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Economica (2003) 70, 405–422
The Malleability of Undiscounted Utilitarianism
as a Criterion of Intergenerational Justice
By GEIR B. ASHEIM{ and WOLFGANG BUCHHOLZ{
{ University of Oslo
{ University of Regensburg
Final version received 9 January 2002.
Discounting future utilities is often justified by the ethically motivated objective of protecting
earlier generations from the excessive saving that seems to be implied by undiscounted
utilitarianism in productive economies. We question this justification of discounting by
showing that undiscounted utilitarianism has sufficient malleability within important classes
of technologies: any efficient and non-decreasing allocation can be the unique optimum
according to an undiscounted utilitarian criterion for some choice of utility function.
INTRODUCTION
In the theory of economic growth it is quite common to determine optimal
growth programmes by means of a discounted utilitarian criterion, where the
positive discount rate reflects pure time preference. Future utilities of
consumption measured on a cardinal scale are transformed, through
discounting, into present values, the sum of which one seeks to maximize.
Thus, discounted utilitarianism provides a class of objective functions that are
often used for evaluating intertemporal choice.
Even though discounting has sometimes been considered as ‘irrational’ in
the context of individual decision-making, such impatience has still been
accepted as a natural part of exogenously given individual preferences.
However, when discounting refers to future generations, some fundamental
ethical problems seem to arise. In the case of intergenerational discounting, it is
natural to question whether it is fair to value the utility of future generations
less than that of the present one. This criticism against discounting has a long
tradition in economics, dating back at least to Pigou (1932). It was revitalized
in the ongoing debate on sustainable development, which — it is sometimes
claimed —is in danger when discounting is applied. The proponents of
intergenerational discounting, however, also turn to ethical reflections when
they try to justify this procedure. Thus, it appears that a deep ethical conflict is
present in the debate of the discounting issue.
It is the aim of this paper to evaluate the use of discounting in utilitarian
social criteria for choice between intergenerational allocations. As a first step,
we retrace the ethical arguments of the opponents and the proponents of
intergenerational discounting. As a second step, we show that this debate might
be considered misplaced, as there is not much room for a serious controversy
on the discounting question. In particular, in important classes of technologies
the undiscounted utilitarian criterion is sufficiently malleable to avoid excessive
saving and thus allow for the choices that proponents of intergenerational
discounting seek to implement by discounting future utilities.
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ECONOMICA
I. INTERGENERATIONAL DISCOUNTING
AS AN
ETHICAL PROBLEM
The position of the opponents of intergenerational discounting is based
primarily on the view that it is not a priori justified to give different generations
unequal weight in social evaluations as they do not seem to be fundamentally
different, at least if population size is constant and there is no uncertainty. The
only obvious distinction between members of different generations is that they
do not appear simultaneously on the time axis, which, however, does not
provide an ethically compelling reason for unequal treatment. If, under
otherwise identical circumstances, welfare comparisons are to be made in a
static context with a finite number of agents living within the same generation,
one would usually not deny that equal individuals should get equal
consideration in social welfare functions. Therefore, it is quite standard in
welfare economics to adopt an anonymity principle by which discrimination
against particular agents is excluded.
However, discounting seems questionable not only from its normative basis
but also from its consequences under certain technological assumptions. For
example, when the level of production depends not only on man-made capital
and labour, but also on the input of an exhaustible natural resource (like oil),
applying a discounted utilitarian criterion for any time-invariant and strictly
positive discount rate will, in the long run, force the consumption level to
approach zero, even though positive and non-decreasing consumption is
technically feasible. Thus, in such a Dasgupta – Heal –Solow technology,
constant and positive utility discounting leads to an outcome that does not
appeal to commonly shared ethical intuitions, and is not compatible with
sustainable development —i.e. with no generation enjoying a level of well-being
that cannot be shared by future generations.
As appealing as these ethically motivated arguments against intergenerational discounting may look, there are also important arguments in favour of
the position of the proponents of discounting.
The first of these arguments is of a technical nature. If the world does not
come to its end at some predetermined date, it is an ethical imperative to take
all future generations into consideration, which is well in line with the
advocates of sustainability. But it is just the fact that the number of generations
are modelled to be infinite that creates specific problems in making welfare
comparisons. In the infinite case, the existence of a socially most preferred
intergenerational allocation cannot be ensured in the same way as is usually
done in the finite case, where the Weierstrass theorem easily applies. This
theorem says that on a compact domain a continuous real valued function will
have at least one maximum. In the finite case, not very demanding assumptions
are needed to ensure that the premisses of this theorem hold so that the
existence of maximal elements is ensured. In the infinite case, however, it is not
possible— under relevant technological assumptions —to find a topology that
leads to compactness of the set of feasible allocations and, at the same time, to
a continuity of social preferences that are sensitive to the interest of each
generation and treat all generations equally (cf. Koopmans 1960; Diamond
1965; Svensson 1980; and, for a general discussion, Lauwers 1997). Sensitivity
here refers to the ‘Strong Pareto’ axiom (which we will also call ‘Efficiency’),
meaning that social preferences must deem one allocation superior to another
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if at least one generation is better off and no generation is worse off; while
equal treatment refers to the ‘Weak anonymity’ axiom (which we will also call
‘Equity’), meaning that social preferences must leave the social valuation of an
allocation unchanged when the consumption levels of any two generations
along the allocation are permuted. That this way of establishing existence
cannot be generalized to the infinite case if ‘Equity’ is postulated explains some
of the scepticism people have with this normative precept in the infinite number
case. From such a perspective, adhering to this axiom may seem rather
pointless if it is difficult to make use of it.
The second argument, however— for the widespread refusal of the ‘Equity’
axiom in the intergenerational context — flows from ethical reservations. It is
claimed that, even if future consumption is perfectly certain to be realized,
equal treatment is in conflict with finding an acceptable balance between the
interests of different generations. In particular, it is often suggested that the
application of undiscounted utilitarianism leads to a distributional imbalance
by impairing the earlier generations to an unacceptable degree. For example,
Rawls (1971, p. 287) argues 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’; while Dasgupta and Mäler
(1995, p. 2395) refer to calculations by Mirrlees (1967) and Chakravarty (1969)
showing, in plausible economic models, that the present generation would
be asked to save and invest around 50% of GNP under undiscounted
utilitarianism. Thus, with a very productive economy, the danger exists that
maximizing the sum of undiscounted utilities will lead to a growth pattern that
requires high savings rates initially and thereby imposes excessive hardships on
earlier generations. Following such arguments, Rawls (1971, p. 297) reluctantly points out that ‘[t]his consequence can be to some degree corrected by
discounting the welfare of those living in the future’; and 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’. To avoid consumption patterns that are too much to
the disadvantage of earlier generations, the incorporation of a positive discount
rate into the utilitarian criterion is thus considered inevitable even in the case of
certainty. Put another way, this justification of utility discounting entails that
discounting is required not only in order to be able to make any choices at all,
but also to ensure that the chosen distributions are ethically acceptable.
