One common argument against allowing new housing in popular cities is that as long as rich foreigners use up the housing supply prices will never go down.

Every so often I read an argument that says something like this: "There's no way that rents can ever go down in expensive cities like New York and San Francisco, because rich foreigners are buying up everything, and as a result demand is essentially infinite. So there's really no reason to allow new apartments because the foreigners will just buy them all."
This argument could certainly be true in theory: for example, if an entire city were so expensive that only wealthy foreign investors could afford to live there. But it doesn't seem to fit the social reality of the expensive city I am most familiar with (New York).
If demand was essentially infinite, there would be some evidence of this fact other than high real estate prices. For example, if New York was growing faster than any city in the United States, there might be a credible argument that new demand is truly insatiable. In fact, many low-cost cities are growing far more rapidly than New York and yet have cheaper rents. Between 2000 and 2010, Austin and Raleigh (both of which are far cheaper than New York) grew by 20 and 46 percent, respectively, while New York grew by less than 10 percent.
It could be argued that New York is dissimilar to other high-growth cities, because the number of wealthy foreigners is so huge that it makes New York a high-growth city. For example, if New York had to accommodate not only half a million new residents per decade but also three million wealthy foreigners seeking second homes, its housing-consuming population would have grown by about 45 percent over the past decade, about the same level of population increase as Raleigh. But are there really that many wealthy foreigners in the New York housing market?
I suspect not. According to a New York Times article about the evils of foreign investment in New York housing, "About $8 billion is spent each year for New York City residences that cost more than $5 million each." Using the magic of long division, I calculate that even if each residence cost only $5 million and not a penny more, there would be 1600 such residences. In fact, some residences cost far more, so the actual number if super-expensive residences is lower (and of course, the number of such residences purchased by foreigners is lower still). In a city with 8 million people (and thus a few million households), a thousand or so really rich people seems to be like a drop in the bucket, even if their wealth does give them disproportionate influence and notoriety.
Moreover, it seems to me that if New York was really flooded with millions of foreign billionaires, housing prices would be even more expensive than they are. During my last year in New York (2013-14), I lived in a 448-square-foot studio in Midtown and paid $2330 a month in rent (The same apartment would cost $2680 today). This rent is very expensive by the standards of Planet Earth—but by the standards of Planet Foreign Billionaire Oligarch (FBO), it is nothing. What self-respecting FBO lives in a 450 square foot studio? And what self-respecting FBO pays less than $3000 in rent? I don't know how much FBOs make, but I'm guessing that the median FBO income was at least $10 million per year (1 percent interest of $1 billion in wealth)—which means that any self-respecting FBO should be able to afford $60,000 per month (the price of the most expensive New York apartment I found on zillow.com).
And I wasn't even living in the neighborhood's cheap housing: had I chosen to live in a walkup, I could have paid around $1700-1800 per month—not exactly billionaire rents. And in the supposedly oligarch-ridden Upper East Side, rents are even cheaper.
And these non-FBO-friendly rents are in the nice parts of Manhattan; rents are even cheaper in most of the outer boroughs. Thus, it is pretty clear that to live in New York (even in the nice parts of Manhattan) you don't have to be a FBO.
In sum, the overwhelming majority of people who are living in, and bidding up the price of, New York real estate are not FBOs. There is no reason, other than government regulation (and the technical difficulties of building housing in a city that is already "built out") why supply could not rise to meet demand more effectively.
Having said that, there is two grains of truth in the "insatiable demand" argument: first, even if New York only has a thousand FBOs, it has lots of domestic rich people- not just billionaires, but "ordinary" people earning in the high six figures, people who cannot afford to pay $60,000 per month but can afford to pay $5,000. The more rich people a city has, the more money that will be used to bid up real estate prices- which in turn means the city has to work harder to keep supply growing.
Second, even if an adequate building supply keeps rents down, it might be that no politically feasible policies will increase supply enough to restrain rents. But as readers of this blog are probably aware by now, "policies I favor" and "politically feasible policies" are two concepts that rarely intersect.

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