The Principle of Strong Diminishing Transfer

Transcription

1 The Priciple of Strog Dimiishig Trasfer Alai Chateaueuf, Thibault Gajdos, Pierre-Hery Wilthie To cite this versio: Alai Chateaueuf, Thibault Gajdos, Pierre-Hery Wilthie. The Priciple of Strog Dimiishig Trasfer. Joural of Ecoomic Theory, Elsevier, 2002, 103 2, pp <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 The Priciple of Strog Dimiishig Trasfer Alai Chateaueuf, Thibault Gajdos, Pierre-Hery Wilthie September 2000 Abstract We recosider the priciples of dimiishig trasfer itroduced by Kolm 1976 ad dual dimiishig trasfer itroduced by Mehra It appears that if a Rak Depedet Expected Utility RDEU maximizer respects the priciple of dimiishig resp. dual dimiishig trasfer, the he behaves i accordace with the Expected Utility model resp. Yaari s dual model. This leads us to defie the priciple of strog dimiishig trasfer, which is a combiatio of the priciples of dimiishig ad dual dimiishig trasfer. We give ecessary coditios for a RDEU maximizer to respect this priciple. These results are applied to the problem of iequality measuremet. Joural of Ecoomic Literature Classificatio Number: D63 Key words : Iequality Measuremet, Priciple of Dimiishig Trasfers. 1 Itroductio The well-kow Pigou-Dalto trasfer priciple requires that ay trasfer of icome from a richer to a poorer idividual, which does ot reverse which of the two is richer, reduces iequality. I a semial paper, Kolm 1976 goes a step further, ad itroduces the priciple of dimiishig trasfer. This priciple requires that oe values more such a trasfer betwee persos with give icome differece if these icomes are lower tha if they are higher Kolm, 1976, p.87. It is well-kow that the Atkiso, Kolm ad Theil idices respect this priciple. It is obviously ot the case for the Gii idex, sice its sesitivity depeds ot o the icome levels of the idividuals cocered by the trasfer, but o their raks. This has bee see as a limit of the Gii idex 1. But, o the other had, oe may argue that the Gii idex respects aother priciple, which is ot satisfied by the Atkiso, Kolm ad Theil idices: The Gii We thak a aoymous referee of the Joural of Ecoomic Theory for very helpful commets ad suggestios. Correspodig author. CERMSEM, Uiversité Paris 1 Pathéo Sorboe, Boulevard de L Hôpital Paris cedex chateau@uiv-paris1.fr EUREQua Uiversité Paris 1 Pathéo-Sorboe ad ESSEC, boulevard de L Hôpital Paris cedex gajdos@uiv-paris1.fr CERMSEM, Uiversité Paris 1 Pathéo Sorboe, Boulevard de L Hôpital Paris cedex ph@wilthie.com 1 See, e.g., Se

3 idex is more sesitive for Pigou-Dalto trasfers betwee persos with give rak differece if these raks are lower tha if they are higher. Let us call this priciple, itroduced by Mehra 1976 ad Kakwai 1980, the priciple of dual dimiishig trasfer. At first sight, there is o obvious reaso to prefer oe priciple to the other, ad oe may eve wat both to be satisfied. Admittedly, this is a matter of taste. Nevertheless, ad surprisigly eough, whereas a umber of papers are devoted to the priciple of dimiishig trasfer see, amog others, Kolm 1976, Shorrocks ad Foster 1987, Davies ad Hoy 1994, little have bee writte about the priciple of dual dimiishig trasfer 2. Obviously, the priciple of dual dimiishig trasfer is oly relevat if oe cares about idividuals raks. Therefore, these cocepts have o meaig i the vo Neuma-Morgester model. O the cotrary, they are relevat i Yaari s dual model 1987 ad its geeralizatio, Quiggi s Rak Depedet Expected Utility Model The aim of this paper is hece to characterize the priciples of dimiishig trasfer ad of dual dimiishig trasfer i Yaari s ad Quiggi s models. We give ecessary ad sufficiet coditios for a decisio maker who behaves i accordace with Yaari s dual model to respect the priciple of dual dimiishig trasfer. Ufortuately, it appears that if a decisio maker who behaves i accordace with the Rak Depedet Expected Utility model respects the priciple of dimiishig trasfer, the he behaves i accordace with the Expected Utility model. Similarly, if such a decisio maker respects the priciple of dual dimiishig trasfer, the he behaves i accordace with Yaari s dual model. This leads us to defie the priciple of strog dimiishig trasfer, which states that a trasfer from a idividual with rak i ad icome x to oe with rak i p ad icome x has a greater impact the lower i ad x are. Whereas it is uclear if oe should prefer the priciple of dimiishig trasfer or the priciple of dual dimiishig trasfer, it seems that, at least, oe should agree with the priciple of strog dimiishig trasfer obviously, if the decisio maker respects the priciple of dimiishig trasfer or the priciple of dual dimiishig trasfer, he also respects the priciple of strog dimiishig trasfer the coverse, however, is ot true. Hece, it seems that the priciple of strog dimiishig trasfer does really make sese. Furthermore, we give ecessary ad sufficiet coditios for a decisio maker who behaves i accordace with the Rak Depedet Expected Utility model to respect the priciple of trasfer ad the priciple of strog dimiishig trasfer. It should be emphazised that, although the various trasfer priciples defied i this paper oly apply to comparisos of distributios with the same populatio size, the results we establish are variable-populatio theorems. Actually, 2 See however the related otio of favourable double trasfer i Chateaueuf ad Wilthie 1999, ad Zoli s idepedat paper

