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Transfer principles and relative inequality aversion a majorization approach
Mathematical Social Sciences 45 (2003) 299–311
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Transfer principles and relative inequality aversion a majorization approach Ronny Aboudi a , Dominique Thon b , * a
Department of Management Science, University of Miami, P.O. Box 248237, Coral Gables, FL 33124, USA b Instituto Inter-Universitario de Macau, NAPE, Lote 18, Rua de Londres-P, Edf Tak Ip Plaza, Macau, China Received 30 November 2001; received in revised form 31 August 2002; accepted 31 October 2002
Abstract This paper characterizes two preorders over vectors, representing here income distributions, which, for the case of an order-preserving additive welfare function W 5 o n1 u(x i ), correspond to the increasing concave functions u which are more concave than u(x) 5 ln x and than u(x) 5 2 1 /x, respectively. We provide a characterization of those W functions that is quite distinct from the one recently provided by Fleurbaey and Michel, Mathematical Social Sciences, 2001. The two sets of results are shown to be complementary. 2003 Elsevier Science B.V. All rights reserved. Keywords: Majorization; Stochastic dominance; Income inequality; Risk aversion JEL classification: D63; D81; I31
1. Introduction It is well known that if the sum of the k lowest incomes in allocation y is larger than the same for allocation x, each allocation containing n income recipients, for k 5 1, . . . , n, then all additive welfare functions W(x) 5 o n1 u(x i ), with u increasing concave, will unanimously pronounce y preferable to x. Noting the fact that a concave function is a function which is more concave than a linear function, this preorder can be thought of as corresponding to the unanimous ranking given by all additive welfare functions *Corresponding author. Tel.: 186-853-796-4507; fax: 186-853-72-5517. E-mail address: [email protected] (D. Thon). 0165-4896 / 03 / $ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016 / S0165-4896(03)00003-9
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W(x) 5 o n1 u(x i ), with the increasing linear function u(x) 5 x (or any increasing linear function) as the ‘benchmark’, in the sense that the functions u considered are the ones more concave than this particular function. One might then think of using some increasing function other than the linear one as the benchmark function. In this paper we consider two particular cases: the function u(x) 5 ln x and the function u(x) 5 2 1 /x. A recent paper by Fleurbaey and Michel (2001) deals with such classes of functions, and we offer some complementary results. By choosing a strictly concave function as the benchmark, one does in effect allow for some leakage in a reallocation of income. Few redistributive schemes are costless, and intuition suggests that if one has strict inequality aversion, then one might prefer y to x even if y contains less total income. This is equivalent to the buying of less-than-fair insurance by a strictly risk averse agent in expected utility theory. Section 2 sets the stage. Our main results are given in Sections 3 and 4. For each of the two cases, we essentially provide the analogue of a T-transform, and a description of the set of income vectors which dominate some initial vector. Our results are logically distinct from the ones recently presented by Fleurbaey and Michel (2001); in Section 5, we discuss the relationship between the two sets of results. Section 6 concludes.
2. Preliminaries In what follows, x (i ) represents the increasing re-arrangement of the vector x [ R n ; for convenience, we abbreviate Marshall and Olkin (1979) to M&O, and Fleurbaey and Michel (2001) to F&M. We say that the function f is more concave than the function g if f is a concave function of g. A well-known preorder over (income) vectors in R n is defined as follows: Definition 2.1. Let x, y [ R n ; we say that x a 2 y if o k1 x (i ) # o k1 y (i ) , k 5 1, . . . , n. Consider the following transformation of the vector x into the vector y: two of the elements of x, namely i and j, are transformed as follows and the other elements are left untouched: x i ⇒ u x i 1 (1 2 u )x j 5 y i with 0 # u # 1. x j ⇒ (1 2 u )x i 1 u x j 5 y j
(2.1)
The following equivalences are well-known. Lemma 2.1. For x, y [ R n , the following are equivalent: x a 2y
(2.2)
O u(x ) # O u( y ) for all u continuous, increasing and concave
(2.3)
y can be reached from x through a sequence of transfers such as in (2.1) and increases in the elements of x.
(2.4)
n 1
n
i
1
i
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The preorder in Definition 2.1 corresponds to the income distribution y dominating income distribution x by second degree stochastic dominance, hence our symbol a 2 . In the terminology of M&O, p. 10, y is said to be weakly supermajorized by x; in M&O, x a 2 y would be written as y a w x. The kind of transformation represented by (2.1) corresponds to a transfer from the originally richer of the two income recipients to the other, which transfer does not go beyond permuting their original incomes. The bistochastic matrices constructing such transfers are known as T-transforms and their relationship to majorization is described in Muirhead’s Lemma (M&O, p. 21). The equivalence between (2.2) and (2.3) is a mild extension of a part of Hardy, Littlewood and Polya’s Theorem, known as Tomic’s Theorem (see M&O, p. 109). The equivalence between (2.2) and (2.4) is an extension of Muirhead’s Lemma, due to Mirsky; see M&O, A.9.a, p. 123, and C.6., p. 28. The preorder a 2 was introduced in economics by Kolm (1969); see also Shorrocks (1983). Our analysis is based on the preorder a 2 rather than the majorization ordering (the opposite of x a 2 y, with the condition S x 5 S y) because, as will become clear, we do not wish (and in fact cannot) assume a prescribed total income in our analysis. Paradoxically, building on a 2 will allow us to pronounce an allocation with less total income to be preferred.
3. The logarithmic case We first propose, as a variation on Definition 2.1: ln
n Definition 3.1. Let x, y [ R 11 ; we say that x ay if o 1k ln x (i ) # o k1 ln y (i ) , k 5 1, . . . , n.
Consider now the following kind of transformation, with x i # x j : (a) x i ⇒ x i /l 5 y i with x i /x j # l # 1. (3.1) (b) x j ⇒ l x j 5 y j ]] Note that y i 5 y j 5œx] i x j for l 5œx i /x j , and that for l 5 x i /x j , its lower bound, x i and x j are permuted. Theorem 3.1. For x, y [ R n11 , the following are equivalent: ln
x ay
(3.2)
O u(x ) # O u( y ) for all u continuous, increasing and more concave n 1
n
i
1
i
than ln x y can be reached from x through a sequence of transformations such as in (3.1) and increases in the elements of x. ln
(3.3) (3.4)
Proof. Obviously x ay is equivalent to [ln x i ] 1n a 2 [ln y i ] 1n ; thus we are basically dealing with a 2 with a change of variables. The equivalence between (3.2) and (3.3) follows from simply performing this change of variable in the expressions (2.2) and (2.3) which
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are equivalent to each other by Lemma 2.1, and noting that ln x is increasing. Now, (3.2) is equivalent to y being reachable from x through income increases and the following T-transforms applied to the vector [ln x i ] 1n , see (2.1) and Lemma 2.1: (a) ln x i ⇒ u ln x i 1 (1 2 u ) ln x j 5 ln y i with 0 # u # 1. (b) ln x j ⇒ (1 2 u ) ln x i 1 u ln x j 5 ln y j
(3.5)
We now show that there is a one-to-one correspondence between u [[0, 1] in (3.5) and l [ [xi /xj , 1] in (3.1). Combining (3.1a) with (3.5a), and assuming that xi # x j , one has that
u ln x i 1 (1 2 u ) ln xj 5 ln yi 5 ln [x i /l]. u Simple transformations give: ln [x ui 3 x j12u ] 5 ln [x i /l]; x iu 3 x j12u 5 x i /l; x iu 21 3 x 12 5 j 1 /l, or:
(x i /x j )12u 5 l.