The first of these arguments —the problem of existence when the axioms of
‘Efficiency’ and ‘Equity’ are imposed— has been considered in a previous work
(Asheim et al. 2001). There we show that the axioms of ‘Efficiency’ and
‘Equity’ are not incompatible with the existence of maximal allocations, given
that one considers technologies that are productive in the sense of satisfying the
following two conditions: ‘Immediate productivity’, which means that there are
negative transfer costs from the present to the future if the future is worse off
than the present, and ‘Eventual productivity’, which means that there exist
efficient and completely egalitarian allocations. In fact, we show that any
efficient and non-decreasing allocation is maximal if the social preferences over
infinite intergenerational allocations are generated by the axioms of
‘Efficiency’ and ‘Equity’.
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In the present paper we consider the second of these arguments. We
show that (i) under a general assumption undiscounted utilitarianism is so
flexible that any efficient and non-decreasing allocation can be the unique
optimal intergenerational allocation under the utilitarian criterion, provided that the utility function is appropriately chosen, and that (ii) this
assumption is satisfied within three important classes of productive
technologies. This means that, for any member of these classes of
technology, any outcome consistent with ‘Efficiency’ and ‘Equity’ can be
realized under undiscounted utilitarianism. Hence the problem with
undiscounted utilitarianism is neither that it does not allow for optimal
allocations, nor that it leads to unequal distributions imposing a too heavy
burden on the present generation. Rather, the problem is that, as a class of
orderings, it does not limit the set of optimal allocations more than the
axioms of ‘Efficiency’ and ‘Equity’ do. Thus, undiscounted utilitarianism
has no bite beyond these axioms.
Throughout, the well-being of any generation will be measured by a onedimensional indicator, ‘consumption’, which comprises everything that affects
a generation’s livelihood. It is appropriate to think of ‘consumption’ as the
money metric utility that a generation derives from the goods and services at its
disposal, with the money metric utility function being identical for all
generations. Even though we prove as our main result that any efficient and
non-decreasing allocation is compatible with zero utility discounting, the
underlying assumption requires that any such allocation is characterized by a
positive consumption interest rate that cannot decrease too rapidly. Our
analysis is therefore not an argument against positive discounting in social
cost – benefit analysis, even for decisions relating to long-term environmental
policy (e.g. controlling the greenhouse effect). However, the consumption
interest rate —which will reflect net capital productivity— will be of a
magnitude, and have a time structure,1 such that the decisions taken will not
undermine the livelihood of future generations.
In the analysis of the present paper there is no uncertainty about whether
future consumption will be realized. Hence, u is not a von Neumann –
Morgenstern utility function, where the corresponding concavity would be an
expression of risk aversion. Rather, the concavity of the utility function u
represents the aversion in social evaluation towards inequality between
generations. The concerns about the unfavourable consequences for the earlier
generations of using undiscounted utilitarianism can be seen as an expression
of inequality aversion. It seems appropriate therefore to respond to these
concerns by making the utility function more concave.
If risk and uncertainty were introduced into our analysis, it would still be
appropriate in principle to distinguish between inequality aversion and risk
aversion (cf. Kreps and Porteus 1979), although this is not often done in
practical applications. Furthermore, even if one follows Harsanyi (1953) in
arguing that undiscounted utilitarianism derives its justification from
hypothetical decisions under uncertainty in the original position, the risk
aversion in his setting relates to decision problems that are not faced by
actual decision-makers. We therefore claim that the concavity of the utility
function u in the undiscounted utilitarian criterion is not observable in a
market economy.
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The paper is organized as follows. Section II contains a simple numerical
illustration, while Section III presents the main result. Section IV shows that
the assumption underlying this result is satisfied in the three classes of
technologies that we explicitly consider. Section V contains a discussion of the
results, while analysis of more technical nature is included in the two
appendices.
II. A NUMERICAL ILLUSTRATION
The main argument presented in this paper is that, within certain classes of
technology, any efficient and non-decreasing allocation can be realized under
undiscounted utilitarianism for some choice of utility function. We now
illustrate this possibility within a simple setting where there are only two
generations. Let c1 and c2 denote consumption (or well-being) of generations 1
and 2, and let the set of feasible allocations be given by
(c1 ; c2 ) 0 is feasible if and only if c1 þ 14 c2 ˘ 2:
This linear technology implies that the net capital productivity equals 3
(¼ 4 1). Let u be the time-invariant utility function, and let (2 (0; 1]) be the
discount factor. This means that the utility discount rate equals 1= 1. An
allocation (c 01 ; c 02 ) is a utilitarian optimum if and only if (c 01 ; c 02 ) solves
max
c1 0; c2 0
u(c1 ) þ u(c2 )
s:t: c1 þ 14 c2 ˘ 2:
This means that, if (c 01 ; c 02 ) 0, then (c 01 ; c 02 ) satisfies u 0 (c 01 ) ¼ 4u 0 (c 02 ).
We can now verify that (c 01 ; c 02 ) ¼ (1; 4) is a utilitarian optimum under both
of the following constellations of utility function and discount factor:
u(c) ¼ c 1=2
and
¼ 1=2
u(c) ¼ ln c
and
¼ 1:
This illustrates how a more concave utility function can substitute for
discounting along an allocation where consumption is strictly increasing.
III. MAIN RESULT
In a general setting, assume that consumption ct in period t ¼ 0; 1; 2; ::: is onedimensional. Write 0 c ¼ (c0 ; c1 ; :::) and correspondingly for other sequences.
Refer to 0 c as an allocation. An allocation 0 c is said to be non-decreasing if
ct þ 1 ct for all t 0 and stationary if ct þ 1 ¼ ct for all t 0. Given a set of
feasible allocations, a feasible allocation 0 c ¼ (c0 ; c1 ; :::) is said to be efficient if
there exists no alternative feasible allocation 0 c 0 ¼ (c 00 ; c 01 ; :::) with c 0t ct for all
t 0, with strict inequality for some t.
