An impossibility result

Ideally there would be a single price index that satisfies the five axioms and three tests of the previous sections. This would mean that there is only one index-number formula that should ever be used, at least for the purposes of creating national statistics. However, it can be shown that there is no price index that satisfies the identity axiom, the circularity test, and the product test (Balk 1995, Theorem 1), and so a fortiori there is no index that satisfies all five axioms and all three tests. This is because there is a trade-off between having an index that can be chained across successive periods (circularity test), and an index that fits in a national accounting framework (product test). Thus, the usefulness of a particular test depends on the use of the index, and this matters for selecting a best index in the axiomatic framework.