Transfers Principles in a Generalized Utilitarian Framework

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1 Transfers Principles in a Generalized Utilitarian Framework Mickaël Beaud Lameta Université Montpellier I Stéphane Mussard Lameta Université Montpellier I Marc Dubois Lameta Université Montpellier I Abstract In a Generalized Utilitarian framework, it is possible to represent formally the decisionmaker s value judgments about inequality. In this framework, the necessary and sufficient conditions to fulfill the Pigou-Dalton and the Diminishing Transfers Principle are provided. The normative contents of these principles depend on assumptions on the individuals utility function. On the other hand, these principles display weaker value judgments about inequality than it is usually claimed in a pure utilitarian framework. We investigate more precisely the normative contents of stochastic dominance results at the orders 2 and 3. Classification: D3; D6; H2. Keywords: Social Welfare; Transfers Principles; Inequality Aversion. We thank Rolf Aaberge and Lucio Esposito for their helpful comments. Previous versions of this paper were presented at PET 2013 Lisbon and ECINEQ 2013 Bari conferences. The usual disclaimer applies. Lameta Université Montpellier I, Faculté d Economie, Av. Raymond Dugrand, Site de Richter C.S , Montpellier Cedex 2, France. mbeaud@univ-montp1.fr. Lameta Université Montpellier I, Faculté d Economie, Av. Raymond Dugrand, Site de Richter C.S , Montpellier Cedex 2, France. dubois2@lameta.univ-montp1.fr. I wish to thank the Grédi s members of the Université de Sherbooke, in which some part of this work was written. I thank finally Saghar Saïdi for her suggestions. Lameta Université Montpellier I, Faculté d Economie, Av. Raymond Dugrand, Site de Richter C.S , Montpellier Cedex 2, France. Tel: 33 (0) / Fax : 33 (0) smussard@adm.usherbrooke.ca, Associate researcher at Grédi, Université de Sherbrooke. 1

2 1 Introduction According to Blackorby, Bossert and Donaldson (2002, p. 545): a family of principles whose value functions have the same additively separable mathematical structure as the utilitarian value function is the Generalized Utilitarian family of principles [...]. Each of these principles employs transformed utilities and some exhibit aversion to utility inequality. In an Utilitarian framework, the concept of inequality aversion is related to inequality of utilities since only individuals utilities are taken into account to evaluate social states. The so-called Minimal Equity axiom exhibits this inequality aversion. Starting from an initial state, it postulates that a decision maker prefers a social policy in which the utility is improved for a worse-off person compared with another social program in which the utility has been improved for a better-off individual. 1 Minimal Equity seems to be close to the Pigou-Dalton Transfers Principle. The latter postulates that a rank-preserving transfer of income from an individual to a poorer one should increase the social welfare. In a Generalized Utilitarian framework, both principles do not have the same involvements. If the transformation applied to individual utilities is (strictly) concave, the resulting ordering represents (strict) inequality aversion i.e. it satisfies Minimal Equity but it does not imply the Pigou-Dalton Transfers Principle, see Blackorby, Bossert and Donaldson (2002, p. 556). Fishburn and Willig (1984) and Foster and Shorrocks (1987, 1988a), among others, have analyzed the Transfers Principles within the field of normative inequality and poverty measurement, with a social welfare function (henceforth referred to as SWF ) directly inspired from Utilitarianism in Welfare economics. The SWF usually employed is the sum of individuals utility over income. Blackorby, Bossert and Donaldson (2002, p. 556) explain that any utilitarian SWF is a special case of Generalized Utilitarianism in which the transformation [of utilities] is affine, i.e., it displays inequality neutrality. Indeed, if a Generalized Utilitarian decision-maker assumes that the utility function is increasing and strictly concave over all incomes, then the income transfer improves the utility of the income recipient in a more valuable way than it decreases the utility of the income donor. This transfer strictly improves the amount of social welfare, nevertheless it does not imply any inequality aversion. In other words, decreasing marginal utility of individuals is the necessary and sufficient condition to fulfill the Pigou-Dalton Transfers Principle as long as the decision-maker is inequality neutral. Kaplow (2010, p. 27) argues that it seems very hard to capture the normative content about attitude towards inequality and individuals utility by using an Utilitarian approach only. He asserts that this process yields incomplete and potentially misleading interpretations. The aim of this paper is to provide precise interpretations of the normative content of income Transfers Principles. For this purpose, we study the necessary and sufficient conditions to fulfill such principles 1 This axiom may be interpreted with a transfer of utility form a better-off person to a worse-off one, so that the social welfare increases. This axiom is presented in D Aspremont and Gevers (1977) and Blackorby, Bossert and Donaldson (2002) among others. In this paper, we interpret it as a Transfers Principle in order to link it with the Pigou-Dalton Transfers Principle. In the same manner that Fleurbaey and Michel (2001), we assume that utilities are perfectly measurable and interpersonally comparable. 2