One problem when applying the undiscounted utilitarianism to a situation
with an infinite number of generations is that the sum of undiscounted utilities
will generally diverge in the infinite-horizon case. To resolve this, we invoke an
‘overtaking’ criterion, under which one allocation is better than another if, as T
goes to infinity, the lim inf of the sum up to time T of the difference between
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the utilities generated by the allocations is positive:
(P)
0c
0
Pu0 c 00 u lim inf
T21
T
X
(u(c 0t ) u(c 00t )) > 0:
t¼0
We will use this criterion to show how, under a given technological
assumption, any efficient and non-decreasing allocation is the unique
optimum according to undiscounted utilitarianism for some choice of utility
function.
The technological assumption —which as shown in Section IV is satisfied
for any member of three important classes of technologies — is the following.
Assumption 1. If a feasible allocation 0 c ¼ (c0 ; c1 ; :::) is efficient and nondecreasing, then there exists a sequence of consumption discount factors
0 p ¼ (p0 ; p1 ; :::) satisfying 0 < pt þ 1 < pt for all t 0 such that
1>
1
X
pt c t t¼0
1
X
pt c 0t
t¼0
for any feasible allocation 0 c 0 ¼ (c 00 ; c 01 ; :::).
We can now state the main result of this paper.
Proposition 1. Under Assumption 1, if a feasible allocation 0 c ¼ (c0 ; c1 ; :::) is
efficient and non-decreasing, then there exists a continuous, concave and nondecreasing utility function u: Rþ 2 R such that 0 cPu0 c 0 for any feasible
allocation 0 c 0 ¼ (c 00 ; c 01 ; :::) that does not coincide with 0 c.
Proof. By Assumption 1, there exists a sequence of consumption discount
factors
p1 ; :::) satisfying 0 < pt þ 1 < pt for all t 0 such that
P 0 p ¼ (p0 ; P
1
0
0
1> 1
p
c
t
t
t¼0
t ¼ 0 pt c t for any feasible allocation 0 c . Construct a
continuous, concave and non-decreasing utility function u as follows. Let
u0 2 (p0 ; 1)
and
ut 2 (pt ; pt 1 ) for t 1:
If 0 ˘ c < sup{ct j t 2 N}, write t c :¼ min{t 2 N j ct c}, and let
u(c) ¼
tc
X
ut (ct ct 1 ) ut c (ct c c);
t¼0
where c1 :¼ 0. Note that u is a piecewise-linear, continuous and concave
function, with u(0) ¼ 0 and with the slope on each linear segment (ct ; ct þ 1 )
lying strictly between pt þ 1 and pt . If sup{ct j t 2 N} ¼ 1, then u has been
constructed. Otherwise, write Zc :¼ sup{ct j t 2 N}. Since u is continuous and
non-decreasing on [0; c), limc 9 Zc u(c) exists; write Yu :¼ limc 9 c u(c). Complete
the construction of u by setting u(c) ¼ u for c sup{ct j t 2 N}.
Note that u is constructed so that, for each t 0,
u(ct ) pt ct u(c) pt c
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for any c 0, with the inequality being strict if c 6¼ ct . This implies that
lim infT 2 1
T
X
t¼0
(u(ct ) u(c 0t )) 1
X
pt (ct c 0t ) 0
t¼0
for any feasible allocation 0 c 0 , where the first inequality is strict if 0 c 6¼ 0 c 0 and
the second inequality follows from Assumption 1. This means that 0 cPu0 c 0 if
0
&
0 c 6¼ 0 c .
The construction of the continuous and piecewise-linear utility function u is
shown in Figure 1, thereby illustrating the proof of Proposition 1. An
extension of the preceding analysis shows that a differentiable utility function
can be obtained if and only if 0 c is strictly increasing.
As illustrated by the figure, the utility function may have to be chosen such
that marginal utility is zero beyond some consumption level. If so,
undiscounted utilitarianism will not satisfy the axiom of ‘Efficiency’ (or
‘Strong Pareto’; cf. Section I) unless some amendment is introduced. In
Appendix A we show how an extended undiscounted utilitarian criterion can
be constructed, having the properties that it
1. yields a (strict) preference for 0 c 0 over 0 c 00 whenever 0 c 0 Pu0 c 00 ,
2. is complete in the sense of deeming two allocations indifferent whenever
there is not a (strict) preference, and
3. satisfies the axioms of ‘Efficiency’ and ‘Equity’.
Note that the first property means that Proposition 1 holds if such an extended
undiscounted utilitarian criterion is substituted for Pu as defined by (P).
FIGURE 1. Illustration of the proof of Proposition 1.
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IV. HOW STRONG
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ASSUMPTION 1?
Assumption 1 provides a general condition that, if fulfilled, yields a
justification for applying undiscounted utilitarianism. In this section we
illustrate the generality of Assumption 1 by considering three standard classes
of technology. We show that any technology within these classes satisfies
Assumption 1. Hence, by invoking Proposition 1 this in turn implies that an
arbitrary efficient and non-decreasing allocation for a given initial endowment
in such a technology is the unique optimum according to undiscounted
utilitarianism for an appropriate choice of utility function.
We consider the classes of linear, Ramsey, and Dasgupta – Heal – Solow
technologies. For the technologies of these classes we assume that gross
output yt , available at the beginning of period t ¼ 0; 1; 2; :::, is onedimensional. Furthermore, kt denotes invested man-made capital, i.e. the
gross output not consumed in period t. Any member of the three classes of
technology satisfies the conditions of ‘Immediate productivity’ and ‘Eventual
productivity’ (in the terminology of Asheim et al. 2001) under given
assumptions.
Immediate productivity means that, if 0 c ¼ (c0 ; c1 ; :::; ct ; ct þ 1 ; :::) is feasible
and ct > ct þ 1 , then 0 c 0 ¼ (c0 ; c1 ; :::; ct þ 1 ; ct ; :::) is feasible and inefficient.
Eventual productivity means that, for any initial endowment, there exists an
efficient and stationary allocation 0 c ¼ (c; c; :::). For further comparison, one
can associate ‘Immediate productivity’ with positive net capital productivity
leading to decreasing consumption discount factors, and ‘Eventual productivity’ with the existence of an efficient
and non-decreasing allocation having
P
‘finite consumption value’ (i.e. 1
p
t ¼ 0 t ct < 1).
