Contributions to the measurement of relative p-bipolarisation Marek Kosny, Wroclaw University of Economics Gaston Yalonetzky, University of Leeds IEA World Congress, Mexico, June 22, 2017
Background Bipolarisation indices have gained traction as methods to measure the growth or disappearance of middle-classes, since the foundational work of Foster and Wolfson (2010; based on a 1992 working paper) and Wolfson (1994). Bipolarisation measurement (Foster and Wolfson 2010) requires partitioning distributions into two groups, and then distinguishing between transfers within one group or between groups). Like in inequality measurement , a progressive transfer across the dividing percentile reduces the spread of mean attainment between the two groups, thereby reducing bipolarisation. Unlike in inequality measurement , a progressive transfer within any one group increases clustering, in the limit leading to bimodality; hence these progressive transfers increase bipolarisation. Contributions to the measurement of relative p-bipolarisation
Motivation Relative bipolarisation indices can be classified into median- dependent , i.e. whenever their formula includes the median, or median-independent otherwise. Examples of the former include the Foster-Wolfson index, but also the class 𝑄 𝑂 (𝑦) by 2 Wang and Tsui (2000). Examples of the latter include proposals by Wang and Tsui (2000), e.g. the class 𝑄 𝑂 (𝑦) , and by Rodriguez 1 and Salas (2003). Median-dependent indices violate the key transfer axioms of bipolarisation (Yalonetzky 2017b), unless the median remains unaltered by transfers, which is not guaranteed in practice. Hence we can effectively rely only on median-independent indices. To date, all median-independent indices of relative bipolarisation proposed in the literature are rank-dependent what results in the axiom trade-offs. Contributions to the measurement of relative p-bipolarisation
Main contribution Main methodological contribution is the introduction of the first class of indices of relative bipolarisation which are both median (percentile)-independent and partially rank- independent. These indices are based on normalised differences of generalised means. We derive a partial ordering for relative bipolarisation measurement , a framework which relies on two benchmarks of extreme bipolarisation (i.e. minimum and maximum). We seek to popularise the idea that relative bipolarisation assessments can be performed for any partition of distributions into two groups (i.e. not just identical halves using the median). We compare bipolarisation level for the US and Germany . Based on PSID and SOEP data, income bipolarisation proves to be higher among individuals in the US, but higher among households in Germany. Contributions to the measurement of relative p-bipolarisation
