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You are making a big assumption that all tradesmen are VAT registered. You need a turnover in excess of 77K to be VAT registered, so many fall well below that. (Its not difficult so long as you get your punters to buy the large materials themselves eg the bathroom suite). I think you will find many VAT companies would not be interested in cash (apart from when people need to pay that way for personal reasons - and three are quite a few still about)
As for immigrants, in actually it is well documented that many immigrants work cash in hand, epsecially the illegal ones !!!! So please, stop trying to drag the debate down with racism.