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Time in Economics, part I

Equality of when?

Égalité de quand ?
Giorgos Galanis et Roberto Veneziani
p. 25-59


Cet article analyse l’unité de temps retenue dans les analyses égalitaristes. Dans un contexte intertemporel, les différences entre des visions égalitaristes peuvent s’apprécier, non seulement en termes d’analyse des inégalités, mais aussi relativement à la distribution égalitaire à réaliser. Trois principes d’égalité inter-générationnelle sont analysés dans cet article : L’égalitarisme sur toute la vie (Complete Life Egalitarianism), l’égalitarisme des segments correspondants (Corresponding Segments Egalitarianism) et l’égalitarisme des segments simultanés (Simultaneous Segments Egalitarianism). On défend le deuxième principe (CSE) comme critère approprié. On analyse également la relation entre les trois principes et d’autres idéaux moraux, soit le maximin et l’utilitarisme. On démontre que les principes de CLE et CSE sont compatibles avec une préoccupation pour les plus démunis, et partiellement compatibles avec une approche utilitariste. En revanche, l’adoption du principe de SSE impose un compromis entre l’égalitarisme et les autres idéaux moraux.

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1The publication of John Rawls’ A Theory of Justice (1971) has marked a renewal of interest in egalitarian theories, and has sparked a large literature. One of the central issues of contention has been the choice of the appropriate equalisandum: income, wealth, utility, primary goods or, more recently, capabilities, functionings, and opportunities. As Amartya Sen (1980) forcefully put it, one of the central questions for egalitarians raised by Rawls’ theory, and by his forceful criticism of utility as the relevant equalisandum, is equality of what?

2The analysis of the appropriate unit of egalitarian concern– and indeed other debates on the foundations of egalitarian theory following Rawls’ seminal contribution—has been carried out in the main within the confines of an essentially static environment. The choice of the appropriate equalisandum, for example, has been explored abstracting from the time structure of individual lives and focusing on a single generation (e.g., Sen, 1980; Dworkin, 1981a; 1981b; Arneson, 1989; Cohen, 1989; Roemer, 1998). Similarly, the analysis of different measures of inequality has typically focused on the distribution of the relevant variable in a single period (e.g., Sen, 1973; 1992; Temkin, 1993).

  • 1 For a discussion see Mariotti et al. (2012) and Lombardi et al. (2016).
  • 2 See, for example, the classic papers by Arrow (1973) and Dasgupta (1974). For a more recent discuss (...)

3To be sure, time and dynamics have played a significant role in egalitarian debates. Many empirical studies have analysed the dynamics of inequality over time or across generations, and the intergenerational and intertemporal impact of different distributive policies. Theoretically, a large literature has emerged on intergenerational justice and the evaluation of infinite utility streams,1and the issues that a dynamic approach poses to egalitarianism, including the trade-offs between distribution and growth, have been highlighted almost immediately after the publication of A Theory of Justice.2

4Yet, even when distributive dynamics, growth and time have been considered, the complex economic and philosophical implications of the fact that agents’ lives develop over time have often been overlooked by focusing either on the analysis of a sequence of time slices (e.g., in the analysis of the evolution of income distribution over time) or by implicitly assuming that the relevant unit of distributive concern can be reduced to a single variable measuring the lifetime attainment of the relevant equalisandum. As Milton Friedman (1957, 38) famously put it, “the identification of low measured income with ‘poor’ and high measured income with ‘rich’ is justified only if measured income can be regarded as an estimate of expected income over a lifetime or a large fraction thereof”. A similar view is held by most political and moral philosophers: “the subject of an egalitarian principle is not the distribution of particular rewards to individuals at some time, but the prospective quality of their lives as a whole, from birth to death” (Nagel, 1991, 69).

5In a seminal article, Dennis McKerlie (1989) questioned the standard view—defined as complete lives egalitarianism (CLE)— according to which agents’ lives, taken as a whole, are the proper unit of egalitarian concern. For, McKerlie argued, CLE is insensitive to inequalities occurring within particular segments of agents’ lives, no matter how severe such inequalities are, provided they cancel out over the course of agents’ lives. He suggested that egalitarians focus instead on inequalities occurring between contemporaries at a given time—a view called simultaneous segments egalitarianism (SSE). More generally, McKerlie (1989) argued that because agents’ lives extend over time, a sound egalitarian analysis requires the definition of the proper unit of egalitarian concern, i.e. whole lives or selected parts of them. Egalitarian principles based on different temporal units incorporate different moral concerns and have different policy implications.

  • 3 See, for example, Temkin (1992, 1993); Kappel(1997); McKerlie (2001a,b, 2012); Huemer (2003); Carte (...)

6The issues raised by McKerlie (1989) are deep, go beyond the boundaries of liberal egalitarian approaches, and have sparked a growing literature across the disciplinary borders of philosophy and economics.3Veneziani (2007, 2013) has shown, for example, that the temporal structure of individual lives has relevant implications in exploitation theory: a capitalist economy in which individuals switch roles over time so that everyone is exploited for an equal amount of time is not necessarily a just economy. Moreover, according to various authors, the fact that individual lives are structured into normatively relevant segments raises difficult, if not intractable issues for egalitarianism.

  • 4 A similar point is made by Attas (2008), who argues that the temporal structure of agents’ lives cr (...)

7In a series of contributions, for example, Temkin (1992, 1993) has analysed the three main intertemporal egalitarian principles, namely CLE, SSE and also corresponding segments egalitarianism(CSE), which holds that inequalities must be measured between corresponding stages of agents’ lives—e.g., childhood, middle age, old age, etc. According to him, in the analysis of inequalities no principle is entirely satisfactory: “several views are possible, each of which seems plausible in some cases and implausible in others” (Temkin , 1993, 291). Indeed, the absence of a single approach properly capturing egalitarian views suggests that we should reconsider our intuitions and acknowledge that equality is inevitably a complex notion. So much so that it may sometimes be difficult to formulate clear and consistent egalitarian considered judgements.4

8Kappel (1997) has taken Temkin’s (1992; 1993) conclusion further, suggesting that we should actually abandon egalitarianism. In his view, none of the egalitarian views is completely satisfactory because it is normatively irrelevant and even misleading to compare the relative attainments of the relevant variable across individuals. Once we consider the temporal structure of individual lives, it becomes clear that “what matters in our egalitarian judgement is giving priority to the worse off” (Kappel, 1997, 204).

9More radically, Huemer (2003) has argued that the conflicting intuitions arising in the intertemporal context reveal a deep conceptual problem for egalitarianism. According to him, once we acknowledge that individual lives are divided into temporal stages, egalitarianism cannot be coherently defined: intuitions concerning complete lives and about temporal segments inevitably conflict leading to inconsistencies.

  • 5 To be sure, CLE, CSE and SSE do not exhaust the set of logically conceivable intertemporal egalitar (...)
  • 6 We focus in particular on the arguments advanced by Temkin (1992, 1993) and Kappel (1997). For a th (...)

10This paper analyses the issues that time and the temporal structure of individual lives pose in normative economics, and in particular for egalitarianism. To be specific, we reconsider the three main intertemporal egalitarian principles proposed in the literature in order to identify the appropriate temporal unit of egalitarian concern—that is, to answer the question posed in the title: equality of when?5 We argue that, although critics have highlighted some important and often neglected issues, their arguments do not pose any intractable problems for egalitarians.6 It may be true that no principle fully captures our egalitarian intuitions when evaluating inequalities, but this does not mean that egalitarianism is incoherent or that no satisfactory egalitarian benchmark can be defined.

11Indeed, we argue that an important distinction has been overlooked in the literature, which is a peculiar feature of the intertemporal context. Unlike in the static setting, apart from differing in the analysis of unequal distributions, intertemporal egalitarian principles also define different egalitarian states to reach. The two issues are connected but they should be kept conceptually distinct in the choice of the appropriate principle. This is even more evident for policy purposes,—e.g., from the viewpoint of a government concerned with equality,—since the definition of the ideal “steady-state” egalitarian distribution and the design of the transition process to that state raise different problems. In order to implement an egalitarian strategy, in addition to a correct analysis of the status quo, it is necessary to define the appropriate egalitarian benchmark.

12This paper focuses on the latter issue. Section 1 briefly reviews the main results of the existing literature on the properties of the three egalitarian views in the evaluation of unequal distributions. Then, the distinction between the evaluation of existing inequalities and the definition of the appropriate egalitarian distribution is introduced, and it is argued that, as regards the distribution to establish, CSE defines the appropriate intertemporal egalitarian benchmark.

  • 7 For a discussion of the relation between the maximin and egalitarianism, see Temkin (1993).