Only linear technologies explicitly incorporate technological progress.
Technological progress corresponds to increasing net capital productivity
and makes it easier to satisfy ‘Eventual productivity’. Dasgupta – Heal –
Solow technologies illustrate the case where resource depletion plays an
important role for future development and leads to decreasing net capital
productivity. If production depends on inputs of depletable natural
resources, Ramsey technologies can be seen to represent the case where
technological progress just ‘makes up’ for diminishing resource availability,
thus avoiding the decline in net capital productivity that would otherwise
occur.
We assume throughout that population is constant. With a non-constant
population, and interpreting ct as per capita consumption, ‘Immediate
productivity’ entails, in the context of linear and Ramsey technologies, that
net capital productivity exceeds the rate of population growth.
(a) Linear technologies
A technology is linear if, in any period, the ratio of gross output and invested
man-made capital is fixed and equal to at . On the other hand, this gross
productivity factor can vary between periods. We assume positive net capital
productivity, i.e.;
(A1)
at > 1
for all
t > 0:
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This assumption means that the condition of ‘Immediate productivity’ is
satisfied. Let the transformation set T 1t in period t be given by
T 1t ¼ {(k; y) j 0 ˘ y ˘ at k}:
Let y > 0 be the exogenously given endowment of output available at the
beginning of period 0. A programme (0 y; 0 k) is said to be feasible given y if
y0 ¼ y
and
kt ˘ yt
(kt ; yt þ 1 ) 2 T 1t
and
for all t 0:
An allocation 0 c is said to be feasible if there is a feasible programme (0 y; 0 k)
such that ct ¼ yt kt for all t 0.2
A linear technology determines a unique sequence of consumption discount
factors 0 p ¼ (p0 ; p1 ; :::) as follows:
p0 ¼ 1
pt þ 1 at ¼ pt
and
for all t 0:
Note that it holds, for all t 0, that 0 < pt þ 1 < pt since at > 1. It follows from
Lemma 1 of Appendix B that the set of efficient and non-decreasing feasible
allocations is non-empty if the following condition holds:
(E1)
1
X
pt < 1;
t¼0
P
since then the stationary allocation 0 c ¼ (c; c; :::) with c ¼ y=( 1
t ¼ 0 pt ) > 0 is
feasible given y. If on the other hand (E1) does not hold, then this set is empty.
This means that a linear technology satisfies ‘Eventual productivity’ if and only
if (E1) is satisfied. An increasing sequence of gross productivity factors — i.e.
0 a ¼ (a0 ; a1 ; :::) satisfies at þ 1 > at for all t 0 —can be interpreted as
exogeneous technological progress. Condition (E1) clearly holds if a0 > 1 and
0 a is non-decreasing.
(b) Ramsey technologies
A Ramsey technology (cf. Ramsey 1928) is determined by a stationary
production function g : Rþ 2 Rþ that satisfies
(A2)
g is concave; continuous for k 0; and twice differentiable for k > 0;
g(0) ¼ 0;
g0 > 0
0
g (k) 2 1
g 0 (k) Y 0
for k > 0;
as k Y 0; and
as k 2 1:
This assumption implies that the condition of ‘Immediate productivity’ is
satisfied. Let the gross output function f be given by f(k) ¼ g(k) þ k for all k 0.
The transformation set T 2 is time-invariant and is given by
T 2 ¼ {(k; y) j 0 ˘ y ˘ f(k); k 0}:
Let y > 0. A programme (0 y; 0 k) is said to be feasible given y if
y0 ¼ y
and
kt ˘ yt
and
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(kt ; yt þ 1 ) 2 T 2
for all t 0:
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An allocation 0 c is said to be feasible if there is a feasible programme (0 y; 0 k)
where 0 k is bounded above, such that ct ¼ yt kt for all t 0. That 0 k is
bounded above means that, given 0 k, there exists Mk such that kt ˘ Mk for all t 0.
This assumption is made in order to show in Lemma 2 that efficient and nondecreasing allocations have ‘finite consumption value’, and thereby enable
Proposition 2 below to be established. All programmes promoted by
proponents of discounted utilitarianism in Ramsey technologies are included,
as under (A2) 0 k is bounded above in any discounted utilitarian optimum.
The set of efficient and non-decreasing feasible allocations is non-empty,
since the stationary allocation 0 c ¼ (c; c; :::), where c > 0 solves y ¼ f(y c), is
feasible given y. Hence any Ramsey technology satisfies ‘Eventual productivity’.
(c) Dasgupta – Heal –Solow technologies
A Dasgupta –Heal – Solow technology (cf. Dasgupta and Heal 1974; Solow
1974) is determined by a stationary production function G : R 3þ 2 Rþ that
satisfies
(A3)
G is concave, non-decreasing, homogeneous of degree 1 and
continuous for (k; r; ‘) 0;
G is twice differentiable and satisfies (Gk ; Gr ; G‘ ) 0 for (k; r; ‘) 0;
G(k; 0; ‘) ¼ 0 ¼ G(0; r; ‘); and, given any (k 0 ; r 0 ) 0, there is 0 > 0 such
that, for all (k; r) satisfying k k 0 , 0 < r ˘ r 0 , we have [rGr (k; r; 1)]=
G‘ (k; r; 1) 0
This assumption implies that the condition of ‘Immediate productivity’ is
satisfied. Note that both capital kt and resource extraction rt are essential in
production. The available labour force is assumed to be stationary and equal to
1. Still, labour needs to be explicitly considered in order to state the latter part
of (A3), namely that the ratio of the share of the resource in net output to the
share of labour in net output is assumed to be bounded away from zero. Let
the gross output function F be given by F(k; r) ¼ G(k; r; 1) þ k for all (k; r) 0.
The transformation set T 3 is time-invariant and is given by
T 3 ¼ {[(k; m); (y; m 0 ) j 0 ˘ y ˘ F(k; r); 0 ˘ r ¼ m m 0 ; (k; m 0 ) 0}:
Let (y; m) 0, where m is the resource stock available at the beginning of
period 0. A programme (0 y; 0 m; 0 k) is said to be feasible given (y; m) if
y0 ¼ y
and
m0 ¼ m;
kt ˘ yt
and
[(kt ; mt ); (yt þ 1 ; mt þ 1 )] 2 T 3
for all t 0:
An allocation 0 c is said to be feasible if there is a feasible programme
(0 y; 0 m; 0 k) such that ct ¼ yt kt for all t 0.