(3.6)
Since 0 # u # 1, it is clear that l in (3.6) ranges from x i /x j to 1. Conversely, given any l [ [xi /xj , 1], then there exists a unique u [ [0, 1] that satisfies (3.6). Thus the one-to-one equivalence is established. It remains to show that l in (3.6) satisfies (3.5b). This is easily established, as ln y j 5 ln lx j 5 ln [(x i /x j )12u x j ] 5 (1 2 u ) ln x i 1 u ln x j . The equivalence between (3.2) and (3.4) then follows from the equivalence between (2.2) and (2.4) in Lemma 2.1, noting that the notion of ‘increase in ln x’ is equivalent to the one of ‘increase in x’. h Example. If x 5 (5, 10) and l 5 0.8 in (3.1), then y 5 (6.25, 8). Now the u in (3.5a) which corresponds to this l can be calculated from (3.6), with here x i /x j 50.5, as follows: 0.5 12u 5 0.8; (1 2 u ) ln 0.5 5 ln 0.8, and thus u 5 0.67807. Calculating (3.5) with this u indeed gives the same y as using l 5 0.8 in (3.1). The following two elementary results provide a perspective on the distinction between ln a 2 and a. Let x (i ) be the increasing rearrangement of x, as before, and let x [i ] be its decreasing rearrangement. Theorem 3.2. Let x, a [ R n . Then [x (i ) 1 a (i ) ] n1 a 2 [x i 1 a i ] 1n a 2 [x (i ) 1 a [i ] ] 1n . n
n ln
n ln
n
Theorem 3.3. Let x, a [ R 11 . Then [x (i ) 3 a (i ) ] 1 a[x i 3 a i ] 1 a[x (i ) 3 a [i ] ] 1 . Theorem 3.2 shows that if one thinks of x as being an original income vector and a as being a set of prescribed additions of income, each a i to the individual income x i , then the ordering a 2 can be characterized by the fact that the highest level of welfare is reached by adding to each element of x one of the elements of a in such a way that the smallest x i receives the largest a i , etc., rather than according to any other form of
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ln
pairing. Theorem 3.3 shows that the equivalent is true for x ay if a positive vector a is now used to perform a multiplicative rather than an additive transformation of x. Theorem 3.2 provides an interesting economic interpretation of majorization over incomes, yet does not seem to have appeared in the economic literature. The first a 2 follows from as Proposition A2, p. 140 in M&O, and the second one from Proposition A1, p. 139 in M&O. Theorem 3.3 follows directly from Theorem 3.2, by replacing in the ln latter x i by ln x i , a i by ln a i , and a 2 by a. The two above theorems suggest an interpretation ofln a 2 as being a preorder dealing with absolute (i.e., additive) income changes and x ay as dealing with relative (i.e., multiplicative) ones, in the sense of Theorems 3.2, 3.3. Related results and further references about functions less concave than ln (the mirror-image case) can be found in Thon and Thorlund-Petersen (1993). ln 2 Replacing a by a as a criterion of welfare dominance allows one to pronounce y to be better than x in some cases where S y , S x, as the condition S ln x # S ln y is milder than the condition S x # S y. Note that stochastic dominance of any degree requires, just like a 2 , that the dominating income vector have at least as large an income sum than the dominated one. A transformation of incomes such as (3.1) can, on the other hand, be thought of as a leaky transfer from a rich to a poor. This is readily illustrated by considering, for n 5 2, the better-than-x set, that is the set ln 2 of vectors y that dominate a prescribed x, according to a and a, respectively. We call ln those sets B(x, a 2 ), and B(x, a). The set B(x, a 2 ), is represented in Fig. 1 by the hatched area. The segment between x 5 (x 1 , x 2 ), the original vector, and its permutation, represents redistributions of income at constant sum as in (2.1). This segment is defined by: x (1) # y ( 1) and x 1 1 x 2 5 y 1 1 y 2 . With x 1 , x 2 taken to be constant, then this segment can be represented by a function expressing y 2 as a function of y 1 : y 2 5 x 1 1 x 2 2 y 1 ; y 1 [ [x (1 ) , x ( 2) ].
(3.7) ln
A similar construction is easily made for a. Here the redistributions accomplished by
Fig. 1. The set B(x, a 2 ).
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2hyp
ln
Fig. 2. The sets B(x, a) and B(x, a ).
(3.5), or equivalently (3.1) are defined by ln x ( 1) # ln y (1) and ln x 1 1 ln x 2 5 ln y 1 1 ln y 2 . We obtain the analogue of (3.7): ln y 2 5 ln x 1 1 ln x 2 2 ln y 1 ; y 1 [ [x ( 1) , x (2) ],
(3.8)
which, for x 1 , x 2 taken to be constant, gives the following hyperbolic relationship between y 1 and y 2 : y 2 5 x 1 x 2 /y 1 . This is the continuous convex curve in Fig. 2. From ln this, we can easily construct B(x, a), represented as the hatched area. It consists of B(x, a 2 ), as in Fig. 1, plus the lens-shaped set below B(x, a 2 ), representing distributions whose total income is less than the one of x. The ‘bended’ part of the ln closure of B(x, a) is precisely what (3.8) describes; it corresponds to the redistributions which preserve the sum of the logarithms of income, up to a permutation of incomes.
4. The hyperbolic case We now define: 2hyp
n Definition 4.1. Let x, y [ R 11 ; we say that x a y if o 1k 2 1 /x (i ) # o 1k 2 1 /y (i ) , k 5 1, . . . , n.
Consider now the following kind of transformation, with x i # x j : (a) x i ⇒ x i /l 5 y i xj (b) x j ⇒ ]]]] 5 y j with x i /x j # l # 1. xj 1 1 ] [1 2 l] xi
(4.1)
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Note that y i 5 y j 5 2x i x j /(x i 1 x j ) for l 5 (x i 1 x j ) / 2x j , and that for l 5 x i /x j , its lower bound, x i and x j are permuted. n
Theorem 4.1. For x, y [ R 11 , the following are equivalent: 2hyp
x a y
(4.2)
O u(x ) # O u( y ) for all u continuous, increasing and more n 1
n
i
1
i
concave than 2 1 /x y can be reached from x through a sequence of transformations such as in (4.1) and increases in the elements of x.
(4.3) (4.4)
2hyp
Proof. Obviously x a y is equivalent to [21 /x i ] n1 a 2 [21 /y i ] n1 ; thus we are basically dealing with a 2 with a change of variables. The equivalence between (4.2) and (4.3) follows from simply performing a change of variable in the expressions (2.2) and (2.3) which are equivalent to each other, by Lemma 2.1, and noting that 2 1 /x is increasing. Note that (4.2) is equivalent to y being reachable from x through income increases and the following T-transforms applied to [21 /x i ] n1 , see (2.1) and Lemma 2.1: 21 21 21 21 (a) ]] ⇒ u ]] 1 (1 2 u ) ]] 5 ]] xi xi xj yi 21 21 21 21 (b) ]] ⇒ (1 2 u ) ]] 1 u ]] 5 ]] xj xi xj yj
with 0 # u # 1.
