We stablish a general relationship between the standard form of the individualistic social-welfare function and the "reduced-form" version that is expressed in terms of inequality and mean income. This shows the relationship between the property of monotonicity and the slope of the equity-efficient trade-off. Particularly simple results are available for a large class of inequality measures that includes the Gini. These results do not require differentiability of the social-welfare function

Amiel, Yoram

Cowell, Frank

English

LSE Research Online

Inequality, Welfare and Monotonicity  by   Yoram Amiel Ruppin Institute, Israel  and  Frank Cowell STICERD, London School of Economics and Political Science                     The Toyota Centre      Suntory and Toyota International Centres for      Economics and Related Disciplines      London School of Economics and Political Science Discussion Paper   Houghton Street No.DARP/29    London WC2A 2AE February 1997   Tel.: 020-7955 6678   This research was partially supported by an ESRC Grant No.R000235725. We are grateful to Leonardo Gasparini and Peter Lambert for comments.  Abstract  We stablish a general relationship between the standard form of the individualistic social-welfare function and the "reduced-form" version that is expressed in terms of inequality and mean income. This shows the relationship between the property of monotonicity and the slope of the equity-efficient trade-off. Particularly simple results are available for a large class of inequality measures that includes the Gini. These results do not require differentiability of the social-welfare function.  Keywords: Inequality, social welfare, monotonicity. JEL Nos.: C13, D63.         © by the authors. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.         Contact address: Contact address: Professor Frank Cowell, STICERD, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK. Email: f.cowell@lse.ac.uk     1 IntroductionInequality indices embody explicitly or implicitly social values on income distrib-ution. So it is common for those who are concerned with comparing distributionalchanges over time or between economies to draw conclusions from comparisonsof estimates of average income and inequality. However the relationship betweenintuitive comparisons in terms of say, mean income and the Gini coe±cient, andformal principles of social welfare is not always made clear. The present paperfocuses upon this relationship by characterising the shape of the equity-e±ciencytrade-o®..This trade-o® is at the heart of much theoretical and applied research. Forexample, it is become standard practice to use the obvious two-way relationshipbetween certain classes of inequality indices and their associated social-welfarefunctions to infer propositions about distributional rankings and the ethical mean-ing of inequality measures.1 In addition there is an interest in constructing social-welfare functions for inequality measures with principally pragmatic appeal: forexample, there have been a number of attempts to formalise a social-welfare func-tion on the basis that the Gini coe±cient is the appropriate index of inequality.2The precise method of doing this formalisation does not concern us here, exceptas illustrations of our approach. We want to examine in a general way what may1See for example Atkinson (1970), Blackorby and Donaldson (1978, 1980), Kolm (1976a,1976b).2See for example Chipman (1974), Katz (1972), Kondor (1975), Lambert (1985), Newbery(1970), Sheshinski (1972).1be said in terms of standard welfare criteria if we use well-known practical indicesas basic tools of distributional analysis.2 The ApproachFirst, some notation and de¯nitions.De¯nition 1 The set of income distributions X is the set of all ordered ¯nite-dimensioned non-negative vectors, excluding the zero vector.In implementing this de¯nition we adopt the convention that for any x 2 X:x1 · x2 · x3 · :::. Also, for any x 2 X, denote the number of persons inthe distribution as n(x) and the arithmetic mean of the distribution as ¹(x).Social welfare and inequality are given, respectively, by functions W : X ! <,I : X ! <. Economic