Counter examples

For the most part, it is difficult to write down a half-way decent price that does not satisfy the five axioms in the previous section. Pathological examples abound, but it is more interesting to look at reasonable cases where an index-number formula does not satisfy all the axioms.12 Two interesting examples of reasonable price indices that do not satisfy the five axioms are the Dutot index and the median-value index (i.e., the index that takes the median price relative).

The Dutot index satisfies all axioms except dimensional invariance, whereas the mean-value index satisfies all axioms except monotonicity. To see why the Dutot index does not satisfy dimensional invariance, suppose there are only two goods (i.e., \(n = 2\)) with \(p_{1t} = 1\), \(p_{2t} = 2\), and \(p_{10} = p_{20} = 1\), and with \[\begin{align*} A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}. \end{align*}\] In this case \[\begin{align*} I(Ap_{t}, Ap_{0}, A^{-1}q_{t}, A^{-1}q_{0}) = \frac{1 \times 1 + 2 \times 2}{1 \times 1 + 2 \times 1} = 5 / 3 \end{align*}\] and \[\begin{align*} I(p_{t}, p_{0}, q_{t}, q_{0}) = \frac{1 + 2}{1 + 1} = 3 / 2 \neq I(Ap_{t}, Ap_{0}, A^{-1}q_{t}, A^{-1}q_{0}), \end{align*}\] thus failing the dimensional invariance axiom. This simply formalizes the well known issue with the Dutot index that all goods and services must be measured in the same units for it to be useful.

To see that the median price index does not satisfy monotonicity, suppose there are 3 goods such that \(p_{1t} = 1\), \(p_{2t} = 2\), \(p_{3t} = 3\), and \(p_{10} = p_{20} = p_{30} = 1\). The median index returns the value 2 (the median price relative). Now if \(p'_{1t} = 1\), \(p'_{2t} = 2\), and \(p'_{3t} = 4\), the median index still returns 2; prices have increased, but the index value remains the same.

  1. Two interesting pathological cases have to do with failure of monotonicity for the geometric Laspeyres and Paasche indices. Although the geometric Laspeyres is always increasing in period-\(t\) prices, it can also increase in period-0 prices. Necessary for this is that \(p_{i1} / p_{i0} \geq \exp(1) I(p_{1}, p_{0}, q_{t}, q_{0}) \approx 2.72 I(p_{1}, p_{0}, q_{t}, q_{0})\) for at least one good \(i\), so that at least some goods experience a change in price that is considerably larger than the change in price for the other goods. Similarly, while the geometric Paasche is always decreasing in period-0 prices, it can also decrease in period-\(t\) prices. Necessary for this to be the case is that \(p_{i1} / p_{i0} \leq I(p_{1}, p_{0}, q_{t}, q_{0}) / \exp(1) \approx I(p_{1}, p_{0}, q_{t}, q_{0}) / 2.72\) for at least one good \(i\). See Balk (2008, sec. 3.12).↩︎