Elementary price indices under the GBM price model

Abstract

Countries use either the Dutot, Carli or Jevons indices for the calculation of their Consumer Price Index (CPI) at the lowest level of aggregation. In this paper, we compare expected values and variances of sample elementary indices under the assumption that prices are described by a geometric Brownian motion (GBM). We consider two situations, i.e., the case with only one homogeneous product group and the case with a heterogeneous group of products consisting of homogeneous subgroups. We provide formulas for their biases, variances and mean-squared errors. We confirm the utility of the presented continuous time approach via a simulation study.

Keywords:

• Consumer Price Index
• geometric Brownian motion
• Dutot index
• Carli index
• Jevons index

JEL CLASSIFICATION:

• E1
• E2
• E3

Notes

1 The formulas of population indices can be formally justified by introducing an additional discrete random variable $\chi$ with the following probability distribution: $P(\chi =i)=1/M,$ for $i\in \{1,2,\dots ,M\}.$ This random variable describes the process of selecting the homogeneous product subgroup (each subgroup has the same chance to be included in the sample). Assuming that $E({\tilde{p}}^t/\chi =i)=E({\tilde{p}}_i^t),$ we obtain the expected value of the price of products from the elementary group as: $E({\tilde{p}}^t)=E(E({\tilde{p}}^t/\chi ))={\sum }_{i=1}^{N}E({\tilde{p}}_i^t)\cdot P(\chi =i)=\frac{1}{M}{\sum }_{i=1}^{M}E({\tilde{p}}_i^t).$

Additional information

Funding

The paper is financed by the National Science Center in Poland (grant no. 2017/25/B/HS4/00387).