13Because the evaluation of a distribution, e.g., for policy purposes, is influenced by more than one normative concern, in section 2, a formal analysis of the trade-offs between the different egalitarian principles and other normative views is presented, which aims to provide a formal basis for all things considered judgements (Temkin, 1993). The relation of CLE, CSE, and SSE with two non-primarily egalitarian normative concerns, Rawls’s difference principle and utility,7 are analysed. A stylised model is set up, which generalises Arrow (1973) and Dasgupta (1974) by considering overlapping generations, so that at each date there are two types of individual, young and old, rather than a representative agent. This allows us to analyse intertemporal as well as intratemporal equality. It is proved that the maximin solution yields CSE and CLE, but not SSE, and if the assumptions of the model are relaxed, CLE remains the egalitarian principle that can best accommodate Rawlsian or utilitarian concerns, and it is easier to reconcile these concerns with CSE than with SSE.

14Finally, it is worth noting that our formal analysis yields some interesting insights on a vexed issue in normative economics, namely the well-known trade-off between Rawlsian distributive justice and growth. For we show that, once the temporal structure of individual lives is explicitly modelled, then contrary to Arrow’s (1973) and Dasgupta’s (1974) classic papers, the dynamics of the economy is not completely stationary, even though the application of the maximin principle precludes permanent growth. We briefly elaborate on this point in the concluding section. The proofs of all formal results are relegated to the Appendix.

1 Three Egalitarian Principles Compared

Let Image 100002010000000D00000019C53FF49CFA5EE394.png be the relevant equalisandum, which shall be called ‘welfare’ (Image 100002010000000D00000019C53FF49CFA5EE394.png could be income, utility, opportunities, primary goods, etc.). Assume that agents’ lives can be divided into an equal number Image 10000201000000120000001910130F5273CE4508.png of well-defined periods of equal length. Let Image 1000020100000041000000197802E24D0BF58E1B.png Image 100002010000013900000019242201FE6EA61503.png be the set of vectors describing the attainment of Image 1000020100000033000000196FD7D02D8F041BBD.png by agent Image 100002010000000700000019B7E0A6A2E240FCE3.png in period Image 1000020100000073000000197388BA6FDCB63390.png of her life, at date Image 100002010000006400000019044B369417ED8941.png , where Image 100002010000000800000019DF4FF4CCC5496A39.png is the date of birth of Image 100002010000000700000019B7E0A6A2E240FCE3.png . For the sake of simplicity, assume Image 100002010000000D00000019C53FF49CFA5EE394.png to be interpersonally and intertemporally comparable, and additive along agents’ lives, so that Image 100002010000007C000000195621DE53243B9FA9.png is the lifetime attainment of Image 100002010000000D00000019C53FF49CFA5EE394.png by agent Image 100002010000000700000019B7E0A6A2E240FCE3.png born in Image 100002010000000800000019DF4FF4CCC5496A39.png . These assumptions make the analysis comparable with McKerlie (1989) and Temkin (1993) and they are quite natural if Image 100002010000000D00000019C53FF49CFA5EE394.png is a variable such as income or an index of primary goods. On the other hand, if a subjective variable like utility is considered, these assumptions give the opportunity to compare the egalitarian principles in vitro, as a first step towards a more satisfactory and realistic analysis.

The three egalitarian principles can be interpreted as different ways of evaluating distributions of the Image 100002010000001400000019CEE40BDE95CF6AA0.png vectors. Let Image 100002010000001C000000192F878F47591FF987.png , Image 100002010000001E0000001980193AF628970955.png and Image 100002010000001D00000019551588D1738C7409.png denote inequality measures associated with CLE,

CSE, and SSE, respectively. Formally, Image 10000201000000CD0000001962F9C284D06DE04A.png Image 100002010000001F00000019C4EB6B579111E648.png for Image 100002010000005F000000195E7A96BD2F7FD7A7.png . If Image 100002010000003F00000019F4240FB10DFE29BA.png , then a distribution is egalitarian according to the relevant principle Image 100002010000005A00000019CBD2913CA3BEACA2.png . Given the definitions in the previous section, Image 100002010000004700000019C090DD6E15369D81.png if and only if Image 100002010000004000000019C9D247C7ED16F342.png for all agents Image 100002010000000700000019B7E0A6A2E240FCE3.png , Image 100002010000000E000000191662AD734F5F12A5.png and dates of birth Image 100002010000005F000000194C62102C8CBDEE20.png  if and only if Image 100002010000007A00000019AD043CE934A850D8.png for all agents Image 100002010000000700000019B7E0A6A2E240FCE3.png , Image 100002010000000E000000191662AD734F5F12A5.png , dates Image 10000201000000C900000019C22C29F6F4F2F6A4.png , and corresponding life stages Image 100002010000000C0000001967B1E1C456C0AB7E.png ; and Image 100002010000004F0000001969337E82190FA61A.png if and only if Image 100002010000006F000000192924FA1DB4793547.png for all agents Image 100002010000000700000019B7E0A6A2E240FCE3.png , Image 100002010000000E000000191662AD734F5F12A5.png , life stages Image 100002010000000C0000001967B1E1C456C0AB7E.png , Image 100002010000000C000000194772E5F77FFDB89C.png and simultaneous dates Image 10000201000000DE000000192C55348EF86ABAC9.png

In order to focus on the implications of the three egalitarian principles,—rather than on the features of specific measures,— no further restrictions are imposed on the Image 100002010000002D0000001937E7327A31CAD7E3.png . As in the static setting, where the problems of inequality measurement are reflected into the existence of several measures capturing different aspects of inequality (e.g., Gini index, Atkinson’s measure, etc.; see Temkin, 1993) in principle there are many possible ways of measuring inequalities according to each criterion, that is, there are various specifications of every Image 100002010000001C000000196F2BA17091F29271.png . Actually, in the intertemporal context the choice of the appropriate inequality measure associated to each criterion is more complex, since the Image 100002010000002D0000001937E7327A31CAD7E3.png should rank distributions of vectors rather than distributions of real numbers.

As convincingly argued by Temkin (1993), however, one of the specific features of intertemporal analysis is that, unlike in the atemporal context, even assuming a unique possible Image 100002010000001C000000196F2BA17091F29271.png associated to each principle, the issue of inequality measurement would not be solved: different egalitarian principles highlight different kinds of inequalities and no principle, CLE, CSE, or SSE, seems completely satisfactory in the analysis of unequal distributions.

  • 8 In all examples in this section, we suppose that agents’ lives are divided into four stages, so tha (...)

The main problem of CLE is that it leads to “changing places egalitarianism” (McKerlie , 1989). If whole lives are the unit of egalitarian concern, in a “situation involving differential treatment of equally deserving people—no matter how significant, ... and even perverse those differing treatments are—there can be no egalitarian objection as long as the roles of the equally deserving people are interchanged so that each receives an equivalent share of the treatments meted out” (Temkin, 1993, 236). According to CLE, for example, an allocation where Image 100002010000007A000000192678C096294C89A7.png , for each agent Image 1000020100000007000000198844C49B6624CD8B.png in a set Image 10000201000000A400000019566EF38BF6760FE1.png born in Image 1000020100000008000000198610D736B2CA12A4.png , is equivalent to the following distribution:8

15Example 1.

Image 10000201000001AB0000004BA19AD148FCEE78B3.png

  • 9 See, for example, Sen (1980); Dworkin (1981a,b); Arneson (1989); Cohen (1989) and Roemer (1998).

It may be objected that it is difficult to judge the distribution in Example 1 without additional contextual considerations. For instance, an anonymous referee has argued that Example 1 does not necessarily involve “differential treatment of equally deserving people” if the different intertemporal distributions reflect the free optimizing choices of fully rational adults based on equal ex ante endowments. This objection raises a normatively crucial issue concerning the role of circumstances and responsibility in egalitarian theories, but it does not really question our conclusion. For we have left the equalisandum Image 100002010000000D00000019C53FF49CFA5EE394.png unspecified and therefore “changing places egalitarianism”, and Example 1, continue to hold if Image 100002010000000D00000019C53FF49CFA5EE394.png is interpreted as primary goods, or endowments, or more generally resources that are beyond the agents’ control.9In fact, in order to clearly separate the analysis of the appropriate intertemporal egalitarian benchmark from the issue of the appropriate currency of egalitarian concern, all of our examples can be interpreted as capturing the choices of a social planner, consistently with a standard social choice theoretic perspective and the analysis developed in section 2 below.

