Journal of Agricultural and Environmental Ethics (2005) 18: 237–257
DOI 10.1007/s10806-005-1491-8
Ó Springer 2005
YEW-KWANG NG
INTERGENERATIONAL IMPARTIALITY:
REPLACING DISCOUNTING BY PROBABILITY WEIGHTING
(Accepted in revised form December 10, 2004)
ABSTRACT. Intergenerational impartiality requires putting the welfare of future
generations at par with that of our own. However, rational choice requires weighting
all welfare values by the respective probabilities of realization. As the risk of nonsurvival of mankind is strictly positive for all time periods and as the probability of
non-survival is cumulative, the probability weights operate like discount factors,
though justified on a morally justifiable and completely different ground. Impartial
intertemporal welfare maximization is acceptable, though the welfare of people in the
very far future has lower effects as the probabilities of their existence are also lower.
However, the effective discount rate on future welfare values (distinct from monetary
values) justified on this ground is likely to be less than 0.1% per annum. Such
discounting does not compromise environmental protection and sustainability unduly. The finiteness of our universe implies that the sum of our expected welfare to
infinity remains finite, solving the paradox of having to compare different infinite
values in optimal growth/conservation theories.
KEY WORDS: discounting, environmental ethics, impartiality, intergenerational,
intertemporal, probability, sustainable development, welfare
1.
INTRODUCTION
Environmental values, caused either by agricultural or non-agricultural
activities, typically involves costs and benefits long into the future, generations
from now. Thus the issues of intergenerational equity is usually involved. In
particular, should future values be discounted? The use of a high discount rate
favors the present generations at a possibly much bigger cost on the future
generations. Using a low or zero discount rate may be giving undue weights to
the far future. Especially for problems with an infinite time horizon, not using
any discounting may mean the absence of a solution. While even the whole
human species is unlikely to live forever, we do not know when will be our
extinction year. It is true that the Earth may become uninhabitable in less than
two billion years as a gradually warming Sun produces a runaway greenhouse
effect. But despite these, we cannot be absolutely certain that we cannot live
beyond 2 or even 20, 200 billion years. The possible scope of scientific advance
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YEW-KWANG NG
in just the next few thousand years is unimaginable to us now. Imagine
someone just 3 or 4 hundred years ago. His expectation of what is possible by
the year 2005 would be very, very wide off the mark. Thus, how could we
completely rule out the possibility that, after some thousand or hundred of
thousand years from now, we may no longer have to rely on the Earth, the Sun,
or even the Milky Way for our survival. We may estimate that this is unlikely
even in a million years, but we cannot validly rule it out completely. In view of
this, an infinite time horizon with uncertainty is an appropriate one. On the
other hand, we cannot be absolutely certain of our survival even for the next
century. There is always a very small but positive probability that we may be
destroyed by a celestial collision, a nuclear war, or some causes we may not be
able to imagine. Thus, neither a finite nor an infinite time horizon with certainty is appropriate. Rather, an infinite time horizon with uncertainty is an
appropriate one. In other words, the probability of our surviving into the
future decreases with time but will never quite reach absolute zero for any
finite time.1
Ramsey (1928), the first mathematical analyst of optimal savings, regarded the discounting of future utilities as arising from the ‘‘weakness of
imagination’’ (p. 543) about the future. However, without discounting, the
sum/integral may go to infinity. How does one compare different paths that
all have an infinite present value of utility through time? (For philosophical
discussion of this see Segerberg, 1976; Vallentyne and Kagan, 1997). Ramsey
adopted an ingenious method of minimizing the short-fall from a posited bliss
level of utility. This was later overtaken by the overtaking criterion of Weizsacker (1965). (If the cumulated utility value, i.e., instantaneous utility
integrated to any given time, of stream A overtakes that of stream B at any
finite time and remains higher than that of B forever thereafter, stream A is to
be preferred to stream B.) However, both the Ramsey criterion and the
overtaking criterion are not complete (see Chichilnisky, 1996). Thus, studies
of optimal savings/growth typically use a positive discount rate.
The discounting of the future has been objected to by many ecologists,
economists, environmentalists, and sustainable development analysts (e.g.,
Doeleman, 1980; Broome, 1992; Cline, 1992; Peterson, 1993; Price, 1993;
Beckerman, 1995; Portney and Weyant, 1999; Islam, 2001).2 Many objections to discounting are caused by confusing the discounting of future dollar
1
The problem of intergenerational justice is a complex one (see, e.g., Beekman
(2004) and references therein). In this paper, the problem is viewed from the perspective of appropriate time discouting. Elsewhere (Ng, 2004), I also argue that the
problem of sustainability is more one of the disregard of the welfare of others now in
environmental disruption rather than the disregard of the welfare of future generations.
2
See also Spash (1993) on compensatory transfers.