Assumption (A3) does not ensure the existence of a stationary allocation
with positive consumption. Therefore assume the following condition.
(E3)
There exists, given (y; m) 0, a feasible stationary allocation with
positive consumption.
Dasgupta and Mitra (1983) show, within the setting of Dasgupta– Heal – Solow
technologies, that this implies the existence of an efficient and stationary
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allocation. This means that a Dasgupta –Heal –Solow technology satisfies
‘Eventual productivity’ and has a non-empty set of efficient and nondecreasing feasible allocations if (E3) holds. We state (E3) in its reduced form
since it is outside the scope of the present paper to provide primitive
technological conditions; instead we refer to Cass and Mitra (1991), who give
necessary and sufficient conditions on the production function G for (E3) to
hold.
The importance of Dasgupta– Heal – Solow technologies in a discussion of
intergenerational justice derives from the property that under (A3) net capital
productivity Gk (k; r; 1) converges to zero along any efficient and nondecreasing allocation.3 This occurs in a stationary technology setting owing
to the dwindling availability of the resource.
(d) A characterization result for efficient and non-decreasing allocations
Consider a member of the classes of linear, Ramsey or Dasgupta –Heal –Solow
technologies. Assume (A1)– (A3). In Appendix B we prove that, if a feasible
allocation 0 c ¼ (c0 ; c1 ; :::) is efficient and non-decreasing, there exists a
sequence of consumption discount factors 0 p ¼ (p0 ; p1 ; :::) satisfying
0 < pt þ 1 < pt for all t 0 such that
1>
1
X
t¼0
pt c t 1
X
pt c 0t
t¼0
for any feasible allocation 0 c 0 ¼ (c 00 ; c 01 ; :::).
Proposition 2. Under (A1) –(A3), any linear, Ramsey or Dasgupta –Heal –
Solow technology satisfies Assumption 1.
V. DISCUSSION
We have earlier (in Asheim et al. 2001) showed that the rather uncontroversial
ethical axioms of ‘Efficiency’ and ‘Equity’ rule out any allocation that is not
efficient and non-decreasing in technologies that satisfy the conditions of
‘Immediate productivity’ and ‘Eventual productivity’. All members of the three
classes of technologies that we have considered in Section IV satisfy
‘Immediate productivity’ under the assumptions of (A1)–(A3), and they
satisfy ‘Eventual productivity’ provided that (E1) and (E3) are satisfied. Hence,
‘Efficiency’ and ‘Equity’ alone rule out any allocation that is not efficient and
non-decreasing. Here we have showed that, within these classes of technology,
any efficient and non-decreasing allocation can be a unique undiscounted
utilitarian optimum for a given initial endowment, provided that the utility
function is appropriately chosen. Hence the class of undiscounted utilitarian
criteria — any member of which satisfies ‘Efficiency’ and ‘Equity’ —is
sufficiently malleable to allow for any efficient and non-decreasing allocation
to be the unique choice for a given initial endowment. This may be taken to
mean that no efficient and non-decreasing allocation can be ruled out solely
from ethical consideration, at least in the three classes of technologies that we
consider.
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This is a negative conclusion, in the sense that adopting undiscounted
utilitarianism as a class of social preferences does not indicate how to provide a
sharper prescription for choice among feasible allocations — unless one has a
specific ethical intuition concerning the concavity of the utility function, which,
however, seems hard to obtain on the basis of ethical axioms alone. In Ramsey
technologies discounted utilitarianism with a positive and time-invariant
discount rate might be seen as an alternative that, even as an entire class of
criteria, yields sharper prescriptions. Suppose that the discount rate is chosen
to be smaller than or equal to g 0 (k(y)), where k(y) solves y ¼ f(k(y)) for given y.
Then discounted utilitarianism applied to the Ramsey model will, quite
independently of the shape of the utility function, lead to an efficient and nondecreasing allocation. In fact, the stock of capital cannot grow to a size that
exceeds Mk, where g 0 (k) equals the utility discount rate.4 This means that
consumption cannot grow beyond g(k), and thus, the introduction of such a
‘small’ utility discount rate excludes allocations with a large inequality between
the present and unfortunate generations and the future and fortunate
generations. Hence, even though discounted utilitarianism as a criterion for
social decision-making is inconsistent with the axiom of ‘Equity’, in Ramsey
technologies it leads to outcomes that are consistent with ‘Efficiency’ and
‘Equity’ as long as the discount rate is sufficiently ‘small’.
This attractive feature of discounted utilitarianism arises in Ramsey
technologies since an efficient and stationary allocation has constant net
capital productivity g 0 (k(y)). In a Dasgupta –Heal –Solow (D –H –S) technology, however, the net capital productivity Gk (k; r; 1) converges to zero along
any efficient and non-decreasing allocation. This means that discounted
utilitarianism will force consumption to approach zero in the long run,
independently of how ‘small’ the discount rate is, even if (E3) is satisfied so that
the set of efficient and non-decreasing allocations is non-empty. This in turn
implies that discounted utilitarianism is what Page (1977, p. 198) calls a ‘fair
weather criterion’, which necessarily leads to outcomes inconsistent with the
axioms of ‘Efficiency’ and ‘Equity’ in the more severe context of a Dasgupta –
Heal – Solow technology. These results are summarized in Table 1.
These conclusions mean that, although discounted utilitarianism appeals to
our ethical intuitions in the simple context of Ramsey technologies,
introducing positive discounting is not in general an appropriate way of
achieving ethically desirable outcomes. In contrast, undiscounted utilitarianism
is sufficiently malleable that any ethically desirable outcome can be realized as
RESULTS
ON
TABLE 1
UNDISCOUNTED AND DISCOUNTED UTILITARIANISM
Ramsey
technologies
D–H–S
technologies
Undiscounted
utilitarianism
All allocations consistent with
‘Efficiency’ & ‘Equity’
All allocations consistent with
‘Efficiency’ & ‘Equity’
Discounted
utilitarianism
More equal alloc. consistent with
‘Efficiency’ & ‘Equity’
No allocation consistent with
‘Efficiency’ & ‘Equity’
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long as such an allocation is consistent with ‘Efficiency’ and ‘Equity’. Hence, in
the context of utilitarianism, there appears to be no ethical argument in favour
of discounting if there is no uncertainty. In the end, such a conclusion may
appear uncontroversial, but — as documented in Section I— it contradicts
views expressed by many influential scholars.
