Constant-quality price indices

Motivating a constant-quality price index proceeds exactly as it did in the stochastic framework, except now potential prices are used in place of transaction prices. The idea is that there is a distribution of potential price relatives, one for each good, and a price index acts as a best predictor of the change in prices over time. By using potential prices, however, the exact same goods are being compared over time, even if a product as it appears in the market changes between period 0 and period 1. Formally, a constant-quality (geometric) price index is given by \[\begin{align*} I^{Q} &= \exp\left(E\left[\log\left(\frac{p(1)}{p(0)}\right)\right]\right) \\ &= \prod_{i = 1}^{n} \left(\frac{p_{i}(1)}{p_{i}(0)}\right)^{P_{i}}, \end{align*}\] where \(P_{i}\) is the probability of observing good \(i\) in the population of goods. A constant-quality index is a generalization of the standard geometric index that explicitly allows goods, as well as prices, to change over time.

In order to simplify the notation, let \(\rho(t) = \log(p(t))\). With this new notation, a constant-quality price index is given by \[\begin{align*} \log(I^{Q}) = E(\rho(1)) - E(\rho(0)). \end{align*}\] This notation is convenient because it allows a constant-quality price index to be written as a difference in average potential prices, although the index is still a geometric average of potential price relatives.31

  1. This is a Törnqvist-like index, as it gives the change in price for all goods and services transacted between period 0 and period 1. A Laspeyres-like index uses only the distribution of goods that transact in period 0, so that \(\log(I^{Q}) = E(\rho(1) | t = 0) - E(\rho(0) | t = 0)\); a Paasche-like index uses the distribution of goods that transact in period 1, so that \(\log(I^{Q}) = E(\rho(1) | t = 1) - E(\rho(0) | t = 1)\). In the language of an experiment, a Törnqvist-like constant-quality index is the average treatment effect, whereas a Laspeyres-like index is the average treatment effect on the untreated, and a Paasche-like index is the average treatment effect on the treated.↩︎