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The class of absolute decomposable inequality measures
Economics Letters 109 (2010) 154–156
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
The class of absolute decomposable inequality measures Kristof Bosmans a,⁎, Frank A. Cowell b a b
Department of Economics, Maastricht University, Tongersestraat 53, 6211 LM Maastricht, The Netherlands STICERD, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 25 August 2010 Accepted 15 September 2010 Available online 24 September 2010
a b s t r a c t We provide a parsimonious axiomatisation of the complete class of absolute inequality indices. Our approach uses only a weak form of decomposability and does not require a priori that the measures be differentiable. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D63 H20 H21 Keywords: Inequality measures Decomposability Translation invariance
1. Introduction Absolute inequality indices have the property that, for any income distribution, if any given number of dollars is added to (or subtracted from) every income, inequality remains unchanged. This means that such indices have the property that they are defined for negative incomes: this is particularly important in empirical applications where one is interested in applying inequality indices to the distribution of wealth (net worth may often be substantially negative) or to specific components of income (for example business income). Decomposability of inequality indices requires that, for an arbitrary subgroup of the population, if inequality in the subgroup increases then, ceteris paribus, inequality overall increases. Previous descriptions and characterizations of the class of indices that has these properties have invoked additional assumptions about structure — see Section 4 below. Our approach (in Sections 2 and 3) is deliberately minimalist in that it uses just the axioms required to define (a) an inequality measure (b) the “absolutist” property and (c) the population structure. 2. Setting The income of individual i is a real number xi, and the income distribution for a population of n individuals is a vector x := ðx1 ; x2 ; …; xn Þ in ℝn. The set of income distributions for m or more ⁎ Corresponding author. Tel.: +31 43 3883942; fax: +31 43 3884878. E-mail addresses: [email protected] (K. Bosmans), [email protected] (F.A. Cowell). 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.09.007
n individuals is Xm : = ∪∞ n = m ℝ . For an income distribution x, we write nðxÞ for the dimension and μ ðxÞ for the mean. A vector of dimension n of which all components are equal to 1 is denoted by 1n. An inequality measure is a continuous function I : X1 → ℝ with the property that IðxÞ = 0 if x is an equal income distribution. The higher the value of the function I, the higher income inequality. To give meaning to I it is standard to assume the following two properties:
Anonymity. For every x ∈X1 , IðxÞ = Iðx′Þ if x′ is obtained from x by rearrangement of components. Transfer principle. For every x ∈ X2 and any positive real number δ, we have that if xi b xi + δ ≤ xj − δ b xj, then I(x1, …, xi, …, xj, …, xn) N I(x1, …, xi + δ, …, xj − δ, …, xn). The essential property for an absolute inequality measure is this: Translation invariance. For every x ∈X1 and any real number δ, IðxÞ = Iðx + δ1n Þ:
ð1Þ
Finally, the following two axioms describe the relationship between population structure and inequality: Decomposability. There exists a function A such that, for all x; y ∈X2 , Iðx; yÞ = AðIðxÞ; I ðyÞ; μ ðxÞ; μ ðyÞ; nðxÞ; nðyÞÞ; where A is continuous and strictly increasing in its first two arguments. Replication invariance. For every x ∈ X1 , Iðx; xÞ = IðxÞ.
