A Note on the Additive Decomposition of GEKS Indexes

Abstract

It is often useful to decompose an index number into the contribution of each product toward the total index, and consequently there are several well-known decompositions for bilateral indexes. In this note, I extend these decompositions to cases where bilateral indexes are made into multilateral GEKS indexes. Although the result is primarily of theoretical interest, it shows how decompositions based on a bilateral index can be extended to a multilateral index, and highlights the challenge of decomposing GEKS indexes.

Keywords

price indexes index decomposition GEKS index

1. Introduction

It is often useful to decompose an index number into the contribution of each product toward the total index. I focus on temporal price indexes in the interest of simplicity, but everything applies equally to both temporal quantity indexes and spatial price/quantity indexes. Following Balk (2008, Chapter 4), a price index is said to admit an additive (multiplicative) decomposition if there are weights such that it can be represented as an arithmetic (geometric) mean of price relatives, and there are several well-known decompositions for bilateral price indexes.

The purpose of this note is to show how additive decompositions for bilateral price indexes, principally the Fisher and Törnqvist indexes, can be extended when such indexes are used to make multilateral GEKS indexes. Webster and Tarnow-Mordi (2019) derive a multiplicative decomposition for the GEKS-Törnqvist index, with Białek (2026) extending this to the GEKS-Fisher and other variants, but these decompositions do not represent the GEKS index as a geometric mean of price relatives. Consequently, these decompositions are more difficult to interpret and compare across index-number formulas than the multiplicative decompositions for the corresponding bilateral indexes, and the usual avenue to derive additive and percent-change decompositions by Balk (2008, Chapter 4) is closed (although see the approached based on relative impacts by Białek (2026) for an alternative method). This is in contrast to the Geary-Khamis and repeat sales (Shiller 1991) multilateral indexes that can be directly represented as a mean of (imputed) price relatives and therefore readily decomposed. Instead, I augment the weights for the additive decomposition of a bilateral index to represent the GEKS index as the arithmetic mean of all combinations of price relatives, mirroring the way in which the GEKS index extends a bilateral index by considering all combinations of bilateral comparisons.

2. Decomposing the GEKS Index

The GEKS index between periods $a$ and $b$ , with $0 \leq a \leq b \leq T$ , is a (potentially weighted) geometric mean of all bilateral comparisons for some index-number function $P$ between period 0 and $T$

\frac{\prod_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}}}{\prod_{t = 0}^{T} P {(p_{a}, p_{t}, q_{a}, q_{t})}^{ω_{t}}},

where $p_{t}$ and $q_{t}$ are positive vectors of prices and quantities for $N$ products in period $t$ . The weight $ω_{t}$ is normally $1 / (T + 1)$ for a temporal index, but can vary across locations for a spatial index. The price-index function $P$ is usually a Fisher or Törnqvist index, and consequently there is an additive decomposition for this function such that it can be represented as an arithmetic mean of price relatives for all $N$ products

\begin{matrix} P (p_{b}, p_{a}, q_{b}, q_{a}) \equiv \sum_{i = 1}^{N} w_{i} (b, a) \frac{p_{ib}}{p_{ia}} . \end{matrix}

(1)

See Balk (2008, Chapter 4) or Reinsdorf et al. (2002) for the details of these decompositions—importantly, for each of these decompositions, the weights $w_{i} (b, a)$ are all positive and sum to $1$ . The weights $w_{i} (b, a)$ generally depend on prices and quantities for all products in periods $b$ and $a$ , but this dependence is suppressed to simplify notation. There are decompositions that do not represent $P$ as an arithmetic mean, as noted by Balk (2008, Chapter 4), and these are excluded in what follows.

Martin (2021) generalizes the decompositions for the Fisher and Törnqvist indexes to any price index that is based on nested generalized means, so little is lost by starting from an additive decomposition for the price-index function $P$ . In particular, this covers all the GEKS variants, such as those where $P$ is a Laspeyres or Lloyd-Moulton index, discussed by Białek (2026). Note that this assumes prices are strictly positive, something that does not always hold in practice, and so any products with zero prices are excluded from the calculation of the index.

The goal is to extend the additive decomposition for the bilateral price-index function $P$ in Equation (1) into an additive decomposition for the multilateral GEKS index. Just as the GEKS index generalizes a bilateral index by considering the geometric mean of all pairs of bilateral comparisons, the goal is to decompose the GEKS index into an arithmetic mean of all pairs of price relatives.

