Prices Analytical Series
A Note on General Decompositions for Price Indexes

Abstract

The additive and multiplicative decompositions for geometric, arithmetic, and Fisher price indexes that make use of the logarithmic mean are generalized by replacing the logarithmic mean with the more general extended mean. The result is a simple decomposition that takes existing results as special cases, and applies to other index numbers like the harmonic, Lloyd-Moulton, and superlative quadratic mean indexes, and their sample counterparts.

1 Introduction

It is often useful to be able to decompose a price index into an additive or multiplicative form to evaluate how each input to the index affects its value. Following Balk (2008), a price index is said to admit an additive decomposition if there exist weights that allow it to be represented as an arithmetic mean of price relatives, and is said to admit a multiplicative decomposition if there are weights that allow it to be represented as a geometric mean. Switching prices for quantities gives the analogous statements for a quantity index, and nothing is lost by focusing on price indexes.

There are a number of well-known decompositions for the most common types of bilateral price indexes. Balk (2008, equation 4.13) gives an additive decomposition for any index based on the geometric mean by transmuting the weights in the geometric mean with the logarithmic mean. This is the same decomposition derived by Reinsdorf et al. (2002, equation 20) for the Törnqvist index. A similar approach yields a multiplicative decomposition for any index based on the arithmetic mean, again using the logarithmic mean (Balk, 2008, equation 4.8). Combining these results gives additive and multiplicative decompositions for the Fisher index (Reinsdorf et al., 2002, section 6).^Note  Each of these decompositions results in weights that are positive and sum to one, as required to represent an index as an arithmetic or geometric mean.^Note 

The purpose of this note is to show that the additive and multiplicative decompositions for geometric, arithmetic, and Fisher indexes that use the logarithmic mean can be consolidated and made more general by switching out the logarithmic mean for the more general extended mean. The main result is a function that transmutes the weights in a generalized mean of a given order so that it can be represented as a generalized mean of any other order. This covers additive and multiplicative decompositions for indexes that do not belong to the arithmetic or geometric families, like harmonic indexes or the Lloyd-Moulton index, and allows both additive and multiplicative decompositions to be covered by a single equation, rather than treating them as different cases. Expressing a generalized index as a generalized mean of any other order also allows for the decomposition of indexes that are nested generalized means, like the family of superlative quadratic mean indexes that includes the Fisher index, or the AG mean index by Lent and Dorfman (2009).

The result in this note is partly of theoretical interest, as it shows how existing decompositions can be extended to a broader range of index-number formulas, and gives decompositions for indexes like the Lloyd-Moulton and AG mean index that are not covered by existing results. It is also applicable to situations where only prices and weights are known, rather than prices and quantities, such as with sample data. This makes it easier to use a larger collection of index-number formulas in settings where there is an expectation of having data for the contribution of each product towards the value of an index. One practical use of this more general decomposition is developing software to calculate price indexes, as additive and multiplicative decompositions for a large collection of indexes can be implemented by simply implementing the extended mean rather than many special cases.

2 Decomposing generalized-mean indexes

A natural extension to the decompositions for indexes based on the arithmetic and geometric means is to derive weights that transform an index based on a generalized mean of order $\rho \in ℝ$ into one based on a generalized mean of order $\varsigma \in ℝ$ . To fix notation, let $r=\left(r_1,r_2,\dots ,r_n\right)\in {ℝ}_{++}^n$ be a vector of price relatives for $n$ products and let $w=\left(w_1,w_2,\dots ,w_n\right)\in {\text{Δ}}^{n-1}$ be the corresponding weights, where ${\text{Δ}}^{n-1}=\left\{w\in {ℝ}_{+}^n|{\displaystyle \sum }_{i=1}^n{w}_i=1\right\}$ is the unit simplex. The goal is to find a vector-valued function

$$v\left(r,w;\rho ,\varsigma \right)=\left(v_1\left(r,w;\rho , \varsigma \right),v_2\left(r,w;\rho , \varsigma \right),\dots ,v_n\left(r,w;\rho , \varsigma \right)\right)$$

mapping into ${\text{Δ}}^{n-1}$ such that

$${\mathfrak{M}}_{\rho }\left(r,w\right)\equiv {\mathfrak{M}}_{\varsigma }\left(r, v\left(r,w;\rho ,\varsigma \right)\right),$$

where ${\mathfrak{M}}_{\rho }$ is the generalized mean of order $\rho$

$${\mathfrak{M}}_{\rho }(r,w)=\{\begin{array}{ll}{\left({\displaystyle \sum _{i=1}^n{w}_i}{r}_i^{\rho }\right)}^{1/\rho }\hfill & \text{if }\rho \ne 0\hfill \\ \exp \left({\displaystyle \sum _{i=1}^n{w}_i}\log (r_i)\right)\hfill & \text{if }\rho =0.\hfill \end{array}$$

