Single Crossing Lorenz Curves and Inequality Comparisons

Transcription

1 Sigle Crossig Lorez Curves ad Iequality Comparisos Thibault Gajdos To cite this versio: Thibault Gajdos. Sigle Crossig Lorez Curves ad Iequality Comparisos. Mathematical Social Scieces, Elsevier, 2004, 47 1), pp < /S ) >. <halshs > HAL Id: halshs Submitted o 17 Jul 2006 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Sigle Crossig Lorez Curves ad Iequality Comparisos Thibault Gajdos Jue 4, 2003 Abstract Sice the order geerated by the Lorez criterio is partial, it is a atural questio to woder how to exted this order. Most of the literature that is cocered with that questio focuses o local chages i the icome distributio. We follow a differet approach, ad defie uiform α spreads, which are global chages i the icome distributio. We give ecessary ad sufficiet coditios for a Expected Utility or Rak-Depedet Expected Utility maximizer to respect the priciple of trasfers ad to be favorable to uiform α spreads. Fially, we apply these results to iequality idices. Keywords ad Phrases: Iequality measures, Itersectig Lorez Curves, Spreads. JEL Classificatio Number: D63. 1 Itroductio The Pigou-Dalto priciple of trasfers plays a cetral role i the ormative measuremet of iequality. This criterio simply says that a rak preservig) icome trasfer from a richer to a poorer perso reduces iequality. This priciple is equivalet to the Lorez criterio, applied to distributios with the same total icome ad populatio size: If the Lorez curve associated to a icome distributio Y is owhere below the oe associated to the distributio X, ad X has the same total icome ad populatio size tha Y, the Y ca be obtaied from X by a fiite sequece of Pigou-Dalto trasfers, ad therefore I wish to thak Michèle Cohe, Marco Scarsii ad Jea-Marc Tallo for helpful discussios ad commets. I am especially edebted to Alai Chateaueuf, Mark Machia, ad my referees for helpful commets ad suggestios. Ay remaiig errors are of course mie. Crs-Crest, 15 Bd Gabriel Péri, 92245, Malakoff Cedex, Frace, gajdos@esae.fr 1

3 Y is less uequal tha X. Furthermore, the Lorez criterio is also equivalet to secoddegree stochastic domiace for distributios with equal meas. Fially, a social welfare fuctio is compatible with the priciple of trasfers if, ad oly if, this fuctio is S cocave see e.g. Atkiso 1970), Dasgupta, Se ad Starrett 1973)). Obviously, the weak order geerated by the Lorez criterio is partial. It is therefore a atural questio to woder how to exted the set of distributios that ca be ordered. Most of the literature that is cocered with that questio focuses o the priciple of composite trasfers, i.e., o the combiatio of a progressive trasfer ad a regressive trasfer. Such a approach focuses o local chages, sice these trasfers cocer at most) four idividuals. We follow here a differet approach, sice we restrict our attetio to some global chages i the icome distributio. Although this is certaily less geeral, it turs out to be eough to derive some eat characterizatios of social welfare fuctios ad iequality idices. More precisely, we itroduce the otio of uiform α spreads. Cosider a icome distributio amog agets. Now, assume that aget with rak k + 1) i the icome distributio pays a tax that is uiformly distributed amog the remaiig agets icludig himself), without perturbig idividuals rak i the distributio. The resultig distributio is obtaied from the iitial oe through a uiform α spread, with α = k. Obviously, these two distributios caot be ordered with the Lorez criterio, sice a uiform spread is a combiatio of progressive ad regressive trasfers. Moreover, the Lorez curves associated to these two distributios cross oce. It turs out that a decisio maker who behaves i accordace with the Expected Utility model is favorable to uiform α spreads if, ad oly if, his utility idex is liear, whatever the value of α is. O the other had, we fid some ecessary ad sufficiet coditios for a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model to respect the priciple of trasfers ad to be favorable to uiform α spreads. Fially, sice ormative iequality idices rely o social welfare fuctios, it is the possible to apply these characterizatios for iequality idices more precisely the Atkiso- Kolm-Se idices correspod to a utilitaria social welfare fuctios, whereas the Gii idex ad its geeralizatios correspod to rak-depedet social welfare fuctios). 2

