The Crown, and therefore the Queen, has an arrangement with Parliament to surrender the profits from the Crown Estate (the Queen's personally-owned lands) to Parliament in exchange for an annual salary (roughly 15% of the profits for the year that started two years before, or about £30–40 million).
A common argument given by republicans against the royal family is that this costs the each taxpayer 81p (closer to 62p) per year, but actually it's the opposite; the Crown Estate earns Parliament around £200 million per year, meaning £160 million goes to Parliament, directly saving each citizen £2.50 per year.
If Parliament were to stop paying this annual salary, the Crown would take all of the profits for itself and it would cost the taxpayer £3.12 per year, because while Parliament earns the profits from the estate, it is still owned by the Crown, and so control of the profits would revert back to the royal family if Parliament didn't hold up their end of the deal.
Furthermore, the UK earns an estimated £7 billion per year from tourists visiting royal sites like the Tower of London and Buckingham Palace, both of which, while owned by Parliament, are closely associated with the royal family. The royal family are a major part of the reason Great Britain's castles are more visited by tourists than, say, Germany's or France's; they are still being used by real-life royalty and nobility, a concept which is fascinating to tourists from the monarch-free Americas.
In short, the royal family gets paid because to not do so would actually cost the UK greatly. Keep'emPeeled.
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