INTERGENERATIONAL IMPARTIALITY
239
values and the discounting of future subjective values (utility or welfare), as
pointed out by Dasgupta (1995, pp. 119–120). The future generations should
be treated equally as our own generation, at least from the viewpoint of
intergenerational impartiality. However, this need not mean that future
economic values including consumption, incomes, and wealth should not be
discounted. One valid reason to discount future objective economic values
(in contrast to the subjective values like utility or welfare) is based on the
fact that the marginal productivity of capital is usually positive. A dollar of
product now can be invested to become more than a dollar in the future.3
Thus, a dollar now is worth more than a dollar in the future. Equivalently,
a dollar in the future is worth less than a dollar now and should thus be
discounted even if we treat the same unit of subjective utility or welfare as
the same whether it occurs (assumed to happen with perfect certainty) in the
present or in the future. This valid reason for discounting future dollars does
not justify the discounting of future subjective values, which is the concern
of this paper. Nevertheless, the absence of discounting future subjective
values (while allowing for the discounting of future dollar values) may be
sufficient to give rise to paradoxes of no solutions and incompleteness, when
the time horizon is infinite. Even where solutions exist, they may be rather
unacceptable, such as requiring the present generations to save more than
50% of the GNP under plausible parameters (Mirrlees, 1967). Adopting a
finite time horizon without discounting is not acceptable either.4 While the
extinction of our species is possible, we do not know when it may occur.
A definite terminal date is thus not reasonable. For any terminal date T, the
welfare of people who would or might live after T is ignored and the welfare
of people who might live in T minus epsilon is given a full weight. This
abrupt difference in treatment cannot be justified unless we know for sure
that mankind will be terminated precisely at time T. Even for those who
believe that the probability of our surviving 100 trillion years from now is
absolutely zero, so that ignoring the welfare of people more than 100 trillion
years from now is reasonable, it is still unreasonable to give full weight to
the welfare of people 99 or 90 or 9 trillion years from now whose probability
3
Another reason for discounting future dollar values is that a dollar of future
consumption is likely to have lower welfare value as the income levels in the future
will likely be larger. With optimal savings/capital accumulation, the two reasons give
the same discount rate. Newell and Pizer (2003) point out that the discount rate on
dollar values should be significantly lower than that suggested by the current practice
if the future path of the discount rate is uncertain and highly correlated.
4
Apart from the point discussed in the text, it also violates the non-dictatorship of
the present (Chichilnisky, 1996). ‘‘We shall say that a welfare function W. . . is a
dictatorship of the present, if W. . . disregards the utility levels of all generations from
some generation on.’’
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YEW-KWANG NG
of survival is close to zero or very small. So how could we avoid the above
paradoxes and also avoid the possibly morally unacceptable discrimination
against the future generations. This paper suggests a solution.
The next section sets out a set of reasonable axioms and shows that
intertemporal impartial welfare maximization satisfies these axioms, that the
use of probability (of survival) weighting is consistent with intertemporal/
intergenerational impartiality and, due to the similar cumulative nature of
discounting and the survival probability, probability weighting can be used
in place of discounting to avoid the well-known paradoxes associated with
an infinite time horizon. This avoidance is not necessarily possible even with
probability weighting if the population size may increase. Section 3 argues
that the allowance for a variable population size does not change our
conclusions in Section 2. Despite some possibly misleading appearance at
certain places in the argument and my personal strong support for a utilitarian social welfare function (that the society should maximize the unweighted sum of individual welfares), the argument of this paper is not
based on utilitarianism.
2.
THE IMPARTIALITY OF PROBABILITY-WEIGHTED
WELFARE MAXIMIZATION
Consider different states of the world at various time period
t ¼ 0; 1; 2; . . . ; 1. As we may define the relevant time period as fine as we
like (instead of year, we may use second or micro-second), this periodic
treatment of time is not really restrictive. Moreover, making the time period
instantaneous and hence going into the continuous time approach only involves replacing the various summations below by integrations, without
changing the substance. For the present (i.e., for t ¼ 0), we start with a given
state x0 . (Starting with t ¼ 1 makes no difference. The state x need not be
confined to consumption and may include all relevant factors including the
stock of natural capital; cf. Heal, 1998.) For any given time t in the future
(i.e., for t > 0), alternative states xst may prevail. Let the number of alternative states that may prevail at time t be nt . Let the welfare value (or
expected welfare value if uncertainty is present) associated with x st be denoted as wðxst Þ. It is not necessary to assume that, given the state xst , the
welfare value wðxst Þ must also be known with certainty. For each xst , we may
have a probability distribution
of the different values of welfare v associated
R1
with it. Then wðxst Þ ¼ 1 fðvÞ:vðxst Þdv is the expected welfare of xst . (This is
equals to the sum over all possible states of the welfare value in each possible
state weighted by the probability of that state.) Let the probability associated with xst be denoted pst . Then the (unconditional) expected welfare at t is
INTERGENERATIONAL IMPARTIALITY
wut
¼
nt
X
pst :wðxst Þ;
241
ð1Þ
s¼1
Pt s
where ns¼1
pt ¼ 1 and the superscript u to wt just indicates that it is the
unconditional value. (See below on the conditional value.)