K. Bosmans, F.A. Cowell / Economics Letters 109 (2010) 154–156
Because c(λθ) = c(λ)c(θ) for all positive λ and θ, we have c(θ) = θc (Aczél, 2006, Theorem 4, p. 144). Consequently
3. Result Our result characterizes the class of inequality measures satisfying the five properties listed in Section 2. Theorem. An inequality measure I satisfies anonymity, the transfer principle, replication invariance, decomposability and translation invariance if and only if there exists a real number c and a continuous and strictly increasing function f : ℝ → ℝ, with f(0) = 0, such that, for all x ∈ X1 ,
f ðIðxÞÞ =
155
8 o > 1 nðxÞ n c½xi −μ ðxÞ > > e −1 if c≠0; > < nðxÞ i∑ =1 > > 1 nðxÞ > 2 > ∑ ½x −μ ðxÞ : nðxÞ i = 1 i
ð2Þ
cδ 0 = Fˆ ðIðx + δ12 Þ; μ ðxÞ + δÞ−e Fˆ ðIðxÞ; μ ðxÞÞ;
u+v +δ 0 = ϕðu + δÞ + ϕðv + δÞ−2ϕ 2 u+v cδ −e ϕðuÞ + ϕðvÞ−2ϕ : 2
cδ
ψðu; δÞ := ϕðu + δÞ−e ϕðuÞ;
ð7Þ
ð4Þ
u+v ;δ : 0 = ψðu; δÞ + ψðv; δÞ−2ψ 2 This is a Pexider equation with solution (Aczél, 2006, p. 142) ψðu; δÞ = aðδÞu + bðδÞ:
ð8Þ
From Eq. (7), c½μ ψðu; μ + δÞ = ϕðu + μ + δÞ−e
Define λ := eδ and θk := eμk. Then λθk = eμk + δ, and Eqs. (4) and (1) imply
J ðGðI1 ; θ1 Þ + GðI2 ; θ2 Þ; θ1 ; θ2 Þ = J ðGðI1 ; λθ1 Þ + GðI2 ; λθ2 Þ; λθ1 ; λθ2 Þ: From Lemma 1 in Shorrocks (1984): GðI1 ; λθ1 Þ = cGðI1 ; θ1 Þ; GðI2 ; λθ2 Þ = cGðI2 ; θ2 Þ;
independent of I and θ. Therefore GðI; λθÞ = cðλÞGðI; θÞ = cðλÞcðθÞGðI; 1Þ = cðλθÞGðI; 1Þ:
c½μ + δ
ð9Þ
ϕðuÞ
cδ
= ψðu + μ; δÞ + e ψðu; μ Þ: Substitute from Eq. (8) into Eq. (9) to get cδ
aðμ + δÞu + bðμ + δÞ = aðδÞðu + μ Þ + bðδÞ + e ½aðμ Þu + bðμ Þ; which implies cδ
cμ
aðμ + δÞ = aðδÞ + e aðμ Þ = aðμ Þ + e aðδÞ; cδ
ð10Þ cμ
bðμ + δÞ = aðδÞμ + bðδÞ + e bðμ Þ = aðμ Þδ + bðμ Þ + e bðδÞ:
ð11Þ
There are two cases to consider. Case 1. (c ≠ 0). In this case Eq. (10) gives aðδÞ ecμ −1 = aðμ Þ ecδ −1 ; aðμ Þ =
GðI; λθÞ = cðλÞ; GðI; θÞ
ϕðuÞ
= ψðu + μ; δÞ + e ϕðu + μ Þ−e
from which it follows that c=
+ δ
cδ
Fˆ ðH ðz; μ1 ; μ2 Þ; μ12 Þ = z + 2ϕðμ1 Þ + 2ϕðμ2 Þ−4ϕðμ12 Þ:
or, with appropriate definitions of G from Fˆ and J from H,
ð6Þ
If we define
where H is a function implicitly defined by
H Fˆ ðI1 ; lnðθ1 ÞÞ + Fˆ ðI2 ; lnðθ2 ÞÞ; lnðθ1 Þ; lnðθ2 Þ = H Fˆ ðI1 ; lnðλθ1 ÞÞ + Fˆ ðI2 ; lnðλθ2 ÞÞ; lnðλθ1 Þ; lnðλθ2 Þ ;
ð5Þ
i=1
then Eq. (6) becomes ð3Þ
where ϕ is continuous and strictly convex, and where F is continuous in I and μ, strictly increasing in I, with F(0, μ) = 0. Define Fˆ ðI; μ Þ : = nF ðI; μ Þ. Choose any x1 and x2 such that nðx1 Þ = nðx2 Þ = 2, and write x12 := ðx1 ; x2 Þ, Ik := Iðxk Þ and μk := μ ðxk Þ. We may write Iðx12 Þ = H Fˆ ðI1 ; μ1 Þ + Fˆ ðI2 ; μ2 Þ; μ1 ; μ2 ;
n
−μc ∑ ½ϕðxi Þ−ϕðμ Þ: Fˆ ðIðxÞ; 0Þ = e
and writing x := ðu; vÞ this becomes
if c = 0:
1 nðxÞ ∑ ½ϕðxi Þ−ϕðμ ðxÞÞ; nðxÞ i = 1
and Eq. (3) yields