4 it is importat for some of the proofs that the set of icome distributios is ot restricted to a fixed populatio size. Fially, these results are applied to the measuremet of iequality. The orgaizatio of this paper is as follows. I sectio 2 we itroduce otatio ad some defiitios. Sectio 3 cotais our mai results, which are applied to the problem of iequality measuremet i sectio 4. 2 Notatio ad defiitios Let Ω be the set of rak-ordered uiform ad discrete icome distributios, with values i R +. A icome distributio X Ω is defied by: X = x 1, 1 ; x 2, 1 ;...; x, 1 with 0 x 1 x 2... x. 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 k i=1 p i = 1 there exists m 2 such that Y = y 1, 1 m ; y 2, 1 m ;...; y m, 1 m. For simplicity, we let X = x 1,x 2,..., x. X deotes the mea of the icome distributio X, ad e i deotes the -tuple 0,...,0, 1, 0,...,0 whose oly o-zero elemet occurs i the i-th positio. 2.1 Prefereces Let be the decisio maker s preferece relatio over Ω. We call here a decisio maker aybody behid the veil of igorace. This assumptio is stadard i the field of ormative iequality measuremet. A decisio maker behaves i accordace with the Expected Utility model EU model if there exists a strictly icreasig utility fuctio u : R + R such that is represeted by: U X = i=1 1 u x i. A decisio maker behaves i accordace with Yaari s dual model if there exists a strictly icreasig cotiuous probability trasformatio f : [0, 1] [0,1] with f 0 = 0 ad f 1 = 1 such that is represeted by: W X = i=1 [ f i + 1 f i ] x i. Fially, a decisio maker behaves i accordace with Quiggi s Rak Depedet Expected Utility Model RDEU model if there exists a strictly icreasig utility fuctio u : R + R ad a strictly icreasig cotiuous probability trasformatio f : [0,1] [0,1] with f 0 = 0 3

5 ad f 1 = 1 such that is represeted by: W ux = i=1 [ f i + 1 f = f i+1 i ] u x i. f i. I the sequel, we let for ay i {1,..., }: Ψ i Followig Kolm 1969, Atkiso 1970 ad Se 1973, oe ca derive a iequality measure from a social welfare fuctio. Let ΞX be the per capita icome which, if distributed equally, is idifferet to X accordig to the social welfare fuctio W. This equally distributed equivalet icome is implicitly defied by the relatio: W X = W ΞX e, where e deotes the uit vector of R. It is the possible to defie a relative iequality idex: ad a absolute iequality idex: I X = 1 ΞX X, J X = X ΞX. The Gii idex which may be either absolute or relative ad its geeralizatios are based o a rak depedet social welfare fuctio. 2.2 The priciple of strog dimiishig trasfers Let us ow defie the well kow priciple of trasfer, the priciple of dimiishig trasfer ad the priciple of dual dimiishig trasfer. Defiitio 1 A decisio maker respects the priciple of trasfer if for all X belogig to Ω, i 1 < i 2,ad ε > 0 such that x i1 + ε x i1 +1 ad x i2 1 x i2 ε : X 1 = X + e i1 e i2 ε X. Defiitio 2 A decisio maker respects the priciple of dimiishig trasfer if for all X belogig to Ω, i 1 < i 2, i 3 < i 4, i 1 < i 3 ad all ε > 0 such that x i1 + ε x i1 +1, x i2 1 x i2 ε, x i3 + ε x i3 +1, x i4 1 x i4 ε, x i2 x i1 = x i4 x i3, X 1 = X + ε e i1 e i2 X + εe i3 e i4 = X 2. 4