Notation (1) Let 𝑧 𝑗 ≥ 0 denote the income of individual i . 𝑂 is the income distribution with mean 𝜈 𝑍 > 0 , and size 𝑂 ≥ 4 𝑍 ∈ ℝ + (individuals are ranked in non-decreasing order: 𝑧 1 ≤ ⋯ ≤ 𝑧 𝑂−1 ≤ 𝑧 𝑂 ). 𝑞 ∈ [0,1] ⊂ ℝ + denotes a percentile and 𝑧(𝑞) – quantile functions (for instance, 𝑧(0.5) is the median of 𝑍 ). denotes the bottom part of the 𝑍 = 𝑍 𝑞 = {𝑧 𝑗 ∈ 𝑍: 𝑧 𝑗 ≤ 𝑍(𝑞)} distribution 𝑍 , as well as 𝑍 = 𝑍 𝑞 = {𝑧 𝑗 ∈ 𝑍: 𝑧 𝑗 > 𝑍(𝑞)} the top part . Transfers involving incomes 𝑧 𝑗 < 𝑧 𝑘 and a positive amount 𝜀 > 0 such that 𝑧 𝑗 + 𝜀 ≤ 𝑧 𝑘 − 𝜀 will be referred to as rank-preserving Pigou- Dalton progressive transfer , analogous transfers in the opposite direction will be called regressive transfer . Contributions to the measurement of relative p-bipolarisation
Notation (2) Minimum relative bipolarisation benchmark (set 𝓕 ) is made of distributions exhibiting equal non-negative incomes: 𝑂 : 𝑧 1 = 𝑧 2 = ⋯ = 𝑧 𝑂 = 𝑧 > 0 . ℰ = 𝑍 ∈ ℝ ++ Maximum relative bipolarisation benchmark (set 𝓒 𝒒 ) is made of a bottom 𝑞 of null incomes and a top 1 − 𝑞 of equal incomes: 𝑂 : 𝑧 1 = ⋯ = 𝑧 𝑞𝑂 = 0 ∧ 𝑧 𝑞𝑂+1 = ⋯ = 𝑧 𝑂 = 𝑧 > 0 . ℬ 𝑞 = 𝑍 ∈ ℝ + Generalised means of the bottom and top parts: 1 𝑂𝑞 𝑧 𝑗 𝛽 , ∀𝛽 ≠ 0 1 𝛽 𝑂𝑞 σ 𝑗=1 𝜈 𝑍; 𝑞, 𝛽 ≡ 1 𝛾 ∀𝛾 ≠ 0 1 𝛾 𝑂 𝑂 1−𝑞 σ 𝑗=𝑂𝑞+1 𝜈 𝑍; 𝑞, 𝛾 ≡ 𝑧 𝑗 Contributions to the measurement of relative p-bipolarisation
Bipolarisation properties (1) Axiom 1: Symmetry (SY) 𝐶 𝑌; 𝑞 = 𝐶 𝑍; 𝑞 if 𝑌 = 𝑊𝑍 where 𝑊 is an 𝑂 × 𝑂 permutation matrix Axiom 2: Population principle (PP) 𝜇𝑂 is obtained from 𝑍 ∈ ℝ + 𝑂 through an 𝐶 𝑌; 𝑞 = 𝐶 𝑍; 𝑞 if 𝑌 ∈ ℝ + equal replication of each individual income, 𝜇 times. Axiom 3: Scale invariance (SC) 𝐶 𝑌; 𝑞 = 𝐶 𝑍; 𝑞 if 𝑌 = 𝜄𝑍 and 𝜄 > 0 . Contributions to the measurement of relative p-bipolarisation
Bipolarisation properties (2) Axiom 4: Spread-decreasing Pigou-Dalton transfers (SD) If 𝑌 is obtained from 𝑍 through PD transfers across the 𝑧(𝑞) quantile , which do not make any affected income switch the part of the distribution (bottom or top) to which they initially belonged, then 𝐶 𝑌; 𝑞 < 𝐶(𝑍; 𝑞) . Axiom 4a: Spread-increasing regressive transfers (SR) If 𝑌 is obtained from 𝑍 through regressive transfers across the 𝑧(𝑞) quantile then 𝐶 𝑌; 𝑞 > 𝐶(𝑍; 𝑞) . Contributions to the measurement of relative p-bipolarisation
Bipolarisation properties (3) Axiom 5: Clustering-increasing Pigou-Dalton transfers (CI) If 𝑌 is obtained from 𝑍 through PD transfers on one side of the 𝑧(𝑞) quantile then 𝐶 𝑌; 𝑞 > 𝐶(𝑍; 𝑞) . Axiom 5a: Clustering-decreasing regressive transfers (CR) If 𝑌 is obtained from 𝑍 through regressive transfers on one side of the 𝑧(𝑞) quantile, which do not make any affected income switch the part of the distribution (bottom or top) to which they initially belonged, then 𝐶 𝑌; 𝑞 < 𝐶 𝑍; 𝑞 . Contributions to the measurement of relative p-bipolarisation
