how do you incentivize building, and yet have a vacancy tax? If you invest in building, but couldn't lent it out (as it's been incentivized so lots of new competition), it would get hit with a vacancy tax! So the risk of building becomes high under a vacancy tax - therefore, it would cost more to incentivize.

A vacancy tax would make it less attractive to investors but people purchasing it to live there as their primary residence wouldn't mind. I dont see how forcing a landlord to rent a property out at market rates or pay a tax is a massive burden. It doesn't have to be a real tax, but it would help stop landlords withholding units for no reason except they aren't making enough profit.