To start, the geometric mean of indexes across all base periods in the numerator of the GEKS index (i.e., the GEKS “price level” in period $b$ ) can be represented as an arithmetic mean using the usual decomposition based on the logarithmic mean. For two positive numbers $x$ and $y$ , the logarithmic mean is $L (x, y) = (x - y) / \log (x / y)$ when $x \neq y$ and $L (x, x) = x$ otherwise, so that

\prod_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}} \equiv \sum_{t = 0}^{T} v (b, t) P (p_{b}, p_{t}, q_{b}, q_{t}),

where

\begin{matrix} v (b, t) \propto ω_{t} / L (P (p_{b}, p_{t}, q_{b}, q_{t}), Π_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}}) . \end{matrix}

See again Balk (2008, Chapter 4) or Reinsdorf et al. (2002) for details. Combining with Equation (1) then yields the numerator of the GEKS index as an arithmetic mean of price relatives across all base periods

\prod_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}} \equiv \sum_{t = 0}^{T} \sum_{i = 1}^{N} v (b, t) w_{i} (b, t) \frac{p_{i b}}{p_{i t}} .

(2)

The same approach can be taken to represent the denominator of the GEKS index as an arithmetic mean of price relatives across all base periods. The only difference is that the goal is to represent the denominator as a harmonic mean of price relatives (i.e., decompose the inverse of the GEKS “price level” in period $a$ ), not an arithmetic mean, and this requires transmuting the weights to turn the arithmetic mean into a harmonic mean,

1 / \prod_{t = 0}^{T} P {(p_{a}, p_{t}, q_{a}, q_{t})}^{ω_{t}} \equiv \sum_{t = 0}^{T} \sum_{i = 1}^{N} u_{i} (a, t) \frac{p_{i t}}{p_{i a}},

(3)

where

\begin{matrix} u_{i} (a, t) \propto v (a, t) w_{i} (a, t) \frac{p_{ia}}{p_{it}} . \end{matrix}

Combining Equation (2) and Equation (3) then gives the GEKS index as an arithmetic mean of all combinations of price relatives that link period $a$ to period $b$

\begin{matrix} \frac{Π_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}}}{Π_{t = 0}^{T} P {(p_{a}, p_{t}, q_{a}, q_{t})}^{ω_{t}}} \equiv \sum_{τ = 0}^{T} \sum_{j = 1}^{N} u_{j} (a, τ) \frac{p_{j τ}}{p_{ja}} (\sum_{t = 0}^{T} \sum_{i = 1}^{N} v (b, t) w_{i} (b, t) \frac{p_{ib}}{p_{it}}) \\ = \sum_{τ = 0}^{T} \sum_{t = 0}^{T} \sum_{j = 1}^{N} \sum_{i = 1}^{N} u_{j} (a, τ) v (b, t) w_{i} (b, t) (\frac{p_{ib}}{p_{it}} \frac{p_{j τ}}{p_{ja}}) . \end{matrix}

(4)

Compared to the decomposition of the bilateral index-number formula $P$ in (1) that contains $N$ terms, the decomposition in Equation (4) contains $(N (T + 1 {))}^{2}$ terms as it is based on the outer product of all price relatives between period $0$ and period $T$ . For a given product $i$ , there are $T + 1$ terms in Equation (4) that give the direct contribution of the price relative $p_{ib} / p_{ia}$

\begin{matrix} \sum_{t = 0}^{T} u_{i} (a, t) v (b, t) w_{i} (b, t) \frac{p_{ib}}{p_{ia}} . \end{matrix}

The weight for product $i$ in the decomposition is the total product of weights from the bilateral decompositions that come from the numerator and denominator of the GEKS index between period 0 and period $T$ . However, these direct contributions do not add up to the value of the index—instead, the GEKS index is decomposed into the sum of these $T + 1$ direct contributions for each of the $N$ products and $N (T + 1) (N (T + 1) - 1)$ interaction terms that take into account all pairs of price relatives across different products and time periods.

