Equitable intergenerational preferences and sustainability
GEIR B. ASHEIM
Department of Econonmics, University of Oslo
December 27, 2012
[7120 words]
1. Introduction
There are about 7 billion people currently alive. Just above 100 billion people have ever
lived (Haub, 2011). Hence, the ratio of people who have ever lived in the past to people living
today is only 15 to 1. With 500 million years left of the earth as acceptable habitat for
humans, population being stable at a sustainable level of 1 billion with an average life-length
of 70 years, the ratio of people who will potentially live in the future to people living now is
more than 1 million to 1. Thus, there are many people that might potentially live in the future,
and this observation might justify modeling the future as consisting of infinitely many
generations.
In spite of the development that accumulated reproducible and human capital has lead to
during the recent past, there are clear conflicts of interest between generations: the wellbeing
of future generations might be undermined unless we take costly action today. Abating
greenhouse gas emissions, which reduces future climate change, is a prime example of such
costly current action with long-term future benefits. This conflict has been the subject of
much recent attention (Stern, 2007; Nordhaus, 2008). However, other conflicts with similar
characteristics include preserving biodiversity (which widens options for future generations),
exploiting soil and water resources with caution (which increases the potential for future food
production) and using antibiotics with care (which reduces future health problems).
How should such conflicts be resolved from an impartial perspective? Does resolution of
intergenerational conflicts lead to sustainable development? These are questions posed in the
field of intertemporal social choice. Contributions in this field consider an infinite but
countable number of generations that follow each other in sequence. To each generation is
assigned a level of generational well-being that indicates the situation under which people
within this generation live. Hence, future development is given by an infinite stream of wellbeing. One seeks to answer the following normative question: How should different streams
of well-being be ranked when the interests of all generations are taken into account?
1 The notion of well-being should be thought of as including much more than material
consumption, capturing also the importance of health, culture, and nature. However, to
distinguish the discussion of intergenerational equity from the forces that can motivate our
generation to act in an equitable manner, it is natural to impose two restrictions: Well-being
does not include the welfare that people derive from their children’s well-being. Likewise,
well-being includes only nature’s instrumental value (i.e., recognized value to humans), not
its intrinsic value (i.e., value in its own right regardless of human experience); i.e., an
anthropocentric perspective is taken.i
2. Can an infinite number of generations be treated equally?
Following Diamond (1965), Svensson (1980), Basu & Mitra (2003, 2007b), Zame (2007) and
others, let [0,1] be the interval of admissible well-being levels. Hence, the set of admissible
streams consists of infinite intergenerational well-being streams where, for each generation t,
its well-being xt is between 0 and 1. I assume that the level of well-being can be compared
across generations.
A social welfare relation (SWR) is a reflexive (i.e., it satifies that any stream is as good as
itself) and transitive (i.e., it satisfies the coherency property that if a first stream is as good as
a second stream, and the second stream is as good as a third stream, then the first stream is as
good as the third stream) binary relation on the set of admissible well-being streams. If an
SWR is complete (i.e., it satisfies the richness property that any two different streams are
comparable), then it is called a social welfare order (SWO). A social welfare function (SWF)
assigns to any admissible stream a real number. That an SWF numerically represents an SWR
means that one stream is as good as another according to the SWR if and only if the number
assigned to the former by the SWF is as high as the number assigned to the latter.
An uncontroversial ethical axiom on an SWR is that one well-being stream must be deemed
strictly better than another if at least one generation is better off and no generation is worse
off. This axiom is called Strong Pareto (SP), and it captures efficiency concerns and ensures
that the social evaluation is sensitive to an increase in well-being for any one generation.
Another basic ethical axiom imposes equal treatment of all generations by requiring that any
well-being stream must be deemed equally good as the stream obtained after permuting the
well-being levels of generations. There are two versions of this axiom: Finite Anonymity
(FA), where the social evalution is required to be unchanged if the well-being of only a finite
number of generations is permuted, and Strong Anonymity (SA), where the social evaluation
is required to be unchanged even if the well-being of an infinite number of generations is
permuted. This axiom captures equity concerns in a setting in which there is no uncertainty
2 about the existence of future generations. It is a basic fairness norm as it ensures that everyone
counts the same in social evaluation.
There is a basic conflict between treating an infinite number of generations equally, and being
sensitive to the interests of each one of them. The simplest way to illustrate this dilemma is to
consider the following two well-being streams:
0
1
1
1
0
0
1
1
0
0
1
1
…
…
The second of these streams is a permutation of the first: move the second location to the first,
all other even locations two periods backwards, and all odd locations two periods forward.
Hence, by axiom SA they are equally good. However, by axiom SP the second is strictly
better than the first. Hence, this shows that axioms SP and SA are incompatible: an SWR can
satisfy one, but not both.
The example above relies on a permutation of an infinite number of generations. Therefore, it
does not show that there is conflict between axioms FA and SP. However, even if axiom SA
is weakened to FA, combining such equal treatment with sensitivity in the form of axiom SP
leads to impossibilities. There are two sets of impossibility results, both of which concern the
problems of combining axioms FA and SP with an SWR that is also complete (i.e., is an
SWO). The first of these might be called the Diamond-Basu-Mitra Impossibility Result:
Proposition 1 (Basu & Mitra 2003, theorem 2): Assume that an SWO satisfies axioms SP and
FA. Then there does not exist any SWF numerically representing this SWO.