Some of the nt states xst at any given t may involve non-survival of the
human species. We may thus define the expected welfare at t conditional
upon our survival as
a
wt ¼
nt
X
s¼1
a
pst :wðxst Þ=
nt
X
pst ;
ð2Þ
s¼1
where the convention of listing all those states involving our survival first is
used, with nat standing for the number of states we do survive at period t (a
for ‘alive’). In other words, given the condition that we do survive at period
t, our expected welfare is given by wt . If the probability of our non-survival
at period t (as seen from time 0) is strictly positive and wt is strictly positive,
wt > wut if Axiom 1 below is accepted.
Axiom 1 (Zero welfare of non-survival): If xst involves our non-survival,
wðxst Þ ¼ 0:
This axiom is compelling if we agree that we are doing welfare calculus
for this human world. With the welfare of non-human beings (including
animals, on which see Ng, 1995, and life after death, on which I have no
expertise) not considered, if we (i.e., the whole human species) do not survive to period t, our welfare at period t must be zero.
It may be noted that this axiom does not imply that the welfare of people
who do survive is necessarily positive. If their welfare is negative, it can be
treated accordingly.
It is now well established that no reasonable social choice involving
interpersonal (including inter-generational) tradeoffs could be made without
using interpersonally comparable cardinal utility/welfare levels. (See Mueller, 1989, chapter 19 and Ng, 2000, chapter 2 that discuss the results of Sen,
1970 on the necessity of interpersonal comparisons and Kemp and Ng, 1976;
Parks, 1976; Roberts, 1980 and others on the necessity of cardinal utility).5
Most welfare economic analysts accept what Bergson (1938) calls ‘‘the
5
In fact, we need what may be called fully cardinal utility/welfare levels, i.e., ratioscale measurability (where only the unit of measurement used is arbitrary) rather
than just functions unique up to increasing affine (linear) transformations that do not
have a well-defined zero point. It may also be noted that it is only economists
unfamiliar with the necessity of interpersonal cardinal utility who have hesitation/
objection to its use. Philosophers (e.g., Vallentyne and Kagan, 1997, p. 7) use
interpersonal cardinal welfare indices with well-defined zeros readily.
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YEW-KWANG NG
fundamental value premise of individual preference’’ (called welfarism by
John Hicks and Amartya Sen) that the objective variables are only important, at least ultimately speaking, only through their effects on individual
preferences (represented by utilities) or welfare (the differences between
utility and welfare are ignored here; on which see Ng, 2000, chapter 4).
Then, without some interpersonal comparisons of utility (representing
preference) or welfare, we cannot say, e.g., that this generation has not left
enough natural or man-made capital for the future generations. No intertemporal (or interpersonal at a given point in time for that matter) ethics is
really possible. In fact, this is still largely true even if we accept welfarism
only partially; that welfare is important, even if not the only important
thing. Thus, I have no need to apologize for the use of cardinal welfare (also
used by other analysts in this area) in the next axiom. (Ordinalism is however sufficient for the objective theories of consumption, demand, and
general equilibrium.)
Axiom 2 (Intertemporal parity): Once expressed in the interpersonally comparable cardinal indices, the welfare level at all times should be treated at par
with that at any other.
This requires that the welfare of people in say the 300th century
should not be treated as less (or more) important than the welfare of
people now. This axiom is sufficient to ensure intertemporal (including
inter-generational) equity. The only qualification to this axiom is that, if
the welfare level refers to representative welfare of an individual, a
complication may arise if the number of people at different times differ.
We will have to find some aggregate welfare measure for a changing total
population that is intertemporally comparable. But this may be regarded
as a separate problem of how to evaluate social welfare when the population is changing, a problem really beyond the main focus of this
paper. Social optimality involving different sets of population raises wellknown paradoxes. As I have discussed these in Ng (1989) (which argues
in favor of classical utilitarianism maximizing the equally weighted sum
of individual welfares at the ideal moral level but allowing for our possible bias in favor of our own welfares; but see Blackorby et al., 1997 and
Carlson, 1998 on other attempts at the problem and the associated difficulties), I will abstract from this issue by ignoring possible changes in
population size here. (Section 3 shows that that allowing for a variable
population size does not affect our conclusions.) In this section, we take
the population size as unchanged until the time of extinction when the
population size abruptly drops to zero. This makes Axiom 2 compelling
INTERGENERATIONAL IMPARTIALITY
243
as the same welfare value of the same number of people should be
treated equally, as otherwise there is no intertemporal parity.
