¹ The literature on price indexes is extensive. The intention of this chapter is to present a broad overview of the theory drawing heavily on documents that are in many cases overviews themselves as well as presenting an ABS perspective. For a detailed consolidation of producer price index theory and internationally recommended practices, see the Producer Price Index Manual, Theory and Practice (International Labour Organization (ILO), International Monetary Fund (IMF), Organisation for Economic Cooperation and Development (OECD), Statistical Office of the European Communities (Eurostat), United Nations Economic Commission for Europe (UNECE), and the World Bank, 2004). Available online: http://www.imf.org/external/np/sta/tegppi/. This chapter draws heavily on material from that manual.

² By convention, the initial value for an index series is made equal to 100.0

³ To quote Fisher (1922, p. 45) "… any index number implies two dates, and the values by which we are to weight the price ratios for those two dates will be different at the two dates. Constant weighting (the same weight for the same product in different years) is, therefore, a mere makeshift, never theoretically correct, and not even practically admissible when values change widely."

⁴ The use of the geometric mean of the Laspeyres and Paasche indexes was first proposed by Pigou in 1920 and given the title 'ideal' by Fisher (1922).

⁵ Footnote 5 See Diewert (1993) for a discussion of symmetrical averages.

⁶ Footnote 6 The relationship between the Laspeyres and Paasche indexes holds while ever there is a 'normal' relationship (negative correlation) between prices and quantities, that is, quantity falls (rises if price rises (falls) between the two periods.

⁷ Footnote 7 Use of the RAP approach was first suggested by Dutot in 1738, the APR approach by Carli in 1764 and the geometric mean by Jevons in 1865 (see Diewert (1987)). Fisher (1922) described the RAP approach as the 'simple aggregative'. These are not the only possible formulae – another formula often mentioned in the literature is the harmonic mean. The harmonic mean of price relatives is given by the inverse of the arithmetic averages of the inverses of the relatives of the individual product prices, that is: \(\frac{1}{\frac1n\sum^n_\limits {i=1}\frac{p^0_i}{p^t_i}}\). The harmonic mean is equal to or lower than the geometric mean. Fisher (1922) also discusses use of the median and mode.

⁸ The implicit weights applied by the three formulae are equal reference period quantities (RAP), equal reference period values (quantities inversely proportional to reference period prices) (APR) and equal value shares in both periods (GM).

⁹ The geometric mean of \(n \) numbers is the nth root of the product of the numbers. For example, the geometric mean of 4 and 9 is 6\((6=\sqrt{4\times9})\), while the arithmetic mean is 6.5 \((6.5=(4+9/2)\). Although the geometric mean has been presented in terms of price relatives, the same result is obtained by taking the ratio of the geometric means of prices in each period, that is: \(\frac{\big(\Pi P_{it}\big)^{\frac1N}}{\big(\Pi p_{io}\big)^{\frac1N}}\)

¹⁰ For a mathematical proof of this see Diewert (1995). The unweighted indexes will all produce the same result if all prices move in the same proportion (have the same relative). In addition, the RAP and APR will produce the same index number if all reference period prices are equal. In general, the RAP formula is expected to produce index numbers above but reasonably close to the GM. Diewert also refers to other studies that compare real world results for elementary aggregate formulae.

¹¹ For example, Woolford (1994) calculated these indexes for 23 fresh fruit and vegetable elementary aggregates of the Australian CPI over the period June 1993 to June 1994. He found that the GM produced the lowest increase in 16 of the 23 elementary aggregates and the APR produced the highest increase for 19 of the elementary aggregates. The RAP formula produced the middle estimate for 13 of the elementary aggregates. Combining the elementary aggregates to produce the fresh fruit and vegetables index, the index compiled using the APR estimates was 4.7% higher than the index based on GM estimates and the RAP was 1.7% higher than the index based on GM.

¹² See Diewert(1995) for further discussion of unit values.

¹³ Szulc (1983) applied the term “price bouncing” to situation 3.