Setting $\varsigma =1$ then yields an additive decomposition for any index based on a generalized mean of order $\rho$ , such that

$${\mathfrak{M}}_{\rho }\left(r,w\right)\equiv {\displaystyle \sum _{i=1}^n{v}_i}(r,w;\rho ,1)r_i,$$

and setting $\varsigma =0$ yields a multiplicative decomposition,

$${\mathfrak{M}}_{\rho }\left(r,w\right)\equiv {\displaystyle \prod }_{i=1}^n{r}_i{}^{v_i(r,w;\rho ,0)}.$$

The results by Balk (2008) and Reinsdorf et al. (2002) show how to derive $v\left(r,w;\rho ,\varsigma \right)$ when $\rho = 1$ and $\varsigma = 0$ (multiplicative decomposition of an arithmetic index) and $\rho = 0$ and $\varsigma = 1$ (additive decomposition of a geometric index), using the logarithmic mean. Generalizing these results follows from replacing the logarithmic mean with the more general extended mean (Bullen, 2003, p. 393), defined for any $a,b \in {ℝ}_{++}$ as

$${\mathfrak{E}}_{\rho \varsigma }(a,b)=\{\begin{array}{ll}{\left(\frac{\varsigma (a^{\rho }-b^{\rho })}{\rho (a^{\varsigma }-b^{\varsigma })}\right)}^{1/(\rho -\varsigma )}\hfill & \text{if }\rho \ne \varsigma ,\rho \ne 0,\varsigma \ne 0,a\ne b\hfill \\ {\left(\frac{a^{\rho }-b^{\rho }}{\rho \log (a/b)}\right)}^{1/\rho }\hfill & \text{if }\rho \ne 0,\varsigma =0,a\ne b\hfill \\ {\left(\frac{a^{\varsigma }-b^{\varsigma }}{\varsigma \log (a/b)}\right)}^{1/\varsigma }\hfill & \text{if }\rho =0,\varsigma \ne 0,a\ne b\hfill \\ \frac{1}{\exp {(1)}^{1/\rho }}{\left(\frac{a^{a^{\rho }}}{b^{b^{\rho }}}\right)}^{1/(a^{\rho }-b^{\rho })}\hfill & \text{if }\rho =\varsigma \ne 0,a\ne b\hfill \\ \sqrt{ab}\hfill & \text{if }\rho =\varsigma =0,a\ne b\hfill \\ a\hfill & \text{if }a=b.\hfill \end{array}$$

The extended mean reduces to the logarithmic mean when either $\rho = 0$ and $\varsigma = 1$ , or $\rho = 1$ and $\varsigma = 0$ . But using the extended mean in place of the logarithmic mean allows for decompositions of indexes based on other types of means, like harmonic indexes ( $\rho = -1$ ) and the Lloyd-Moulton index ( $\rho = 1 - \sigma$ , where $\sigma$ is an elasticity of substitution).

The key to transforming the weights in a generalized mean of order $\rho$ into the weights for a generalized mean of order $\varsigma$ comes from noting that the extended mean is always positive and satisfies the identity^Note 

$$(1){\displaystyle \sum _{i=1}^n{w}_i}{\mathfrak{E}}_{\rho \varsigma }{(r_i,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }{d}_i(r,w;\rho ,\varsigma )\equiv 0,$$

where $d_i$ is the deviation from the generalized mean

$$d_i(r,w;\rho ,\varsigma )=\{\begin{array}{ll}{r}_i^{\varsigma }-{\mathfrak{M}}_{\rho }{(r,w)}^{\varsigma }\hfill & \text{if }\varsigma \ne 0\hfill \\ \log (r_i)-\log ({\mathfrak{M}}_{\rho }(r,w))\hfill & \text{if }\varsigma =0.\hfill \end{array}$$