6 Hece a decisio maker respects the priciple of dimiishig trasfer if a trasfer from a idividual with icome x to oe with icome x with > 0 has a greater impact o social welfare the lower x is. This priciple is illustrated o Figure 1. We start from the distributio represeted by a cotiuous lie. We the costruct two distributios: Oe is obtaied usig a Pigou-Dalto trasfer from the secod poorest idividual to the poorest oe thi dotted lie, ad the other is obtaied usig a Pigou-Dalto trasfer of the same amout ε from the richest idividual to the third richest idividual thick dashed lie. Here, x 5 x 3 = x 2 x 1. Hece, if oe agrees with the priciple of dimiishig trasfer, the first distributio is preferred to the secod oe. Iclude figure 1 : The priciple of dimiishig trasfer The priciple of dimiishig trasfer raises a problem, which is illustrated by Figure 2. Iclude figure 2 : First objectio to the priciple of dimiishig trasfer Here, we start from the icome distributio represeted by a thick lie. Now, we costruct two distributios: Oe is obtaied usig a Pigou-Dalto trasfer of amout ε from the secod poorest idividual to the poorest oe thi dashed lie, ad the other is obtaied usig to a Pigou-Dalto trasfer of amout ε from the richest idividual to the secod poorest oe thick dashed lie. Sice x 8 x 2 = x 2 x 1, if oe agrees with the dimiishig trasfer priciple, oe should prefer the first trasfer to the secod oe. But, i the first case, we take moey from the secod poorest idividual, whereas i the secod case we take moey from the richest oe. Moreover, i the first case, the share of the two poorest idividuals icome i the total icome remais costat, whereas it icreases i the secod case. Assume, ow, that we evaluate icome distributios through a additively separable evaluatio fuctio which respects the priciple of dimiishig trasfer e.g., a vo Neuma- Morgester fuctioal with a utility fuctio u such that u x > 0 ad u x > 0. The a ew problem arises, as show by Figure 3. Iclude figure 3 : secod objectio to the priciple of dimiishig trasfer Here, there are two distributios, oe represeted by a thick lie X ad the other represeted by a thi lie Y. I both distributios, we cosider a Pigou-Dalto trasfer of amout ε, from x 3 to x 1 ad from y 7 to y 5, respectively. Sice x 3 x 1 = y 7 y 5 ad y 5 < x 1, the gai evaluated by a evaluatio fuctio which respects the dimiishig trasfer priciple of the trasfer i the distributio Y is higher tha the oe of the trasfer i the distributio X. But i 5

7 the distributio X the trasfer cocers the poorest idividual, whereas it cocers the fourth richest oe i the distributio Y. All this leads us to itroduce the priciple of dual dimiishig trasfer, which is defied as follows: Defiitio 3 A decisio maker respects the priciple of dual dimiishig trasfer if for all Y belogig to Ω, i 1 < i 3, i 1 i 2 = i 3 i 4 < 0, ad all ε > 0 such that y i1 +ε y i1 +1, y i2 1 y i2 ε, y i3 + ε y i3 +1, y i4 1 y i4 ε, Y 1 = Y + ε e i1 e i2 Y + εe i3 e i4 = Y 2. Hece, a decisio maker respects the priciple of dual dimiishig trasfer if a trasfer from a idividual with rak i to oe with rak i p has a greater impact o social welfare the lower i is. This priciple is illustrated o the followig Figure. Iclude Figure 4 : The priciple of dual dimiishig trasfer Of course, the priciple of dual dimiishig trasfer is also debatable, sice it does ot take ito accout the gap of icome betwee idividuals cocered by the trasfer. Hece, there is o obvious reaso to prefer oe priciple to the other. Admittedly, that is a matter of taste. Therefore, we defie the priciple of strog dimiishig trasfer as follows: Defiitio 4 A decisio maker respects the priciple of strog dimiishig trasfer if for all Z belogig to Ω, i 1 < i 3, i 1 i 2 = i 3 i 4 < 0 ad all ε > 0 such that z i1 +ε z i1 +1, z i2 1 z i2 ε, z i3 + ε z i3 +1, z i4 1 z i4 ε, z i2 z i1 = z i4 z i3, Z 1 = Z + ε e i1 e i2 Z + ε e i3 e i4 = Z 2. Whereas it is uclear if oe should prefer the priciple of dimiishig trasfer or the priciple of dual dimiishig trasfer, it seems that, at least, oe should agree with the priciple of strog dimiishig trasfer obviously, if the decisio maker respects the priciple of dimiishig trasfer or the priciple of dual dimiishig trasfer, he also respects the priciple of strog dimiishig trasfer the coverse, however, is ot true. Hece, it seems that the priciple of strog dimiishig trasfer does really make sese. By defiitio, X 1 X 2 resp. Y 1 Y 2 or Z 1 Z 2 is equivalet to W X 1 W X 2 resp. W Y 1 W Y 2, or W Z 1 W Z 2 which, sice X 1 = X 2, Ȳ1 = Ȳ2, ad Z 1 = Z 2 is equivalet to I X 1 I X 2 ad J X 1 J X 2 resp. I Y 1 I Y 2 ad J Y 1 J Y 2, ad I Z 1 I Z 2 ad J Z 1 J Z 2. We will therefore, i the sequel, restrict our attetio to the social welfare fuctios W. 6