Diamond’s (1965, p. 176) original theorem, a result he attributes to M.E. Yaari, establishes
that no SWO can satisfy both continuity in the supnorm topology (see Section 4) and axioms
SP and FA. Because any supnorm continuous and strongly Paretian SWO is numerically
representable, Basu & Mitra (2003) strengthen the Diamond-Yaari theorem. Fleurbaey &
Michel (2003, theorem 2) and Basu & Mitra (2007a, theorem 4) offer strengthenings of the
Diamond-Yaari and Basu-Mitra theorems by weakening the strong Pareto axiom to the
weaker Paretian axiom (axiom WP) requiring that one well-being stream be deemed superior
to another if all generations are better off.
The second of these might be called the Lauwers-Zame Impossibility Result. Fleurbaey &
Michel (2003, p. 794) conjecture that no SWO satisfying axioms SP and FA can be explicitly
defined. This question was resolved by Lauwers (2010) and Zame (2007), who essentially
confirm the Fleurbaey-Michel conjecture. Zame’s (2007) result is stated as follows.
Proposition 2: No definable SWO can be proved to satisfy axioms SP and FA.
3 The Diamond-Basu-Mitra and Lauwers-Zame impossibility results constitute an ethical
dilemma. If we consider only SWRs that can be explicitly defined, then we are forced to make
a choice between the following alternatives: either require that the SWR satisfy axioms SP
and FA or require that the SWR be complete. In the former case, the SWR must be
incomplete (and, hence, not numerically representable). In the latter case, efficiency and
equity concerns must be captured by conditions other than the combination of axioms SP and
FA.
The next section is devoted to equitable and strongly Paretian SWRs, for which the term
equitable means that the finite anonymity axiom is satisfied. The subsequent Section 4
presents equitable and numerically representable SWOs that either do not satisfy axiom FA
(hence, equity is captured by alternative axioms) or do not satisfy even the weaker Paretian
axiom (axiom WP).
In relation to the Diamond-Basu-Mitra and Lauwers-Zame impossibility results, it is worth
noting that axioms SP and FA are not in conflict by themselves. And it is of interest to
explore the consequences of imposing these axioms. Asheim et al. (2001) show that axioms
SP and FA have far-reaching implications in technological environments that satisfy the
following productivity condition.
Condition of immediate productivity: A set of feasible well-being streams satisfies
immediate productivity if, for all feasible well-being with xt  xt+1 for some t, the well-being
stream (x1, x2, … , xt-1, xt+1, xt, xt+2, … ) is feasible and inefficient.
With this domain restriction, we can make the following argument. If a well-being stream is
not nondecreasing — i.e., there exists some t such that xt  xt+1 — then it is feasible to save
the additional well-being of generation t for the benefit of generation t+1 such that t+1’s gain
is larger than t’s sacrifice. By axiom FA the new stream would have been socially indifferent
to the old one even if the additional utility of t were simply transferred to t+1 so that t+1’s
gain would have been the same as t’s sacrifice (as this would have amounted to a permutation
of the well-being levels of generations t and t+1). By axiom SP it follows that the new stream
is (strictly) preferred as the condition of immediate productivity implies that t+1’s gain is
larger than t’s sacrifice. This argument means that only nondecreasing well-being streams are
undominated by an SWR satisfying axioms SP and FA in technological environments
satisfying the condition of immediate productivity. Because any nondecreasing utility stream
is sustainable — according to any common definition of the notion of sustainable
development — axioms SP and FA justify sustainability.
4 3. Equitable and Paretian preferences
Following Sidgwick (1907), Pigou (1932), and Ramsey (1928), there is a long tradition in
economics of considering the unfavorable treatment of future generations as ethically
unacceptable. The quote from Pigou (1932, part I, chapter 2) in which he explains the
preference for present pleasure over future pleasure by our defective telescopic faculty is
well-known. Likewise, Ramsey (1928, p. 543) assumes that “we do not discount later
enjoyment in comparison to earlier ones, a practice which is ethically indefensible and arises
merely from the weakness of imagination.” These positions also invalidate unequal treatment
of generations (Collard 1996).
Undiscounted utilitarianism, as suggested by Ramsey (1928), can be adapted to the setting
with an infinite but countable number of generations as follows (Basu & Mitra 2007b):
Consider two well-being streams (x1, x2, … , xt, … ) and (y1, y2, … , yt, … ). The former is as
good as the latter according to the undiscounted utilitarian SWR if and only if there exists T
such that x1  x2  …  xT  y1  y2  …  yT and xt  yt for all t  T.
This criteron is characterized by adding to axioms SP and FA a partial translation scale
invariance (PTSI) axiom requiring that the ranking of two well-being streams (x1, x2, … , xt,
… ) and (y1, y2, … , yt, … ), with xt  yt for all t beyond some T, not change by adding to each
generation the same amount of well-being in both streams. Combined with axiom FA, axiom
PTSI implies that a transfer from one generation to another leaves the stream equally good if
the former’s sacrifice in terms of well-being equals the latter’s benefit. This entails that any
inequality aversion has already been captured by the scale used to measure well-being.