CSE and SSE rule out changing places egalitarianism insofar as they do not allow present inequalities to compensate for past ones: in Example 1, both Image 100002010000001D0000001942AA077786FF05B5.png and Image 100002010000001D000000193E86BD488FCDFC77.png would be different from zero. Yet, they do not represent entirely satisfactory alternatives in the evaluation of unequal distributions. By focusing only on inequalities in selected portions of agents’ lives, both principles can lead to the paradoxical conclusion that a distribution exhibiting changing places egalitarianism is as objectionable as one in which the agents’ roles are not interchanged and one agent is worse off in every relevant segment, as the next example shows:

Example 2. Consider again the set of agents Image 10000201000000A100000019CBF15D922ECD9DDF.png born in Image 1000020100000008000000198610D736B2CA12A4.png :

Image 10000201000001BF0000004BC54B89E059BE6770.png

Although we have not imposed any restrictions on the measures Image 100002010000001C000000196F2BA17091F29271.png when evaluating unequal distributions, under a large set of reasonable functional specifications—including most notably the additive specifications typically (albeit often implicitly) used in the literature (see, e.g., Bidadanure, 2016)—according to Image 100002010000001D0000001942AA077786FF05B5.png and Image 100002010000001D000000193E86BD488FCDFC77.png , the latter distribution would be as unequal as that in Example 1. Hence, Temkin (1993, 291) concludes that in the evaluation of existing inequalities, “several views are possible, each of which seems plausible in some cases and implausible in others”, and it may be opportune to use the information conveyed by all principles rather than focusing only on one of them.

  • 10 We are grateful to Robert Sugden for this suggestion.

It may be objected that the distribution in Example 1 is egalitarian and we find it intuitively objectionable because it fails on other criteria, e.g. fraternity (equality between contemporaries promotes feelings of solidarity), or welfare (because of diminishing marginal utility).10 It is difficult to address this objection without more information on the nature of x and on the meaning of the welfare levels in Example 1. But, granting that information on the temporal structure of the attainment of Image 100002010000000D00000019C53FF49CFA5EE394.png is at least prima facie relevant, then one may argue that the distribution in Example 1 is not equivalent to one in which Image 100002010000007A000000192678C096294C89A7.png , for all Image 100002010000000800000019E447C527444B08FF.png , and agents in Example 1 are not equally treated.

  • 11 See, e.g., the measures discussed in Sen (1973, 1992) and in Temkin (1993, chapter 5).
  • 12 The distinction is also ignored in Lippert-Rasmussen’s (2003) otherwise insightful analysis.

We need not adjudicate this issue here. For, even granting the relevance of changing places egalitarianism, we argue that this does not raise intractable problems for egalitarians. The key point to note is that Temkin’s (1993) (and indeed Kappel’s 1997) arguments do not extend to the choice of the appropriate intertemporal egalitarian benchmark, which is quite a different issue from the analysis of past and present inequalities. In the static context, while the measurement of inequalities can be controversial, the definition of egalitarian states is uncontroversial: different inequality measures give the same answer if the distribution is egalitarian—that is, when all agents attain the same level of the relevant variable Image 100002010000000D00000019C53FF49CFA5EE394.png .11 In contrast, in the intertemporal context, it is misleading to say that different views can “be regarded as built around ways of measuring the inequality between lives” (McKerlie , 1989, 487). The three principles stress different aspects of existing inequalities, but they also define different egalitarian states to reach, as shown by the fact that, unlike in the static context, Image 100002010000004100000019F669D6A75A766BAD.png does not necessarily imply Image 100002010000009800000019A59F11B01FB70D47.png . The two issues are connected, but should be kept conceptually distinct in the choice of the appropriate egalitarian principle.12

16The difference between the two perspectives is particularly evident for policy purposes, since the definition of the ideal “steady-state” egalitarian distribution and the design of the transition process to that state raise different issues. In order to implement an egalitarian strategy, in addition to a correct analysis of the status quo (involving the evaluation of existing inequalities and claims for compensation of past ones), the proper intertemporal egalitarian benchmark must be defined.

17This distinction has been largely overlooked in the literature, and so the scope of the conclusions reached is sometimes unclear. For example, McKerlie (1989) discusses the choice of the egalitarian benchmark, but his arguments are based mainly on the analysis of the claims for compensation of past inequalities implied by the different views. Similarly, Temkin (1993) criticises CLE because it entails changing places egalitarianism but he finds SSE and CSE faulty based on the analysis of unequal distributions, which provides little information as to the features of the egalitarian distributions associated with them. In the rest of this section, we show that if the distinction is rigorously drawn, then even though no principle may fully capture our egalitarian intuitions when evaluating unequal distributions, this does not mean that egalitarianism is incoherent or that no satisfactory egalitarian benchmark can be defined.

Consider the three principles from the point of view of the distributions with Image 10000201000000F200000023C9FE9CA69AA856F9.png . As noted above, changing places egalitarianism may raise doubts on CLE as the intertemporal egalitarian benchmark. SSE is not a satisfactory alternative.

  • 13 Consider, for example, an economy with N identical agents with the same, strictly increasing, welfa (...)
  • 14 Indeed, Bidadanure (2016) has highlighted a conceptually analogous problem of D3, which arises from (...)

A first puzzling feature of SSE is reflected in the time dependence of D3 and in particular in its sensitivity to small changes in the agents’ date of birth. Suppose that agents’ lives are divided into relatively short time periods. Then, it is easy to construct examples in which for given allocations of Image 100002010000000D00000019C53FF49CFA5EE394.png , a “slight” shift in the date of birth of an agent is sufficient to change dramatically the value of Image 100002010000001D000000193E86BD488FCDFC77.png and the egalitarian judgement.13However, it is hard to see why if an agent is born, say, a few months later, or earlier, the judgement about an otherwise identical (and possibly CL- and CS-egalitarian) distribution should change. This problem may be partly circumvented by dividing agents’ lives into longer stages. However, the determination of the appropriate length of agents’ life stages raises a number of problems for SSE. For one, a trade-off arises between the robustness of the results (which tends to increase with the length of periods) and their relevance (since in the limit only whole lives matter). For another, as Bidadanure (2016) has forcefully noted, if the emphasis is on simultaneity then there is no obvious criterion to choose a relevant finite time span for the normatively relevant simultaneous segment and one could always advocate the use of shorter segments.14

Second, according to SSE, only inequalities between contemporaries are ethically relevant, and therefore Image 100002010000004B00000019811428661EB3FB8D.png whenever agents’ lives do not overlap. However, consider a set of agents Image 10000201000000A200000019CCC10155E5C21A88.png born at different points in time and with the following welfare profiles.

18Example 3.

Image 1000020100000224000000239BEAFD56EB65806E.png

  • 15 An anonymous referee has argued that this premise may be extreme and therefore raise doubts on the (...)

In Example 3, Image 1000020100000094000000195000B4A3A56AA59B.png , while Image 100002010000001D00000019D62EA56D3DEF75B5.png would definitely be positive. Suppose next that the only available action to reach Image 100002010000004B00000019811428661EB3FB8D.png is the construction of a nuclear plant that will explode in Image 10000201000000420000001956748989865A51B1.png yielding the following distribution.15

19Example 4.

Image 100002010000025A000000B9BE376F15E772AF82.png

20According to SSE, if future generations’ welfare is uniformly affected in each period, no other egalitarian consideration is necessary to evaluate a policy: the distribution in Example 4 is strictly preferable to that in Example 3 and raises no egalitarian objection. The nuclear plant should be built. This conclusion would be rejected by most egalitarians and it raises serious doubts on SSE as the appropriate egalitarian benchmark.

21It is important to note that Example 4 does not represent a variant of the so-called ‘levelling down objection’, which is problematic for all egalitarian principles, including CLE or CSE. It is indeed well-known that the enforcement of any egalitarian distribution may require a welfare loss. In the above scenario, what is objectionable is not that SSE leads to a lower welfare level in Example 4 than in Example 3 but rather that according to SSE, Example 4 must be considered better than Example 3 from an egalitarian viewpoint.

Examples 3 and 4 also show a more general point: the requirement of CL-equality cannot be abandoned without generating unappealing results (from an egalitarian perspective). This suggests that the analysis of intertemporal egalitarian benchmarks should focus on the choice of the most appropriate restriction on CLE. Indeed, only in the context of inequality analysis the “views are independent of each other, in the sense that each of their judgments may be in agreement or disagreement depending on the particular case in question” (Temkin, 1993, 242). If egalitarian distributions are analysed, then it is misleading to ask whether “the whole lives view [should] be rejected entirely, and replaced by some combination of the simultaneous and corresponding segments views” (Temkin, 1993, 238). Neither CSE nor a simultaneous segments restriction on CLE (discussed below) ‘replaces’ the latter: they imply it. Actually, in order to avoid changing places egalitarianism, any restriction on CLE should require all agents belonging to the same generation to have identical patterns of Image 100002010000000D00000019763540D213A2D889.png during their lives. Hence, for a given total amount of welfare Image 100002010000001400000019800EF17BD2A269EF.png equal for all agents Image 100002010000000800000019C05910B61290DC05.png , alternative restrictions will differ only in the admissible patterns of Image 100002010000000D00000019763540D213A2D889.png for agents born at different dates Image 10000201000000080000001958814A1535B9992F.png and thus belonging to different generations.