4 The orgaizatio of the paper is as follows. I sectio 2 we defie the otio of uiform α spreads, ad discuss some of its properties. I sectio 3, we give ecessary ad sufficiet coditios for a a decisio maker who behaves i accordace with the Expected Utility model or with the Rak-Depedet Expected Utility model to respect the priciple of trasfer ad to be favorable to uiform α spreads. Fially, a last sectio is devoted to the applicatio of the precedig results to the problem of iequality measuremet. 2 Uiform Spreads Let D be a arbitrary iterval of R, ad D be the iterior of D. We deote by D the set of rak-ordered discrete icome distributios of size N where N = N\{0}) with values i D. A icome distributio X D is defied by: X = x 1, 1 ; x 2, 1 ;... ; x, 1 ), with x 1 x 2... x. Therefore, X deotes the icome distributio where a fractio 1 of the total populatio has a icome equal to x i, for all i {1,..., }. Note that for ay icome distributio Y = y 1, p 1 ; y 2, p 2 ;...; y k, p k ), where the p i are ratioal umbers ad ki=1 p i = 1, there exists m 2 such that Y = y 1, 1 ; y m 2, 1 ;...; y ) m m, 1 m. For simplicity, we let X = x 1,..., x ). Furthermore, we will deote D = N D. We deote by F X the probability distributio fuctio associated to X, ad by F 1 X the iverse distributio fuctio defied by F 1 X p) = if {x : F x) p}. Fially, X = i=1 1 x i deotes the mea of X D. Let be the decisio maker s preferece relatio over D. We say that a decisio maker behaves i accordace with the Expected Utility model see vo Neuma ad Morgester 1947)) if there exists a cotiuous ad strictly icreasig utility fuctio u : D R, bouded 1 o D, such that is represeted by: U X) = i=1 1 u x i). 1 This assumptio is required i order to avoid a super St. Petersburg paradox of the Meger type. The same restrictio applies for the Rak-Depedet Expected Utility model. 3

5 A decisio maker behaves i accordace with Yaari s Dual model see Yaari 1987)) if there exists a strictly icreasig cotiuous frequecy trasformatio fuctio f : [0, 1] [0, 1] with f 0) = 0 ad f 1) = 1, such that is represeted by: V X) = i=1 [ f ) i + 1 i f )] x i. Fially, a decisio maker behaves i accordace with Quiggi s Rak-Depedet Expected Utility model see Quiggi 1982)) if there exist a cotiuous ad strictly icreasig utility fuctio u : D R, bouded o D, ad a strictly icreasig cotiuous frequecy trasformatio fuctio f : [0, 1] [0, 1] with f 0) = 0 ad f 1) = 1, such that is represeted by: V u X)) = i=1 [ f We will deote for ay i {1,..., }: Ψ i ) i + 1 i f ) = f i+1 )] u x i ). ) ) f i. I the sequel, we iterpret U ad V ad V u.)) as social welfare fuctios. Obviously, U correspods to a utilitaria social welfare fuctio, whereas V correspods to what we call a liear rak-depedet social welfare fuctio, ad V u.)) correspods to a rak-depedet social welfare fuctio. Now, let us recall the well-kow otio of Lorez order. Defiitio 1. Let X, Y belog to D. Y is less uequal tha X i the sese of the Lorez order, deoted Y L X iff: L F Y, ξ) = ξ 0 F Y 1 t) dt Ȳ ξ 0 F X 1 t) dt X = L F X, ξ), ξ [0, 1], i.e., if the Lorez fuctio L F X, ξ) of X is owhere below the Lorez fuctio L F Y, ξ) of Y. We say that a decisio maker respects the Lorez order iff for all X, Y i D, Y L X Y X. The Lorez order which is a partial order) plays a cetral role i the field of iequality measuremet. Ideed, it had bee proved see Hardy, Littlewood ad Pólya 1934)) that 4

