  Over the past five decades a number of multilateral index number systems have
been proposed for spatial and cross-country price comparisons. These
multilateral indexes are usually expressed as solutions to systems of linear or
nonlinear equations. In this paper, we provide general theorems that can be
used to establish necessary and sufficient conditions for the existence and
uniqueness of the Geary-Khamis, IDB, Neary and Rao indexes as well as potential
new systems including two generalized systems of index numbers. One of our main
results is that the necessary and sufficient conditions for existence and
uniqueness of solutions can often be stated in terms of graph-theoretic
concepts and a verifiable condition based on observed quantities of
commodities.