(4.5)
We now show that there is a one-to-one correspondence between u [[0, 1] in (4.5) and l [[xi /x j , 1] in (4.1). Combining (4.1a) with (4.5a), and assuming that x i # xj , one has that u (21 /x i ) 1 (1 2 u )(21 /x j ) 5 2 1 / [x i /l] 5 2 l /x i , or: xi u 1 (1 2 u ) ] 5 l. (4.6) xj Note that the left side of (4.6) is a convex combination of 1 and x i /x j # 1, and thus the one-to-one correspondence is established. It remains to show that l satisfying (4.6) also satisfies (4.5b). We have that 2 1 /y j
2 1 2 (x j /x i )[1 2 l] 5 ]]]]]] 5 2 1 /x j 1 (21 /x i )[1 2 l] xj 5 2 1 /x j 1 (21 /x i )[1 2 u 2 (1 2 u )(x i /x j )] 5 (1 2 u )(21 /x i ) 1 u (21 /x j ),
which is (4.5b). The equivalence between (4.2) and (4.4) then follows from the equivalence between (2.2) and (2.4), noting that the notion of ‘increase in 2 1 /x’ is equivalent to the one of ‘increase in x’. h Example. If x 5 (5, 10) and l 50.8 in (4.1), then y 5 (6.25, 7.143). Now the u in (4.5a) that corresponds to this l is, from (4.6), with here x i /x j 50.5, such that 0.85u 1 (1 2 u )0.5, or u 50.6. Calculating (4.5) with this u indeed gives the same y as using
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l 50.8 in (4.1). 2hyp
Transformations such as (4.1) — and the associated dominance concept a — allow an ln even larger leakage than a, illustrated below, as suggested by the fact that the function 2hyp 2 1 /x is more concave than the function ln(x). For a , one obtains the following analogue to (3.7) and (3.8): 1 /y 2 5 1 /x 1 1 1 /x 2 2 1 /y 1 ; y 1 [ [x (1 ) , x ( 2) ], 2hyp
from which B(x, a ) can easily be constructed. It appears on Fig. 2, as the set of points 2hyp ln to the N–E of the dotted curve. Obviously, B(x, a 2 ) # B(x, a) # B(x, a ).
5. Relationship to Fleurbaey and Michel In the paper that inspired the present one, F&M provide two main results dealing with welfare functions of the form W(x) 5 S u(x i ) using the two classes of u functions more concave than ln x and 21 /x, respectively (they also provide an application to economic growth that will not concern us). Unlike us they do not start with the definition of a preorder (in fact, they do not consider preorders at all, see below), but with particular types of transformations affecting a pair of incomes. In the logarithmic case, their starting point is to consider leaky transfers constructed such that, as a proportion of the final incomes, the gain to the (poorer) receiver is equal to the loss of the (richer) donor. Unlike ours, their transfers are bounded so as not to invert the relative positions, as in a Dalton transfer; as a result in this Section we assume x i # x j , for the sake of comparability. They call such transfers ‘proportional ex-post transfers’, PEP-transfers for short. Definition 5.1. (PEP-transfer principle) Consider a vector (x 1 , x 2 , . . . , x n ), two individuals i and j, and a positive number d. If the vector (z 1 , z 2 , . . . , z n ) is such that for all k ± i, j, z k 5 x k and z i 5 x i /(1 2 d ) # z j 5 x j /(1 1 d ),
(5.1)
then W(x) , W(z). In their Proposition 2, F&M establish the following result, which we shall compare to our own Theorem 3.1. Proposition 2. (F&M, p. 7) W satisfies the proportional ex-post transfer principle if and only if there is a concave function v: R → R such that u(x) 5 v(ln x). Note that a sequence of PEP-transfers is not such a transfer itself, as any example will establish. In fact, as we now show, a sequence of small PEP-transfers gives at the limit
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our transformations (3.1), and there is thus a connection between PEP-transfers and our ln
better-than-x set B(x, a). We first establish a technical lemma. Lemma 5.1. Let 0 , l , 1 and let m be a natural number. Then (A): [2 2 l 1 / m ] m , 1 /l. Furthermore, (B): [2 2 l 1 / m ] m is strictly increasing in m. Finally, (C): lim [2 2 l 1 / m ] m 5 1 /l. m→` 1/m
1/m
1/m
Proof. (A) is equivalent to [2 2 l ] , 1 /l . Then, multiplying by l and collecting terms, we have: 2 [ l 1 / m 2 1] 2 , 0, which holds for all positive l ± 1. h (B) Let us assume that m is continuous and m $ 1. Then [2 2 l 1 / m ] m is strictly positive. Now, [2 2 l 1 / m ] m is strictly increasing in m iff ln([2 2 l 1 / m ] m) is. Now let f(m)5 ln ([2 2 l 1 / m ] m)5m ln (2 2 l 1 / m). Then f 9(m)5ln (2 2 l 1 / m)1 l 1 / m ln l /(m[2 2 l 1 / m ]). Let a 5 l 1 / m ; thus l 5 a m and ln l 5 m ln a. Furthermore, since 0 , l , 1 and m $ 1, we have 0 , a , 1. Substituting in f 9(m), we have: f 9(m)5ln (2 2 a ) 1 a ln a /(2 2 a ). Since 2 2 a . 0, f 9(m).0 iff g(a )5(2 2 a ) ln (2 2 a ) 1 a ln (a ) . 0 for all 0 , a , 1. We have g(1)50, and g9(a )5ln a 2 ln (2 2 a ) 5 ln (a /(2 2 a )) , 0, where the inequality follows from: a /(2 2 a ) , 1 for all 0 , a , 1. Thus g(a ) is strictly decreasing on (0, 1). But since g(1)50, we have that g(a ).0 for all a in (0, 1). So f 9(m).0 for all m $ 1. h (C) We proceed by showing that for m → `, ln[2 2 l 1 / m ] m → 2 ln l, which is equivalent to (C). Now, ln(2 2 l 1 / m) 1/m ]]]] lim m ln (2 2 l ) 5 lim m→` m →` 1 /m (1 /(2 2 l 1 / m))(2ln l)l 1 / m (21 /m 2 ) 5 lim ]]]]]]]]]] 5 2 ln l. m →` 2 1 /m 2 ˆ The second equality is obtained by applying L’Hopital’s rule, and the third equality by 1/m 1/m noting that m→` lim (2 2 l ) 5 1, and that mlim l 5 1, for l . 0. h →` Theorem 5.1. Let the pair hy i , y j j be reached from hx i , x j j, with 0 , x i , x j , by a single ]] transformation such as in (3.1) with œx i /x j # l , 1. Starting from the same hx i , x j j, for any natural number m, consider a finite number m of successive transformations such as (m) (m) in (5.1), each with the same d, and which produce the pair hz (m) 5 yi . i , z j j with z i (m) (m) Then, (A): z j . y j . Furthermore, (B): z j is strictly decreasing in m. Finally, (C): lim m→` z j(m) 5 y j . Proof. Let the transformation (3.1) give y i 5 x i /l and y j 5 lx j . Let the result of the (m) m (m) m sequence of m transformations (5.1) be z i 5x i /(1 2 d ) and z j 5x j /(1 1 d ) , with, m (m) for a given l, d chosen such that y i 5x i /l 5x i /(1 2 d ) 5z i . This implies l 5(1 2 d )m or d 5 1 2 l 1 / m . Thus we have: y j 5 lx j and z j(m) 5 x j /(1 1 d )m 5 x j / [2 2 l 1 / m ] m .
