Reader Bruce Charlton points me to a post by Half Sigma.
Unskilled low-IQ immigrants who come to the U.S. contribute slightly to higher total GDP, but they lower the GDP per capita.
Letting in a low-skilled immigrant to mow my lawn will indeed lower average productivity in this country. But I will be better off.
To the extent that low-skilled immigrants are significant, standard productivity indicators can be misleading. For example, suppose that France lets in a bunch of low-skilled immigrants and puts them on welfare, while we let in a bunch of low-skilled immigrants and put them to work. Then our GDP per capita will be higher, but France's GDP per worker will be higher and their GDP per hour of work will be higher. If you use the latter as a measure of productivity, you would say that France is following a brilliant strategy, and we should copy it.
To get closer to a relevant measure of national economic performance, you might invent something like endowment-adjusted consumption per working-age adult. By adjusting for endowment, you correct for the lower skills of immigrants. By using consumption, you get a measure that is closer to the quality of life. And by using working-age adults, you don't treat high unemployment of low-skilled workers as a good thing.
But it's important to remember that you personally don't get to consume GDP per capita. You get to consume based on the purchasing power of what you produce. If immigrants lower the cost of goods and services for us, then bully for immigrants.