Decomposing price indexes

This vignette presents an extended version of Martin (2021), and shows how to use the tools in this package to make these decompositions.

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, Diewert, and Ehemann (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, Diewert, and Ehemann 2002, sec. 6). The van IJzeren additive decomposition for the Fisher index (Balk 2008, equation 4.18) is an alternative that does not use the logarithmic mean. 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.1

I show how 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, the AG mean index by Lent and Dorfman (2009), or GEKS indexes (Webster and Tarnow-Mordi 2019).

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 \(\rho\) into one based on a generalized mean of order \(\varsigma\). To fix notation, let \(\mathbf{r} = (r_{1}, r_{2}, \ldots, r_{n}) \in \mathbb{R}^{n}_{++}\) be a vector of price relatives for \(n\geq2\) products and let \(\mathbf{w} = (w_{1}, w_{2}, \ldots, w_{n}) \in \Delta^{n - 1}\) be the corresponding weights, where \(\Delta^{n - 1} = \{\mathbf{w} \in \mathbb{R}_{+}^{n} | \sum_{i = 1}^{n} w_{i} = 1\}\) is the unit simplex. The goal is to find a vector-valued function \[ \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) = (v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma), v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma),\ldots, v_{n}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)) \] mapping into \(\Delta^{n - 1}\) such that \[ \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) \equiv \mathfrak{M}_{\varsigma}(\mathbf{r}, \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)), \] where \(\mathfrak{M}_{\rho}\) is the generalized mean \[ \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) = \begin{cases} \left(\sum_{i = 1}^{n} w_{i} r_i^{\rho}\right)^{1 / \rho} & \text{if } \rho \neq 0 \\ \prod_{i = 1}^{n} r_{i}^{w_{i}} & \text{if } \rho = 0. \end{cases} \]

Setting \(\varsigma = 1\) then yields an additive decomposition for any index based on a generalized mean of order \(\rho\), such that \[ \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) = \sum_{i=1}^{n} r_{i} v_i(\mathbf{r}, \mathbf{w}; \rho, 1), \] and setting \(\varsigma = 0\) yields a multiplicative decomposition, such that \[ \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) = \prod_{i=1}^{n} r_{i}^{ v_i(\mathbf{r}, \mathbf{w}; \rho, 0)}. \] Note that any index that admits an additive decomposition can be used to derive percent-change contributions for each price relative, \(v_i(\mathbf{r}, \mathbf{w}; \rho, 1) (r_i - 1)\), that sum up to \(\mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) - 1\). I examine these types of contributions in more detail in Section 2.

Balk (2008) and Reinsdorf, Diewert, and Ehemann (2002) show how to derive \(\mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)\) when \(\rho = 1\) and \(\varsigma = 0\) (multiplicative decomposition of an arithmetic index) \[ v_{i}(\mathbf{r}, \mathbf{w}; 1, 0) = \frac{w_{i} \mathfrak{L}(r_{i}, \mathfrak{M}_{1}(\mathbf{r}, \mathbf{w}))}{\sum_{j=1}^{n} w_{j} \mathfrak{L}(r_{j}, \mathfrak{M}_{1}(\mathbf{r}, \mathbf{w}))} \] and \(\rho = 0\) and \(\varsigma = 1\) (additive decomposition of a geometric index) \[ v_{i}(\mathbf{r}, \mathbf{w}; 0, 1) = \frac{w_{i} / \mathfrak{L}(r_{i}, \mathfrak{M}_{0}(\mathbf{r}, \mathbf{w}))}{\sum_{j=1}^{n} w_{j} / \mathfrak{L}(r_{j}, \mathfrak{M}_{0}(\mathbf{r}, \mathbf{w}))}, \] using the logarithmic mean \[ \mathfrak{L}(a, b) = \begin{cases} \frac{a - b}{log(a / b)} & a \neq b\\ a & a = b. \end{cases} \]

