Okay, just making sure.
I see what you mean, but most people in a city are not trying to buy a house anyway, especially since most of the houses in a city are segregated into a basically suburban area far away from downtown, and there wouldn’t be enough of them for the population of the city, anyway. Apartments are much more effective. I would probably try and condemn those older, rusting building and build affordable housing there instead.
The easiest thing to do would be to restrict who is allowed to buy in what neighborhood, but that is central planning, and would probably backfire anyway.
Or switch from a property tax to a land tax, which taxes the underlying value of the land rather than the property on it. I.e a parking lot is taxed at the same assessment as the skyscraper next door. What land use do you think derives more income and value to the owner now?