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The three concluding statements of the Theorem follow by applying in turn the three statements of Lemma 5.1: (A) [2 2 l 1 / m ] m , 1 /l → z j(m) .y j (B) [2 2 l 1 / m ] m strictly increasing in m → z (m) is strictly decreasing in m j (C) lim [2 2 l 1 / m ] m 51 /l → lim z (m) 5y . h j j m→`
m→`
Before illustrating Theorem 5.1, we make some remarks on its assumptions. The assumptions l , 1 and x i , x j are meant to avoid trivialities ( l 5 1 means ‘zero ]] transfer’). The condition œx i /x j # l in the Theorem insures that y i # y j , which is necessary to be able to compare our result to F&M’s, as the latter assume rank(m) (m) (m) (m) preservation. Note that, as we have z (m) j .y j , then z j .y j $ y i 5z i . Thus z j .z i , as it should be (for a finite m). Note, that 0 , x i implies, in this rank-preserving situation, (m) that 0 , z (m) j , y j , y i , z i . The assumptions of Lemma 5.1 have been chosen to cover the ones of Theorem 5.1, but some of the results in Lemma 5.1 are more general, for example part (A). In our two-dimensional examples, we write x 1 for x i , x 2 for x j . Example. Consider first the fact that (3.1) with x 5 (5, 10) and l 50.8 gives y 5 (6.25, 8) while (5.1) with d 50.2 gives z 5 (6.25, 8.333) and one has z 1 5y 1 but z 2 . y 2 . Thus a single transformation (5.1) results in a pair with a second element larger than with a single transformation (3.1), at constant first element. Consider now d 9 chosen such that z (12 ) 5x 1 /(1 2 d 9)2 56.255y 1 . This gives d 950.1056. Thus this sequence of two transformation (5.1) gives z (12 ) 5z 1 5y 1 ; now z (22 ) 5x 2 /(1 1 d 9)2 58.181, and z 5 (6.25, 8.181). Comparing to the one-step procedure, we have that the value of the second element is reduced. Now Theorem 5.1 tells us that the thinner one slices one PEP-ex-post transfer into m smaller and equal such transfers that keep z (m) 1 5z 1 constant, the closer z (m) comes to y . Furthermore, at the limit, a sequence of PEP-transfers can 2 2 reproduce any income pair reached by one application of our transformations (3.1), and PEP-transfers plus income increases (F&M assume monotonicity) can generate, at the ln
limit, the same better-than-x set B(x, a) as we have, Fig. 2, if anonymity [symmetry] is further assumed. The point can be illustrated graphically, see Fig. 3. Note that for x 1 , x 2 treated as constants, (5.1) gives a relationship between z 1 and z 2 which can easily be calculated to be: z 2 5 x 2 /(2 2 x 1 /z 1 ); z 1 [ [x 1 , (x 1 1 x 2 ) / 2]. This is represented by the convex curve joining x to A in Fig. 3. The other convex curve in the figure is the same as the continuous one on Fig. 2, which represents the lower ln bound of our better-than-x set B(x, a), see Section 3. Consider finally the fact that a single PEP-transfer producing equality, for x 5 (5, 10), gives z 5 (7.5, 7.5), implying no leakage, while the point of our better-than-x set with equal incomes is the pair y 5 (7.0711, 7.0711), with some leakage. It is easy to calculate that a single PEP-transfer that gives z 1 57.0711 implies z 2 57.7346, and that a sequences of three PEP-transfers
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Fig. 3. The logarithmic case.
can be constructed, see above, to give (z (13 ) , z (23 ) )5(7.0711, 7.3297). As Theorem 5.1 indicates, increasing the number of such transfers (m) while keeping z 1(m) 5 z 1 57.0711 decreases z (m) 2 , and, at the limit, will produce the pair (7.0711, 7.0711). In their Proposition 1 (F&M, p. 5), dealing with functions u(x) more concave than 2 1 /x, F&M consider the following kind of transformation, called ‘proportional transfers’, P-transfers for short, which are constructed so as to make, in terms of proportion to the original incomes, the gain of the (poorer) receiver equal to the loss of the (richer) donor. Again the transfers are bounded to preserve the relative positions. Definition 5.2. P-Transfer Principle. Consider a vector (x 1 , x 2 . . . , x n ), two individuals i and j, and a positive number d. If the vector (z 1 , z 2 , . . . , z n ) is such that for all k ± i, j, z k 5 x k and z i 5 x i (1 1 d ) # z j 5 x j (1 2 d )
(5.2)
then W(x) , W(z). They prove the following result (note the strict concavity of v); compare to our own Theorem 4.1: Proposition 1. (F&M, p. 5) W satisfies the proportional transfer principle if and only if there is a strictly concave function v: R 2 2 → R such that u(x) 5 v(21 /x). The situation here is the same as in the logarithmic case, as regards the comparison of the effect of one of our transformations (4.1) to one of F&M (5.2): at constant first element in the resulting pair, (5.2) gives a larger second element: Taking again x 5 (5, 10), and d 50.2 in (5.2) gives the income vector z 5 (6, 8), while with l 55 / 6,
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expression (4.1) gives y 5 (6, 7.5), with the same first element but a smaller second element. F&M’s Definition 5.2 gives:
F
G
2x 1 x 2 z 2 5 2x 2 2 (x 2 /x 1 )z 1 ; z 1 [ x 1 , ]] , x1 1 x2 that is, a linear relationship between z j and z i , which is represented in Fig. 4. Note that both Definition 5.2 and our own (4.1) imply that equality of income is achieved for those incomes for which y 1 5y 2 5z 1 5z 2 52x 1 x 2 /(x 1 1 x 2 ). Otherwise, this line is entirely above the dotted curve, reproduced from Fig. 2, which is the lower bound of our 2hyp better-than-x set B(x, a ). In contradistinction to the logarithmic case, performing a sequence of small P-transfers rather than a single one does not decrease the value of the resulting z j , for a prescribed z i ; it actually increases it. This is readily established. Consider z 1 5x 1 (1 1 d ) 5 x 1 (1 1 d1 )(1 1 d2 ), which requires d 5 d1 1 d2 1 d1d2 and z 2 5 x 2 (1 2 d ), z 29 5 x 2 (1 2 d1 )(1 2 d2 ). Elementary calculations show that for d1 , d2 . 0, z 2 ,z 29 . This is the opposite of what we have in the logarithmic case. Thus here, a succession of small P-transfers cannot be used to approximate our better-than-x set 2hyp B(x, a ). The results of F&M are of an entirely different nature from ours. This should be clear from the fact that the concept of a pre-order, the notion of a sequence of transformations and the one of a better-than-x set appear nowhere in their paper and are not their concern. Their point of departure is to consider transfers with well-defined properties, PEP- and P-transfers and they ‘look for’ the largest class of W functions preserving such single transfers. We, on the other hand, move in the opposite direction. We start with two benchmark functions, and on their basis define two preorders that are variations on the theme of majorization. From there we characterize the analogues of the T-transforms, and the better-than-x set spanned by a sequence of them. Thus, in spite of the consideration of the same classes of functions, our respective two main theorems should
Fig. 4. The hyperbolic case.
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be expected to have no logical relationship to each other. In the light of this it is, if anything, very surprising that in the logarithmic case, there is actually a connection between the two results, Theorem 5.1, and that, in some sense, our better-than-x set can be constructed using PEP-transfers.
6. Conclusion We have characterized the preorders obtained essentially from the standard second degree stochastic dominance ordering through both a logarithmic and a negative hyperbolic transformation of the variables appearing in the order-preserving functions of the form W(x) 5 o n1 u(x i ). We have identified the corresponding analogues to Ttransforms, and described the corresponding better-than-x sets. Note that if one werelnto n compare two given income vectors x, y [ R 11 , and wanted to check whether x ay holds, then a straightforward algorithm is provided by the very Definition 3.1. Likewise 2hyp
for x a y. It is well known that the functions ln x and 2 1 /x have a constant relative risk aversion, of 1 and 2, respectively, see Pratt (1964). This obviously suggests the possibility of a generalization considering as the benchmark function any member of the class of (increasing concave) constant relative risk aversion utility functions. This also suggests an independent generalization of F&M’s results.
Acknowledgements Thanks are due to M. Fleurbaey, L. Thorlund-Petersen and two referees for useful comments.
