Unit-Consistent Decomposable Inequality Measures: Some Extensions

Transcription

1 WORKING PAPER SERIES NOVEMBER 2005 WORKING PAPER No Unit-Consistent Decomposable Inequality Measures: Some Extensions by Buhong Zheng DEPARTMENT OF ECONOMICS UNIVERSITY OF COLORADO AT DENVER AND HEALTH SCIENCES CENTER DENVER, CO

2 Unit-Consistent Decomposable Inequality Measures: Some Extensions Buhong Zheng Department of Economics University of Colorado at Denver and HSC Denver, CO, USA November 2005 Abstract: This note provides three extensions to Zheng (2005): it replaces the differentiability assumption with continuity; it generalizes the main results to the conditions that are weaker than decomposability namely, aggregability and subgroup consistency; and it shows that a unitconsistent social welfare function defines a relative inequality measure. 1

3 I. Replacing Differentiability by Continuity In this part, we characterize the entire class of unit-consistent decomposable inequality measures without using the differentiability assumption. To do so, we need to first restate a result of Shorrocks (1984) which can be used to further specify the structure of a decomposable inequality measure beyond (2.1). In his 1984 paper, Shorrocks assumed a weaker aggregation condition requiring that the overall inequality level is some general function of the subgroup means, population sizes, and inequality values. To distinguish from the decomposable inequality measures that we are considering, inequality measures satisfying this weaker aggregation condition are referred to as being aggregative. Clearly, a decomposable inequality measure is also aggregative. Proposition 2 (Shorrocks, 1984, Theorem 4): An aggregative inequality measure I(x) satisfies symmetry, normalization and strict Schur-concavity ifandonlyifthere exist functions F ( ) and φ( ) such that F [I(x),µ(x),] = 1 X [φ(x i ) φ(µ(x))] (3.1) where φ( ) is continuous and strictly convex; and where F (I,µ,n) is continuous in I and µ, strictly increasing in I, withf (0,µ,n)=0. In addition, I is replication invariant if and only if F is independent of n. The following result documents the implication of unit-consistency on decomposable inequality measures (2.1). Proposition 3: IfI(x) satisfies symmetry, normalization, decomposability, strict Schur-concavity and unit-consistency, then I(θx) =θ τ I(x) for all θ R ++ and some constant τ R. Proof: Following Shorrocks (1984), for any distribution x D, letω(x) = (µ(x),) be a parameter vector for the distribution x. Income distributions drawn from D with a common parameter vector ω constitute the set S(ω) {x D ω(x) =ω}. For each feasible vector ω, S(ω) is a connected, open subset of D containing more than one element. Hence, by symmetry, normalization and strict Schur-concavity, I[S(ω)] = {I(x) x S(ω)} =[0, ξ(ω)) where ξ(ω) is strictly positive and may be finite and infinite. Define Ω = {ω(x) x D}. For each ω =(µ, n) Ω, letx and y be any two distributions drawn from S(ω). By definition, µ(x) =µ(y) =µ and =n(y) =n. 2