6 if X = Ȳ, ad X ad Y have the same populatio size, the Y L X if ad oly if Y ca be derived from X through a fiite sequece of rak-preservig icome trasfers from richer to poorer idividuals Pigou-Dalto trasfers). Although the Lorez criterio is ormatively very appealig, it suffers from a serious drawback, sice the weak order geerated by this criterio is obviously partial. It is therefore a atural questio to woder how to exted the set of distributios that ca be ordered. Most of the literature that is cocered with that questio focuses o the priciple of composite trasfers, i.e., o the combiatio of a progressive trasfer ad a regressive trasfer. More precisely, two kids of composite trasfers are cosidered: The composite trasfers that preserve the variace ad the mea of the iitial distributio, ad the oes that preserve the mea ad the value of the Gii idex of the iitial distributio. The first oe is associated with third-degree stochastic domiace see, e.g., Shorrocks ad Foster 1987), Foster ad Shorrocks 1988), Davies ad Hoy 1994)), whereas the secod oe is associated with iverse third-degree stochastic domiace see, e.g., Muliere ad Scarsii 1989), Moyes 1990), Chateaueuf ad Wilthie 1998), Zoli 1999)). I both cases, ecessary ad sufficiet coditios for a social welfare fuctio to respect both the priciple of trasfers ad the priciple of composite trasfer uder cosideratio have bee idetified. Both approaches focus o local spreads, i.e., spreads cocerig oly four at most) idividuals. Our approach is somewhat differet, sice we restrict our attetio to global chages i the distributio. The mai idea is the followig. Cosider a distributio X with idividuals, ad assume that x k < x k+1. What would be the cosequece of taxig the idividual occupyig the k + 1) th positio i the ladder, without perturbig the orderig, ad the redistributig the collected tax uiformly amog the remaiig agets? We will call such a chage i a distributio a uiform α spread, with α = k. More formally, we have the followig defiitio. Defiitio 2. Let X, Y belog to D. Y is obtaied from X through a uiform α spread, deoted Y u α X, if there exist 1 k 1, with k N ad α = k, 0 < ε x k+1 x k 5

7 such that: { yi = x i + ε, i k + 1) y k+1 = x k+1 1) ε. Obviously, if Y u α X, these two distributios caot be ordered by the Lorez criterio. Furthermore, a simple ispectio of Defiitio 2 shows that the Lorez curves associated with Y ad X cross oly oce, ad that the curve associated with Y is above the oe associated with X for ξ k, ad below for ξ > k. It the follows that if X ad Y are two distributios with the same total icome ad populatio size, a ecessary coditio for Y to be obtaied from X by a sequece of Pigou-Dalto trasfers ad/or uiform α spreads, with α α, is that LF Y, ξ) LF X, ξ) for all ξ < α. I other words, the partial order associated with fiite sequeces of Pigou-Dalto trasfers ad/or uiform α spreads with α α, does ot allow oe to rak icome distributios with itersectig Lorez curves if a itersectio occurs at ξ < α. Defiitio 3. A decisio maker is favorable to uiform α spreads if Y X wheever Y u α X. We say that a decisio maker satisfies the priciple of uiform α spreads if he is favorable to uiform α spreads. Observe that, if Y is obtaied from X by a uiform α spread, ad Y is obtaied from X by a uiform α spread of the same amout, with α > α, the Y is obtaied from Y by a Pigou-Dalto trasfer from the aget i positio α to that i positio α. This leads us to the followig Propositio. Propositio 1. If a decisio maker respects the priciple of trasfer ad is favorable to uiform α spreads, the he is favorable to uiform α spreads, for all α α. Propositio 1 leads us to a atural defiitio of a decisio maker s sesitivity to uiform spreads. Defiitio 4. The degree of sesitivity to uiform spreads of a decisio maker who respects the Lorez order is defied by: α u = 1 if {α : the decisio maker is favourable to uiform α spreads}. 6

