This type of tax (see also 'roof tax', 'Planning Gains Supplement') is vaguely heading-in-the-right-direction, but it starts from the wrong end (if that's not a contradiction in itself).
1. The most fundamental question is whether local councils should be able to control planning/building this tightly. I am all in favour of rabid liberalisation, but by the same token, I am in favour of returning as many powers as possible to local councils, and if local voters are Greenies/NIMBYs, then I suppose it's only fair to let them stifle developments/improvements.
2. Yes, more people in an area impose greater 'infrastructure' costs. But by the same token, those increased costs are shared between a larger number of people. And you should never overlook the benefits of agglomeration.
3. Further, who's to say that the man in the story wanted to use both additional rooms as bedrooms? How would the council respond if they had instead caught him chucking out his beds and buying half-a-dozen sets of bunk-beds to go in the existing rooms? (Answer here!)
4. Let's assume, that local voters are happy with the idea that everybody can extend his house by 25%. For sure, on the one hand, that makes the area a bit more crowded (depressing land values) but, on the other, the fact that there is implied permission to build an extension improves the value of each individual house (increasing land values); there is such a thing as a happy medium.
5. Then let's assume that we had a sensible system of local taxation, i.e. Land Value Tax to pay for all locally controlled expenditure and ...
5 a) Council #1 has very liberal local planning by-laws. A lot of people build extensions, so local population increases. There's more pressure on local infrastructure, so the council then does a cost-benefit-analysis: would widening roads, increasing frequency of buses etc. make the area more desirable again, thereby increasing land values?
If the additional tax revenues from the incremental increase in land values would cover the additional costs, then that extra spending is worth doing and home-owners do not lose out (else not, plans shelved). Bluntly speaking, local residents and businesses are getting what they pay for. And as mentioned, a larger local population means that the total money to be raised is borne by more people, so pro rata, nobody has to lose out.
5 b) Council #2 imposes a total ban on any local developments/extensions etc. Local property owners thereby enhance the value of their land with existing buildings(scarcity value), so they pay more in Land Value Tax and they are paying for the artifical scarcity value that they themselves have created (and the cost of those public amenities like railway stations etc. is shared between a smaller number of people), and also compensating owners of those un- or underimproved sites for the fact that their land is rendered nigh worthless (and who thus would pay much less LVT).
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