Arithmetic price indices
An arithmetic price index takes price relatives for a collection of goods and services over two periods and combines them together as a weighted average. That is, an arithmetic index is simply the average change in price between two points in time. Letting goods be enumerated by \(i = 1,\ldots, n\), an arithmetic index between period 0 and period 1 has the general form2 \[\begin{align*} I^{A} = \sum_{i = 1}^{n} \omega_{i} \frac{p_{i1}}{p_{i0}}, \end{align*}\] where \(p_{it}\) is the price of good \(i\) in period \(t = 0,1\), and \(\omega_{i} \geq 0\) is the weight that good \(i\) receives in the index calculation, such that \(\sum_{i = 1}^{n} \omega_{i} = 1\).
Different arithmetic price indices correspond to special cases of the general arithmetic index, depending on the choice of the weights. Most of these choices make use of information about the quantity of a good sold, so that the weights give a measure of the economic importance of a good. Denote by \(q_{it}\) the amount of good \(i\) consumed/produced in period \(t = 0,1\).
Before examining specific arithmetic indices, it is worth noting that an arithmetic index can always be written as the ratio of expenditure/revenue for a “basket” of goods and services at two points in time, so that for any set of weights there are implied “quantities” such that \[\begin{align*} I^{A} = \frac{\sum_{i = 1}^{n} p_{i1} \tilde{q}_{i}}{\sum_{i = 1}^{n} p_{i0} \tilde{q}_{i}}, \end{align*}\] where \(\tilde{q}_{i} = \alpha \omega_{i} / p_{i0}\) for some factor of proportionality \(\alpha\).3 Thus, an arithmetic index can always be interpreted as the ratio of the expenditure required to purchase a fixed basket of goods at two points in time (or the revenue from a fixed basket of goods at two points in time). The choice of basket is linked one-to-one with the choice of weights used to aggregate price relatives. Both representations of the arithmetic index get used, as some indices are easier to represent in one form than the other.
The capital letter sigma \(\Sigma\) is the summation operator. For a collection of numbers \(x_{1}, x_{2},\ldots,x_{n}\), \(\sum_{i=1}^{n} x_{i}\) means \(x_{1} + x_{2} + \ldots + x_{n}\).↩︎
The value of \(\alpha\) has no impact on the index, but is needed to properly interpret \(\tilde{q}_{i}\) as a quantity. For example, if \(\omega_{i}\) is the period-0 expenditure share on good \(i\), \(p_{i0} q_{i0} / \sum_{j = 1}^{n} p_{j0} q_{j0}\), then \(\alpha = \sum_{j = 1}^{n} p_{j0} q_{j0}\) so that \(\tilde{q}_{i} = q_{i0}\).↩︎