References Fleurbaey, M., Michel, P., 2001. Transfer principles and inequality aversion, with an application to optimal growth. Mathematical Social Sciences 42, 1–11. Kolm, S., 1969. The optimal production of social justice. In: Guitton, H., Margolis, J. (Eds.), Public Economics. Macmillan, London, pp. 145–200. Marshall, A., Olkin, I., 1979. Inequalities: Theory of Majorization and its Applications. Academic Press, New York. Pratt, J., 1964. Risk aversion in the small and in the large. Econometrica 32 (1–2), 122–136. Shorrocks, A., 1983. Ranking income distributions. Economica 50, 3–18. Thon, D., Thorlund-Petersen, L., 1993. The value of information in a simple investment problem. Information Economics and Policy 5, 51–71.
www.elsevier.com / locate / econbase
Transfer principles and relative inequality aversion a majorization approach Ronny Aboudi a , Dominique Thon b , * a
Department of Management Science, University of Miami, P.O. Box 248237, Coral Gables, FL 33124, USA b Instituto Inter-Universitario de Macau, NAPE, Lote 18, Rua de Londres-P, Edf Tak Ip Plaza, Macau, China Received 30 November 2001; received in revised form 31 August 2002; accepted 31 October 2002
Abstract This paper characterizes two preorders over vectors, representing here income distributions, which, for the case of an order-preserving additive welfare function W 5 o n1 u(x i ), correspond to the increasing concave functions u which are more concave than u(x) 5 ln x and than u(x) 5 2 1 /x, respectively. We provide a characterization of those W functions that is quite distinct from the one recently provided by Fleurbaey and Michel, Mathematical Social Sciences, 2001. The two sets of results are shown to be complementary. 2003 Elsevier Science B.V. All rights reserved. Keywords: Majorization; Stochastic dominance; Income inequality; Risk aversion JEL classification: D63; D81; I31
1. Introduction It is well known that if the sum of the k lowest incomes in allocation y is larger than the same for allocation x, each allocation containing n income recipients, for k 5 1, . . . , n, then all additive welfare functions W(x) 5 o n1 u(x i ), with u increasing concave, will unanimously pronounce y preferable to x. Noting the fact that a concave function is a function which is more concave than a linear function, this preorder can be thought of as corresponding to the unanimous ranking given by all additive welfare functions *Corresponding author. Tel.: 186-853-796-4507; fax: 186-853-72-5517. E-mail address: [email protected] (D. Thon). 0165-4896 / 03 / $ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016 / S0165-4896(03)00003-9
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W(x) 5 o n1 u(x i ), with the increasing linear function u(x) 5 x (or any increasing linear function) as the ‘benchmark’, in the sense that the functions u considered are the ones more concave than this particular function. One might then think of using some increasing function other than the linear one as the benchmark function. In this paper we consider two particular cases: the function u(x) 5 ln x and the function u(x) 5 2 1 /x. A recent paper by Fleurbaey and Michel (2001) deals with such classes of functions, and we offer some complementary results. By choosing a strictly concave function as the benchmark, one does in effect allow for some leakage in a reallocation of income. Few redistributive schemes are costless, and intuition suggests that if one has strict inequality aversion, then one might prefer y to x even if y contains less total income. This is equivalent to the buying of less-than-fair insurance by a strictly risk averse agent in expected utility theory. Section 2 sets the stage. Our main results are given in Sections 3 and 4. For each of the two cases, we essentially provide the analogue of a T-transform, and a description of the set of income vectors which dominate some initial vector. Our results are logically distinct from the ones recently presented by Fleurbaey and Michel (2001); in Section 5, we discuss the relationship between the two sets of results. Section 6 concludes.
2. Preliminaries In what follows, x (i ) represents the increasing re-arrangement of the vector x [ R n ; for convenience, we abbreviate Marshall and Olkin (1979) to M&O, and Fleurbaey and Michel (2001) to F&M. We say that the function f is more concave than the function g if f is a concave function of g. A well-known preorder over (income) vectors in R n is defined as follows: Definition 2.1. Let x, y [ R n ; we say that x a 2 y if o k1 x (i ) # o k1 y (i ) , k 5 1, . . . , n. Consider the following transformation of the vector x into the vector y: two of the elements of x, namely i and j, are transformed as follows and the other elements are left untouched: x i ⇒ u x i 1 (1 2 u )x j 5 y i with 0 # u # 1. x j ⇒ (1 2 u )x i 1 u x j 5 y j
(2.1)
The following equivalences are well-known. Lemma 2.1. For x, y [ R n , the following are equivalent: x a 2y
(2.2)
O u(x ) # O u( y ) for all u continuous, increasing and concave
(2.3)
y can be reached from x through a sequence of transfers such as in (2.1) and increases in the elements of x.
(2.4)
n 1
n
i
1
i
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The preorder in Definition 2.1 corresponds to the income distribution y dominating income distribution x by second degree stochastic dominance, hence our symbol a 2 . In the terminology of M&O, p. 10, y is said to be weakly supermajorized by x; in M&O, x a 2 y would be written as y a w x. The kind of transformation represented by (2.1) corresponds to a transfer from the originally richer of the two income recipients to the other, which transfer does not go beyond permuting their original incomes. The bistochastic matrices constructing such transfers are known as T-transforms and their relationship to majorization is described in Muirhead’s Lemma (M&O, p. 21). The equivalence between (2.2) and (2.3) is a mild extension of a part of Hardy, Littlewood and Polya’s Theorem, known as Tomic’s Theorem (see M&O, p. 109). The equivalence between (2.2) and (2.4) is an extension of Muirhead’s Lemma, due to Mirsky; see M&O, A.9.a, p. 123, and C.6., p. 28. The preorder a 2 was introduced in economics by Kolm (1969); see also Shorrocks (1983). Our analysis is based on the preorder a 2 rather than the majorization ordering (the opposite of x a 2 y, with the condition S x 5 S y) because, as will become clear, we do not wish (and in fact cannot) assume a prescribed total income in our analysis. Paradoxically, building on a 2 will allow us to pronounce an allocation with less total income to be preferred.
3. The logarithmic case We first propose, as a variation on Definition 2.1: ln
n Definition 3.1. Let x, y [ R 11 ; we say that x ay if o 1k ln x (i ) # o k1 ln y (i ) , k 5 1, . . . , n.
Consider now the following kind of transformation, with x i # x j : (a) x i ⇒ x i /l 5 y i with x i /x j # l # 1. (3.1) (b) x j ⇒ l x j 5 y j ]] Note that y i 5 y j 5œx] i x j for l 5œx i /x j , and that for l 5 x i /x j , its lower bound, x i and x j are permuted. Theorem 3.1. For x, y [ R n11 , the following are equivalent: ln
x ay
(3.2)
O u(x ) # O u( y ) for all u continuous, increasing and more concave n 1
n
i
1
i
than ln x y can be reached from x through a sequence of transformations such as in (3.1) and increases in the elements of x. ln
(3.3) (3.4)
Proof. Obviously x ay is equivalent to [ln x i ] 1n a 2 [ln y i ] 1n ; thus we are basically dealing with a 2 with a change of variables. The equivalence between (3.2) and (3.3) follows from simply performing this change of variable in the expressions (2.2) and (2.3) which
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are equivalent to each other by Lemma 2.1, and noting that ln x is increasing. Now, (3.2) is equivalent to y being reachable from x through income increases and the following T-transforms applied to the vector [ln x i ] 1n , see (2.1) and Lemma 2.1: (a) ln x i ⇒ u ln x i 1 (1 2 u ) ln x j 5 ln y i with 0 # u # 1. (b) ln x j ⇒ (1 2 u ) ln x i 1 u ln x j 5 ln y j
(3.5)
We now show that there is a one-to-one correspondence between u [[0, 1] in (3.5) and l [ [xi /xj , 1] in (3.1). Combining (3.1a) with (3.5a), and assuming that xi # x j , one has that
u ln x i 1 (1 2 u ) ln xj 5 ln yi 5 ln [x i /l]. u Simple transformations give: ln [x ui 3 x j12u ] 5 ln [x i /l]; x iu 3 x j12u 5 x i /l; x iu 21 3 x 12 5 j 1 /l, or:
(x i /x j )12u 5 l.
