author's proof - DARP

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AUTHOR'S PROOF Reference distributions and inequality measurement JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13 Q2 sensitive inequality where, in terms of “distance from maximum inequality,” the 253 change in the distribution is inconclusive. 254 Theoretical distribution Finally let us consider how inequality changed using a con- 255 Q4 tinuous reference distribution F ∗ . The last panel of Table 2 tabulates the results for 256 three F ∗ from the Beta distribution family. Did UK income inequality, interpreted 257 as a distance from a Beta-family reference distribution increase We can see that 258 the values of J α are not statistically different between 1979 and 1988 when the 259 Beta(1,1) (uniform) or Beta(2,5) (unimodal, right skewed) is used as the reference 260 distribution distribution, while they are statistically different when the Beta(2, 2) 261 (unimodal symmetric) is used as the reference distribution. 262 The estimates of the standard GE inequality measures J α,1,1 and of those of J α,k,p 263 and J α in Table 2 provide us with different information about divergence of the 264 empirical distribution from the chosen reference distribution. By varying the values 265 of k and p, one can specify the exact skewness of the reference distribution one would 266 like to measure distance of the empirical distribution from. Likewise, by varying the 267 values of α one can focus on different parts of the income distribution. A large value 268 of α implies a greater weight on parts of the distribution where the observed incomes 269 are vastly different from the corresponding values in the reference distribution. 270 Finally, one can choose specific parametric distributions which correspond to the 271 relevant reference distribution that the researcher is interested in. 272 5 Conclusion 273 The problem of comparing pairs of distributions is a widespread one in distributional 274 analysis. It is often treated on an ad-hoc basis by invoking the concept of norm 275 incomes and an arbitrary inequality index. 276 Our approach to the issue is a natural generalisation of the concept of inequal- 277 ity indices where the implicit reference distribution is the trivial perfect-equality 278 distribution. Its intuitive appeal is supported by the type of axiomatisation that 279 is common in modern approaches to inequality measurement and other welfare 280 criteria. The axiomatisation yields indices that can be interpreted as measures of 281 divergence and are related to the concept of divergence entropy in information 282 theory [10]. Furthermore, they offer a degree of control to the researcher in that 283 the J α indices form a class of measures that can be calibrated to suit the nature of the 284 economic problem under consideration. Members of the class have a distributional 285 interpretation that is close to members of the well-known generalised-entropy class 286 of inequality indices. 287 In effect the user of the J α -index is presented with two key questions: 288 UNCORRECTED PROOF 1. the income discrepancies underlying inequality are with reference to what 289 2. to what kind of discrepancies do you want the measure to be particularly 290 sensitive 291 As our empirical illustration has shown, different responses to these two key ques- 292 tions provide different interpretations from the same set of facts. 293
AUTHOR'S PROOF JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13 F. A. Cowell et al. 294 Appendix: Axiomatic foundation 295 For convenience we work with z :=(z 1 , z 2 , ..., z n ), where each z i is the ordered pair 296 (x i , y i ), i = 1, ..., n and belongs to a set Z , which we will take to be a connected 297 subset of R + × R + . For any z ∈ Z n denote by z (ζ,i) the member of Z n formed 298 by replacing the ith component of z by ζ ∈ Z . The divergence issue focuses on the 299 discrepancies between the x-values and the y-values. To capture this introduce a 300 discrepancy function d : Z → R such that d (z i ) is strictly increasing in |x i − y i |.The 301 solution is in two steps: (1) characterise a weak ordering ≽ on Z n (z ≽ z ′ means “the 302 income pairs in z are at least as close according to ≽ as the income pairs in z ′ ”) (2) 303 use the function representing ≽ to generate the index J. 304 Axiom 1 (Continuity) ≽ is continuous on Z n . 305 Axiom 2 (Monotonicity) If z, z ′ ∈ Z n differ only in their ith component then d (x i , y i ) < d ( 306 x i ′, i) y′ ⇐⇒ z ≻ z ′ . 307 Axiom 3 (Symmetry) For any z, z ′ ∈ Z n such that z ′ is obtained by permuting the 308 components of z: z ∼ z ′ . 309 Axiom 3 implies that we may order the components of z such that x 1 ≤ x 2 ≤ ... ≤ 310 x n and y 1 ≤ y 2 ≤ ... ≤ y n . 311 Axiom 4 (Independence) For z, z ′ ∈ Z n such that: z ∼ z ′ and z i = z i ′ for some i then z (ζ,i) ∼ z ′ (ζ,i) for all ζ ∈ [ ] [ ] 312 z i−1 , z i+1 ∩ z ′ i−1 , z i+1 ′ . 313 Axiom 5 (Zero local discrepancy) Let z, z ′ ∈ Z n be such that, for some i and j, x i = 314 y i , x j = y j , x i ′ = x i + δ, y i ′ = y i + δ, x ′ j = x j − δ, y ′ j = y j − δ and, for all k ̸= i, j, x ′ k = 315 x k , y ′ k = y k.Thenz ∼ z ′ . 316 Axiom 6 (Income scale irrelevance) For any z, z ′ ∈ Z n such that z ∼ z ′ , tz ∼ tz ′ for 317 all t > 0. UNCORRECTED PROOF 318 Axiom 7 (Discrepancy scale irrelevance) Suppose there are z 0 , z ′ 0 ∈ Z n such that z 0 ∼ z ′ 0 . Then for all t > 0 and z, z′ such that d (z) = td (z 0 ) and d ( z ′) = td ( 319 z 0) ′ : z ∼ z ′ . 320 321 322 323 324 325 326 327 Lemma 1 Given Axioms 1 to 5 (a) ≽ is representable by ∑ n i=1 φ i (z i ) , ∀z ∈ Z n where, for each i, φ i : Z → R is a continuous function that is strictly decreasing in |x i − y i | and (b) φ i (x, x) = a i + b i x. Proof Axioms 1 to 5 imply that ≽ can be represented by a continuous function : Z n → R that is increasing in |x i − y i |, i = 1, ..., n. Part (a) of the result follows from Axiom 4 and Theorem 5.3 of [23]. Now take z ( ′ and z as specified in Axiom 5: z ∼ z ′ if and only if φ i (x i + δ, x i + δ) − φ i (x i , x i ) − φ j x j + δ, x j + δ ) ( + φ j x j + δ, x j + δ ) = 0 which implies φ i (x i + δ, x i + δ) − φ i (x i , x i ) = f (δ) 328 for any x i and δ. The solution to this Pexider equation implies (b).
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