3 (Pigou-Dalton transfers and Kolm s diminishing transfers) in a Generalized Utilitarian framework. Kaplow (2010) is the first claiming that both value judgments about inequality and individuals utility are necessary to determine the social marginal value of income redistribution. Here we demonstrate that there is no straightforward relationship between attitude towards inequality and the fulfillment of any Transfers Principle insofar this fulfillment depends on assumptions about individuals utility. We deepen Kaplow s (2010) results by displaying that the Pigou-Dalton Transfers Principle is relevant with a SWF exhibiting moderate inequality-loving as long as one assumes that individuals have increasing utility over income which is marginally decreasing. Moreover, Kolm s (1976) Diminishing Transfers Principle is relevant with a SWF exhibiting inequality aversion and which prescribes (moderately) more sensibility to transfers at the top of the distribution of utilities. Our article defends Ng s (1972) prescription to make clear and precise interpretations about the normative contents of stochastic dominance analyses at the orders 2 and 3. In Section 2, we present notations and definitions to expose Fishburn and Willig s (1984) theorem. The results are provided in Section 3. In Section 4, we discuss the prescriptions underlying our results. Precisely, in Kaplow (2010), the SWF contains the individuals utility function, which is defined to be an empirical phenomenon. We suggest another interpretation insofar we are aware that Utilitarianism relies on an introspective method aiming at ranking social states. 2 Notations and definitions Let P be a finite population of n individuals. Let Ω be the set of all rank-ordered income distributions (by ascending order). Let Y := (y 1, y 2,..., y n ) Ω be an income distribution such that y i R +, where R + is the non-negative part of the real line [R n + the n dimensional nonnegative Euclidean space]. We denote by e i = (0,..., 1, 0,..., 0) the n tuple whose only nonzero element occurs at the i th income level. The set of all social welfare functions is denoted by W. The reduced-form social welfare functions in W are defined as: SW (Y ) = n w(y i ), (1) i=1 where w is the individual s utility, being a real-valued and continuous function. The celebrated Pigou-Dalton Principle is the following. Definition 2.1. Pigou-Dalton Transfers Principle (Order 2). Consider two individuals indexed i and j, with i < j, and two income distributions Y, X Ω. Let X be obtained from Y by a progressive rank-preserving transfer of amount δ > 0 from agent j to i, such that: x i = y i + δ x i+1 (PDa) ; x j = y j δ x j 1 (PDb) ; x l = y l l i, j (PDc). SW satisfies the Pigou-Dalton Transfers Principle if, and only if: SW (X) := SW (Y + (e i e j ) δ) SW (Y ). (PD) 3

4 Given that all income distributions are rank-ordered by ascending order, then it turns out that the classical definition of the Pigou-Dalton Transfers Principle coincides with the definition introduced above, which defines rank-preserving progressive transfers of incomes. 2 The determination of the second-order derivative of w is derived from the Pigou-Dalton Principle, which provides w (2) 0. It means that a positive increment δ to an i th income-person, compared with a j th income-person (j > i), yields a better impact on the social welfare, the lower is the income recipient in the income distribution (the individuals ranking remains unchanged after such transfers). For this reason, w (1) 0 is a necessary condition to the fulfillment of the Pigou-Dalton Transfers Principle. That is, SW satisfies this principle if, and only if, w (1) 0 and w (2) 0. This Transfers Principle is the cornerstone of the theory of inequality measurement and it is well-known to display inequality aversion in an utilitarian framework. Kolm s (1976) Diminishing Transfers Principle is the following. Definition 2.2. Diminishing Transfers Principle (Order 3). The income distribution X is obtained from Y Ω by a rank-preserving progressive transfer of income from agent j to i and the distribution Z is obtained from Y by a rank-preserving progressive transfer of income from agent l to k, such that: y k < y l < y i < y j (KLMa) ; y i y j = y k y l < 0 (KLMb). SW satisfies the Diminishing Transfers Principle if, and only if: SW (X) := SW (Y + (e i e j )δ) SW (Y + (e k e l )δ) =: SW (Z). (KLM) The non-negative third derivative w (3) and w (1) 0 are the necessary and sufficient conditions for the fulfillment of the Diminishing Transfers Principle and no condition on the concavity of w is necessary to prove this equivalence (see Chateauneuf, Gajdos and Wilthien (2001), p. 319). 3 The interpretation is that the (KLM) principle exhibits a preference for a progressive rank-preserving transfer of income insofar the transfer is the lowest possible in the income distribution. This assertion is valid with a strictly inequality-averse decision-maker, i.e. w (2) < 0. If the social planner is inequality-loving, i.e. w (2) > 0, she respects the (KLM) principle if, and only if, she prefers a regressive rank-preserving transfer of income insofar the transfer is the highest possible in the distribution of incomes. Fishburn and Willig (1984) postulate an s-order Transfers Principle (s = 2, 3,...), the so-called generalized transfer principle associated with s-order stochastic dominance. For s 2, the idea is to compare a transfer of order s 1 among some individuals with another transfer of order s 1 among richer individuals. The implication of such generalized transfers is that we constrain 2 The classical definition of the Pigou-Dalton Transfers Principle only demands that a donor cannot become poorer than the receiver of a progressive transfer of income, see Pigou (1912) and Dalton (1920). The rank-preserving transfer s condition was introduced by Fields and Fei (1978). 3 More precisely, the authors postulate that w (1) > 0 throughout the paper but w (1) 0 does not change the results of Theorem 1, p

