Fairness and the proportionality principle

Transcription

1 Fairness and the proportionality principle Alexander W. Cappelen and Bertil Tungodden November 12, 2016 Abstract How should income be distributed in a way that respects both the egalitarian ideal that inequalities due to differences in opportunities should be eliminated and the liberal ideal that people should be free to pursue their own idea of the good life without interference from society? We show that reasonable interpretations of the egalitarian and the liberal ideal characterize what we refer to as the generalized proportionality principle. This principle states that an individual should have the share of total income that he or she would have had if everyone had the same opportunities and these opportunities were given by the average of the pre-tax income functions of all individuals in society. We argue that a redistribution mechanism based on this principle would eliminate unfair inequalities and preserve fair inequalities, and discuss when the generalized proportionality principle is equivalent to the simple proportionality principle. 1 Introduction Inequalities in earned income are a result of differences in both the opportunities people have and the choices they make. A theory of distributive justice needs to address how society should handle such inequalities. The liberal Both authors: Norwegian School of Economics, Bergen, Norway. alexander.cappelen@nhh.no and bertil.tungodden@nhh.no. We have received extremely useful comments and suggestions from two anonymous referees. The project was financed by support from the Research Council of Norway, research grant and administered by The Choice Lab. 1

2 egalitarian theories of distributive justice answer this question by combining the egalitarian ideal that inequalities due to differences in opportunities should be eliminated and the liberal ideal that people should be free to pursue their own idea of the good life without interference from society (see among others Dworkin, 1981; Arneson, 1989; Cohen, 1989; Le Grand, 1991; Roemer, 1993, 1996, 1998; Fleurbaey, 1994, 1995a-d, 2008; Bossert, 1995; Bossert and Fleurbaey, 1996; Iturbe-Ormaetxe, 1997; Sprumont, 1997; Boadway et al., 2002; Cappelen and Tungodden, 2002, 2003, 2006, 2009; Tungodden, 2005; Luttens, 2010; and Fleurbaey and Maniquet, 2011). In line with liberal egalitarian theories of justice, recent papers in behavioral economics have shown that a majority of people seem to accept income inequalities as fair if they reflect differences in choices, but reject income inequalities due to luck (Konow, 1996, 2000, 2001; Frohlich et al., 2004; Cappelen et al., 2007, 2010, 2013, 2014). A key challenge for a liberal egalitarian redistribution mechanism is how to eliminate income inequalities due to differences in opportunities and at the same time preserve income inequalities due to choice. An answer to this challenge, with a history going back to Aristotle, is provided by the proportionality principle. Applied to income distribution, this principle is commonly interpreted as saying that income should be distributed proportionally to individual effort. An appealing feature of the proportionality principle is that it eliminates income inequalities between individuals who choose the same effort level. A less appealing feature of this principle, as we demonstrate in this paper, is that it may violate the liberal ideal that society should be neutral with respect to how individuals choose to live their lives. However, we show that the generalized proportionality principle, closely related to principles suggested by Bossert (1995) and Konow (1996), satisfies both the egalitarian ideal of equalization and the liberal ideal of neutrality. 1 We argue that a liberal egalitarian redistribution mechanism must satisfy two basic requirements, a liberal requirement and an egalitarian requirement. In the formulation of these requirements, we refer to factors within individual control as effort and factors beyond individual control as talent. 2 First, we argue that the egalitarian ideal is captured by the requirement of equalization for equal effort, which states that the share of the total income that a 1 Our work is inspired by Iturbe-Ormaetxe (1997) and Sprumont (1977), who study related redistribution mechanisms. 2 See Fleurbaey and Maniquet (2011) for an excellent survey of alternative interpretations of liberal egalitarianism in the social choice literature. 2

3 person receives should only depend on the person s chosen level of effort and not on the talent profile in society. Second, we argue that the liberal ideal of neutrality is captured by the requirement of no equalization for uniform distribution of talent, which implies that there should be no redistribution between individuals at different effort levels if there is a uniform distribution of talent across effort levels. This requirement captures the intuition that any redistribution between individuals who make different choices must be justified by reference to differences in the distribution of talent among those who make different choices. We show that these two requirements jointly characterize the generalized proportionality principle. This principle states that each individual s fair share of the total income in any given situation is determined by considering a hypothetical situation which is identical to the actual situation with respect to the distribution of effort, but where everyone has the same opportunities and these opportunities are given by the average of the pre-tax income function of all members of society. The generalized proportionality principle assigns a share of the total income to each individual that is equal to this person s share of the total income in the hypothetical situation. In his seminal contribution to the analysis of liberal egalitarian justice, John Roemer often assumes that the distribution of effort is the same at all talent levels and he discusses what would constitute a liberal egalitarian distribution when the effort level of each individual is unobservable (Roemer 1993, 1998). In this paper we consider what would be the liberal egalitarian distribution in a first best setting where both talent and effort levels are observable, and we argue that the generalized egalitarian principle follows from attractive formulations of the liberal and egalitarian ideals even without assuming that effort is uniformly distributed within each talent group. The paper is organized as follows: Section 2 presents the basic framework, while Section 3 introduces the formal liberal and egalitarian requirements. In Section 4, we characterize the generalized proportionality principle, and in Section 5 we provide a formal analysis of how it relates to the simple proportionality principle. Section 6 provides some concluding remarks. In an online appendix, we report some further results on how the conditions used in the paper relate to other liberal and egalitarian conditions. 3

