IMO that's the real key to the real estate problem and it should be an escalating process.
Property you live in = 1 rate
1 property you own but don't live in = slightly higher rate (family vacation home perhaps)
All other property you own but don't live in = much higher rate (this is the business tier)
I'm not sure what the solution is. There's probably a tax component to it, but I think second order effects need to be carefully considered.
If it forces sales, you get more home inventory and lower prices.
This is in aggregate, of course; individual landlords may charge less than what individual tenants can bear but across the rental market, landlords extract monopoly rents already.
That's why you will find rents increase proportionately to incomes, not landlord costs. This is highly apparent in New Zealand, where mortgages are not fixed rates for more than 5 years; interest rate changes have little to no impact on rents, but supply and tenant incomes do.
> ...individual landlords may charge less than what individual tenants can bear...
I lived in a supply-constrained city once and my landlord, a nationwide public company, could have charged me significantly more without prompting me to move. I am one individual though and you called out that exception.
However, had I been priced out of the unit in which I lived I would have found a smaller and/or less desirable unit at lower cost. Rental units vary in size and quality.
The existence of individuals who can pay more and the non-fungibilty of rental stock makes me suspect that such taxes will simply be passed on.