22One possibility, suggested by McKerlie (1989, 484) is to impose SS-equality in addition to CL-equality. We find this proposal unconvincing. This version of SSE (hereafter SSE2) is subject to the same time-sensitivity problem faced by the unconstrained SSE (hereafter, SSE1). Moreover, the emphasis on simultaneity as the relevant egalitarian restriction on the allocation of x along agents’ lives is not entirely compelling. SSE2 removes changing places egalitarianism between agents belonging to the same generation, but the requirement of equality in the overlapping segments of the lives of agents belonging to different generations seems less convincing. According to SSE2, the distribution in Example 3—in which agents are treated identically regardless of the generation they belong to—is definitely non-egalitarian, while the following distribution is SSE2-egalitarian.

23Example 5.

Image 100002010000025D00000096105BD1909F569933.png

In Example 5, only agents born every four periods have the same pattern of attainment of x during their lives. However, unless agents are assumed to be extremely myopic and to care only about the inequalities that they can actually observe in every Image 10000201000000080000001958814A1535B9992F.png , it is hard to see why a distribution exhibiting such a cyclical pattern—in which welfare is very low in periods 0,4,8,... and very high in periods 3,7,11,...—should be desirable from an egalitarian perspective, and indeed why it should be strictly preferable to that in Example 3. Notice that the egalitarian intuition behind SSE2 is not the same as that behind SSE1: according to SSE1, inequalities between contemporaries are worse than inequalities between removed generations—e.g., between the present generation and people living in the middle age. Instead, given the same total level of Image 100002010000000D00000019763540D213A2D889.png , the only role played by simultaneity in SSE2 is to constrain the allocation of Image 100002010000000D00000019763540D213A2D889.png during agents’ lives.

  • 16 If the duration of agents’ lives is uncertain, neither CSE nor SSE2 necessarily implies CLE ex-post(...)

Another possibility is to adopt CSE: since the distributions with Image 100002010000005600000019D954716222096E9D.png are a strict subset of those with Image 100002010000005500000019B4063AC99CE55AE9.png for all Image 100002010000004C000000191C06EA241AD40D10.png , CSE can be naturally interpreted as a restriction on CLE.16Moreover, unlike CLE and SSE2, CSE fully incorporates the egalitarian intuition that identical agents should be treated exactly in the same way, since in CS-egalitarian distributions they have an identical welfare allocation along their lives. Formally, unlike Image 100002010000001D00000019C78B3C11D7902FF8.png and Image 100002010000007B000000193CCC0F67442DAFA5.png if and only if Image 1000020100000048000000194C71A0FF44236495.png , for all agents Image 100002010000000800000019C05910B61290DC05.png , Image 100002010000000E000000191662AD734F5F12A5.png and all dates of birth Image 10000201000000080000001958814A1535B9992F.png , Image 100002010000000E00000019E9CEEF371112493D.png that is, Image 100002010000005600000019D954716222096E9D.png if and only if the vectors describing the pattern of the egalitarian variable along agents’ lives are identical. Thus, all distributions in the class with Image 100002010000005600000019D954716222096E9D.png can be simply described as follows.

24Example 6.

Image 100002010000021A0000002369077BAFA55F2A9B.png

Unlike the distributions with Image 100002010000005500000019B4063AC99CE55AE9.png or Image 100002010000004B0000001993D4444BDDC04030.png ,—as changing places egalitarianism and Example 4 respectively show,— those belonging to the class with Image 100002010000005600000019D954716222096E9D.png are equivalent from an egalitarian point of view, as the comparison of any pair of CSegalitarian distributions shows.

It is worth taking stock to summarise our argument here. Consider different welfare distributions from under a veil of ignorance, removing all normatively irrelevant information about personal identities. Equality requires that any agent randomly drawn from the economy be indistinguishable from any other in the normatively relevant (welfare) dimensions. In the static framework, every agent is simply defined by her level of Image 100002010000000D00000019763540D213A2D889.png (a scalar). In an egalitarian distribution, every agent picked up at random from the economy would indeed be indistinguishable from every other agent as they all reach the same level of Image 100002010000000D00000019763540D213A2D889.png .

In the intertemporal framework, agents are defined by a vector describing the attainment of Image 100002010000000D00000019763540D213A2D889.png in each period of their lives. In a CL- or SS-egalitarian distribution, an agent picked up randomly from the economy would not necessarily be identical to any other agent. Assuming that all of the information contained in the vector Image 10000201000000170000001958D6DA4153049510.png is normatively relevant in principle (as it is natural to assume in the intertemporal context), under either CLE or SSE equally deserving agents would not necessarily be treated identically. In contrast, at a CS-egalitarian distribution any agent randomly drawn from the economy would have exactly the same vector Image 10000201000000170000001958D6DA4153049510.png . Discarding the (morally irrelevant) information about birth dates, a CS-egalitarian distribution indeed implies identical agents.

To be sure, it is not necessarily true that Image 10000201000000950000001996EDB22470462674.png , and thus CSE allows potentially great inequalities between people living in the same period and belonging to different age cohorts. For instance, a CS-egalitarian distribution could imply that in every period there are happy young people, while the elderly live in despair. As noted by an anonymous referee, CSE permits contemporaneous inequalities across cohorts—so long as cohorts are treated equally—and one could object to such unequal treatment on the grounds that age does not provide a morally legitimate exception to access x in any particular period. Three points should be made here to suggest that this objection is not entirely compelling.

  • 17 This conclusion may be disputed if one adopts a relational perspective, as proposed by Bidadanure ( (...)
  • 18 If identity changed during an agent’s life, there might be an egalitarian objection to the distribu (...)

Firstly, however undesirable an unbalanced distribution of Image 100002010000000D00000019763540D213A2D889.png along agents’ lives may be, if Image 100002010000005600000019D954716222096E9D.png —that is, if, when they were young, the elderly were treated as the current young— there should be no egalitarian objection to it, since identical people have an identical pattern of x during their lives. From the distributive perspective adopted in this paper, and in most of the literature, CSE clearly embodies the principle of equal moral worth of individuals.17 Secondly, it is unclear that a principle of ‘no discrimination on the basis of age’ can be defended in general. There are numerous age-based provisions in modern legal systems that treat citizens differentially based on age, and they are widely considered morally sound and in line with our normative intuitions. As Daniels notes, concerning health care: “Since health needs vary with age, and the opportunity range for people does as well, a scheme that protected agerelative opportunity range at each stage of life would be treating people differently by age” (Daniels, 2008, 483). There is no reason to consider this as an unacceptable form of age bias and it is difficult to object to such a differential treatment of agents on egalitarian grounds.18

  • 19 See, for example, the approach developed by Gosseries (2014) discussed in section 3 below.

Thirdly, at least in a distributive perspective, the objection to the unequal treatment of simultaneous cohorts is likely driven by intuitions that are outside of the stock of egalitarian principles. For example, we may object to examples in which one age group lives in luxury while another leads a miserable life because we implicitly adopt a sufficientarian view applied to each segment of agents’ lives, whereby any permissible allocation should be such that all agents attain at least a minimum level of Image 100002010000000D00000019763540D213A2D889.png in each period of their lives.19 More generally, there may be many non-egalitarian objections to an unbalanced allocation of x along agents’ lives and in general, distributions with Image 100002010000005600000019D954716222096E9D.png are not equivalent all things considered. For instance, distributions with a higher overall welfare or without unbalanced welfare allocations along agents’ lives may be preferred. As shown by Example 6, if CSE is adopted, egalitarian and non-egalitarian concerns can be clearly distinguished in the evaluation of a distribution. The former reduce to the requirement Image 100002010000003E000000194ED8335F21112EB4.png , for all agents Image 100002010000000700000019EC879E71082C6BA1.png and dates of birth Image 10000201000000080000001958814A1535B9992F.png , while the latter are related to the features of Image 100002010000000F000000193D797CE1CBBB4868.png , that is, the desirable pattern of the egalitarian variable along agents’ lives. All things considered a distribution with, say, Image 10000201000000AD00000019A263A1BCEAD11014.png may be rejected because of the unbalanced welfare allocation. However, this is an argument regarding the welfare pattern along an agent’s life and not how she fares relative to others and therefore it is not an egalitarian reason to reject the distribution. A smoother welfare profile would probably be preferable but this would be justified on prudential, or utilitarian or maybe sufficientarian grounds.