(3.6)
Since 0 # u # 1, it is clear that l in (3.6) ranges from x i /x j to 1. Conversely, given any l [ [xi /xj , 1], then there exists a unique u [ [0, 1] that satisfies (3.6). Thus the one-to-one equivalence is established. It remains to show that l in (3.6) satisfies (3.5b). This is easily established, as ln y j 5 ln lx j 5 ln [(x i /x j )12u x j ] 5 (1 2 u ) ln x i 1 u ln x j . The equivalence between (3.2) and (3.4) then follows from the equivalence between (2.2) and (2.4) in Lemma 2.1, noting that the notion of ‘increase in ln x’ is equivalent to the one of ‘increase in x’. h Example. If x 5 (5, 10) and l 5 0.8 in (3.1), then y 5 (6.25, 8). Now the u in (3.5a) which corresponds to this l can be calculated from (3.6), with here x i /x j 50.5, as follows: 0.5 12u 5 0.8; (1 2 u ) ln 0.5 5 ln 0.8, and thus u 5 0.67807. Calculating (3.5) with this u indeed gives the same y as using l 5 0.8 in (3.1). The following two elementary results provide a perspective on the distinction between ln a 2 and a. Let x (i ) be the increasing rearrangement of x, as before, and let x [i ] be its decreasing rearrangement. Theorem 3.2. Let x, a [ R n . Then [x (i ) 1 a (i ) ] n1 a 2 [x i 1 a i ] 1n a 2 [x (i ) 1 a [i ] ] 1n . n
n ln
n ln
n
Theorem 3.3. Let x, a [ R 11 . Then [x (i ) 3 a (i ) ] 1 a[x i 3 a i ] 1 a[x (i ) 3 a [i ] ] 1 . Theorem 3.2 shows that if one thinks of x as being an original income vector and a as being a set of prescribed additions of income, each a i to the individual income x i , then the ordering a 2 can be characterized by the fact that the highest level of welfare is reached by adding to each element of x one of the elements of a in such a way that the smallest x i receives the largest a i , etc., rather than according to any other form of
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ln
pairing. Theorem 3.3 shows that the equivalent is true for x ay if a positive vector a is now used to perform a multiplicative rather than an additive transformation of x. Theorem 3.2 provides an interesting economic interpretation of majorization over incomes, yet does not seem to have appeared in the economic literature. The first a 2 follows from as Proposition A2, p. 140 in M&O, and the second one from Proposition A1, p. 139 in M&O. Theorem 3.3 follows directly from Theorem 3.2, by replacing in the ln latter x i by ln x i , a i by ln a i , and a 2 by a. The two above theorems suggest an interpretation ofln a 2 as being a preorder dealing with absolute (i.e., additive) income changes and x ay as dealing with relative (i.e., multiplicative) ones, in the sense of Theorems 3.2, 3.3. Related results and further references about functions less concave than ln (the mirror-image case) can be found in Thon and Thorlund-Petersen (1993). ln 2 Replacing a by a as a criterion of welfare dominance allows one to pronounce y to be better than x in some cases where S y , S x, as the condition S ln x # S ln y is milder than the condition S x # S y. Note that stochastic dominance of any degree requires, just like a 2 , that the dominating income vector have at least as large an income sum than the dominated one. A transformation of incomes such as (3.1) can, on the other hand, be thought of as a leaky transfer from a rich to a poor. This is readily illustrated by considering, for n 5 2, the better-than-x set, that is the set ln 2 of vectors y that dominate a prescribed x, according to a and a, respectively. We call ln those sets B(x, a 2 ), and B(x, a). The set B(x, a 2 ), is represented in Fig. 1 by the hatched area. The segment between x 5 (x 1 , x 2 ), the original vector, and its permutation, represents redistributions of income at constant sum as in (2.1). This segment is defined by: x (1) # y ( 1) and x 1 1 x 2 5 y 1 1 y 2 . With x 1 , x 2 taken to be constant, then this segment can be represented by a function expressing y 2 as a function of y 1 : y 2 5 x 1 1 x 2 2 y 1 ; y 1 [ [x (1 ) , x ( 2) ].
(3.7) ln
A similar construction is easily made for a. Here the redistributions accomplished by
Fig. 1. The set B(x, a 2 ).
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2hyp
ln
Fig. 2. The sets B(x, a) and B(x, a ).
(3.5), or equivalently (3.1) are defined by ln x ( 1) # ln y (1) and ln x 1 1 ln x 2 5 ln y 1 1 ln y 2 . We obtain the analogue of (3.7): ln y 2 5 ln x 1 1 ln x 2 2 ln y 1 ; y 1 [ [x ( 1) , x (2) ],
(3.8)
which, for x 1 , x 2 taken to be constant, gives the following hyperbolic relationship between y 1 and y 2 : y 2 5 x 1 x 2 /y 1 . This is the continuous convex curve in Fig. 2. From ln this, we can easily construct B(x, a), represented as the hatched area. It consists of B(x, a 2 ), as in Fig. 1, plus the lens-shaped set below B(x, a 2 ), representing distributions whose total income is less than the one of x. The ‘bended’ part of the ln closure of B(x, a) is precisely what (3.8) describes; it corresponds to the redistributions which preserve the sum of the logarithms of income, up to a permutation of incomes.
4. The hyperbolic case We now define: 2hyp
n Definition 4.1. Let x, y [ R 11 ; we say that x a y if o 1k 2 1 /x (i ) # o 1k 2 1 /y (i ) , k 5 1, . . . , n.
Consider now the following kind of transformation, with x i # x j : (a) x i ⇒ x i /l 5 y i xj (b) x j ⇒ ]]]] 5 y j with x i /x j # l # 1. xj 1 1 ] [1 2 l] xi
(4.1)
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Note that y i 5 y j 5 2x i x j /(x i 1 x j ) for l 5 (x i 1 x j ) / 2x j , and that for l 5 x i /x j , its lower bound, x i and x j are permuted. n
Theorem 4.1. For x, y [ R 11 , the following are equivalent: 2hyp
x a y
(4.2)
O u(x ) # O u( y ) for all u continuous, increasing and more n 1
n
i
1
i
concave than 2 1 /x y can be reached from x through a sequence of transformations such as in (4.1) and increases in the elements of x.
(4.3) (4.4)
2hyp
Proof. Obviously x a y is equivalent to [21 /x i ] n1 a 2 [21 /y i ] n1 ; thus we are basically dealing with a 2 with a change of variables. The equivalence between (4.2) and (4.3) follows from simply performing a change of variable in the expressions (2.2) and (2.3) which are equivalent to each other, by Lemma 2.1, and noting that 2 1 /x is increasing. Note that (4.2) is equivalent to y being reachable from x through income increases and the following T-transforms applied to [21 /x i ] n1 , see (2.1) and Lemma 2.1: 21 21 21 21 (a) ]] ⇒ u ]] 1 (1 2 u ) ]] 5 ]] xi xi xj yi 21 21 21 21 (b) ]] ⇒ (1 2 u ) ]] 1 u ]] 5 ]] xj xi xj yj
with 0 # u # 1.