In the case of two individuals,
Proof. The inequality measures given in Eq. (2) satisfy anonymity, the transfer principle, replication invariance, decomposability and translation invariance. We focus on the reverse implication. If I satisfies anonymity, the transfer principle, replication invariance and decomposability, then (Shorrocks, 1984, Theorem 4) there exist functions F and ϕ such that F ðI ðxÞ; μ ðxÞÞ =
c
GðI; θÞ = θ GðI; 1Þ; μc Fˆ ðI; μ Þ = e Fˆ ðI; 0Þ;
aðδÞ cμ e −1 ; ecδ −1
so that, for any given value of δ and all μ, aðμ Þ =
k1 cμ e −1 ; k2
where k1 := a(δ) and k2 :=ecδ − 1 can be taken as constants (conditional on the chosen δ and the arbitrary value of c). So the only solution is cμ aðμ Þ = α e −1 :
156
K. Bosmans, F.A. Cowell / Economics Letters 109 (2010) 154–156
Note that, from Eq. (7), we have:
Therefore, from Eq. (11) cμ cδ cδ cμ bðμ + δÞ = μα e −1 + bðδÞ + e bðμ Þ = δα e −1 + bðμ Þ + e bðδÞ;
so that
4. Concluding remarks
δα + bðδÞ cμ e −1 −μα: cδ e −1
Since this must be true for arbitrary δ and c, the solution is cμ bðμ Þ = β e −1 −αμ:
ð12Þ
Case 2. (c = 0). Here Eq. (10) becomes aðμ + δÞ = aðμ Þ + aðδÞ; and the solution to this Cauchy equation is a(μ) = κμ (Aczél, 2006, Theorem 2, pp. 35–36). Therefore, from Eq. (11) bðμ + δÞ = κμδ + bðδÞ + bðμ Þ:
ð16Þ
However the constant ϕ(0) is arbitrary. Setting ϕ(0) = 0 and substituting from Eqs. (15) and (16) into Eq. (5) produces Eq. (2).□
μα ecδ −1 + bðμ Þ ecδ −1 = δα ecμ −1 + bðδÞ ecμ −1 ; bðμ Þ ecδ −1 = ½δα + bðδÞ ecμ −1 −μα ecδ −1 ; bðμ Þ =
cx
ϕðxÞ = ψð0; xÞ + e ϕð0Þ:
ð13Þ
Noting that b(0) = 0 and letting δ = − μ and κ = 2γ, we get
While it is well known that the measures in Eq. (2) belong to the class of absolute decomposable measures (Chakravarty and Tyagarupananda, 1998, 2009), it has not previously been demonstrated that only these measures belong to the class: previous treatments have introduced a priori additional restrictions such as differentiability and have used a stronger decomposability property. Letting c = 0, we obtain the variance and, letting c b 0, the family in Eq. (2) is ordinally equivalent to the Kolm (1976) family of measures. As the parameter c decreases, the corresponding measures are more sensitive to transfers between incomes at the bottom of the income distribution. Note that the Kolm measures are the only members of the class of absolute decomposable measures satisfying transfer sensitivity (Shorrocks and Foster, 1987), which is the requirement that a rich-to-poor transfer combined with a simultaneous poor-torich transfer at a higher income level (such that there is no effect on the variance) decreases inequality. Acknowledgment
2
bðμ Þ + bð−μ Þ = 2γμ ;
We thank an anonymous referee for useful suggestions.