Rearranging (1) gives that

$${\mathfrak{M}}_{\rho }(r,w)\equiv \{\begin{array}{ll}{\left({\displaystyle \sum _{i=1}^n\frac{w_i{\mathfrak{E}}_{\rho \varsigma }{(r_i,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }}{{\displaystyle \sum _{j=1}^n{w}_j}{\mathfrak{E}}_{\rho \varsigma }{(r_j,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }}}{r}_i^{\varsigma }\right)}^{1/\varsigma }\hfill & \text{if }\varsigma \ne 0\hfill \\ \exp \left({\displaystyle \sum _{i=1}^n\frac{w_i{\mathfrak{E}}_{\rho \varsigma }{(r_i,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }}{{\displaystyle \sum _{j=1}^n{w}_j}{\mathfrak{E}}_{\rho \varsigma }{(r_j,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }}}\log (r_i)\right)\hfill & \text{if }\varsigma =0,\hfill \end{array}$$

so that setting

$$(2)v_i(r,w;\rho ,\varsigma )=w_i{\mathfrak{E}}_{\rho \varsigma }{(r_i,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }/{\displaystyle \sum _{j=1}^n{w}_j}{\mathfrak{E}}_{\rho \varsigma }{(r_j,{\mathfrak{M}}_{\rho }(r,w))}^{\rho -\varsigma }$$

gives a suitable function to find weights that turn an index based on a generalized mean of order $\rho$ into one based on a generalized mean of order $\varsigma$ . (It is obvious that the function $v$ defined by (2) maps into ${\Delta }^{n-1}$ .) Setting $\rho = 0$ and $\varsigma = 1$ , or $\rho = 1$ and $\varsigma = 0$ , gives the special cases in Balk (2008) and Reinsdorf et al. (2002).

It is worth noting that the function given by (2) is necessarily not unique, and there are other functions that decompose indexes based on the generalized mean.^Note  Despite this, any function that decomposes the generalized mean when $n = 2$ must agree with (2) whenever $r_1\ne r_2$ , and (2) is the only such function that always returns $w$ when $r_1 = r_2$ .^Note  It must be that $n \ge 3$ for any other function to successfully transform the weights in a generalized mean when all price relatives are not equal, and this implies that any other function $u$ that decomposes the generalized mean can be written as an extension of (2)

$$u(r,w;\rho ,\varsigma ,n)=\{\begin{array}{ll}v(r,w;\rho ,\varsigma )\hfill & \text{if }n=2\hfill \\ u(r,w;\rho ,\varsigma )\hfill & \text{if }n\ge 3.\hfill \end{array}$$

3 Decomposing superlative indexes

The additive and multiplicative decompositions for the Fisher index by Reinsdorf et al. (2002, section 6) can be generalized in the same way as the decompositions for the arithmetic and geometric indexes by noting that the Fisher index is simply a nested generalized mean of indexes based on the generalized mean. For a pair of generalized means $({\mathfrak{M}}_{{\rho }_1}(r,w_1),{\mathfrak{M}}_{{\rho }_2}(r,w_2))$ mapping into ${ℝ}_{++}^2$ with weights $({\omega }_1,{\omega }_2)\in {\Delta }^1$ , an index based on nested generalized means is written as

$$(3){\mathfrak{M}}_{\rho }(({\mathfrak{M}}_{{\rho }_1}(r,w_1),{\mathfrak{M}}_{{\rho }_2}(r,w_2)),({\omega }_1,{\omega }_2)).$$

The general family of superlative quadratic mean of order $\tau$ indexes come from setting $\rho = 0$ , ${\rho }_1 = \tau /2$ , and ${\rho }_2 = -\tau /2$ when ${\omega }_1={\omega }_2 = 1/2$ , $w_1$ are base-period expenditure/revenue shares, and $w_2$ are current-period expenditure/revenue shares. In particular, setting $\rho = 0$ , ${\rho }_1 = 1$ , and ${\rho }_2 = -1$ gives the Fisher index. But (3) covers other types of indexes as well; for example, setting each element of $w_1$ and $w_2$ to $1/n$ gives the Carruthers-Sellwood-Ward-Dalén index that serves as an estimator for the Fisher index, whereas setting $\rho = -1$ gives the harmonic analogue of the Fisher index. Setting $\rho = {\rho }_1 = {\rho }_2$ and $w_1 = w_2$ gives an index based on a generalized mean of order $\rho$ , so that the decomposition of an index based on the generalized mean is a special case of the decomposition for (3).  

