There's some misunderstanding about housing markets illustrated by this comment which seems intuitive to me. Given the luxury pricing, not all the units will be rented (or not for long) because they don't need to be to keep the property profitable. Of those individuals that could afford the price, a large amount will move into the more affordable housing out of personal self-interest. The trickle down doesn't work in the manner described, in any meaningful way. Historically (rather than regionally) it has been working out this way for at least my lifetime.
One could of course tax empty housing. In Hamburg, Germany, you can leave an appartment empty for up to 3 months after that, you are taxed on the base of its rental value, whether you get that paid or not.
