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I want to calculate the real terms rise in UK private rent prices since 2011 using the CPIH measure. The indexes of nominal rent prices and the CPIH rose by 14.7% and 13.5% respectively, so I can say rent prices rose by 1.2% more than inflation did.

But do I need to account for the fact that rent prices are a component in the basket of goods used to calculate CPIH? It seems circular to say 'Rents didn't rise that much faster than the CPIH' if the CPIH would have risen by less without rents being included. Is that worry correct? If so, how do economists adjust for it? Many thanks

Edit: Note this question doesn't actually have anything to do with rent/ housing prices. It's about how to measure the real times rise in the price of any item that is part of the basket of goods used to calculate the CPIH.

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