Decomposing indexes with Euler weights
Steve Martin 
I’ve always liked Hallerbach (2005)’s decomposition of the Fisher index. It’s such a simple and straightforward way to decompose the Fisher index that generalizes to a wide range of index-number formulas. The latter has, to my knowledge, not been explored before, and Hallerbach’s approach of using Euler weights to decompose an index number is instead associated with the Fisher index. What I want to do here is explore this decomposition in more detail and show a couple properties that aren’t obvious when decomposing the Fisher index alone.
Let’s start with the family of generalized-mean indexes. For an index based on a generalized mean of order \(\rho\),
\[ \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) = \begin{cases} \left(\sum_{i = 1}^{n} w_{i} r_i^{\rho}\right)^{1 / \rho} & \text{if } \rho \neq 0 \\ \prod_{i = 1}^{n} r_{i}^{w_{i}} & \text{if } \rho = 0, \end{cases} \] the derivative with respect to \(r_{i}\) is \[ \frac{\partial \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w})}{\partial r_{i}} = w_{i} \left(\frac{r_{i}}{\mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w})}\right)^{\rho - 1}. \tag{1}\] Multiplying each of these terms by \(r_{i}\) and summing gives the value of \(\mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w})\).
Equation 1 is similar in form to the generalized additive decomposition by Martin (2021) \[ w_{i} \mathfrak{L}_{\rho}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - 1} \Bigg/ \sum_{j=1}^{n} w_{j} \mathfrak{L}_{\rho}(r_j, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - 1}, \tag{2}\] where \(\mathfrak{L}\) is the generalized logarithmic mean (Balk 2008, chap. 4). As with Equation 1, multiplying each of these terms by \(r_{i}\) and summing returns the values of the generalized mean. In both cases, these decompositions work by shifting weight to the larger \(r_{i}\)s when \(\rho > 1\) and shifting weight to the smaller \(r_{i}\)s when \(\rho < 1\). The break point is particularly clean for Equation 1, as an \(r_{i}\) above the mean gets more (less) weight whenever \(\rho > 1\) (\(\rho < 1\)), whereas this need not be true for Equation 2.
Except when \(\rho = 1\) (additive decomposition for an arithmetic mean) or all \(r_{i}\) are the same, Equation 1 and Equation 2 give different decompositions. In particular, the terms in Equation 2 always sum to 1, whereas Equation 1 is greater than 1 whenever \(\rho < 1\) and less than 1 whenever \(\rho > 1\). To see this, note that the sum of Equation 1 is
\[ \frac{\sum_{i=1}^{n} w_{i}r_{i}^{\rho - 1}}{\mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w})^{\rho - 1}}. \] If \(\rho > 1\) then this must be less than 1 because the generalized mean in an increasing function of \(\rho\). If \(\rho < 1\) then the inequality is reversed. This is in contrast to decomposition by Hallerbach (2005) where the Euler weights for Fisher index are always greater than 1 (unless the Laspeyres and Paasche indexes are equal, in which case the weights sum to 1).
A stark example of the difference between the two decompositions is the additive decomposition of the geometric mean: each \(i\) in Equation 1 gets the same value when multiplied by \(r_{i}\), whereas Equation 2 is increasing with respect to \(r_{i}\).1 This seems like a counter-intuitive property for the decomposition of an index based on the geometric mean. Equation 2 transfers weight from the larger \(r_{i}\)s to the smaller ones—so that the arithmetic mean equals the geometric one—to determine the contribution of each \(r_{i}\) while preserving the overall ranking. Instead, Equation 1 divides the value of the geometric mean into equal parts so that the sum equals the geometric mean, independent of the value of each \(r_{i}\).