8 Note that all the preset discussio may be exteded to the problem of decisio uder risk, iterpretig W as a idividual utility fuctio ad Ω as a set of lotteries. I this cotext, respectig the priciple of dimiishig trasfer is similar to prudece see, e.g., Kimball 1990, Eeckhoudt, Gollier ad Scheider Similarly respectig the priciple of dual dimiishig trasfer may be traslated i the field of risk aversio. 3 Characterizatio of the priciples of dimiishig trasfer ad dual dimiishig trasfer I the sequel, we cosider a utility fuctio u, cotiuous o R +, strictly icreasig o R + ad three times differetiable o R ++, ad a frequecy trasformatio fuctio f, cotiuous o [0, 1], strictly icreasig o [0, 1], three times differetiable o ]0, 1[ ad such that f 0 = 0 ad f 1 = 1. Our mai results are as follows. Theorem 1 For a decisio maker who behaves i accordace with the Expected Utility model, the two followig propositios are equivalet: i the decisio maker respects the priciple of dimiishig trasfer, ii u x 0 for all x i R ++. Proof: See the appedix. This result is already kow whe u is cocave see, e.g., Kolm 1976 ad Eeckhoudt, Gollier ad Scheider It appears that ideed o coditio o the cocavity of u is eeded to prove this result. Theorem 2 For a decisio maker who behaves i accordace with Yaari s dual model, the two followig propositios are equivalet: i the decisio maker respects the priciple of dual dimiishig trasfer, ii f p 0 for all p i ]0, 1[. Proof: See the appedix. Note that a similar result ca be foud i Mehra 1976 for cotiuous icome distributios. Oe may woder uder what coditios a decisio maker who behaves i accordace with the RDEU model respects the priciple of dimiishig trasfer or the priciple of dual dimiishig trasfer. Ufortuately, as show by the two followig theorems, it appears that if such a decisio 7

9 maker respects the priciple of dimiishig trasfer, the he behaves i accordace with the EU model. O the other had, if he respects the priciple of dual dimiishig trasfer, the he behaves i accordace with Yaari s dual model. Theorem 3 If a decisio maker who behaves i accordace with the RDEU model respects the priciple of dual dimiishig trasfer, the ux = x for all x i R +, up to a icreasig affie trasformatio. Proof: See the appedix. Theorem 4 If a decisio maker who behaves i accordace with the RDEU model respects the priciple of dimiishig trasfer, the f p = p for all p i ]0, 1[. Proof: See the appedix. It immediately follows from Theorems 3 ad 4 that if a decisio maker who behaves i accordace with the RDEU model respects the priciple of dimiishig trasfer ad the priciple of dual dimiishig trasfer, the his social welfare fuctio reduces to W X = 1 i=1 x i. This result motivates the itroductio of the priciple of strog dimiishig trasfer ad the followig results. Theorem 5 If a decisio maker who behaves i accordace with the RDEU model respects the priciple of strog dimiishig trasfer, the f p 0 for all p i ]0, 1[ ad u x 0 for all x i R ++. Proof: See the appedix. Remark: The coverse of Theorem 5 does ot hold, as show by the followig example. Example Let us choose the utility fuctio u x = x 2 for all x i R +, ad the frequecy distortio fuctio f p = p 2 for all p i [0, 1]. Obviously, u x 0 for all x i R + ad f p 0 for all p i [0, 1]. Note, furthermore, that u x = f x for all x i [0, 1]. Now, defie Z 1 ad Z 2 as i Defiitio 4, choosig = 8, i 1 = 1, i 2 = 3, i 3 = 5, i 4 = 7, ε = 1 ad: 0 < z i1 = z = i 4 < z i2 = z + α = i 3 < z i3 = y = i 2 < z i4 = y + α = i 1 < 1. Thus, α = 2. With these requiremets, the expressio δ = W u Z 1 W u Z 2 becomes: δ = [uz i1 + ε u z i1 ][u z i4 + ε u z i4 ] [u z i2 u z i2 ε][u z i3 + ε u z i3 ] [u z i3 + ε u z i3 ][u z i3 + ε u z i2 ] + [u z i4 u z i4 ε][u z i1 + ε u z i1 ]. 8

10 Therefore, δ = [uz i1 + ε u z i1 ][u z i4 + ε u z i4 ε] [u z i3 + ε u z i3 ][uz i2 + ε u z i2 ε] = [uz + ε u z][u y + α + ε u y + α ε] [uy + ε u y][u z + α + ε u z + α ε] [ = z + ε 2 z 2][ y + α + ε 2 y + α ε 2] [y + ε 2 y 2][ z + α + ε 2 z + α ε 2] = 4ε 2 y zε 2α < 0. Hece, there are f ad u with f 0 ad u 0 such that a decisio maker who behaves i accordace with the RDEU model does ot respect the priciple of strog dimiishig trasfer. Theorem 6 For a decisio maker who behaves i accordace with the RDEU model, the two followig propositios are equivalet: i the decisio maker respects the priciple of trasfer ad the priciple of strog dimiishig trasfer, ii f p 0, u x 0, f p 0 ad u x 0 for all p i ]0, 1[ ad all x i R ++. Proof: See the appedix. It may be of iterest to ote that a couterpart of Theorem 6 exists whe the decisio maker does t like ay Pigou-Dalto trasfer. Of course, such a assumptio does t really make sese i the framework of iequality measuremet. O the other had, it may be of iterest i the cotext of a decisio maker s attitude towards risk. I this case, we say that ay decisio maker who respects the priciple of trasfer is risk averse i.e., he respects secod order stochastic domiace, whereas a decisio maker who does t like ay Pigou-Dalto trasfer is a risk lover. We the have the followig result. Theorem 7 For a decisio maker who behaves i accordace with the RDEU model, the two followig propositios are equivalet: i the decisio maker is a risk lover ad respects the priciple of strog dimiishig trasfer, ii f p 0, u x 0, f p 0 ad u x 0 for all p i ]0, 1[ ad all x i R ++. Proof: Similar to the proof of Theorem 6. As we ca see, the coditios o the sig of the third derivatives of u ad f are the same whe the decisio maker is risk averse ad whe he is risk lover. Furthermore, it should be oted that, ulike i Example 1, u ad f have opposite sigs i Theorems 6 ad 7. 9