4 Now consider a new distribution z which is the pooling of x and y, i.e., z =(x, y). Employing the decomposability condition (2.1), we have I(z) =w 1 (µ, n)i(x)+w 2 (µ, n)i(y), (3.2) where w 1 (µ, n) is the weight for distribution x and w 2 (µ, n) is the weight for distribution y. Notethatherewehavesimplified the notation of the weights since µ(z) =µ and n(z) =2n. Also in (3.2), the between-group inequality term vanishes because the smoothed distribution for the term is (µ, µ,..., µ) and, by normalization, I(µ, µ,..., µ) =0. Now multiplying all incomes in x, y and z by a factor θ R ++,(2.2)becomes I(θz) =w 1 (θµ, n)i(θx)+w 2 (θµ, n)i(θy). (3.3) Recalling the implication of unit-consistency (2.6), there exists a continuous function f(, ) such that f[θ,i(z)] = w 1 (θµ, n)f[θ,i(x)] + w 2 (θµ, n)f[θ,i(y)]. (3.4) Substituting (3.2) into (3.4), we further have f[θ,w 1 (µ, n)i(x)+w 2 (µ, n)i(y)] (3.5) = w 1 (θµ, n)f[θ,i(x)] + w 2 (θµ, n)f[θ,i(y)]. Denoting f( ) =f(θ, ), I(x) =k, I(y) =l, w j = w j (µ, n) and w j = w j (θµ, n) for j =1, 2, (3.5) becomes f(w 1 k + w 2 l)= w 1 f(k)+ w2 f(l) (3.6) for all k, l [0, ξ(ω)). This functional equation is a special case considered in Aczél (1966, p.66) and through a series of transformation, as Aczél shows, the solution also satisfies the standard Cauchy equation 1 The nontrivial solution to the Cauchy equation is f(k + l) = f(k)+ f(l). (3.7) f(k) =ak for some constant a 6= 0. (3.8) Substituting (3.8) into (3.6), we obtain aw 1 k + aw 2 l = a w 1 k + a w 2 l which entails w 1 = w 1 and w 2 = w 2. Replacing in (3.8) f(k) with f(θ,k) and k with I(x), we obtain I(θx) =f[θ,i(x)] = a(θ)i(x) (3.9) 1 Aczél s equation is f(ax + by + c) =Af(x)+Bf(y)+C. Our equation here is a special case with c = C =0. Also due to the normalization assumption, f(0) = 0, Aczél s equation (4) of page 66 is further simplified to our equation (3.7). 3

5 for all x D and some positive function a( ). The proof is completed by further noting that for any two factors θ, ϑ R ++ and from (3.8) we have: a(θϑ) =a(θ)a(ϑ) and the solution to this standard Cauchy equation is a(θ) =θ τ for some constant τ. Thus, a decomposable inequality measure satisfying (2.1) is unit-consistent if and only if it is a homogenous function of degree τ in incomes. Clearly, for the generalized entropy class, τ =0; and for the variance, τ =2. Proposition 4: I(x) satisfies symmetry, normalization, replication-invariance, continuity, decomposability, strict Schur-concavity, andunit-consistency if and only if it is a positive multiple of the form or or I(x) = 1 1 X [x α α(α 1) nµ(x) β i µ(x)α ] with α 6= 0, 1, I(x) = 1 X nµ(x) β 1 I(x) = x i µ(x) ln 1 X ln µ(x) nµ(x) β x i x i µ(x), (3.10) for α, β R. Proof: Since a decomposable inequality measure is also aggregative, then if I(x) satisfies symmetry, normalization, decomposability, andstrict Schur-concavity Proposition 2 implies that there exist functions F ( ) and φ( ) such that equation (3.1) holds. If I(x) also satisfies replication invariance, then we further have F [I(x),µ(x)] = 1 X [φ(x i ) φ(µ(x))]. (3.1a) Now consider the same set of three distributions x, y and z =(x, y) as they were considered in the proof of Proposition 3. Applying (3.1a) to the same set of distributions, we have F [w 1 (µ, n)i(x)+w 2 (µ, n)i(y),µ]=0.5f [I(x),µ]+0.5F [I(y),µ]. (3.11) 4