Proposition 3 (Basu & Mitra 2007b, theorem 1): The following two statement are equivalent:
(1) An SWR satisfies axioms SP, FA, and PTSI.
(2) An SWR has the property of deeming (x1, x2, … , xt, … ) strictly better than (equally good
as) (y1, y2, … , yt, … ) if the undiscounted utilitarian SWR deems (x1, x2, … , xt, … ) strictly
better than (equally good as) (y1, y2, … , yt, … ).
Maximin, the principle of maximizing the well-being of the worst-off generation, also
satisfies axiom FA and is thus an alternative way of treating generations equally. Maximin is
often identified with Rawls’s (1971) difference principle, although Rawls applied this
principle to an index of primary goods (which cannot necessarily be identified with wellbeing) and did not recommend its use in the intergenerational setting. Solow (1974) is, in his
own words, “plus Rawlsien que le Rawls” by applying the maximin principle for finding
optimal intergenerational distributions. I follow Sen (1970) by considering maximin in its
5 lexicographic form, referred to as leximin, as this makes the principle compatible with axiom
SP too.
Leximin can be adapted to the setting with an infinite but countable number of generations as
follows (Bossert et al. 2007): Consider two well-being streams (x1, x2, … , xt, … ) and (y1, y2,
… , yt, … ). For each T, let (x[1], x[2], … , x[T]) and (y[1], y[2], … , y[T]) denote the rank-ordered
permutation of (x1, x2, … , xT) and (y1, y2, … , yT), so that x[r]  x[r+1] and y[r]  y[r+1] for all
ranks r  1, … , T1. Finite-dimensional leximin deems (x1, x2, … , xT) strictly better than (y1,
y2, … , yT) if and only if there is R  T such that x[R]  y[R] and x[r]  y[r] for all r  1, … , R1,
and (x1, x2, … , xT) equally good as (y1, y2, … , yT) if and only if (x[1], x[2], … , x[T])  (y[1], y[2],
… , y[T]). The stream (x1, x2, … , xt, … ) is as good as the stream (y1, y2, … , yt, … ) according
to the leximin SWR if and only if there exists T such that finite-dimensional leximin deems
(x1, x2, … , xT) as good as (y1, y2, … , yT) and xt  yt for all t  T.
The leximin SWR can be characterized by combining axioms SP and FA with Hammond’s
(1976) equity (HE) axiom, requiring that (x1, x2, … , xt, … ) is as good as (y1, y2, … , yt, … )
if there are times i and j such that yj  xj  xi  yi and xt  yt for all t  i, j. Axiom HE means
that a stream cannot be made worse by making a transfer from a richer to a poorer generation,
independently of the size of the richer generation’s sacrifice and the size of the poorer
generation’s benefit.
Proposition 4 (Bossert et al. 2007, theorem 2): The following two statement are equivalent:
(1) An SWR satisfies axioms SP, FA, and HE.
(2) An SWR has the property of deeming (x1, x2, … , xt, … ) strictly better than (equally good
as) (y1, y2, … , yt, … ) if the leximin SWR deems (x1, x2, … , xt, … ) strictly better than
(equally good as) (y1, y2, … , yt, … ).
Both the undiscounted utilitarian and leximin SWRs are incomplete criteria that satisfy
axioms SP and FA and thus lead to sustainable streams in productive economies. In
particular, in the one-sector Ramsey model and the DHSS model (Dasgupta & Heal, 1974,
Solow, 1974, Stiglitz, 1974) of capital accumulation and resource depletion, the undiscounted
utilitarian SWR leads to strictly increasing consumption (if an undominated stream exists),
while the leximin SWR leads to constant consumption.
The undiscounted utilitarian and leximin SWRs also satisfy other axioms often invoked in
intertemporal social choice:
 Separable present (SEP): If the tails of two streams beyond time T coincide, then the
ranking of the streams does not depend on what this common tail is.
6  Separable future (SEF): If the head of two streams up to time T coincide, then the
ranking of the streams does not depend on what this common head is.
 Stationarity (ST): If two streams concide at time 1 (so that generation 1 has the same
well-being level in both streams), then the ranking of the streams remains the same if
generation 1 gets the well-being level of generation 2, generation 2 gets the well-being
level of generation 3, and so forth.
However, an obvious problem with the undiscounted utilitarian and leximin SWRs, as
presented above, is their incompleteness. In particular, if the tails of two streams beyond some
finite time do not coincide or Pareto-dominate each other, then the streams are incomparable.
There is a series of contributions (e.g., Lauwers, 1997; Fleurbaey & Michel, 2003; Asheim &
Tungodden, 2004; Banerjee, 2006; Kamaga & Kojima 2009, 2010; Asheim & Banerjee,
2010) that investigate how more comparability can be achieved by adding additional axioms,
keeping in mind that the Lauwers-Zame impossibility result rules out completeness of any
explicitly defined SWR. These versions of undiscounted utilitarianism and leximin invoke
overtaking and catching-up procedures which are beyond the scope of the current chapter.
[FIGURE 1 ABOUT HERE]
Even though both the undiscounted utilitarian and leximin SWRs treat generations equally,
they lead to quite different (and perhaps undesirable) consequences in a class of simple
present-future conflicts illustrated in Figure 1. Consider an egalitarian stream where every
generation’s well-being equals z, where 0  z  1. Consider an alternative stream (x1, x, x, x,
… ) where generation 1 makes a sacrifice, leading to a uniform gain for all future generations;
i.e., 0  x1  z and z  x  1. Should generation 1 make such a sacrifice, when the evaluation is
made having the interests of all generations in mind? The undiscounted utilitarian and leximin
SWRs answer this question in opposite ways.