meaning is given to W and I by a number of assumptionsabout their mathematical properties, many of which have come to be acceptedas standard. Foremost among these areDe¯nition 2 The social welfare function is monotonic if, for any x;x0 2 X,x0 > x implies W (x0) > W (x).3De¯nition 3 The social welfare function (resp. an inequality measure) satis¯esthe principle of transfers if, for any x;x0 2 X and ± > 0 such that x0i = xi + ±,3\ x0 > x " means \x0 ¸ x, x0 6= x".2x0j = xj ¡ ±, x0k = xk; k 6= i; j and xi < x0i · x0j < xj W (x0) ¸ W (x) (resp.I(x0) · I(x)).The property of monotonicity is of particular interest in view of its relationshipto the Pareto principle - see Amiel and Cowell (1994b) - although it does notenjoy universal support as a welfare principle - see Amiel and Cowell (1994a),McClelland and Rohrbaugh (1978).De¯nition 4 The class W1 of social welfare functions consists of functions W :X ! < that satisfy the principles of monotonicity and transfers. If, for somex;x0 2 X, it is the case that W (x) > W (x0) for all W 2 W1 we shall say thatdistribution x is W1-preferable to x0.We shall establish results for a fairly broad class of inequality indices. Tointroduce this we need the following two de¯nitions:De¯nition 5 For any x 2 X, the income share of person i is:si(x) :=xin(x)¹(x)(1)De¯nition 6 An additive distributional-shares inequality index can be writtenasn(x)Xi=1wiÁ (si(x)) (2)3increasingpreferencemIFigure 1: The Reduced-Form Social Welfare Functionwhere wi is a weight and Á : [0; 1] 7! < is monotonic and convex.4Examples of the ADSI class of indices include all the generalised-entropyindices, the Gini index, the relative mean deviation and the logarithmic variance.5The results which follow will also apply to monotonic a±ne transforms of (2).4The approach is based upon the so-called relative inequality aggregation property Blackorbyand Donaldson (1984). Notice the essential di®erence from the standard welfare-theoreticapproach to the de¯nition of an inequality measure as in Atkinson (1970), Kolm (1969), Sen(1973) where the speci¯cation of the welfare function or of equally-distributed equivalent incomeusually forces monotonicity on the part of the social welfare function - see also Ebert (1988).5This is distinct from the variance of logarithms - see Cowell (1995).42.1 The Reduced-form SWFIt is common to ¯nd the social welfare function written in a \reduced form" -as a function of mean income and inequality.6 To formalise this approach let usintroduce the concept of a distributional pro¯le z := (¹; I), a mean-income andinequality pair. In this interpretation the admissible set of pro¯les Z µ <2+ maybe writtenZ :=©(¹; I) : ¹ ¸ 0; 0 · I · ¹Iª (3)where ¹I is the least upper bound on I. Then the reduced-form SWF H(W ) : Z !< is given byW (x) = H(W ) (¹(x); I(x)) (4)where H(W ) is non-decreasing in its ¯rst argument and non-increasing in itssecond argument.Where there is no problem of ambiguity we will drop the superscript on H.Notice that the adoption of this approach implies that two distributions withsame z-pro¯le will be regarded as equivalent in welfare terms, and that the trade-o® implicit in the use of H is only well-de¯ned for a particular cardinalisationof the inequality index I. We shall assume that the issue of the appropriatecardinalisation has been settled by criteria which are not of concern to the presentstudy.6This is the term used by Champernowne and Cowell (1997). See also the term \abbreviatedsocial-welfare function" used in Lambert (1993), Chapter 5.53 Di®erentiable Social-welfare FunctionsThe ADSI class of measures contains a number of well known indices and yieldssome readily interpretable results. To see this let us endow I with the elementarycardinalisation of (2). Substituting from (2) into (4) we haveW (x) = H0@¹(x); n(x)Xi=1wiÁ (si(x))1A (5)Assume further that W is di®erentiable. Then di®erentiating (5) with respect toxi we have:n(x)@W (x)@xi= H¹ +HI"wiÁs (si(x))¹(x)¡Pn(x)j=1 wjsj(x)Ás (sj(x))¹(x)#(6)where the relevant partial derivatives are denoted by subscripts. From (6) thefollowing result is immediateTheorem 1 Given a di®erentiable reduced form social-welfare function