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Balanced egalitarian redistribution of income
Mathematical Social Sciences 33 (1997) 185–201
Balanced egalitarian redistribution of income Yves Sprumont* ´ ´ ´ Centre de Recherche et developpement en economique ( C.R.D.E.) and Departement de Sciences ´ ´ , Quebec ´ , Canada H3 C Economiques , Universite´ de Montreal, C.P. 6128, succursale Centre-ville, Montreal 3 J7 Received 1 March 1995; revised 1 October 1996; accepted 1 November 1996
Abstract This paper reconsiders the issue of how income should be redistributed when people endowed with different levels of talent exert different levels of effort. Two new schemes, called balanced egalitarian equivalence and balanced conditional egalitarianism, are proposed and characterized. 1997 Elsevier Science B.V. Keywords: Redistribution; Egalitarianism
1. Introduction How should income be redistributed among people endowed with different skills, handicaps or needs? The ideal of equal opportunity, advocated in a broader context by Arneson (1989) and Cohen (1989), is based on the postulate that some determinants of an individual’s income are within his control while others are not. Throughout the paper I will use the term effort to refer to the former category and talent to refer to the latter. I assume that both are real variables; this tremendous simplification does not destroy the interest of the analysis. Equalizing opportunities, it may be argued, requires that income be redistributed so as to offset inequalities that are due to differences in talent while preserving those that arise from differences in effort. As it turns out, there is a tension between these two desiderata. In fact, it is generally impossible to guarantee an equal income to individuals exerting; the same effort, while performing identical income transfers to those: who are equally talented. This remarkable impossibility was first pointed out (in a more general framework) by Fleurbaey (Fleurbaey, 1994a,b) and later reinforced by Bossert (1995). *Correspondence address. Tilburg, University CentER, P.O. Box 90153, 5000 Tilburg, LE Netherlands. 0165-4896 / 97 / $17.00 1997 Published by Elsevier Science B.V. All rights reserved PII S0165-4896( 96 )00835-9
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Weakening the defining requirements of equal opportunity opens the way to a variety of interesting redistribution schemes. Egalitarian equivalence and conditional egalitarianism are among the most appealing ideas. Egalitarian equivalence insists that the principle of equal income for equal effort be preserved. The simplest egalitarianequivalent schemes work as follows. First, a reference level of talent is fixed. Then, each individual receives a post-transfer income equal to the pre-transfer income he would earn were he endowed with the reference talent level, plus a uniform amount dictated by the budget constraint. These schemes were introduced in a slightly different setting and given an axiomatic characterization by Fleurbaey (1995). An alternative axiomatization was proposed by Bossert and Fleurbaey (1994). The main difficulty lies in the arbitrary nature of the reference level of talent. In an effort to overcome this difficulty, Moulin (1994) suggested the average egalitarian-equivalent scheme: successively choose each individual’s level of talent as the reference level compute the corresponding egalitarianequivalent post-transfer income distributions, and take the average. That method was recently axiomatized by Bossert and Fleurbaey (1994). Yet another solution, mentioned in Moulin (1994), is to choose the median individual talent as the reference. Conditional egalitarianism is essentially dual to egalitarian equivalence. Here the principle of equal transfer for equal talent is given priority. The simplest conditionally egalitarian schemes are based on a fixed reference level of effort. The first component of every individual’s post-transfer income is the average income that would be generated if all individuals exerted the reference level of effort. To that component is added the income increment created by the individual’s deviation from the reference level. Median conditional egalitarianism selects the median value of the individual efforts as the reference level. Average conditional egalitarianism averages the conditionally egalitarian post-transfer income distributions obtained by successively choosing each individual’s effort as the reference level. Conditionally egalitarian schemes and their average version were studied by Fleurbaey (1995) and Bossert and Fleurbaey (1994). Roemer’s (Roemer, 1993) approach, though somewhat different, also makes use of reference levels of effort. This paper proposes and defends two new income redistribution schemes to which I will refer by the phrase balanced egalitarianism. They are motivated by weaknesses of the average and median egalitarian-equivalent and conditionally egalitarian approaches that I shall now describe. Consider first the median and average egalitarian-equivalent schemes. By the aforementioned impossibility result of Fleurbaey, the transfers received by two equally talented individuals under either of these two schemes will generally differ. What is more disturbing, however, is that one individual may be taxed while the other is subsidized. As it turns out, the gap between the tax and the subsidy is unbounded. This is a rather extreme violation of the principle of equal transfer for equal talent. I will suggest a simple scheme, called balanced egalitarian-equivalent, that guarantees that equally talented individuals never receive transfers of opposite signs, and delivers equal income for equal effort. I will show that these properties are characteristic of the balanced egalitarian-equivalent scheme in the limit case where all configurations of effort and talents are present in society. Average, or median, conditional egalitarianism suffers from a weakness that is
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technically dual to that of average or median egalitarian equivalence, though maybe conceptually less serious: individuals exerting the same effort may enjoy post-transfer incomes that are on opposite sides of society’s average income. The balanced conditionally egalitarian scheme defined in this paper avoids that difficulty. Since the two schemes that I propose are dual, and because I find the property of equal income for equal effort more fundamental than equal transfer for equal talent, I shall analyze balanced egalitarian equivalence in some detail but content myself with a brief description of balanced conditional egalitarianism. The beginning of Section 2, which is a bit informal, serves as a motivation for introducing balanced egalitarian equivalence. I recall the definition of average egalitarian equivalence in (the onedimensional version of) Bossert’s (Bossert, 1995) model and offer an example where two equally talented individuals receive transfers of opposite signs. A similar example could be constructed for the median egalitarian-equivalent scheme. Next, I set up the basic model used in the paper, which is a continuous version of Bossert’s. I define and characterize balanced egalitarian equivalence. (The formal proof is in Appendix A.) Resorting to a continuum of individuals is necessary for the characterization result but balanced egalitarian equivalence is, of course, well defined in the finite case as well. A shortcoming of Bossert’s model is that an individual’s income does not depend on the talent and effort of other members of society. This assumption is relaxed in Section 3. The definition of balanced egalitarian equivalence is generalized to a richer setting and conditions under which its characterization remains valid are discussed. Section 4 defines and characterizes balanced conditional egalitarianism. Finally, Section 5, Appendix B and Appendix C discuss the limitations of the paper, some possible extensions, another characterization of balanced egalitarian equivalence, and an alternative approach to the redistribution problem.
2. A basic formulation of balanced egalitarian equivalence Consider a society composed of n agents. Agent i’s effort is e i , his talent is t i , and his income before redistribution is f(e i , t i ). The function f is increasing in both arguments. Let us fix a reference talent t 0 . Under the t 0 -egalitarian-equivalent redistribution scheme, agent i’s post-transfer income is
O
1 n x ti 0 5 f(e i , t 0 ) 2 ] [ f(e k , t 0 ) 2 f(e k , t k )]. n k 51 Under average egalitarian equivalence, agent i receives the post-transfer income
O
1 n tj xi 5 ] x n j 51 i Simple computations show that the corresponding income transfer, or compensation, takes the form
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FO
O
1 n 1 n c i : 5 x i 2 f(e i , t i ) 2 ] f(e j , t j ) 2 ]2 f(e k , t j ) n j 51 N j, k51
FO n
G
G
1 1 ] f(e i , t j ) 2 f(e i , t i ) . n j 51 Suppose now that n53, e 1 .e 2 .e 3 , and t 1 5t 2 5:t,t 3 . Thus, agents 1 and 2 are equally talented. Assume that f(e 1 , t) 5 5, f(e 1 , t 3 ) 5 10, f(e 2 , t) 5 2, f(e 2 , t 3 ) 5 3, f(e 3 , t) 5 1, f(e 3 , t 3 ) 5 2. Then c 1 511 / 9.0. 21 / 95c 2 : agent 1 is subsidized whereas agent 2 is taxed. In fact, if f(e 1 , t 3 )551a, where a.2, then c 1 5(112a) / 9 and c 2 5(42a) / 9: agent 1’s subsidy and agent 2’s tax are unbounded. There is a natural solution to the above problem. Choose the talent t* such that o nk 51 f(e k , t*)5o nk 51 f(e k , t k ). Agent i’s post-transfer income under the t*-egalitarianequivalent scheme is merely f(e i , t*). No uniform amount is needed to balance the budget: each agent simply gets the pre-transfer income he would earn if he were endowed with the talent t*. The transfer to agent i being f(e i , t*)2f(e i , t), it is obvious that equally talented agents cannot receive transfers of opposite signs. Let us call the scheme just described balanced egalitarian equivalent. It turns out that when all combinations of effort and talent are present in society, balanced egalitarian equivalence is the only way to guarantee transfers of equal sign to equally talented agents and equal income for equal effort. The rest of this section is devoted to a formal statement and proof of this assertion. From now on and until Section 5, I postulate a continuum of agents. Formally, the agent space is the probability space (I, @, ,), where I5[0, 1], @ is the Borel s -algebra of I, and , is Lebesgue’s measure on @. Effort ´ and talent t are random variables on the agent space taking values in E5T5(0, 1). Society is a joint probability density function s of effort and talent. The corresponding distribution function is denoted by s. The marginal density functions of effort and talent are s´ and st , with associated ¯ distribution functions s´ and st . Mean effort and mean talent in society s are e(s)5e E es´ (e) de and ¯t(s)5eT tst (t) dt. Society s is called regular if for every measurable subset A of E 3T, s (A).0 if and only if l(A).0, where l is Lebesgue’s measure on the Borel s -algebra of E 3T. Loosely speaking, this means that all effort–talent combinations are found in society. The (pre-transfer) income of an agent who exerts the effort level e and is endowed with the talent level t in society s is denoted f(e, t). Note that this quantity does not vary with s: in this model, an agent’s income is entirely determined by his own characteristics. Think of a society of farmers who all cultivate corn on their own identical parcel of land. This assumption that all agents are completely independent of each other will be
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relaxed in the next section. I suppose that the income function f : E 3T →R 1 satisfies the following conditions: (A.12 )
for each t[T, f(e, t) is continuous and increasing in e, and lim e →0 f(e, t)50;
(A.2)
for each e[E, f(e, t) is continuous and increasing in t, and lim t→0 f(e, t)50.
