More general price indices

The arithmetic and geometric indices are special cases of a broader class of price indices that solve the following prediction problem: \[\begin{align*} \min_{I} E\left[\left(\left(\frac{p_{t}}{p_{0}}\right)^{r} - I^{r} \right)^{2}\right], \end{align*}\] for some \(r \neq 0\). The solution to this problem is \(I = E((p_{t} / p_{0})^{r})^{1 / r}\). Setting \(r = 1\) gives an arithmetic index, whereas taking \(r \rightarrow 0\) gives a geometric index (Bullen 2003, III 1 Theorem 2).

What’s useful about this approach is that many other types of price indices correspond to different choices of \(r\), and these in turn can be motivated as solving a type of prediction problem. Setting \(r = -1\), for example, yields a harmonic price index \[\begin{align*} I^{H} = \left(\sum_{i = 1}^{n} \frac{P_{i}}{p_{it} / p_{i0}} \right)^{-1}. \end{align*}\] When the probability of observing a good is given by its period-\(t\) expenditure share, then this is the Paasche index. Setting \(r = 1 - \sigma\), where \(\sigma\) is the elasticity of substitution, results in the Lloyd-Moulton price index \[\begin{align*} I^{LM} = \left(\sum_{i = 1}^{n} P_{i} \left(\frac{p_{it}}{p_{i0}}\right)^{1 - \sigma}\right)^{1 / (1 - \sigma)} \end{align*}\] when the probabilities are period-0 expenditure/revenue shares.

An interesting point about these more general types of price indices is that, for a given set of weights, the index value is larger when the parameter \(r\) increases (Bullen 2003, III 3.1 Theorem 1). This gives the familiar result that an arithmetic index is larger than a geometric index, which in turn is larger than a harmonic index, but it can also be used to rank more exotic types of indices like the Lloyd-Moulton index.