2 Egalitarianism, Utility and the Maximin

25In the static context, given the relevant equalisandum, different egalitarian views can “be regarded as built around ways of measuring the inequality between lives” (McKerlie, 1989, 487), but the egalitarian state to reach is unambiguously defined. As a result, the differences between the various views in relation to other normative principles can be shown in unequal distributions, but not if one evaluates the desirability of reaching the common egalitarian state in relation, say, to utilitarian concerns. This is not true in the intertemporal context: different principles yield different trade-offs between egalitarian and non-egalitarian concerns also in egalitarian distributions. Since the evaluation of a distribution is influenced by more than one ethical concern, it is important to analyse these tradeoffs in a systematic way.

26In this section, CLE, CSE, and SSE are analysed in relation to two non-primarily-egalitarian normative principles, namely Rawls’s (1971) difference principle and utilitarianism. If, as argued in section 1, it is appropriate to impose a restriction on CLE, then it is important to analyse whether this implies a welfare loss, whether different restrictions have different effects on welfare, and what are the consequences for the worst-off generation.

The problem is modelled in a stark way. We generalise Arrow (1973) and Dasgupta (1974), in which the maximin criterion is examined in a dynamic framework. Society exists for an infinite number of generations. There is a single good that can be consumed or invested. Population is stationary and there is no technical progress. In this section, for the sake of notational simplicity, let Image 10000201000000080000001958814A1535B9992F.png denote a specific period in time, rather than the date of birth of an agent. We assume that agents’ welfare in every period can be captured by a standard utility function Image 100002010000008400000019F51FA088BDC3C4B6.png . Thus Image 10000201000000900000001956EE623DDAA23266.png , where Image 100002010000001400000019C980E0E90B384DC9.png denotes the welfare obtained by agents in the j-th age cohort at date Image 10000201000000080000001958814A1535B9992F.png and the subscript Image 100002010000000800000019C3C3A5039B2BAE96.png denoting different agents belonging to the same age cohort is dropped in order to focus on intergenerational inequalities. Further, we assume that Image 100002010000004300000019A9A0E127B941600E.png and Image 100002010000005300000019DB499BDB5C21D609.png (youth and old age), and agents have identical additively-separable utility functions:

Image 100002010000013400000023459F0150C0CA0DA9.png

  • 20 We also suppose that marginal utility becomes unboundedly large at very low levels of consumption l (...)

where Image 100002010000001400000019ACCEFB9CA0A4A356.png is consumption of the young in Image 100002010000003500000019B0AE44DDCF312792.png is the consumption of the elderly in Image 10000201000000A9000000195269F2F9384F4A3E.png is the subjective discount factor capturing agents’ time preference and the function Image 100002010000001000000019BB211BF8E8B6BA58.png is normalised such that if an agent does not consume, her utility is zero Image 100002010000005D000000194AD9374367D9DA25.png . Furthermore, we assume that Image 100002010000001000000019BB211BF8E8B6BA58.png is twice differentiable and welfare is increasing in consumption, Image 100002010000004800000019A97078F642013BDD.png Image 100002010000006100000019589F10E71E5F907E.png , but at a decreasing rate, Image 10000201000000DB00000019230BAC30A85099B5.png .20

In every period Image 1000020100000009000000194DD76D8E306ACA09.png , production possibilities can be represented by a neoclassical aggregate production function Image 100002010000005F000000195338B5A93E8B1EA0.png where Image 100002010000001B000000194E2F7F46DDEF3929.png and Image 1000020100000016000000199E451F4742213205.png are, respectively, the stock of capital and the labour supply at Image 1000020100000009000000194DD76D8E306ACA09.png . Image 100002010000001300000019C99E5A5A8164FB5E.png is continuous and production displays constant returns to scale—formally, Image 100002010000001300000019C99E5A5A8164FB5E.png is homogeneous of degree one. The labour supply Image 1000020100000016000000199E451F4742213205.png is proportional to population and normalised to one. Thus, we can define the ratio Image 1000020100000032000000190591BC0566CEBB2F.png Image 100002010000003C000000195E266117E5CB9592.png , capturing capital per capita and because Image 100002010000001300000019C99E5A5A8164FB5E.png is homogeneous of degree one we can identify the production function

  • 21 We also assume that (...)

Image 10000201000001440000001971F01E4F4C1AA54F.png which describes output per capita. We assume that the function Image 100002010000000E000000192E3E98E92A612445.png is twice differentiable and capital is necessary to produce output with Image 10000201000000580000001961E8306CE5318A20.png . Furthermore, output is increasing in capital per capita albeit a decreasing rate: formally, Image 100002010000000E000000192E3E98E92A612445.png satisfies Image 100002010000003E00000019578E4422A1B27270.png and Image 100002010000004300000019108D9F49BD893FDC.png .21

  • 22 Most notably in Daniels’ (1988; 1993; 2008) prudential lifespan account, whose basic intuitions are (...)

27It may be argued that the assumptions of identical preferences and stationary technology are unrealistic and miss some important dynamic features of capitalist economies. Yet our purpose is not to explain the determinants of capital accumulation and welfare inequalities. Ours is a normative analysis of alternative egalitarian benchmarks that aims to identify the appropriate temporal unit of egalitarian concern. For this purpose, it is appropriate to abstract from the complications arising from technical progress and, more generally, (irreversible) structural changes in the economy. Similarly, assuming all individuals to be fundamentally alike is both analytically and normatively appropriate, because it allows us to separate the issue of the appropriate unit of egalitarian concern from questions about the currency of egalitarian justice. Furthermore, this assumption is standard in the literature on intergenerational justice (see, for example, Roemer and Veneziani, 2004; 2007, and the references therein) and it is usually—albeit often implicitly—made in the debate on CLE.22

For any variable Image 100002010000000C00000019878E6E679CF3AE5C.png , let Image 100002010000006200000019B8923F1B23A6AC0E.png denote an infinite sequence of values of Image 100002010000000C00000019878E6E679CF3AE5C.png . We follow Rawls and assume that complete lives are the relevant temporal unit of normative concern for the difference principle. For, “Justice as fairness focuses on inequalities in citizens’ life prospects—their prospects over a complete life” (Rawls, 2001, 55). Therefore the maximin programme (MP) can be written as follows.

Image 100002010000013600000032F62B3DF74C8A8537.png

subject to Image 10000201000000E600000019207B9403B9F1A457.png , all Image 100002010000003700000019234E6F6DF28AE95B.png , given Image 100002010000001400000019A543EF34D6053DEA.png and Image 100002010000001200000019262CE23F1A9804FB.png.

In other words, given the initial capital stock Image 100002010000001400000019A543EF34D6053DEA.png and the level of consumption of the generation born in Image 1000020100000058000000193BE251D70011A942.png , the Rawlsian social planner should choose the lifetime consumption profile of all generations—and therefore the optimal intertemporal path of aggregate capital—in order to maximise the welfare of the worst off generation.

28Propositions 1 and 2 provide necessary conditions for a maximin solution.

Image 100002010000033B0000004B2A6DA05CB6E1C3E7.png

29In other words, a welfare distribution must satisfy CLE in order to be the maximin solution. In this sense the maximin criterion poses an efficiency restriction on CLE: the maximin solution is the CL-egalitarian distribution with the highest level of equal welfare.

Image 10000201000003310000004B333DD05BD25656AB.png

By Proposition 2, at the maximin solution, for all dates t, agents born in t must attain the highest Image 100002010000006600000019525555A6E5A49EF5.png given Image 1000020100000031000000194C36594A76A3267A.png , Image 100002010000000F000000198ECF289C7868524C.png and Image 1000020100000068000000195ED6CDDF9B469581.png . In other words, for all Image 1000020100000009000000194DD76D8E306ACA09.png , given the level of capital and consumption of all other generations, the difference principle requires that agents born in Image 1000020100000009000000194DD76D8E306ACA09.png  attain the highest possible lifetime welfare. This suggests that the condition in Proposition 2 can be derived as the first order condition of a constrained optimisation problem, defining the optimal consumption allocation along an agent’s life. Given the assumptions on Image 100002010000001000000019BB211BF8E8B6BA58.png and Image 100002010000000E000000192E3E98E92A612445.png , Proposition 2 implies that the maximin solution is unique, while none of the egalitarian criteria identifies per se a particular welfare distribution. However, the main implication of Proposition 2 for our analysis is that in general Image 100002010000008F00000019232D7A601FB67094.png : at a given period, the young and the old will attain different welfare levels, so that the maximin allocation will not be SS-egalitarian.