to which the solution is
References
2
bðμ Þ = γμ + pðμ Þ; where p(μ) := Π(μ, − μ) with the property p(μ) = − p(− μ) (Polyanin and Manzhirov, 1998). Plugging this into Eq. (13) gives pðμ + δÞ = pðμ Þ + pðδÞ; to which the solution is p(μ) = ημ. Therefore 2
bðμ Þ = γμ + ημ:
ð14Þ
From Eq. (8), we have ψ(0, x) = b(x); so from Eqs. (12) and (14) we find
ψð0; xÞ =
cx β e −1 −αx if c≠0; if c = 0: γx2 + ηx
ð15Þ
Aczél, J., 2006. Lectures on Functional Equations and Their Applications. Dover Publications, Inc., Mineola. Chakravarty, S.R., Tyagarupananda, S., 1998. The subgroup decomposable absolute indices of inequality. In: Chakravarty, S.R., Coondoo, D., Mukherjee, R. (Eds.), Quantitative Economics: Theory and Practice, Essays in Honor of Professor N. Bhattacharya. Allied Publishers Limited, New Delhi, pp. 247–257. Chakravarty, S.R., Tyagarupananda, S., 2009. The subgroup decomposable intermediate indices of inequality. Spanish Economic Review 11, 63–93. Kolm, S.C., 1976. Unequal inequalities I. Journal of Economic Theory 12, 416–442. Polyanin, A.D., Manzhirov, A.V., 1998. Handbook of Integral Equations: Exact Solutions. Faktorial, Moscow. Shorrocks, A.F., 1984. Inequality decomposition by population subgroups. Econometrica 52, 1369–1385. Shorrocks, A.F., Foster, J.E., 1987. Transfer sensitive inequality measures. Review of Economic Studies 54, 485–497.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
The class of absolute decomposable inequality measures Kristof Bosmans a,⁎, Frank A. Cowell b a b
Department of Economics, Maastricht University, Tongersestraat 53, 6211 LM Maastricht, The Netherlands STICERD, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 25 August 2010 Accepted 15 September 2010 Available online 24 September 2010
a b s t r a c t We provide a parsimonious axiomatisation of the complete class of absolute inequality indices. Our approach uses only a weak form of decomposability and does not require a priori that the measures be differentiable. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D63 H20 H21 Keywords: Inequality measures Decomposability Translation invariance
1. Introduction Absolute inequality indices have the property that, for any income distribution, if any given number of dollars is added to (or subtracted from) every income, inequality remains unchanged. This means that such indices have the property that they are defined for negative incomes: this is particularly important in empirical applications where one is interested in applying inequality indices to the distribution of wealth (net worth may often be substantially negative) or to specific components of income (for example business income). Decomposability of inequality indices requires that, for an arbitrary subgroup of the population, if inequality in the subgroup increases then, ceteris paribus, inequality overall increases. Previous descriptions and characterizations of the class of indices that has these properties have invoked additional assumptions about structure — see Section 4 below. Our approach (in Sections 2 and 3) is deliberately minimalist in that it uses just the axioms required to define (a) an inequality measure (b) the “absolutist” property and (c) the population structure. 2. Setting The income of individual i is a real number xi, and the income distribution for a population of n individuals is a vector x := ðx1 ; x2 ; …; xn Þ in ℝn. The set of income distributions for m or more ⁎ Corresponding author. Tel.: +31 43 3883942; fax: +31 43 3884878. E-mail addresses: [email protected] (K. Bosmans), [email protected] (F.A. Cowell). 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.09.007
n individuals is Xm : = ∪∞ n = m ℝ . For an income distribution x, we write nðxÞ for the dimension and μ ðxÞ for the mean. A vector of dimension n of which all components are equal to 1 is denoted by 1n. An inequality measure is a continuous function I : X1 → ℝ with the property that IðxÞ = 0 if x is an equal income distribution. The higher the value of the function I, the higher income inequality. To give meaning to I it is standard to assume the following two properties:
Anonymity. For every x ∈X1 , IðxÞ = Iðx′Þ if x′ is obtained from x by rearrangement of components. Transfer principle. For every x ∈ X2 and any positive real number δ, we have that if xi b xi + δ ≤ xj − δ b xj, then I(x1, …, xi, …, xj, …, xn) N I(x1, …, xi + δ, …, xj − δ, …, xn). The essential property for an absolute inequality measure is this: Translation invariance. For every x ∈X1 and any real number δ, IðxÞ = Iðx + δ1n Þ:
ð1Þ
Finally, the following two axioms describe the relationship between population structure and inequality: Decomposability. There exists a function A such that, for all x; y ∈X2 , Iðx; yÞ = AðIðxÞ; I ðyÞ; μ ðxÞ; μ ðyÞ; nðxÞ; nðyÞÞ; where A is continuous and strictly increasing in its first two arguments. Replication invariance. For every x ∈ X1 , Iðx; xÞ = IðxÞ.