Equation 1 can be extended to the family indexes based on nested generalized means. For a pair of generalized means \(\mathfrak{N}_{\rho_{1}\rho_{2}}(\mathbf{r}, \mathbf{w}_{1}, \mathbf{w}_{2})=\left(\mathfrak{M}_{\rho_1}(\mathbf{r}, \mathbf{w}_1), \mathfrak{M}_{\rho_2}(\mathbf{r}, \mathbf{w}_2)\right)\) with weights \(\mathbf{\omega}=(\omega_1, \omega_2)\), an index based on nested generalized means is written as \[ \mathfrak{M}_{\rho}\left(\mathfrak{N}_{\rho_{1}\rho_{2}}(\mathbf{r}, \mathbf{w}_{1}, \mathbf{w}_{2}), \mathbf{\omega}\right). \] The Euler weights for this index are then
\[ \begin{align*} \frac{\partial \mathfrak{M}_{\rho}\left(\mathfrak{N}_{\rho_{1}\rho_{2}}(\mathbf{r}, \mathbf{w}_{1}, \mathbf{w}_{2}), \mathbf{\omega}\right)}{\partial r_{i}} &= \omega_{1} \left(\frac{\mathfrak{M}_{\rho_{1}}(\mathbf{r}, \mathbf{w_{1}})}{\mathfrak{M}_{\rho}\left(\mathfrak{N}_{\rho_{1}\rho_{2}}(\mathbf{r}, \mathbf{w}_{1}, \mathbf{w}_{2}), \mathbf{\omega}\right)}\right)^{\rho - 1} w_{i1} \left(\frac{r_{i}}{\mathfrak{M}_{\rho_{1}}(\mathbf{r}, \mathbf{w_{1}})}\right)^{\rho_{1} - 1}\\ &+ \omega_{2} \left(\frac{\mathfrak{M}_{\rho_{2}}(\mathbf{r}, \mathbf{w_{2}})}{\mathfrak{M}_{\rho}\left(\mathfrak{N}_{\rho_{1}\rho_{2}}(\mathbf{r}, \mathbf{w}_{1}, \mathbf{w}_{2}), \mathbf{\omega}\right)}\right)^{\rho - 1} w_{i2} \left(\frac{r_{i}}{\mathfrak{M}_{\rho_{2}}(\mathbf{r}, \mathbf{w_{2}})}\right)^{\rho_{2} - 1}. \end{align*} \tag{3}\]
For the Fisher index (\(\rho = 0\), \(\rho_{1} = 1\), \(\rho_{2} = -1\), \(\omega_{1} = \omega_{2} = 1 / 2\)), Equation 3 simplifies to
\[ \frac{1}{2}\sqrt{\frac{\mathfrak{M}_{-1}(\mathbf{r}, \mathbf{w_{2}})}{\mathfrak{M}_{1}(\mathbf{r}, \mathbf{w_{1})}}} w_{i1} + \frac{1}{2}\sqrt{\frac{\mathfrak{M}_{1}(\mathbf{r}, \mathbf{w_{1}})}{\mathfrak{M}_{-1}(\mathbf{r}, \mathbf{w_{2})}}} w_{i2} \left(\frac{r_{i}}{\mathfrak{M}_{-1}(\mathbf{r}, \mathbf{w_{2}})}\right)^{-2}. \]
Note that the second term is not the same as in Equation 13 by Hallerbach (2005). Applying Equation 2 to transmute the weights \(\mathbf{w}_{2}\) to represent the harmonic mean as an arithmetic mean (i.e., hybrid weights for the Paasche index) yields the decomposition due to Hallerbach (2005). But this alternate form is interesting because it gives larger weights that are strictly greater than 1 even when the Laspeyres and Paasche indexes are the same (unless all \(r_{i}\) are the same). This suggests that Equation 1 can be combined with Equation 2 to produce even more decompositions based on Euler weights.
References
Footnotes
To see this, note that \(\partial (x / \mathfrak{L}(x, y))/ \partial x > 0\) if \(x - y > y \log(x / y)\). This is true whenever \(x \neq y\) because \(\min(x, y) < \mathfrak{L}(x , y) < \max(x, y)\); otherwise, \(\partial (x / \mathfrak{L}(x, y))/ \partial x = 0\).↩︎
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Citation
@online{martin2026,
author = {Martin, Steve},
title = {Decomposing Indexes with {Euler} Weights},
date = {2026-08-07},
url = {https://marberts.github.io/blog/posts/2026/euler/},
langid = {en}
}