(4.5)
We now show that there is a one-to-one correspondence between u [[0, 1] in (4.5) and l [[xi /x j , 1] in (4.1). Combining (4.1a) with (4.5a), and assuming that x i # xj , one has that u (21 /x i ) 1 (1 2 u )(21 /x j ) 5 2 1 / [x i /l] 5 2 l /x i , or: xi u 1 (1 2 u ) ] 5 l. (4.6) xj Note that the left side of (4.6) is a convex combination of 1 and x i /x j # 1, and thus the one-to-one correspondence is established. It remains to show that l satisfying (4.6) also satisfies (4.5b). We have that 2 1 /y j
2 1 2 (x j /x i )[1 2 l] 5 ]]]]]] 5 2 1 /x j 1 (21 /x i )[1 2 l] xj 5 2 1 /x j 1 (21 /x i )[1 2 u 2 (1 2 u )(x i /x j )] 5 (1 2 u )(21 /x i ) 1 u (21 /x j ),
which is (4.5b). The equivalence between (4.2) and (4.4) then follows from the equivalence between (2.2) and (2.4), noting that the notion of ‘increase in 2 1 /x’ is equivalent to the one of ‘increase in x’. h Example. If x 5 (5, 10) and l 50.8 in (4.1), then y 5 (6.25, 7.143). Now the u in (4.5a) that corresponds to this l is, from (4.6), with here x i /x j 50.5, such that 0.85u 1 (1 2 u )0.5, or u 50.6. Calculating (4.5) with this u indeed gives the same y as using
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l 50.8 in (4.1). 2hyp
Transformations such as (4.1) — and the associated dominance concept a — allow an ln even larger leakage than a, illustrated below, as suggested by the fact that the function 2hyp 2 1 /x is more concave than the function ln(x). For a , one obtains the following analogue to (3.7) and (3.8): 1 /y 2 5 1 /x 1 1 1 /x 2 2 1 /y 1 ; y 1 [ [x (1 ) , x ( 2) ], 2hyp
from which B(x, a ) can easily be constructed. It appears on Fig. 2, as the set of points 2hyp ln to the N–E of the dotted curve. Obviously, B(x, a 2 ) # B(x, a) # B(x, a ).
5. Relationship to Fleurbaey and Michel In the paper that inspired the present one, F&M provide two main results dealing with welfare functions of the form W(x) 5 S u(x i ) using the two classes of u functions more concave than ln x and 21 /x, respectively (they also provide an application to economic growth that will not concern us). Unlike us they do not start with the definition of a preorder (in fact, they do not consider preorders at all, see below), but with particular types of transformations affecting a pair of incomes. In the logarithmic case, their starting point is to consider leaky transfers constructed such that, as a proportion of the final incomes, the gain to the (poorer) receiver is equal to the loss of the (richer) donor. Unlike ours, their transfers are bounded so as not to invert the relative positions, as in a Dalton transfer; as a result in this Section we assume x i # x j , for the sake of comparability. They call such transfers ‘proportional ex-post transfers’, PEP-transfers for short. Definition 5.1. (PEP-transfer principle) Consider a vector (x 1 , x 2 , . . . , x n ), two individuals i and j, and a positive number d. If the vector (z 1 , z 2 , . . . , z n ) is such that for all k ± i, j, z k 5 x k and z i 5 x i /(1 2 d ) # z j 5 x j /(1 1 d ),
(5.1)
then W(x) , W(z). In their Proposition 2, F&M establish the following result, which we shall compare to our own Theorem 3.1. Proposition 2. (F&M, p. 7) W satisfies the proportional ex-post transfer principle if and only if there is a concave function v: R → R such that u(x) 5 v(ln x). Note that a sequence of PEP-transfers is not such a transfer itself, as any example will establish. In fact, as we now show, a sequence of small PEP-transfers gives at the limit
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our transformations (3.1), and there is thus a connection between PEP-transfers and our ln
better-than-x set B(x, a). We first establish a technical lemma. Lemma 5.1. Let 0 , l , 1 and let m be a natural number. Then (A): [2 2 l 1 / m ] m , 1 /l. Furthermore, (B): [2 2 l 1 / m ] m is strictly increasing in m. Finally, (C): lim [2 2 l 1 / m ] m 5 1 /l. m→` 1/m
1/m
1/m
Proof. (A) is equivalent to [2 2 l ] , 1 /l . Then, multiplying by l and collecting terms, we have: 2 [ l 1 / m 2 1] 2 , 0, which holds for all positive l ± 1. h (B) Let us assume that m is continuous and m $ 1. Then [2 2 l 1 / m ] m is strictly positive. Now, [2 2 l 1 / m ] m is strictly increasing in m iff ln([2 2 l 1 / m ] m) is. Now let f(m)5 ln ([2 2 l 1 / m ] m)5m ln (2 2 l 1 / m). Then f 9(m)5ln (2 2 l 1 / m)1 l 1 / m ln l /(m[2 2 l 1 / m ]). Let a 5 l 1 / m ; thus l 5 a m and ln l 5 m ln a. Furthermore, since 0 , l , 1 and m $ 1, we have 0 , a , 1. Substituting in f 9(m), we have: f 9(m)5ln (2 2 a ) 1 a ln a /(2 2 a ). Since 2 2 a . 0, f 9(m).0 iff g(a )5(2 2 a ) ln (2 2 a ) 1 a ln (a ) . 0 for all 0 , a , 1. We have g(1)50, and g9(a )5ln a 2 ln (2 2 a ) 5 ln (a /(2 2 a )) , 0, where the inequality follows from: a /(2 2 a ) , 1 for all 0 , a , 1. Thus g(a ) is strictly decreasing on (0, 1). But since g(1)50, we have that g(a ).0 for all a in (0, 1). So f 9(m).0 for all m $ 1. h (C) We proceed by showing that for m → `, ln[2 2 l 1 / m ] m → 2 ln l, which is equivalent to (C). Now, ln(2 2 l 1 / m) 1/m ]]]] lim m ln (2 2 l ) 5 lim m→` m →` 1 /m (1 /(2 2 l 1 / m))(2ln l)l 1 / m (21 /m 2 ) 5 lim ]]]]]]]]]] 5 2 ln l. m →` 2 1 /m 2 ˆ The second equality is obtained by applying L’Hopital’s rule, and the third equality by 1/m 1/m noting that m→` lim (2 2 l ) 5 1, and that mlim l 5 1, for l . 0. h →` Theorem 5.1. Let the pair hy i , y j j be reached from hx i , x j j, with 0 , x i , x j , by a single ]] transformation such as in (3.1) with œx i /x j # l , 1. Starting from the same hx i , x j j, for any natural number m, consider a finite number m of successive transformations such as (m) (m) in (5.1), each with the same d, and which produce the pair hz (m) 5 yi . i , z j j with z i (m) (m) Then, (A): z j . y j . Furthermore, (B): z j is strictly decreasing in m. Finally, (C): lim m→` z j(m) 5 y j . Proof. Let the transformation (3.1) give y i 5 x i /l and y j 5 lx j . Let the result of the (m) m (m) m sequence of m transformations (5.1) be z i 5x i /(1 2 d ) and z j 5x j /(1 1 d ) , with, m (m) for a given l, d chosen such that y i 5x i /l 5x i /(1 2 d ) 5z i . This implies l 5(1 2 d )m or d 5 1 2 l 1 / m . Thus we have: y j 5 lx j and z j(m) 5 x j /(1 1 d )m 5 x j / [2 2 l 1 / m ] m .