Determining an intergenerational allocation amounts to nothing more than
resolving the distributional conflict between the different generations.
Following Atkinson (1970), it is quite familiar in economics to have
distributional objectives incorporated into symmetric additive social welfare
functions where the consumption (or income) of every economic agent is
evaluated by some utility function u. Then the degree of inequality aversion
contained in the social welfare function is expressed by the degree of concavity
of the utility function u. (Cf. Cowell 2000, for a general overview, and Collard
1994, and Fleurbaey and Michel, 2001, for specific applications in the
intergenerational context.) In the static case this leads to a completely
egalitarian distribution unless there are costs of redistribution. In the present
intergenerational context, the condition of ‘Immediate productivity’ means
that there are negative transfer costs from the present to the future. Referring
to Okun’s well-known ‘leaking bucket’, Schelling (1995, p. 396) vividly
describes such a situation by using the term ‘incubation bucket’, in which
‘the good things multiply in transit so that more arrives at the destination than
was removed from the origin’. Hence, if such productivity of the technology is
assumed, any efficient and non-decreasing allocation may be consistent with
the maximization of a symmetric additive welfare function. This means that the
familiar welfarist approach might be attractive to apply also for evaluating
intergenerational allocations, even in the infinite case.5
That a more concave u can be used for skewing the intergenerational
distribution in favour of earlier generations, thus slowing economic growth,
has been observed for a long time by interpreting the well-known Ramsey rule
for optimal economic growth (see e.g. Dasgupta and Heal 1979, p. 292;
Fleurbaey and Michel 1994, pp. 294; 1999, p. 723). Here we have shown, in the
context of linear and Ramsey technologies, that whatever decreased inequality
is desirable can be obtained under undiscounted utilitarianism by choosing a
more concave utility function, since the utility function can always be adjusted
so that a given efficient and non-decreasing allocation is obtained as the
undiscounted utilitarian optimum. Such adjustment is possible even in
Dasgupta – Heal –Solow technologies, where discounted utilitarianism, for
any positive and constant discounting, leads to unacceptable treatment of
generations in the distant future. Thus, this malleability of undiscounted
utilitarianism makes it possible in the intergenerational context to determine
optimal allocations that appeal to our ethical intuitions through applying
complete and transitive social preferences that satisfy reasonable ethical
axioms.
However, the conclusion that discounting is not needed for ethically
appealing choices to be made in the intergenerational context does not imply
that intergenerational discounting is unjustified in any case. Discounting may
well provide an instrument for taking uncertainty into account. This aspect
makes generations inherently different and therefore has ethical relevance for
intergenerational discounting. Referring to Sidgwick (1890, p. 412), Dasgupta
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and Heal (1979, p. 262) clearly state: ‘The point then is that one might find it
ethically reasonable to discount future utilities at positive rates ... because
there is a positive chance that future generations will not exist.’ A discussion of
such more legitimate reasons for intergenerational discounting has not been the
subject here. Rather, our purpose has been to clarify the ethical role of
discounting in the case of certainty.
APPENDIX A: EXTENDING UNDISCOUNTED UTILITARIANISM
To construct an extended undiscounted utilitarian criterion that yields a (strict)
preference for 0 c 0 over 0 c 00 whenever 0 c 0 Pu0 c 00 (cf. (P) of Section III), is complete, in the
sense of deeming two allocations indifferent whenever there is not a (strict) preference,
and satisfies the axioms of ‘Efficiency’ and ‘Equity’, we introduce the following
components:
1. a first-order utility function u : Rþ 2 R that is continuous, concave and nondecreasing, and is applied in a ‘catching up’ criterion, under which one allocation is
as good as another if the lim inf, as T goes to infinity, of the sum up to time T of the
difference between the utilities generated by the allocations is non-negative;
2. a second-order utility function : Rþ 2 R that is continuous and strictly increasing,
and is applied lexicographically to resolve ties so that the axiom of ‘Efficiency’ is
satisfied; for this purpose we will assume throughout that is given by (c) ¼ c;
3. given u, a complete and transitive binary relation R 00u that is a completion of ‘catching
up’ utilitarianism when these utility functions are used lexicographically, and which
invokes a result due to Szpilrajn (1930).
We first show how an incomplete reflexive and transitive binary relation R 0u can be
derived from u when ‘catching up’ utilitarianism is lexicographically extended. We then
consider the issue of completing R 0u to R 00u .
Let 0 c 0 ¼ (c 00 ; c 01 ; :::) and 0 c 00 ¼ (c 000 ; c 001 ; :::) be two allocations. Let u be given, and
determine R 0u as follows:
8
PT
0
00
>
lim infT 2 1
t ¼ 0 (u(c t ) u(c t )) 0;
>
<
PT
0 0
00
0
00
(R)
and lim infT 2 1
0 c R u0 c u
t ¼ 0 ((c t ) (c t )) 0
>
>
P1
:
whenever t ¼ 0 (u(c 0t ) u(c 00t )) ¼ 0:
It is clear that R 0u is reflexive and transitive. Let P 0u and I 0u denote the asymmetric and
symmetric parts of R 0u , respectively; i.e. P 0u denotes (strict) preference, while I 0u denotes
indifference. It follows from (R) and (P) that 0 c 0 P 0u0 c 00 whenever 0 c 0 Pu0 c 00 . To show that
R 0u satisfies ‘Efficiency’ (or ‘Strong Pareto’), let 0 c 0 be derived from 0 c 00 by having
c 0s ¼ c 00s þ ", " > 0, and c 0t ¼ c 00t for t 6¼ s; then 0 c 0 P 0u0 c 00 , since, even if u(c 0s ) ¼ u(c 00s ), we
have that (c 0s ) > (c 00s ). This means that R 0u satisfies the axiom of ‘Efficiency’. To show
00
that R 0u satisfies ‘Equity’ (or ‘Weak anonymity’), let 0 c 0 be derived fromP
0 c by having
1
0
c 0s 0 ¼ c 00s 00 , c 0s 00 ¼ P
c 00s 0 and c 0t ¼ c 00t for t 6¼ s 0 , s 00 ; then 0 c 0 I 0u0 c 00 , since
t ¼ 0 (u(c t ) 1
0
00
0
u(c 00t )) ¼ 0 and
t ¼ 0 ((c t ) (c t )) ¼ 0. This means that R u satisfies the axiom of
‘Equity’.
