Shapley Inequality Decomposition by Factor Components: Some Methodological Issues

Abstract

The purpose of the paper is to focus on three methodological issues regarding ''the Shapley inequality decomposition'' by factor components which cannot be solved from a theoretical point of view. First, should we use either zero income decomposition, or equalized income decomposition? Second, should we favour the Nested-Shapley or the Owen decomposition? Third, can we structure the set of income components in some bliss tree? The empirical evidence displayed by results of several variants of the Shapley decomposition using the LIS database regarding the British and American income distributions help us to propose an answer to these questions: confine the analysis on gross income decomposition and select the equalized Nested Shapley method. In the absence of an ideal tree we propose three trees among which a choice has to be made. This document was completed when Mercedes Sastre was visiting THEMA. The support of the European Commission through the Contract n ERBFMRXCT980248 is grateful acknowledged. We thank LIS for computing assistance and Frederic Chantreuil and Steve Jenkins for helpful comments. This article is forthcoming in Living Standards, Inequality and Taxation. P. Moyes, C.Seidl et A.Shorrocks eds, Kluwer 2000.

Shapley-value decompositions of changes in wage distributions: a note

“…A non-exhaustive list of recent applications include Israeli[9], Mussard and Terraza[14], and Sastre and Trannoy[16].…”

Source decomposition of changes in income inequality: the integral-based approach and its approximation by the chained Shapley-value approach

“…also has the property that an evenly distributed income source contributes negatively, which creates the same problem as the zero Shapley in Example 1. Sastre and Trannoy [17] pointed out that the contributions of income sources obtained by the zero Shapley are highly volatile against income source classification, and are substantially inconsistent among level of aggregation, which is also observed in Section 5.…”

“…Row and the 'Female' column equals the sum of the contributions of female earnings and 'other incomes' in the three-source case minus the contribution of the collapsed category in the two-source case. The source contributions obtained by Z-Shapley tend to be strongly affected by the choice of source classification, as pointed out by Sastre et al [17]. E-Shapley generally shows relatively small discrepancies in the contributions among the three choices of source classification.…”

“…Similarly, products of the terms for a pair of ε m (t l ) and ε j (t l ), ε k (t l ) and ε m (t l ), or ε m (t l ) and ε m (t l ) (m = k, m = j) vanish when L → ∞. Thus, only products of the terms at T and T remain and converge to the first term in (17). In k j , among products between the first two terms and the last two terms of Eq.…”

“…However, the zero Shapley decomposition, as well as the Shorrocks decomposition of the Theil index [18], which corresponds to the natural decomposition of SCV, has an adverse effect on decomposition of the change in inequality in some cases. High volatility against income source classification, as pointed out by Sastre and Trannoy [17], is another drawback. Thus, the existing decomposition methods are less than fully satisfactory.…”