Intergenerational Fairness Lecture 4 PDF

Recap Discounting in detail Inequality and Discounting Intergenerational Fairness Fall 2023, Lecture 4 Paolo G. Piacquadio1 1 FGN-HSG October 16, 2024 Paolo G. Piacquadio Intergenerational Fairness ...

Recap Discounting in detail Inequality and Discounting Intergenerational Fairness Fall 2023, Lecture 4 Paolo G. Piacquadio1 1 FGN-HSG October 16, 2024 Paolo G. Piacquadio Intergenerational Fairness  Recap Discounting in detail Inequality and Discounting Recap of discounting Paolo G. Piacquadio Intergenerational Fairness  Recap Discounting in detail Inequality and Discounting Main points: SDR The Social Discount Rate is the rate at which we (locally) trade off benefits and losses across time. Consider the standard exponentially discounted utilitarian criterion ∞ ! W (C0 , C1 ,...) = exp {−δt} U (Ct ) , t=0 C1−η where U (Ct ) = t 1−η for η ≥ 0. Then, the SDR is given by the Ramsey rule: SDR = δ + ηg, where g is the growth rate of consumption. Paolo G. Piacquadio Intergenerational Fairness  Recap Discounting in detail Inequality and Discounting Main points As we discussed, δ is about discounting; ηg is about inequality. There is a large disagreement about: how and where to take these parameters; whether this is the right formulation; how to extend them in different directions; how to find a consensual view; etc. Paolo G. Piacquadio Intergenerational Fairness  Recap Discounting in detail Inequality and Discounting Main points As we discussed, δ is about discounting; ηg is about inequality. There is a large disagreement about: how and where to take these parameters; whether this is the right formulation; how to extend them in different directions; how to find a consensual view; etc. Paolo G. Piacquadio Intergenerational Fairness  Recap Discounting in detail Inequality and Discounting Open questions Discounting: Why exponential discounting? What are the alternatives? Inequality: Why weighted sum of CES utilities? What are the alternatives? How restrictive is it to think in terms of “discounting” and “inequality”? Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting in detail Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Most philosophers are against discounting. So why do economists do so? Sidgwick (1907, p. 414): “How far we are to consider the interests of posterity when they seem to conflict with those of existing human beings? It seems...clear that the time at which a man exists cannot affect the value of his happiness from a universal point of view; and that the interests of posterity must concern a Utilitarian as much as those of his contemporaries, except in so far as the effect of his actions on posterity—and even the existence of human beings to be affected—must necessarily be more uncertain.” Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Most philosophers are against discounting. So why do economists do so? Sidgwick (1907, p. 414): “How far we are to consider the interests of posterity when they seem to conflict with those of existing human beings? It seems...clear that the time at which a man exists cannot affect the value of his happiness from a universal point of view; and that the interests of posterity must concern a Utilitarian as much as those of his contemporaries, except in so far as the effect of his actions on posterity—and even the existence of human beings to be affected—must necessarily be more uncertain.” Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Ramsey (1928) starts out agreeing with Sidgwick: “One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination;...” then he continues: “...we shall, however,... include such a rate of discount in some of our investigations.” Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Ramsey (1928) starts out agreeing with Sidgwick: “One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination;...” then he continues: “...we shall, however,... include such a rate of discount in some of our investigations.” Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Arrow (1999) changed his view on the issue based on two key points. First, “Imagine initially that output consists of a constant stream of completely per- ishable goods. There can be no investment by definition. Now imagine that an investment opportunity occurs, available only to the first generation. For each unit sacrificed by them, a perpetual stream of per unit time is generated. If there were no time preference,...we can say that given any investment, short of the entire income, a still greater investment would be preferred.” Thus, Arrow concludes that without impatience the optimal saving rate could become arbitrarily close to 100%. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Arrow (1999) changed his view on the issue based on two key points. First, “Imagine initially that output consists of a constant stream of completely per- ishable goods. There can be no investment by definition. Now imagine that an investment opportunity occurs, available only to the first generation. For each unit sacrificed by them, a perpetual stream of per unit time is generated. If there were no time preference,...we can say that given any investment, short of the entire income, a still greater investment would be preferred.” Thus, Arrow concludes that without impatience the optimal saving rate could become arbitrarily close to 100%. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Next, as we shall analyze, he relates to some results in the literature: “Koopmans in several classic papers (1960, 1964) gave a crushing answer; see also Brown and Lewis (1981) for a more general treatment. The argument seems recondite. Koopmans considers a world which lasts forever. Therefore choice (including ethically-based choice) is based on a preference ordering over infinite-dimensional consumption streams. He argues that if the ordering is continuous and also sensitive (i.e., if one stream is never worse than another and is better at one or more time points, then it must be strictly preferred), it must display impatience.” Thus, Arrow considers discounting necessary. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: ethical defensible? Next, as we shall analyze, he relates to some results in the literature: “Koopmans in several classic papers (1960, 1964) gave a crushing answer; see also Brown and Lewis (1981) for a more general treatment. The argument seems recondite. Koopmans considers a world which lasts forever. Therefore choice (including ethically-based choice) is based on a preference ordering over infinite-dimensional consumption streams. He argues that if the ordering is continuous and also sensitive (i.e., if one stream is never worse than another and is better at one or more time points, then it must be strictly preferred), it must display impatience.” Thus, Arrow considers discounting necessary. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: a justification The strongest defense of discounting in general and exponentially-discounted utilitarianism in particular are: Koopmans (1960); and Diamond (1965). Briefly on Diamond (1965): he shows that there exists no social welfare function that satisfies: efficiency -> more consumption is better; and finite anonymity -> reshuflling consumptions is a matter of indifference. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discounting: a justification The strongest defense of discounting in general and exponentially-discounted utilitarianism in particular are: Koopmans (1960); and Diamond (1965). Briefly on Diamond (1965): he shows that there exists no social welfare function that satisfies: efficiency -> more consumption is better; and finite anonymity -> reshuflling consumptions is a matter of indifference. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans (1960) The key novelty was the following: rather than constructing a social welfare function by summing up each generation’s (positive) utility, Koopmans inverts the analysis and derives the utility functions of each generation and the constant discount factor from axioms on the planner’s preferences. He adopts the axiomatic approach to rank infinite streams of consumption. He proposes and studies the implication of 5 postulates. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans (1960) The key novelty was the following: rather than constructing a social welfare function by summing up each generation’s (positive) utility, Koopmans inverts the analysis and derives the utility functions of each generation and the constant discount factor from axioms on the planner’s preferences. He adopts the axiomatic approach to rank infinite streams of consumption. He proposes and studies the implication of 5 postulates. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 1 Continuity for any given consumption stream C, if social welfare satisfies M < W (C) < M̄, then there is a distance d such that M < W (C′ ) < M̄ for any C′ such that ∥C − C′ ∥1 ≤ d. Continuity means that a small variation in the consumption stream does not lead to drastic changes in the level of social welfare. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 1 Continuity for any given consumption stream C, if social welfare satisfies M < W (C) < M̄, then there is a distance d such that M < W (C′ ) < M̄ for any C′ such that ∥C − C′ ∥1 ≤ d. Continuity means that a small variation in the consumption stream does not lead to drastic changes in the level of social welfare. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 2 Sensitivity Consider any two consumption streams C ≡ (C1 ,2 C) and C′ ≡ (C′1 ,2 C). There must exist values of first period consumptions C1 and C′1 such that W (C) ̸= W (C′ ). Sensitivity tells that the level of consumption of the first generation matters. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 2 Sensitivity Consider any two consumption streams C ≡ (C1 ,2 C) and C′ ≡ (C′1 ,2 C). There must exist values of first period consumptions C1 and C′1 such that W (C) ̸= W (C′ ). Sensitivity tells that the level of consumption of the first generation matters. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 3 Non-complementarity If W (C1 ,2 C) = (C1 ,2 C′ ) then W (C′1 ,2 C) = (C′1 ,2 C′ ); and if W (C1 , 2 C) = (C′1 , 2 C) then W (C1 , 2 C′ ) = (C′1 , 2 C′ ). Non-complementarity imposes that the assessment of the future is independent of a common present and that the assessment of the present is independent of a common future. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 3 Non-complementarity If W (C1 ,2 C) = (C1 ,2 C′ ) then W (C′1 ,2 C) = (C′1 ,2 C′ ); and if W (C1 , 2 C) = (C′1 , 2 C) then W (C1 , 2 C′ ) = (C′1 , 2 C′ ). Non-complementarity imposes that the assessment of the future is independent of a common present and that the assessment of the present is independent of a common future. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 4 Stationarity W (C1 ,2 C) ≥ W (C1 ,2 C′ ) holds if and only if W (2 C) ≥ W (2 C′ ). Stationarity asks that when you to “eliminate” a common first period consumption, the ranking does not change. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 4 Stationarity W (C1 ,2 C) ≥ W (C1 ,2 C′ ) holds if and only if W (2 C) ≥ W (2 C′ ). Stationarity asks that when you to “eliminate” a common first period consumption, the ranking does not change. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 5 Boundedness There exist constant "consumption # streams C and C̄ such that W (C) ≤ W (C) ≤ W C̄. Boundedness is a technical assumption and means that growing streams cannot be worse/better than all constant streams. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Postulate 5 Boundedness There exist constant "consumption # streams C and C̄ such that W (C) ≤ W (C) ≤ W C̄. Boundedness is a technical assumption and means that growing streams cannot be worse/better than all constant streams. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Koopmans’ Result Theorem A social welfare function satisfies the 5 postulates if and only if it can be written as ! W (C) = β t U (Ct ) t for some β ∈ (0, 1) and a continuous and strictly increasing function U. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion The equivalence result is very elegant and strong. If you “dislike” the social welfare function, you must “dislike” some of the axioms (postulates). Each axiom can be debated: which ones do you find too restrictive? Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion The equivalence result is very elegant and strong. If you “dislike” the social welfare function, you must “dislike” some of the axioms (postulates). Each axiom can be debated: which ones do you find too restrictive? Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion The equivalence result is very elegant and strong. If you “dislike” the social welfare function, you must “dislike” some of the axioms (postulates). Each axiom can be debated: which ones do you find too restrictive? Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion I Arguably, a quite controversial one is stationarity. Consider the consumption stream (1, 1, 1, 1,...). Now, add X to the first period or Y to the second period so that these are indifferent: W(1 + X, 1, 1, 1,...) = W(1, 1 + Y, 1, 1,...). Stationarity tells that W(1, 1 + X, 1, 1, 1,...) = W(1, 1, 1 + Y, 1, 1,...). So, what the stationarity axiom effectively says is that intertemporal preferences between two periods remain the same even if we shunt those two periods further ahead in time. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion I Arguably, a quite controversial one is stationarity. Consider the consumption stream (1, 1, 1, 1,...). Now, add X to the first period or Y to the second period so that these are indifferent: W(1 + X, 1, 1, 1,...) = W(1, 1 + Y, 1, 1,...). Stationarity tells that W(1, 1 + X, 1, 1, 1,...) = W(1, 1, 1 + Y, 1, 1,...). So, what the stationarity axiom effectively says is that intertemporal preferences between two periods remain the same even if we shunt those two periods further ahead in time. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion II Another controversial axiom is non-complementarity. Consider the consumption stream (4, 2, 10, 8, 15,...). There is an equally desirable stream (4, x, x, 8, 15,...), for some choice of x. For example, W(4, 2, 10, 8, 15,...) = W(4, 5, 5, 8, 15,...). By non-complementarity, this should be true for any streams W(x, 2, 10, y, z,...) = W(x, 5, 5, y, z,...) So, the non-complementarity axiom says that inequality attitudes between any two periods cannot depend on what happens in some other period. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion II Another controversial axiom is non-complementarity. Consider the consumption stream (4, 2, 10, 8, 15,...). There is an equally desirable stream (4, x, x, 8, 15,...), for some choice of x. For example, W(4, 2, 10, 8, 15,...) = W(4, 5, 5, 8, 15,...). By non-complementarity, this should be true for any streams W(x, 2, 10, y, z,...) = W(x, 5, 5, y, z,...) So, the non-complementarity axiom says that inequality attitudes between any two periods cannot depend on what happens in some other period. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Discussion II Another controversial axiom is non-complementarity. Consider the consumption stream (4, 2, 10, 8, 15,...). There is an equally desirable stream (4, x, x, 8, 15,...), for some choice of x. For example, W(4, 2, 10, 8, 15,...) = W(4, 5, 5, 8, 15,...). By non-complementarity, this should be true for any streams W(x, 2, 10, y, z,...) = W(x, 5, 5, y, z,...) So, the non-complementarity axiom says that inequality attitudes between any two periods cannot depend on what happens in some other period. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions What alternatives are available? Hyperbolic discounting has been suggested to model individuals’ behavior. Thaler (1981) finds that individuals are indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. This means that the annual discount rates decline from 277% to 139% to 63% as delays got longer. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions What alternatives are available? Hyperbolic discounting has been suggested to model individuals’ behavior. Thaler (1981) finds that individuals are indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. This means that the annual discount rates decline from 277% to 139% to 63% as delays got longer. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions What alternatives are available? Hyperbolic discounting has been suggested to model individuals’ behavior. Thaler (1981) finds that individuals are indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. This means that the annual discount rates decline from 277% to 139% to 63% as delays got longer. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions What alternatives are available? Hyperbolic discounting has been suggested to model individuals’ behavior. Thaler (1981) finds that individuals are indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. This means that the annual discount rates decline from 277% to 139% to 63% as delays got longer. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A particularly relevant case is that of quasi-hyperbolic discounting (Laibson, 1997). The weighting function is then $ 1 t=0 bt = βδ t t≥1 When β = 1, exponentially-discounted utilitarianism emerges. When β < 1, first period discounting is βδ, later period discounting is δ > βδ. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A particularly relevant case is that of quasi-hyperbolic discounting (Laibson, 1997). The weighting function is then $ 1 t=0 bt = βδ t t≥1 When β = 1, exponentially-discounted utilitarianism emerges. When β < 1, first period discounting is βδ, later period discounting is δ > βδ. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A particularly relevant case is that of quasi-hyperbolic discounting (Laibson, 1997). The weighting function is then $ 1 t=0 bt = βδ t t≥1 When β = 1, exponentially-discounted utilitarianism emerges. When β < 1, first period discounting is βδ, later period discounting is δ > βδ. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A particularly relevant case is that of quasi-hyperbolic discounting (Laibson, 1997). The weighting function is then $ 1 t=0 bt = βδ t t≥1 When β = 1, exponentially-discounted utilitarianism emerges. When β < 1, first period discounting is βδ, later period discounting is δ > βδ. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A possible issue is that any non-exponential discounting function violates time consistency. This is the requirement that if a consumption stream is better than an alternative one,... it remains better as time elapses. Can social welfare by time inconsistent? Two possible answers: no, it doesn’t make sense; yes, just require that the planner is “smart” and anticipates this change. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A possible issue is that any non-exponential discounting function violates time consistency. This is the requirement that if a consumption stream is better than an alternative one,... it remains better as time elapses. Can social welfare by time inconsistent? Two possible answers: no, it doesn’t make sense; yes, just require that the planner is “smart” and anticipates this change. Paolo G. Piacquadio Intergenerational Fairness  Recap The debate Discounting in detail An axiomatic justification Inequality and Discounting Alternative discounting functions Alternative discounting functions A possible issue is that any non-exponential discounting function violates time consistency. This is the requirement that if a consumption stream is better than an alternative one,... it remains better as time elapses. Can social welfare by time inconsistent? Two possible answers: no, it doesn’t make sense; yes, just require that the planner is “smart” and anticipates this change. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Inequality and Discounting Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Inequality vs Discounting In a recent contribution with Frikk Nesje, we show how social welfare functions can disentangle discounting from inequality: a “modular” theory of intergenerational justice. Under some specific conditions, social attitudes towards discounting are orthogonal to social attitudes towards inequality. The key advantage is that these attitudes can be chosen separatly: many new criteria (some might better fit individuals views); more clarity on policy implications. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Inequality vs Discounting In a recent contribution with Frikk Nesje, we show how social welfare functions can disentangle discounting from inequality: a “modular” theory of intergenerational justice. Under some specific conditions, social attitudes towards discounting are orthogonal to social attitudes towards inequality. The key advantage is that these attitudes can be chosen separatly: many new criteria (some might better fit individuals views); more clarity on policy implications. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Inequality vs Discounting In a recent contribution with Frikk Nesje, we show how social welfare functions can disentangle discounting from inequality: a “modular” theory of intergenerational justice. Under some specific conditions, social attitudes towards discounting are orthogonal to social attitudes towards inequality. The key advantage is that these attitudes can be chosen separatly: many new criteria (some might better fit individuals views); more clarity on policy implications. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The disentanglement The central step of our representation is to transform consumption streams in “calendar time” to consumption streams in “equivalent time”. The duration of each generation is expanded/ compressed up to the point where all consumption has the same moral value; Since each instant of consumption in equivalent time has the same value, we can apply any standard anonymous inequality averse aggregation of consumption. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting A consumption stream We study the problem of ranking consumption streams c : T → R+ in infinite continuous time T ≡ [0, ∞). The set of all consumption streams is C. A theory of intergenerational justice is represented by a social welfare function W : C → R+. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting A consumption stream We study the problem of ranking consumption streams c : T → R+ in infinite continuous time T ≡ [0, ∞). The set of all consumption streams is C. A theory of intergenerational justice is represented by a social welfare function W : C → R+. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The time-discounting function An instant of time t ∈ T is expressed in calendar time. An alternative representation of a consumption stream is on the unit interval I ≡ [0, 1]. In fact, for any continuous and strictly increasing function φ : T → I, y (i) = y (φ(t)) ≡ c (t) is an equivalent representation of the same streams. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The time-discounting function An instant of time t ∈ T is expressed in calendar time. An alternative representation of a consumption stream is on the unit interval I ≡ [0, 1]. In fact, for any continuous and strictly increasing function φ : T → I, y (i) = y (φ(t)) ≡ c (t) is an equivalent representation of the same streams. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The time-discounting function An instant of time t ∈ T is expressed in calendar time. An alternative representation of a consumption stream is on the unit interval I ≡ [0, 1]. In fact, for any continuous and strictly increasing function φ : T → I, y (i) = y (φ(t)) ≡ c (t) is an equivalent representation of the same streams. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The time-discounting function We say that consumption streams admit a representation in equivalent time, if there exists a function ω : T → I, such that permuting the transformed consumption is a matter of social indifference. W(c) = W(c′ ) whenever c = y(ω), c′ = y′ (ω), y = Πy′. In other words, all intervals in equivalent time have the same ethical value. We call ω the time-discounting function. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The time-discounting function We say that consumption streams admit a representation in equivalent time, if there exists a function ω : T → I, such that permuting the transformed consumption is a matter of social indifference. W(c) = W(c′ ) whenever c = y(ω), c′ = y′ (ω), y = Πy′. In other words, all intervals in equivalent time have the same ethical value. We call ω the time-discounting function. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Illustration Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Illustration Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Illustration Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Time-discounting functions Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The aggregator function How to aggregate consumption streams in equivalent time? Given the social welfare function W and the time-discounting function ω, the aggregator function V : Y → R+ immediately follows by setting V (y) = W (y (ω)). Note that the aggregator function is anonymous by definition of equivalent time. Thus, V could be any standard (atemporal) social welfare function. Examples: Additive, rank-weighted, and rank-dependent. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The aggregator function How to aggregate consumption streams in equivalent time? Given the social welfare function W and the time-discounting function ω, the aggregator function V : Y → R+ immediately follows by setting V (y) = W (y (ω)). Note that the aggregator function is anonymous by definition of equivalent time. Thus, V could be any standard (atemporal) social welfare function. Examples: Additive, rank-weighted, and rank-dependent. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The aggregator function How to aggregate consumption streams in equivalent time? Given the social welfare function W and the time-discounting function ω, the aggregator function V : Y → R+ immediately follows by setting V (y) = W (y (ω)). Note that the aggregator function is anonymous by definition of equivalent time. Thus, V could be any standard (atemporal) social welfare function. Examples: Additive, rank-weighted, and rank-dependent. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The aggregator function Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The modular social welfare function We say that W is modular if consumption streams can be represented in equivalent time and, thus, there exist an ω and V induced by W. Importantly, all the most common welfare criteria adopted in the literature are modular. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The modular social welfare function We say that W is modular if consumption streams can be represented in equivalent time and, thus, there exist an ω and V induced by W. Importantly, all the most common welfare criteria adopted in the literature are modular. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The modular social welfare function It immediately follows that: W is uniquely identified by ω and V. Thus, one can construct a wide range of criteria by selecting and combining alternatives within these two modules. The opposite is also true: Assume W is modular. W uniquely pins down ω and (up to an order-preserving transformation) V. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting The modular social welfare function It immediately follows that: W is uniquely identified by ω and V. Thus, one can construct a wide range of criteria by selecting and combining alternatives within these two modules. The opposite is also true: Assume W is modular. W uniquely pins down ω and (up to an order-preserving transformation) V. Paolo G. Piacquadio Intergenerational Fairness  Recap The disentanglement Discounting in detail Discounting and equivalent time representation Inequality and Discounting Some modular theories of intergenerational justice Paolo G. Piacquadio Intergenerational Fairness