¯ The mean income in society s is y(s)5e E eT f(e, t) s(e, t) de dt. In what follows, every uncountable subset of an Euclidean space is endowed with the induced Euclidean topology and every finite set is endowed with the discrete topology. The measurability of a mapping between two topological spaces is always understood with respect to the Borel s -algebras of these spaces. A redistribution (of income in society s) is a measurable mapping F: E 3T →R 1 satisfying the feasibility condition eE ¯ eT F(e, t) s(e, t) de dt5y(s). The number F(e, t) is the post-transfer income of any agent exerting effort e and endowed with talent t in society s.1 The compensation mapping C associated with F is defined by C(e, t)5F(e, t)2f(e, t) for all (e, t)[E 3T. In order to define balanced egalitarian equivalence, consider a society where the distribution of effort is the same as in society s but where all agents are endowed with a same level of talent. The level ensuring that the mean income in this society is the same as in s is called the balanced talent level (in society s) and denoted by t*. Formally, t* satisfies the condition ¯ Ef(e, t*)s (e) de 5 y(s). ´
E
Under assumptions (A.1) and (A.2), t* exists, is unique, and belongs to T. A redistribution F in society s is balanced egalitarian-equivalent if for s -a.e. (e, t) [ ExT, F(e, t) 5 f(e, t*). Balanced egalitarian-equivalent redistributions are unique up to differences on sets of zero s -measure. As already mentioned, they guarantee transfers of equal sign to equally talented agents and equal income for equal effort. Formally, a redistribution F satisfies the principle of Equal Income for Equal Effort (EIEE) if there exists a measurable mapping a : E →R 1 such that for s -a.e. (e, t), F(e, t)5 a (e). It satisfies the principle of Fairly Signed Transfers (FST) if there exists a measurable mapping b : T →h21, 11j such that for s -a.e. (e, t), b (t) C(e, t)$0. Proposition 1. Suppose that the income function f : E 3T →R 1 satisfies assumptions (A.1 ) and (A.2 ) and let s be a regular society. Then a redistribution of income in society s satisfies the principles of Equal Income for Equal Effort and Fairly Signed Transfers if and only if it is balanced egalitarian-equivalent. 1
Of course, there need not exist such agents. This is irrelevant, however, since the behavior of F on sets of zero s -measure is unconstrained by the feasibility condition.
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Fig. 1. Illustration of the proof of Proposition 1.
The ‘‘if’’ part of Proposition 1 is obvious. The principle of the proof of the ‘‘only if’’ statement is also very simple. Consider Fig. 1. By EIEE, (almost) all agents on a same vertical line receive the same post-transfer income. If the redistribution is not balanced egalitarian-equivalent, I may assume that all agents on the e 1 vertical line get more than f(e 1 , t*) and, by feasibility, all agents on the e 2 line get less than f(e 2 , t*). But this means that (e 1 , t*) agents are subsidized while (e 2 , t*) agents are taxed. The measure-theoretic details of the proof are worked out in Appendix A. The regularity assumption in Proposition 1 is restrictive but can be slightly weakened. One might think that it could be replaced with the assumption that the support of s, hereafter denoted A s , is a convex set. The following example shows that this is not true. Let s be the uniform probability density function over the convex hull of the four points (1 / 8, 15 / 16), (7 / 8, 3 / 16), (7 / 8, 1 / 16), (1 / 8, 13 / 16). Suppose that f(e, t)5e1t for all ¯ e, t. Check that the mean income in this society is y(s)51 and the balanced talent level is t*51 / 2. Define the redistribution F by e/3 1 5/8 F(e, t) 5 e 1 1 / 2 e / 3 1 25 / 24
5
if e # 3 / 16, if 3 / 16 , e , 13 / 16, if 13 / 16 # e,
which clearly satisfies the feasibility condition but differs from the balanced egalitarian equivalent redistribution. Obviously, F satisfies EIEE and it is easily checked that it also meets FST. There is nothing pathological in this example; in particular, A s has positive Lebesgue measure and F is continuous and strictly increasing in e. The difficulty arises from the
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fact that all the agents who exert a low effort (e#3 / 16) have talent well above the balanced level while all those exerting a high effort (e$13 / 16) have talent well below that level. One set of conditions that can be substituted for the regularity assumption without affecting the validity of Proposition 1—and hardly its proof—consists of the following two restrictions: i) for every measurable set B , A s , s (B).0 if and only if l(B).0, and ii) there is a neighborhood T * around t* such that T *3E9, A s for every subset E9 of E having positive s´ -measure. Observe that these restrictions do not imply the convexity of A s ; yet, they remain demanding. While Bossert (1995), and Bossert and Fleurbaey (1994) do not need any restriction of this sort, their results hold for redistribution rules defined over sufficiently rich domains of societies and rely on axioms making comparisons across societies. To reach a characterization result in a fixed-society framework, enough richness must be assumed on society’s support.
3. Balanced egalitarian equivalence in more complex environments This section allows each agent’s pre-transfer income to depend not only on his own characteristics, but also on the characteristics of other members of society. The agent space, effort, talent and society are defined as in the previous section. The (pre-transfer) income of an agent who exerts effort e and is endowed with talent t in society s is now written f(e, t, s). Letting S stand for the set of all societies, a ( generalized) income function f is defined on E 3T 3S. Assumptions (A.1) and (A.2) are reexpressed as follows: (A.3 )
for each t[T and s[S, f(e, t, s) is continuous and increasing in e, and lim e →0 f(e, t, s)50;
(A.4)
for each e[E and s[S, f(e, t, s) is continuous and increasing in t, and lim t →0 f(e, t, s)50.
These are restrictions on the pre-transfer income function within any society. In addition, I shall impose two restrictions on how income varies euchres societies. Consider a society in which the distribution of talent is concentrated at a single level. Provided that the distribution of effort remains unchanged, a small increase of the uniform talent level is likely to induce a small increase in income at every level of effort. Formally, let s t denote the society where talent is concentrated at level t and effort is distributed as in s: s t (e, t9) 5
H
0 s´ (e)
for all e [ E and t9 [ T • ht j, for all e [ E and t9 5 t.
The assumption is: (A.5 )
for each e[E and s[S, f(e, t, s t ) is continuous and increasing in t.
This requirement is fairly weak because it bears only on very comparable societies. By
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contrast, the last assumption involves societies that may be quite different from each other. Recall that the balanced talent level t* in society s was defined in Section 2 by the condition that shifting everyone’s talent to that level does not affect mean income. Since society itself is now an argument of the income function, the proper formulation of that condition becomes ¯ E f(e, t*, s * )s (e) de 5 y(s), t
´
E
¯ where y(s):5e E eT f(e, t, s) s(e, t) de dt is the mean income in society s. Assumptions (A.3) to (A.5) ensure that t* exists, is unique and belongs to T. I now assume: (A.6 )
for each e[E, t[T and s[S, f(e, t*, s)5f(e, t*, s t * ).
This is a limited neutrality property. It says that bringing all talents to the balanced level does not affect the agents who are endowed with precisely the balanced level of talent. Of course, assumptions (A.3) to (A.6) are satisfied if f is constant in s and f(?,?, s) satisfies (A.1) and (A.2) for any s: the present model encompasses the no-interaction model of Section 2. Assumptions (A.3) to (A.6), however, allow for interesting patterns of influence among the agents, as the following example illustrates. Example. Suppose that each agent combines effort and talent to supply some standardized input through the mapping x, say, x(e, t)5et. Suppose that the mean input ¯ in society s, x(s), determines the mean income through some aggregate production function w : (0,1)→R 1 which is continuous, increasing, and such that lim x→0 w (x)50. Finally, suppose that each agent’s income is proportional to his input contribution: ¯ w (x(s)) ¯ f(e, t, s) 5 (x(e, t) /x(s)) for all e, t, s.
(1)
This technological environment is considered in Mirrlees (1974); Roemer and Silvestre (1987); Moulin (1990); Fleurbaey and Maniquet (1994a, (1994b); Roemer (1994a). The traditional story is that of a group of fishers catching fish in a lake. The proportionality constraint (1) simply means that each fisher eats his own catch. In that context, it is natural to assume that w (x) /x is decreasing in x: individual income then decreases with society’s mean input. I shall not make that assumption here, thereby allowing for a greater variety of possible interactions among agents. It is obvious that assumptions (A.3) and (A.4) are satisfied. Next, since ¯ w (e(s)t) ¯ f(e, t, s t ) 5 (e /e(s)) for all e, t, s, assumption (A.5) is met as well. Finally, since the balanced level of talent in society s ¯ /e(s), ¯ t* 5 x(s) ¯ ¯ ¯ it follows that f(e, t*, s)5(x(e, t*) /x(s)) w (x(s))5(e /e(s)) w (e(s) t*)5f(e, t*, s t * ), in agreement with assumption (A.6). Returning to the general model, define a redistribution F in society s as in the
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previous section. The compensation mapping associated with F is C5F 2f(?,?, s) and the EIEE and FST principles are unchanged. A redistribution F in s is balanced egalitarian-equivalent if for s -a.e. (e, t), F(e, t) 5 f(e, t*, s t * ). (Of course, under assumption (A.6), this amounts to imposing F(e, t)5f(e, t*, s) for s -a.e. (e, t). Yet, the latter condition may contradict feasibility when (A.6) is violated whereas (2) never does). Obviously, a balanced egalitarian-equivalent redistribution F satisfies the EIEE principle. To check the FST property, observe that for s -a.e. (e, t), C(e, t) $ 0⇔ f(e, t*, s) $ f(e, t, s)⇔t* $ t. If b is defined on T by b (t)51 if t#t* and b (t)5 21 otherwise, then, b (t) C(e, t)$0 for s -a.e. (e, t), as required. Conversely, it is a simple matter to modify the proof of Proposition 1 to obtain: Proposition 2. Suppose that the generalized income function f : E 3T 3S →R 1 satisfies assumptions (A.3 ) to (A.6 ), and let s be a regular society. Then a redistribution of income in s satisfies the principles of Equal Income for Equal Effort and Fairly Signed Transfers if and only if it is balanced egalitarian-equivalent.
4. Balanced conditional egalitarianism The symmetric structure of the model laid down in Section 3 suggests a dual approach to balanced egalitarian equivalence. Denoted by s e the society in which effort is concentrated at level e and talent is distributed as in society s. Maintain assumptions (A.3) and (A.4) and replace (A.5) by the following assumption: for each t[T and s[S, f(e, t, s e) is continuous and increasing in e.