11 4 Applicatio : geeralized Gii idices ad super-geeralized Gii idices We will ow apply these results to the problem of iequality measuremet. Let us cosider the large class of iequality idices, itroduced by Yaari These idices are defied as 3 : Applyig Theorem 2, it follows that: I GG X = 1 W X X. Theorem 8 A Yaari iequality idex satisfies the priciple of dual dimiishig trasfer if ad oly if f p 0 for all p ]0, 1[. 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 4 : I SG X = 1 i=1 [ i+1 δ i ] δ x i X with δ 1. These idices are based o the followig social welfare fuctio: [ ] i + 1 i W SG X = f f x i. i=1 with f p = p δ. Note that for δ = 2, I SG is othig but the Gii idex. Furthermore, the decisio maker is more iequality averse the higher δ is. Applyig Theorem 2 agai, we the obtai: Theorem 9 A S-Gii idex respects the priciple of dual dimiishig trasfer if ad oly if δ 2. Hece, the Gii idex correspods to the lowest value of δ for which I SG respects the priciple of dual dimiishig trasfer. Oe may ask if it is possible to defie a iequality idex which respects the priciple of dimiishig trasfer ad the priciple of dual dimiishig trasfer. We have o aswer to this questio. Nevertheless, the very geeral iequality idex let us call it a super-geeralized Gii idex, I SSG X = 1 u 1 [ i=1 f i+1 f i ] u xi. X 3 We will cosider, without loss of geerality, relative Yaari idices. 4 Ideed, these idices are coceived by Doaldso ad Weymark 1980 ad Bossert 1990 as a extesio of Weymark s idices But retrospectively these idices appear as a particular case of Yaari idices. 10

12 cosidered by Ebert 1988 ad Chateaueuf 1996 is based o a RDEU-like social welfare fuctio, with a utility fuctio u ad a frequecy trasformatio fuctio f. Applyig Theorem 3, it immediately follows that a super-geeralized Gii idex respects the priciple of trasfer ad the priciple of strog dimiishig trasfer if ad oly if u x 0, u x 0 for all x i R ++ ad f p 0, f p 0 for all p i ]0, 1[. This may be see as a argumet i favor of this particular class of idices. 5 Appedix The proofs rely o the followig lemma: Lemma Let g be a cotiuous real fuctio o I = [0,a], a R + resp. I = R + three times differetiable o ]0,a[ resp. o R ++. The g x 0 for all x i ]0,a[ resp: i R ++ if ad oly if 3 x; a 1,a 2,a 3 0 for all x, a 1, a 2, a 3 i I such that x + a 1 + a 2 + a 3 belogs to I, where 3 x;a 1,a 2,a 3 is defied by: 3 x; a 1,a 2,a 3 = g x + a 1 + a 2 + a 3 g x + a 1 + a 2 g x + a 1 + a 3 g x + a 2 + a 3 +g x + a 1 + g x + a 2 + g x + a 3 g x. Such a result is proved i Choquet 1954 p. 149 i the case where I = R + ad g is three times differetiable o I = R +, from which ca be readily deduced the slight geeralizatio above. Proof of Theorem 1 ii = i A decisio maker who behaves i accordace respects the priciple of dimiishig trasfer if ad oly if for all X 1 ad X 2 as i Defiitio 2, U X 1 U X 2 0. A simple computatio leads to: U X 1 U X 2 = u x i4 u x i4 ε u x i3 + ε u x i2 1 +u x i3 + u x i2 ε + u x i1 + ε u x i1. Let x = x i1, a 1 = ε, a 2 = x i2 x i1 ε, a 3 = x i3 x i1. Note that x, a 1, a 2 ad a 3 belog to R +. Oe easily checks that: U X 1 U X 2 = 3 x; a 1, a 2, a 3 11