According to the undiscounted utilitarian SWR, the sacrifice should always be made, also
when z  x1 is large and x  z is small. From a utilitarian point of view, any sacrifice, however
large, by the current generation should be made to ensure the uniform gain, however small,
for the infinite number of future generations. This observation illustrates the argument of
Rawls (1971, p. 287) that “the utilitarian doctrine may direct us to demand heavy sacrifices of
the poorer generations for the sake of greater advantages for the later ones that are far better
off.” Following such arguments, Arrow (1999, p. 16) concludes “that the strong ethical
requirement that all generations be treated alike, itself reasonable, contradicts a very strong
intuition that it is not morally acceptable to demand excessively high savings rates of any one
generation, or even of every generation.”
7 According to the leximin SWR, the sacrifice should never be done, even when z  x1 is small
and x  z is large. From a leximin point of view, no sacrifice, however small, by the current
generation should be made to ensure the uniform gain, however large, for the infinite number
of future generations. In particular, as pointed out by Solow (1974, p. 41), the principle of
maximizing the well-being of the worst-off generation may perpetuate poverty.
Are any of these conclusions consistent with commonly held ethical intuitions? Most of us
will probably claim that the conclusion should depend on the circumstances. We may hold the
position that generation 1 should make the sacrifice for the benefit of the infinite number of
future and better-off generations if its sacrifice is small relative to their uniform gain, but not
if its sacrifice is relatively large. Owing to the infinite number of generations, the
undiscounted utilitarian and egalitarian criteria yield extreme and opposite conclusions,
neither of which might be defendable.
4. Equitable and complete preferences
The previous section considers incomplete SWRs satisfying axioms SP, FA, SEP, SEF, and
ST. The Diamond-Basu-Mitra impossibility result implies that there exists no numerically
representable SWR satisfying both axioms SP and FA, whereas the Lauwers-Zame
impossibility result entails that we cannot explicitly define any complete SWR satisfying
these two axioms. This section explores numerically representable (and, thus, complete)
SWRs obtained by dropping either axiom SP or FA.
If an SWO satisfies the following two axioms,
 Continuity (C) in the supnorm topology (meaning that two streams are close if all
elements are close) as a robustness property,
 Montonicity (M), requiring that one stream be deemed as good as another if no
generation is worse off,
then it is numerically represented by an SWF. Since axiom SP implies axiom M, it follows
that any SWO satisfying axioms C and SP is also numerically representable.
If an SWO satisfies axioms C, SP, SEP, SEF, and ST, then we obtain a characterization of
the time-discounted utilitarian (TDU) SWO. This is the commonly applied criteron both in the
theory of economic growth and in the practical evaluation of economic policy with long-term
effects (e.g., climate policies).
Proposition 5: The following two statement are equivalent:
(1) An SWO satisfies axioms C, SP, SEP, SEF, and ST.
8 (2) An SWO is numerically represented by an SWF W satisfying, for some strictly increasing
and continuous utility function U, and utility discount factor , with 0 <  < 1,
W ( x1 , x2 , , xt ,)  (1   ) t 1  t 1U ( xt ) .

In the present setting, in which well-being for every generation is a scalar in the unit interval,
this characterization of the TDU SWO is quite close to Koopmans’ (1960) original
axiomatization. Multiplying by 1   ensures that the utility weights 1 – , (1 – ), … , (1 –
)t–1, … add up to one. The TDU SWO is well-defined on the set of streams where each
generations well-being is between 0 and 1, and the corresponding SWF satisfies:
W ( x1 , x2 , , xt ,)  (1   )U ( x1 )  W ( x2 , x3 , , xt ,) ,
W ( z, z , , z,)  U ( z ) .
Since the TDU SWO does not satisfy axiom FA, it need not lead to sustainable streams even
in productive economies. Indeed, consumption is strictly decreasing in the Ramsey model if
the initial capital stock exceeds the stock corresponding to the modified Golden Rule, and
eventually decreasing in the DHSS model for any vector of initial stocks. To test the TDU
SWO in choice situations in which there is conflict between the present generation and the
equally well-off future generations, consider two kinds of alternative streams to which an
egalitarian stream where every generation’s well-being equals z is compared.
If, as illustrated in Figure 1, in an alternative stream (x1, x, x, x, … ) generation 1 makes a
sacrifice, leading to a uniform gain for all future generations, i.e., 0  x1  z and z  x  1, then
the class of TDU SWOs leads to the appealing conclusion that the ranking depends on the
gain/sacrifice ratio. Any TDU SWO is consistent with the position that generation 1 should
make a sacrifice increasing the well-being of the infinite number of future generations if its
sacrifice is small relative to their uniform gain, but not if its sacrifice is relatively large.