K. Bosmans, F.A. Cowell / Economics Letters 109 (2010) 154–156
Because c(λθ) = c(λ)c(θ) for all positive λ and θ, we have c(θ) = θc (Aczél, 2006, Theorem 4, p. 144). Consequently
3. Result Our result characterizes the class of inequality measures satisfying the five properties listed in Section 2. Theorem. An inequality measure I satisfies anonymity, the transfer principle, replication invariance, decomposability and translation invariance if and only if there exists a real number c and a continuous and strictly increasing function f : ℝ → ℝ, with f(0) = 0, such that, for all x ∈ X1 ,
f ðIðxÞÞ =
155
8 o > 1 nðxÞ n c½xi −μ ðxÞ > > e −1 if c≠0; > < nðxÞ i∑ =1 > > 1 nðxÞ > 2 > ∑ ½x −μ ðxÞ : nðxÞ i = 1 i
ð2Þ
cδ 0 = Fˆ ðIðx + δ12 Þ; μ ðxÞ + δÞ−e Fˆ ðIðxÞ; μ ðxÞÞ;
u+v +δ 0 = ϕðu + δÞ + ϕðv + δÞ−2ϕ 2 u+v cδ −e ϕðuÞ + ϕðvÞ−2ϕ : 2
cδ
ψðu; δÞ := ϕðu + δÞ−e ϕðuÞ;
ð7Þ
ð4Þ
u+v ;δ : 0 = ψðu; δÞ + ψðv; δÞ−2ψ 2 This is a Pexider equation with solution (Aczél, 2006, p. 142) ψðu; δÞ = aðδÞu + bðδÞ:
ð8Þ
From Eq. (7), c½μ ψðu; μ + δÞ = ϕðu + μ + δÞ−e
Define λ := eδ and θk := eμk. Then λθk = eμk + δ, and Eqs. (4) and (1) imply
J ðGðI1 ; θ1 Þ + GðI2 ; θ2 Þ; θ1 ; θ2 Þ = J ðGðI1 ; λθ1 Þ + GðI2 ; λθ2 Þ; λθ1 ; λθ2 Þ: From Lemma 1 in Shorrocks (1984): GðI1 ; λθ1 Þ = cGðI1 ; θ1 Þ; GðI2 ; λθ2 Þ = cGðI2 ; θ2 Þ;
independent of I and θ. Therefore GðI; λθÞ = cðλÞGðI; θÞ = cðλÞcðθÞGðI; 1Þ = cðλθÞGðI; 1Þ:
c½μ + δ
ð9Þ
ϕðuÞ
cδ
= ψðu + μ; δÞ + e ψðu; μ Þ: Substitute from Eq. (8) into Eq. (9) to get cδ
aðμ + δÞu + bðμ + δÞ = aðδÞðu + μ Þ + bðδÞ + e ½aðμ Þu + bðμ Þ; which implies cδ
cμ
aðμ + δÞ = aðδÞ + e aðμ Þ = aðμ Þ + e aðδÞ; cδ
ð10Þ cμ
bðμ + δÞ = aðδÞμ + bðδÞ + e bðμ Þ = aðμ Þδ + bðμ Þ + e bðδÞ:
ð11Þ
There are two cases to consider. Case 1. (c ≠ 0). In this case Eq. (10) gives aðδÞ ecμ −1 = aðμ Þ ecδ −1 ; aðμ Þ =
GðI; λθÞ = cðλÞ; GðI; θÞ
ϕðuÞ
= ψðu + μ; δÞ + e ϕðu + μ Þ−e
from which it follows that c=
+ δ
cδ
Fˆ ðH ðz; μ1 ; μ2 Þ; μ12 Þ = z + 2ϕðμ1 Þ + 2ϕðμ2 Þ−4ϕðμ12 Þ:
or, with appropriate definitions of G from Fˆ and J from H,
ð6Þ
If we define
where H is a function implicitly defined by
H Fˆ ðI1 ; lnðθ1 ÞÞ + Fˆ ðI2 ; lnðθ2 ÞÞ; lnðθ1 Þ; lnðθ2 Þ = H Fˆ ðI1 ; lnðλθ1 ÞÞ + Fˆ ðI2 ; lnðλθ2 ÞÞ; lnðλθ1 Þ; lnðλθ2 Þ ;
ð5Þ
i=1
then Eq. (6) becomes ð3Þ
where ϕ is continuous and strictly convex, and where F is continuous in I and μ, strictly increasing in I, with F(0, μ) = 0. Define Fˆ ðI; μ Þ : = nF ðI; μ Þ. Choose any x1 and x2 such that nðx1 Þ = nðx2 Þ = 2, and write x12 := ðx1 ; x2 Þ, Ik := Iðxk Þ and μk := μ ðxk Þ. We may write Iðx12 Þ = H Fˆ ðI1 ; μ1 Þ + Fˆ ðI2 ; μ2 Þ; μ1 ; μ2 ;
n
−μc ∑ ½ϕðxi Þ−ϕðμ Þ: Fˆ ðIðxÞ; 0Þ = e
and writing x := ðu; vÞ this becomes
if c = 0:
1 nðxÞ ∑ ½ϕðxi Þ−ϕðμ ðxÞÞ; nðxÞ i = 1
and Eq. (3) yields
In the case of two individuals,
Proof. The inequality measures given in Eq. (2) satisfy anonymity, the transfer principle, replication invariance, decomposability and translation invariance. We focus on the reverse implication. If I satisfies anonymity, the transfer principle, replication invariance and decomposability, then (Shorrocks, 1984, Theorem 4) there exist functions F and ϕ such that F ðI ðxÞ; μ ðxÞÞ =
