The Sato–Vartia index and the monotonicity axiom

The Sato–Vartia index is believed to be on a par with the Fisher ideal index in terms of its ability to satisfy index number axioms or tests. Yet we show that this index fails to satisfy the monotonicity axiom, and we provide formulas for the point at which a Sato–Vartia price index becomes non-monotonic in prices. In fact, requiring monotonicity for initial as well as final prices rules out not only the Sato–Vartia index but all log-change price indexes whose weights depend upon initial or final expenditure shares. Since cost of living index theory leads naturally to the Sato–Vartia index, as well as to other log-change indexes, there is an irreconcilable tension between the standard axiomatics for indexes, and cost of living index theory.