[FIGURE 2 ABOUT HERE]
If, as illustrated in Figure 2, in an alternative stream (x1, x, x, x, … ) all future generations
make a uniform sacrifice, leading to a gain for generation 1, i.e., z  x1  1 and 0  x  z, then
again the class of TDU SWOs leads to the conclusion that the ranking depends on the
gain/sacrifice ratio. Any TDU SWO is consistent with the position that the infinite number of
future generations should make a uniform sacrifice increasing the well-being of the present
generation if their uniform sacrifice is small relative to its gain, but not if their sacrifice is
relatively large.
9 However, in the latter case, the conclusion might not be deemed appealing. If the infinite
number of future generations makes a uniform sacrifice increasing the well-being of the
present generation, then, compared with the egalitarian stream, inequality is increased, and the
undiscounted sum of utilities is reduced (independently of the cardinal scale chosen). Hence,
both from an egalitarian and undiscounted utilitarian perspective, the egalitarian stream is
socially preferred to the stream where all future generations make a uniform sacrifice, leading
to a gain for generation 1, independently of the gain/sacrifice ratio. In particular, both the
undiscounted utilitarian and leximin SWRs considered in Section 3 lead to this conclusion.
As observed by Chichilnisky (1996), the TDU SWO is a dictatorship of present: if one stream
is strictly preferred to another, then what happens after some finite time does not matter for
the strict ranking. To take into account the interests of the generations in the infinite future,
Chichilnisky (1996) suggests a no dictatorship of the present (NDP) axiom, ruling out a
such a dictatorial role for the present. By dropping ST from the list of axioms characterizing
the TDU SWO and adding NDP, one obtains the following characterization.
Proposition 6: The following two statement are equivalent:
(1) An SWO satisfies axioms C, SP, SEP, SEF, and NDP.
(2) An SWO is numerically represented by an SWF W satisfying, for some sequence of strictly
increasing and continuous utility functions, Ut, t  1, 2, … , and some asymptotic part, ,
which is an integral with respect to a purely finitely additive measure,
W ( x1 , x2 , , xt ,)   t 1U t ( xt )   ( x1 , x2 , , xt ,) .

This characterization is based on Chichilnisky (1996, theorem 2), except that her
independence assumption has been replaced by the separability axioms SEP and SEF. By
choosing, for each t ≥ 1, U t ( xt )   t 1 xt for some discount factor , with 0 <  < 1, and
 ( x1 , x2 ,..., xt ,...)  lim inft  xt , it follows that Proposition 6 ensures the existence of an SWO
satisfying axioms C, SP, SEP, SEF, and NDP.
Chichilnisky (1996, definition 6) uses the term sustainable preference for a numerically
representable SWO satisfying axioms SP and NDP as well as a no dictatorship of the future
axiom. Because axiom SP implies the no dictatorship of the future axiom, the SWO
characterized by Proposition 6 is a sustainable preference. By comparing Propositions 5 and
6, it follows that an SWO satisfying axioms C, SP, SEP, SEF, and NDP does not satisfy
axiom ST because the sensitivity for what happens in the infinite future, as captured by
 ( x1 , x2 ,..., xt ,...) , rules out that the SWO is TDU. This means that such an SWO is not timeconsistent if the social evaluation is time-invariant.
10 When testing the class of sustainable preferences by its performance in applications, the
verdict is mixed. In simple present-future conflicts, its qualitative behavior is the same as that
for the class of TDU SWOs: The ranking depends on the gain/sacrifice ratio, both in
situations in which the present generation makes a sacrifice for the infinite number of betteroff future generations and in situations in which the infinite number of future generations
makes a sacrifice for the better-off present generation.
When a sustainable preference, in the class characterized by Proposition 6, is applied to
models of economic growth, there is a generic nonexistence problem, as welfare is increased
by delaying the response to the interests of the infinite far future, whereas welfare is
decreased if delay is infinite. This nonexistence problem has spurred an interest in how to
adapt Chichilnisky’s sustainable preferences to ensure applicability (e.g., see Heal 1998, Li &
Löfgren 2000, Alvarez-Cuadrado & Long 2009, Figuieres & Tidball 2012).
The performance of undiscounted utilitarian and leximin SWRs, as discussed in the previous
section, and the TDU SWO, as discussed in this section, motivates the following question:
Are there sets of axioms leading to classes of SWOs allowing for trade-off in present-future
conflicts in which the present generation makes a sacrifice for the infinite number of betteroff future generations, while giving priority to the future in situations in which the infinite
number of future generations makes a sacrifice for the benefit of the better off present
generation?
[FIGURE 3 ABOUT HERE]
One possibility, suggested by Asheim et al. (2012) and illustrated in Figure 3, is to introduce a
Hammond equity for the future (HEF) axiom, which requires that (x1, x, x, …) is deemed
as good as (y1, y, y, …) if y1  x1  x  y. For streams in which well-being is constant from the
second period on, axiom HEF captures the idea of giving priority to the infinite number of
future generations in the choice between alternatives in which the future is worse off than the
present.
Contrary to the standard Hammond equity axiom, the transfer from the better-off present to
the worse-off future specified in axiom HEF leads to an infinite increase in the sum of
wellbeing, independently of what cardinal scale is used to measure well-being. Hence, axiom
HEF is satisfied by both the undiscounted utilitarian and leximin SWRs considered in Section
3. In particular, it is weaker and more compelling than the standard Hammond equity axiom.
The sustainable discounted utilitarian (SDU) SWO, introduced by Asheim & Mitra (2010),
satisfies axiom HEF. The SDU SWO modifies TDU by requiring that the SWO not be
11 sensitive to the interests of the present generation if the present generation is better off than
the future:
(1   )U ( x1 )   W ( x2 , x3 , , xt ,) if U ( x1 )  W ( x2 , x3 , , xt ,)
W ( x1 , x2 , , xt ,)  
W ( x2 , x3 , , xt ,) if U ( x1 )  W ( x2 , x3 , , xt ,)

W ( z , z , , z ,)  U ( z ) .