4 2 The basic framework Consider a population N = {1,..., n}, n 2, where each person i is characterized by his talent level, a T i, and effort level, a E i. 3 Let Ω E = {e 1, e 2...} be the set of possible effort levels and Ω T = {t 1, t 2...} be the set of possible talent levels, where Ω E, Ω T R. 4 Each individual can thus be represented by a vector a i = (a T i, a E i ) Ω = Ω E Ω T, where the pre-tax income function, f : Ω R ++, is assumed to be strictly increasing in effort and talent. 5 A society can be characterized by a vector a = (a 1,..., a n ) Ω N, where Ω N = Ω Ω... Ω is the set of possible characteristics profiles of society. Let Ω E (a), Ω T (a) be the effort levels and talent levels present in a, and a E and a T be the vectors of effort levels and talent levels in a. We now define the average population pre-tax income at effort level e j in a as g(e j, a T ) i N f(ej, a T i )/N, which is equal to what would have been the average income in a population with talent profile a T if everyone exercised effort level e j. In the core part of the paper we assume restricted domain richness, i.e., we assume that for any situation a there exists another situation ã established by permuting the effort levels in a such that the talent composition is the same at each effort level in ã. To formalize this domain restriction, let N j (a) be the set of individuals who choose effort level e j Ω E (a), i.e., N j (a) = {i N a E i = e j } and n j (a) be the cardinality of this set. Further, let E(a) represent the distribution of effort in a, i.e. the vector consisting of the fraction of the population exercising each possible effort level. We can now define restricted domain richness as follows: Restricted Domain Richness. The set of possible characteristics profiles of society is given by Ω N = {a Ω N there exists ã Ω N, where a T = ã T, 3 We thus focus on cases where talent and effort are unidimensional, but it is straightforward to extend the main analysis to the multidimensional cases. Further, this formal framework also presupposes that we can clearly distinguish between talent and effort, which may not always be the case in practice where some factors affecting income (for example education) may be shaped by both talent and effort. 4 Our results do not depend on the set of possible effort and talent levels being the set of real numbers. All the results in the paper can be established as long as there is more than one element in Ω E, Ω T. 5 In this framework, all effort levels are available for all talent levels. One might argue that for some interpretations of effort, low talented people have a more restricted set of available effort levels than high talented people. 4

5 E(a) = E(ã), and i N j (ã) f(ã i) n j (ã) = g(e j, ã T ) for all j Ω E (ã)}. This should be a straightforward assumption to make when studying redistribution in large societies, and nothing of importance seems to be lost by restricting the domain in this way. Since the aim of our analysis is to describe what constitutes a fair income distribution, we assume that people s choice of effort is insensitive to the ex post distribution of income. This implies that all allocations of income will be Pareto-optimal (as long as we assume that people have self-interested preferences and a positive marginal utility of income). We pay attention only to information about effort and talent levels when choosing among Paretooptimal allocations, and thus our object of study is a redistribution mechanism, F : Ω N R n, that for any distribution of talent and effort assigns a post-tax income to each person in society. We assume that F satisfies the no-waste condition i N F i(a) = i N f(a i), a Ω N. 3 The two ideals Liberal egalitarianism aims to combine an egalitarian ideal and a liberal ideal. In our framework, the egalitarian ideal implies that people should not be held responsible for their talent, while the liberal ideal implies that society should be neutral with respect to what effort level people choose to exercise. The basic moral intuition underlying the egalitarian ideal is that inequalities due to talent are unfair and should be eliminated. A very weak interpretation of this ideal is the minimal condition demanding that people who make the same choices in a particular situation face the same consequences (Bossert and Fleurbaey, 1996; Fleurbaey, 2008; Fleurbaey and Maniquet, 2011). In other words, if a low talented and a high talented person exercise the same level of effort in a particular situation, then they should end up with the same post-tax income. This condition, however, is silent about which factors should affect how we share total income more generally. For example, consider two situations, a and ā, that only differ in the effort levels of two individuals l and k being permuted. The minimal condition places no restriction on how the distribution of total income changes when we move from one of these situations to the other. It is consistent with giving nothing to l in a and everything to k in ā, even though the effort level exercised by 5