(A.7 )
Define the balanced effort level e* in society s by the condition ¯ E f(e*, t, s * )s (t) dt 5 y(s), e
t
T
and replace (A.6) by (A.8 )
for each e[E, t[T and s[S, f(e*, t, s)5f(e*, t, s e * ).
Call a redistribution F in society s balanced conditionally egalitarian if its associated compensation mapping satisfies the condition: ¯ 2 f(e*, t, s e * ). for s -a.e. (e, t), C(e, t) 5 y(s) Such a redistribution satisfies the principle of Equal Transfer for Equal Talent
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(ETET): there exists a measurable mapping g : T →R such that for s -a.e. (e, t), C(e, t)5g (t). Moreover, agents who exert the same effort receive post-transfer incomes that are almost never on opposite sides of the mean income: either almost all of them are richer than average, on almost all are poorer. This I call the principle of Fairly Ranked Incomes (FRI): there is a measurable mapping d : E →h21, 11j such that for s -a.e. (e, ¯ t), d (e) (F(e, t)2y(s))$0. These two principles are characteristic of balanced conditional egalitarianism: Proposition 3. Suppose that the generalized income function f : E 3T 3S →R 1 satisfies assumptions (A.3 ), (A.4 ), (A.7 ) and (A.8 ), and let s be a regular society. Then a redistribution of income in s satisfies the principles of Equal Transfer for Equal Talent and Fairly Ranked Incomes if and only if it is balanced conditionally egalitarian. The proof of Proposition 3 mimics those of Propositions 1 and 2 and is therefore omitted.
5. Discussion
5.1. The dimensional restrictions A weakness of the approach to income redistribution outlined in this paper is the assumption that talent and effort are summarized by real numbers. This assumption could be defended on the ground that the simplest case is clarifying as a benchmark for a more complex analysis. Balanced egalitarian equivalence, however, does not extend in any obvious way to the case when talent is multi-dimensional and balanced conditional egalitarianism is restricted to those environments where effort is one-dimensional. Note that this shortcoming is (essentially) shared by the median versions of egalitarian equivalence and conditional egalitarianism discussed in the introduction, which require a preorder structure on the set of talents or efforts. The average approach, by contrast, is well defined regardless of the structure of those sets. As it turns out, Equal Income for Equal Effort and Fairly Signed Transfers are generally incompatible when talent is multi-dimensional, unless very stringent restrictions are imposed on the income function. A proof of this incompatibility is given in Appendix B. A similar incompatibility can be established between Equal Transfer for Equal Talent and Fairly Ranked Incomes.
5.2. The continuum assumption The continuum assumption is important for the validity of the results in this paper. Consider the finite version of the model described in Section 2. Now, N5h1, . . . , nj is the set of agents, e5(e 1 , . . . , e n )[E N is the vector of their effort levels, t N 5(t 1 , . . . , t n )[T N is the vector of their talents, f(e i , t i ) is agent i’s pre-transfer income Fi (e N , t N ) is his post-transfer income. It is straightforward to redefine our axioms in this framework. Unfortunately, the characterization results no longer hold. EIEE and FST, for instance,
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do not characterize balanced egalitarian equivalence in society (e N , t N ). In fact, these axioms do not place any restriction on the redistribution of income if all effort and talent levels differ. A variant of Proposition 1 can nevertheless be established if the axioms are properly reinforced. Let us first replace EIEE with the following requirement. Higher Income for Higher Effort (HIHE): Let i, j [ N. If e i , e j , then Fi (e N , t N ) # Fj (e N , t N ). In the same spirit, substitute the following axiom for FST. Fairly Signed Transfers 2 (FST 2): Let i, j [ N be such that t i # t j . If Ci (e N , t N ) # 0, then Cj (e N , t N ) # 0; if Cj (e N , t N ) $ 0, then Ci (e N , t N ) $ 0. It turns out that every redistribution of income satisfying HIHE and FST2 must be ‘‘close’’ to the balanced egalitarian equivalent redistribution if the income function f is ‘‘well-behaved’’ and society (e N , t N ) is ‘‘sufficiently diverse’’. A precise statement and a proof of this fact are given in Appendix C. An analogous result can be proved for balanced conditional egalitarianism.
5.3. Incentives and efficiency As in Bossert (1995) and Bossert and Fleurbaey (1994), I have treated effort levels as exogenously given. This is a serious shortcoming: indeed, an agent’s optimal choice of effort clearly depends on the redistribution rule itself. Roemer (1994b, chap. 8) addresses this problem by postulating an ‘‘effort response function’’ which specifies, for each talent, how the level of effort is affected by the; redistribution rule. This response function may (and presumably does) result from a preference-maximization process. Endowing agents with preferences, if useful to discuss incentives issues, is absolutely necessary to address efficiency concerns. Suppose that an agent is characterized by a preference u over effort–income pairs, and a talent level. To fix ideas, suppose that u satisfies the usual assumptions and is also separable and linear in income. Call a density function r on preference–talent pairs an economy. How can balanced egalitarianism be generalized to this richer setting? The answer depends on the technological environment. Consider the example discussed in Section 3. If the aggregate production function w is linear, decentralized preference maximization is unambiguous. It yields for every economy r a society, i.e., a density s over effort–talent pairs. Moreover, since effort is efficiently distributed in that society, it is natural to operate balanced egalitarianism on s to redistribute the corresponding income. If w is not linear, decentralized preference maximization may be interpreted in different ways. If we assume Nash behavior, the equilibrium density function of effort will generally be inefficient. In that case, any income redistribution would perpetuate that inefficiency. It seems to me that this does not destroy the relevance of balanced
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egalitarianism: even if effort is inefficiently distributed, it is of interest to fairly redistribute income. But balanced egalitarianism is compatible with efficiency. To each preference–talent pair corresponds an efficient effort level. Using this correspondence, it is straightforward to transform an economy r into a society s* whose marginal s ´* is efficient. Balanced egalitarianism may now be applied to s*.
5.4. An alternative characterization There is an alternative characterization of balanced egalitarian equivalence which uses a pair of axioms different from those in Proposition 1 and does not require the continuum assumption. Consider again the finite model introduced in Section 5.2. No restriction is placed on society (e N , t N ); in particular, perfectly homogeneous societies where all talents coincide are admissible. The first axiom is adapted from Bossert (1995) and Bossert and Fleurbaey (1994). It is a much weakened version of ETET and is actually implied by FST: No Transfer for Uniform Talent: For every e N [ E N and t [ T, Fi (e N , (t, . . . ,t)) 5 f(e i , t) for all i [ N. The second axiom requires that all post-transfer incomes be similarly affected by changes in talents: Solidarity in Talents: For every e N [ E N and t N ,t N9 [ T N , either Fi (e N , t N ) # Fi (e n , t 9N ) for all i [ N or Fi (e N , t N ) $ Fi (e n , t 9N ) for all i [ N. Under Equal Treatment of Equals (Fi (e N , t N )5Fj (e N , t N ) if (e i , t i )5(e j , t j )), Solidarity in Talents implies EIEE. Solidarity axioms have been used in Bossert (1995); Bossert and Fleurbaey (1994) and Fleurbaey and Maniquet (1994b). In the first two papers, the authors require that everybody’s change in income be equal when an agent’s ‘‘irrelevant characteristics’’ (talent, in our terminology) are modified. This ‘‘cardinal’’ axiom is stronger than the ordinal requirement I suggest. It is not satisfied by balanced egalitarian equivalence. Fleurbaey and Maniquet’s model incorporates preferences. In that richer setting, their solidarity axiom is the natural counterpart of mine: it requires that all agents experience welfare changes of equal sign when talents change. It is almost obvious that balanced egalitarian equivalence is characterized by No Transfer for Uniform Talent and Solidarity in Talents. Suppose that there exists a society (e N , t N ) where F(e N , t N ) is not balanced egalitarian equivalent. There must then exist two agents, say, 1 and 2, such that F1 (e N , t N ),f(e 1 , t*) and F2 (e N , t N ).f(e 2 , t*), where t* is the balanced talent level. By Solidarity in Talents, however, F(e N , t N )5F(e N , (t*, . . . , t*)). It follows that F1 (e N , (t*, . . . , t*)),f(e 1 , t*) and F2 (e N , (t*, . . . , t*)).f(e 2 , t*), contravening No Transfer for Uniform Talent. The case for Solidarity in Talents is that if talents are really outside the agents’
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control, they should be considered a commonly owned resource. This argument strikes me as perfectly valid, but it certainly goes all the way to the most radical consequences of our assumption about the scope of the agents’ control. The characterization just presented thus combines a very mild axiom with a very controversial one. By contrast, Proposition 1 rests on two middle-of-the-road axioms. It should also be noted that Solidarity in Talents involves comparisons across societies, while both axioms in Proposition 1 are intrasociety properties.
5.5. An alternative approach If one insists on Equal Income for Equal Effort and sets out to weaken Equal Transfer for Equal Talent to avoid the basic incompatibility between these axioms, two routes can be taken. Either one fully equalizes for equally talented agents something as close as possible to the transfers they receive, or one approximately equalizes their actual transfers. I have chosen the first route in Proposition 1. The second route, though attractive, leads to the difficult problem of measuring the ‘‘ETET gap’’, i.e., the extent to which an income redistribution falls short of fully equalizing the transfers to equally talented agents. ¯ denote the mean transfer to the agents endowed One possibility is as follows. Let C(t) ¯ 5eE C(e, t)s´ (e) de. Since one wishes to (but cannot) make sure with talent t, i.e., C(t) ¯ for almost every (e, t), it makes sense to minimize that C(e, t)5C(t) ¯ EE (C(e, t) 2 C(t)) s(e, t) de dt. 2
T E
¯ But why not rather minimize eT eE uC(e, t)2C(t)u s(e, t) de dt? Or maybe sup t[T eE (C(e, 2 ¯ ¯ rather than the ratios t)2C(t)) s´ (e) de? And why focus on the differences C(e, t)2C(t) ¯ ¯ ¯ C(e, t) /C(t)? Or perhaps the relative differences (C(e, t)2C(t)) /C(t)? Identifying simple axioms on the redistribution F or its associated compensation mapping C that could recommend a particular measure of the ETET gap seems to be exceedingly difficult.
Acknowledgments I thank W. Bossert, M. Fleurbaey, B. MacLeod, F. Maniquel, H. Moulin, J. Roemer, an associate editor, and two referees for useful comments on an earlier draft. Financial ´ support from the FCAR of Quebec and the SSHRC of Canada is gratefully acknowledged.
