The tests
In addition to the five axioms, there are three important tests that may be desirable for a price index to satisfy. These tests are nice-to-have properties of an index, but are not as fundamental as the axioms. They act as a way to further narrow the set of price indices that satisfy the five axioms.
Stating these tests requires the concept of a quantity index, a function \(Q\) that is identical to a price index except that the role of prices and quantities is reversed. A quantity index maps quantities and prices to produce a number, although instead of giving the change in price over time, a quantity index gives the change in physical quantities over time. A quantity index should also satisfy the five axioms (switching prices with quantities).
With a quantity index in hand, the three key tests are as follows.
Circularity test \(I(p_{t}, p_{0}, q_{t}, q_{0}) = I(p_{k}, p_{0}, q_{k}, q_{0}) I(p_{t}, p_{k}, q_{t}, q_{k})\).
Product test \(I(p_{t}, p_{0}, q_{t}, q_{0}) Q(q_{t}, q_{0}, p_{t}, p_{0}) = \frac{\sum_{i = 1}^{n} p_{it}q_{it}}{\sum_{i = 1}^{n} p_{i0}q_{i0}}\).
Consistency in aggregation For any partition of \(n\) goods into \(k = 1, \ldots, m\) distinct groups, there is a function \(\psi\) such that \(\psi(I(p_{t}, p_{0}, q_{t}, q_{0}), V_{0}, V_{t}) = \sum_{k = 1}^{m} \psi(I(p_{kt}, p_{k0}, q_{kt}, q_{k0}), V_{k0}, V_{kt})\), where \(V_{kt} = \sum_{i = 1}^{n_{k}} p_{ikt}q_{ikt}\) is the value of the \(n_{k}\) goods for group \(k\) in period \(t\), with \(V_{t} = \sum_{k = 1}^{m} V_{kt}\) being the total value in period \(t\), and if \(I(p_{kt}, p_{k0}, q_{kt}, q_{k0}) = c\) for all \(m\) groups, then \(I(p_{t}, p_{0}, q_{t}, q_{0}) = c\).
The circularity test is straightforward, and expresses the idea that an index between two periods can be calculated by chaining the index for consecutive periods together. The circularity test legitimizes the use of chaining when calculating an index, and using period-over-period changes in price to calculate an index with a fixed base period. It also legitimizes rebasing an index.
The product test is similarly straightforward, and says that the change in aggregate value between two periods can be decomposed into a price index and a quantity index. The product tests legitimizes the use of price indices for deflating aggregate values in a national accounting framework.
The consistency in aggregation test looks complicated, but expresses a simple idea: a price index should be an aggregate of sub-indices, where aggregation depends on only the index itself and the total value of the goods for each sub-index. Consistency in aggregation legitimizes the hierarchical calculation of a price index, where price indices are calculated for increasingly broad categories of goods using the same index-number formula throughout, with value shares as weights.
As an example, the Laspeyres index satisfies the product test and the consistency in aggregation test.13 For the product test, let the quantity index be \[\begin{align*} Q(q_{t}, q_{0}, p_{t}, p_{0}) = \frac{\sum_{i = 1}^{n} p_{it}q_{it}}{\sum_{i = 1}^{n} p_{it}q_{i0}}, \end{align*}\] (this is the Paasche quantity index) so that \[\begin{align*} I(p_{t}, p_{0}, q_{t}, q_{0}) Q(q_{t}, q_{0}, p_{t}, p_{0}) &= \frac{\sum_{i = 1}^{n} p_{it}q_{i0}}{\sum_{i = 1}^{n} p_{i0}q_{i0}} \frac{\sum_{i = 1}^{n} p_{it}q_{it}}{\sum_{i = 1}^{n} p_{it}q_{i0}} \\ &= \frac{\sum_{i = 1}^{n} p_{it}q_{it}}{\sum_{i = 1}^{n} p_{it}q_{i0}}. \end{align*}\]
For the consistency in aggregation test, let \(\psi(I(p_{t}, p_{0}, q_{t}, q_{0}), V_{0}, V_{t}) = V_0 I(p_{t}, p_{0}, q_{t}, q_{0})\), so that the test becomes \[\begin{align*} I(p_{t}, p_{0}, q_{t}, q_{0}) = \sum_{k = 1}^{m} \frac{V_{k0}}{V_0} I(p_{kt}, p_{k0}, q_{kt}, q_{k0}). \end{align*}\] The Laspeyres index obviously satisfies this condition as \(V_{k0} / V_0\) is the period-0 expenditure/revenue share for the products in group \(k\).
The Laspeyres index does not satisfy the circularity test, as a Laspeyres index would need to be multiplied with a Lowe index to return a Laspeyres index.↩︎