6 l in a is the same as the effort level exercised by k in ā and, importantly, the overall effort profiles are the same in the two situations. Hence, it is consistent with letting the share received by any particular person depend on whether he is a low talented person or a high talented person. This clearly shows that there is a need for a stronger condition to capture the egalitarian ideal. We argue that the appropriate way of capturing the egalitarian ideal is to require that the share of total income that a person receives in any particular situation only should depend on the effort profile in this situation. Clearly, whether it is a talented or less talented that is exercising any particular level of effort will be important in determining total income, but it should not be relevant for how the total income is distributed among individuals. In particular, if the effort profile is the same in two situations, then the share of the total income given to an individual exercising a particular effort level in one of the situations should be equal to the share of the total income given to another individual exercising the same effort level in the other situation. Formally, this requirement can be captured by the following condition. 6 Equalization for Equal Effort (EEE). For any a, ã Ω N and k, l N, if a T = ã T, E(a) = E(ã), and ã E l = a E k, then F l (ã) = F k(a) i N f(ã i) i N. f(a i) It follows straightforwardly that if a redistribution mechanism satisfies EEE, then it also satisfies the weaker demand that people who make the same choices in a particular situation should face the same consequences. The liberal ideal addresses the question of how income should be distributed between individuals who exercise different levels of effort; i.e., what is a fair inequality between individuals who make different choices? In a situation where people only differ with respect to their effort and not with respect to their talent, the liberal ideal has a straightforward interpretation. If everyone has the same talent, then there is no reason for a liberal society to redistribute between individuals since any difference in pre-tax income is only a result of a difference in choices. The liberal ideal then requires that we should preserve the pre-tax income inequality (Bossert and Fleurbaey, 1996; Fleurbaey, 2008; Fleurbaey and Maniquet, 2011). But how should we interpret the liberal ideal in situations where people differ in their talent? We argue that there is an attractive strengthening 6 In an online appendix, we discuss the relationship between the conditions introduced in this section and alternative formulations of the egalitarian and the liberal ideal. 6

7 of the minimal liberal requirement that captures the essence of the liberal ideal more generally: there should be no redistribution between individuals at different effort levels in situations where the average pre-tax income at any chosen effort level is equal to the average population pre-tax income at this effort level. This would be the case if there is a uniform distribution of talent across all chosen effort levels (or if the average pre-tax income at any effort level is equal to what would have been the case if there were a uniform distribution of talent at all effort levels). We can formalize this requirement as follows: No Equalization for Uniform Distribution of Talent (NEUDT). For any a Ω N : if i N j (a) f(a i)/n j (a) = g(e j, a T ) for all e j Ω E (a), then i N j (a) F i(a) = i N j (a) f(a i) for all e j Ω E (a). NEUDT implies that any redistribution between individuals at different effort levels must be justified on the basis of differences in the talent composition at the different effort levels. In particular, if the distribution of talent is the same across all chosen effort levels, then the liberal ideal of neutrality requires no redistribution between effort levels. 4 The generalized proportionality principle We will now show that the egalitarian ideal and the liberal ideal jointly characterize the generalized proportionality principle. F GP k (a) = g(ae k,at ) i N g(ae i,at ) i N f(a i), k N, a Ω n. According to the generalized proportionality principle, the fair share of the total income a person should receive in a given situation should be equal to the share of the total income that the person would have earned in a hypothetical situation where everyone exercised the same effort as in the actual situation, but where the pre-tax income function of each person is given by g(e j, a). In other words, a person s fair share of the total income is determined by considering the counterfactual situation where everyone has the same opportunities, and these opportunities are given by the average of the pre-tax income function of all members of society. The generalized proportionality principle is egalitarian in the sense that the share of the total income that an individual receives only depends on the 7