Let Image 100002010000006F00000019FC23961E1040F699.png denote the total consumption of the young and the elderly at time Image 1000020100000009000000194DD76D8E306ACA09.png . Let Image 1000020100000046000000191B937E4D90ADF496.png , for all Image 1000020100000009000000194DD76D8E306ACA09.png , denote a generic constant aggregate consumption level and let Image 1000020100000092000000191B2207C233F8A237.png is the aggregate level of consumption that can be attained in a given period if (per capita) capital is Image 100002010000001400000019A543EF34D6053DEA.png and there is no accumulation (or decumulation).

Lemma 1. Image 100002010000001A0000001909AFD6BDEE32B3D0.png is the maximum sustainable aggregate constant consumption.

  • 23 Alternatively, the benchmark path could be the solution to the following problem:

Lemma 1 provides a natural benchmark for the maximin path. Let Image 10000201000000560000001971613E8D77F62B9B.png and Image 10000201000000A000000019EABBE86EC6D81224.png . By Lemma 1, it is always feasible to choose Image 100002010000005E0000001955CFCECE12C8CC3B.png and Image 100002010000005000000019BEA04E56117A87F8.png , all Image 1000020100000009000000194DD76D8E306ACA09.png , and thus no intertemporal path in which any generation attains a welfare level lower than Image 10000201000000AC000000195BC7D9B19549AA35.png can be the maximin path. Formally, for any given Image 100002010000001200000019307C2AF37C7A4A59.png , no distribution in which Image 1000020100000102000000194D189D644FA7EBEA.png , for some Image 1000020100000009000000194DD76D8E306ACA09.png , can solve (MP).23

Hence, let Image 10000201000000BA000000193ED59EEE24D71375.png denote the resources available to the generation born in Image 1000020100000009000000194DD76D8E306ACA09.png , given the capital they have inherited, Image 100002010000001400000019C7D08C16B28BABB2.png , and the consumption of the elderly of the previous generation, Image 100002010000001200000019098A3FD3BA98B23D.png . If Image 1000020100000063000000194C6DCDF5CDD5D05A.png then all generations from Image 1000020100000009000000194DD76D8E306ACA09.png  onwards can reach at least Image 100002010000002A000000195F33665FFC42D893.png . Consider the following sequence of maximisation programmes Image 1000020100000027000000190F5371EB07F8BB10.png .

Image 1000020100000254000000466EC83CFB12E2D72D.png

subject to

Image 1000020100000242000000D2F98DFCD9283DBEA7.png

The programmes Image 10000201000000160000001900AE712EBAE1B4FD.png allow us to transform the infinite-horizon programme (MP) into an infinite sequence of one-generation programmes in which the welfare of each generation Image 1000020100000009000000194DD76D8E306ACA09.png is maximised given a certain path of resources inherited and bequeathed Image 100002010000004C000000196C397DF0A20E0FED.png . The important choice in the solution of the original maximin programme lies precisely in the choice of the sequence Image 100002010000004100000019AC8E388D0520B08A.png.

Let Image 100002010000005A00000019C0A1EDD18DBE38D6.png be the solution of any given Image 10000201000000160000001900AE712EBAE1B4FD.png with Image 100002010000008500000019B3ADC4AB3A5438AE.png , where in general Image 10000201000000580000001903FE551B3AD6546E.png . Let Image 100002010000006E000000194B94A484BEF84C92.png denote the maximum function associated with Image 100002010000009000000019C3868C89FF3CD4FD.png denotes the maximum level of welfare that generation Image 1000020100000009000000194DD76D8E306ACA09.png can attain, given the resources Image 100002010000004C000000191ADEC93DA5F49D35.png that it inherited and that it has to bequeath, respectively. Let

Image 100002010000010600000019A27AF2E0B3491067.png Image 100002010000002100000019531A984EDABE2E8E.png  denotes the maximum welfare level that can be attained by a generation at a stationary path of resources with Image 1000020100000055000000199FA093CA5DAA94FC.png all Image 1000020100000009000000194DD76D8E306ACA09.png . The main theorem can now be proved.

Image 10000201000002CB0000004B56D4322E4CD86604.png

  • 24 With a finite horizon this is not necessarily true. However, the adoption of the infinite horizon h (...)
  • 25 An anonymous referee has suggested that this result immediately follows from the assumption of equa (...)

Theorem 1 states that although the maximin principle and CSE represent different restrictions on CLE, they coincide in the economy described, since at the solution to MP, all agents have the same consumption Image 10000201000000390000001933A21A76560031DF.png —and welfare Image 100002010000007000000019B139E6948C8691BB.png —allocation during their lives.24Thus, if the egalitarian social planner also adopts an intergenerational maximin criterion, Theorem 1 proves that the two objectives are not in contradiction if CLE or CSE are adopted, while if SSE1 (or SSE2) is chosen, a trade-off between the two concerns arises.25

  • 26 However, while Theorem 1 is more sensitive to changes in the assumptions, heterogeneous, non additi (...)

Moreover, since the maximin solution coincides with the allocation that maximises agents’ utility under a CLE constraint, the model allows us to introduce some utilitarian concern in the analysis. Consider, for instance, classical (average or total) utilitarianism. By Proposition 2, it is more difficult to reconcile a utilitarian concern with SSE2 than with CLE or CSE, since SSE2 does not allow for a constrained welfare-maximising allocation along agents’ lives. Instead, if SSE1 is adopted, in principle it is possible for infinitely many generations to reach a higher welfare level than at the maximin, with only a finite number of generations falling below it in order to start capital accumulation. Thus, due to the infinite gain in utility, a utilitarian would prefer the latter distribution to the maximin/CSE solution. In general, such a distribution might be appealing (as opposed to CSE or CLE distributions) not because it is SSegalitarian but because some CS, or even CL inequalities can be outweighed by an infinite gain in utility, all things considered. In this sense, SSE1 is the only intertemporal egalitarian principle compatible with sustained welfare growth and thus the principle that can best accommodate utilitarian concerns (although SS- equality could still imply some welfare loss with respect to unconstrained utility maximisation). However, this result derives from the exclusive focus of SSE1 on intratemporal inequalities, and thus it should not be seen as a solution to the equality/growth dilemma, but rather as a way of bypassing it. The model presented is highly stylised and some caution is necessary in interpreting the results. While the analysis of SSE1 does not depend on any particular assumptions, in more general settings, CLE and CSE will not be equivalent as concerns their relations with other normative principles and the maximin solution will be neither CS- nor CL-egalitarian.26 However, despite its simplified structure, the model does capture in vitro some inherent features of the egalitarian views. As concerns utilitarianism, since CSE and SSE2 distributions are strict subsets of those with Image 100002010000004B00000019940C9A6C6FB96823.png , the CLE lifetime welfare level will always be at least as high as the SSE1 and CSE levels. Moreover, from Proposition 2, it is legitimate to infer that even in more general settings CLE lifetime welfare would be at least as high as the SSE2 welfare, since SSE2 does not allow agents to allocate consumption optimally along their lives. Similarly, as regards Rawlsian concerns, the above results suggest that in a more general setting, if the maximin solution was not egalitarian, the CLE lifetime welfare level would be at least as close to it as the CSE level, and the latter in turn would be at least as close to the maximin as the SSE2 level.

3 Relation with the Literature

30Our defence of CSE echoes the elaborations of a small but growing literature on equality through time, including Gosseries (2014), Bidadanure (2016), and Daniels (1988, 1993, 2008). In this section, we briefly discuss their proposals and compare them to our approach.

31Like us, these authors argue that CLE is the essential building block of any intertemporal egalitarian approach, even though it should be supplemented with additional restrictions to deal with ‘changing places egalitarianism’. Gosseries (2014) has proposed a sufficiency restriction on CLE, according to which at every point in time along the life of a person, each should have enough to cover at least their basic needs. For him, if people are equal over their complete lives but fall below a given welfare threshold at some point in their lives, the demands of temporal justice are not met. This approach likely captures some of our intuitions in rejecting extreme simultaneous inequalities among agents, but it does not properly address the issues raised by changing places egalitarianism in scenarios with less extreme, but still morally relevant inequalities. This is not surprising because, unlike our corresponding segments restriction, sufficientarianism is not meant to capture egalitarian intuitions. Indeed, sufficientarianism has explicitly been proposed as an alternative to egalitarianism and embodies the intuition that “equality is not, as such, of particular moral importance” (Frankfurt , 1987, 21).

  • 27 See, for example, Anderson (1999).

32According to Bidadanure (2016), the strong dystopian feeling created by extreme examples of changing places egalitarianism cannot really be captured in a standard welfarist, distributive framework. She argues against SSE and claims that by moving away from the paradigm of distributive egalitarianism we can find a non-arbitrary complement to CLE focusing on a relational perspective, according to which people should be treated as equals.27 She therefore endorses a relational egalitarian complement to CLE which limits the scope of acceptable synchronic inequalities.

