Basic Statistics of Jevons and Carli Indices under the GBM Price Model

Most countries use either the Jevons or Carli index for the calculation of their Consumer Price Index (CPI) at the lowest (elementary) level of aggregation. The choice of the elementary formula for inflation measurement does matter and the effect of the change of the index formula was estimated by the Bureau of Labor Statistics (2001). It has been shown in the literature that the difference between the Carli index and the Jevons index is bounded from below by the variance of the price relatives. In this article, we extend this result, comparing expected values and variances of these sample indices under the assumption that prices are described by a geometric Brownian motion (GBM). We provide formulas for their biases, variances and mean-squared errors.