Index Numbers and Decomposition Spaces

A basic property of a mean, such as the arithmetic or geometric, for example, is that a grand mean may always be calculated as a suitably weighted mean of submeans. An analogous property of index numbers is called consistency in aggregation in the relevant literature. In this article we show that this property under weak regularity conditions implies that such index numbers may be represented as convex combinations in a suitable linear space. Linear algebra may thus be applied to their study, a completely new approach to the subject.To illustrate the strength of our approach, we prove an equivalence theorem of index formulas and additive decompositions of value change. We argue that this shows that consistency in aggregation may be interpreted as consistency between the treatment of ratios and differences. Also, many of the classical axioms of index number theory are shown to have a linear algebraic structure