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The three concluding statements of the Theorem follow by applying in turn the three statements of Lemma 5.1: (A) [2 2 l 1 / m ] m , 1 /l → z j(m) .y j (B) [2 2 l 1 / m ] m strictly increasing in m → z (m) is strictly decreasing in m j (C) lim [2 2 l 1 / m ] m 51 /l → lim z (m) 5y . h j j m→`
m→`
Before illustrating Theorem 5.1, we make some remarks on its assumptions. The assumptions l , 1 and x i , x j are meant to avoid trivialities ( l 5 1 means ‘zero ]] transfer’). The condition œx i /x j # l in the Theorem insures that y i # y j , which is necessary to be able to compare our result to F&M’s, as the latter assume rank(m) (m) (m) (m) preservation. Note that, as we have z (m) j .y j , then z j .y j $ y i 5z i . Thus z j .z i , as it should be (for a finite m). Note, that 0 , x i implies, in this rank-preserving situation, (m) that 0 , z (m) j , y j , y i , z i . The assumptions of Lemma 5.1 have been chosen to cover the ones of Theorem 5.1, but some of the results in Lemma 5.1 are more general, for example part (A). In our two-dimensional examples, we write x 1 for x i , x 2 for x j . Example. Consider first the fact that (3.1) with x 5 (5, 10) and l 50.8 gives y 5 (6.25, 8) while (5.1) with d 50.2 gives z 5 (6.25, 8.333) and one has z 1 5y 1 but z 2 . y 2 . Thus a single transformation (5.1) results in a pair with a second element larger than with a single transformation (3.1), at constant first element. Consider now d 9 chosen such that z (12 ) 5x 1 /(1 2 d 9)2 56.255y 1 . This gives d 950.1056. Thus this sequence of two transformation (5.1) gives z (12 ) 5z 1 5y 1 ; now z (22 ) 5x 2 /(1 1 d 9)2 58.181, and z 5 (6.25, 8.181). Comparing to the one-step procedure, we have that the value of the second element is reduced. Now Theorem 5.1 tells us that the thinner one slices one PEP-ex-post transfer into m smaller and equal such transfers that keep z (m) 1 5z 1 constant, the closer z (m) comes to y . Furthermore, at the limit, a sequence of PEP-transfers can 2 2 reproduce any income pair reached by one application of our transformations (3.1), and PEP-transfers plus income increases (F&M assume monotonicity) can generate, at the ln
limit, the same better-than-x set B(x, a) as we have, Fig. 2, if anonymity [symmetry] is further assumed. The point can be illustrated graphically, see Fig. 3. Note that for x 1 , x 2 treated as constants, (5.1) gives a relationship between z 1 and z 2 which can easily be calculated to be: z 2 5 x 2 /(2 2 x 1 /z 1 ); z 1 [ [x 1 , (x 1 1 x 2 ) / 2]. This is represented by the convex curve joining x to A in Fig. 3. The other convex curve in the figure is the same as the continuous one on Fig. 2, which represents the lower ln bound of our better-than-x set B(x, a), see Section 3. Consider finally the fact that a single PEP-transfer producing equality, for x 5 (5, 10), gives z 5 (7.5, 7.5), implying no leakage, while the point of our better-than-x set with equal incomes is the pair y 5 (7.0711, 7.0711), with some leakage. It is easy to calculate that a single PEP-transfer that gives z 1 57.0711 implies z 2 57.7346, and that a sequences of three PEP-transfers
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Fig. 3. The logarithmic case.
can be constructed, see above, to give (z (13 ) , z (23 ) )5(7.0711, 7.3297). As Theorem 5.1 indicates, increasing the number of such transfers (m) while keeping z 1(m) 5 z 1 57.0711 decreases z (m) 2 , and, at the limit, will produce the pair (7.0711, 7.0711). In their Proposition 1 (F&M, p. 5), dealing with functions u(x) more concave than 2 1 /x, F&M consider the following kind of transformation, called ‘proportional transfers’, P-transfers for short, which are constructed so as to make, in terms of proportion to the original incomes, the gain of the (poorer) receiver equal to the loss of the (richer) donor. Again the transfers are bounded to preserve the relative positions. Definition 5.2. P-Transfer Principle. Consider a vector (x 1 , x 2 . . . , x n ), two individuals i and j, and a positive number d. If the vector (z 1 , z 2 , . . . , z n ) is such that for all k ± i, j, z k 5 x k and z i 5 x i (1 1 d ) # z j 5 x j (1 2 d )
(5.2)
then W(x) , W(z). They prove the following result (note the strict concavity of v); compare to our own Theorem 4.1: Proposition 1. (F&M, p. 5) W satisfies the proportional transfer principle if and only if there is a strictly concave function v: R 2 2 → R such that u(x) 5 v(21 /x). The situation here is the same as in the logarithmic case, as regards the comparison of the effect of one of our transformations (4.1) to one of F&M (5.2): at constant first element in the resulting pair, (5.2) gives a larger second element: Taking again x 5 (5, 10), and d 50.2 in (5.2) gives the income vector z 5 (6, 8), while with l 55 / 6,
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expression (4.1) gives y 5 (6, 7.5), with the same first element but a smaller second element. F&M’s Definition 5.2 gives:
F
G
2x 1 x 2 z 2 5 2x 2 2 (x 2 /x 1 )z 1 ; z 1 [ x 1 , ]] , x1 1 x2 that is, a linear relationship between z j and z i , which is represented in Fig. 4. Note that both Definition 5.2 and our own (4.1) imply that equality of income is achieved for those incomes for which y 1 5y 2 5z 1 5z 2 52x 1 x 2 /(x 1 1 x 2 ). Otherwise, this line is entirely above the dotted curve, reproduced from Fig. 2, which is the lower bound of our 2hyp better-than-x set B(x, a ). In contradistinction to the logarithmic case, performing a sequence of small P-transfers rather than a single one does not decrease the value of the resulting z j , for a prescribed z i ; it actually increases it. This is readily established. Consider z 1 5x 1 (1 1 d ) 5 x 1 (1 1 d1 )(1 1 d2 ), which requires d 5 d1 1 d2 1 d1d2 and z 2 5 x 2 (1 2 d ), z 29 5 x 2 (1 2 d1 )(1 2 d2 ). Elementary calculations show that for d1 , d2 . 0, z 2 ,z 29 . This is the opposite of what we have in the logarithmic case. Thus here, a succession of small P-transfers cannot be used to approximate our better-than-x set 2hyp B(x, a ). The results of F&M are of an entirely different nature from ours. This should be clear from the fact that the concept of a pre-order, the notion of a sequence of transformations and the one of a better-than-x set appear nowhere in their paper and are not their concern. Their point of departure is to consider transfers with well-defined properties, PEP- and P-transfers and they ‘look for’ the largest class of W functions preserving such single transfers. We, on the other hand, move in the opposite direction. We start with two benchmark functions, and on their basis define two preorders that are variations on the theme of majorization. From there we characterize the analogues of the T-transforms, and the better-than-x set spanned by a sequence of them. Thus, in spite of the consideration of the same classes of functions, our respective two main theorems should
Fig. 4. The hyperbolic case.
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be expected to have no logical relationship to each other. In the light of this it is, if anything, very surprising that in the logarithmic case, there is actually a connection between the two results, Theorem 5.1, and that, in some sense, our better-than-x set can be constructed using PEP-transfers.
6. Conclusion We have characterized the preorders obtained essentially from the standard second degree stochastic dominance ordering through both a logarithmic and a negative hyperbolic transformation of the variables appearing in the order-preserving functions of the form W(x) 5 o n1 u(x i ). We have identified the corresponding analogues to Ttransforms, and described the corresponding better-than-x sets. Note that if one werelnto n compare two given income vectors x, y [ R 11 , and wanted to check whether x ay holds, then a straightforward algorithm is provided by the very Definition 3.1. Likewise 2hyp
for x a y. It is well known that the functions ln x and 2 1 /x have a constant relative risk aversion, of 1 and 2, respectively, see Pratt (1964). This obviously suggests the possibility of a generalization considering as the benchmark function any member of the class of (increasing concave) constant relative risk aversion utility functions. This also suggests an independent generalization of F&M’s results.
Acknowledgements Thanks are due to M. Fleurbaey, L. Thorlund-Petersen and two referees for useful comments.
References Fleurbaey, M., Michel, P., 2001. Transfer principles and inequality aversion, with an application to optimal growth. Mathematical Social Sciences 42, 1–11. Kolm, S., 1969. The optimal production of social justice. In: Guitton, H., Margolis, J. (Eds.), Public Economics. Macmillan, London, pp. 145–200. Marshall, A., Olkin, I., 1979. Inequalities: Theory of Majorization and its Applications. Academic Press, New York. Pratt, J., 1964. Risk aversion in the small and in the large. Econometrica 32 (1–2), 122–136. Shorrocks, A., 1983. Ranking income distributions. Economica 50, 3–18. Thon, D., Thorlund-Petersen, L., 1993. The value of information in a simple investment problem. Information Economics and Policy 5, 51–71.