The aggregate cost of living index requires averaging across household indices. But what if the aggregate index was constructed for the ‘representative’ household as is usually done? The paper examines the resulting bias in the Tornqvist index, widely used for constructing superlative indices as well as for the Cobb–Douglas index, which has a similar functional form. We show that the difference between the two consists of a ‘plutocratic’ bias and a ‘curvature’ bias. The former is well known but the latter has not been recognized earlier. In empirical applications, the curvature bias is small and orders of magnitude smaller than the plutocratic bias. This suggests much of the overall bias would be removed by constructing the representative agent by democratic averages of budget shares.