An index of form (3) can be transformed into an index based on the generalized mean of order $\rho$ using the weights in (2), as it can be written as

$${\mathfrak{M}}_{\rho }(r,{\omega }_{1}v(r,w_1;{\rho }_1,\rho )+{\omega }_{2}v(r,w_2;{\rho }_2,\rho )).$$

The transformation in (2) then applies as before, just replacing $w$ with the more complicated weights ${\omega }_{1}v(r,w_1;{\rho }_1,\rho )+{\omega }_{2}v(r,w_2;{\rho }_2,\rho )$ .

4 Numerical example

It is easy to see how these decompositions work with a small numerical example using values from tables 19.1 and 19.2 of the Producer Price Index Manual (ILO et al., 2004). These values are reproduced in table 1, with table 2 giving the additive decompositions ( $\varsigma = 1$ ) for the Fisher ( $\rho = 0, {\rho }_1 = 1, {\rho }_2 = -1$ ), Törnqvist ( $\rho = 0$ ), Lloyd-Moulton ( $\rho = 1.75$ ), and Carruthers-Sellwood-Ward- Dalén ( $\rho = 0, {\rho }_1 = 1, {\rho }_2 = -1$ ) indexes, along with the base-period and current-period revenue shares.

 

The decompositions for the Fisher and Törnqvist indexes are simply those from Reinsdorf et al. (2002), and tell the same story. The weights to turn these indexes into arithmetic means are fairly similar to the base-period revenue shares, with these shares getting brought down for products with larger price relatives, and brought up for products with a smaller price relatives. The decomposition for the Lloyd-Moulton index is largely the same, as the value of the index is fairly similar to the Fisher and Törnqvist indexes, and follows closely the base-period revenue shares. With this decomposition, however, base-period revenue shares are brought down for products with smaller price relatives and brought up for products with larger relatives. The decomposition for the Carruthers-Sellwood-Ward-Dalén index is quite different, as this index uses only price data; consequently, the weights are much more uniform than for the other indexes. But the mechanics of the decomposition are similar to the Fisher and Törnqvist indexes, as products with smaller price relatives get larger weight.

5 Conclusion

Decompositions for price indexes are useful to understand how each price relative affects the value of an index, and there are several well-known decompositions that cover the most important cases. This note develops a simple generalization that consolidates existing decompositions for bilateral indexes, and applies them to a larger range of price indexes. Although this new decomposition covers a large collection of prices indexes, it is not exhaustive, and there are non-linear bilateral indexes (like those based on the median, rather than the mean) that require a different approach. One interesting avenue for future work comes from noting that the decomposition in (2) can be used to express a generalized mean as other types of means, such as a Lehmer mean (Bullen, 2003, p. 245), suggesting that these types of means could be relevant for constructing price indexes.

Acknowledgements

This work has benefited from helpful comments by Justin Francis, Zachary Glazier, Xin Ha, Brett Harper, Dragos Ifrim, Klaus Kostenbauer, and Clément Yélou.

References

Balk, B. M. (2008). Price and Quantity Index Numbers. Cambridge University Press.

Bullen, P. S. (2003). Handbook of Means and their Inequalities. Springer Science+Business Media.

Diewert, W. E. (2002). The quadratic approximation lemma and decompositions of superlative indexes. Journal of Economic and Social Measurement, 28(1-2):63–88.

Hallerbach, W. G. (2005). An alternative decomposition of the Fisher index. Economics Letters, 86(2):147–152.

ILO, IMF, OECD, Eurostat, UN, and World Bank (2004). Producer Price Index Manual: Theory and Practice. International Monetary Fund.

Lent, J. and Dorfman, A. H. (2009). Using a weighted average of base period price indexes to approximate a superlative index. Journal of Official Statistics, 25(1):139–149.

Reinsdorf, M. B., Diewert, W. E., and Ehemann, C. (2002). Additive decompositions for Fisher, Törnqvist and geometric mean indexes. Journal of Economic and Social Measurement, 28(1-2):51–61.