Say that R 0 is a subrelation to R 00 if (i) 0 c 0 R00 c 00 implies 0 c 0 R000 c 00 and (ii) 0 c 0 P00 c 00
implies 0 c 0 P000 c 00 , with P 0 and P 00 denoting the asymmetric parts of R 0 and R 00 ,
respectively. Svensson’s (1980) Theorem 2 states that any reflexive and transitive binary
relation that satisfies the axioms of ‘Efficiency’ and ‘Equity’ is a subrelation to a
complete and transitive binary relation (i.e. an ordering). In proving this result,
Svensson refers to a general mathematical lemma by Szpilrajn (1930). By invoking this
result, there exists a complete and transitive binary relation R 00u which has R 0u as a subrelation. Note that it follows, from the definition of a subrelation, that R 00u also satisfies
the axioms of ‘Efficiency’ and ‘Equity’. For the purpose of Proposition 1 we need not be
concerned about how R 0u is completed to R 00u , because Proposition 1 shows that the
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considered efficient and decreasing allocation Pu-dominates (and thus P 0u -dominates)
any alternative feasible allocation for an appropriately chosen u. Then it follows that the
considered efficient and decreasing allocation P 00u -dominates any alternative allocation,
independently of how R 0u is completed to R 00u . Hence, Proposition 1 obtains also if P 00u is
substituted for Pu.
APPENDIX B: PROOF OF PROPOSITION 2
The proof of Proposition 2 is based on three lemmas. We start with Lemma 1, which
states the following result for linear technologies.
Lemma 1. An allocation
0 c in a linear technology is feasible given y if and only if ct 0
P
for all t 0 and 1
t ¼ 0 pt ct ˘ y.
Proof. Assume that 0 c is feasible. Then there exists a feasible programme (0 y; 0 k)
such that
yt kt 0 for
t 0. In particular, yt þ 1 ˘ at kt for all t 0. This means
P Tct ¼
P Tall
1
1
that
this implies that
t þ 1 at ¼ pt ,
P T 1 t ¼ 0 pt þ 1 yt þ 1 ˘ t ¼ 0 pt þ 1 at kt . Since pP
1
t ¼ 0 pt (yt kt ) ˘ p0 y0 pT yT . We now obtain
t ¼ 0 pt ct ˘ y as pt ct ¼ pt (yt kt ) 0, p0 y0 ¼ y and pT yT 0.
P1
Assume that ct 0 for all t 0 and
t ¼ 0 pt ct ˘ y. Construct (0 y; 0 k) by y0 ¼ y,
and kt ¼ yt ct and yt þ 1 ¼ aP
t kt for all t 0. Then pt ct þ pt þ 1 yt þ 1 ¼ pt yt pt kt þ
1
pt þ 1 at kt ¼ pt yt , implying that
s ¼ t ps cs ˘ pt yt . As ps cs 0 for all s t, it follows that
yt 0 for arbitrary t, and (0 y; 0 k) is feasible.
&
Turn now to the class of Ramsey technologies. A feasible programme (0 y; 0 k) is
competitive if there is a non-null sequence of non-negative prices 0 p such that, for t 0,
pt þ 1 yt þ 1 pt kt pt þ 1 y pt k
for all (k; y) 2 T 2
In other words, along a competitive programme intertemporal profits are maximized at
each point in time. A competitive programme is said to satisfy the transversality
condition at the price sequence 0 p if
limT 2 1 pT kT ¼ 0:
Lemma 2. Under (A2), if a feasible and non-decreasing allocation 0 c in a Ramsey
technology is efficient, then there exists a feasible program (0 y; 0 k) with ct ¼ yt kt for
all t 0 that is competitive and satisfies the transversality condition at prices 0 p given by
pt þ 1 f 0 (kt ) ¼ pt
for t 0:
Furthermore, it holds that, for all t 0, 0 < pt þ 1 < pt , and
P1
t¼0
pt ct < 1.
Proof. Assume that the feasible allocation 0 c is efficient and non-decreasing.
Construct (0 y; 0 k) by y0 ¼ y, and kt ¼ yt ct and yt þ 1 ¼ f(kt ) for all t 0. Since 0 c is
feasible and efficient, it follows that there exists Mk such that 0 < kt < Mk for all t 0. Since,
in addition, 0 c is non-decreasing, it follows that kt þ 1 kt 0 for all t 0, because
otherwise kt would have been reduced to zero in a finite number of periods. This in turn
implies that ct ¼ g(kt 1 ) (kt kt 1 ) ˘ g(k) for all t 1. Determine a sequence of
consumption discount factors 0 p ¼ (p0 ; p1 ; :::) as follows:
p0 ¼ 1
and
pt þ 1 f 0 (kt ) ¼ pt
for all t 0:
Note that it holds for all t 0 that 0 < pt þ 1 < pt , since f 0 (kt ) ¼ g 0 (kt ) þ 1 > 1. By the
concavity of g, it follows that (0 y; 0 k) is competitive. Since kt < Mk for all t 0, it follows
that (0 y; 0 k) satisfies the transversality condition: pT kT ˘ T Mk, where P
:¼ 1=f 0 (k).
1
Finally, since c0 ˘ c1 and ct ˘ g(k) for all t 1, it follows that
t ¼ 0 pt ct ˘
P
1
t
g(k)
<
1.
&
t¼0
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Finally, consider the class of Dasgupta – Heal – Solow technologies. A feasible
programme (0 y; 0 m; 0 k) is competitive if there is a non-null sequence of non-negative
prices (0 p; 0 q) such that, for t 0,
pt þ 1 yt þ 1 þ qt þ 1 mt þ 1 pt kt qt mt pt þ 1 y þ qt þ 1 m 0 pt k qt m
for all [(k; m)(y; m 0 )] 2 T 3 :
A competitive programme is said to satisfy the transversality condition at the price
sequence (0 p; 0 q) if
limT 2 1 (pT kT þ qT mT ) ¼ 0:
Lemma 3. Under (A3), if a feasible and non-decreasing allocation 0 c in a Dasgupta –
Heal – Solow technology is efficient, then there exists a feasible programme (0 y; 0 m; 0 k),
with ct ¼ yt kt for all t 0, that is competitive and satisfies the transversality condition
at prices (0 p; 0 q) given by
pt þ 1 Fk (kt ; mt mt þ 1 ) ¼ pt
qt ¼ q > 0
and
for t 0
pt þ 1 Fr (kt ; mt mt þ 1 ) ¼ q
for t 0:
P1
Furthermore, it holds that, for all t 0, 0 < pt þ 1 < pt , and t ¼ 0 pt ct < 1.