Appendix A Here is the formal proof of the ‘‘only if’’ part of Proposition 1. Fix a regular society s and a redistribution F satisfying EIEE and FST. By EIEE, there exists a measurable
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mapping a : ER 1 such that F(e, t)5 a (e) for s -a.e. (e, t). Extend the income function f 2 to R 1 in such a way that f(e, ?) is continuous, increasing and unbounded for each e[E and f(?, t) is continuous for each t[R 1 . For each e[E there is a unique number t 1 (e) such that f(e, t 1 (e))5 a (e). The mapping t 1 : E →R 1 is such that for s -a.e. (e, t), F(e, t) 5 f(e, t 1 (e)).
(3)
This mapping is measurable. (To check this assertion, it suffices to verify that the set E(t 0 ):5he[Eut 1 (e),t 0 j is measurable for each t 0 [R 1 . But for every e[E, t 1 (e), t 0 ⇔ f(e, t 1 (e)),f(e, t 0 )⇔a (e),f(e, t 0 ). Therefore E(t 0 )5he[Eua (e)2f(e, t 0 ),0j, which is clearly measurable for each t 0 since a is measurable and f(?, t 0 ) is continuous, hence measurable as well). Define A2 5 (e, t) [ ExT uF(e, t) , f(e, t*), E 2 5 e [ Eut 1 (e) , t*, and define A1 and E 1 by reversing the inequality signs in the above definitions. All these sets are measurable because F, f, and t 1 are measurable mappings. By (3) and because f is increasing in t, A2 and A1 are almost equal to E 2 3T and E 1 3T respectively. Formally, if D denotes the symmetric difference operator, s (A2 D(E 2 3 T ))5 s (A1 D(E 1 3T ))50. We must prove that s (A2)5 s (A1 )50. Since s is regular, this is equivalent to
l´ (E 2) 5 l´ (E 1 ) 5 0,
(4)
where l´ is Lebesgue’s measure on the Borel s -algebra of E. Suppose, by way of contradiction, that
l´ (E 2) 1 l´ (E 1 ) . 0.
(5)
By the condition defining t* and the feasibility constraint on F,
EE F(e, t)s(e, t) de dt 5E f(e, t*)s (e) de ´
E T
E
E f(e,t (e))s (e) de 5E f(e, t*)s (e) de
⇒
1
´
⇒
(6)
´
E
E
E [ f(e, t*) 2 f(e, t (e))]s (e) de 5 E [ f(e, t (e)) 2 f(e, t*)]s (e) de.
E2
1
´
1
´
E1
From (5) and (6) follows that l´ (E 2).0 and l´ (E 1 ).0. This conclusion can be slightly 1 strengthened. For n51, 2, . . . , define E 2 n 5he[Eut 1 (e),t*21 /nj and define E n 2 2 similarly. There exists a positive integer N such that l´ (E N 2 ).0 since otherwise ` 1 l´ (E 2)2 l´ (< n51 E n2 )#o n251 l´ (E 2 n )#0. Likewise, there exists a positive integer N 2 1 such that l´ (E 1 N 1 ).0. Letting N5max [N , N j,
Y. Sprumont / Mathematical Social Sciences 33 (1997) 185 – 201 1 l´ (E 2 N ) . 0 and l´ (E N ) . 0.
199
(7)
Define T N 5[t*21 /N, t*11 /N]. Since s is regular, (7) implies
s (E N2 3 T N ) . 0 and s (E N1 3 T N ) . 0. 2
2 N
(8) 1
1 N
Now, for s -a.e. (e , t)[E 3T N and s -a.e. (e , t)[E 3T N ,
S D 1 F(e , t) 2 f(e , t (e )) . fSe , t* 1 ]D $ f(e , t). N 1 F(e 2 , t) 2 f(e 2 , t 1 (e 2)) , f e 2 , t* 2 ] # f(e 2 , t), N 1
1
1
1
1
1
1 2 1 This means that there exist full s´ -measure subset; of E 2 N and E N , say E N and E N themselves, and a full st -measure subset of T N , say T N itself, such that
C(e 2 , t) , 0 , C(e 1 , t) 1 1 for all e 2 [E 2 N , e [E N , and t[T N . It is easily seen that this contradicts the FST principle. Indeed, let b be any measurable function from T into h21, 11j. For all t[T N , one of the following statements holds:
b (t)C(e 2 , t) , b (t)C(e 1 , t) for all e [ E N2 and e 1 [ E N1 b (t)C(e 1 , t) , b (t)C(e 2 , t) for all e [ E N2 and e 1 [ E N1 Let B 2 5h(e, t)[E N2 3T N u b (t) C(e 2 , t),0j and define B 1 similarly. The measurability of F and f ensures that these sets are measurable. I claim that s (B 2
Appendix B Throughout this Appendix, I consider a variant of the continuous model of Section 2 in which talent is multidimensional. It will be convenient to keep our earlier notation even though this constitutes some abuse; in particular, T now stands for (0, 1)m , where m.1, and t is a m-dimensional vector. Assumptions (A.1) and (A.2) and the regularity condition are adapted accordingly. I claim that for all regular societies there exist income functions such that no redistribution of income satisfies EIEE and FST. Here is a sketch of the argument. The first two steps follow the proof of Proposition 1; Step 3 is the crucial point. Step 1. Fix a regular society s and a redistribution F satisfying EIEE and FST. By the former axiom, we can define a correspondence T : E →R m1 such that
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F(e, t) 5 f(e, t 1 (e)) for s -a.e. (e, t),
(9)
where f(e, t 1 (e))5y means that f(e, t)5y for all t[T 1 (e). Step 2. Let t* be any talent vector such that ¯ E f(e, t*)s (e) de 5 y(s). ´
(10)
E
Along every interior ray of R m1 , there is a (unique) t* satisfying this condition. Contrary to the one-dimensional case, there is thus an infinite set of ‘‘balanced talent vectors’’; we call it T *. For each t* in T *, define A- (t*)5h(e, t)[E 3T uF(e, t),f(e, t*)j and define A1 (t*) by reversing the strict inequality in the above definition. Essentially the same argument as in the proof of Proposition 1 shows that
s (A2 (t*)) 5 s (A1 (t*)) 5 0.
(11)
(One only needs to replace E 2 by E 2 (t*):5heu f(e, T 1 (e)),f(e, t*)j and E 2 by n 1 1 E2 (t*):5heu f(e, T (e)),f(e, (121\n)t*)j and define E and E (t*) similarly. The rest n 1 n n of the argument is virtually unchanged.) Step 3. Since (11) must hold for every t* in T *, it follows that F(e, t)5f(e, t*) for s -a.e. (e, t) and all t* in T *. Hence, for s´ -a.e. e, there is a number k(e) such that f(e;t*) 5 k(e) for all t* [ T *.
(12)
But this imposes severe separability restrictions on f. For instance, if m52 and f(e, t)5f(e, (t 1 , t 2 ))5et 1 1t 2 for all e, t, (12) demands that for almost every e there be a number k(e) such that et *1 1t 2 5k(e) for each (t *1 , t *2 ) in T *. Since T * has a unique 2 intersection with every interior ray in r 1 , it must be a line segment of slope 2e: this, however, cannot hold true for almost all e simultaneously.
Appendix C Suppose that the income function f is well-behaved in the following sense: it is differentiable and there exists a positive real number c such that (≠f / ≠e) (e,t),c(≠f / ≠t) (e,t) for every (e,t). Assume c51: this assumption entails no loss of generality since it can always be met by changing the units in which effort and talent are measured. Suppose next that society (e N , t N ) is sufficiently diverse in the following sense: there is a (small) number d, 0,d ,1, such that if(d, d ) , (e, t) , (1 2 d, 1 2 d ), there exist agents i, j in N such that (e 2 d, t 2 d ) , (e i , t i ) , (e, t) , (e j , t j ) , (e 1 d, t 1 d ). Under these assumptions, I claim that every redistribution satisfying the axioms HIHE and FST2 (stated in Section 5.2) must be close to the balanced egalitarian equivalent
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redistribution. Formally: if F satisfies HIHE and there exist two agents, say, 1 and 2, with d ,e 1 , e 2 ,12d, such that F1 (e N ,t N ) , f(e 1 ,t* 2 2d ) and F2 (e N ,t N ) . f(e 2 ,t* 1 2d ), then F must violate FST2. To see why this is true, let 1 and 2 be two agents such that d ,e 1 , e 2 ,12d and F1 (e N , t N )5f(e 1 , t*2d1 ), F2 (e N , t N ) 5 f(e 2 , t* 1 d2 ) with d1 , d2 .2d. This implies 2d ,d1 ,t*,12d2 ,122d. By the diversity assumption, there exists an agent i [N for whom e 1 2 d , e i , e 1 and t* 2 d , t i , t*. Since f is well-behaved, f(e i , t i ) . f(e 1 , t* 2 d1 ). Invoking HIHE yields Fi (e N , t N ),F1 (e N , t N ), hence, Fi (e N , t N ),f(e 1 , t*2d1 ),f(e i , t i ), meaning that agent i is taxed.On the other hand, there exists an agent j [N for whom e 2 , e j , e 2 1 d and t* , t j , t* 1 d. Therefore f(e j , t j ),f(e 2 , t*1d2 ) and HIHE yields f(e j , t j ),Fj (e N , t N ). Thus, agent j is subsidized, a contradiction to FST2.