Generalizing these results follows from replacing the logarithmic mean with the more general extended mean (Bullen 2003, 393), defined for any \(a,b > 0\) as \[ \mathfrak{E}_{\rho\varsigma}(a, b) = \begin{cases} \left(\frac{\varsigma(a^\rho - b^\rho)}{\rho(a^\varsigma - b^\varsigma)}\right)^{1 / (\rho - \varsigma)} & \rho \neq \varsigma, \rho \neq 0, \varsigma \neq 0, a \neq b \\ \left(\frac{a^\rho - b^\rho}{\rho\log(a / b)}\right)^{1 / \rho} & \rho \neq 0, \varsigma = 0, a \neq b \\ \left(\frac{a^\varsigma - b^\varsigma}{\varsigma\log(a / b)}\right)^{1 / \varsigma} & \rho = 0, \varsigma \neq 0, a \neq b \\ \frac{1}{\exp(1 / \rho)} \left(\frac{a^{a^\rho}}{b^{b^\rho}}\right)^{1 / (a^\rho - b^\rho)} & \rho = \varsigma \neq 0, a \neq b \\ \sqrt{ab} & \rho = \varsigma = 0, a \neq b \\ a & a = b. \end{cases} \] 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 (\(\varsigma = -1\)) and the Lloyd-Moulton index (\(\varsigma = 1 - \sigma\), 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 strictly positive and satisfies the identity \[ \sum_{i=1}^{n} w_{i} \mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \varepsilon(\mathbf{r}, \mathbf{w}; \rho, \varsigma) \equiv 0, \tag{1}\] where \[ \varepsilon(\mathbf{r}, \mathbf{w}; \rho, \varsigma) = \begin{cases} r_i^\varsigma - \mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w})^\varsigma & \text{if } \varsigma \neq 0, \\ \log(r_i) - \log(\mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w})) & \text{if } \varsigma = 0. \end{cases} \] Equation 1 uses the extended mean to keep the weighted deviation from the mean constant for each price relative (up to a common factor of proportionality) when changing the order of the mean from \(\rho\) to \(\varsigma\), without changing the value of the mean. Rearranging then gives that \[ v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) = w_{i} \mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \Bigg/ \sum_{j=1}^{n} w_{j} \mathfrak{E}_{\rho\varsigma}(r_j, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \tag{2}\] is 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 \(\varsigma\).

The decomposition in Equation 2 has a number of useful properties. Transmuting the weights to turn a generalized mean of order \(\rho\) into a generalized mean of order \(\varsigma\), and then transmuting these weights again to turn a generalized mean of order \(\varsigma\) into a generalized mean of order \(\rho\) returns the original weights.

Proposition 1 (Invariance) The decomposition given by Equation 2 has the following properties.

  1. \(\mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \rho) = \mathbf{w}\).
  2. \(\mathbf{v}(\mathbf{r}, \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma); \varsigma, \rho) = \mathbf{w}\).
  3. \(w_{k} v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) = w_{i} v_{k}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)\) when \(r_{i} = r_{k}\).

Proof.

  1. Since the extended mean is strictly positive, \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \rho) = w_{i} / \sum_{j=1}^{n} w_{j} = w_{i}\) because \(\sum_{j=1}^{n} w_{j}=1\).
  2. From the definition of the extended mean, \(\mathfrak{E}_{\rho\varsigma}(a, b) = \mathfrak{E}_{\varsigma\rho}(a, b)\) for all \(a, b\). This means that \(\mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma}\mathfrak{E}_{\varsigma\rho}(r_i, \mathfrak{M}_{\varsigma}(\mathbf{r}, \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)))^{\varsigma - \rho} = 1\) for each \(i=1,...,n\) because \(\mathfrak{M}_{\rho}(\mathbf{r}, \mathbf{w}) = \mathfrak{M}_{\varsigma}(\mathbf{r}, \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma))\). Plugging into Equation 2 gives the result.
  3. If \(r_{i} = r_{k}\) then \[\begin{align*} w_{k} v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) &= w_{k}w_{i}\mathfrak{E}_{\rho\varsigma}(r_k, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \Bigg/ \sum_{j=1}^{n} w_{j} \mathfrak{E}_{\rho\varsigma}(r_j, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma}\\ &= w_{i} v_{k}(\mathbf{r}, \mathbf{w}; \rho, \varsigma). \end{align*}\]

An implication of invariance is the Equation 2 is the unique decomposition for indexes based on the generalized mean.

Proposition 2 (Uniqueness) The decomposition in Equation 2 is the only such decomposition that is invariant in the sense of Proposition 1.

Proof. When \(n=2\), any other function \(\mathbf{u}\) that decomposes the generalized mean must satisfy \((v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma))(r_{1}^\varsigma - r_{2}^\varsigma) \equiv 0\) for \(i=1,2\), or \((v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma))(\log(r_{1}) - \log(r_{2})) \equiv 0\) when \(\varsigma = 0\), so that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) \equiv u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)\) whenever \(r_{1} \neq r_{2}\). This means that, for some \(\rho\), there is an \(\mathbf{r}\) and \(\mathbf{w}\) such that \(\mathbf{u}(\mathbf{r}, \mathbf{w}; \rho, \rho) \neq \mathbf{w}\).

