Future generations: A prioritarian view

George Washington Law Review

Volumn 77, Issue 5-6, 2009, Pages 1478-1520

  Adler, Matthew D ^a  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

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EID: 70449115985     PISSN: 00168076     EISSN: None     Source Type: Journal    
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References (211)

1. See MATTHEW D. ADLER, WELL-BEING AND EQUITY: A FRAMEWORK FOR POLICY ANALYSIS (forthcoming 2010) [hereinafter, ADLER, WELL-BEING AND EQUITY];
2. Matthew D. Adler & Chris William Sanchirico, Inequality and Uncertainty: Theory and Legal Applications, 155 U. PA. L. REV. 279 (2006);
3. Matthew D. Adler, Risk Equity: A New Proposal, 32 HARV. ENVTL. L. REV. 1 (2008);
4. Matthew D. Adler, Well-Being, Inequality and Time: The Time-Slice Problem and Its Policy Implications (Univ. of Pa. Law Sch. Inst, for Law & Econ., Working Paper No. 07-17,2007), available at http://ssrn.com/abstractsl006871.
5. This is the appropriate formula for what I call the "core case," where the very same N individuals exist in all outcomes. Variations in the formula for other cases are considered below. See infra Part III.
6. Formally, a SWF is sensitive to equity if it satisfies the "Pigou-Dalton" principle. Pri-oritarian SWFs are a subset of the SWFs that satisfy this principle. See sources cited supra note 1.
7. By "possible outcomes," I mean all the outcomes in the outcome set, O, which is relevant to the choice at hand. See infra text accompanying notes 13-15.
8. "Iff' means "if and only if."
9. If individual j exists in outcome x but not y, and has utility ut(x) in x, the effect of the above formula is to assign i a utility level of zero in y, and to take full account of the difference between u.(x) and zero in ranking the two outcomes, as much as if i existed in outcome y.
10. For reviews of this literature, see Walter Bossert & John A. Weymark, Utility in Social Choice, in 2 HANDBOOK OF UTILITY THEORY 1099 (Salvador Barbera et al. eds., 2004);
11. and Claude d'Aspremont & Louis Gevers, Social Welfare Functional and Interpersonal Comparability, in 1 HANDBOOK OF SOCIAL CHOICE AND WELFARE 459 (Kenneth J. Arrow et al. eds., 2002).
12. For reviews, see MATTI TUOMALA, OPTIMAL INCOME TAX AND REDISTRIBUTION (1990);
13. Christopher Heady, Optimal Taxation as a Guide to Tax Policy, in THE ECONOMICS OF TAX POLICY 23 (Michael P. Devereux ed., 1996);
14. Nicholas Stern, The Theory of Optimal Commodity and Income Taxation: An Introduction, in THE THEORY OF TAXATION FOR DEVELOPING COUNTRIES 22 (David Newbery & Nicholas Stern eds., 1987).
15. See Ian W.H. Parry et al., The Incidence of Pollution Control Policies 26-28 (Nat'l Bureau of Econ. Research, Working Paper No. 11438,2005).
16. NICHOLAS STERN, THE ECONOMICS OF CLIMATE CHANGE (2007).
17. WILLIAM NORDHAUS, A QUESTION OF BALANCE (2008).
18. See NORDHAUS, supra note 11, at 165-91;
19. David Weisbach & Cass R. Sunstein, Climate Change and Discounting the Future: A Guide for the Perplexed (Reg-Markets Ctr., Working Paper No. 08-19,2008), available at http://ssrn.com/abstract=1223448.
20. See sources cited supra note 1. The articles cited there are agnostic as between priori-tarianism and other types of equity-regarding SWFs, but ray forthcoming book defends priori-tarianism.
21. See id.
22. See ADLER, WELL-BEING AND EQUITY, supra note 1.
23. Theorists who are concerned about equalizing opportunity for well-being, rather than well-being itself, may see prioritarianism as a rough approximation of the correct moral theory. Hybrid theorists who believe that morality requires the maximization of good consequences, within deontological constraints, may see prioritarianism as one component of a broader moral theory-as may theorists who believe that morality requires attention both to human well-being, and to other considerations, e.g., animal well-being or intrinsic environmental values. See id.