8 Because it is ot assumed that the size of the populatio is fixed, ad because it ca be arbitrarily large, the degree of sesitivity to uiform spreads ca take ay value i the iterval [0, 1]. Assume, for istace, that the decisio maker is favorable to uiform 1 spreads. The, whe teds to, the ifimum of α such that the decisio maker is favorable to uiform α spreads is equal to lim 1 = 0, ad therefore the decisio maker s degree of sesitivity to uiform spreads is equal to 1. Observe that, i this case, the decisio maker is favorable to ay uiform spread. O the other had, a decisio maker who respects the priciple of trasfers, must at least be favorable to uiform 1 spreads, sice these spreads are actually a sequece of Pigou-Dalto trasfers. However, assume that the decisio maker is favorable oly to uiform 1 spreads ad to Pigou-Dalto trasfers. The, his degree of sesitivity to uiform spreads is equal to 1 lim 1 = 0. Observe that a uiform k spread ca be see as the combiatio of a sequece of progressive trasfers from the idividual occupyig the k + 1) th positio i the ladder to the k poorest idividuals, ad a sequece of regressive trasfers from the same idividual to the k 1) richest idividuals. The results of these trasfers are a reductio of iequality amog the k + 1) poorest idividuals, ad a icrease of iequality amog the k) richest idividuals. Therefore, a uiform k spread seems appealig for large values of k whe the size of the populatio is large, ad the decisio maker respects the Lorez order: It meas that the decisio maker is ready to accept a icrease of iequality amog the very rich persos, provided that it is accompaied by a decrease of iequality amog the rest of the populatio. Roughly speakig, the decisio maker s degree of sesitivity to uiform spreads measures the size of the populatio amog which the reductio of iequality is ot see as a priority by the decisio maker. The extreme case is that of a Rawlsia decisio maker, who is maily cocered by the poorest idividual: His degree of sesitivity to uiform spreads is the equal to 1. This does ot mea, however, that such a decisio maker is ot favorable to Pigou-Dalto trasfers amog richer idividuals. But a policy that icreases the poorest idividual s icome is the see as favorable, eve if the cost of such a policy is a icrease of iequality amog the rest of the populatio. A atural iterpretatio is that, if the decisio maker is favorable to uiform k -spreads, 7

9 he cosiders the k poorest idividuals as poor idividuals. However, it does ot imply, ulike to the focusig axiom used i poverty measuremet, that the decisio maker is ot cocered with richer idividuals. Hece, the priciple of uiform α spreads lies somewhere betwee the priciple of trasfers ad the focusig priciple. 3 Uiform Spreads ad Social Welfare Fuctios We give here ecessary ad sufficiet coditios for a decisio maker who respects the Lorez order to be favorable to uiform α spreads. We successively focus o decisio makers who behave i accordace with the Expected Utility model, with the Rak-Depedet Expected Utility model, ad with Yaari s dual model, which is a particular case of the Rak-Depedet Expected Utility model. 3.1 Uiform Spreads ad the Expected Utility model Our first result is, at first sight, strikig: A decisio maker who behaves i accordace with the Expected Utility model is favorable to uiform α spreads if, ad oly if, his Social Welfare Fuctio reduces to the mathematical expectatio of the icome distributio, whatever the value of α is. Theorem 1. Let α ]0, 1[ Q. For a decisio maker who behaves i accordace with the Expected Utility model, with a utility fuctio two times cotiuously differetiable o D, the followig two propositios are equivalet: i) The decisio maker is favorable to uiform α spreads. ii) u x) = x, x D up to a icreasig affie trasformatio). Proof. Fix k ad > 2 such that 1 k 1 ad α = k. The decisio maker is favorable to uiform α spreads if for ay X = x 1,..., x ) i D such that x k < x k+1 ad ε such that 0 < ε x k+1 x k ad x + ε D, i k+1 u x i + ε) + u x k+1 1) ε) i u x i ), 8