The SDU SWO means that future utilities are not discounted (the discount factor is set to 1; the
utility discount rate is set to 0) if the present is better off than the future. In this case, present
utility is given zero weight. The utility weights are still of the form 1 – , (1 – ), … , (1 –
)t–1, … if generations with zero utility weight are left out, implying that the utility weights
add up to one also for the SWF representing the SDU SWO. This means that the utility of
each generation is comparable to the welfare of the stream and makes the comparison
between U ( x1 ) and W ( x2 , x3 , , xt ,) meaningful. In particular, the welfare of an egalitarian
stream is equal to the utility of the constant level of well-being. The SDU SWO is welldefined and unique on the set of streams where each generation’s well-being is between 0 and
1, with the corresponding SWF determined by the algorithm of Asheim & Mitra (2010, proof
of theorem 1).
The SDU SWO satisfies axioms C, M, RSP, RSEP, SEF, ST, NDP, and HEF, where axioms
RSP and RSEP are restricted versions of axioms SP and SEP applied only to the domain of
non-decreasing streams. Imposing axiom HEF comes at the cost of weakening the strong
Pareto axiom and the axiom of separable present. In addition to being insensitive to the wellbeing of the current generation if its well-being exceeds future welfare (thereby not satisfying
axiom SP on a non-restricted domain), the SDU SWO fails to satisfy axiom SEP in this
circumstance. However, if one accepts the intuition that the stream  0, 34 ,1,1,1, is socially
preferred to  14 , 14 ,1,1,1, , while  0, 34 , 14 , 14 , 14 , is not socially preferred to
 14 , 14 , 14 , 14 , 14 , , then one supports this weakening of axiom SEP. It is not obvious that we
should treat the conflict between the worst-off and the second worst-off generation in the first
comparison in the same manner as the conflict between the worst-off and the best-off
generation in the second comparison.
The idea of restricting axioms to the domain of non-decreasing streams opens interesting
possibilities. Restricting also axioms SEF and ST to the set of non-decreasing streams
(leading to axioms RSEF and RST) and combining these axioms with axioms C, M, RSP,
and RSEP, leads to a characterization of TDU on the set of non-decreasing streams (Zuber &
Asheim, 2012, prop. 2). Adding the strong axiom of equal treatment, axiom SA, results in a
characterization of the rank-discounted utilitarian (RDU) SWO.
12 Under RDU, streams are first reordered into a non-decreasing stream, so that discounting
becomes according to rank, not according to time. For any streams (x1, x2, … , xt, … ), let
( x[1] , x[2] , , x[ r ] ,) denote the rank-ordered permutation of all elements xt, so that x[ r ]  x[ r 1]
for all ranks r, taking into account that streams with elements of infinite rank, like (1, 0, 0, 0,
…), cannot be reordered into a non-decreasing stream.ii
Proposition 7 (Zuber & Asheim, 2012, theorem 1): The following two statement are
equivalent:
(1) An SWO satisfies axioms C, M, RSP, RSEP, RSEF, RST, and SA.
(2) An SWO is numerically represented by an SWF W satisfying, for some strictly increasing
and continuous utility function U, and utility discount factor , with 0 <  < 1,
W ( x1 , x2 , , xt ,)  (1   ) r 1  r 1U ( x[ r ] ) .

The RDU SWO fills out the gap between the undiscounted utilitarian and leximin SWR,
approaching the former as  goes to 1 and the latter as  goes to 0. In addition to the axioms
listed under point (1), the RDU SWO satisfies also axioms NDP and HEF, thereby satisfying
the requirements suggested by Chichinisky (1996) and Asheim et al. (2012) to take into
account the interests of future generations.
The SDU and RDU SWOs allow for trade-offs in present-future conflicts in which the present
generation makes a sacrifice for the infinite number of better-off future generations, while
giving priority to the future in situations in which the infinite number of future generations
makes a sacrifice for the benefit of the better-off present generation. Moreover, Asheim &
Mitra (2010) and Zuber & Asheim (2012, Section 6) show that the SDU and RDU SWOs lead
to sustainable streams in the Ramsey and DHSS models, while Dietz and Asheim (2012)
apply the SDU SWO to numerical evaluation of climate policies.
Both SDU and RDU discount future utility as long as the future is better off than the present,
thereby trading off current sacrifice and future gain. In this case, the future's higher well-being
is discounted because, at a higher level, added well-being contributes less to utility (if the
utility function U is stricly concave), and being better off, its utility is assigned less weight.
Hence, if well-being is perfectly correlated with time, these criteria work as the ordinary TDU
criterion which economists usually promote. The important difference is that, in the criteria of
SDU and RDU, the future is discounted because priority is given to the worse-off earlier
generations.