24. See JOHN C. HARSANYI, RATIONAL BEHAVIOR AND BARGAINING EQUILIBRIUM IN GAMES AND SOCIAL SITUATIONS 48-83 (1977);
25. John A. Weymark, A Reconsideration of the Harsanyi-Sen Debate on Utilitarianism, in INTERPERSONAL COMPARISONS OF WELL-BEING 255,289-97 (Jon Elster & John E. Roemer eds., 1991).
26. See ADLER, WELL-BEING AND EQUITY, supra note 1;
27. See also MATTHEW D. ADLER & ERIC A. POSNER, NEW FOUNDATIONS OF COST-BENEFIT ANALYSIS 47-50 (2006).
28. This means that v(.) produces the very same ranking of life histories, lotteries over life histories, and comparisons of each life history to nonexistence as u(.) iff v(.) = ku(.), where A: is a positive constant.
29. "At least as good as" means either better than or equally good as.
30. See Karsten Klint Jensen, What Is the Difference between (Moderate) Egalitarianism and Prioritarianism?, 19 ECON. & PHIL. 89,99 (2003);
31. John Broome, Equality Versus Priority: A Useful Distinction, in FAIRNESS AND GOODNESS IN HEALTH (Daniel Wikler et al. eds., forthcoming) (manuscript at 2, on file with author);
32. Nils Holtug, Prioritarianism, in EGALITARIANISM: NEW ESSAYS ON THE NATURE AND VALUE OF EQUALITY 125,133 (Nils Holtug & Kasper Lippert-Rasmussen eds., 2007);
33. Wlodek Rabinowicz, Prioritarianism and Uncertainty: On the Interpersonal Addition Theorem and the Priority View, in EXPLORING PRACTICAL PHILOSOPHY: FROM ACTION TO VALUES 139,148 (Dan Egonsson et al. eds., 2001);
34. Campbell Brown, Matters of Priority 99-105 (Mar. 2005) (unpublished Ph.D. thesis, The Australian National University) (on file with author).
35. "The value of the SWF" means complex math equations
36. See Brown, supra note 21 at 117-18.
37. See THOMAS NAGEL, EQUALITY AND PARTIALITY 63-74 (1991) (presenting a conception of equality that sees each individual as having a distinct claim to well-being, and that gives priority to worse-off individuals);
38. THOMAS NAGEL, MORTAL QUESTIONS 106-27 (1979) (same).
39. Of course, the individual is at different levels in the two outcomes, so to be more precise, we should say that the strength of the claim is a decreasing function of the average well-being level.
40. That is, ul(x)-ul(y) = ui(y)-uj(x).
41. That is, u(x) > ut(y) > ufy) > ut(x).
42. Louis Kaplow and Steven Shavell use the term "fairness" to mean non-welfarism."
43. See Louis KAPLOW & STEVEN SHAVELL, FAIRNESS VERSUS WELFARE 38-45 (2002). However, if one views fairness (as I do) as a matter of the equitable satisfaction of individuals' claims to well-being, there is no inconsistency between fairness and welfarism.
44. In the case of infinite populations, we need to distinguish between different variants of the anonymity axiom. See infra note 84 and accompanying text. But in the finite case, anonymity is straightforward: utility vectors with the same, finite number of entries that are permutations of each other must be ranked as equally good.
45. Leximin says to rank two outcomes equally if the utility vectors corresponding to the two are permutations of each other. Otherwise, x is better than y if the worst-off individual in x is better off than the worst-off individual in y; if their well-being levels are equal, then x is better than y if the 2nd-worst-off individual in x is better off than the 2nd-worst-off individual in y; and if their well-being levels are equal, then x is better than y if the 3rd-worst-off individual in x is better off than the 3rd-worst-off individual in y; and so forth.
46. These are the only prioritarian SWFs which are invariant to multiplication of individual utilities by a common positive constant. More precisely, Atkinsonian SWFs are thus invariant if all utilities are nonnegative. No prioritarian SWF is invariant to multiplication of utilities by a common positive constant once negative individual utilities are allowed into the domain of utility vectors. And, quite apart from the invariance issue, the Atkinsonian SWF is not suitable for use with negative utilities because-with negative utilities-the function o(u.(x)) = ()u*~r ' i-r ' is either undefined or, if defined, not both strictly increasing and strictly concave.
