The aim of this research is to estimate the extent of substitution bias in the annual inflation figures derived from the UK Retail Price Index (RPI). Substitution bias refers to differences between a fixed weight price index and an index that measures the true cost-of-living. The RPI tracks the cost of buying a basket of commodities over time. Annual inflation figures use fixed weights for the commodities. Substitution bias might arise because fixed weights ignore the possibility that spending patterns may adjust when relative prices change, as consumers substitute towards relatively cheaper goods. A true cost-of-living index would measure the changing cost of achieving a fixed level of economic welfare rather than of buying a fixed basket of commodities. The research shows that the direction of substitution bias in the RPI is not known a priori - it could under- or over-state the true increase in cost-of-living. A non-parametric method based on the axioms of revealed preference is then used to place a bound on the possible extend of substitution bias in the RPI. This works by placing bounds on the set of consumption baskets that would yield the same level of welfare as the base basket.