I'd love to see some kind of "use it or lose it" law on unoccupied residences, where they would be subject to large tax penalties or worse. Unfortunately, the kind of people who would be hurt by this are also the kind of people who can and will hire experts to skirt, game, or flout the law. So there's your problem.
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Yes, it's basically a residential property tax as it's based (very loosely) on property value and is payed per-property, not per-resident. But it falls on the resident, not the property owner.
Called council tax because it funds council services like street maintenance and rubbish collection.
True, I was imagining (though I see I never actually said this) a situation though where money raised through these properties would go to making other housing cheaper. I'm not an expert in these matters, but it seems to me like using it for rent subsidies might be the way to do it.