References R.J. Arneson, Equality of Opportunity for Welfare, Phil. Studies 56 (1989) 77–93. W. Bossert, Redistribution Mechanisms based on Individual Characteristics, Math. Social Sci. 29 (1995) 1–17. W. Bossert and M. Fleurbaey, Redistribution and Compensation, mimeo (1994). G.A. Cohen, On the Currency of Egalitarian Justice, Ethics 99 (1989) 906–944. M. Fleurbaey, On fair Compensation, Theory Decision 36 (1994a) 277–307. M. Fleurbaey, The Requisites of Equal Opportunity, in: W.A. Barnett et al., eds., Advances in Social Choice Theory and Cooperative Games (Cambridge University Press, Cambridge: forthcoming, 1994b). M. Fleurbaey, Three Solutions for the Compensation Problem, J. Econ. Theory 65 (1995) 505–521. M. Fleurbaey and F. Maniquet, Fair Allocation with Unequal Production Skills: The No-Envy Approach to Compensation, INSEE Working Paper 9433 (1994a). M. Fleurbaey and F. Maniquet, Fair Allocation with Unequal Production Skills: The Solidarity Approach to Compensation, THEMA Working Paper 9419 (1994b). J. Mirrlees, Notes on Welfare Economics, Information and Uncertainty, in: C.C. Balch et al., eds., Essays on Economic Behavior under Uncertainty, (North-Holland, Amsterdam, 1974). H. Moulin, Joint Ownership of a Convex Technology: Comparison of three Solutions, Rev. Econ. Studies 57 (1990) 439–452. ´ ´ H. Moulin, La presence d’envie: comment s’en accomoder?, Recherches Economiques de Louvain 60 (1994) 63–72. J.E. Roemer, A Pragmatic Theory of Responsibility for the Egalitarian Planner, Phil. Public Affairs 22 (1993) 146–166. J.E. Roemer, Egalitarian Perspectives (Cambridge University Press, Cambridge, 1994a). J.E. Roemer, Theories of Distributive Justice (Harvard University Press, forthcoming, 1994b). J.E. Roemer and J. Silvestre, Public Ownership: Three Proposals for Resource Allocation, mimeo, University of California at Davis, 1987).
Balanced egalitarian redistribution of income Yves Sprumont* ´ ´ ´ Centre de Recherche et developpement en economique ( C.R.D.E.) and Departement de Sciences ´ ´ , Quebec ´ , Canada H3 C Economiques , Universite´ de Montreal, C.P. 6128, succursale Centre-ville, Montreal 3 J7 Received 1 March 1995; revised 1 October 1996; accepted 1 November 1996
Abstract This paper reconsiders the issue of how income should be redistributed when people endowed with different levels of talent exert different levels of effort. Two new schemes, called balanced egalitarian equivalence and balanced conditional egalitarianism, are proposed and characterized. 1997 Elsevier Science B.V. Keywords: Redistribution; Egalitarianism
1. Introduction How should income be redistributed among people endowed with different skills, handicaps or needs? The ideal of equal opportunity, advocated in a broader context by Arneson (1989) and Cohen (1989), is based on the postulate that some determinants of an individual’s income are within his control while others are not. Throughout the paper I will use the term effort to refer to the former category and talent to refer to the latter. I assume that both are real variables; this tremendous simplification does not destroy the interest of the analysis. Equalizing opportunities, it may be argued, requires that income be redistributed so as to offset inequalities that are due to differences in talent while preserving those that arise from differences in effort. As it turns out, there is a tension between these two desiderata. In fact, it is generally impossible to guarantee an equal income to individuals exerting; the same effort, while performing identical income transfers to those: who are equally talented. This remarkable impossibility was first pointed out (in a more general framework) by Fleurbaey (Fleurbaey, 1994a,b) and later reinforced by Bossert (1995). *Correspondence address. Tilburg, University CentER, P.O. Box 90153, 5000 Tilburg, LE Netherlands. 0165-4896 / 97 / $17.00 1997 Published by Elsevier Science B.V. All rights reserved PII S0165-4896( 96 )00835-9
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Weakening the defining requirements of equal opportunity opens the way to a variety of interesting redistribution schemes. Egalitarian equivalence and conditional egalitarianism are among the most appealing ideas. Egalitarian equivalence insists that the principle of equal income for equal effort be preserved. The simplest egalitarianequivalent schemes work as follows. First, a reference level of talent is fixed. Then, each individual receives a post-transfer income equal to the pre-transfer income he would earn were he endowed with the reference talent level, plus a uniform amount dictated by the budget constraint. These schemes were introduced in a slightly different setting and given an axiomatic characterization by Fleurbaey (1995). An alternative axiomatization was proposed by Bossert and Fleurbaey (1994). The main difficulty lies in the arbitrary nature of the reference level of talent. In an effort to overcome this difficulty, Moulin (1994) suggested the average egalitarian-equivalent scheme: successively choose each individual’s level of talent as the reference level compute the corresponding egalitarianequivalent post-transfer income distributions, and take the average. That method was recently axiomatized by Bossert and Fleurbaey (1994). Yet another solution, mentioned in Moulin (1994), is to choose the median individual talent as the reference. Conditional egalitarianism is essentially dual to egalitarian equivalence. Here the principle of equal transfer for equal talent is given priority. The simplest conditionally egalitarian schemes are based on a fixed reference level of effort. The first component of every individual’s post-transfer income is the average income that would be generated if all individuals exerted the reference level of effort. To that component is added the income increment created by the individual’s deviation from the reference level. Median conditional egalitarianism selects the median value of the individual efforts as the reference level. Average conditional egalitarianism averages the conditionally egalitarian post-transfer income distributions obtained by successively choosing each individual’s effort as the reference level. Conditionally egalitarian schemes and their average version were studied by Fleurbaey (1995) and Bossert and Fleurbaey (1994). Roemer’s (Roemer, 1993) approach, though somewhat different, also makes use of reference levels of effort. This paper proposes and defends two new income redistribution schemes to which I will refer by the phrase balanced egalitarianism. They are motivated by weaknesses of the average and median egalitarian-equivalent and conditionally egalitarian approaches that I shall now describe. Consider first the median and average egalitarian-equivalent schemes. By the aforementioned impossibility result of Fleurbaey, the transfers received by two equally talented individuals under either of these two schemes will generally differ. What is more disturbing, however, is that one individual may be taxed while the other is subsidized. As it turns out, the gap between the tax and the subsidy is unbounded. This is a rather extreme violation of the principle of equal transfer for equal talent. I will suggest a simple scheme, called balanced egalitarian-equivalent, that guarantees that equally talented individuals never receive transfers of opposite signs, and delivers equal income for equal effort. I will show that these properties are characteristic of the balanced egalitarian-equivalent scheme in the limit case where all configurations of effort and talents are present in society. Average, or median, conditional egalitarianism suffers from a weakness that is
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technically dual to that of average or median egalitarian equivalence, though maybe conceptually less serious: individuals exerting the same effort may enjoy post-transfer incomes that are on opposite sides of society’s average income. The balanced conditionally egalitarian scheme defined in this paper avoids that difficulty. Since the two schemes that I propose are dual, and because I find the property of equal income for equal effort more fundamental than equal transfer for equal talent, I shall analyze balanced egalitarian equivalence in some detail but content myself with a brief description of balanced conditional egalitarianism. The beginning of Section 2, which is a bit informal, serves as a motivation for introducing balanced egalitarian equivalence. I recall the definition of average egalitarian equivalence in (the onedimensional version of) Bossert’s (Bossert, 1995) model and offer an example where two equally talented individuals receive transfers of opposite signs. A similar example could be constructed for the median egalitarian-equivalent scheme. Next, I set up the basic model used in the paper, which is a continuous version of Bossert’s. I define and characterize balanced egalitarian equivalence. (The formal proof is in Appendix A.) Resorting to a continuum of individuals is necessary for the characterization result but balanced egalitarian equivalence is, of course, well defined in the finite case as well. A shortcoming of Bossert’s model is that an individual’s income does not depend on the talent and effort of other members of society. This assumption is relaxed in Section 3. The definition of balanced egalitarian equivalence is generalized to a richer setting and conditions under which its characterization remains valid are discussed. Section 4 defines and characterizes balanced conditional egalitarianism. Finally, Section 5, Appendix B and Appendix C discuss the limitations of the paper, some possible extensions, another characterization of balanced egalitarian equivalence, and an alternative approach to the redistribution problem.
2. A basic formulation of balanced egalitarian equivalence Consider a society composed of n agents. Agent i’s effort is e i , his talent is t i , and his income before redistribution is f(e i , t i ). The function f is increasing in both arguments. Let us fix a reference talent t 0 . Under the t 0 -egalitarian-equivalent redistribution scheme, agent i’s post-transfer income is
O
1 n x ti 0 5 f(e i , t 0 ) 2 ] [ f(e k , t 0 ) 2 f(e k , t k )]. n k 51 Under average egalitarian equivalence, agent i receives the post-transfer income
O
1 n tj xi 5 ] x n j 51 i Simple computations show that the corresponding income transfer, or compensation, takes the form
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FO
O
1 n 1 n c i : 5 x i 2 f(e i , t i ) 2 ] f(e j , t j ) 2 ]2 f(e k , t j ) n j 51 N j, k51
FO n
G
G
1 1 ] f(e i , t j ) 2 f(e i , t i ) . n j 51 Suppose now that n53, e 1 .e 2 .e 3 , and t 1 5t 2 5:t,t 3 . Thus, agents 1 and 2 are equally talented. Assume that f(e 1 , t) 5 5, f(e 1 , t 3 ) 5 10, f(e 2 , t) 5 2, f(e 2 , t 3 ) 5 3, f(e 3 , t) 5 1, f(e 3 , t 3 ) 5 2. Then c 1 511 / 9.0. 21 / 95c 2 : agent 1 is subsidized whereas agent 2 is taxed. In fact, if f(e 1 , t 3 )551a, where a.2, then c 1 5(112a) / 9 and c 2 5(42a) / 9: agent 1’s subsidy and agent 2’s tax are unbounded. There is a natural solution to the above problem. Choose the talent t* such that o nk 51 f(e k , t*)5o nk 51 f(e k , t k ). Agent i’s post-transfer income under the t*-egalitarianequivalent scheme is merely f(e i , t*). No uniform amount is needed to balance the budget: each agent simply gets the pre-transfer income he would earn if he were endowed with the talent t*. The transfer to agent i being f(e i , t*)2f(e i , t), it is obvious that equally talented agents cannot receive transfers of opposite signs. Let us call the scheme just described balanced egalitarian equivalent. It turns out that when all combinations of effort and talent are present in society, balanced egalitarian equivalence is the only way to guarantee transfers of equal sign to equally talented agents and equal income for equal effort. The rest of this section is devoted to a formal statement and proof of this assertion. From now on and until Section 5, I postulate a continuum of agents. Formally, the agent space is the probability space (I, @, ,), where I5[0, 1], @ is the Borel s -algebra of I, and , is Lebesgue’s measure on @. Effort ´ and talent t are random variables on the agent space taking values in E5T5(0, 1). Society is a joint probability density function s of effort and talent. The corresponding distribution function is denoted by s. The marginal density functions of effort and talent are s´ and st , with associated ¯ distribution functions s´ and st . Mean effort and mean talent in society s are e(s)5e E es´ (e) de and ¯t(s)5eT tst (t) dt. Society s is called regular if for every measurable subset A of E 3T, s (A).0 if and only if l(A).0, where l is Lebesgue’s measure on the Borel s -algebra of E 3T. Loosely speaking, this means that all effort–talent combinations are found in society. The (pre-transfer) income of an agent who exerts the effort level e and is endowed with the talent level t in society s is denoted f(e, t). Note that this quantity does not vary with s: in this model, an agent’s income is entirely determined by his own characteristics. Think of a society of farmers who all cultivate corn on their own identical parcel of land. This assumption that all agents are completely independent of each other will be
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relaxed in the next section. I suppose that the income function f : E 3T →R 1 satisfies the following conditions: (A.12 )
for each t[T, f(e, t) is continuous and increasing in e, and lim e →0 f(e, t)50;
(A.2)
for each e[E, f(e, t) is continuous and increasing in t, and lim t→0 f(e, t)50.