6 Here we have used the conditions that µ(x) =µ(y) =µ(z) =µ and =n(y) =n. Denote F ( ) =F (,µ), I(x) =k, I(y) =l, andw j = w j (µ, n) for j =1, 2, (3.10) becomes F (w 1 k + w 2 l)=0.5 F (k)+0.5 F (l). (3.12) for all k, l [0, ξ(ω)). Resorting to Aczél (1966, p.66) once again, we know that the solution to (3.12) also satisfies and whose nontrivial solution is F (k + l) = F (k)+ F (l) F (k) =λk for some constant λ 6= 0. (3.13) Replacing in (3.13) F ( ) with F (,µ) and k with I(x) and using (3.1), we have I(x) = 1 X [φ(x i ) φ(µ(x))] (3.14) λ[µ(x)] for some continuous function λ( ). Proposition 3 has shown that I(x) must be a homogenous function if the axiom of unit-consistency is to be satisfied. Suppose I(θx) =θ τ I(x). Define J(x) = [µ(x)] τ I(x). It is easy to see that J(x) is also decomposable (and, of course, aggregative) and is homogenous of degree zero. That is, J(x) satisfies the condition of scale-invariance. As such, Theorem 5 of Shorrocks (1984) can be applied to J(x): J(x) is a positive multiple of the generalized entropy measure (2.2). It follows immediately that I(x) must be a positive multiple of the measures in (3.10) with β = α τ. This proves the necessity of the proposition. The sufficiency of the proposition is obvious. II. Replacing Decomposability by Subgroup Consistency Shorrocks (1984, 1988) proposed two consistency conditions and characterized the generalized entropy measures in these broader settings. The aggregability axiom of Shorrocks (1984) requires that the overall inequality value be obtained from information concerning the size, arithmetic mean, and inequality value of each population subgroup. The subgroup consistency axiom of Shorrocks (1988) requires that the overall inequality level increase when the inequality level of each population subgroup increaseswhilethesizeandthemeanofeachsubgroupareheldfixed. With either condition replacing decomposability, Shorrocks showed that the satisfactory inequality measure must be of the form J(x) =g[i(x)] (4.1) for some continuous and strictly increasing function g( ) and some decomposable inequality measure I(x). 5

7 Applying unit-consistency to (4.1), we obtain for all θ R ++ there exists a continuous function which is also strictly increasing in the second argument such that J(θx) =f[θ,j(x)] (4.2) for all x D. Now for each ω =(µ, n) Ω, choose two distributions x and y from S(ω) {x D ω(x) =ω}. Letz =(x, y), then (4.2) implies g[w 1 (θµ, n)i(θx)+w 2 (θµ, n)i(θy)] = f{θ,g[w 1 (µ, n)i(x)+w 2 (µ, n)i(y)]}. (4.3) Further denoting g = g 1 f(θ, ) g and using other notations from Proposition 3, we obtain g(w 1 k + w 2 l)= w 1 g(k)+ w 2 g(l) (4.4) for all k, l [0, ξ(ω)). The solution to this functional equation includes or g(k) =ak for some constant a, g[i(θx)] = g[a(θ)i(x)] for some continuous function a(θ). Because g( ) is a strictly increasing function, it follows that I(θx) =a(θ)i(x). This is equation (3.8) and, hence, the remainder of the characterization is the same as that in Propositions 3 and 4. Thus, if decomposability is replaced with either aggregability or subgroup-consistency, the corresponding (aggregative or subgroupconsistent) unit-consistent inequality measures would be g[i(x)] with I(x) being the unit-consistent decomposable class characterized in Proposition 4. III. Unit-Consistent Social Welfare Function Defines a Relative Inequality Measure Suppose a social welfare function w(x) is continuous and monotonically increasing in x. If w(x) is unit consistent, then by Proposition 1 of Zheng (2005), there exists a continuous function f such that w(θx) =f[θ,w(x)]. (5.1) The inequality measure we consider is the Kolm-Atkinson-Sen type: I(x) =1 x x (5.2) where x is EDEI (equally distributed equivalent income) and x is the mean income. 6

8 Clearly, for a homogenous social welfare function I(x) is relative. In (5.1), for any given θ > 0, sincew(x) =w( xu ) by the definition of EDEI. It follows that f[θ,w(x)] = f[θ,w( xu )] = w[θ xu ] which equals w( f θx) again by the definition of EDEI. Thus, since w(x) is monotonically increasing in x, wehave f θx = θ x and θx I(θx) =1 f θ x =1 θx θ x = I(x). References Aczél, J. (1966): Lectures on Functional Equations and Their Applications, Academic Press, New York. Shorrocks, A. (1980): The Class of Additively Decomposable Inequality Measures, Econometrica 48, Shorrocks, A. (1984): Inequality Decomposition by Population Subgroups, Econometrica 52, Shorrocks, A. (1988): Aggregation Issues in Inequality Measurement, in W. Eichhorn (ed), Measurement in Economics, Heidelberg: Physica-Verlag, Zheng, B. (2005): Unit-Consistent Decomposable Inequality Measures, Economica (forthcoming). 7