If you subsidize housing you create increased demand for housing, ultimately leading to rent going up for all.

So, as I said, I'm not an advocate of subsidizing housing out of taxes. I'm just saying that people who are arguing for rent control are arguing for a policy that tends to exacerbate the problem in the long run.

Subsidizing housing doesn't normally run into that, because it's normally possible to build more housing.

It is true that that's not always the case, and one very real way in which that can not be the case is where there have been restrictions placed on constructing more housing. If housing prices are high, the first thing I would look at is "why can't developers build more housing, and are there regulatory restrictions preventing them from doing so". It is quite common to place height restrictions on new constructions, which prevents developers from building property to meet that demand, which drives up housing prices (and rents). In London, there are restrictions placed that disallow building upwards such that a building would be in line-of-sight between several landmarks. That restricts construction in London and makes housing prices artificially rise. Getting planning permission may also be a bottleneck. I agree with you that that sort of thing is the thing that I would tend to look at first as well: removing restrictions on housing construction is the preferable way to solve a housing problem.

I remember an article from Edward Glaeser some time back talking about how much restrictions on construction -- he particularly objected to the expanding number of protected older, short buildings -- have led to cost of housing going up.

How Skyscrapers Can Save the City

Besides making cities more affordable and architecturally interesting, tall buildings are greener than sprawl, and they foster social capital and creativity. Yet some urban planners and preservationists seem to have a misplaced fear of heights that yields damaging restrictions on how tall a building can be. From New York to Paris to Mumbai, there’s a powerful case for building up, not out.

By Edward Glaeser

It looks like it's paywalled, so here:

https://archive.is/jRQIm