¯ The mean income in society s is y(s)5e E eT f(e, t) s(e, t) de dt. In what follows, every uncountable subset of an Euclidean space is endowed with the induced Euclidean topology and every finite set is endowed with the discrete topology. The measurability of a mapping between two topological spaces is always understood with respect to the Borel s -algebras of these spaces. A redistribution (of income in society s) is a measurable mapping F: E 3T →R 1 satisfying the feasibility condition eE ¯ eT F(e, t) s(e, t) de dt5y(s). The number F(e, t) is the post-transfer income of any agent exerting effort e and endowed with talent t in society s.1 The compensation mapping C associated with F is defined by C(e, t)5F(e, t)2f(e, t) for all (e, t)[E 3T. In order to define balanced egalitarian equivalence, consider a society where the distribution of effort is the same as in society s but where all agents are endowed with a same level of talent. The level ensuring that the mean income in this society is the same as in s is called the balanced talent level (in society s) and denoted by t*. Formally, t* satisfies the condition ¯ Ef(e, t*)s (e) de 5 y(s). ´
E
Under assumptions (A.1) and (A.2), t* exists, is unique, and belongs to T. A redistribution F in society s is balanced egalitarian-equivalent if for s -a.e. (e, t) [ ExT, F(e, t) 5 f(e, t*). Balanced egalitarian-equivalent redistributions are unique up to differences on sets of zero s -measure. As already mentioned, they guarantee transfers of equal sign to equally talented agents and equal income for equal effort. Formally, a redistribution F satisfies the principle of Equal Income for Equal Effort (EIEE) if there exists a measurable mapping a : E →R 1 such that for s -a.e. (e, t), F(e, t)5 a (e). It satisfies the principle of Fairly Signed Transfers (FST) if there exists a measurable mapping b : T →h21, 11j such that for s -a.e. (e, t), b (t) C(e, t)$0. Proposition 1. Suppose that the income function f : E 3T →R 1 satisfies assumptions (A.1 ) and (A.2 ) and let s be a regular society. Then a redistribution of income in society s satisfies the principles of Equal Income for Equal Effort and Fairly Signed Transfers if and only if it is balanced egalitarian-equivalent. 1
Of course, there need not exist such agents. This is irrelevant, however, since the behavior of F on sets of zero s -measure is unconstrained by the feasibility condition.
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Fig. 1. Illustration of the proof of Proposition 1.
The ‘‘if’’ part of Proposition 1 is obvious. The principle of the proof of the ‘‘only if’’ statement is also very simple. Consider Fig. 1. By EIEE, (almost) all agents on a same vertical line receive the same post-transfer income. If the redistribution is not balanced egalitarian-equivalent, I may assume that all agents on the e 1 vertical line get more than f(e 1 , t*) and, by feasibility, all agents on the e 2 line get less than f(e 2 , t*). But this means that (e 1 , t*) agents are subsidized while (e 2 , t*) agents are taxed. The measure-theoretic details of the proof are worked out in Appendix A. The regularity assumption in Proposition 1 is restrictive but can be slightly weakened. One might think that it could be replaced with the assumption that the support of s, hereafter denoted A s , is a convex set. The following example shows that this is not true. Let s be the uniform probability density function over the convex hull of the four points (1 / 8, 15 / 16), (7 / 8, 3 / 16), (7 / 8, 1 / 16), (1 / 8, 13 / 16). Suppose that f(e, t)5e1t for all ¯ e, t. Check that the mean income in this society is y(s)51 and the balanced talent level is t*51 / 2. Define the redistribution F by e/3 1 5/8 F(e, t) 5 e 1 1 / 2 e / 3 1 25 / 24
5
if e # 3 / 16, if 3 / 16 , e , 13 / 16, if 13 / 16 # e,
which clearly satisfies the feasibility condition but differs from the balanced egalitarian equivalent redistribution. Obviously, F satisfies EIEE and it is easily checked that it also meets FST. There is nothing pathological in this example; in particular, A s has positive Lebesgue measure and F is continuous and strictly increasing in e. The difficulty arises from the
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fact that all the agents who exert a low effort (e#3 / 16) have talent well above the balanced level while all those exerting a high effort (e$13 / 16) have talent well below that level. One set of conditions that can be substituted for the regularity assumption without affecting the validity of Proposition 1—and hardly its proof—consists of the following two restrictions: i) for every measurable set B , A s , s (B).0 if and only if l(B).0, and ii) there is a neighborhood T * around t* such that T *3E9, A s for every subset E9 of E having positive s´ -measure. Observe that these restrictions do not imply the convexity of A s ; yet, they remain demanding. While Bossert (1995), and Bossert and Fleurbaey (1994) do not need any restriction of this sort, their results hold for redistribution rules defined over sufficiently rich domains of societies and rely on axioms making comparisons across societies. To reach a characterization result in a fixed-society framework, enough richness must be assumed on society’s support.
3. Balanced egalitarian equivalence in more complex environments This section allows each agent’s pre-transfer income to depend not only on his own characteristics, but also on the characteristics of other members of society. The agent space, effort, talent and society are defined as in the previous section. The (pre-transfer) income of an agent who exerts effort e and is endowed with talent t in society s is now written f(e, t, s). Letting S stand for the set of all societies, a ( generalized) income function f is defined on E 3T 3S. Assumptions (A.1) and (A.2) are reexpressed as follows: (A.3 )
for each t[T and s[S, f(e, t, s) is continuous and increasing in e, and lim e →0 f(e, t, s)50;
(A.4)
for each e[E and s[S, f(e, t, s) is continuous and increasing in t, and lim t →0 f(e, t, s)50.
These are restrictions on the pre-transfer income function within any society. In addition, I shall impose two restrictions on how income varies euchres societies. Consider a society in which the distribution of talent is concentrated at a single level. Provided that the distribution of effort remains unchanged, a small increase of the uniform talent level is likely to induce a small increase in income at every level of effort. Formally, let s t denote the society where talent is concentrated at level t and effort is distributed as in s: s t (e, t9) 5
H
0 s´ (e)
for all e [ E and t9 [ T • ht j, for all e [ E and t9 5 t.
The assumption is: (A.5 )
for each e[E and s[S, f(e, t, s t ) is continuous and increasing in t.
This requirement is fairly weak because it bears only on very comparable societies. By
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contrast, the last assumption involves societies that may be quite different from each other. Recall that the balanced talent level t* in society s was defined in Section 2 by the condition that shifting everyone’s talent to that level does not affect mean income. Since society itself is now an argument of the income function, the proper formulation of that condition becomes ¯ E f(e, t*, s * )s (e) de 5 y(s), t
´
E
¯ where y(s):5e E eT f(e, t, s) s(e, t) de dt is the mean income in society s. Assumptions (A.3) to (A.5) ensure that t* exists, is unique and belongs to T. I now assume: (A.6 )
for each e[E, t[T and s[S, f(e, t*, s)5f(e, t*, s t * ).
This is a limited neutrality property. It says that bringing all talents to the balanced level does not affect the agents who are endowed with precisely the balanced level of talent. Of course, assumptions (A.3) to (A.6) are satisfied if f is constant in s and f(?,?, s) satisfies (A.1) and (A.2) for any s: the present model encompasses the no-interaction model of Section 2. Assumptions (A.3) to (A.6), however, allow for interesting patterns of influence among the agents, as the following example illustrates. Example. Suppose that each agent combines effort and talent to supply some standardized input through the mapping x, say, x(e, t)5et. Suppose that the mean input ¯ in society s, x(s), determines the mean income through some aggregate production function w : (0,1)→R 1 which is continuous, increasing, and such that lim x→0 w (x)50. Finally, suppose that each agent’s income is proportional to his input contribution: ¯ w (x(s)) ¯ f(e, t, s) 5 (x(e, t) /x(s)) for all e, t, s.
(1)
This technological environment is considered in Mirrlees (1974); Roemer and Silvestre (1987); Moulin (1990); Fleurbaey and Maniquet (1994a, (1994b); Roemer (1994a). The traditional story is that of a group of fishers catching fish in a lake. The proportionality constraint (1) simply means that each fisher eats his own catch. In that context, it is natural to assume that w (x) /x is decreasing in x: individual income then decreases with society’s mean input. I shall not make that assumption here, thereby allowing for a greater variety of possible interactions among agents. It is obvious that assumptions (A.3) and (A.4) are satisfied. Next, since ¯ w (e(s)t) ¯ f(e, t, s t ) 5 (e /e(s)) for all e, t, s, assumption (A.5) is met as well. Finally, since the balanced level of talent in society s ¯ /e(s), ¯ t* 5 x(s) ¯ ¯ ¯ it follows that f(e, t*, s)5(x(e, t*) /x(s)) w (x(s))5(e /e(s)) w (e(s) t*)5f(e, t*, s t * ), in agreement with assumption (A.6). Returning to the general model, define a redistribution F in society s as in the
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previous section. The compensation mapping associated with F is C5F 2f(?,?, s) and the EIEE and FST principles are unchanged. A redistribution F in s is balanced egalitarian-equivalent if for s -a.e. (e, t), F(e, t) 5 f(e, t*, s t * ). (Of course, under assumption (A.6), this amounts to imposing F(e, t)5f(e, t*, s) for s -a.e. (e, t). Yet, the latter condition may contradict feasibility when (A.6) is violated whereas (2) never does). Obviously, a balanced egalitarian-equivalent redistribution F satisfies the EIEE principle. To check the FST property, observe that for s -a.e. (e, t), C(e, t) $ 0⇔ f(e, t*, s) $ f(e, t, s)⇔t* $ t. If b is defined on T by b (t)51 if t#t* and b (t)5 21 otherwise, then, b (t) C(e, t)$0 for s -a.e. (e, t), as required. Conversely, it is a simple matter to modify the proof of Proposition 1 to obtain: Proposition 2. Suppose that the generalized income function f : E 3T 3S →R 1 satisfies assumptions (A.3 ) to (A.6 ), and let s be a regular society. Then a redistribution of income in s satisfies the principles of Equal Income for Equal Effort and Fairly Signed Transfers if and only if it is balanced egalitarian-equivalent.