Bipolarisation properties (4) Axiom 6: Normalisation (N) (a) 𝐶(𝑍; 𝑞) > 𝐶(𝑌; 𝑞) = 0 if and only if 𝑌 ∈ ℰ and 𝑍 ∉ ℰ , and (b) 𝐶 𝑍; 𝑞 < 𝐶(𝑌; 𝑞) = 1 if and only if 𝑌 ∈ ℬ 𝑞 and 𝑍 ∉ ℬ 𝑞 . Axiom 7: Independence (IN) (a) 𝐶 𝑍, 𝑍; 𝑞 ≥ 𝐶 𝑌, 𝑍; 𝑞 ↔ 𝐶 𝑍, 𝑌; 𝑞 ≥ 𝐶 𝑌, 𝑌; 𝑞 and (b) 𝐶 𝑍, 𝑍; 𝑞 ≥ 𝐶 𝑍, 𝑌; 𝑞 ↔ 𝐶 𝑌, 𝑍; 𝑞 ≥ 𝐶 𝑌, 𝑌; 𝑞 . Axiom 8: Within-group consistency (WC) 𝐶 𝑍; 𝑞 > (<)𝐶 𝑌; 𝑞 if 𝑍 is obtained from 𝑌 by increasing (decreasing) some values in 𝑌 , and/or if 𝑍 is obtained from 𝑌 by decreasing (increasing) some values in 𝑌 . Contributions to the measurement of relative p-bipolarisation
Bipolarisation properties (5) Axiom 9: Standardisation (ST) (a) 𝐶 𝑍, 𝑍; 𝑞 = 𝐶 𝜈 𝑍; 𝑞, 1 , 𝜈 𝑍; 𝑞, 1 ; 𝑞 whenever 𝑧 𝑗 = 𝜈 𝑍; 𝑞, 1 ∀𝑧 𝑗 ≤ 𝑍(𝑞) and 𝑧 𝑘 = 𝜈 𝑍; 𝑞, 1 ∀𝑧 𝑘 > 𝑍(𝑞) ; (b) 𝐶 𝑍, 𝑍; 𝑞 = 𝐶 𝜈 𝑍; 𝑞, 1 , 𝑍; 𝑞 whenever 𝑧 𝑗 = 𝜈 𝑍; 𝑞, 1 ∀𝑧 𝑗 ≤ 𝑍(𝑞) ; (c) 𝐶 𝑍, 𝑍; 𝑞 = 𝐶 𝑍, 𝜈 𝑍; 𝑞, 1 ; 𝑞 whenever 𝑧 𝑘 = 𝜈 𝑍; 𝑞, 1 ∀𝑧 𝑘 > 𝑍(𝑞) . Axiom 10: Linear homogeneity (LH) = Φ[𝜇 1 𝜚 𝑍 , 𝜇 2 𝜚 𝑍 ] . Φ 𝜚 𝜇 1 𝑍 , 𝜚 𝜇 2 𝑍 Contributions to the measurement of relative p-bipolarisation
Index of relative p -bipolarisation Class of indices of relative bipolarisation : 𝑪 𝒁; 𝒒, 𝜷, 𝜸 ≡ 𝟐 − 𝒒 𝝂 𝒁; 𝒒, 𝜸 − 𝝂 𝒁; 𝒒, 𝜷 𝝂 𝒁 Proposed class of indices utilizes new concept of relative p -bipolarisation. Class 𝐶(𝑍; 𝑞, 𝛽, 𝛾) is median-independant and partially rank-dependent in the sense that its computation requires splitting the population into a bottom and a top part. Proposition 1: 𝐶(𝑍; 𝑞, 𝛽, 𝛾) fulfils axioms SR (spread-increasing regressive transfers), CI (clustering-increasing Pigou-Dalton transfers), and N (normalisation) if and only if 𝛽 > 1 > 𝛾 . Moreover, 𝐶 𝑍; 𝑞, 𝛽, 𝛾 fulfils SY (symmetry), PP (population principle) and SC (scale invariance) for 𝛽, 𝛾 > 0 . Contributions to the measurement of relative p-bipolarisation
Axiomatic characterization Theorem 1: A bipolarisation index fulfils axioms SY (symmetry), PP (population principle), SC (scale invariance), SR (spread- increasing regressive transfers), CI (clustering-increasing Pigou- Dalton transfers), N (normalisation), IN (independence), WC (within-group consistency), ST (standardisation), and LH (linear homogeneity), if and only if it is a member of the class 𝐶(𝑍; 𝑞, 𝛽, 𝛾) with 𝛽 > 1 > 𝛾 . Existing indices of relative bipolarisation in the literature are absent from the characterisation. It is not due to the imposition of some of the more `technical’ axioms, like ST or LH: existing bipolarisation indices do not satisfy all the `core’ axioms of relative bipolarisation (e.g. PP, SC, SR, CI, N) simultaneously. Contributions to the measurement of relative p-bipolarisation
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