Exploring the GK price index

The present paper intends to explore the Geary-Khamis price index and to consider some variants. This exploration was primarily driven by the interests of the author. The goal was simply to better understand this price index. For the present author this price index has great appeal because it is based on a reasoning that seems quite natural. Various refinements are possible, some of which are rather obvious.
The GK price index is introduced in terms of a system of equations. These equations are not symmetric and also not linear in its parameters, which are vectors of prices of goods and a vector with price index values. By using different parameters it is possible to rewrite the defining equations in such a way that the defining equations are both symmetric and linear in its parameters. The convergence of these equations is then investigated. The defining equations are modified in various ways, thus yielding defining equations of variants of the GK-price index. Then the nontransitivity of the GK-price index is considered. A method (related to the cycle method) called transitive completion is proposed to produce transitive price indices from a GK-price index. Also chaining methods are considered that are not restricted to GK-(type) price indices.