10 which is equivalet to: i<k+1 [u x i + ε) u x i )]+ i>k+1 We first prove that i) u x) 0 for all x D. [u x i + ε) u x i )] u x k+1 ) u x k+1 1) ε). 1) Let y ad x, i D with y < x be arbitrarily chose, ad let x 1 = x 2 =... = x k = y, x k+1 = x k+2 =... = x = x. The 1) implies, for all ε 0, x y ] such that x + ε D: k [u y + ε) u y)] + k 1) [u x + ε) u x)] u x) u x 1) ε). Divide this expressio by 1)ε: ) k u y + ε) u y) + k 1 1 ε 1 ) u x + ε) u x) ε u x) u x 1) ε). 1)ε Now let ε ted to 0. Oe obtais: k 1 u y) + k 1 u x) u x), 1 ad therefore, u y) u x) for all y ad x i D such that y < x. Hece u x) 0 for all x i D. We ow prove that i) u x) 0 for all x D. Let x, y ad β be arbitrarily chose such that x < y, β > 0, ad x β ad y belog to D. Let x 1 =... = x k = x β, x k+2 =... = x = y, ad x k+1 = x. If the decisio maker is favorable to uiform α spreads, the, for ε 0, β ] such that y + ε D: k [u x β + ε) u x β)] + k 1) [u y + ε) u y)] u x) u x 1) ε). Divide this expressio by 1)ε: ) k u x β + ε) u x β) + k 1 1 ε 1 ) u y + ε) u y) ε u x) u x 1) ε). 1)ε Now let ε ted to 0. We obtai: k 1 u x β) + k 1 u y) u x). 1 Now let β ted to 0. We get: u y) u x) for ay x ad y i D such that x < y. Hece u x) 0 for ay x i D. 9

11 Sice u x) 0 ad u x) 0 for ay x i D, ad sice u is cotiuous o D, u x) = x, up to a icreasig affie trasformatio, for all x i D. We have hece proved that i) implies ii). That ii) implies i) is trivial, ad the proof is completed. Note that Theorem 1 does t deped o ay assumptio about the decisio maker s attitude toward Pigou-Dalto trasfers. Actually, this result does t really come as a surprise. Ideed, a uiform spread is a combiatio of progressive ad regressive trasfers. I the Expected Utility model, the size of the impact of a trasfer depeds o the icome distace betwee the idividuals cocered by this trasfer, ad o the size of this trasfer: The size of the impact of a small trasfer ε > 0 from a idividual with icome x to a idividual with icome y is give by [u y) u x)] ε. Assume that u is strictly cocave o some iterval [a, b]. Cosider the, for a arbitrarily chose 1 k < the distributio i which i) x k is arbitrarily close to a with x k > a, ii) x i a, x) for all i < k, so that the impact of each progressive trasfer is as close to 0 as oe would like, ad iii) x i is arbitrarily close to b with x i < b for all i > k, so that the impact of each regressive trasfer is as close to [u b) u a)]ε as oe would like. Sice by strict cocavity of u, [u b) u a)] < 0, it the follows that the et impact of these trasfers ca be egative, ad therefore the decisio maker caot be favorable to uiform k spreads. A similar argumet applies i the strictly covex case. Therefore, it must be the case that u is liear. 3.2 Uiform Spreads ad Rak-Depedet Expected Utility model Now, let us cosider a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model. First, we recall the followig result see Chew, Kari ad Safra 1987)). Theorem 2. For a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model, with a frequecy trasformatio f differetiable o [0, 1], the followig two propositios are equivalet: 10

12 i) The decisio maker respects the Lorez order. ii) u is cocave ad f is covex. We also eed to defie the idex of thriftiess of a utility fuctio, itroduced by Chateaueuf, Cohe ad Meilijso 1997). This idex is defied by: T u = u x) sup {x,y D x<y} u y). The followig theorem gives ecessary ad sufficiet coditios for a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model to respect the Lorez order ad to be favorable to uiform α spreads. Theorem 3. Let α ]0, 1[ Q. For a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model, with a frequecy trasformatio fuctio differetiable o [0, 1], ad with a utility fuctio u cotiuously differetiable o D, the followig two propositios are equivalet: i) The decisio maker respects the Lorez order ad is favorable to uiform α spreads. ii) f is covex, u is cocave o D ad f 1 α) 1 f1 α) Proof. i) ii) 1 Tu T u. We kow from Theorem 2 that a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model respects the Lorez order if ad oly if u is cocave ad f is covex. Let α = l r ]0, 1[ Q be fixed. Let = rm ad k = lm, where m N is arbitrarily chose. Assume that the decisio maker is favorable to uiform α spreads, ad let X = x 1,..., x ) D with x k < x k+1. The, for ay 0 < ε x k+1 x k such that x + ε D: ) i k + 1 i Ψ u x i + ε) + Ψ u x k+1 1) ε) Ψ u x i ), i k+1 ) i=1 ) which is equivalet to: i<k+1 i Ψ [u x i + ε) u x i )] + ) i>k+1 ) k + 1 Ψ [u x k+1 ) u x k+1 1) ε)]. 11 i Ψ [u x i + ε) u x i )] 2) )