3. Example

Consider the following example to illustrate the decomposition in Equation (4). There are two products, each transacted at three points in time according to the data in Table 1, and the goal is to make a GEKS-Fisher index. See Supplemental Material for the accompanying R code for the step-by-step computations. With these data the GEKS-Fisher index between period 0 and 2 is 1.15. Decomposing this index involves first finding the outer product of the price data to get all combinations of price relatives between periods 0 and 2, shown in Table 2. This includes all the terms in the parentheses in Equation (4). Associated with each pair of price relatives in Table 2 is a weight, such that all weights sum to 1, given in Table 3. Calculating these weights involves calculating each of the three weights in Equation (4) for each pair of price relatives. Multiplying each weight by the corresponding price relative pair in Table 2 and summing gives the GEKS-Fisher index of 1.15.

Table 1.

Prices and Quantities for Two Products at Three Points in Time.

Time
Product	0	1	2
Prices
1	1.00	1.30	1.20
2	1.00	0.80	1.10
Quantities
1	10	8	9
2	10	12	11

Table 2.

All Combinations of Price Relatives Between Periods 0 and 2.

Relative	$p_{10} / p_{10}$	$p_{20} / p_{20}$	$p_{11} / p_{10}$	$p_{21} / p_{20}$	$p_{12} / p_{10}$	$p_{22} / p_{20}$
$p_{12} / p_{10}$	1.20	1.20	1.56	0.96	1.44	1.32
$p_{22} / p_{20}$	1.10	1.10	1.43	0.88	1.32	1.21
$p_{12} / p_{11}$	0.92	0.92	1.20	0.74	1.11	1.02
$p_{22} / p_{21}$	1.38	1.38	1.79	1.10	1.65	1.51
$p_{12} / p_{12}$	1.00	1.00	1.30	0.80	1.20	1.10
$p_{22} / p_{22}$	1.00	1.00	1.30	0.80	1.20	1.10

Table 3.

Arithmetic Weights for the GEKS-Fisher Index.

Relative	$p_{10} / p_{10}$	$p_{20} / p_{20}$	$p_{11} / p_{10}$	$p_{21} / p_{20}$	$p_{12} / p_{10}$	$p_{22} / p_{20}$
$p_{12} / p_{10}$	0.026	0.026	0.023	0.029	0.023	0.026
$p_{22} / p_{20}$	0.029	0.029	0.026	0.032	0.026	0.029
$p_{12} / p_{11}$	0.031	0.031	0.027	0.033	0.027	0.030
$p_{22} / p_{21}$	0.025	0.025	0.023	0.028	0.023	0.025
$p_{12} / p_{12}$	0.028	0.028	0.025	0.031	0.025	0.028
$p_{22} / p_{22}$	0.031	0.031	0.028	0.034	0.028	0.031

It is worth noting that, despite being the smallest possible example, the GEKS-Fisher index is decomposed into thirty-six terms. Terms along the diagonal give the direct contribution of each product to the index (about 8% for each product), with the off-diagonal terms giving the (numerous) interaction terms.

4. Discussion

Unlike the decompositions by Webster and Tarnow-Mordi (2019) and Białek (2026), Equation (4) represents the GEKS index as an arithmetic mean of price relatives. This means that Equation (4) can also be used to generate a multiplicative decomposition, whereby the GEKS index is represented as geometric mean of all combinations of price relatives, as well as a percent-change decomposition, both for temporal and spatial indexes.

In practice, there are usually missing data when making a GEKS index, either due to product imbalance between period $0$ and $T$ or incomplete data at a point in time, and the set of matched products used for $P$ changes over time. Missing data can be directly accommodated by treating products with missing data as having a weight of 0 in Equation (1), and consequently a weight of $0$ in Equations (2) and (3). Equation (4) continues to hold, but certain weights will be zero when there are missing data for some products in order to balance the set of products over time.

Although the goal is to decompose a GEKS index, the result is actually slightly more general. Martin (2021) generalizes the additive decomposition of the geometric mean based on the logarithmic mean presented by Balk (2008, Chapter 4), and this can be used to generalize (4) when the index is based on a generalized mean instead of a geometric mean. That is, for an index of the form

\begin{matrix} {\begin{matrix} \frac{{(\sum_{t = 0}^{T} ω_{t} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ρ})}^{1 / ρ}}{{(\sum_{t = 0}^{T} ω_{t} P {(p_{a}, p_{t}, q_{a}, q_{t})}^{ρ})}^{1 / ρ}} & ρ \neq 0 \\ \frac{Π_{t = 0}^{T} P {(p_{b}, p_{t}, q_{b}, q_{t})}^{ω_{t}}}{Π_{t = 0}^{T} P {(p_{a}, p_{t}, q_{a}, q_{t})}^{ω_{t}}} & ρ = 0 \end{matrix}, \end{matrix}

Martin (2021) provides a generalization of the function $v$ in Equations (2) and (3) to express the numerator of this index as an arithmetic mean and the denominator as a harmonic mean. This makes the decomposition in (4) directly extendable to some of the generalizations of the GEKS index noted by Balk (2008, Chapter 7).

