Index number problem

The first decomposition was applied to the index number problem of price indices, which are used to determine inflation. Price indices try to answer the question of how much of the change of a given basket of goods can be explained with price changes and how much with changing commodity weights. Since the interest is on the relative change of prices, multiplicative decomposition is used throughout inflation calculation. Thus attribute r in Equation (4.5) are goods and the vectors are the price vector P and the quantity vector Q, which form the value index V

Comparing the wealth of Louis XII in the 16th century and Louis XV of France in the 18th century, Dutot (1738, p. 120) was the first to propose a price index:

(4.9)

This price index is a ratio of the sum of all prices at the point in time T and the point in time 0. The unsuitability of this price index to calculate inflation becomes clear, if one imagines that the price index will change merely with a change in the unity of measurement of one good.

The earliest price index that was the first to be used in index decomposition analysis of the energy sector is the Laspeyres index (Laspeyres 1871) given by:

(4.10)

At the time when Laspeyres proposed his price index, consumption and price statistics were so badly developed that his approach was only of little use. Nevertheless, price indices of this type are still used around the world as an indicator for inflation. Both, the Harmonised Index of Consumer Prices (HICP) of the European Union, as well as the United States Consumer Price Index (CPI) use the Laspeyres index. The characteristic of the Laspeyres index is that the goods‟ quantities are fixed in the base situation. The opposite, where the quantities are fixed in the observed situation, is called after Herrmann Paasche (1874), who applied this index to prices on the Hamburg exchange:

(4.11)

A combination of the Laspeyres and Paasche approach, i.e. taking the arithmetic average of the quantities in the base situation and the observed situation, represents the Edgeworth-Marshall index (4.12) This index was independently suggested by Marshall (1887) and Edgeworth (1925). Another milestone in the development of index numbers was the work by Fisher (1922), who developed the „ideal‟ index, a geometric mean of the Paasche and Laspeyres index. As this index is only used in multiplicative decomposition, it is not discussed here in

more detail. The last type of index to be proposed in the context of index numbers and heavily used in decomposition analysis is the Divisia index (Divisia 1925; Divisia 1928). Divisia based his reasoning on the equation of exchange. In this way, he defines the change of the value index V in Equation (4.8) as a total differential

(4.13) where the first summand is the price index and the second one the quantity index. Extracting the price index and dividing by P(t) yields

(4.14)

Transforming the right hand side results in (4.15)

Incorporating (with f being an arbitrary function) gives (4.16) Integrating (4.17)

As a ratio the price index looks the following:

(4.18) Divisia was aware of the fact that his index was nothing more than a curvilinear integral (Divisia 1925, p. 1004) so that the calculation of the price index depends not only on the values in the base and observed situation, but on all values in between. As the Divisia index is an integral, it needs to be approximated in order to be used as a price index. Vogt (1978) concluded that if it is assumed that the relationship, which is being decomposed, is continuous, each index represents a time path between two discrete

points. This is illustrated in Figure 4.2, which presents different paths for price indices with the commodity vector on the ordinate and the price vector on the abscissa.

Figure 4.2: Representation of the Index Problem in the 2n-dimensional Quantity-Price Space

Source: adapted from Vogt (1978)1

The path in a) represents the Laspeyres index, b) corresponds to the Paasche index, c) corresponds to the Edgeworth-Marshall index and e) to the exponential Divisia index. The path in d) was called the “natural” index or the Divisia index on a straight line in the context of index number and became better known under the term Refined Laspeyres in the index decomposition context (see 4.3.3).

Montgomery (1937, p.51f) was the first to propose an approximation of the Divisia index using the logarithmic mean (without using the term logarithmic mean), defined as

, resulting in the following Divisia index

(4.19)

This approach has several desirable properties that will be discussed in section 4.4. Other averages include the arithmetic mean (Hulten 1973; Törnqvist et al. 1985), geometric means (Theil 1973; Sato 1974) and rediscoveries of the logarithmic mean (Sato 1976; Vartia 1976; Törnqvist et al. 1985). In the context of decomposition analysis one speaks of the logarithmic mean I and II according to Vartia‟s (1976) definitions. All of the presented indices are possible approaches to the index number problem.

Series expansion is another way of explaining the nature of the residual term in decomposition analysis. Referring back to Equation (4.13), the total differential can be approximated in the following way, representing a first-order Taylor expansion

(4.20) The assumption is that the changes are very small so that the differential can be approximated. Nevertheless, infinitesimal calculus is only an approximation of differential calculus and therefore leaves a residual if only first order changes are considered as it is the case in decomposition analysis. The approximation on the right hand side is a series expansion that is truncated after the first order terms. Higher order terms or interaction terms form the residual in the case where it is not redistributed to the variables. Proops et al. (1992, Appendix A4) give an explanation on the use of differences instead of differentials.