Axiom 3 (probability weighting): In the presence of uncertainty, any welfare
level should be weighted by the probability of its realization.6
This is just the requirement of rationality for choices involving uncertainty and is familiar to all economists and decision theorists as it is used in
the calculation of expected utility of any action where the utility of any
possible state has to be multiplied by the probability of realization of that
state. This is also done at the beginning of this section for the calculation of
expected welfare and is widely accepted as a compelling condition for rational choice.7 A commentator pointed out that, when some outcomes involve some very bad results, they may be avoided irrespective of the
expected gain. For example, insurance companies do not sell insurances if it
means that the company would go bankrupt should the worse outcome be
realized, no matter how improbable it is. Since bankruptcy probably means
the unemployment of most of the employees of the company and since
unemployment has very high disutility or negative welfare (Winkelmann
and Winkelmann, 1998), the above practice of many insurance companies
may be a good rule-of-thumb method to approximate expected welfare
maximization. Similarly, some environmentalists may find it morally
appealing to eliminate alternatives that involve extinction or terrible suffering for the future generations. If my argument below that the annual risk
of extinction that is beyond our control is very small for a long time to
come, the impartial maximization of expected welfare does not differ significantly from this somewhat overprudent (from the viewpoint of impartial
expected welfare maximization) policy. In other words, both policies would
imply that measures that could (even at very low, though not vanishing,
probability) cause enormous harm or extinction should be avoided even at
very big costs on the present generations.
6
A reviewer of this paper suggests that the essence, though not the precise
statement, of this axiom may be traced to Kant who wrote ‘‘human nature is such
that it cannot be indifferent even to the most remote epoch which may eventually
affect our species, so long as this epoch can be expected with certainty’’ (see Reiss,
1970/1991, p. 50).
7
Some economists regard the use of expected utility as just a convention and has
no normative significance and that the utility indices used need not be the subjective
cardinal utility of the Neoclassical economists. Under some compelling axioms, I
have shown the normative significance and the subjective (fully) cardinal utility/
welfare nature in Ng (1984).
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YEW-KWANG NG
Axiom 4 (Positive risk): The probability of our non-survival at any future time
is strictly positive.
This is reasonable for both an individual and the whole species; the
axiom here is meant for the species. For an individual, she may be certain
of her survival now. However, for the future, no one can be sure for
certain. Fatal accidents may strike at any time. Consider the victims in
the World Trade Center on 11 September 2001. The same is true for our
species as celestial collisions and other unexpected events may happen,
though the time scale involved is much larger and the probability of nonsurvival is much smaller. The probability of my non-survival 100 years
from now is close to 100%, at least over 90%, but the same for the
whole human species is close to zero (but strictly positive), at least less
than 10%.
Axiom 5 (Non-absolute expected risk): The expected risk of our non-survival
at any year, given survival to the previous year, is smaller than one.
While the risk of non-survival for any given non-degenerate (i.e., not
just a point in time but a discrete period like a year) period is strictly
positive as required in the previous axiom, it is compelling that it is not
absolute in the sense of absolute certainty of non-survival for a particular
year, even given survival to the previous year. It is true that, given the
likely non-perpetual survival, we will become extinct in some future year.
However, we do not know when this particular year will be. Moreover,
we cannot rule out the (even if only slight) possibility of our future
ingenuity in postponing extinction very dramatically such as by moving
to another planet in another solar system. Thus, in terms of expected risk,
this axiom is compelling.
Axiom 6 (cumulative risk): The probability of non-survival is cumulative over
time.
If the probability of my not surviving the first decade from now is 10%
(I hope that this is an exaggeration for ease of calculation rather than
reflecting the true probability), the probability of my surviving this first
decade is 90%. Given that I do survive the first decade, suppose that the
probability of my not surviving the second decade would be 20% and my
survival probability would be 80%. However, looking from now before we
know whether I will survive even the first decade or not, my surviving the
second decade is only 90% times 80% ¼ 72% or my non-survival probability is 28%. This is what is meant by the cumulative nature of
INTERGENERATIONAL IMPARTIALITY
245
non-survival. For the simple case of a constant rate of per-period nonsurvival risk of r, the probability of survival decreases with time t at ð1 rÞt .
(For the instantaneous/continuous case, it is ert .) However, the cumulative
nature of non-survival is general and applies also to variable rates of nonsurvival risk.
Figure 1
For the next axiom, we need a definition of semi-constant or nondecreasing risk (of non-survival). The case of constant risk r is a special case
of semi-constancy in risk. However, semi-constant or non-decreasing risk is
more general and allows the risk to vary with t but with the probability of
survival eventually decreases to no larger than the case of constant risk for
some positive and constant risk r > 0. In Figure 1, curve A depicts the
probability (measured along the vertical axis) of survival (expected at time 0)
to the respective time (on the horizontal axis) for a case of constant risk.
Curve B depicts a case of non-constant risk but with the probability of
survival eventually decreases to a level no higher than A after some point
t ¼ T and remains lower through to infinity. This is called semi-constant or
non-decreasing risk.
246
YEW-KWANG NG
Definition. If the probability of survival decreases (not necessarily at a
constant rate) and eventually stays no higher than the probability of survival
of a constant (positive) risk at all time after some T through to infinity, it is
called a case of semi-constant or non-decreasing risk.