33The relational approach proposed by Bidadanure is interesting and innovative, and it may be an essential component of a complex, multifaceted approach to egalitarianism. As she aptly notes, distributive and relational approaches complement each other as they “simply appeal to different kinds of reasons to care about inequalities” (Bidadanure, 2016, 238). Nonetheless, an emphasis on the relational dimension of egalitarianism does not provide a complete answer to the equally important distributive questions. Indeed, her approach leaves the central questions raised in the literature on the distributive dimension of temporal inequalities largely unanswered. While acknowledging the relevance of relational considerations for egalitarianism, our paper focuses precisely on the appropriate distributive benchmark.

34In a series of seminal contributions, Daniels (1988, 1993, 2008) has argued that both lifetimes and temporal stages of lives should be taken into account within a Prudential Lifespan Approach. According to him, complete lives remain the key unit of egalitarian concern but CLE should be constrained by a prudential procedure to allocate resources between young and old. To be precise, in the Prudential Lifespan Approach, the best way of synchronically distributing resources among people of different ages should be identified by thinking prudentially about a diachronic distribution across the different temporal stages of a single complete lifetime. As a result, there may be significant simultaneous inequalities, but this is “not by itself a form of age-bias. This differential treatment would not be morally objectionable, ..., if it made each life go as well as possible (it was a ‘prudent’ allocation) and if all people were treated similarly over their whole lives” (Daniels, 2008, 483).

  • 28 For a thorough critique, see McKerlie (2012).

35Our results in section 2 are consistent with Daniels’ (1988; 1993; 2008) account of intertemporal justice: under the assumptions of our model, the prudential lifetime allocation will coincide with the maximin solution. More generally, if agents are fundamentally alike, and the economic environment does not undergo major structural changes, then the prudential lifespan approach will yield a CS-egalitarian allocation. Nonetheless, some important differences should be noted. First, Daniels’ can be considered as a mixed account, which imposes a prudential —and thereby non-egalitarian—constraint on CLE. In contrast, in our account, CSE emerges as the appropriate intertemporal benchmark, based on purely egalitarian considerations. Maximin, or prudential considerations are clearly distinguished, and theoretically subsidiary in our analysis, and therefore— unlike in Daniels’ (1988; 1993; 2008) account—the defence of CSE is independent on the specific formulation of maximin, or prudential accounts.28 In fact, formally, under our assumptions, while CSE identifies a set of egalitarian allocations, the prudential lifespan account picks up one (or a strict subset) of such allocations. Second, in our model, we do not assume that the agents, or the social planner, actually act prudentially subject to a CLE constraint: the fact that the allocation in Theorem 1 corresponds with the allocation advocated by the prudential lifetime account is a result of a more basic—and normatively well founded, at least within an egalitarian approach— principle, namely Rawls’s maximin.

36In addition to the specific points mentioned above, it is worth briefly mentioning two more general methodological features that differentiate our paper from the literature reviewed here. First, we clearly distinguish the identification of the appropriate egalitarian benchmark from the choice of a suitable inequality measure. As we have argued in Section 1, this distinction is important and it is not always properly spelled out. Second, we clearly distinguish egalitarian and non-egalitarian concerns. Our defence of CSE lies entirely on egalitarian principles and intuitions, and although we do bring non-egalitarian considerations to bear in Section 2, they are only meant to provide additional support for CSE.

4 Conclusion

37In this paper three egalitarian views are analysed in the intertemporal context. Once the static setting is abandoned, egalitarian principles—apart from differing in the analysis of existing inequalities,—also define different ideal egalitarian distributions. While it may be important to use the different information conveyed by every criterion in the analysis of existing inequalities, when the egalitarian distributions associated with them are analysed, CLE and SSE have undesirable features while CSE represents the appropriate egalitarian benchmark.

38The relations between the three egalitarian principles and other moral ideals, namely maximin and utilitarianism, are also analysed. As regards the maximin principle, Propositions 1-2 and Theorem 1 show that, unlike with CLE and CSE, the adoption of SSE implies a trade off between egalitarianism and a concern for the worst off. As regards utility, the same conclusion holds if one interprets SSE as a restriction on CLE, since it yields a lower egalitarian lifetime welfare level. This is not true if SSE is analysed per se, but this is just because in this case the SSE is a strictly intratemporal principle.

39In closing this paper, it is worth noting that our formal analysis yields some interesting insights on a vexed issue in normative economics, namely the well-known trade-off between equality and growth. Arrow (1973) and Dasgupta (1974) proved that if agents are selfish, live for one period, and their lives do not overlap, then Rawls’ maximin principle implies a stationary path of consumption, capital and welfare. Our model yields a more nuanced conclusion and suggests some interesting directions for further research. In our framework with overlapping generations, at the maximin solution the path of capital must be chosen so as to maximise lifetime welfare and, under quite general assumptions, this implies growth in at least one period. In other words, although the application of the maximin principle precludes permanent growth in the economy, as in Arrow (1973) and Dasgupta (1974), the dynamics of the economy is not completely stationary.

  • 29 A similar point is made by Silvestre (2002), albeit in a rather different formal setting.

40Our conjecture is that an explicit and more realistic analysis of the temporal structure of agents’ lives (which span over many periods), and the overlaps across generations, together with the introduction of uncertainty and irreversibility of investments may alter the justice/growth trade-off.29 This indicates a promising line for further research on intertemporal and intergenerational justice.

We are grateful to Meghnad Desai, Ben Fine, John Lane, Marco Mariotti, Wlodek Rabinowicz, Max Steuer, Robert Sugden, Alessandro Vercelli, three anonymous referees, and audiences at the LSE, the University of Siena, Warwick University, the Free University Bolzano, and the University of Milan for comments and suggestions. Special thanks go to Larry Temkin for extensive and detailed comments on an earlier draft. This paper substantially extends and improves on a previous paper titled "Intertemporal Egalitarian Principles", and it supersedes it. The usual disclaimer applies.

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For any variable Image 100002010000000C00000019352974AB7F3C55B7.png , let Image 1000020100000077000000194FE8F394BDD83F7E.png denote a change in Image 100002010000000C00000019352974AB7F3C55B7.png .

Proof of Proposition 1.

Let Image 1000020100000024000000192E5F5A8DC27B2672.png be the value of MP and suppose that, contrary to the statement, Image 100002010000008900000019744C74097EA41424.png . By continuity, there is a sufficiently small Image 1000020100000051000000197306D26C70C04E40.png , such that Image 100002010000011700000019C8F0715D9CBCBED5.png and the amount of resources available in Image 100002010000004000000019D23D537BD7BD60E8.png increases by Image 100002010000002B00000019F3D4A4AA2A2BACA5.png Image 100002010000005A00000019614150A24B031C00.png . Let Image 10000201000000B800000019C860451E6F90618D.png and Image 100002010000005F0000001934E6F90D27B308AC.png and repeat the procedure for all Image 100002010000003900000019269C25D6862F189F.png so that Image 10000201000000B9000000197DFEEA2E8232882B.png , Image 100002010000004A00000019598D86B539096E52.png Image 100002010000004700000019E17069C9CAB85F5F.png and Image 100002010000009C000000197E4A7FAE799A9BB9.png , all Image 100002010000000B000000190A0638DDB5D5C4ED.png , a contradiction. The proof of the case with Image 1000020100000097000000193183B2042A217CB4.png , some Image 100002010000003600000019540ECE6F40F65C82.png , is similar.

Proof of Proposition 2.

Suppose not. Then there is Image 100002010000005100000019D919AD0056C1E4AE.png such that Image 100002010000007A000000198824CA62D102CFC0.png Image 100002010000006800000019B37562B533C8525F.png and Image 100002010000010E00000019BFB2203DB9D4A5A8.png By the concavity of Image 100002010000001A0000001938B09CBB010A4822.png , this implies Image 10000201000000EE00000019F94D745B4B2877C0.png leaving unmodified Image 100002010000001100000019FD72D02441F09EF1.png , all Image 100002010000003A00000019CA0368A47BDD6DC9.png and Image 100002010000002F00000019541883A778A08658.png , all Image 100002010000006400000019A2B9943D39D40EB3.png , violating Proposition 1.

Proof of Lemma 1.

Consider Image 100002010000004600000019BF69897DA09DD27D.png . At Image 1000020100000035000000197044F3FF9DB92D21.png , Image 100002010000004E00000019FF4F48DFBB2CF285.png and thus Image 10000201000000BE00000019352A81259ABD99E7.png Image 10000201000000A300000019A3D35532BFE35820.png and Image 10000201000000A6000000199C41CC3FBDABA8A8.png , i.e. Image 10000201000000C700000019D9EBDEA45E2129FD.png , and, by induction Image 10000201000000F80000001910B89D0620229443.png . Therefore Image 1000020100000042000000197F1F06AB5FB5B776.png for Image 100002010000000B000000190A0638DDB5D5C4ED.png finite, and Image 100002010000000C000000194B0BA9927BA37D27.png is not sustainable.

