The stochastic approach
The axiomatic and economic approaches to motivating a price index take prices as fixed. As the name suggests, the stochastic approach does not; instead, prices are treated as random, at least from the point of view of whoever is compiling the price index. This is not to say that prices are chosen or determined at random, but simply that there is a distribution of prices for goods and services transacted, and observing any given price is akin to a random draw from this distribution. The obvious benefit of this approach is that it agrees with the usual practice of calculating a price index using a sample of data, allowing for an explicit treatment of the statistical properties of a price index.21
At a conceptual level, however, the stochastic approach motivates a price index by one of its main uses: deflating and inflating prices across time. The act of deflating/inflating prices is essentially one of prediction—what would the price of a good or service be in another period?—and a price index can be motivated by examining how to pick a single value that bests predicts a change in price for a population of goods sold between two periods.
This section of the course concludes the journey into price index theory by giving a somewhat more thorough introduction to the stochastic approach than is seen elsewhere. This is because the stochastic approach is probably the most practical of the three approaches. While not as lofty as the axiomatic or economic approaches, it offers the most useful model for actually producing a price index as a macroeconomic statistic.
📖 PPI Manual: Chapter 1, section D; Chapter 16, section D.
Formally, in the stochastic approach, goods and services are modeled as belonging to a probability space \((\Omega, \mathcal{F}, P)\), where \(\Omega\) is the (finite) set of goods and services that are transacted, \(\mathcal{F}\) is the power set of \(\Omega\), and \(P\) is a probability measure that gives the likelihood of observing a good or service in \(\Omega\) transacted. There is then a mapping from \(\Omega\) to \(\mathbb{R}_{+}^{2}\) that gives the price of a good in period 0 and period \(t\).↩︎