13 Hece, from the Lemma, if u x 0 for all x i R ++ the U X 1 U X 2. i = ii Let y, x be fixed, with y > x > 0. We have to prove that u y u x. Let x i4 = y + α, x i3 = y, x i2 = x + α ad x i1 = x, where α > 0 is arbitrarily chose such that x i3 x i2. Divide both sides of equatio 1 by ε > 0, ad let ε ted to 0. Oe the obtais: u y + α u y u x + α u x. Divide both sides of this iequality by α ad let α ted to 0. It follows that u y u x. Proof of Theorem 2 ii = i Assume that f p 0 for all p belogig to ]0, 1[ ad let us show that W Y 1 W Y 2 with Y 1 ad Y 2 as i Defiitio 3. A simple computatio gives: i1 i2 W Y 1 W Y 2 = y i1 + εψ + y i2 εψ + y i3 Ψ i1 i2 i3 y i1 Ψ y i2 Ψ y i3 + εψ i3 + y i4 Ψ i4 y i4 εψ Hece, W Y 1 W Y 2 = εd, with: i1 + 1 i1 i2 + 1 i3 + 1 d = f f f f i2 i3 i4 + 1 i4 +f + f + f f. Let x = i 4, a 1 = 1, a 2 = i 4 i 3, a 3 = i 4 i 2 ; x ad a i belogig to [0, 1] ad d = 3 x;a 1,a 2,a 3. Hece, from the Lemma, d 0, which gives the desired result. i = ii i4 From cotiuity of f o ]0, 1[, it is eough to prove that for ay give p 1, p 2 belogig to Q ]0, 1[ with p 2 > p 1, we obtai f p 2 f p 1. Let us choose a sufficietly small q belogig to the set Q ++ of positive ratioals, such that p 1 + q < p 2 ad p 2 + q < 1 ad let i the set N of positive itegers be chose such that p 2 = m 2, p 1 = m 1, q = m. 12.

14 Cosider ow Y 1 ad Y 2 of size k, with k belogig to N, defied as i defiitio 3 ad with i 1 = k k m + m 2, i 2 = k km 2, i 3 = k k m 1 + m, i 4 = k km 1. W Y 1 W Y 2 0 therefore implies: Divide by 1 k [ f p 2 + q + 1 k [ f p 1 + q + 1 k ] f p 2 + q ] f p 1 + q [ ad let k ted to +. It follows that: f p k p k [ f f p 2 + q f p 2 f p 1 + q f p 1. ] f p 2 ] f p 1. Dividig ow by q ad lettig q tedig to 0 gives the desired result: f p 2 f p 1. Proof of Theorem 3 We first prove that u x 0 for all x i R ++ Let 1 p q belogig to Q ]0,1[ be such that f 1 p q > 0. Cosider Y = y 1,..., y, ad Y 1 ad Y 2 as i Defiitio 3. We assume that = qrm, where r > 3 ad m are arbitrarily chose i N. Let i 1 = p r 3 m, i 2 = p r 2m, i 3 = p r 1m, i 4 = prm. Fially, let y i1, y i2, y i3, y i4 ad α > 0 be arbitrarily chose such that 0 < y i1 < y i2 = y i1 + α < y i3 < y i4. The, W u Y 1 W uy 2 0 with ε sufficietly small implies: i4 i3 [u y i4 u y i4 ε]ψ [u y i3 + ε u y i3 ] Ψ [u y i1 + α u y i1 + α ε]ψ i2 [u y i1 + ε u y i1 ]Ψ i1. 2 First, let r be costat. Divide expressio 2 by 1 qrm [u y i4 u y i4 ε]f 1 p q [uy i1 + α u y i1 + α ε]f 1 p q + 2 p qr Now, let r ted to +. It follows that: f 1 p q ad let m ted to +. It follows that: [u y i3 + ε u y i3 ]f 1 p q + p qr [u y i1 + ε u y i1 ]f 1 p q + 3 p qr δ 0, where. δ = [u y i4 u y i4 ε] [u y i3 + ε u y i3 ] [u y i1 + α u y i1 + α ε] + [u y i1 + ε uy i1 ]. 13

15 Hece, δ 0. Now, divide δ by ε > 0 ad let ε ted to 0. Oe obtais: u y i4 u y i3 u y i1 + α u y i1. Fially, let α ted to 0. It follows that: u y i4 u y i3 0 for all 0 < y i3 < y i4. Hece, u x 0 for all x i R ++. We ow prove that u x 0 for all x i R ++ With the same otatio as i the first part of the proof, let y i1, y i2, y i3, y i4 ad β > 0 be arbitrarily chose such that 0 < y i1 < y i2 < y i3 = y i4 β < y i4. The, W u Y 1 W uy 2 0 with ε sufficietly small implies: i4 i3 [u y i4 u y i4 ε]ψ [u y i4 β + ε u y i4 β]ψ i2 i1 [u y i2 u y i2 ε]ψ [u y i1 + ε u y i1 ]Ψ. The same argumet as i the first part of the proof leads to: Now, let β ted to 0. It follows that: u y i4 u y i4 β u y i2 u y i1. u y i2 u y i1 0 for all 0 < y i1 < y i2. Hece, u x 0 for all x i R ++. Hece, u x 0 ad u x 0 for all x i R ++. Therefore, u is affie o R ++. By cotiuity of u o R +, this implies that u is also affie o R +. Hece, ux = x o R +, up to a positive affie trasformatio. Proof of Theorem 4 We first prove that f p 0 for all p i ]0,1[ 14