c
GðI; θÞ = θ GðI; 1Þ; μc Fˆ ðI; μ Þ = e Fˆ ðI; 0Þ;
aðδÞ cμ e −1 ; ecδ −1
so that, for any given value of δ and all μ, aðμ Þ =
k1 cμ e −1 ; k2
where k1 := a(δ) and k2 :=ecδ − 1 can be taken as constants (conditional on the chosen δ and the arbitrary value of c). So the only solution is cμ aðμ Þ = α e −1 :
156
K. Bosmans, F.A. Cowell / Economics Letters 109 (2010) 154–156
Note that, from Eq. (7), we have:
Therefore, from Eq. (11) cμ cδ cδ cμ bðμ + δÞ = μα e −1 + bðδÞ + e bðμ Þ = δα e −1 + bðμ Þ + e bðδÞ;
so that
4. Concluding remarks
δα + bðδÞ cμ e −1 −μα: cδ e −1
Since this must be true for arbitrary δ and c, the solution is cμ bðμ Þ = β e −1 −αμ:
ð12Þ
Case 2. (c = 0). Here Eq. (10) becomes aðμ + δÞ = aðμ Þ + aðδÞ; and the solution to this Cauchy equation is a(μ) = κμ (Aczél, 2006, Theorem 2, pp. 35–36). Therefore, from Eq. (11) bðμ + δÞ = κμδ + bðδÞ + bðμ Þ:
ð16Þ
However the constant ϕ(0) is arbitrary. Setting ϕ(0) = 0 and substituting from Eqs. (15) and (16) into Eq. (5) produces Eq. (2).□
μα ecδ −1 + bðμ Þ ecδ −1 = δα ecμ −1 + bðδÞ ecμ −1 ; bðμ Þ ecδ −1 = ½δα + bðδÞ ecμ −1 −μα ecδ −1 ; bðμ Þ =
cx
ϕðxÞ = ψð0; xÞ + e ϕð0Þ:
ð13Þ
Noting that b(0) = 0 and letting δ = − μ and κ = 2γ, we get
While it is well known that the measures in Eq. (2) belong to the class of absolute decomposable measures (Chakravarty and Tyagarupananda, 1998, 2009), it has not previously been demonstrated that only these measures belong to the class: previous treatments have introduced a priori additional restrictions such as differentiability and have used a stronger decomposability property. Letting c = 0, we obtain the variance and, letting c b 0, the family in Eq. (2) is ordinally equivalent to the Kolm (1976) family of measures. As the parameter c decreases, the corresponding measures are more sensitive to transfers between incomes at the bottom of the income distribution. Note that the Kolm measures are the only members of the class of absolute decomposable measures satisfying transfer sensitivity (Shorrocks and Foster, 1987), which is the requirement that a rich-to-poor transfer combined with a simultaneous poor-torich transfer at a higher income level (such that there is no effect on the variance) decreases inequality. Acknowledgment
2
bðμ Þ + bð−μ Þ = 2γμ ;
We thank an anonymous referee for useful suggestions.
to which the solution is
References
2
bðμ Þ = γμ + pðμ Þ; where p(μ) := Π(μ, − μ) with the property p(μ) = − p(− μ) (Polyanin and Manzhirov, 1998). Plugging this into Eq. (13) gives pðμ + δÞ = pðμ Þ + pðδÞ; to which the solution is p(μ) = ημ. Therefore 2
bðμ Þ = γμ + ημ:
ð14Þ
From Eq. (8), we have ψ(0, x) = b(x); so from Eqs. (12) and (14) we find
ψð0; xÞ =
cx β e −1 −αx if c≠0; if c = 0: γx2 + ηx
ð15Þ
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