Now suppose \(n\geq3\) and \(r_1\neq r_2 \neq r_3 = \ldots = r_n\) and \(w_{3} = \ldots = w_{n}\). Any other weights that successfully decompose the generalized mean of order \(\rho\) and are invariant must satisfy \[ \mathfrak{M}_\rho(\mathbf{r}, \mathbf{w})^\varsigma = (v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_1) r_1^\varsigma + (v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_2) r_2^\varsigma + (n - 2) (v_{3}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_3) r_3^\varsigma \] for some \((\delta_1, \delta_2, \delta_3)\). This implies that \[ \delta_1 r_1^\varsigma + \delta_2 r_2^\varsigma = -(n -2)\delta_3r_3^\varsigma \] It also must be that these weights sum to 1, so that \[ \delta_1 + \delta_2 = -(n - 2)\delta_3. \] Therefore \[ \delta_2 = \delta_1 \frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma}. \] Decomposing these weights again requires that \[ v_{1}(\mathbf{r}, v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_1; \varsigma, \rho) + \eta_1 = w_{1} \] and \[ v_{2}\left(\mathbf{r}, v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_1 \frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma}; \varsigma, \rho\right) + \eta_1\frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma} = w_2 \] for some \(\eta_1\). Using the invariance of Equation 2 implies that \[ \delta_1 \mathfrak{E}_{\varsigma\rho}(r_2, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w})) = \eta_1 \mathfrak{E}_{\varsigma\rho}(r_1, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w})). \] As \(r_1 \neq r_2\), this can only be true if \(\delta_1=0\).

The function in Equation 2 has a monotonicity property where the weights increase (decrease) for large (small) price relatives if and only if \(\rho > \varsigma\).

Proposition 3 (Monotonicity) Assuming \(\mathbf{r}\) is ordered from smallest to largest (and does not contain all the same value), if \(\rho > \varsigma\) then there is a pair of integer \(k,l\), with \(k\leq l\), such that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) > w_{i}\) for \(i\geq l\) and \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) < w_{i}\) for \(i\leq k\), with these equalities reversed if \(\rho < \varsigma\).

Proof. To start, note that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) \geq w_{i}\) if and only if \[ \mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \geq \sum_{j=1}^{n} w_{j} \mathfrak{E}_{\rho\varsigma}(r_j, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma}. \] Since the extended mean is strictly increasing in its first argument (Bullen 2003, theorem 16), the right-hand side is an increasing function if \(\rho > \varsigma\). This inequality is not satisfied for \(i=1\) and is strictly satisfied for \(i=n\), so \(k\) can then be set to the largest value such that this inequality is not satisfied, and \(l\) can be set to the smallest value such that it is strictly satisfied. If \(\rho < \varsigma\) then the right-hand side is a decreasing function, so the inequality is not satisfied for \(i=n\) and is strictly satisfied for \(i=1\). As before, \(k\) can be set to the largest value such that the inequality is strictly satisfied, and \(l\) can be set to the smallest value such that it is not satisfied.

The function given by Equation 2 takes on all existing decompositions that I know of as special cases. Setting \(\rho = 0\) and \(\varsigma = 1\), or \(\rho = 1\) and \(\varsigma = 0\), gives the special cases by Balk (2008) and Reinsdorf, Diewert, and Ehemann (2002) for decomposing indexes based on arithmetic and geometric means (because the extended mean reduces to the logarithmic mean). Similarly, setting \(\rho = -1\) and \(\varsigma = 1\) reduces Equation 2 to \[ v_{i}(\mathbf{r}, \mathbf{w}; -1, 1) = \frac{w_{i} / r_{i}}{\sum_{i=1}^{n}w_{i} / r_{i}} ; \] if \(\mathbf{w}\) is a vector of current-period expenditure/revenue shares then these are the hybrid weights that allow a Paasche index to be calculated as an arithmetic mean of price relatives. Similarly, setting \(\rho = 1\) and \(\varsigma = -1\) reduces Equation 2 to \[ v_{i}(\mathbf{r}, \mathbf{w}; 1, -1) = \frac{w_{i} r_{i}}{\sum_{i=1}^{n}w_{i} r_{i}}. \] If \(\mathbf{w}\) is a vector of base-period expenditure/revenue shares then these are the hybrid weights that allow a Laspeyres index to be calculated as a harmonic mean of price relatives.2 As should be expected, the weights are unchanged if \(\rho = \varsigma\) or each element of \(\mathbf{r}\) takes on the same value.