Axiom 7 (semi-constant or non-decreasing risk): The risk of non-survival is
semi-constant or non-decreasing with time.8
First, consider the non-survival risk faced by a typical individual. After
the (life) birth of an individual, the probability that she is alive right then
is 100%. The probability that she will die within a year is the infant
mortality rate. The probability of death at any period is strictly positive
but very small between the age of a few years old and middle age. After
about age 60 or 70 (depending on the public health condition of the
country concerned), the risk increases significantly. Looking at the time of
birth, the probability of survival to the respective future age/time decreases strictly throughout due to the positive risk of death at any time.
(See Figure 2.) However, the rate of decrease may increase or decrease.
Typically, this rate is very high soon after birth but decreases to a very
low figure around youth. It increases after middle/old age. The absolute
risk is likely to decrease again at some very old age, due to the fact that
when the survival probability (expected at birth) has decreased to a very
low figure, say 5% for age 90, there is not much scope for further
absolute decrease. Even if, given survival to age 90, there is as high as
20% of non-survival in the year of age 90, the absolute decrease is only
1% point from 5% to 4%. In contrast, the absolute decrease at age 75
could be 4% points from 40% to 36%, implying however a lower relative
decrease of just 10% in comparison to the relative decrease of 20% at age
90. Put it differently (though not strictly equivalently except under the
additional assumption of no changes in age structure and the profile of
life expectancy), an average person aged 90 is more likely to die within a
year than a person aged 75, but there are more people dying at aged 75
than at age 90 simply because there are less 90 years-old around. (The
figures are all hypothetical, though selected to be reasonable.) For our
purpose here, it is the relative (or conditional) risk that is relevant that is
likely to keep increasing with age after middle age. Although the rate of
absolute decrease in the survival probability decreases at very old age, the
probability of survival still decreases very fast, approaching zero well
before the age of 200. Thus, the risk faced by an individual satisfies
Axiom 7.
8
Axioms 5 and 7 are needed only for Proposition 4.
INTERGENERATIONAL IMPARTIALITY
247
Figure 2
For the whole human species, the risk of non-survival is much smaller
than that of an average (in fact any) person. Whether that risk increases or
decreases is also debatable and a general pattern like that for an average
individual discussed above may be absent. In the absence of more specific
knowledge except that the risk is strictly positive at any period and rather
small, the expectation that the risk is roughly constant may be acceptable,
making the survival probability decreases at ð1 rÞt or ert as already noted.
This is represented by curve A in Figure 1 and is taken as a special case
satisfying Axiom 7. However, our requirement is more general than this,
allowing cases like curve B, which involves some sections of decreasing risk
but not strongly and persistently enough to make the curve staying above any
curve of constant positive risk throughout (i.e., through to infinity). In reality,
one may argue that we will sooner or later be faced with increasing risk when
our sun is expected to run out of energy, though this is likely to occur only
billions of years away. Thus, while we cannot rule out some periods of
decreasing risk, the decrease in risk cannot be expected to be persistent enough to make Axiom 7 inapplicable. Thus, Axiom 7 is compelling.
Axiom 8 (Finite welfare): The welfare level at any time is finite.
Common sense and biological limits suggest that no one can have infinite
welfare. Under our current premise of a constant population size (but see
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YEW-KWANG NG
the next section for a variable population size), it is also compelling that for
the whole species, the welfare level is also finite at any one time.. Moreover,
allowing for infinite welfare gives rise to paradoxes (including the wellknown St. Petersburg paraxdox; see Arrow, 1964). Other analysts (e.g.,
Chichilnisky, 1996; Vallentyne and Kagan, 1997, p. 7) also reasonably assumes finite utility/welfare (or ‘‘goodness’’) levels.
Axiom 8 allows us to follow Chichilnisky in normalizing welfare levels to
be bounded above by a constant that, without loss of generality, is taken to be
one. For example, if we take the upper welfare limit to be 10 to the power of
999,999,999,999 quintrillion times that of our present welfare value (assumed
positive and non-infinitesimal; or replace ‘‘our present welfare value’’ by ‘‘the
highest welfare value that we would presently enjoy if everyone on earth is as
happy as the happiest person’’), we have the slight inconvenience of having to
use small numbers in the order of 0.000 . . . (999,999,999,999 quintrillion of
them) 1 to describe the size of our present welfare. However, conceptually, this
is not a problem at all. If one is unsure that a welfare limit that large is
unsurpassable, one may just replace the ‘‘quintrillion’’ above by ‘‘quintrillion
to the power of [quintrillion to the power of (quintrillion to . . . and so on for a
quintrillion times’’ and also use the original English (where a billion is a
million millions and a trillion is a billion billions and so on) rather than the
American (where a billion is a thousand millions) definition of quintrillion.
And of course we may complicate the above process for another quintrillion
times, if necessary to satisfy skeptics, without reaching infinity.
Axiom 9 (Welfare relevance): The welfare levels now and in the future are
relevant to our choice.
