The theory presented so far applies to aggregates consisting of a finite set of commodities. Two basic assumptions are that the set of commodities does not change between the two time periods compared, and that all the price and quantity data which are necessary for the computation of an index are available. In this chapter we are concerned with what to do when the second of these assumptions is not or cannot be fulfilled. There are, of course, various kinds of unavailability of data. The situation we will consider in particular in this chapter is that we have only data for a sample of commodities. These data might consist of prices and quantities, but more often only prices are available.

Since such a situation materializes at the very first stage of the computation of any official price index, such as a consumer price index (CPI) or a producer price index (PPI), we are dealing here with an issue of great practical significance.

The usual approach to the problem of unavailable quantity data is to consider price indices that are functions of prices only. The main formulas discussed in the literature and used in practice are

• the ratio of arithmetic average prices; that is, the formula of Dutot; see expression (1.1);

• the arithmetic average of price relatives; that is, the formula of Carli; see expression (1.2);

• the geometric average of price relatives=the ratio of geometric average prices; that is, the formula of Jevons; see expression (1.5).

In the literature, the suitability of these formulas has been studied by various methods.