5 the sign of the sth derivative of w. Accordingly, letting C s be the set of all s-time differentiable functions, it is convenient to define { } Γ s := f C s ( 1) s f (s) (x) := ( 1) s f (s 1) (x) 0, x R x as the class of real-valued functions, which share the property of having their sth derivative nonpositive (non-negative) at an even (odd) order s, where s is a positive integer. In addition, we consider a more restrictive set Γ s := { f C s f Γ l, l = 1,..., s }, which is the set of all s-time differentiable functions for which their first s successive derivatives alternate in sign. Formally, Γ s = Γ 1 Γ 2 Γ s. Hence, we can state the following Theorem. Theorem 2.1. For any positive integer s, the two following statements are equivalent: (i) SW satisfies all Transfers Principles up to the sth order. (ii) w Γ s. Proof. Fishburn and Willig (1984). There is a straightforward relationship between the shape of w and the fulfillment of some income Transfers Principle. In order to interpret precisely the shape of w, we now define it as a composition of functions. This enables one to capture two things: the attitude towards inequality and the assumption on individuals utility to be made by a decision-maker. 3 A generalized utilitarian framework An extended-form of SWF in W, directly inspired by Generalized Utilitarianism, is: n SW G (U) = g(u(y i )), (2) where u is the utility common to each individual with U := (u(y 1 ),..., u(y n )) (u 1,..., u n ), where g represents the attitude towards inequality of utilities of the social planner. Throughout the Section, we assume that g is completely monotone, such that g Γ 1, moreover u (1) > 0 and U stands for the set utility distributions ranked by ascending order. Blackorby, Bossert and Donaldson (2002) assert that if g (2) 0 then the function SW G respects the minimal equity principle. Definition 3.1. Minimal Equity. U, V U, such that: i=1 Consider two individuals i, j and two utility distributions v j > u j > u i > v i (MEa) ; u l = v l l i, j (MEb). SW G satisfies the Minimal Equity Principle if, and only if: SW G (U) SW G (V ). (ME) 5

6 The minimal equity can be viewed as a transfer of utility from a better-off person to a worse-off one as long as the utility recipient does not become better-off compared with the person who loses utility after the transfer. Kaplow (2010, p. 26) claims that a greater social preference for equality [than inequality neutrality] implies that g is strictly concave in u. The Minimal Equity presented above implies the use of ordinal utilities. Actually, (ME) outlines some comparisons of utility levels but it does not deal with individual losses or gains of utility. In our cardinal framework, we postulate utility variations (or transfers) in order to clear implications between (ME) and the concavity of g. Claim 3.1. Let u (1) > 0 and g {Γ 1 C 2 }, then the following statements hold. (i) If u i v i =: α β := v j u j > 0 and if SW G respects (ME), then g (2) g (2) + 0. (ii) If u i v i =: α β := v j u j > 0 and if g (2) 0, then (ME) is respected. Proof. See the Appendix. Claim 3.1 shows, on the one hand (i), that the extended social welfare functions SW G respecting the minimal equity does not always provide a concave function g. It can be locally convex if an additional condition is not stated. This condition postulates that a social reform improves the utility of worse-off agents compared with better-off ones, i.e., the condition α β holds. On the other hand (ii), any social reform respecting α β for which the function g is concave yields the respect of the minimal equity. As a consequence, setting α = β yields the idea of a mean preserving transfer of utilities (see the conditions PDa and PDb for income transfers). In this case, the concavity of g implies and is implied by (ME). Then, whenever g (1) 0 (Pen s criterion 4 ) we consider that [strict] concavity of g is equivalent to [strict] inequality aversion whenever the condition on worse-off agents is at least guaranteed: α = u i v i = v j u j = β 0. (MEc) In the remainder of the paper, the cardinal condition (MEc) will be included in the Minimal Equity principle in order to get a clear equivalence between inequality aversion and the concavity of g. In what follows, we aim at outlining that (PD) and (KLM) principles, based on income transfers resulting for instance from income tax reforms, do not always coincide with inequality aversion as defined above. Starting from the fact that w(y i ) could be a reduced form of the composite function g(u(y i )), then there is no straightforward relationship between attitude towards inequality and the fulfillment of a Transfers Principle in an extended-form framework. In the reduced utilitarian layout, w (2) [<] 0 is defined to be equivalent to [strict] aversion to inequality (whenever w (1) 0) and also equivalent to (PD). The second derivative of w is as follows: w (2) = g (1) u (2) + g (2) [ u (1)] 2. (3) 4 Roughly speaking, Pen criterion demands that if one person improves his utility level by a change of income, ceteris paribus, then the social welfare should increase. This also demands the symmetry of the SWF: the individuals have to be treated symmetrically (impartially). 6