8 effort profile, E(a), and not on how talent is distributed across effort levels. The generalized proportionality principle is liberal in the sense that it does not view any choice as more or less deserving and only redistributes income in order to eliminate inequalities that result from an unequal distribution of non-responsibility factors. It turns out that the generalized proportionality principle is closely related to EEE and NEUDT. Proposition 1: Given restricted domain richness, a redistribution mechanism F satisfies EEE and NEUDT iff F = F GP. Proof. The if-part is trivial, and thus we only show the only-if-part (1) Consider any a Ω N and k N, where a E k = ek. By restricted domain richness we know that there exists ã Ω N such that a T = ã T, E(a) = E(ã), and i N j (ã) f(ã i)/n j (ã) = g(e j, ã T ), e j Ω E (ã), and m N such that ã E m = a E k. (2) By (1) and EEE, F k (a) i N f(a i) = Fm(ã) i N f(ã i). (3) By (1) and NEUDT, i N k (ã) F i(ã) = i N k (ã) f(ã i). By EEE, F m (ã) = F i (ã), i N k. Hence, F m (ã) = 1 n k (ã) i N k (ã) f(ã i). (4) From (2) and (3), we find that F k (a) = i N k (ã) f(ã i) 1 n k (ã) i N f(ã i) i N f(a i). 1 From the definition of ã in (1), it follows that n k (ã) i N k (ã) f(ã i) = g(e k, ã T ) and i N f(ã i) = i N g(ãe i, ã T ). Thus, the result follows by noticing that g(e k, a T ) = g(e k, ã T ) and i N g(ãe i, ã T )= i N g(ae i, ã T ). Within a more general domain, there are also other redistribution mechanisms consistent with EIEE and NEUDT. In particular, the class of subgroup solidarity mechanisms proposed by Cappelen and Tungodden (2002, 2003) satisfies both these requirements. 5 The simple proportionality principle In applying the idea of proportionality to fairness, the more traditional approach has been to adopt a straightforward version of the proportionality 8

9 principle, where each person receives a share of the total income that is proportional to his or her share of the total effort exercised in society (Konow 1996, 2000, 2001; Cappelen et al., 2007, 2013). F P k (a) = ae k i N ae i i N f(a i), k N, a Ω N. Given the popularity of the basic proportionality principle, it is interesting to study both in which economic environments it is an appealing approach and its shortcomings as a general principle of fairness. It follows straightforwardly that Fk P satisfies the egalitarian idea as formulated in EEE, but it is less likely to satisfy the liberal ideal Consider the following weak version of the liberal requirement (Bossert and Fleurbaey, 1996; Fleurbaey, 2008; Fleurbaey and Maniquet, 2011), which is restricted to situations where everyone has the same talent. No Equalization for Uniform Talent (NEUT): For all a Ω N, if a T j = a T k, j, k N, then F i(a) = f(a i ), i N. The proportionality principle is only consistent with NEUT in an economic environment where pre-tax income is proportional to effort. 7 Proposition 2: The redistribution mechanism F P f(e j,t k ) = f(el,t k ) for all e j, e l Ω E and for all t k Ω T. e j e l satisfies NEUT iff Proof. The if-part. (1) Consider any a Ω N, j N, and t j Ω T, where a T i = a T j = t j, i N. By assumption, f(a E i, t j ) = ae i a E j i N ae i i N a E i a E j The only-if part. a E j f(a E j, t j ), i N. Hence, F P j (a) = f(a E j, t j ) = f(a E j, t j ), and thus NEUT is satisfied. (2) Suppose that there exist e j, e l Ω E and t j Ω T such that f(ej,t j ) e j f(e l,t j ). Consider any a Ω N, where a T e l i = t j, i N and for some r, s N, 7 It follows straightforwardly that NEUDT implies NEUT. 9

10 a E r = e j and a E s = e l. By NEUT, F i (a) = f(a E i, t), i N. Hence, Fr(a) F s(a) = f(ar). However, F r P (a) = ae f(a s) Fs P r f(ar) (a) a E s f(a s), and thus the supposition is violated. The restriction of the economic environment in Proposition 2, however, is not sufficient to make the basic proportionality principle in line with our preferred interpretation of the liberal ideal (NEUDT). This will only be the case if we impose the further restriction that the average pre-tax income at each effort level is proportional to effort. Proposition 3: The redistribution mechanism Fk P satisfies NEUDT iff g(e j,a T ) = g(el,a T ) for all e j, e l Ω E and for all a Ω N. e j e l Proof. The if-part. (1) Consider any a Ω N, where i N j (a) f(a i)/n j (a) = g(e j, a T ), e j Ω E (a). By assumption, g(e i, a T ) = ei g(e j, a T ), e i Ω E (a), and e j by definition, i N f(a i) = i N g(ae i, a T ). Hence, i N j (a) F i P (a) = n j (a)g(e j, a T ) = i N j (a) f(a i), e j Ω E (a), and the result follows. The only-if-part. (4) Suppose that for some a Ω N and e r, e s Ω E (a), g(er,a T ) Consider some ã Ω N, where n r (ã) = n s (ã) and all j Ω E (ã). It now follows that i N r (ã) F i P (ã) i N s (ã) F i P er = (ã) (5) From the supposition, it follows that g(er,a T ) it follows that the result follows. i N r (ã) F i P (ã) i N i N r (ã) f(ã i) s (ã) F i P (ã) i N s (ã) f(ã i) g(es,a T ). e r e s i N j (ã) f(ã i) = g(e j, ã) for n j (ã) e s and g(e s,a T ) er i N r (ã) f(ã i) i N s (ã) f(ã i) = g(er,a T ) g(e s,a T ). e s, and hence from (1). But this violates NEUDT, and It follows directly that the generalized proportionality principle is equivalent to the basic proportionality principle when the average pre-tax income at each effort level is proportional to effort. Proposition 4: Given restricted domain richness, Fk P for all j, l Ω E (a), where e j > 0, g(ej,a) = g(el,a). e j e l GP (a) = Fk (a) iff 10

