This paper proposes an empirically feasible method for correcting the path-dependence bias of chain indices of the cost of living. Chain indices are discrete approximations to Divisia indices and it is well known that the latter are path-dependent: the level of a Divisia index is affected not just by the level of prices at the two endpoints but also by the path between the endpoints. It is also well-known that a Divisia index of the cost of living is path-independent if and only if all income elasticities are equal to one, a restriction that is decisively rejected by studies of consumer demand. In theory, the true cost of living index (or Konüs price index) could be derived by estimating the expenditure function. But this seems impractical due to data limitations: the number of independent parameters rises roughly in proportion to the square of the number of commodities and consumer price indices contain hundreds of items. This paper shows how this problem can in fact be overcome empirically using a flexible model of demand like the "Quadratic Almost Ideal Demand System". The proposed method requires data only on prices, aggregate budget shares and aggregate expenditure. The method is applied to estimate Konüs price indices for 70 products covering nearly all the UK's Retail Prices Index over 1974-2004, with each year in turn as the base. The choice of base year for utility is found to have a significant effect on the index, even in the low inflation period since 1990.