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By knowingly paying a tradesperson in cash without a VAT receipt for the sole purpose of getting a cheaper quote by avoiding VAT it is the person paying that is effectively not paying the VAT.
But the tradesperson being paid in cash might also miss that job from their annual return. My brother-in-law is a carpet fitter. He pays an accountant to prepare his tax return in a way that doesn't arouse the suspicion of HMRC but substantially reduces his tax bill. Anything paid in cash makes that process easier.
(there are other issues with my brother-in-law which are probably not for a public forum)
Some would argue that it's the person paying cash that is "actively practicing tax evasion".
By knowingly paying a tradesperson in cash without a VAT receipt for the sole purpose of getting a cheaper quote by avoiding VAT it is the person paying that is effectively not paying the VAT.
In most cases the tradesperson just passes on the VAT that you have paid onto HMRC, so if you're not paying it in the first place it's not the builder that is evading tax (although they are the ones enabling it by offering to do the work off the books).
As TW says, invoicing and charging VAT but not paying HMRC is a different thing, choice of payment method doesn't really affect this.
Some trades people want to keep stuff off the books to keep corporation tax bills down too.