IMO, the ideal situation is a steeply progressive income tax, along with a reasonable property tax that has a limit to annual increases for primary residences. If one wanted to make the argument that there should be a cap on other types of property, it might be reasonable to have a cap that is higher than for primary residences.
Property taxes cannot be passed on to tenants unless the landlords are willing to accept much higher densities in their rental house. Renters have to pay out of pocket, so there is an upper limit on what can be charged. Of course, one might suggest that if higher property taxes should be passed on to renters, then the subsidy should be passed on to renters as well. With a few exceptions, most LLs will charge the highest that the market will bear, though a few smart ones will charge somewhat less in an effort to attract premium tenants.
A variable property tax will also help correct market inefficiencies since the higher taxes will hit higher-end homes that likely are in high demand and short supply. This will have the effect of putting more homes on the market at precisely the time (and place) when they are needed to keep property prices in line.
Just so you know, Prop 13 is absolutely a subsidy in every way, shape, and form. It is the very essence of a subsidy.
