Change in Immigrant Stock is NOT the Same as Net Migration

[This is a post in my series on free migration. I will keep an overview with short summaries updated that you can find here: Synopsis: Arguments for Free Migration.]

Apparenty I need to write this up as a reference because I see this mistake on a regular basis. The latest entry is from Noah Smith, and a few days back it was Kevin Drum (at least it seems as if the same error were in the background). The context is this: Someone wants to disprove that there is a crisis at the Southern border of the US. As far as it goes, I fully agree. This is a made-up panic. Still it seems as if it also lets loose a lot of motivated reasoning.

To show that there is no crisis, the following argument is deployed: The stock of immigrants, resp. the stock of illegal immigrants, has been shrinking for a decade or so. As is obvious, still further immigrants — both legally and illegally — cross the border. So there is gross immigration to the US. Hence the conclusion is that even more people must go in the other direction because the change in stock is negative.

Kevin Drum even claimed that there is net *illegal* migration with Mexico. I tripped him up with the question whether this means there is a lot of *illegal* migration from the US to Mexico, even more than in the other direction. Of course, that seems patently false. Drum then adjusted it to “illegal immigrants return,” presumably legally as Mexican citizens do not have to pay coyotes to cross the border from the US to Mexico. Yet, I don’t know whether that is warranted either.

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The main point here is simple accounting:

Net migration is the number of people who enter the country (immigrants) minus the number people who leave the country (emigrants or returnees). If the change in stock were the same as net migration — and that is apparently the underlying assumption — you could conclude from shrinking numbers plus obviously positive numbers for new immigrants that the number of emigrants or returnees must be positive, too, and even higher.

However, here is why this is wrong:

People leave the stock also in other ways: One way is that someone dies. Only if immigrants enjoyed eternal life could you disregard this. But obviously you can’t. As for illegal immigrants, there is also another way: They can obtain some legal status and leave the stock for that reason. If you look at two countries like the US and Mexico, it is also possible that the stock of immigrants from Mexico goes down because they exit the US for some other country, eg. Canada, not Mexico. Even if the logic above worked, you could only conclude that immigrants from Mexico leave the US, not where they are going. Maybe this is negligible as an effect, but in any event, you would have to address this point before jumping to conclusions.

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But then the logic is wrong and the main point is that obviously also immigrants die as everybody else. Let me work through this in a stylized example. I developed it in some tweets in a different context, namely for immigration to Sweden. The mistake at hand is often linked with another mistake where the implicit assumption is also eternal life for immigrants.

Suppose you have a country of 10 million people like Sweden. To get confusing details out of the way, I assume that there has been no migration up until now. Then annual immigration of 1% of the initial population sets in, ie. of 100,000 people per year. I assume there is no e/trans/remigration to make things more transparent.

After one year, you have 100,000 immigrants, after two you have 200,000, after three 300,000. The related mistake is now to extrapolate this and assume that the number of immigrants will increase as a linear function. If that were so, their share in the total population would keep rising. After 100 years (hundred times 1% of the initial population), there would be 10 million immigrants. If the rest of the population remains stable at 10 million, that means immigrants would now make up half the population. Afterwards, they would become a majority.

The operative word here is “would” because this is false. The number of immigrants contines to rise, but only if you keep counting also those who have already died. There would be 10 million immigrants in a century, but many of them would also be dead. The number of immigrants alive must be lower because the first 100,000 immigrants will surely be dead by then, the next 100,000, too, etc.

If immigrants come at an age of perhaps 25 years — not unrealistic in such situations — they have about half a century left to live on average. So after about half a century, a cohort (those who come in one year) will die on average per year. In reality, not everyone drops dead at the same time and they are not of the same age initially, so this is somewhat smoothed out. But that is roughly what happens.

This means that after about half a century of continuing immigration, one cohort of 100,000 dies each year and a new cohort of 100,000 arrives. In other words, the stock of immigrants will now remain stable and not grow any further. It will level off at something like 5 million (fifty times 100,000). And it is just not possible with this setup to ever have more than a share of one third for immigrants in the total population: 5 million out of 15 million. They will never become a majority although always new immigrants arrive.

This estimate is actually an overestimate. The 5 million immigrants in the population will also have children. Those come on top, so-called population momentum. For a stable population (ie. with just replacement fertility!), that would be roughly another 5 million people (it depends on how many children immigrants bring along). So the population after a century will be greater than 15 million, more like 20 million including the second generation. Therefore, the share of immigrants will not go beyond even a quarter.

Ironically, the descendants of previous immigrants — who are natives — make sure that immigrants stay even further away from a majority than with the crude first stab: a quarter, not a third at most. And then the children have children of their own. Those come on top as well. Hence, the share of immigrants in the population will not go much beyond 20% ever with this setup, no matter how long you have immigration of 100,000 per year. This is not intuitive because it is tempting to conclude from “there are always more and more immigrants” to “their share will keep rising to 100%.”

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Now back to the case at hand: Immigration from Mexico to the US has been going on for a long time. With changes in immigration law in the 1960s, there is a backstory of about half a century, just what you need to reach a stable level. That’s why you have to expect that the stock should be stagnating by now — even with regular positive net migration of further Mexicans. In principle that could be so with zero migration back. A stable stock does not have to mean that as many return as come. It could also be that as many die. People leave the stock in this case, too, but for the graveyard, not for Mexico (where it is not clear anyway that it has to be Mexico per my remark above).

The stock of immigrants can also decrease with positive net migration. There are 12 million or so illegal immigrants in the US, most of them from Mexico as far as I know. That has been going on for decades. If you divide by 50, you get a net inflow of about 240,000 per year that would lead to such a stable stock after half a century.

Now suppose there used to be more immigration, eg. 300,000 a year in the early phase. But that has gone down to 200,000 by now. Older cohorts are stronger than younger cohorts. So a cohort of 300,000 dies annually after half a century, and only a cohort of 200,000 arrives. The net effect is that the stock decreases by 100,000 per year. Actually, that is roughly what has happened over the past decade.

However, if you look at it: You still have net migration of 200,000 annually by assumption. It is false to conclude that net migration has to be -100,000 now because the stock decreases by that number, and that 300,000 go back to Mexico per year. It is actually awfully wrong. Of course, my assumptions may be mistaken and net migration is lower. It is also entirely possible that there is indeed remigration and a net outflow. But it seems easily possible that net migration could be positive with lower, but still positive net migration. I have not checked what it is because I expect people who make such claims to do this before they rush to conclusions.

The least you should do in this situation is to address this important point. But that raises the question why you argue with the stock of immigrants at all, and do not refer your readers to estimates for net migration outright. The only reason I can see why so many people make this mistake seems to be motivated reasoning. You want an argument against a panic that the US is being “swamped” by illegal immigrants — fair enough — and you see that the stock of immigrants (or illegal immigrants) decreases, and so it is good enough and you jump to a conclusion that is just what the doctor prescribed. A less generous explanation would be that you expect your readers to make the mistake above.

Now that I have written it up, I am sure no one will ever get this wrong again.