47. See Adler, Risk Equity, supra note 1, at 41 n.122.
48. See, e.g., Peter J. Hammond, Utilitarianism, Uncertainty and Information, in UTILITARIANISM AND BEYOND 85,90-96 (Amartya Sen & Bernard Williams eds., 1994);
49. Rabi-nowicz, supra note 21;
50. Marc Fleurbaey, Assessing Risky Social Situations (Society for Social Choice and Welfare, Working Paper, Feb. 2008), http://www.accessecon.com/pubs/ SCW2008/SCW2008-08-00028S.pdf.
51. See, e.g., Elchanan Ben-Porath et al., On the Measurement of Inequality under Uncertainty, 75 J. ECON. THEORY 194 (1997);
52. Peter A. Diamond, Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility: Comment, 75 J. POL. ECON. 765 (1967);
53. Larry G. Epstein & Uzi Segal, Quadratic Social Welfare Functions, 100 J. POL. ECON. 691 (1992).
54. The dispute about the appropriate application of a social welfare function under conditions of uncertainty is fully reviewed in ADLER, WELL-BEING AND EQUITY, supra note 1.
55. See, e.g., CHARLES BLACKORBY ET AL., POPULATION ISSUES IN SOCIAL CHOICE THEORY, WELFARE ECONOMICS, AND ETHICS 253-71 (2005);
56. John Creedy & Ross Guest, Discounting and the Time Preference Rate, 84 ECON. REC 109 (2008);
57. Geoffrey Heal, Discounting: A Review of the Basic Economics, 74 U. CHI. L. REV. 59 (2007).
58. There is a vast literature about discounting.
59. See, e.g., Weisbach & Sunstein, supra note 12, at 3 n.7.
60. For a discussion of discounting and cost-benefit analysis,
61. See ADLER & POSNER, supra note 18, at 173-77. One argument for discounting in the context of cost-benefit analysis-that it is a way to compensate for the fact that future generations will be richer and have a lower marginal utility of consumption-is unavailing in the SWF context, because the marginal utility of consumption is already reflected in the utility function u(x). See infra text accompanying notes 46-48.
62. Ricky Revesz draws a similar distinction. See Richard L. Revesz, Environmental Regulation, Cost-Benefit Analysis, and the Discounting of Human Lives, 99 COLUM. L. REV. 941, 948 (1999) (arguing that "discounting raises analytically distinct issues in the cases of latent harms and harms to future generations").
63. As already explained, I am assuming the completeness and convergence of extended preferences in this Article to simplify the analysis.
64. See supra text accompanying note 18.
65. This could be a vector of consumptions of different goods or a single consumption amount.
66. See ADLER, WELL-BEING AND EQUITY, supra note l.
67. See DEREK PARFIT, REASONS AND PERSONS 117-95 (1984).
68. For a full discussion of formulas for social choice that take account of individuals' lifetime utility, birth dates, and length of life,
69. See BLACKORBY ET AL., supra note 34, at 253-71. A variation on this formula puts the discount rate inside rather than outside the g-N function, i.e., 'V g(D(b(x' i))u. (x)) ? Other variations make the discount factor a function of some date other than the individual's birth date, e.g., his death date, or the date midway between birth and death. My arguments against interpersonal discounting apply to all these variations.
70. See Tyler Cowen, Consequentialism Implies a Zero Rate of Intergenerational Discount, in JUSTICE BETWEEN AGE GROUPS AND GENERATIONS 162 (Peter Laslett & James S. Fishkin eds., 1992);
71. JOHN BROOME, WEIGHING LIVES 127 (2004);
72. BLACKORBY ET AL., supra note 34, at 253-59.
73. See Cowen, supra note 43, at 164.
74. To be sure, the deliberates who wanted to be impartial vis-a-vis some proper subset of the world's population would presumably want to be sensitive to the fact that harms and benefits to persons outside the community of interest might, in turn, affect the well-being of individuals inside the community of interest-for example, because these insiders care about the outsiders. However, such impacts would show up in the utility function of the insiders.
