Approximating an economic index

An economic price index cannot be directly calculated in practice. There is no hope of calculating either an input-price or output-price index without knowledge of the firm’s technology of production, because the form of the firms’ expenditure or revenue function is unknown. The problem is even more severe for a cost-of-living index, as it requires knowledge of consumer preferences to construct the consumer’s expenditure function. The best that can be done is to approximate an economic index using data on observable prices and quantities. Both the arithmetic Laspeyres and geometric Laspeyres indices offer a good approximation to either the input price, output-price index, or cost-of-living index, at least to a first-order approximation.

The argument for approximating an economic index with an arithmetic Laspeyres or geometric Laspeyres is the same for an input-price index, an output-price index, and a cost-of-living index, and so attention is restricted to approximating an input-price index. To show that the Laspeyres index approximates the input-price index, recall that \[\begin{align*} e(p_{0}, y_{0}) = \sum_{i = 1}^{n} p_{i0} q_{i0}. \end{align*}\] Now, it can be shown that the best linear approximation to the expenditure function with period \(t\) prices is \[\begin{align*} e(p_{t}, y_{0}) \approx \sum_{i = 1}^{n} p_{it}q_{i0} \end{align*}\] whenever period-\(t\) prices are not too different from period-0 prices.19 Therefore \[\begin{align*} \frac{e(p_{t}, y_{0})}{e(p_{0}, y_{0})} \approx \frac{\sum_{i = 1}^{n} p_{it} q_{i0}}{\sum_{i = 1}^{n} p_{i0} q_{i0}}, \end{align*}\] at least for small price changes.

Similarly, for the geometric Laspeyres, the best linear approximation is based on \[\begin{align*} \log(e(p_{t}, y_{0})) - \log(e(p_{0}, y_{0})) \approx \sum_{i = 1}^{n} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} \log\left(\frac{p_{it}}{p_{i0}} \right). \end{align*}\] Therefore \[\begin{align*} I{^C} &= \exp(\log(e(p_{t}, y_{0})) - \log(e(p_{0}, y_{0}))) \\ &\approx \exp\left( \sum_{i = 1}^{n} \frac{p_{i0} q_{i0}}{\sum_{j = 1}^{n} p_{j0} q_{j0}} \log\left(\frac{p_{it}}{p_{i0}} \right) \right) \\ &= \prod_{i = 1}^{n} \left(\frac{p_{it}}{p_{i0}} \right)^{\omega_{i0}} \end{align*}\] where \(\omega_{i0} = p_{i0} q_{i0} / \sum_{j = 1}^{n} p_{j0} q_{j0}\) is the period-0 expenditure weight for good \(i\). As a geometric index is always smaller than its corresponding arithmetic index, and the Laspeyres index is too large relative to the input-price index (i.e., the approximation error is always positive due to substitution bias), the geometric Laspeyres index will give a better approximation to the input-price index than the Laspeyres index whenever the approximation error for the geometric index is positive.

In many cases the Fisher or Törnqvist index offers an even better approximation to an input-price index than the Laspeyres or geometric Laspeyres index. The argument for why these indices can be a better approximation are more complex—see ILO et al. (2004a, chap. 17) and ILO et al. (2004b, chap. 17) for details. The cost for this better approximation, however, is the need for more information about period \(t\) quantities, which may or may not be available at the time when the index is actually calculated.20

  1. This follows from Shephard’s lemma (McFadden 1978), which implies that \(e(p_{t}, y_{0}) = \sum_{i = 1}^{n} p_{it}q_{i0} + \eta(p_{t} - p_{0})||p_{t} - p_{0}||\), where \(\eta(p_{t} - p_{0}) \rightarrow 0\) when \(p_{t} \rightarrow p_{0}\).↩︎

  2. In certain cases these approximations are exact. A Laspeyres index will be a true input-price index if the firm has a Leontief production function; a Fisher index will also be exact in this case. Similarly, a geometric Laspeyres will be a true input-price index if the firm has a Cobb-Douglas production functions; a Törnqvist index will also be exact in this case. The Lloyd-Moulton price index is an interesting case as it is exact whenever the firm has a constant elasticity of substitution production function, but it only requires period-0 quantity information.↩︎