There are economists and philosophers (e.g., Amartya Sen, being both)
who regard that there are principles important over and above and independent of their contributions to welfare, while I regard welfare as ultimately the only important thing (Ng, 2000, chapter 3). Injustice is the denial
of due welfare. In a short-term analysis, the allowance for other things is
important as the violation of certain desirable principles now may be detrimental to welfare in the longer run. In our present analysis of an infinite
time horizon, ultimate exclusive concern with welfare is defensible. However, Axiom 9 only requires some, rather than exclusive, concern with
welfare and hence is non-controversial. (The proofs of all propositions below are provided in Appendix A.)
Proposition 1 (Intertemporal impartial welfare maximization). Where nonwelfare considerations are unchanged, the criterion of maximizing the unweighted or equally weighted sum/integral of (unconditional and comparable)
welfare
249
INTERGENERATIONAL IMPARTIALITY
W¼
1
X
wut
ð3Þ
t¼0
satisfies the above set of axioms.
Proposition 2 (Impartiality of probability-weighted welfare maximization).
The maximization of (3) is equivalent to the maximization of
W¼
1
X
ð4Þ
pt :wt
t¼0
where wt is the (conditional) welfare defined in (2) and pt probability of our survival at t (expected at t = 0).
Pnat
s¼1
pst is the
The above two propositions means that weighting future welfare levels
by the (lower) probabilities of their realization is consistent with impartiality
between the present and the future generations.
Proposition 3 (Discounting replaced by probability weighting). We may
achieve the ‘‘effective’’ discounting of future welfare levels by the appropriate
use of probability weighting without violating the impartiality between the
present and the future.
Proposition 4 (Insignificance of the very distant future). In intertemporal
impartial welfare maximization (3) or (4), there exists a sufficiently large
positive integer N such that the welfare levels of people at and after year N
from now have no significant effects on current decisions.
Insignificance here does not mean irrelevance. (Thus, Proposition 4 does
not violate Axiom 9.) Where the welfare comparison for the periods before
N happens to be virtually identical, the welfare levels of people at the year N
and thereafter forever do affect the relative size of intertemporal sums of any
two welfare streams. Thus, the welfare of people very far in the future is still
relevant. If it is not relevant, it will not affect such comparisons.
Some conservationists/environmentalists may find Proposition 4 disturbing as it implies that there exists a large number N such that the
welfare of people after year N from now should not affect current decisions significantly. However, this N could be extremely large. In fact, if
my argument that the probability of our extinction in any century in the
next several hundreds if not thousands of centuries will be very low, N
should be extremely large. Provided environmentalists continue to work
to enlighten voters and governments in the world to achieve international
cooperation for greater environmental protection (on a way to help
achieve this, see Ng and Liu, 2003), we are very unlikely to face a high
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YEW-KWANG NG
probability (but definitely positive) of extinction for a long time to come.
Nevertheless, however small the expected probabilities, they cumulate to
become a large figure after a long time. Since our solar system and even
our universe will not last forever, this is what it should be. We may live
for a trillion (for the optimists, replace that with quintrillion in the
English sense) years, but the probability of that is rather small. If the
welfare of people quintrillion years from now has a very low chance to be
realized, it is proper that it should not significantly affect current decisions
that have significant effects on the welfare of current people (who certainly exist) and people in the nearer future (who will more likely exist).
Environmentalists should worry more about the environmental disruption
that may threaten our survival or impose great suffering or welfare forgone even just in the next few decades, centuries, millenniums, and possibly millions of years rather than about the welfare of people trillions of
years from now after we might have possibly migrated to other planets in
other solar systems or universes. The former more important concern is
not compromised by Proposition 4. For example, consider the concrete
example of the storage of nuclear waste that will likely leak after X years.
If the probability that there will be people X years from now to suffer the
leakage is not negligible, we should take that expected costs seriously,
though some allowance could be made for the possible development of
methods to deal with the leak before X years. If X = one quintrillion
such that the probability that there will be people then to suffer the leak
is negligible, we should not be too much concerned.
3.
THE CASE OF A VARIABLE POPULATION SIZE
For simplicity and to concentrate on the pure intertemporal problem, the
previous section abstracts away the complication of a variable population
size. This makes Axiom 8 (Finite Welfare) clearly compelling. This section
discusses informally the more general case where the population size may
change.
If the population size increases, the acceptability of Axiom 8 may be
suspect. In particular, if the population size increases exponentially at a rate
higher than the rate of risk of extinction (i.e., non-survival), then the expected welfare may increase with time. This will in fact be so if (but not only
if) the welfare of the representative individual is non-decreasing and the
total or aggregate welfare evaluation function is classical utilitarian. The
classical utilitarian welfare function w (dropping the time subscript) is
the equally weighted (or unweighted) sum (over all individuals) of individual
welfares and equals the representative individual welfare value wr (where r
INTERGENERATIONAL IMPARTIALITY
251
stands for ‘‘representative’’) times the number of individual np (p for population), or w ¼ np wr . Of course, classical utilitarianism need not be the only
objective function acceptable (I strongly argue in its favor in Ng (2000,
chapter 5). However, the basic arguments of this paper is not based on this
utilitarian ethics.) However, this is the one that is most likely (out of all
reasonable ones proposed) to cause the violation of Finite Welfare, our case
of defending Axiom 8 using it as a reference point, if valid, must apply a
fortiori if some other reasonable aggregate welfare evaluation function is
used instead, as discussed later.