75. See ADLER & POSNER, supra note 18, at 173-77.
76. Whether the prioritarian SWF would, in fact, approve the policy would depend on the utility levels of the future and present individuals.
77. I assume here that the ranking of actions in the choice set A is a function of probability numbers expressing the probability that a given action a in this set would yield a given outcome x in the outcome set O-probability numbers of the form jt (x) -as well as the utility function M(.) that maps each outcome onto a utility vector, and the SWF X"1 g(u(x)) tnat produces a ranking of those vectors. The probability term n (x) is what I am calling, roughly, the probability of an outcome "conditional" on a choice. Whether jr (*) is, strictly, a conditional probability implicates the debate between so-called "causal" and "evidential" decision theory, which I need not further discuss here.
78. See ADLER, WELL-BEING AND EQUITY, supra note 1. Further, my assumption that the ranking of actions in the choice set is a function of the n (x) values, the utility function, and the SWF is more generic than the specific claim that this information is integrated via the EU formula-an issue I leave open in this Article.
79. See supra text accompanying notes 32-33.
80. See supra note 48.
81. One variation on the uncertainty argument for discounting points to an extinction risk. See STERN, supra note 10, at 50-54;
82. Antoine Bommier & Stephane Zuber, Can Preferences for Catastrophe Avoidance Reconcile Social Discounting with Intergenerational Equity?, 31 SOC. CHOICE & WELFARE 415 (2008);
83. Partha Dasgupta, Discounting Climate Change, 37 J. RISK & UNCERTAINTY 141,160 (2008). This argument moves outside the core case. Instead, the date after which individuals no longer exist, and the number of individuals who exist, can vary across outcomes. If the existence of a given generation is likelier than succeeding generations, and less likely than preceding generations, then an argument for ranking outcomes with a discount factor to reflect this extinction risk arises. However, a more transparent and flexible approach to taking account of extinction risk would seem to be ranking outcomes without a discount factor, and incorporating the risk into the representation of each action as a probability distribution across outcomes. The flexibility of this approach would be particularly advantageous, it would seem, if we wish to allow the extinction risk to vary depending on which policy is undertaken (e.g., whether or not we take steps to mitigate a catastrophic event), rather than being exogenous.
84. There certainly can be societies that are immoral in some fundamental sense, e.g., racist, and are also seen by some of the members of these societies to be immoral.
85. See Geoffrey Brennan, Discounting the Future, Yet Again, 6 POL., PHIL. & ECON. 259, 268 (2007).
86. This premise is to simplify the analysis, and might mean that without consumption individuals die prematurely and end up with a life no better than nonexistence; or that they subsist but end up with a life no better than nonexistence.
87. Take any two individuals, one better off, one worse off. Consider the ratio between the change in the value of the SWF produced by giving a small increment of utility to the worse-off individual, and the change produced by giving a small increment to the better-off individual. That ratio increases without limit as y increases.
88. See ADLER, WELL-BEING AND EQUITY, supra note 1;
89. Adler, Risk Equity, supra note 1, at 40-45.
90. To show this formally, let us denote by S the stock of resources held by the current generation, and N the number of individuals in each generation. The leader of the current generation chooses some amount c for the first-generation individuals to consume. With an interest rate of r, this yields (5-c)(l+r) for the future generation. If the leader makes this decision using the Atkinsonian SWF, without discounting the future generation's utility, she chooses c so as to maximize complex math equations
91. See, e.g., JOHN BROOME, The Value of a Person, in ETHICS OUT OF ECONOMICS 228 (1999) [hereinafter BROOME, Value];
92. BROOME, supra note 43;
93. DAVID HEYD, GENETHICS (1992);
94. PARFIT, supra note 41, at 351-79;
95. MELINDA A. ROBERTS, CHILD VERSUS CHILDMAKER (1998);
96. Krister Bykvist, The Benefits of Coming into Existence, 135 PHIL. STUD. 335 (2007);
97. Caspar Hare, Voices from Another World: Must We Respect the Interests of People Who Do Not, and Will Never, Exist?, 117 ETHICS 498 (2007);