16 Let a b < a b belogig to Q ]0, 1[, ad x > 0 be such that u x > 0. Cosider X = x 1,..., x ad X 1, X 2 as i Defiitio 2. We assume that = bb rm, where r > 2 ad m are arbitrarily chose i N. Let i 1 = b a b r 2 m, i 2 = b ab r 1 m, i 3 = b a b rm ad i 4 = bb a rm. It is immediate that i 1 < i 2 < i 3 < i 4. Fially, let α > 0 be arbitrarily chose, ad let x i1, x i2, x i3, x i4 be such that x i1 = x < x i2 = x i1 + α < x i3 = x i1 + 2α < x i4 = x i1 + 3α. The, W u X 1 W u X 2 0 with ε sufficietly small implies: i4 i3 [u x i1 + 3α u x i1 + 3α ε]ψ [u x i1 + 2α + ε u x i1 + 2α]Ψ 3 i2 i1 [u x i1 + α u x i1 + α ε]ψ [u x i1 + ε ux i1 ] Ψ. Divide 3 by ε ad let ε ted to 0. It follows that: u i4 x i1 + 3αΨ u i3 x i1 + 2α Ψ u x i1 + αψ Now let α ted to 0. We obtai: [ u x i1 Ψ i4 Ψ i3 Ψ i2 + Ψ i2 u x i1 Ψ ] i1 0. i1. Sice u x i1 > 0, this implies that [ Ψ i 4 Ψ i3 Ψ i2 + Ψ i1 ] 0. Hece: [ a f b + 1 ] a bb f rm b [ b ar 2 + f ] b ar 2 br bb f 1 rm br [ b ar 1 f ] b a r 1 br bb f 1 rm br [ a + f b + 1 a bb f rm b ] 4 First, let r be costat. Divide 4 by 1 bb rm f a b + f 1 b ar 2 br Now, let r ted to +. We the have: f a b ad let m ted to +. It follows that: b ar 1 f a 0. br b f 1 f a b for all 0 < a b < a b. Hece, f p 0 for all p i Q ]0,1[, ad by cotiuity o ]0, 1[. 15

17 We ow prove that f p 0 for all p i ]0,1[ Let a b < a b belogig to Q ]0, 1[, ad x > 0 be such that u x > 0. Cosider X = x 1,..., x ad X 1, X 2 as i Defiitio 2. We assume that = bb rm, where r ad m are arbitrarily chose i N, such that 1 a b r r+2. Let i 1 = b ab rm, i 2 = b b a rm, i 3 = b b a r + 1m ad i 4 = b b a r + 2 m. It is immediate that i 1 < i 2 < i 3 < i 4. Fially, let α > 0 be arbitrarily chose, ad let x i1, x i2, x i3, x i4 be such that x i1 = x < x i2 = x i1 + α < x i3 = x i1 + 2α < x i4 = x i1 + 3α. As i the first part of the proof, W u X 1 W u X 2 0 with ε sufficietly small implies that [ Ψ Hece: i4 Ψ i3 Ψ i2 + Ψ ] i1 0. [ f 1 b a r + 2 b + 1 r bb f 1 b a ] r + 2 rm b r [ f 1 b a r + 1 b + 1 r bb f 1 b a ] r + 1 rm b r [ a f b + 1 ] [ a a bb f rm b + f b + 1 a ] bb f 0 rm b 5 First, let r be costat. Divide 5 by 1 bb rm f a + f 1 b a r + 2 b b r Now, let r ted to +. We the have: f a a f b b ad let m ted to +. It the follows that f 1 b a r + 1 a b f r b for all 0 < a b < a b. Hece f p 0 for all p i Q ]0,1[, ad by cotiuity o ]0, 1[. We fially have f p = 0 for all p i ]0,1[. Sice f 0 = 0 ad f 1 = 1, this implies f p = p. Proof of Theorem 5 Cosider x belogig to R ++ ad let us show that u x 0 Let 1 p q 1 belogig to Q ]0, 1[ be such that f p q > 0. Cosider Z 1 ad Z 2 defied as i defiitio 4 of size = q r m, where r ad m are arbitrarily fixed i N, ad i 1 = pr m, 16

18 i 2 = p r + 1 m, i 3 = p r + 2 m, i 4 = p r + 3 m, ad z i1 = z, z i2 = z + α, z i3 = y, z i4 = y + α, where y > x ad α > 0 are arbitrarily chose. W u Z 1 W u Z 2 0 with ε > 0 sufficietly small the implies: i4 [u y + α u y + α ε]ψ [u y + ε u y]ψ i2 [u z + α u z + α ε]ψ [u z + ε u z]ψ i3 i1. 1 First, let r be fixed. Dividig by q r m, ad lettig m ted to +, it follows that [u y + α u y + α ε]f 1 p q 3 p [u y + ε u y]f 1 p q 2 p [u z + α u z + α ε]f 1 p q p qr Now, let r ted to +, it follows that f 1 p q qr [u z + ε u z]f 1 p q δ 0, where: δ = [u y + α u y + α ε] [u y + ε uy] [u z + α u z + α ε]+u z + ε u z.. qr to: Hece, successively dividig by ε > 0, ε tedig to 0, ad by α > 0, α tedig to 0, leads us u y u z, the desired result. Let us ow prove that for give p 1, p 2 belogig to Q ]0, 1[ with p 2 > p 1, we obtai f p 2 f p 1 Defie Y 1 ad Y 2 of size k as i the proof of the ecessary part of Theorem 2, but with y i2 y i1 = y i4 y i3 as i defiitio 4. W u Y 1 W u Y 2 0 implies: [u y i1 + ε u y i1 ]Ψ [u y i3 + ε u y i3 ] Ψ i1 i3 [u y i2 uy i2 ε]ψ + [u y i4 u y i4 ε]ψ Divide the left had side of 6 by ε > 0 ad let ε ted to 0. It follows that u i1 y i1 Ψ u i2 y i2 Ψ u i3 y i3 Ψ + u i4 y i4 Ψ 0 i2 i Let y > 0 be chose belogig to R ++ such that u y > 0. Let y i1, y i2, y i3, y i4 coverge to y. Sice u is cotiuous o R ++ ad u y > 0, oe obtais: i1 i2 i3 i4 Ψ Ψ Ψ + Ψ 0 17

