when you tax things you get less of it.

when you tax supply, you get less supply

many of these second homes are currently on the rental market. if there is less supply of these coming into the rental market, rent prices rise.

I don't know why everybody's brains are so broken when it comes to housing policy.

let's say you own an uninhabitable 2 family home in Brooklyn that was built in 1910 but would require serious renovation to be able to rent it out (not at all a strawman; this is incredibly common). now imagine your incentives as the property owner:

* without the pied-a-terre tax: some risk & upfront cost to renovate, but future cash flows from rental income make this incentivized

* with the tax: same risk & upfront cost, but now the future cash flows are decreased by the amount of the tax (since the assessed value will have increased)

anywhere that difference tips the scales from "renovating" to "not renovating," there is one fewer home on the market.