The generalized decomposition by Martin (2021) also shows that the same approach used to derive Equation (4) can be used to generalize both multiplicative and harmonic decompositions (see Färe and Karagiannis 2025), rather than additive decompositions, for the index-number function $P$ when used in a GEKS index. For example, starting from a harmonic decomposition for the price-index function $P$ , Equation (2) can be replaced with a harmonic decomposition for the numerator of the GEKS index and Equation (3) can be replaced with an additive decomposition for the denominator of the GEKS index to express Equation (4) as a harmonic mean of all pairs of price relatives.

Compared to the additive decomposition of a bilateral index-number formula, the decomposition of the GEKS index in (4) depends on all combinations of price relatives over the window of the index, not just those comparing prices at two points in time. As noted above, this also applies when extending multiplicative and harmonic decompositions, so it is not simply a feature of the additive decomposition. Instead, the decomposition is the sum of two terms: the direct contribution of price relatives comparing prices at two points in time for each product, and a large collection of interaction terms that give the contribution for pairs of price relatives across different products and time periods. Importantly, it is not sufficient to simply sum the weights for each product from the bilateral contributions over the window of the GEKS index—the interaction terms are required to exactly decompose the index as an arithmetic mean. This highlights an important distinction between the decompositions for bilateral indexes and the GEKS index, and one that is not seen when decomposing other types of multilateral indexes (i.e., Geary-Khamis, repeat sales). Although the large number of terms limits the practical use of this decomposition, the result is of theoretical interest as it shows a link between decomposing bilateral and multilateral GEKS indexes, as well as showing the challenges with decomposing GEKS indexes.

Supplemental Material

sj-zip-1-jof-10.1177_0282423X261451318 – Supplemental material for A Note on the Additive Decomposition of GEKS Indexes

Supplemental material, sj-zip-1-jof-10.1177_0282423X261451318 for A Note on the Additive Decomposition of GEKS Indexes by Steve Martin in Journal of Official Statistics

Footnotes

Author Note

All views expressed here are my own and do not necessarily represent those of the Government of Canada. This work has benefited from helpful comments by two anonymous referees and an associate editor. All errors are mine.

ORCID iD

Steve Martin

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online.

Received: February 28, 2026

Accepted: April 29, 2026

References

Balk

B. M.

2008. Price and Quantity Index Numbers. Cambridge University Press.

Białek

2026. “Multiplicative Decompositions of GEKS-Type Indices into the Contribution of Individual Commodities.” Journal of Official Statistics 42 (1): 62–91. DOI: https://doi.org/10.1177/0282423X251394243.

Färe

Karagiannis

2025. “An Alternative Derivation of the van IJzeren Weights and Some New Results for the Additive Decomposition of Fisher Indices.” Journal of Productivity Analysis 64 (3): 363–77. DOI: https://doi.org/10.1007/s11123-025-00774-2.

Martin

2021. “A Note on Generalized Decompositions for Price Indexes.” Prices Analytical Series, Statistics Canada. https://www150.statcan.gc.ca/n1/pub/62f0014m/62f0014m2021012-eng.htm.

Reinsdorf

M. B.

Diewert

W. E.

Ehemann

2002. “Additive Decompositions for Fisher, Törnqvist and Geometric Mean Indexes.” Journal of Economic and Social Measurement 28 (1–2): 51–61. DOI: https://doi.org/10.3233/JEM-2003-0194.

Shiller

R. J.

1991. “Arithmetic Repeat Sales Price Estimators.” Journal of Housing Economics 1 (1): 110–26. DOI: https://doi.org/10.1016/S1051-1377(05)80028-2.

Webster

M. B.

Tarnow-Mordi

R. C.

2019. “Decomposing Multilateral Price Indexes into the Contributions of Individual Commodities.” Journal of Official Statistics 35: 461–86. DOI: https://doi.org/10.2478/jos-2019-0020.

Supplementary Material

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