H theunderlying individualistic welfare function satis¯es monotonicity if and only ifthe following condition on the slope of the welfare-contour is satis¯ed:¡H¹HI> max(wiÁs (si(x))¡Pn(x)j=1 wjsj(x)Ás (sj(x))¹(x))(7)63.1 ApplicationsThe condition (7) can be easily interpreted in the case of the Generalised Entropyindex:¡H¹HI> max[n(x)si(x)]®¡1¡1®¡1 ¡ ®I(x)n(x)¹(x)(8)and the relative mean deviation:¡H¹HI>2m(x)+1n(x)¡ I(x)n(x)¹(x)(9)wherem(x) = #fxi · ¹(x)g. The interpretation of these results will be discussedbelow.3.2 The Gini IndexThe Gini index G(x) can also be obtained as a special case of (7) by rewriting.it as:7G(x) =n(x)Xi=1wisi (10)wherewi =2i¡ n(x)¡ 1n(x)(11)We will call the extended Gini subclass those inequality measures of the form (10)but with the restriction (11) replaced by the mild condition that wi be increasing7See, for example Cowell (1995), Sen (1973).7with i.8 Notice that for the extended Gini and the true Gini Ás = 1 wherever thedi®erential is de¯ned. Also note that for the true Gini maxwi =n(x)¡1n(x)whichfor large populations becomes 1; in this limiting case the right-hand side of (7)becomes:1¡G(x)¹(x)(12)However in the case of the Gini index the di®erentiability assumption used inTheorem 1 will be inappropriate. For this reason and for the reason that -apart from mathematical tractability - di®erentiability has little to recommendit in social-welfare analysis, we seek a more general characterisation of the mainpoint.4 A General ResultTo avoid the over-simpli¯cation associated with the di®erentiability assumptionconsider the general shape properties of the reduced-form social-welfare functionat any arbitrary point in the set of all pro¯les Z.For any distributional pro¯le z0 2 Z it is clear that any point inA (z0) := fz 2 Z : ¹ ¸ ¹0; I · I0; z 6= z0g (13)8See for example the contributions by Donaldson and Weymark (1980, 1983), Weymark(1981), , Yitzhaki (1983). However note that Chakravarty's extended Gini does not ¯t withinthis category (Chakravarty 1988).8will be regarded as W1-preferable to z0. However, we can be more precise aboutthe set of W1-preferable pro¯les. First de¯ne the set of income distributionsconsistent with any pro¯le z0 2 Z:X(z0) := fx : x 2 X;¹ (x) = ¹0; I (x) = I0g (14)and a conditional upper-bound value of inequality, of a given population sizen = n(x):I¤ (¹; z0) := maxx2X(z0)I (x+ ¶nn [¹¡ ¹0]) (15)where ¶n := (0; 0; :::; 0; 1). These constructs then permit the de¯nition of thefollowing subset of A (z0):P (z0) := fz 2 Z : ¹ ¸ ¹0; I · I¤ (¹; z0) ; z 6= z0g (16)Theorem 2 8W 2 W1; 8z1 2 P (z0) :H(W )(z1) > H(W ) (z0) : (17)Proof. For some z0 2 Z and some z1 := (¹1; I1) 2 P (z0), given that ¹0 < ¹1by construction, if I0 ¸ I1 then it is trivial to show that (17) holds. So take the9_Im 01mm 1z0I1I0zIz3I2 2zFigure 2: The set P (z0)more interesting subcase where I0 < I1 , and de¯nex2 := arg maxx2X(z0)I (x+ ¶nn [¹1 ¡ ¹0]) (18)x3 :=¹1¹0x0 (19)The pro¯les z2 and z3 corresponding to x2 and x3 are illustrated in Figure 2.Noting that W (x2) > W (x0) we have8® 2 [0; 1] : W (®x2 + [1¡ ®]x3) > W (x0) ; (20)8® 2 [0; 1] : ¹ (®x2 + [1¡ ®]x3) = ¹1 > ¹ (x0) = ¹0; (21)10and by the principle of transfers I¤ (¹; z0) = I (x2) ¸ I1 ¸ I (x3) = I0. Bycontinuity there must be some ® 2 [0; 1] such that9I (®x2 + [1¡ ®]x3) = I1 (22)in which case it is immediate from (20) that (17) holds.Notice that, by construction, P (z0) is the largest set of pro¯les which areregarded as W1-preferable to z0, and that it is bounded by the graph of I¤:the set of points of the form ¹P (z0) := fz : z = (¹; I¤(¹; z0)) ; ¹ ¸ ¹0g provides aboundary conditional on a particular distributional pro¯le z0. For the ADSI classwe haveI¤ (¹; z0) :=n¡1Xi=1wi·Áµxin¹¶¸+ wn·Áµxn + n [¹¡ ¹0]n¹¶¸(23)which, in the case of the subclass of extended Gini measures, reduces toI¤ (¹; z0) := ¹I ¡£¹I ¡ I0¤ ¹0¹(24)Notice that through each point (pro¯le) z0 2 Z, there is a frontier ¹P and acontour of the reduced-form social welfare function H(W ). We may then state:Theorem 3 For any inequality-averse monotonic SWF and any z0 2 Z, when-9See the pro¯le z1 in Figure 2. Notice that I must be continuous in view of the convexityof Á.11_Immz2I0IH    (z)=H    (z  )0` P(z )0z10z0(W)           (W)Figure 3: A violation of monotonicityever the contour of H(W ) intersects ¹P (z0), it must do so from below.Proof. Suppose to the contrary that H(W ) intersects ¹P (z0) from above asin Figure 3.10 By construction H(W )(z0) = H(W )(z1), z2 is W1-preferable toz0. and pro¯les z1; z2 have the same mean. If W is inequality-averse thenH(W )(z2) > H(W )(z1) = H(W )(z0) which means that W must violate monotonic-ity - a contradiction.This theorem may then be used to give a more general interpretation toTheorem 1 which was obtained under the assumption of uniform di®erentiability:10The H(W)-contour has been drawn with a kink at z0 to illustrate the generality of theresult. Of course it need not be kinked, nor piecewise linear, and it may be convex rather thanconcave.12Corollary 4 IfW satis¯es monotonicity, the limit of the slope of I¤ (¹; z0) as ¹ #¹0 must be less than the limit of the slope of the contour of H(W )(z) = H(W )(z0)as ¹ # ¹0.Substitution from (23) or from (24) will con¯rm the result for the generalADSI class and for the special case of the Gini index respectively. In the lattercase we ¯nd that the condition for monotonicity amounts to:¡H(W )¹H(W )I>1¡ I¹(25)where the slope is to be interpreted in the manner of Corollary 4. This can be re-expressed as a requirement on the reduced-form SWF in terms of the associatedequality index 1 ¡ I: it should have the property that the elasticity of equalitywith respect to mean income be less than unity. The condition (25) can thenbe used to check the properties of reduced-form Gini-SWFs presented in theliterature. Examples of measures satisfying the elasticity condition include n2[1¡I]¹ (Sheshinski 1972)11, log ¹¡ I (Katz 1972) and ¹1+I(Kakwani 1986). However1¡I1+I¹ (Chipman 1974, Dagum 1990) does not satisfy the condition and it is easyto see that this index violates monotonicity: Table 1 shows an example wheredistribution B has a higher mean income and higher inequality than A. B also¯rst-order dominates A. However the Chipman-Dagum social welfare function11A generalisation of this form is to be found in Lambert (1985). See also Ben Porath andGilboa (1994)13A Bx1 4500 4600x2 4000 4000x3 3500 3500¹(x) 4000 4033I(x) 0.056 0.061W (x) = 1¡I1+I¹ 3579 3572Table 1: The Gini coe±cient I and the Chipman-Dagum Welfare Index Windicates that society is worse-o® in B than in A.Similar interpretations can be provided in terms of other members of the ADSIclass (2).5 ConclusionsThe use of ad hoc inequality measures begs the question of whether the empiricaljudgements made using them will be in accordance with conventional welfareproperties. The reduced-form SWF is the appropriate tool to clarify this issue.The transfer principle is inherited automatically by the SWF from an inequalitymeasure; now we have shown that the condition on the reduced-form SWF thatensures consistency with the monotonicity criterion is surprisingly simple. In thecase of the Gini index this condition is particularly appealing, and does not relyon arbitrary assumptions of di®erentiability of the SWF.Finally, a word of caution. Cardinalisations of inequality measures are some-times viewed as irrelevant, but in the analysis of the so-called \equity-e±ciency14trade-o®" this is not quite so. As the discussion of the elasticity of the H(W )-contours makes clear, the results we have obtained are speci¯cally for the \natu-ral" cardinalisation of the index (2). A modi¯ed version of Theorem 1 and relatedresults will be applicable under continuous monotonic transformations of I.15ReferencesAmiel, Y. and F. A. Cowell (1994a). Income inequality and social welfare. InJ. Creedy (Ed.), Taxation, Poverty and Income Distribution, pp. 193{219.Edward Elgar.Amiel, Y. and F. A. Cowell (1994b). Monotonicity, dominance and the Paretoprinciple. Economics Letters 45, 447{450.Atkinson, A. B. (1970). On the measurement of inequality. Journal of EconomicTheory 2, 244{263.Ben Porath, E. and I. Gilboa (1994). Linear measures, the Gini index, and theincome-equality trade-o®. Journal of Economic Theory 64, 443{467.Blackorby, C. and D. 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