13 Let x, y ad β be arbitrarily chose such that x < y, β > 0, x, y D ad x β D. Let x k+1 = x, x 1 = x 2 =... = x k = x β ad x k+2 =... = x = y. The 2) implies for ε 0, β ] such that y + ε D: [ )] ) k k 1 1 f [u x β + ε) u x β)] + f [u y + ε) u y)] [ ) )] k k 1 f f [u x) u x 1) ε)]. Divide this expressio by 1)ε: 1 f ) k ) u x β + ε) u x β) + f ) k 1 ) u y + ε) u y) 1 ε 1 ε [ ) )] k k 1 u x) u x 1) ε) f f. 1)ε Now let ε ted to 0, ad multiply both sides by 1). This leads to: [ )] ) [ ) )] k k 1 k k 1 1 f u x β)+f u y) 1) f f u x). Now let β ted to 0. Oe gets: ) [ ) )] ) k 1 k k 1 k 1 f u y) f f u x)+f u x) u x). Divide both terms by u y): ) [ ) )] k 1 k k 1 u x) f f f u y) +f Hece: f 1 l r 1 ) rm rm [ f 1 l r ) f Let m ted to +. We the have: f 1 l r ) f 1 l r k 1 ) u x) u y) u x) u y). 1 l r 1 )] u x) rm u y) +f 1 l r 1 ) u x) x) rm u y) u u y). ) u x) u y) + f 1 l ) u x) r u y) u x) u y). Hece: f 1 α) ) 1 u x) u y) [f 1 α) 1] u x) u y). 3) 12

14 Therefore: f 1 α) 1 f 1 α) 1 u x) u y). u x) u y) Sice the right had side of this iequality decreases whe u x) u y) is satisfied for all x < y if ad oly if: icreases, this iequality f 1 α) 1 f 1 α) 1 T u T u. ii) i) By Theorem 2, it is sufficiet to show that the decisio maker is favorable to uiform α spreads. Sice u is cocave it is eough to prove that for ay x ad y i D such that x < y, ad ay ε > 0 such that ε y x ad y + ε D, i<k+1 i Ψ [u x + ε) u x)] + ) i>k+1 ) k + 1 Ψ [u x + ε) u x + ε)], for all k, ) for which α = k. Cosider ay such k, ). i Ψ [u y + ε) u y)] ) The cocavity of u implies, for ay ε 0, y x ] such that y + ε D: u x + ε) u x) ε u y + ε) u y) ε u x + ε) u x + ε) 1) ε u x + ε), u y + ε), u x + ε). Hece, it is eough to prove that, for ay y > x i D ad ay ε 0, y x ] such that y + ε D: i Ψ u i<k+1 ) x + ε) + ) i k + 1 Ψ u i>k+1 ) y + ε) 1) Ψ u x + ε). which may be writte as follows: [ 1 f 1 k )] u x + ε) + f 1 k + 1 ) u y + ε) [ 1) f 1 k ) f 1 k + 1 )] u x + ε). 13

15 Dividig both terms by u y + ε) leads to: f 1 k + 1 ) [ f 1 k ) f 1 k + 1 )] u x + ε) u y + ε) + f 1 k + 1 ) u x + ε) u y + ε) u x + ε) u y + ε). Sice f is covex, we have: [ f 1 k ) f 1 k + 1 )] f 1 k It is hece eough to prove: f 1 k + 1 ) [ ] 1 u x + ε) [f 1 k ) ] u x + ε) 1 u y + ε) u y + ε). Sice u is cocave, u x+ε) u y+ε) > 1 for all 0 < x < y ad ε > 0. Sice f is icreasig, the precedig iequality is satisfied wheever: f 1 k ) [ ] 1 u x + ε) [f 1 k ) ] u x + ε) 1 u y + ε) u y + ε), which is equivalet to: ). f 1 ) k 1 f ) 1 u x+ε) u y+ε). 1 k u x+ε) u y+ε) Sice the right had side of this iequality decreases whe u x+ε) u y+ε) iequality is satisfied wheever: icreases, this last f 1 ) k 1 f ) 1 T u, 1 k T u which is the desired result. Note that Theorem 3 implies the followig result. Corollary 1. Let α ]0, 1[ Q. For a decisio maker who behaves i accordace with Yaari s dual model with a frequecy trasformatio fuctio f differetiable o [0, 1], the followig two propositios are equivalet: i) The decisio maker is favorable to uiform α spreads ad respects the Lorez order. ii) f is covex ad f 1 α) 1. 14