If w increases (whether it comes from the increase in population size np
or in the representative welfare wr or a combination, and whether under
the classical utilitarian aggregate welfare evaluation function or some
other variant) at a rate higher than the rate of extinction risk, even the
probability-weighted (or risk-discounted) welfare value increases with
time. Then, the summation of wt (with the time subscript put back)
through to time infinity must then still gives us an infinite value, failing to
provide a solution to the paradox of having to choose between different
infinite values. In fact, even if the probability-weighted welfare value does
not increase but just remains constant or even just fails to decrease sufficiently quickly with time, an infinite value may still be involved. If either
np or wr increases indefinitely with time, then welfare w need not be finite
and our convention of taking a finite value as an upper bound for w can
then not be justified and Axiom 8 (Finite Welfare) is not acceptable. We
thus have to defend Finite Welfare even in the presence of a variable
population.
Our case rests mainly on the observation that our limited world cannot
sustain an infinite number of individuals. Not only is our Earth finite (in
mass, space, and time), our Solar System and even our universe is also
finite. Thus, even if we could eventually colonize other planets, our
number cannot be infinite. Even using the classical utilitarian welfare
function w ¼ np wr , if neither np nor wr can be infinite, w must also be
finite. That the finiteness of the universe implies the finiteness of the
population size is clear; it in fact also implies the finiteness of wr , the
representative level of welfare. Our ability to enjoy is related to the size
and complexity of our brain, which is also limited by the finiteness of our
universe, even allowing for the widest dreams of genetic engineering and
brain stimulation (on the latter, see Ng, 2000, App. A; on the finiteness of
wr , see also the discussion after Axiom 8 in the previous section).While w
may increase at a rate higher than the rate of extinction risk, making even
the probability-weighted welfare value increase with time (in fact, I firmly
believe that this is the case now and for the fairly long foreseeable future,
thanks mainly to the advancement of knowledge, but partly based on the
252
YEW-KWANG NG
possibly debatable presumption9 that countries in the world will eventually
come together to seriously tackle the problems of environmental
protection), this increase cannot be maintained at an undiminished rate
indefinitely simply because of the finiteness of our universe. Moreover,
since we will eventually become extinct, the rate of our extinction risk will
eventually surpass that of the rate of increase in welfare, even if still
positive, making the expected welfare eventually declining. Although, as
discussed in Section 2, we do not know now when our extinction risk will
eventually increase dramatically and may continue to use a model of a
constant rate of extinction risk, the observation above means that, at least
for the analysis of very long-term issues (only for such issues could the
paradox of different infinite values arise), if the rate of extinction risk,
population growth rate, and the rate of increase in the representative
welfare are chosen that make the probability-weighted welfare sum to
infinity, the result is inconsistent with the finiteness of our universe and
our welfare. Some of these rates must be revised to give a finite sum. In
particular, the rate of population growth (or increase in representative
welfare) could be too high for long-term sustainability or the rate of
extinction risk too low. Only combination of values that give a finite sum
could be (though not necessarily be) valid. With this understanding, it
means that any formulation that gives an infinite value is not acceptable.
With this result, the paradox of having to compare different infinite values
is solved.
To see that the finiteness of w must apply with stronger force if some
non-classical utilitarian aggregate welfare evaluation function is used
instead, consider the critical-level utilitarianism of Blackorby and Donaldson (1984; see also Blackorby et al., 1997) where the number of individuals
times the average excess of individual welfare over a positive critical level is
where W
is the given positive critical level.
maximized, i.e., w ¼ np ðwr WÞ
For the case of classical utilitarianism, w increases with np at the rate wr . For
the case of critical-level utilitarianism, w increases with np at the rate
which is smaller than wr . In fact, w is always of a lower value
ðwr WÞ,
under critical-level utilitarianism than under the classical utilitarianism.
Thus, if welfare w is finite under the classical utilitarian aggregate welfare
evaluation function, it must be a fortiori also finite under the critical-level
utilitarian one. Now consider another possible revision (see Ng, 1984, who
is, however, not in favor of it at the level of ideal morality) of classical
utilitarianism that dampen down the effect of number by having a
9
This presumption is particularly challengeable in view of US’s refusal to sign the
Kyoto agreement, which, though imperfect, in an important step in the right
direction.