98. Nils Holtug, On the Value of Coming into Existence, 5 J. ETHICS 361 (2001);
99. Gregory S. Kavka, The Paradox of Future Individuals, 11 PHIL. & PUB. AFF. 93, 93-94 (1981);
100. Jan Narverson, Utilitarianism and New Generations, 76 MIND 62 (1967);
101. Josh Parsons, Axiological Actualism, 80 AUSTRALASIAN J. PHIL. 137 (2002);
102. James Woodward, The Non-Identity Problem, 96 ETHICS 804 (1986).
103. Kavka, supra note 57, at 93-94 (citations omitted).
104. See PARFIT, supra note 41, at 355-56 (distinguishing between "Same People Choices" and "Different People Choices" and, within the latter category, between "Same Number Choices" and "Different Number Choices").
105. For a recent discussion of the person-affecting principle with reference to prioritarian-ism,
106. See Holtug, supra note 21. Seminal discussions of the principle and its implications for population policy include Narveson's and Parfit's.
107. See PARFIT, supra note 41, at 391-417;
108. Narveson, supra note 57.
109. Note that we have reasoned ourselves to this conclusion independent of whether potential nonexistents have very good or very bad lives in outcomes where they exist. 62
110. See BROOME, Value, supra note 57.
111. PARFIT, supra note 41, at 378,447.
112. Nils Holtug argues, in a different way, that an individual can be better or worse off existing or not existing, and therefore that non-identity problems do not jeopardize the person-affecting principle.
113. See Holtug, supra note 57;
114. Holtug, supra note 21;
115. Nils Holtug, Utility, Priority and Possible People, 11 UTILITAS 16 (1999) [hereinafter Holtug, Utility).
116. To be sure, where an outcome set involves potential nonexistents, one needs to ask: whose extended preferences should we look to in comparing life histories? In the core case, one looks to the extended preferences of the N individuals. What happens outside the core case? I lack space to pursue that question here.
117. John Broome denies that a life history can be better or worse for a person than nonexistence. He therefore pursues a different approach to fixing the zero level, which identifies a neutral level for continuing to live, and then defines a neutral life as one which is constantly at this neutral level.
118. See BROOME, supra note 43, at 66-68,233-35,241-53.
119. See PARTHA DASGUFTA, AN INQUIRY INTO WELL-BEING AND DESTITUTION 377-97 (1993). Dasgupta's approach is actually somewhat more complicated than simply ranking the outcomes relative to each population. For a clear description and critical discussion, see BROOME, Value, supra note 57;
120. BROOME, supra note 43, at 157-63.
121. See Parsons, supra note 57.
122. See Hare, supra note 57.
123. Note that this could happen in two ways: (1) by following the approach I recommend (coupling the person-affecting principle with the position that nonexistence can be better or worse for a person than existence), or (2) by following an impersonal approach.
124. See supra text accompanying notes 60-63.
125. See PARFTT, supra note 41, at 381-90. There is a substantial literature on the repugnant conclusion and other population-size problems. Two recent, authoritative contributions, which also cite much of the prior literature, are BLACKORBY ET AL., supra note 34, at 129-208;
126. and BROOME, supra note 43. For discussions with specific reference to prioritarianism, see Campbell Brown, Prioritarianism for Variable Populations, 134 PHIL. STUD. 325 (2007);
127. Holtug, Utility, supra note 64;
128. Brown, supra note 21, at 187-223.
129. Outcome sets are structured to be responsive to decisionmakers' bounded rationality and thus may well exclude some logically possible outcomes. So, strictly, the worry is that there is some possible outcome set which includes both an x in which everyone lives at level L, and a y in which everyone lives at L*, arbitrarily close to zero-and that the "total" formula, applied to this outcome set, ranks y over x.
130. More precisely, the claims in this paragraph as well as the remaining paragraphs in this Section are true if the strictly increasing and concave function used by the prioritarian to trans-N(x) form individual utility-the g(.) function in the formula complex math equations
131. See Brown, supra note 21, at 210-17
132. Holtug, Utility, supra note 64, at 32-35.
133. BLACKORBY ET AL., supra note 34, at 129-208.
134. See id. at 172.
135. Blackorby, Bossert, and Donaldson discuss separability-across-persons under the heading of various "independence" axioms. See id at 159-60.