16 1 T Proof. If u x) = x, u T u = 0. Hece, Theorem 3 implies that a decisio maker who behaves i accordace with Yaari s dual model i.e., a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model with a liear utility idex) respects the Lorez order ad is favorable to uiform α spreads if ad oly if f is covex ad f 1 α) 1. The coditios of Corollary 1 have a atural iterpretatio. Assume that we have f 1 α) = 1, ad that f is covex. This implies that f p) > 1 for all p such that 1 α < p < 1 ad f p) < 1 for all 0 p 1 α. Hece, the α% poorest idividuals are over-weighted i.e., the decisio maker gives them a weight greater tha 1, where is the size of the populatio), ad the 1 α) % richest oes are uder-weighted. Remark 1. The coditio f 1 α) 1 is ecessary for a decisio maker who behaves i accordace with the Rak-Depedet Expected Utility model with a frequecy trasformatio fuctio f differetiable o [0, 1] ad a utility fuctio u cotiuously differetiable o D to respect the Lorez order ad to be favorable to uiform α spreads. Proof. By Theorem 3, if a decisio maker who behaves i accordace with the Rak- Depedet Expected Utility model respects the Lorez order ad is favorable to uiform α spreads, the: f 1 α) 1 f 1 α) 1 T u T u. Theorem 3 also implies that u is cocave. Thus, T u 1. Hece, the precedig iequality implies f 1 α) 1. Let us ow apply our differet results to the problem of iequality measuremet. 4 Iequality Idices ad Uiform Spreads Followig Kolm 1969), Atkiso 1970) ad Se 1973), oe ca derive a iequality measure from a social welfare fuctio. Let Ξ X) be the per capita icome which, if distributed equally, is idifferet to X D accordig to the social welfare fuctio W. This equally distributed equivalet icome is implicitly defied by the relatio: 15

17 W X) = W Ξ X) e), where e deotes the uit vector of R. It is the possible to defie a iequality idex: I X) = 1 ΞX). The Atkiso idex relies o a utilitaria social X welfare fuctio, whereas the Gii idex ad its geeralizatios rely o a rak-depedet social welfare fuctio. We say that a iequality idex respects the priciple of trasfers if for all X ad Y such that Y is obtaied from X through a fiite sequece of Pigou-Dalto trasfers, I Y ) I X). Similarly, we say that a iequality idex respects the priciple of α uiform spreads if for all X ad Y as i Defiitio 2, I Y ) I X). By a slight abuse of otatio, we call degree of sesitivity to uiform spreads of a iequality idex I the degree of sesitivity to uiform spreads of a decisio maker edowed with the social welfare fuctio o which relies I. First, cosider the Atkiso idex defied by I A X) = 1 I A X) = 1 [ i=1 1 ) 1 xi X i=1 ) ] 1 1 ε 1 ε xi X, ε 1, ε = 1. This idex relies o the followig Expected Utility social welfare fuctios: x 1 ε i U A X) = 1, ε 1 i=1 1 ε U A X) = 1 l x i=1 i), ε = 1. The followig propositio immediately follows from Theorem 1: Propositio 2. Let α belog to ]0, 1[ Q. The Atkiso idex respects the priciple of uiform α spreads if ad oly if ε = 0. Hece, the Atkiso idex caot respect the priciple of uiform α spreads, whatever α is, uless the idex reduces to a costat. But it seems difficult to raise ay objectio to the priciple of uiform α spreads, at least for very high values of α. This may be see as a limit of the Atkiso idex from a ormative poit of view. Let us ow cosider the large class of Yaari idices. These idices are defied as follows Yaari 1988), Ebert 1988)): I GG X) = 1 1 X i=1 [ f ) )] ) i + 1 i f x i. 16

