minority in “their” country or that assimilation will break down because there are so many immigrants (every new wave of immigrants comes into a country that is predominantly native).

      And there are also some funny conclusions that go against what restrictionists want to believe. E.g. if immigrants have higher fertility, they have more children and even more grandchildren. So this shifts the balance in the direction of more assimilated people in the group and decreases the relative importance of first-generation immigrants. I will explore all this in further posts. Stay tuned.

  6. To Vipul & Hansjörg, regarding my initial question of condensing your thoughts into a couple of lines, I did this because I think this article would vastly benefit if one were to be inserted at the top.
    It would make this article easier to share if it had an abstract that could be copy/pasted straight from the author to give potential readers an idea of your explanation.

    Unless of course you want curiosity to be the main driver. 🙂

    I should have been more precise with my terminology.
    By “western societies” I was referring to developed countries, that have positive net migration, and are within the western cultural group e.g. UK, USA, Australia, Germany. But looking back it now I don’t think the question is relevant as I am sure that there is not a single answer but rather that fertility among migrant steady-state populations varies due to a large array of factors.

    “In a way, this is not an empirical question: if you make these assumptions, the conclusions follow necessarily. It would be fair to ask whether the assumptions are in accordance with what you have in reality. “

    True, I just find it very helpful for the reader when there are lots of examples at hand.

    1. Thanks a lot for your reply. It would have been a good idea to write a short executive summary as an introduction, but I was so focused on the argument that I forgot to do it. Will try to keep this in mind for my next posts. And again thanks!

  7. Haha! Come on, man, the post tries to make fun of restrictionists, but it’s a fucking disaster for immigrationists (or maybe the other way around). So even if you have a migrant population on replacement fertility levels, due to age structure issues, it takes half a century before it stabilizes, during which it doubles o triples! President Trump is a fact, people.

    1. I don’t know why you think I was trying to make fun of anything. I only explained an effect that many people tend to miss. Anyone can use the argument. It works both ways: It means that focusing on immigrants alone underestimates the effect of immigration. But it also means that extrapolating from the buildup to ever increasing growth is also unwarranted.

      As for Präsident Drumpf: I see you Americans fall for our long-term ploy to take your country over with open borders. Planting Friedrich Drumpf (https://en.wikipedia.org/wiki/Frederick_Trump) was far easier than waging two worldwars. Good news for the German world conspiracy.

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