When people argue about whether immigrants are pulling down our standard of living, they rarely notice a simple but deep arithmetical fact: Everyone in a country can get richer as per-capita income falls. Proof by example:

Suppose the residents of Country A earn $50,000 per year each without immigration, and $60,000 per year with immigration. They benefit from cheaper lawn-mowing. The residents of Country B earn $2,000 per year if they stay at home, or $10,000 per year if they immigrate to Country A to mow lawns.

Now what happens to per-capita income in Country A if immigrants double the population? Per-capita income falls from $50,000 to .5*$60,0000 + .5*$10,000=$35,000. The more immigrants come in, the more steeply per-capita income declines. “Immigrants hurt our standard of living. QED!”

Of course, nothing of the kind has happened. By assumption, immigration makes both natives and immigrants richer. But per-capita income declines, as a matter of pure arithmetic. The numbers don’t lie, but they are very easily misinterpreted.

There are many applications of this simple insight. It also applies nicely to the apparent effect of rising female labor force participation on the pay of the average worker. Proof by example:

Suppose in 1950 the workforce is 90% male. Men earn $10,000, women $3000. In 1975, the workforce is 50% male. Each man now earns $12,000, and each female earns $5000. What happens to average worker earnings? They fall from $9,300 to $8500.

One last example: Average earnings can rise even when they are falling for all educational classes. Proof by example:

Suppose that in 1960 15% of workers have college degrees, 85% only high school. A college degree gets you $20,000/year; a high school diploma gets you $10,000/year. In 1980, the break-down moves to 50/50. A college degree is now worth $18,000, and a high school diploma $9,000. Average earnings before: $11,500. After: $13,500.

This doesn’t prove, of course, that immigrants and women don’t reduce the earnings of other groups. It doesn’t prove that if incomes of all educational classes are falling, that the average is actually rising. What it shows, rather, is that evidence that seems rock-solid can actually be trivial and irrelevant. Before you take an average, you have to think about what you are averaging over.