7 It is clear that u Γ 2 and g Γ 2 imply w Γ 2, but the reverse is false. Then, there is no equivalence between inequality aversion (ME) and the fulfillment of (PD) in a framework with an extended-form of SWF. The third derivative of w is as follows: w (3) = g (1) u (3) + 3g (2) u (1) u (2) + g (3) [ u (1)] 3. (4) In order to fulfill a s-order Transfers Principle, there are two sufficient conditions concerning the shape of u and the attitude towards inequality of a decision-maker (i.e. the shape of g). Lemma 3.1. The following statement is true for any positive integer s: H s : u Γ s and g Γ s together imply w Γ s. Proof. See the Appendix. Lemma 3.1 shows that a SW G which fulfills the Pigou-Dalton Transfers Principle can be an utilitarian one, i.e. it can display inequality neutrality. In this case, g (2) = 0 and u (2) 0: the decision-maker does not care about inequality but a progressive income transfer has a nonnegative impact on the social welfare. Moreover, a SW G which fulfills the Pigou-Dalton Transfers Principle can display some moderate inequality loving whereas Lemma 3.1 does not exhibit it (because the alternating signs for u and g are only sufficient conditions). The normative content about attitude towards inequality of this Transfers Principle is very different in our framework compared with the reduced utilitarian one, in which the concavity of w is interpreted as inequality aversion. More precisely, in the extended-form framework we get: Theorem 3.1. Let u Γ 2 and g {Γ 1 C 2 }. Then, the following statements are equivalent: (i) g (2) g(1) u (2) [u (1) ] 2 =: g (2) max 0. (ii) SW G fulfills the Pigou-Dalton Transfers Principle. Proof. See the Appendix. Whenever u (2) 0, a decision-maker who behaves in accordance with the (ME) principle also behaves in accordance with the Pigou-Dalton one (the reverse being not true). Kaplow (2010) asserts that the assumption on concavity of u has a direct impact on the marginal social value of income redistribution. This assumption has a compensatory role: a decision-maker evaluates a social state from her value judgments (such as g (1) ) as well as she takes into account the redistribution impact on individuals based on her assumptions on u (such as u (1) and the risk aversion of the agents u (2) /u (1) ). The decision-maker respects the Pigou-Dalton Transfers Principle if, and only if, her degree of inequality aversion is no higher than a nonnegative critical value g max, (2) which depends on the ratio social/private marginal utility weighed by the risk aversion of the agents. On this basis, when the decision-maker respects (PD), she may be either inequality-averse or strict inequality-loving, respectively: Wav 2 := { SW G W u Γ 2, g Γ 1, and g (2) 0 } W 2 lov := { SW G W u Γ 2, g Γ 1, and g (2) ( 0, g (2) max]}. 7

8 As a remark, weakening the condition (MEc) into α β could entail some situations in which g (2) + > g max (2) so that a decision maker that behaves in accordance with (PD) is always inequality averse. Also, particular situations such that g (2) + = g max (2) would provide a triple equivalence between: inequality aversion, the respect of (PD) and the respect of (ME). The result outlined in Theorem 3.1 allows us to determine the precise normative content of a dominance analysis at order 2. 5 The order 3 is concerned with the Diminishing Transfers Principle. The literature points out that this transfer is closely related to the third-order derivative of u in a pure utilitarian approach. With extended-form SWFs, we do not know at this stage exactly the decision-maker s sensibility about the location of the transfers of utilities. In a rank-dependent framework, Aaberge (2004, 2009) introduces two kinds of sensibilities. The decision-maker may have a higher [lower] sensibility to downside [upside] transfers if the third derivative of the social weight is positive [negative], providing aversion to downside [upside] inequality. In the extendedform utilitarian framework, the aversion to downside [upside] inequality may be captured by comparing two equalizing variations of utilities (as in (ME)) occurring at different locations of the distribution of utilities. For that purpose, we introduce the following principles. Definition 3.2. Downside [Upside] Minimal Equity. Consider four individuals i, j, k, l. The utility distributions U, V U are each one issued from V U by applying the Minimal Equity respectively between i and j, and between k and l such that: v j > u j > u i > v i > v k > z k > z l > v l (DMEa)[(UMEa)] ; z h = u h = v h h i, j, k, l (DMEb)[(UMEb)]. SW G satisfies the Downside [Upside] Equity Principle if, and only if: SW G (U) [ ] SW G (Z). (DME[UME]) The Downside/Upside Minimal Equity principles imply two types of social planner s behaviors. The Downside principle postulates that the social planner has a preference for an increase of welfare due to a change in utility which is favorable to worse-off agents (Z) compared with another situation where the change in utility is favorable to better-off ones (U). Conversely, if the decisionmaker behaves in accordance with the Upside Minimal Equity principle, then an improvement of social welfare is more valuable if it decreases the utility of better-off people compared with a situation in which the utility is increased for worse-off agents. Then, there is aversion to downside [upside] inequality if the decision-maker prefers equalizing variations of utilities (as in (ME)) the lower [upper] they are in the distribution. 5 Indeed, Pen s criterion is verified by assumption, then the condition about g (2) allows to fulfill the Pigou-Dalton Transfers Principle, as required to establish dominance at order 2. 8