11 Proof. The proof follows from combining Proposition 1 and Proposition 3 and the fact that Fk P satisfies EEE. The basic proportionality principle is attractive because of its simplicity, and it has the appealing feature of satisfying EEE. However, unless we operate in an economic environment where the pre-tax income function is multiplicative, f(a i ) = h(a T i )a E i, which is often the case in experimental studies of fairness, the principle violates what we consider an attractive interpretation of the liberal ideal of neutrality. 6 Conclusion An important point of departure for the axiomatic approach to liberal egalitarian ethics is the impossibility result established in Bossert (1995) and Bossert and Fleurbaey (1996), which shows a potential conflict between the egalitarian ideal and the liberal ideal. A standard response to this result has been to argue that we have to make a trade-off between these two ideals in liberal egalitarian justice (Fleurbaey, 2008). In this paper, however, we argue that the standard minimal interpretations of the egalitarian ideal and the liberal ideal are too weak and too strong, respectively. The minimal egalitarian requirement is too weak because it does not rule out that people s talent determines how the income is distributed; the minimal liberal requirement is too strong because it restricts redistribution in situations where redistribution could be justified by appealing to differences in talent. By strengthening the egalitarian requirement and weakening the liberal requirement, not only do we avoid the impossibility result, but we also characterize a unique liberal egalitarian position, the generalized proportionality mechanism. While this principle has been suggested previously, it has not been given a convincing characterization and it has received far less attention than for example the egalitarian equivalent mechanism and the conditional egalitarian mechanism proposed by Bossert and Fleurbaey (1996). Even if we have shown that there is a way to combine the egalitarian ideal and the liberal ideal in theory, there might still be a tension between these ideals in the design of redistributive policies. Policies aimed at equalizing opportunities will typically involve redistribution between individuals who pursue different ideas of the good life. To illustrate, consider a tax system that redistributes income from those with high income to those with low 11

12 income. Such a progressive tax system would reduce income inequalities due to differences in income opportunities (including talent), but it would at the same time reduce income inequalities between individuals who have the same income opportunities, but who have chosen to pursue different ideas of the good life (including exercising different levels of effort). This problem is unavoidable in practical policy since information required to implement the generalized proportionality mechanism typically is unavailable to tax authorities. A characterization of the liberal egalitarian ideal is still important, since it can be used as a standard for evaluating redistributive policies, including tax policies. Almås et al. (2011) show how the generalized proportionality principle can be used to measure unfair inequality in a population, both pre-tax and post-tax, and demonstrate that such an approach may give a very different picture of the development in a country than more standard inequality measures. This paper has been concerned with the normative question of what constitutes a just redistribution mechanism. However, it is also interesting to ask the positive question of which principle of justice motivates individuals when they make decisions that have distributive implications. In previous research (Cappelen et al. 2007, 2010, 2013, 2014), we have shown that a large majority of people make a distinction between income inequalities due to choice and income inequalities due to luck. Typically, we have focused on economic environments where the pre-tax income function is multiplicative, which implies that the generalized proportionality principle and the simple proportionality coincide. An interesting question for further behavioral research would be to study more complex environments, where the simple proportionality principle has important shortcomings. More research is also needed to understand which inequalities people find fair when individuals make choices on an unlevel playing field, i.e., in situations where the consequences of an individual s choices partly are determined by circumstances beyond individual control. References [1] Almås, I., A. Cappelen, J. T. Lind, E. Ø. Sørensen, and B. Tungodden (2011) Measuring unfair inequality. Theory and evidence from Norwegian data, Journal of Public Economics, 95 :

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