136. They generalize this formula by contemplating a range of critical levels.
137. See id. at 219-21, 248-52. They also consider other sorts of variations, for example "restricted" and "number-sensitive" critical-level views.
138. See id. at 136-51,165-71. John Broome argues for critical-level utilitarianism.
139. See BROOME, supra note 43, at 254-64.
140. For example, the inequality-aversion parameter in the Atkinsonian SWF.
141. BLACKORBY ET AL., supra note 34, at 166 (discussing properties of critical-level generalized utilitarianism, in particular where the critical level is positive).
142. However, it should be noted that difficulties arise in marrying "total" prioritarianism 1 N with the complex math equations
143. If one accepts the argument in this case, then it also surely works where the individuals in x do not exist in y and others take their place.
144. See Peter A. Diamond, The Evaluation of Infinite Utility Streams, 33 ECONOMETRICA 170 (1965);
145. Tjalling C. Koopmans, Stationary Ordinal Utility and Impatience, 28 ECONOMETRICA 287 (1960);
146. F.P. Ramsey, A Mathematical Theory of Saving, 38 ECON. J. 543 (1928).
147. The economic theory literature on infinite utility streams is very large. Recent contributions include: Geir Asheim, Tapan Mitra & Bertil Tungodden, A New Equity Condition for Infinite Utility Streams and the Possibility of Being Paretian, in INTERGENERATIONAL EQUITY AND SUSTAINABILITY 55 (John Roemer & Kotaro Suzumura eds., 2007);
148. Geir B. Asheim & Bertil Tungodden, Resolving Distributional Conflicts Between Generations, 24 ECON. THEORY 221 (2004);
149. Claude d'Aspremont, Formal Welfarism and Intergenerational Equity, in INTERGENERATIONAL EQUITY AND SUSTAINABILITY 113 (John Roemer & Kotaro Suzumura eds., 2007);
150. Kuntal Banerjee & Tapan Mitra, On the Continuity of Ethical Social Welfare Orders on Infinite Utility Streams, 30 SOC. CHOICE & WELFARE 1 (2008);
151. Kuntal Banerjee, On the Equity-Efficiency Trade off in Aggregating Infinite Utility Streams, 93 ECON. LETTERS 63 (2006);
152. Kuntal Banerjee, On the Extension of the Utilitarian and Suppes-Sen Social Welfare Relations to Infinite Utility Streams, 27 SOC. CHOICE & WELFARE 327 (2006) [hereinafter Banerjee, On the Extension of the Utilitarian];
153. Kuntal Banerjee & Tapan Mitra, On the Impatience Implications of Paretian Social Welfare Functions, 43 J. MATH. ECON. 236 (2007);
154. Kaushik Basu & Tapan Mitra, Aggregating Infinite Utility Streams with Intergenerational Equity: The Impossibility of Being Paretian, 71 ECONOMETRICA 1557 (2003) [hereinafter Basu & Mitra, Aggregating Infinite Utility Streams];
155. Kaushik Basu & Tapan Mitra, Possibility Theorems for Equitably Aggregating Infinite Utility Streams, in INTERGENERATIONAL EQUITY AND SUSTAINABILITY 69 (John Roemer & Kotaro Suzumura eds., 2007);
156. Kaushik Basu & Tapan Mitra, Utilitarianism for Infinite Utility Streams: A New Welfare Criterion and its Axiomatic Characterization, 133 J. ECON. THEORY 350 (2007) [hereinafter Basu & Mitra, Utilitarianism for Infinite Utility Streams];
157. Walter Bossert et al., Ordering Infinite Utility Streams, 135 J. ECON. THEORY 579 (2007);
158. Graciela Chichilnisky, An Axiomatic Approach to Sustainable Development, 13 SOC. CHOICE & WELFARE 231 (1996);
159. Juan Alfonso Crespo et al., On the Impossibility of Representing Infinite Utility Streams, 40 ECON. THEORY 47 (2009);
160. Marc Fleurbaey & Philippe Michel, Intertemporal Equity and the Extension of the Ramsey Criterion, 39 J. MATH. ECON. 777 (2003);
161. Chiaki Hara et al., Continuity and Egalitarianism in the Evaluation of Infinite Utility Streams, 31 SOC. CHOICE & WELFARE 179 (2008);