9 Theorem 3.2. Let u (1) > 0 and g {Γ 2 C 3 }. Let α = β = z l v l =: γ = v k z k =: δ, then for any SW G W : (i) g (3) 0 (DME) [aversion to downside inequality]. (ii) g (3) 0 (UME) [aversion to upside inequality]. Proof. See the Appendix. Two families of social welfare functions arise from (DME) and (UME) in the context of extendedform SWFs, those exhibiting downward decision-makers [aversion to downside inequality] and upward decision-makers [aversion to upside inequality], respectively: W 3 down := { SW G W u (1) > 0, g { Γ 2 C 3}, and g (3) 0 } W 3 up := { SW G W u (1) > 0, g { Γ 2 C 3}, and g (3) 0 }. Those two families are not necessarily coincident with Kolm s Diminishing Transfers principle. The condition for the respect of this principle also depends on a critical value. We first check the respect of (KLM) without imposing any restriction on the sign of g (2). Theorem 3.3. Let u Γ 3 and g {Γ 1 C 3 }. Then, the following statements are equivalent: (i) g (3) g(1) u (3) +3g (2) u (1) u (2) [u (1) ] 3 =: g (3) min. (ii) SW G fulfills the Diminishing Transfers Principle. Proof. See the Appendix. It is noteworthy that the sign of the critical value g (3) min cannot be determined without additional assumptions on g (2) (we will concentrate on this point in the next result). The necessary and sufficient conditions to fulfill the Diminishing Transfers Principle do not imply the conditions to fulfill the Pigou-Dalton one. The Diminishing Transfers Principle and the Pigou-Dalton Transfers principle are respected if g (3) g (3) min and g(2) ( ; g (2) max]. The Diminishing Transfers Principle is still respected if g (3) g (3) min and g(2) (g (2) max; + ). From Theorem 3.3, we do not know however the sign of the third derivative of g. Then, we cannot assess whether the Diminishing Transfers Principle is relevant with only Upward decisionmakers or with both Upward and Downward decision-makers. For that purpose, let us consider that a decision-maker is inequality-averse [g (2) 0] and that the decision-maker assumes that u Γ 3 : Theorem 3.4. Let u Γ 3 and g {Γ 2 C 3 }. Then, we get the following: (i) g (3) 0 SW G respects (KLM) [Downward decision-maker]. (ii) g (3) [g (3) min ; 0) SW G respects (KLM) [Upward decision-maker]. Proof. See the Appendix. 9

10 Theorem 3.4 states that the critical value g (3) min is nonpositive. Then, two families of extendedform social welfare functions are relevant with the Diminishing Transfers Principle whenever the decision-maker is inequality-averse [g (2) 0] and u Γ 3 : W 3 down := { SW G W u (1) > 0, u Γ 3, g Γ 2, and g (3) 0 } { } Wup 3 := SW G W u (1) > 0, u Γ 3, g Γ 2, and g (3) [g (3) min ; 0). The extended-form social welfare functions fulfill Kolm s Diminishing Transfers Principle, with the distinction between the decision-maker that exhibits a preference for reducing inequality at the bottom of the income distribution [Wdown 3 ], and the one with a preference for reducing inequality at the top [Wup 3 ]. We now turn to the respect of Transfers Principles simultaneously, up to the order 3, in the same manner than Fishburn and Willig (1984). They have introduced a fundamental link between the respect of successive Transfers Principles and stochastic dominance in a reduced Utilitarian framework, in which the derivatives of w alternate in signs. If we adapt our results to Fishburn and Willig s (1984) finding, then the function g should alternate according to a boundary, i.e.: W 3 := { SW G W u Γ 3, g 1 0, g (2) g (2) max, g (3) g (3) min As a consequence of the Theorems 3.1 and 3.3, the decision-maker may respect both principles: Corollary 3.1. The following statements are equivalent: (i) SW G W 3. (ii) SW G respects (PD) and (KLM). Proof. See the Appendix. The social welfare functions in W 3 fulfill both the Pigou-Dalton and the Diminishing Transfers Principles. However, as we will see in the next section, it is worth to determine the necessary and sufficient conditions about attitude towards inequality whenever the set of the u functions is reduced. If the decision-maker assumes that u (1) is a strictly but not strongly convex function, then: }. Corollary 3.2. Let u Γ 3 such that u (3) 3u(2)2 u (1) statements are equivalent: (i) g (2) g max (2) 0 and g (3) g (3) min 0. (ii) SW G respects (PD) and (KLM). and g {Γ 1 C 3 }. Then, the following Proof. See the Appendix. 10