Invariance and monotonicity properties

The extended mean has two properties that make it useful for decomposing price indexes. First, the order of \(\rho\) and \(\varsigma\) do not matter in the extended mean: \(\mathfrak{E}_{\rho\varsigma}(a, b) = \mathfrak{E}_{\varsigma\rho}(a, b)\). This means the function in Equation 2 has an invariance property such that transmuting the weights to turn a generalized mean of order \(\rho\) into a generalized mean of order \(\varsigma\), and then transmuting these weights again to turn a generalized mean of order \(\varsigma\) into a generalized mean of order \(\rho\) returns the original weights. That is, \[ \mathbf{v}(\mathbf{r}, \mathbf{v}(\mathbf{r}, \mathbf{w}; \rho, \varsigma); \varsigma, \rho) \equiv \mathbf{w}. \] As will be seen later, not all decompositions have this property.

Second, the extended mean is a strictly increasing function in both arguments (Bullen 2003, theorem 16). This means that the function in Equation 2 has a monotonicity property where the weights increase (decrease) for large (small) price relatives if and only if \(\rho > \varsigma\). That is, assuming \(\mathbf{r}\) is ordered from smallest to largest (and does not contain all the same value), if \(\rho > \varsigma\) then there is a pair of integer \(k,l\), with \(k\leq l\), such that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) > w_{i}\) for \(i\geq l\) and \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) < w_{i}\) for \(i\leq k\), with these equalities reversed if \(\rho < \varsigma\).3 Again, not all decompositions satisfy this property.

Uniqueness

The decomposition given by Equation 2 is unique when \(n = 2\) and \(r_{1} \neq r_{2}\), and is the only such function that always returns \(\mathbf{w}\) when \(r_{1} = r_{2}\).4 But Equation 2 is necessarily not unique when \(n\geq3\), and there are infinitely many ways to decompose an index based on the generalized mean.5 What makes Equation 2 unique is that it is the only invariant decomposition of the generalized mean for \(n\geq3\).

To see this, suppose \(n=3\) and \(r_1\neq r_2 \neq r_3\). Any other weights that successfully decompose the generalized mean of order \(\rho\) can be written as \[ \begin{pmatrix} v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_\rho\\ v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_\rho \frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma}\\ 1 - v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - \delta_\rho - v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - \delta_\rho \frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma} \end{pmatrix} \] for some \(\delta_\rho\). Decomposing these weights again requires that \[ v_{1}(\mathbf{r}, v_{1}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_\rho; \varsigma, \rho) + \delta_\varsigma = w_{1} \] and \[ v_{2}\left(\mathbf{r}, v_{2}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) + \delta_\rho \frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma}; \varsigma, \rho\right) + \delta_\varsigma\frac{r_3^\varsigma - r_1^\varsigma}{r_2^\varsigma - r_3^\varsigma} = w_2 \] for some \(\delta_\varsigma\). Using the invariance of Equation 2 implies that \[ \delta_\rho \mathfrak{E}_{\varsigma\rho}(r_2, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w})) = \delta_\rho \mathfrak{E}_{\varsigma\rho}(r_1, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w})). \] As \(r_1 \neq r_2\), this can only be true if \(\delta_\rho=0\).

Decomposing superlative indexes

The additive and multiplicative decompositions for the Fisher index by Reinsdorf, Diewert, and Ehemann (2002, sec. 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 \(\left(\mathfrak{M}_{\rho_1}(\mathbf{r}, \mathbf{w}_1), \mathfrak{M}_{\rho_2}(\mathbf{r}, \mathbf{w}_2)\right)\) mapping into \(R_2^{++}\) with weights \((\omega_1, \omega_2) \in \Delta^1\), an index based on nested generalized means is written as \[ \mathfrak{M}_{\rho}\left((\mathfrak{M}_{\rho_1}(\mathbf{r}, \mathbf{w}_1), \mathfrak{M}_{\rho_2}(\mathbf{r}, \mathbf{w}_2)), (\omega_1, \omega_2)\right). \tag{3}\]