However, if the present is better off than the future, then priority shifts to the future. In this
case, future utility is not discounted, implying that zero relative weight is assigned to present
wellbeing. The criteria of SDU and RDU can therefore capture the intuition that we should be
13 more willing to assist future generations if they are worse off than us, but not to save much for
their benefit if they turn out to be better off.
[TABLE 1 ABOUT HERE]
5. Concluding remarks
In this chapter I have considered six different criteria of intergenerational equity, the
properties of which are summarized in Table 1.
I end my discussion by posing two questions:
1. Can we apply these criteria to resolve the kind of intergenerational conflicts presented
in the introduction; e.g., for evaluating climate policies?
2. Do we want to use criteria that satify equity axioms; e.g., by treating generations
equally?
With respect to the first question, one has to take into account that climate policies do not
only lead to redistribution along time — leading to a generational conflict — but also among
people living at different points in space. Also, such polices have uncertain consequences,
which amounts to a redistribution across different uncertain states. Finally, all time-spacestate combinations may not be inhabited, meaning that climate policies may lead to a different
number of people living in future, and that there is a positive probability of human extinction.
The criteria discussed in this chapter can to a varying degree accommodate these concerns,
but space considerations do not allow a discussion of such generalizations here. I will argue,
though, that the concept of sustainability becomes more problematic as a primititive
conception of equity when not only inequality between generations is considered, but also
inequality across space and uncertain states. Perhaps one should rather base the notion of
equity on an axiomatically based impartial criterion, thereby making explicit the underlying
normative judgements.
As for the second question, decisions of climate change will actually be made through a
bargain between representatives of a variety of countries. Intergenerational equity will only be
taken into account to the extent these representatives care for future generations (cf. Schelling,
1995). And intragenerational equity, as opposed to strategic bargaining power, will only
matter if the representatives are concerned about people living in other countries.
Hence, impartial equity criteria might not be very relevant for the climate change decisions
that actually will be made. However, one might hope that the conclusion of impartial criteria
14 may influence the partial criteria that the decision-makers will use, so that what we ought to
do influences what we prefer to do. In any case, it is important for the analyst to distinguish
whether an evaluation is to be made from an impartial point of view, or whether criteria are to
be used by decision-makers that represent people with partial interests.
References
Alvarez-Cuadrado, F. and Long, N. V. 2009. A mixed Bentham-Rawls criterion for intergenerational
equity: Theory and implications. Journal of Environmental Economics and Management,
58:2, 154–168.
Arrow, K. J. 1999. Discounting, morality, and gaming, in Portney, P. R. and Weyant, J. P. (eds)
Discounting and Intergenerational Equity, Washington, DC: Resources for the Future, 13–21.
Asheim, G. B. and Banerjee, K. 2010. Fixed-step anonymous overtaking and catching-up.
International Journal of Economic Theory, 6:1, 149–165.
Asheim, G. B., Buchholz, W. and Tungodden, B. 2001. Justifying sustainability. Journal of
Environmental Economics and Management, 41:3, 252–268.
Asheim, G. B. and Mitra, T. 2010. Sustainability and discounted utilitarianism in models of economic
growth. Mathematical Social Sciences, 59:2, 148–169.
Asheim, G. B., Mitra, T. and Tungodden, B. 2012. Sustainable recursive social welfare functions.
Economic Theory, 49:2, 267–292.
Asheim, G. B. and Tungodden, B. 2004. Resolving distributional conflicts between generations.
Economic Theory, 24:1, 221–230.
Banerjee, K. 2006. On the extension of utilitarian and Suppes-Sen social welfare relations to infinite
utility streams. Social Choice and Welfare, 27:2, 327–339.
Basu, K. and Mitra, T. 2003. Aggregating infinite utility streams with intergenerational equity: The
impossibility of being Paretian. Econometrica, 71:5, 1557–1563.
Basu, K. and Mitra, T. 2007a. Possibility theorems for aggregating infinite utility streams equitably, in
Roemer, J. and Suzumura, K. (eds) Intergenerational Equity and Sustainability, UK: PalgraveMacmillan, 69–84.
Basu, K. and Mitra, T. 2007b. Utilitarianism for infinite utility streams: a new welfare criterion and its
axiomatic characterization. Journal of Economic Theory, 133:1, 350–373.
Bossert W., Sprumont Y. and Suzumura K. 2007. Ordering infinite utility streams. Journal of
Economic Theory, 135:1, 579–589.
Chichilnisky, G. 1996. An axiomatic approach to sustainable development. Social Choice and
Welfare, 13:2, 231–257.
Collard, D. 1996. Pigou and future generations: a Cambridge tradition. Cambridge Journal of
Economics, 20:5, 585–597.
15 Dasgupta, P. S, and Heal, G. M. 1974. The optimal depletion of exhaustible resources. Review of
Economic Studies, 41:symposium, 3–28.
Diamond, P. 1965. The evaluation of infinite utility streams. Econometrica, 33:1, 170–77.
Dietz, S. and Asheim, G. B. 2012. Climate policy under sustainable discounted utilitarianism. Journal
of Environmental Economics and Management, 63:3, 321–335.
Figuieres, C. and Tidball, M. 2012. Sustainable exploitation of a natural resource: a satisfying use of
Chichilnisky’s criterion, Economic Theory, 49:2, 243–265.