11 The interpretations of Corollary 3.2 rely on the analysis of Chateauneuf et al. s (2002) results: a decision-maker respects the (KLM) principle if, and only if, w (1) 0 and w (3) 0. There is no condition on the sign of the second-order derivative. We have introduced the following possible interpretation (see Section 2): an inequality-loving decision-maker respects the (KLM) principle if, and only if, she prefers a regressive transfer insofar the transfer is the highest possible in the distribution of incomes. In our extended-form framework, without imposing the sign of the second-order derivative of g, from Corollary 3.2, a decision-maker respects both (PD) and (KLM) principles if, and only if: either she is inequality-averse with a preference to equalizing variations of utilities, the lowest they are in the distribution ; or she is inequality-loving with a preference to disequalizing variations of utilities, the highest they are in the distribution. Both interpretations are possible as long as she assumes that u Γ 3 and u (1) is a strictly but not strongly convex function. We could imagine a generalization à la Fishburn and Willig (1984) by generalizing the set W 3 to a set W s for some positive integers s. However, as explained in the next Section, we have to balance normative and positive arguments. 4 Concluding remarks and discussions There is a strong underlying inconsistency between the well-known normative content of a Transfers Principle and the normative content that the Transfers Principle formally displays. From Lemma 3.1, we prove that there is no straightforward relationship between a Transfers Principle and some value judgment about inequality. This normative content depends on the assumption on the shape of the individuals utility function. From Theorem 3.1, there is no equivalence between the Pigou- Dalton Transfers Principle and inequality aversion as long as u is assumed to be increasing and strictly concave. In this case, inequality-averse as well as moderate inequality-loving decisionmakers respect this Transfers Principle. In the same way, Theorem 3.4 shows that inequality averse decision makers with more aversion at the bottom of the distribution of utilities as well as inequality averse decision-makers with moderately more aversion at the top of the distribution are relevant with the Diminishing Transfers Principle as long as u is assumed to be increasing, concave and its third derivative positive. If we consider that the previous assumptions on the shape of u are reasonable, then our results should yield larger social acceptance or, at least, weaker normative contents than those usually exhibited. Behind this assertion, there is a prescription: Economists [should state] explicitly in their recommendations or reports which parts are scientifically established facts, which are judgments of facts partly based on scientific analysis and partly on subjective opinion, and which are pure value judgments. Ng (1972), p

12 The field of inequality measurement could not be so normative if we adopt Ng s (1972, p. 1014) definition of this concept: value judgments, as the term implies, are meant to be contrasted with factual judgments. Factual statements are descriptive, value judgments are evaluative and/or prescriptive. [...] We cannot prove or disprove a value judgment as we verify a factual statement. 6 Ng s position defends the economists legitimacy to make subjective judgments of facts. In order to present this argument in our framework, it was necessary to introduce a SWF with an explicit functional interplay between the value judgments of a decision-maker and the individuals utility function over income. Kaplow (2010) considers the latter as a positive part of the SWF. There is perhaps a contradiction between the Utilitarian point of view and Kaplow s one. According to Blackorby, Bossert and Donaldson (2002, p. 547): utilities are interpreted as indicators of lifetime well-being and measure how good a person s life is from his or her own point of view. This does not mean that the utility function u is a representation of [persons ] actual preferences. However, it seems relevant to consider in our framework that the assessment of u could rely partly on empirical results which give the decision-maker a trend to make assumptions on the individuals utility function. By making subjective judgments of facts partly based on scientific analysis (i.e. empirical results) and partly on subjective opinion (the similarity postulate of Harsanyi). From Deck, Lee and Reyes (2008) and Deck and Schlesinger (2010, p. 11), well-known empirical results report that people respects the nonsatiety hypothesis, they are risk-averse and there is an evidence of prudence, although the degree is not overwhelming. Let us consider that u Γ 3 is a reasonable assumption about the shape of u. The additional condition about u (3) in Corollary 3.2 seems to be close to these results. Then, Theorem 3.1 and Corollary 3.4 should imply weaker value judgments about inequality to fulfill the two already presented Transfers Principles than it is usually claimed. Temperance is not verified empirically (u (4) 0). As value judgments and assumptions on u have compensatory roles, then our approach should display stronger value judgments for the fulfillment of some higher order Transfers Principles. From Theorem 3.1 and Corollary 3.2, the decision-maker s capacity to make realistic assumptions is an important factor in the decisions of contemplating tax reforms. We retrieve this message in Kaplow (2010) and indirectly in Ng (1972). Our model formalizes this assertion within a dominance analysis. Duclos, Makdissi and Wodon (2008) use this process to assess normatively and empirically marginal indirect tax reforms. In our extended-form framework, it would be possible to test whether a tax reform is socially improving with the respect of some Transfers Principles. For that purpose, we have to observe first the decision-maker s assumptions on u. If she does not make any assumption on the decreasing marginal individuals utility, then she is inequality-averse. If she assumes that the individuals exhibit some degrees of prudence, then we can empirically test some tax reforms by assuming she is a downward decision-maker or an upward one. 6 The author adds: not all untestable judgments are value judgments. 12