4. Balanced conditional egalitarianism The symmetric structure of the model laid down in Section 3 suggests a dual approach to balanced egalitarian equivalence. Denoted by s e the society in which effort is concentrated at level e and talent is distributed as in society s. Maintain assumptions (A.3) and (A.4) and replace (A.5) by the following assumption: for each t[T and s[S, f(e, t, s e) is continuous and increasing in e.
(A.7 )
Define the balanced effort level e* in society s by the condition ¯ E f(e*, t, s * )s (t) dt 5 y(s), e
t
T
and replace (A.6) by (A.8 )
for each e[E, t[T and s[S, f(e*, t, s)5f(e*, t, s e * ).
Call a redistribution F in society s balanced conditionally egalitarian if its associated compensation mapping satisfies the condition: ¯ 2 f(e*, t, s e * ). for s -a.e. (e, t), C(e, t) 5 y(s) Such a redistribution satisfies the principle of Equal Transfer for Equal Talent
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(ETET): there exists a measurable mapping g : T →R such that for s -a.e. (e, t), C(e, t)5g (t). Moreover, agents who exert the same effort receive post-transfer incomes that are almost never on opposite sides of the mean income: either almost all of them are richer than average, on almost all are poorer. This I call the principle of Fairly Ranked Incomes (FRI): there is a measurable mapping d : E →h21, 11j such that for s -a.e. (e, ¯ t), d (e) (F(e, t)2y(s))$0. These two principles are characteristic of balanced conditional egalitarianism: Proposition 3. Suppose that the generalized income function f : E 3T 3S →R 1 satisfies assumptions (A.3 ), (A.4 ), (A.7 ) and (A.8 ), and let s be a regular society. Then a redistribution of income in s satisfies the principles of Equal Transfer for Equal Talent and Fairly Ranked Incomes if and only if it is balanced conditionally egalitarian. The proof of Proposition 3 mimics those of Propositions 1 and 2 and is therefore omitted.
5. Discussion
5.1. The dimensional restrictions A weakness of the approach to income redistribution outlined in this paper is the assumption that talent and effort are summarized by real numbers. This assumption could be defended on the ground that the simplest case is clarifying as a benchmark for a more complex analysis. Balanced egalitarian equivalence, however, does not extend in any obvious way to the case when talent is multi-dimensional and balanced conditional egalitarianism is restricted to those environments where effort is one-dimensional. Note that this shortcoming is (essentially) shared by the median versions of egalitarian equivalence and conditional egalitarianism discussed in the introduction, which require a preorder structure on the set of talents or efforts. The average approach, by contrast, is well defined regardless of the structure of those sets. As it turns out, Equal Income for Equal Effort and Fairly Signed Transfers are generally incompatible when talent is multi-dimensional, unless very stringent restrictions are imposed on the income function. A proof of this incompatibility is given in Appendix B. A similar incompatibility can be established between Equal Transfer for Equal Talent and Fairly Ranked Incomes.
5.2. The continuum assumption The continuum assumption is important for the validity of the results in this paper. Consider the finite version of the model described in Section 2. Now, N5h1, . . . , nj is the set of agents, e5(e 1 , . . . , e n )[E N is the vector of their effort levels, t N 5(t 1 , . . . , t n )[T N is the vector of their talents, f(e i , t i ) is agent i’s pre-transfer income Fi (e N , t N ) is his post-transfer income. It is straightforward to redefine our axioms in this framework. Unfortunately, the characterization results no longer hold. EIEE and FST, for instance,
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do not characterize balanced egalitarian equivalence in society (e N , t N ). In fact, these axioms do not place any restriction on the redistribution of income if all effort and talent levels differ. A variant of Proposition 1 can nevertheless be established if the axioms are properly reinforced. Let us first replace EIEE with the following requirement. Higher Income for Higher Effort (HIHE): Let i, j [ N. If e i , e j , then Fi (e N , t N ) # Fj (e N , t N ). In the same spirit, substitute the following axiom for FST. Fairly Signed Transfers 2 (FST 2): Let i, j [ N be such that t i # t j . If Ci (e N , t N ) # 0, then Cj (e N , t N ) # 0; if Cj (e N , t N ) $ 0, then Ci (e N , t N ) $ 0. It turns out that every redistribution of income satisfying HIHE and FST2 must be ‘‘close’’ to the balanced egalitarian equivalent redistribution if the income function f is ‘‘well-behaved’’ and society (e N , t N ) is ‘‘sufficiently diverse’’. A precise statement and a proof of this fact are given in Appendix C. An analogous result can be proved for balanced conditional egalitarianism.
5.3. Incentives and efficiency As in Bossert (1995) and Bossert and Fleurbaey (1994), I have treated effort levels as exogenously given. This is a serious shortcoming: indeed, an agent’s optimal choice of effort clearly depends on the redistribution rule itself. Roemer (1994b, chap. 8) addresses this problem by postulating an ‘‘effort response function’’ which specifies, for each talent, how the level of effort is affected by the; redistribution rule. This response function may (and presumably does) result from a preference-maximization process. Endowing agents with preferences, if useful to discuss incentives issues, is absolutely necessary to address efficiency concerns. Suppose that an agent is characterized by a preference u over effort–income pairs, and a talent level. To fix ideas, suppose that u satisfies the usual assumptions and is also separable and linear in income. Call a density function r on preference–talent pairs an economy. How can balanced egalitarianism be generalized to this richer setting? The answer depends on the technological environment. Consider the example discussed in Section 3. If the aggregate production function w is linear, decentralized preference maximization is unambiguous. It yields for every economy r a society, i.e., a density s over effort–talent pairs. Moreover, since effort is efficiently distributed in that society, it is natural to operate balanced egalitarianism on s to redistribute the corresponding income. If w is not linear, decentralized preference maximization may be interpreted in different ways. If we assume Nash behavior, the equilibrium density function of effort will generally be inefficient. In that case, any income redistribution would perpetuate that inefficiency. It seems to me that this does not destroy the relevance of balanced
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egalitarianism: even if effort is inefficiently distributed, it is of interest to fairly redistribute income. But balanced egalitarianism is compatible with efficiency. To each preference–talent pair corresponds an efficient effort level. Using this correspondence, it is straightforward to transform an economy r into a society s* whose marginal s ´* is efficient. Balanced egalitarianism may now be applied to s*.
5.4. An alternative characterization There is an alternative characterization of balanced egalitarian equivalence which uses a pair of axioms different from those in Proposition 1 and does not require the continuum assumption. Consider again the finite model introduced in Section 5.2. No restriction is placed on society (e N , t N ); in particular, perfectly homogeneous societies where all talents coincide are admissible. The first axiom is adapted from Bossert (1995) and Bossert and Fleurbaey (1994). It is a much weakened version of ETET and is actually implied by FST: No Transfer for Uniform Talent: For every e N [ E N and t [ T, Fi (e N , (t, . . . ,t)) 5 f(e i , t) for all i [ N. The second axiom requires that all post-transfer incomes be similarly affected by changes in talents: Solidarity in Talents: For every e N [ E N and t N ,t N9 [ T N , either Fi (e N , t N ) # Fi (e n , t 9N ) for all i [ N or Fi (e N , t N ) $ Fi (e n , t 9N ) for all i [ N. Under Equal Treatment of Equals (Fi (e N , t N )5Fj (e N , t N ) if (e i , t i )5(e j , t j )), Solidarity in Talents implies EIEE. Solidarity axioms have been used in Bossert (1995); Bossert and Fleurbaey (1994) and Fleurbaey and Maniquet (1994b). In the first two papers, the authors require that everybody’s change in income be equal when an agent’s ‘‘irrelevant characteristics’’ (talent, in our terminology) are modified. This ‘‘cardinal’’ axiom is stronger than the ordinal requirement I suggest. It is not satisfied by balanced egalitarian equivalence. Fleurbaey and Maniquet’s model incorporates preferences. In that richer setting, their solidarity axiom is the natural counterpart of mine: it requires that all agents experience welfare changes of equal sign when talents change. It is almost obvious that balanced egalitarian equivalence is characterized by No Transfer for Uniform Talent and Solidarity in Talents. Suppose that there exists a society (e N , t N ) where F(e N , t N ) is not balanced egalitarian equivalent. There must then exist two agents, say, 1 and 2, such that F1 (e N , t N ),f(e 1 , t*) and F2 (e N , t N ).f(e 2 , t*), where t* is the balanced talent level. By Solidarity in Talents, however, F(e N , t N )5F(e N , (t*, . . . , t*)). It follows that F1 (e N , (t*, . . . , t*)),f(e 1 , t*) and F2 (e N , (t*, . . . , t*)).f(e 2 , t*), contravening No Transfer for Uniform Talent. The case for Solidarity in Talents is that if talents are really outside the agents’
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control, they should be considered a commonly owned resource. This argument strikes me as perfectly valid, but it certainly goes all the way to the most radical consequences of our assumption about the scope of the agents’ control. The characterization just presented thus combines a very mild axiom with a very controversial one. By contrast, Proposition 1 rests on two middle-of-the-road axioms. It should also be noted that Solidarity in Talents involves comparisons across societies, while both axioms in Proposition 1 are intrasociety properties.