18 Applyig Corollary 1, we obtai the followig result: Propositio 3. Let α belog to ]0, 1[ Q. A geeralized Yaari idex with a frequecy trasformatio fuctio f differetiable o [0, 1] respects the priciple of uiform α spreads ad the priciple of trasfers if ad oly if f 1 α) 1 ad f covex. Doaldso ad Weymark 1980) ad Bossert 1990) defie the sub-class of Yaari idices which satisfy a aggregatio axiom. These idices, kow as S-Gii idices, are defied as follows: I SG X) = 1 i=1 [ j=i p j ) δ j=i+1 p j ) δ ] with δ > 1. These idices correspod to the followig social welfare fuctio: V SG X) = f f x i, i=1 p j j=i X p j j=i+1 with f p) = p δ. Note that for δ = 2, I SG is othig but the Gii idex. The followig propositio establishes a lik betwee the degree of sesitivity to uiform spreads see Defiitio 4) of a S Gii idex ad the value of the parameter δ: Propositio 4. For a S-Gii with parameter δ > 1, the degree of sesitivity to uiform spreads of the idex is equal to δ 1 1 δ. Furthermore, the degree of sesitivity to uiform spreads of the idex is greater or equal to 1 e to δ. for all δ > 1, ad is icreasig with respect Proof. Let I SG be a S Gii idex with parameter δ > 1. Accordig to Defiitio 4, the degree of sesitivity to uiform spreads of I SG is defied by: α u δ) = 1 if {α : the decisio maker is favourable to uiform α spreads }, where the decisio maker s prefereces are represeted by δ V SG X) = i=1 p j j=i δ p j xi. j=i+1 Observe that, for δ > 1, fp) = p δ is covex ad f is differetiable o [0, 1]. Therefore, I SG respects the priciple of trasfers for all δ > 1. By Propositio 3, I SG respects 17 x i,

19 the priciple of uiform α spreads, for α ]0, 1[ Q if, ad oly if: f 1 α) 1, i.e., α 1 δ 1 1 δ = 1 φδ), with φδ) = δ 1 1 δ. Observe that φ δ) = exp ξδ)) ξ δ), with ξδ) = l δ 1 δ. Let ψδ) = + l δ. The: ξ δ) = ψδ). Sice ψ δ) = δ 1, we get: 1 δ δ 1 δ) 2 δ 2 ψ δ) > 0 for all δ > 1. Furthermore, ψ1) = 0. Therefore, ψδ) > 0 for all δ > 1. Hece, ξ δ) > 0 for all δ > 1, which etails φ δ) > 0 for all δ > 1. Hece, the degree of sesitivity to uiform spreads of I SG with parameter δ is φδ), ad the greater δ is, the higher is the degree of sesitivity to uiform spreads of the idex. Fially, we have lim δ 1 φδ) = 1 e Fially, the very geeral iequality idex let us call it a super-geeralized Gii idex): I SSG X) = 1 1 X u 1 i=1 [ f ) i + 1 i f )] ) u x i ), cosidered by Ebert 1988) ad Chateaueuf 1996) correspods to a Rak-Depedet Expected Utility-like social welfare fuctio, with a utility fuctio u ad a frequecy trasformatio fuctio f. Applyig Theorem 3 we obtai the followig result. Propositio 5. Let α belog to ]0, 1[ Q. A super-geeralized Gii idex with a frequecy trasformatio fuctio f differetiable o [0, 1] ad a utility fuctio u cotiuously differetiable o D respects the priciple of trasfers ad the priciple of uiform α spreads if ad oly if f 0, u 0 ad f 1 α) 1 f1 α) 1 Tu T u. Refereces Atkiso, A. B. 1970): O the Measuremet of Iequality, Joural of Ecoomic Theory, 2, Bossert, W. 1990): A axiomatizatio of the Sigle-Series Giis, Joural of Ecoomic Theory, 50, Chateaueuf, A. 1996): Decreasig iequalities: a approach through o-additive models, Cahiers Eco & Maths 96-58, Uiversité Paris I. 18

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