Proof of Theorem 1.

  1. The existence and uniqueness of Image 100002010000005F0000001968AFFA7018CBDD18.png is guaranteed by the assumptions on Image 100002010000000E0000001905BF3D3D00D26FF4.png and Image 100002010000000E000000190020BED41528FB25.png . Note also that Image 100002010000005F0000001968AFFA7018CBDD18.png satisfies the condition in Propositions 1 and 2.

  2. Suppose it is possible to raise the welfare of all generations above Image 10000201000000240000001966B2865F27682E19.png . Consider Image 10000201000000200000001987FA3EC540E21AC4.png  by construction the first generation’s welfare can increase over Image 10000201000000240000001966B2865F27682E19.png if and only if Image 100002010000005200000019055233A81221F64B.png . Consider now generation 2: clearly Image 100002010000007000000019C7BDD50E56FCF23B.png Image 10000201000000240000001966B2865F27682E19.png . Moreover Image 100002010000006C00000019F210D0BD3DDE1EE9.png is concave and its iso- welfare contours have slope Image 10000201000000AA0000001917D44C248BDF68AA.png , where Image 100002010000006A0000001924E841AF9B898574.png is the optimum value of Image 1000020100000027000000198034D28AAB9A1846.png  from Image 100002010000001700000019879ECB0627E13EC9.png . Hence, Image 1000020100000067000000198F65008B20E30F29.png Image 1000020100000024000000199B02A565C9CADC17.png implies Image 100002010000005000000019ECA569D4CE610748.png , with Image 100002010000013D0000001903E6EB1BB7A54FC6.png Image 100002010000001E000000193D053DC427847C19.png . Iterating the argument, Image 10000201000000A2000000197DECCA49A34D66BE.png implies Image 100002010000019600000019DE1703C23845B9EA.png , all Image 100002010000000B000000190A0638DDB5D5C4ED.png , and the path violates the non- negativity of Image 10000201000000170000001945B1D509BD163C23.pngat some finite Image 100002010000000B000000190A0638DDB5D5C4ED.png .

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1 For a discussion see Mariotti et al. (2012) and Lombardi et al. (2016).

2 See, for example, the classic papers by Arrow (1973) and Dasgupta (1974). For a more recent discussion see Roemer and Veneziani (2004, 2007).

3 See, for example, Temkin (1992, 1993); Kappel(1997); McKerlie (2001a,b, 2012); Huemer (2003); Carter (2006); Attas (2008); and Bidadanure (2015).

4 A similar point is made by Attas (2008), who argues that the temporal structure of agents’ lives creates some fundamental indeterminacies for egalitarian principles in the Rawlsian tradition which makes them virtually empty.

5 To be sure, CLE, CSE and SSE do not exhaust the set of logically conceivable intertemporal egalitarian views. Yet they do capture the most common egalitarian considered judgements. Lippert-Rasmussen (2003), for example, considers also ranking-order segments egalitarianism according to which egalitarian evaluations should be based on the comparison of equivalent rank-order segments of individual lives—that is, the best segment, the second-best segment, and so on—in terms of the attainment of the relevant variable. However, the ethical foundations of this approach are rather unclear.

6 We focus in particular on the arguments advanced by Temkin (1992, 1993) and Kappel (1997). For a thorough critical analysis of Huemer’s (2003) argument, see Carter (2006).

7 For a discussion of the relation between the maximin and egalitarianism, see Temkin (1993).

8 In all examples in this section, we suppose that agents’ lives are divided into four stages, so that T = 4. This is only for concreteness and yields no loss of generality.

9 See, for example, Sen (1980); Dworkin (1981a,b); Arneson (1989); Cohen (1989) and Roemer (1998).

10 We are grateful to Robert Sugden for this suggestion.

11 See, e.g., the measures discussed in Sen (1973, 1992) and in Temkin (1993, chapter 5).

12 The distinction is also ignored in Lippert-Rasmussen’s (2003) otherwise insightful analysis.

13 Consider, for example, an economy with N identical agents with the same, strictly increasing, welfare profile over their lives. If agents are born on the same date t0 then D3 = 0. If, however, the date of birth of only one of them is slightly shifted forward, or backward, then D3 becomes strictly positive and potentially quite large, especially if N is large.

14 Indeed, Bidadanure (2016) has highlighted a conceptually analogous problem of D3, which arises from its sensitivity to small changes in the temporal partition of agents’ lives. Thus, for example, it is easy to design examples in which “depending on whether we register simultaneous inequalities every 20 years or every 10 years, we will draw substantially different conclusions” (Bidadanure, 2016, 244).

15 An anonymous referee has argued that this premise may be extreme and therefore raise doubts on the relevance of Example 4. This conclusion is unwarranted: Example 4 is meant to illustrate some shortcomings of SSE and a stark scenario makes them particularly vivid. Yet the problems of SSE are structural and the same conclusions can be reached under less extreme assumptions.

16 If the duration of agents’ lives is uncertain, neither CSE nor SSE2 necessarily implies CLE ex-post, but the above arguments still hold ex-ante, if applied to expected welfare.

17 This conclusion may be disputed if one adopts a relational perspective, as proposed by Bidadanure (2015, 2016). We discuss relational egalitarianism in section 3 below.

18 If identity changed during an agent’s life, there might be an egalitarian objection to the distribution. However a similar critique can be moved to any intertemporal egalitarian principle, since it amounts to saying that the principle is analysed in the wrong context. Once the agents’ identity is correctly specified, all the arguments in this paper remain valid.

19 See, for example, the approach developed by Gosseries (2014) discussed in section 3 below.

20 We also suppose that marginal utility becomes unboundedly large at very low levels of consumption lim Image 100002010000008D00000019F6ED30CF1886946C.png .

21 We also assume that  satisfies the so-called Inada conditions.

22 Most notably in Daniels’ (1988; 1993; 2008) prudential lifespan account, whose basic intuitions are based on the idea that individuals are essentially alike.

23 Alternatively, the benchmark path could be the solution to the following problem:
Image 10000201000001AB000000284890FBEAAC2F68EA.png
In this case, the assumption of a given Image 1000020100000010000000147F49B39E5F6C55D9.png would be dropped and the constraint Image 100002010000003300000014F670B963FE29F396.png would be necessary to guarantee equal treatment of the generation born in = −1. This choice would include generation = −1 in the definition of the just path, allowing for an explicit treatment of the transition to justice, instead of taking its past consumption choices as given. However, the main results of this paper would not change.

24 With a finite horizon this is not necessarily true. However, the adoption of the infinite horizon hypothesis is implied by the very nature of the problem, as there is no reason to restrict the analysis of a normative principle to an arbitrary, finite number of generations.

25 An anonymous referee has suggested that this result immediately follows from the assumption of equal utility functions and stationary technology, in which case “the CSE restriction is redundant, since optimizing actors will choose the desired intertemporal allocation given only the CLE condition that each generation receives k”. Three points should be noted about Theorem 1 that cast doubt on these conclusions. First, as is well known in the literature on Rawls’s just savings principle, the assumptions of identical utility functions and stationary technology do not necessarily imply a stationary allocation of consumption, or even welfare (see, for example, Dasgupta, 1974). Second, there is no CSE restriction in the model: that the maximin solution is CS-egalitarian is a result of our analysis. Third, it is misleading to focus on optimising agents in the context of a social planner problem. For it is the solution of the maximin problem—and therefore Theorem 1—that identifies k. CLE alone is not sufficient to pin down any specific level of k.

26 However, while Theorem 1 is more sensitive to changes in the assumptions, heterogeneous, non additive or non concave preferences, technical progress or more general production functions would leave Proposition 1 basically unchanged.

27 See, for example, Anderson (1999).

28 For a thorough critique, see McKerlie (2012).

29 A similar point is made by Silvestre (2002), albeit in a rather different formal setting.

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Giorgos Galanis et Roberto Veneziani, « Equality of when? »Œconomia [En ligne], 7-1 | 2017, mis en ligne le 31 mars 2017, consulté le 08 février 2023. URL : ; DOI :

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Giorgos Galanis

Department of Economics, University of Warwick, Coventry, CV4 7AL, U.K.; and Institute of Management Studies, Gold-smiths, University of London, New Cross, London SE14 6NW, U.K.

Roberto Veneziani

Roberto Veneziani (Corresponding author) School of Economics and Finance, Queen Mary University of London, Mile End Road, London E1 4NS, U.K.

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