5.5. An alternative approach If one insists on Equal Income for Equal Effort and sets out to weaken Equal Transfer for Equal Talent to avoid the basic incompatibility between these axioms, two routes can be taken. Either one fully equalizes for equally talented agents something as close as possible to the transfers they receive, or one approximately equalizes their actual transfers. I have chosen the first route in Proposition 1. The second route, though attractive, leads to the difficult problem of measuring the ‘‘ETET gap’’, i.e., the extent to which an income redistribution falls short of fully equalizing the transfers to equally talented agents. ¯ denote the mean transfer to the agents endowed One possibility is as follows. Let C(t) ¯ 5eE C(e, t)s´ (e) de. Since one wishes to (but cannot) make sure with talent t, i.e., C(t) ¯ for almost every (e, t), it makes sense to minimize that C(e, t)5C(t) ¯ EE (C(e, t) 2 C(t)) s(e, t) de dt. 2
T E
¯ But why not rather minimize eT eE uC(e, t)2C(t)u s(e, t) de dt? Or maybe sup t[T eE (C(e, 2 ¯ ¯ rather than the ratios t)2C(t)) s´ (e) de? And why focus on the differences C(e, t)2C(t) ¯ ¯ ¯ C(e, t) /C(t)? Or perhaps the relative differences (C(e, t)2C(t)) /C(t)? Identifying simple axioms on the redistribution F or its associated compensation mapping C that could recommend a particular measure of the ETET gap seems to be exceedingly difficult.
Acknowledgments I thank W. Bossert, M. Fleurbaey, B. MacLeod, F. Maniquel, H. Moulin, J. Roemer, an associate editor, and two referees for useful comments on an earlier draft. Financial ´ support from the FCAR of Quebec and the SSHRC of Canada is gratefully acknowledged.
Appendix A Here is the formal proof of the ‘‘only if’’ part of Proposition 1. Fix a regular society s and a redistribution F satisfying EIEE and FST. By EIEE, there exists a measurable
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mapping a : ER 1 such that F(e, t)5 a (e) for s -a.e. (e, t). Extend the income function f 2 to R 1 in such a way that f(e, ?) is continuous, increasing and unbounded for each e[E and f(?, t) is continuous for each t[R 1 . For each e[E there is a unique number t 1 (e) such that f(e, t 1 (e))5 a (e). The mapping t 1 : E →R 1 is such that for s -a.e. (e, t), F(e, t) 5 f(e, t 1 (e)).
(3)
This mapping is measurable. (To check this assertion, it suffices to verify that the set E(t 0 ):5he[Eut 1 (e),t 0 j is measurable for each t 0 [R 1 . But for every e[E, t 1 (e), t 0 ⇔ f(e, t 1 (e)),f(e, t 0 )⇔a (e),f(e, t 0 ). Therefore E(t 0 )5he[Eua (e)2f(e, t 0 ),0j, which is clearly measurable for each t 0 since a is measurable and f(?, t 0 ) is continuous, hence measurable as well). Define A2 5 (e, t) [ ExT uF(e, t) , f(e, t*), E 2 5 e [ Eut 1 (e) , t*, and define A1 and E 1 by reversing the inequality signs in the above definitions. All these sets are measurable because F, f, and t 1 are measurable mappings. By (3) and because f is increasing in t, A2 and A1 are almost equal to E 2 3T and E 1 3T respectively. Formally, if D denotes the symmetric difference operator, s (A2 D(E 2 3 T ))5 s (A1 D(E 1 3T ))50. We must prove that s (A2)5 s (A1 )50. Since s is regular, this is equivalent to
l´ (E 2) 5 l´ (E 1 ) 5 0,
(4)
where l´ is Lebesgue’s measure on the Borel s -algebra of E. Suppose, by way of contradiction, that
l´ (E 2) 1 l´ (E 1 ) . 0.
(5)
By the condition defining t* and the feasibility constraint on F,
EE F(e, t)s(e, t) de dt 5E f(e, t*)s (e) de ´
E T
E
E f(e,t (e))s (e) de 5E f(e, t*)s (e) de
⇒
1
´
⇒
(6)
´
E
E
E [ f(e, t*) 2 f(e, t (e))]s (e) de 5 E [ f(e, t (e)) 2 f(e, t*)]s (e) de.
E2
1
´
1
´
E1
From (5) and (6) follows that l´ (E 2).0 and l´ (E 1 ).0. This conclusion can be slightly 1 strengthened. For n51, 2, . . . , define E 2 n 5he[Eut 1 (e),t*21 /nj and define E n 2 2 similarly. There exists a positive integer N such that l´ (E N 2 ).0 since otherwise ` 1 l´ (E 2)2 l´ (< n51 E n2 )#o n251 l´ (E 2 n )#0. Likewise, there exists a positive integer N 2 1 such that l´ (E 1 N 1 ).0. Letting N5max [N , N j,
Y. Sprumont / Mathematical Social Sciences 33 (1997) 185 – 201 1 l´ (E 2 N ) . 0 and l´ (E N ) . 0.
199
(7)
Define T N 5[t*21 /N, t*11 /N]. Since s is regular, (7) implies
s (E N2 3 T N ) . 0 and s (E N1 3 T N ) . 0. 2
2 N
(8) 1
1 N
Now, for s -a.e. (e , t)[E 3T N and s -a.e. (e , t)[E 3T N ,
S D 1 F(e , t) 2 f(e , t (e )) . fSe , t* 1 ]D $ f(e , t). N 1 F(e 2 , t) 2 f(e 2 , t 1 (e 2)) , f e 2 , t* 2 ] # f(e 2 , t), N 1
1
1
1
1
1
1 2 1 This means that there exist full s´ -measure subset; of E 2 N and E N , say E N and E N themselves, and a full st -measure subset of T N , say T N itself, such that
C(e 2 , t) , 0 , C(e 1 , t) 1 1 for all e 2 [E 2 N , e [E N , and t[T N . It is easily seen that this contradicts the FST principle. Indeed, let b be any measurable function from T into h21, 11j. For all t[T N , one of the following statements holds:
b (t)C(e 2 , t) , b (t)C(e 1 , t) for all e [ E N2 and e 1 [ E N1 b (t)C(e 1 , t) , b (t)C(e 2 , t) for all e [ E N2 and e 1 [ E N1 Let B 2 5h(e, t)[E N2 3T N u b (t) C(e 2 , t),0j and define B 1 similarly. The measurability of F and f ensures that these sets are measurable. I claim that s (B 2
Appendix B Throughout this Appendix, I consider a variant of the continuous model of Section 2 in which talent is multidimensional. It will be convenient to keep our earlier notation even though this constitutes some abuse; in particular, T now stands for (0, 1)m , where m.1, and t is a m-dimensional vector. Assumptions (A.1) and (A.2) and the regularity condition are adapted accordingly. I claim that for all regular societies there exist income functions such that no redistribution of income satisfies EIEE and FST. Here is a sketch of the argument. The first two steps follow the proof of Proposition 1; Step 3 is the crucial point. Step 1. Fix a regular society s and a redistribution F satisfying EIEE and FST. By the former axiom, we can define a correspondence T : E →R m1 such that
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F(e, t) 5 f(e, t 1 (e)) for s -a.e. (e, t),
(9)
where f(e, t 1 (e))5y means that f(e, t)5y for all t[T 1 (e). Step 2. Let t* be any talent vector such that ¯ E f(e, t*)s (e) de 5 y(s). ´
(10)
E
Along every interior ray of R m1 , there is a (unique) t* satisfying this condition. Contrary to the one-dimensional case, there is thus an infinite set of ‘‘balanced talent vectors’’; we call it T *. For each t* in T *, define A- (t*)5h(e, t)[E 3T uF(e, t),f(e, t*)j and define A1 (t*) by reversing the strict inequality in the above definition. Essentially the same argument as in the proof of Proposition 1 shows that
s (A2 (t*)) 5 s (A1 (t*)) 5 0.
(11)
(One only needs to replace E 2 by E 2 (t*):5heu f(e, T 1 (e)),f(e, t*)j and E 2 by n 1 1 E2 (t*):5heu f(e, T (e)),f(e, (121\n)t*)j and define E and E (t*) similarly. The rest n 1 n n of the argument is virtually unchanged.) Step 3. Since (11) must hold for every t* in T *, it follows that F(e, t)5f(e, t*) for s -a.e. (e, t) and all t* in T *. Hence, for s´ -a.e. e, there is a number k(e) such that f(e;t*) 5 k(e) for all t* [ T *.
(12)
But this imposes severe separability restrictions on f. For instance, if m52 and f(e, t)5f(e, (t 1 , t 2 ))5et 1 1t 2 for all e, t, (12) demands that for almost every e there be a number k(e) such that et *1 1t 2 5k(e) for each (t *1 , t *2 ) in T *. Since T * has a unique 2 intersection with every interior ray in r 1 , it must be a line segment of slope 2e: this, however, cannot hold true for almost all e simultaneously.
Appendix C Suppose that the income function f is well-behaved in the following sense: it is differentiable and there exists a positive real number c such that (≠f / ≠e) (e,t),c(≠f / ≠t) (e,t) for every (e,t). Assume c51: this assumption entails no loss of generality since it can always be met by changing the units in which effort and talent are measured. Suppose next that society (e N , t N ) is sufficiently diverse in the following sense: there is a (small) number d, 0,d ,1, such that if(d, d ) , (e, t) , (1 2 d, 1 2 d ), there exist agents i, j in N such that (e 2 d, t 2 d ) , (e i , t i ) , (e, t) , (e j , t j ) , (e 1 d, t 1 d ). Under these assumptions, I claim that every redistribution satisfying the axioms HIHE and FST2 (stated in Section 5.2) must be close to the balanced egalitarian equivalent
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redistribution. Formally: if F satisfies HIHE and there exist two agents, say, 1 and 2, with d ,e 1 , e 2 ,12d, such that F1 (e N ,t N ) , f(e 1 ,t* 2 2d ) and F2 (e N ,t N ) . f(e 2 ,t* 1 2d ), then F must violate FST2. To see why this is true, let 1 and 2 be two agents such that d ,e 1 , e 2 ,12d and F1 (e N , t N )5f(e 1 , t*2d1 ), F2 (e N , t N ) 5 f(e 2 , t* 1 d2 ) with d1 , d2 .2d. This implies 2d ,d1 ,t*,12d2 ,122d. By the diversity assumption, there exists an agent i [N for whom e 1 2 d , e i , e 1 and t* 2 d , t i , t*. Since f is well-behaved, f(e i , t i ) . f(e 1 , t* 2 d1 ). Invoking HIHE yields Fi (e N , t N ),F1 (e N , t N ), hence, Fi (e N , t N ),f(e 1 , t*2d1 ),f(e i , t i ), meaning that agent i is taxed.On the other hand, there exists an agent j [N for whom e 2 , e j , e 2 1 d and t* , t j , t* 1 d. Therefore f(e j , t j ),f(e 2 , t*1d2 ) and HIHE yields f(e j , t j ),Fj (e N , t N ). Thus, agent j is subsidized, a contradiction to FST2.
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