Abstract
In this paper, we characterize local circularity for the Laspeyres, Paasche, and Fisher price indexes. In the first two cases, we begin by deriving a sufficient condition for achieving circularity that establishes that at least one of two proposed equalities must hold. We end up showing that the sufficient condition is also necessary. We continue with the Fisher price index that is the geometric mean of the two, and we find a sufficient circularity condition that is a direct consequence of the corresponding sufficient conditions for its two component indexes. However, we also show that, unlike its Laspeyres and Paasche components, this sufficient circularity condition for the Fisher price index is not necessary. We reach different conclusions when we extend our investigation to the circularity properties of the geometric Laspeyres, geometric Paasche, and Törnqvist price indexes, for which none of the proposed sufficient conditions is necessary. Throughout, we distinguish local circularity, which all six price indexes satisfy, from global circularity, which none of the price indexes satisfies.
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Notes
- 1.
Fisher (1922; 241) has an interesting history of the use of the qualifier “ideal” to describe his index.
- 2.
A reviewer has pointed out, correctly, that this is a definition of transitivity, not circularity. However, if a transitive index number satisfies the identity test, then transitivity and circularity are identical properties. Since all six of our classic price indexes satisfy the identity test, we follow Balk and Althin (1996) by referring to the more popular circularity property.
- 3.
Fisher, whose “ideal” index satisfies a large number of desirable properties but fails the circularity property, claimed that the property “…is theoretically a mistaken one…and…a perfect fulfillment of this so-called circular test should really be taken as proof that the formula that fulfils it is erroneous” (1922; 271, emphasis in the original).
- 4.
- 5.
In the index number literature, this result linking a price index with an aggregator function states that a Laspeyres price index is “exact” for a Leontief aggregator function. As Diewert (1981; 182) notes, this relationship has been known for a very long time, although it has not previously been associated with satisfaction of local circularity.
- 6.
Diewert (1981; 184) notes that this result, apart from circularity, is also very old.
- 7.
- 8.
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Acknowledgment
The authors are grateful to an extremely perceptive reviewer, whose comments on our original submission contributed to substantial improvements in the revision.
The authors are also grateful to the financial support from the Spanish Ministry for Economy and Competitiveness (Ministerio de Economía, Industria y Competitividad), the State Research Agency (Agencia Estatal de Investigacion), and the European Regional Development Fund (Fondo Europeo de Desarrollo Regional) under grant MTM2016-79765-P (AEI/FEDER, UE).
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Pastor, J.T., Lovell, C.A.K. (2020). Local Circularity of Six Classic Price Indexes. In: Aparicio, J., Lovell, C., Pastor, J., Zhu, J. (eds) Advances in Efficiency and Productivity II. International Series in Operations Research & Management Science, vol 287. Springer, Cham. https://doi.org/10.1007/978-3-030-41618-8_7
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