162. Luc Lauwers, Intertemporal Objective Functions: Strong Pareto Versus Anonymity, 35 MATH. SOC. SCI. 37 (1998);
163. Tapan Mitra & Kaushik Basu, On the Existence of Paretian Social Welfare Quasi-Orderings for Infinite Utility Streams with Extended Anonymity, in INTERGENERATIONAL EQUITY AND SUSTAINABILITY 85 (John Roemer & Kotaro Suzumura eds., 2007);
164. Lars-Gunnar Svensson, Equity Among Generations, 48 ECONOMETRICA 1 251 (1980);
165. Yongsheng Xu, Pareto Principle and Intergenerational Equity: Immediate Impatience, Universal Indifference, and Impossibility, in INTERGENERATIONAL EQUITY AND SUSTAINABILITY 100 (John Roemer & Kotaro Suzumura eds., 2007);
166. William Zame, Can Intergenerational Equity be Operationalized?, 2 THEORETICAL ECON. 187 (2007);
167. Geir B. Asheim, Claude d'Aspremont & Kuntal Banerjee, Generalized Time-Invariant Overtaking, (Department des sciences 6conomiques de l'Universite' catholique de Louvain, Working Paper No. 2008039, 2008);
168. Chiaki Hara, Kotaro Suzumura & Tomoichi Shinotsuka, On the Possibility of Continuous, Paretian, and Egalitarian Evaluation of Infinite Utility Streams (Andrew Young Sch. of Policy Studies Research Paper Series No. 07-12,2007), available at
169. There is also a smaller philosophical literature of infinity problems. This literature is more general than the economic theory literature just cited, because it discusses the problem of ranking worlds with an infinite number of "locations" of utility-whether or not those are organized in the form of a "stream" that begins with an initial period and never ends. See, e.g., Donniell Fishkind et al., New Inconsistencies in Infinite Utilitarianism: Is Every World Good, Bad or Neutral?, 80 AUSTRALASIAN J. PHIL. 178 (2002);
170. Luc Lauwers & Peter Vallentyne, Infinite Utilitarianism: More Is Always Better, 20 ECON. & PHIL. 307 (2004);
171. Tim Mulgan, Transcending the Infinite Utility Debate, 80 AUSTRALASIAN J. PHIL. 164 (2002);
172. Peter Vallentyne & Shelly Kagan, Infinite Value and Finitely Additive Value Theory, 94 J. PHIL. 5 (1997).
173. Formally, the weak anonymity axiom can be articulated as follows. Consider the set T = (1,2,...), which denotes the periods in a utility stream. Function p is a permutation if it is a bijection (one-to-one, onto mapping) from T onto itself. It is a finite permutation if all but a finite number of elements of T ate mapped onto themselves. complex math equations.
174. Bossert et al., supra note 83, at 581;
175. Fleurbaey & Michel, supra note 83, at 782;
176. Svensson, supra note 83, at 1252. The strong anonymity axiom requires indifference to any permutation, even one that is not finite.
177. See, e.g., Fleurbaey & Michel, supra note 83, at 782-83. For a full discussion of possible anonymity requirements in the infinity context, and their consistency with various forms of the Pareto principle, see id.
178. See Diamond, supra note 82.
179. Throughout the discussion of infinite utility streams, this is what I mean by the principle of Pareto superiority.
180. See Basu & Mitra, Aggregating Infinite Utility Streams, supra note 83.
181. Some scholars in the infinite-utility-stream literature have investigated the possibility of relaxing completeness.
182. See, e.g., Asheim & Tungodden, supra note 83;
183. Basu & Mitra, Utilitarianism for Infinite Utility Streams, supra note 83;
184. Bossert et al., supra note 83;
185. Hara et al., supra note 83.
186. See ADLER, WELL-BEING AND EQUITY, supra note 1;
187. See also ADLER & POSNER, supra note 18, at 161-62 (arguing that outcomes may be incomparable with respect to overall well-being).
