Local Circularity of Six Classic Price Indexes

In this paper, we characterize local circularity for the Laspeyres, Paasche, and Fisher price indexes. In the first two cases, we begin by deriving a sufficient condition for achieving circularity that establishes that at least one of two proposed equalities must hold. We end up showing that the sufficient condition is also necessary. We continue with the Fisher price index that is the geometric mean of the two, and we find a sufficient circularity condition that is a direct consequence of the corresponding sufficient conditions for its two component indexes. However, we also show that, unlike its Laspeyres and Paasche components, this sufficient circularity condition for the Fisher price index is not necessary. We reach different conclusions when we extend our investigation to the circularity properties of the geometric Laspeyres, geometric Paasche, and Tornqvist price indexes, for which none of the proposed sufficient conditions is necessary. Throughout, we distinguish local circularity, which all six price indexes satisfy, from global circularity, which none of the price indexes satisfies.