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Over the past five decades a number of multilateral index number systems havebeen proposed for spatial and cross-country price comparisons. Thesemultilateral indexes are usually expressed as solutions to systems of linear ornonlinear equations. In this paper, we provide general theorems that can beused to establish necessary and sufficient conditions for the existence anduniqueness of the Geary-Khamis, IDB, Neary and Rao indexes as well as potentialnew systems including two generalized systems of index numbers. One of our mainresults is that the necessary and sufficient conditions for existence anduniqueness of solutions can often be stated in terms of graph-theoreticconcepts and a verifiable condition based on observed quantities ofcommodities.

Multilateral Index Number Systems for International Price Comparisons: Properties, Existence and Uniqueness