The general family of superlative quadratic mean indexes of order \(\tau\) comes from setting \(\rho = 0\), \(\rho_1 = \tau / 2\), and \(\rho_2 = -\tau / 2\) when \(\omega_1 = \omega_2 = 1 / 2\), \(\mathbf{w}_1\) are base-period expenditure/revenue shares, and \(\mathbf{w}_2\) are current-period expenditure/revenue shares. In particular, setting \(\tau = 2\) gives the Fisher index and setting \(\tau = 1\) gives the implicit Walsh index. But Equation 3 covers other types of indexes as well; for example, setting each element of \(\mathbf{w}_1\) and \(\mathbf{w}_2\) to \(1 / n\) when \(\tau = 2\) gives the Carruthers-Sellwood-Ward-Dal'en index that serves as an estimator for the Fisher index, whereas \(\tau = 1\) gives the Balk-Walsh index. Setting \(\rho = -1\) gives the harmonic analogue of the Fisher index, which is not a superlative quadratic mean indexes of order \(\tau\). Finally, setting \(\rho = \rho_{1} = \rho_{2}\) and \(\mathbf{w}_{1} = \mathbf{w}_{2}\) 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 Equation 3.

An index of form Equation 3 can be decomposed into an index based on the generalized mean of order \(\rho\) using the weights in Equation 2, as it can be written as the generalized mean \[ \mathfrak{M}_{\rho}(\mathbf{r}, \omega_1 \mathbf{v}(\mathbf{r}, \mathbf{w}_1; \rho_1, \rho) + \omega_2\mathbf{v}(\mathbf{r}, \mathbf{w}_2; \rho_2, \rho)). \] The transformation in Equation 2 then applies as before, just replacing \(\mathbf{w}\) with the more complicated weights \(\omega_1 \mathbf{v}(\mathbf{r}, \mathbf{w}_1; \rho_1, \rho) + \omega_2\mathbf{v}(\mathbf{r}, \mathbf{w}_2, \rho_2, \rho)\), which can be written as \[ \mathbf{v}(\mathbf{r}, \omega_1 \mathbf{v}(\mathbf{r}, \mathbf{w}_1; \rho_1, \rho) + \omega_2\mathbf{v}(\mathbf{r}, \mathbf{w}_2; \rho_2, \rho); \rho, \varsigma). \tag{4}\] The idea is to transmute the weights for both inner means to be of the same order as the outer mean (\(\rho\)) so that they can be added together, and then transmute these weights to represent the outer generalized mean as a mean of order \(\varsigma\). Note that this means the decomposition satisfies the same invariance property as Equation 2.

An alternative approach to decompose Equation 3 is to generalize the (additive) van IJzeren decomposition for the Fisher index. Letting \(\mathbf{m} =\left(\mathfrak{M}_{\rho_1}(\mathbf{r}, \mathbf{w}_1), \mathfrak{M}_{\rho_2}(\mathbf{r}, \mathbf{w}_2)\right)\) and \(\mathbf{\omega} = (\omega_1, \omega_2)\), this can be written as \[ v_{1}(\mathbf{m}, \mathbf{\omega}; \rho, \varsigma) \mathbf{v}(\mathbf{r}, \mathbf{w}_1; \rho_1, \varsigma) + v_{2}(\mathbf{m}, \mathbf{\omega}; \rho, \varsigma) \mathbf{v}(\mathbf{r}, \mathbf{w}_2; \rho_2, \varsigma). \tag{5}\] Note that for a Fisher index, if \(\varsigma=1\) then \(\mathbf{v}(\mathbf{r}, \mathbf{w}_1; \rho_1, \varsigma)\) is a vector of base-period expenditure/revenue shares, \(\mathbf{v}(\mathbf{r}, \mathbf{w}_2; \rho_2, \varsigma)\) is a vector of hybrid Paasche weights, and \(\mathbf{v}(\mathbf{m}, \mathbf{\omega}; \rho, \varsigma)\) are the unique weights that decompose the geometric mean of the Laspeyres and Paasche indexes, which equals the van IJzeren decomposition. The idea here is to transmute the weights for both the inner and outer generalized means so that they are means of order \(\varsigma\), then take the product of these weights. Equation 4 and Equation 5 generally give different decompositions, but reduce to Equation 2 when \(\rho = \rho_{1} = \rho_{2}\) and \(\mathbf{w}_{1} = \mathbf{w}_{2}\).