13 References [1] Aaberge, R. 2000, Characterisations of Lorenz curves and income distributions, Social Choice and Welfare, 17, [2] Aaberge, R. 2004, Ranking intersecting Lorenz curves, CEIS Tor Vergata - Research paper series, 15, no 45. January [3] Aaberge, R. 2009, Ranking Intersecting Lorenz Curves, Social Choice and Welfare, 33, [4] d Aspremont, C., L. Gevers 1977, Equity and the informational basis of collective choice, Review of Economic Studies, 44, [5] Blackorby, C., W. Bossert, D. Donaldson 2002, Utilitarianism and the Theory of Justice, in Handbook of Social Choice and Welfare, vol. 1. K. Arrow, A. Sen and K. Suzumura, eds., Elsevier, Amsterdam, [6] Chateauneuf, A., Gajdos, T. and P.-H. Wilthien 2002, The Principle of Strong Diminishing Transfer, Journal of Economic Theory, 103(2), [7] Dalton, H. 1920, The measurement of the inequality of incomes, Economic Journal, 30, [8] Deck, C., H. Schlesinger 2010, Exploring Higher Order Risk Effects, Review of Economic Studies, 77(4), [9] Deck, C., J. Lee, J. Reyes 2008, Risk attitudes in large stake gambles: evidence from a game show, Applied Economics, 40, [10] Duclos, J-Y., P. Makdissi, Q. Wodon 2008, Socially Improving Tax Reforms, International Economic Review, 49(4), [11] Fields, G., J. Fei 1978, On Inequality Comparisons, Econometrica, 46(2), [12] Fishburn, P., R. Willig 1984, Transfer Principles in Income Redistribution, Journal of Public Economics, 25, [13] Fleurbaey, M., P. Michel 2001, Transfer principles and inequality aversion, with an application to optimal growth, Mathematical Social Sciences, 42, [14] Foster, J., A. Shorrocks 1988a, Poverty Orderings, Econometrica, 56(1), [15] Kaplow, L. 2010, Concavity of utility, concavity of welfare, and redistribution of income, International Tax and Public Finance, 17, [16] Kolm, S-C. (1976a), Unequal Inequalities I, Journal of Economic Theory, 12,

14 [17] Ng Y.-K., 1972, Value judgments and Economists Role in Policy Recommendation, Economic Journal, 82(327), [18] Pigou, T. 1912, Wealth and Welfare, Macmillan, London. [19] Shorrocks, Foster, A. J. 1987, Transfer Sensitive Inequality Measures, Review of Economic Studies, 54(3), [20] Trannoy, A., J. Weymark 2007, Dominance Criteria for Critical-Level Generalized Utilitarianism, Vanderbilt University Department of Economics working paper 0707, A Appendix Proof. of Claim 3.1 (i) From (ME), let v j > u j > u i > v i and SW G (U) SW G (V ), that is, g(u i ) g(v i ) g(v j ) g(u j ). As α β > 0, then: g(v i + α) g(v i ) g(u j + β) g(u j ). Let consider ε 0 such that g(u j + β) = g(u j + α) ε: g(v i + α) g(v i ) g(u j + α) g(u j ) ε. Divide both sides of the last expression by α 0, then: As v i < u j, this implies that: g (v i ) g (v i ) g (u j ) ε α. ε α(u j v i ) =: g(2) + 0. (ii) Let the function g be strictly increasing and concave on its domain, then whenever v j > u j > u i > v i with α β, we always get: g(u i ) g(v i ) g(v j ) g(u j ). Proof. of Lemma 3.1: [ ] We proceed by mathematical induction, i.e., we first prove that the statement H s is true for s = 1 and then we prove that if H s is assumed to be true for any positive integer s, then so is H s+1. It is apparent from w (1)=g(1) u (1) that H 1 is true. Let us now assume that the statement H s is true. Remarking that w (s+1) = [ g (1) u (1)] (s) and remembering Leibniz rule for the sth derivative of the product of two functions we get w (s+1) = s k=0 ( s k ) [g (1) ] (s k) [ u (1) ] (k). 14