188. See Svensson, supra note 83;
189. See also Bossert et al., supra note 83, at 579-80.
190. The overtaking criterion originates with von Weizsacker & Atsumi. See Hiroshi Atsumi, Neoclassical Growth and the Efficient Program of Capital Accumulation, 32 REV. ECON. STUD. 127 (1965);
191. Carl Christian von Weizsacker, Existence of Optimal Programs of Accumulation for an Infinite Time Horizon, 32 REV. ECON. STUD. 85 (1965). For recent discussions, see Asheim & Tungodden, supra note 83, at 226-28;
192. Banerjee, On the Extension of the Utilitarian, supra note 83, at 333-35;
193. Basu & Mitra, Utilitarianism for Infinite Utility Streams, supra note 83, at 356-63;
194. Fleurbaey & Michel, supra note 83, at 786-87;
195. Svensson, supra note 83, at 1253. For a variation on the overtaking criterion, see Asheim et al., supra note 83 ("generalized time-invariant overtaking").
196. See GEOFFREY HEAL, VALUING THE FUTURE: ECONOMIC THEORY AND SUSTAINABILITY 65-67 (1998). complex math equations
197. See Asheim & Tungodden, supra note 83, at 229. Asheim and Tungod-den provide a simpler example of the conflict. The example I provide here, however, involves a case in which one stream is a permutation of another, yet is strictly better in all but a single time period.
198. However, I will assume that an individual is born on the very same date in all outcomes where she exists. Although this assumption is not entailed by the nature of personal identity,
199. see supra text accompanying note 44, relaxing the assumption creates difficulties for the priori-tarian overtaking criterion as presented here. Note that, if some individual is born later in outcome x than outcome y, outcome x might be Pareto superior to y and yet the sum N(x,T) NW > V1 g(u(x)) nught be less than the sum V1 g(u (v)), for values of T in between the in-(=1 /=i dividual's birth date in y and x. This might yield inconsistencies between the prioritarian overtaking criterion and the principle of Pareto superiority if an infinite number of individuals have different birth dates in x and y. Similar difficulties can arise with the Pareto indifference principle. How it might be possible to relax the assumption that an individual is born on the very same date in all outcomes where she exists is a question I leave for another day.
200. See supra Part I.
201. For purposes of applying these criteria to an outcome set with potential nonexistents, who exist in some but not all outcomes, such individuals should be assigned a utility of zero in the outcomes where they do not exist. Then Pareto indifference requires that outcome x be ranked as equally good as y if (1) every individual who exists in both outcomes is equally well off in both, and (2) every individual who exists in one but not both outcomes has a utility of zero in the outcome where she exists. Pareto superiority requires that outcome x be ranked as better than outcome y if the utility assigned each individual in x is at least as great as the utility assigned her in y, and there are some individuals assigned strictly greater utility-where, by "the utility assigned to each individual," I mean that individual's utility if she exists, or zero if she does not. Pigou-Dalton means that if x and y are identical, except that ut(y) = U,(AT)+AU, ufy) = ut(x)-Au, ut(x) > ufy) > u,(y) > ut(x), where these utilities are the utilities assigned to individuals i and / in outcomes x and y, then y is better than x. Separability means that the ranking of x and y does not depend on which utilities are assigned to individuals who are assigned the same utilities in both outcomes.
202. By these terms, in this context, I mean the following. Given an outcome set, there is a set of all individuals who exist in at least one of the outcomes. A permutation is a bijection (one-to-one, onto mapping) from this set onto itself. A finite permutation is a permutation with the property that only a finite number of individuals are mapped onto different individuals. An infinite permutation is a permutation which is not finite. The utilities assigned individuals in outcome y are a permutation of the utilities assigned individuals in outcome x if there is some permutation p such that each individual's utility in x is the utility of individual p(t) in y. If there exists a finite such p, then y is a finite permutation of x. Less formally, y is a finite permutation of x if there is some finite subset of all the individuals (those who exist in at least one outcome), such that the utilities assigned to those individuals in y are a rearrangement of the utilities assigned to them in x, and all individuals not in the subset are assigned the very same utilities in y and x.
203. See supra Part I.
204. See supra Part II.
205. See supra Part III.A.
206. See supra Part III.B.
207. See supra Part III.C.
208. See supra text accompanying notes 46-56.
209. See supra text accompanying notes 42-45.
210. See supra text accompanying notes 64-67,80-81.
211. See supra text accompanying notes 96-97.