INTERGENERATIONAL IMPARTIALITY
253
‘‘dampening’’ function on np , making the aggregate welfare evaluation
function w ¼ fðnp Þwr where f is increasing but concave in np with f 0 <; ¼ 1
everywhere. Then, again, an increase in np increases w less than in the case of
classical utilitarianism, especially at large np . Again, if welfare w is finite
under the classical utilitarian aggregate welfare evaluation function, it must
be a fortiori also finite under the number-dampened utilitarianism. In fact,
even if we were to go the other way (though no one is in favor of that) and
adopt an ‘‘expanding’’ function on number, w would still be finite, provided
that the expanding function reasonably does not give an infinite value to a
finite np (If it does, what value could it then give to an even higher np ?).
From the above discussion, it may be seen that, even when we allow the
population size to be variable, our arguments, including the various propositions, in Section 3, remain valid.
4.
CONCLUDING REMARKS
The main points and implications of our arguments may be briefly summarized.
Impartiality requires putting the welfare of future generations at par
with that of our own. However, rational choice requires weighting all
welfare values by the respective probabilities of realization.
As the risk of non-survival of mankind is strictly positive for all time
periods and as the probability of non-survival is cumulative, the
probability weights operate like discount factors (and the rates of
extinction risk like the discount rates), though justified on a morally
justifiable and completely different ground than that of using discount
rates.
Recognizing the finiteness of our universe in mass, space, and time, and,
hence, the finiteness of our population size, welfare level, and life (of
mankind) expectancy, even though our expected welfare has to be
summed to time infinity due to our lack of information on our extinction
time, the sum of expected welfare to infinity must remain finite, effectively solving the paradox of having to compare different infinite values
in the theories of optimal growth, capital accumulation, and resource
conservation.
Although a morally valid ground for effective discounting of future welfare
values is possible, the per annum discount rate should be very small, as the
probability of our not surviving the next one hundred years is likely to be
much less than 10%, the per annum effective discount rate or probability
adjustment factor should be much less than 0.1%. (However, note that this
254
YEW-KWANG NG
is the discount rate on future welfare, not on future incomes, consumption,
or other dollar values the discount rate on which is more related to the
appropriate rate of interest, as discussed in Section 1.) With such a low rate,
the objective of environmental protection and sustainability will not be
unduly compromised.
APPENDIX
A
Proof of Proposition 1. None of the above axioms are violated by (3) and in
particular Axiom 2 (intertemporal parity) is satisfied by the equal weighting
in (3).10
Proof of Proposition 2. From (4) and (2), we have
W¼
1
X
pt :wt ¼
t¼0
1
X
"
pt
a
nt
X
s¼1
t¼0
#
a
pst :wðxst Þ=
nt
X
pst
;
s¼1
which gives, from the definition of pt ,
W¼
" na
1 X
t
X
t¼0
#
pst :wðxst Þ
which, from Axiom 1, equals
W¼
"
nt
1 X
X
t¼0
;
s¼1
s¼1
#
pst :wðxst Þ
¼
1
X
wut ;
t¼0
where the last equality follows from (1).11
Proof of Proposition 3: Due to the cumulative nature of the probability
of the risk of non-survival (Axiom 6) and the positive risk (Axiom 4), the
probability weights act like the discount factor, decreasing over time at the
rate of non-survival risk. In particular, for the specific case of a constant
risk, it decreases exponentially, being ð1 rÞt or ert where r is the perperiod or instantaneous rate of non-survival risk.
Proof of Proposition 4: From Axiom 3, the welfare levels at all times have
to be weighted by the probability of their realization. From Axiom 4, this
probability decreases at a positive rate at all times. From Axiom 8 and the
normalization wt 1, it is obvious that, for the special case of a constant
10
Axiom 2 requires parity but does not specify a specific form of parity. The
equally weighted sum is clearly a form of parity. For the argument that this is in fact
the appropriate form of parity, see Ng, 2000, chapter 5. Here we do not have to agree
on this specific form by agreeing to take (2) as only a, instead of the, form satisfying
parity.
11
Note that both (1) and (2) are justified by Axiom 3.
INTERGENERATIONAL IMPARTIALITY
255
risk, for every positive e, there exists some t ¼ N, where N is some sufficiently large integer, such that the following is true:
1
X
ð5Þ
pt :wt < e
t¼N
From Axiom 5 (non-absolute risk), pt for t N are strictly positive,
meaning that the existence of people at and after N is possible. From Axiom
7, (5) remains true for the more general cases.12 Now, consider comparisons
where non-welfare factors are unchanged. (This qualification is not needed
for those accepting welfarism.) Then, (5) means that unless the welfare
comparison for the periods before N happens to be virtually identical, the
welfare levels of people at the year N and thereafter forever do not affect the
relative size of intertemporal sums of any two welfare streams and hence do
not affect current decisions based on such comparisons. Since the probability weighted welfare levels for period N and after sum to an extremely
small number, the effect on current decisions is also insignificant.
ACKNOWLEDGMENT
The motivation for writing this paper is sparked by my drafting of the theme
paper on welfare economics and sustainable development with Dr. Ian Wills
for the Encyclopedia of Life Support Systems by UNESCO. I am also
grateful to an anonymous referee for some helpful comments.)
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Dept. of Economics
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