15 Then, multiplying both sides by [ 1] s+1 and rearranging terms yields [ 1] s+1 w (s+1) = s k=0 ( s k ) [ 1] s k+1 g (s k+1) [ 1] k+1 u (k+1). Clearly, the binomial coefficient is positive for all k = 0,..., s. If u Γ s+1 and g Γ s+1, then, for all k = 0,..., s, [ 1] k+1 u (k+1) 0 and [ 1] s k+1 g (s k+1) 0, implying w Γ s+1. Up to this point we have only demonstrated that u Γ s+1 and g Γ s+1 together imply w Γ s+1. Finally, as H s is supposed to be true, if u Γ s+1 and g Γ s+1, then, w Γ s. This concludes the proof. [ ] It is apparent from relations (3) to (4). Proof. of Theorem 3.1: [(ii) (i)]: We first know that: w (2) = g (1) u (2) + g (2) [ u (1)] 2. Furthermore, by Theorem 2.1, to fulfill the Pigou-Dalton Transfers Principle, the condition is w Γ 2, that implies: w (1) 0 and w (2) 0. Here, we assume that g (1) 0 and u (1) 0. Then, by Lemma 3.1, we have w (1) 0. Hence, [(i) (ii)]: g (1) u (2) + g (2) [ u (1)] 2 0 g (2) g(1) u (2) [u (1) ] 2. g (2) [u (1) ] 2 g (1) u (2) g (1) u (2) + g (2) [u (1) ] 2 0 w (2) 0. As a remark, we can show two special cases: g (2) = g(1) u (2) and g (2) = 0 (the utilitarian case). [u (1) ] 2 If g (2) = g(1) u (2), then, replacing g (2) in the w (2) s expression yields: [u (1) ] 2 w (2) = g (1) u (2) + g(1) u (2) [u (1) ] 2 [u (1) ] 2 = 0. If g (2) = 0 < g(1) u (2) [u (1) ] 2, then, replacing g (2) in the w (2) s expression entails: w (2) = g (1) u (2). Proof. of Theorem 3.2 (i): [ ] Let the function g be strictly increasing, concave and g (3) 0 on U. Given that v j > u j > u i > v i > v k > z k > z l > v l, with α = β, we get g(u i ) g(v i ) g(v j ) g(u j ). From γ = δ, we get g(z l ) g(v l ) g(v k ) g(z k ). Given that α β = γ δ = 0, then: g(u i ) + g(u j ) + g(v l ) + g(v k ) g(z l ) + g(z k ) + g(v i ) + g(v j ). 15

16 It turns out that SW G (U) SW G (Z). [ ] From (DME), let v j > u j > u i > v i > v k > z k > z l > v l, SW G (U) SW G (Z), that is g(u i ) + g(u j ) + g(v l ) + g(v k ) g(z l ) + g(z k ) + g(v i ) + g(v j ). Let α = β = γ = δ, we have: g(v i + α) g(v i ) [g(u j + α) g(u j )] g(v l + α) g(v l ) [g(z k + α) g(z k )]. Divide both sides by α 0, then g (z k ) g (v l ) g (u j ) g (v i ). As the gaps between the utilities are equidistant α = β = γ = δ, we get g (v i ) g (v l ) 0 that is g (v l ) 0. (ii): Mutatis mutandis as in (i). Proof. of Theorem 3.3: The proof goes along the lines of Theorem 3.1. We first know that: w (3) = g (1) u (3) + 3g (2) u (1) u (2) + g (3) [ u (1)] 3. Here we assume that g (1) 0, it turns out that w (1) 0. [(ii) (i)]: [(i) (ii)]: g (1) u (3) + 3g (2) u (1) u (2) + g (3) [ u (1)] 3 g (3) g(1) u (3) + 3g (2) u (1) u (2) [u (1) ] 3. g (3) g(1) u (3) + 3g (2) u (1) u (2) [u (1) ] 3 g (1) u (3) + 3g (2) u (1) u (2) + g (3) [ u (1)] 3 w (3) 0. Proof. of Corollary 3.1: It is from the same way that the proof of Theorem 3.3. Proof. of Corollary 3.2: [(ii) (i)]: From Corollary 3.1, it is obvious that g (3) min 0 whenever g(2) 0. However, here the sign of g (2) is unknown. We have g (1) 0, u (1) 0, u (2) 0, u (3) 0. From Theorem 3.1, if the Pigou-Dalton Transfers Principle is respected then: w (2) 0 g (2) g(1) u (2) [u (1) ] 2 = g (2) max 0. Now, if the Diminishing Transfers Principle is respected with the limiting case g (2) = g(1) u (2), [u (1) ] 2 then: g (3) g(1) u (3) + 3 g(1)u(2) u (1) u (2) [u (1) ] 2 = g [u (1) ] min. 3 3 As we have u (3) 3 u(2)2 u (1), we get: [(i) (ii)]: Straightforward. g (3) g(1) u (3) + 3 g(1)u(2) [u (1) ] 2 u (1) u (2) [u (1) ] 3 = g 3 min 0. 16