Numerical example

library(gpindex)

p2 <- price6[[2]]
p1 <- price6[[1]]
q2 <- quantity6[[2]]
q1 <- quantity6[[1]]

rel <- p2 / p1

s2 <- scale_weights(p2 * q1)
s1 <- scale_weights(p1 * q1)
quadratic_mean <- generalized_mean(2)
quadratic_weights <- transmute_weights(2, 1)
quadratic_mean(rel, s1)
[1] 1.531339
v <- quadratic_weights(rel, s1)

arithmetic_mean(rel, v)
[1] 1.531339
transmute_weights(1, 2)(rel, v)
[1] 0.10 0.10 0.20 0.10 0.45 0.05
v[order(rel)] - s1
[1] -0.024395704 -0.060503706 -0.107454243  0.091868099 -0.003049463
[6]  0.103535017
permute <- function(x, w, d) {
  d2 <- d * (x[3] - x[1]) / (x[2] - x[3])
  w + c(d, d2, -d - d2)
}

x <- c(1, 2, 6)
v1 <- permute(x, transmute_weights(0, 1)(x), 0.1)

all.equal(geometric_mean(x), arithmetic_mean(x, v1))
[1] TRUE
u <- transmute_weights(1, 2)(x, v1)
permute(rel, u, 1 / 3 - u[1])
[1] 0.3333333 0.1808477 0.4858190

Percent-change contributions

Contributions for the Fisher index

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): 147–52.
Lent, J., and A. H. Dorfman. 2009. “Using a Weighted Average of Base Period Price Indexes to Approximate a Superlative Index.” Journal of Official Statistics 25 (1): 149–49.
Martin, S. 2021. “A Note on Generalized Decompositions for Price Indexes.” Statistics Canada.
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Footnotes

  1. There are decompositions with “weights” that do not sum to one, such as those for the Fisher index by Reinsdorf, Diewert, and Ehemann (2002, sec. 2) and Hallerbach (2005), and for other superlative indexes by Diewert (2002). Balk (2008, equation 4.28) gives the original multiplicative decomposition of the Fisher index, due to Vartia, for which the weights also do not sum to one. Some of these will be revisited in Section 2.↩︎

  2. To see this, note that \(\mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \equiv \left(\mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}) r_{i}\right)^\rho\) when \(\rho = -1\) and \(\varsigma = 1\), or \(\rho = 1\) and \(\varsigma = -1\), and plug this into Equation 2.↩︎

  3. To see this, note that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) \geq w_{i}\) if and only if \[ \mathfrak{E}_{\rho\varsigma}(r_i, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma} \geq \sum_{j=1}^{n} w_{j} \mathfrak{E}_{\rho\varsigma}(r_j, \mathfrak{M}_{\rho} (\mathbf{r}, \mathbf{w}))^{\rho - \varsigma}. \] Since the extended mean is strictly increasing in its first argument, the right-hand side is an increasing function if \(\rho > \varsigma\). This inequality is not satisfied for \(i=1\) and is strictly satisfied for \(i=n\), so \(k\) can then be set to the largest value such that this inequality is not satisfied, and \(l\) can be set to the smallest value such that it is strictly satisfied. If \(\rho < \varsigma\) then the right-hand side is a decreasing function, so the inequality is not satisfied for \(i=n\) and is strictly satisfied for \(i=1\). As before, \(k\) can be set to the largest value such that the inequality is strictly satisfied, and \(l\) can be set to the smallest value such that it is not satisfied.↩︎

  4. When \(n=2\), any other function \(u\) that decomposes the generalized mean must satisfy \((v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma))(r_{1}^\varsigma - r_{2}^\varsigma) \equiv 0\) for \(i=1,2\), or \((v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) - u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma))(\log(r_{1}) - \log(r_{2})) \equiv 0\) when \(\varsigma = 0\), so that \(v_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma) \equiv u_{i}(\mathbf{r}, \mathbf{w}; \rho, \varsigma)\) whenever \(r_{1} \neq r_{2}\).↩︎

  5. Take, for example, \(\mathbf{r} = (0.5,1,1.5)\) and \(\mathbf{w} = (1/6,1/3,1/2)\). The weighted harmonic mean of \(\mathbf{r}\) is 1, which can be computed as an arithmetic mean with weights \((1/3,1/3,1/3)\) or \((1/4,1/2,1/4)\). Hence there are multiple additive decompositions for the harmonic mean.↩︎