19 ad the proof ca be completed as i Theorem 2. Proof of Theorem 6 i = ii We kow from Chew, Kari ad Safra 1987 that a decisio maker who behaves i accordace with the RDEU model with u ad f twice differetiable respects secod order stochastic domiace if ad oly if u is cocave ad f is covex. But it is well-kow see e.g. Atkiso 1970 that respectig secod order stochastic domiace for distributios with equal meas is equivalet to respectig the priciple of trasfer. Hece, if the decisio maker respects the priciple of trasfer, f p 0 for all p i ]0,1[ ad u x 0 for all x i R ++. A direct applicatio of Theorem 5 completes this part of the proof. ii = i We kow from Chew, Kari ad Safra 1987 that if u is cocave ad f is covex, the the decisio maker respects the priciple of trasfer. Hece, it remais to prove that if u x 0 for all x i R ++ ad f p 0 for all p i ]0, 1[, the the decisio maker respects the priciple of strog dimiishig trasfer. Cosider Z 1 ad Z 2 as i defiitio 4. We have to prove that W uz 1 W u Z 2 0, which reduces to provig that δ 0, where δ is defied by: i1 i2 δ = [u z i1 + ε u z i1 ]Ψ [u z i2 u z i2 ε]ψ i3 i4 [u z i3 + ε u z i3 ]Ψ + [uz i4 u z i4 ε]ψ. By the symmetry of δ i i 2 ad i 3, we may assume without loss of geerality that i 1 < i 2 i 3 < i 4. Hece, z i1 < z i2 z i3 < z i4. Let a j u z ij + ε u z ij for j i {1,3}, aj u z ij u zij ε for j i {2, 4}, ad ij b j Ψ for j i {1,...,4}. Sice u z 0 for all z i R ++, we kow see Theorem 1 that a decisio maker who behaves i accordace with the expected utility model ad whose utility fuctio is u respects the priciple of dimiishig trasfer. Hece, a a 4 a 3 a 2 + a 1 0. Sice f p 0 for all p i ]0, 1[, we deduce from Theorem 2 that a decisio maker who behaves i accordace with Yaari s dual model ad whose frequecy trasformatio fuctio is f respects the priciple of dual dimiishig trasfer. Hece, b b 4 b 3 b 2 + b

20 But δ = a 4 b 4 a 3 b 3 a 2 b 2 + a 1 b 1. Hece, δ = a 4 b + b 1 a + d with d = b 2 b 1 a 4 a 2 + b 3 b 1 a 4 a 3. Nodecreasigess of u ad f imply a 4 0, b 1 0. Hece, a 4 b + b 1 a 0. Covexity of f ad cocavity of u respectively imply b 1 b 2, b 1 b 3, a 2 a 4, a 3 a 4. Hece, d 0 ad therefore W u Z 1 W u Z 2 0. Refereces Atkiso, A. 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 Iequality : a Approach through o-additive Models, Cahiers EcoMaths, 96.58, Uiversité Paris I. Chateaueuf, A., ad P.-H. Wilthie 1999: Third Iverse Stochastic Domiace, Lorez Curves ad Favourable Double Trasfers, Cahiers de la MSE, , Uiversité de Paris I. Chew, S., E. Kari, ad Z. Safra 1987: Risk aversio i the theory of expected utility with rak depedet prefereces, Joural of Ecoomic Theory, 42, Choquet, G. 1954: Théorie des capacités, Aales de l Istitut Fourier, V, Davies, J., ad M. Hoy 1994: The Normative Sigifiace of Usig Third-Degree Stochastic Domiace i Comparig Icome Distributio, Joural of Ecoomic Theory, 64, Doaldso, D., ad J. Weymark 1980: A Sigle-Parameter Geeralizatio of the Gii Idices of Iequality, Joural of Ecoomic Theory, 22, Ebert, U. 1988: Measuremet of iequality: A attempt at uificatio ad geeralizatio, Social Choice ad Welfare, 5, Eeckhoudt, L., C. Gollier, ad T. Scheider 1995: Risk-aversio, prudece ad temperace: A uified approach, Ecoomics Letters, 48, Kakwai, N. C. 1980: O a class of poverty measures, Ecoometrica, 48, Kimball, M. 1990: Precautioary savig i the small ad i the large, Ecoometrica, 58,

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