Fleurbaey, M. and Michel, P. 2003. Intertemporal equity and the extension of the Ramsey criterion.
Journal of Mathematical Economics, 39:7, 777–802.
Hammond, P. J. 1976. Equity, Arrow’s conditions, and Rawls’ difference principle. Econometrica
44:4, 793–804.
Haub, C. 2011. How many people have ever lived on earth? Population Reference Bureau,
www.prb.org/Articles/2002/HowManyPeopleHaveEverLivedonEarth.aspx
Heal, G. M. 1998. Valuing the future: Economic theory and sustainability. New York: Columbia
University Press.
Kamaga, K. and Kojima, T. 2009. Q-anonymous social welfare relations on infinite utility streams.
Social Choice and Welfare, 33:3, 405–413.
Kamaga, K. and Kojima, T. 2010. On the leximin and utilitarian overtaking criteria with extended
anonymity. Social Choice and Welfare 35:3, 377–392.
Koopmans, T. C. 1960. Stationary ordinal utility and impatience. Econometrica, 28:2, 287–309.
Lauwers, L. 1997. Infinite utility: insisting on strong monotonicity. Australasian Journal of
Philosophy, 75:2, 222–233.
Lauwers, L. 2010. Ordering infinite utility streams comes at the cost of a non-Ramsey set. Journal of
Mathematical Economics, 46:1, 32–37.
Li, C.-Z. and Löfgren, K.-G. 2000. Renewable resources and economic sustainability: a dynamic
analysis with heterogeneous time preferences. Journal of Environmental Economics and
Management, 40:3, 236–250.
Nordhaus, W. D. 2008. A Question of Balance: Weighing the Options on Global Warming Policies.
New Haven, CT: Yale University Press.
Pigou, A. C. 1920. The Economics of Welfare. London: Macmillan.
Ramsey, F. P. 1928. A mathematical theory of saving. Economic Journal, 38:152, 543–559.
Rawls, J. 1971. A Theory of Justice. Cambridge, MA: Harvard University Press.
Schelling, T. C. 1995. Intergenerational discounting. Energy Policy, 23:4–5, 395–401.
Sen, A. 1970. Collective Choice and Social Welfare. San Francisco: Holden-Day.
Sidgwick, H. 1907. The Methods of Ethics. London: Macmillan.
Solow, R. M. 1974. Intergenerational equity and exhaustible resources. Review of Economic Studies,
41:symposium, 29–45.
Stern, N. 2007. The Stern Review on the Economics of Climate Change. Cambridge, UK: Cambridge
University Press.
16 Stiglitz, J. 1974. Growth with exhaustible natural resources: Efficient and optimal growth paths.
Review of Economic Studies, 41:symposium, 123–137.
Svensson, L.-G. 1980. Equity among generations. Econometrica, 48:5, 1251–1256.
Zame, W. 2007. Can intergenerational equity be operationalized? Theoretical Economics, 2:2, 187–
202.
Zuber, S. and Asheim, G. B. 2012. Justifying social discounting: the rank-discounted utilitarian
approach. Journal of Economic Theory, 147:4, 1572–1601.
Table 1
Richn & robustn
Separability
Efficiency
Equity
Undisc. U SWR
SEP, SEF, ST
M, SP
NDP, HEF, FA
Leximin SWR
SEP, SEF, ST
M, SP
NDP, HEF, FA
Time-DU SWO
O, C
SEP, SEF, ST
M, SP
Chichilnisky
O, C
SEP, SEF
M, SP
NDP
Sust. DU SWO
O, C
RSEP, SEF, ST
M, RSP
NDP, HEF
Rank-DU SWO
O, C
RSEP,RSEF,RST
M, RSP
NDP,HEF,FA,SA
Axiom O signifies that the SWR satisfies completeness in addition to reflexivity and transitivity. The prefix R
signifies that the axiom in question is imposed on the restricted domain of non-decreasing streams.
17 Figure 1
Figure 2
18 Figure 3
Notes
i
When using the term ’utility’ I will refer to a specific cardinal scale for generational well-being and a
’utilitarian criterion’ will refer to a criterion making use of such a scale. No specific view on what constitutes
individual well-being is therefore implied by this terminology.
ii
Formally, let  ( x1 , x2 ,  , xt , ) denote liminf of (x1, x2, … , xt, … ), and let L(x1, x2, … , xt, … ) be the set of
times t for which xt   ( x1 , x2 ,  , xt , ) . If |L(x1, x2, … , xt, … )|  , let ( x[1] , x[ 2 ] ,  , x[ r ] , ) denote the rankordered permutation of all elements xt with t  L(x1, x2, … , xt, … ), so that x[ r ]  x[ r 1] for all ranks r. If |L(x1, x2,
… , xt, … )|  R  , let ( x[1] , x[ 2 ] ,  , x[ R ] ) denote the rank-ordered permutation of all elements xt with t  L(x1,
x2, … , xt, … ), so that x[ r ]  x[ r 1] for all ranks r = 1, … , R1, and set xr   ( x1 , x2 ,  , xt , ) for all r  R.
Acknowledgement
I thank Simon Dietz for helpful comments. This paper is part of the research activities at the Centre for the Study
of Equality, Social Organization, and Performance (ESOP) at the Department of Economics at the University of
Oslo. ESOP is supported by the Research Council of Norway.
19