Proof. Let s 00 0 be the first period with positive consumption: cs 00 > 0 and ct ¼ 0
for 0 ˘ t < s 00 . Then, by Proposition 2 in Dasgupta and Mitra (1983), kt þ 1 > kt > 0 and
0 < mt þ 1 < mt for t s 00 . Let s 0 , where 0 ˘ s 0 ˘ s 00 , be the first period with positive
extraction: rs 0 ¼ ms 0 ms 0 þ 1 > 0 and rt ¼ mt mt þ 1 ¼ 0 for 0 ˘ t < s 0 . If s 0 > 0, then
(A3) implies that the resulting allocation is inefficient, since a Pareto-dominating
allocation can be constructed by having r 0s 0 1 ¼ " > 0 and r 0s 0 ¼ rs 0 " > 0 (for
sufficiently small ") and reinvesting the additional output at time s 0 1. Thus, s 0 ¼ 0.
Hence, kt > 0 and rt ¼ mt mt þ 1 > 0 for t 0. The result now follows from
Theorem 4.1 and Corollary 4.1 in Mitra (1978).
&
Proof of Proposition 2
Part 1: Linear technologies.
P1
The result follows immediately from
1, since
t ¼ 0 pt ct ¼ y (< 1) if 0 c is
P 1Lemma
0
0
feasible, given y, and efficient, and t ¼ 0 pt c t ˘ y if 0 c is feasible.
Part 2: Ramsey technologies.
Lemma 2 establishes the existence of a feasible programme (0 y; 0 k) with ct ¼ yt kt for
all t 0 that is competitive and satisfies the
P transversality condition at prices 0 p
satisfying 0 < pt þ 1 < pt for all t 0 such that 1
t ¼ 0 pt ct < 1. Furthermore,
T
X
t¼0
pt (c 0t ct ) ¼
T
X
pt [(y 0t k 0t ) (yt kt )]
t¼0
˘ p0 (y 00 y0 ) pT (k 0T kT ) ˘ pT kT ;
where 0 c 0 is any feasible allocation with (0 y 0 ; 0 k 0 ) as a corresponding feasible
programme, since (0 y; 0 k) is competitive, y 00 ¼ y0 ¼ y, and pT k 0T 0. Finally,
P
1
0
0 y; 0 k) satisfies the transversality condition. Note that it
t ¼ 0 pt (c t ct ) ˘ 0, since (P
0
follows from Lemma 2 that 1
t ¼ 0 pt (c t ct ) ˘ 0 holds even if the feasible programme
(0 y 0 ; 0 k 0 ) corresponding to 0 c 0 does not satisfy that 0 k 0 is bounded above.
Part 3: Dasgupta – Heal – Solow technologies.
Lemma 3 establishes the existence of a feasible programme (0 y; 0 m; 0 k), with ct ¼ yt kt
for all t 0, that is competitive and satisfies the transversality condition at prices
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(0 p; 0 q), where
Furthermore,
T
X
t¼0
0p
satisfies 0 < pt þ 1 < pt for all t 0, such that
pt (c 0t ct ) ¼
T
X
P1
t¼0
pt ct < 1.
pt [(y 0t k 0t ) (yt kt )]
t¼0
˘ p0 (y 00 y0 ) þ q0 (m 00 m0 )
[pT (k 0T kT ) þ qT (m 0T mT )]
˘ pT kT þ qT mT ;
where 0 c 0 is any feasible allocation with (0 y 0 ; 0 m 0 ; 0 k 0 ) as a corresponding feasible
0
0
programme, since (0 y; 0 m;
P 0 k) is 0 competitive, y 0 ¼ y0 ¼ y, m 0 ¼ m0 ¼ m, and
pT k 0T þ qT m 0T 0. Finally, 1
p
(c
c
)
˘
0
since
(
y;
m;
k)
satisfies
the transverst
0
0
0
t¼0 t t
ality condition.
&
ACKNOWLEDGMENTS
We have received many helpful comments from two referees, Snorre Kverndokk, Stef
Prost, Eric Rasmusen and Martin Weitzman, as well as seminar participants at
Harvard, ZEW Mannheim, CESifo Munich, Regensburg and Southampton. Financial
support from the Research Council of Norway (Ruhrgas grant) is gratefully
acknowledged.
NOTES
1. It may to some extent decrease over time. This is called ‘slow’ or ‘hyperbolic’ discounting (cf. e.g.
Heal 1998, pp. 62–3; Weitzman 1998, 2001).
2. Writing jt for gross investment, qt for gross product and dt for the rate of depreciation, a linear
and
technology can be described as follows: ct þ jt ¼ qt , 0 ˘ qt ˘ bt 1 kt 1
0 ˘ kt ˘ jt þ (1 dt 1 )kt 1 , where jt satisfies (1 dt 1 )kt 1 ˘ jt ˘ qt and where net capital
productivity is positive at time t 1 if and only if bt 1 > dt 1 .
3. To see that Gk (kt ; rt ; 1) 2 0 as t 2 1, note that kt k0 > 0 and rt > 0 for all t 0 by the proof of
Lemma 3, while rt 2 0 as t 2 1, owing to the finite resource stock m. Hence,
0 < Gk (kt ; rt ; 1) ˘ G(kt ; rt ; 1)=kt ˘ G(k0 ; rt ; 1)=k0 since G(0; rt ; 1) ¼ 0 and G is concave, while
G(k0 ; rt ; 1)=k0 2 0 as t 2 1 since G(k0 ; 0; 1) ¼ 0 and G is continuous.
4. Since the application of such a ‘small’ discount rate leads to kt being bounded above, the
resulting allocation is feasible in the terminology of Section IV(b). Hence, Proposition 2 applies,
and the given discounted utilitarian criterion can be substituted by an undiscounted utilitarian
criterion. (Cf. Joshi, 1994, for a related but different statement in a special case of the Ramsey
model.) Thus, the introduction of pure time preference, which may appear to be without
‘intrinsic ethical appeal’ and therefore ‘purely ad hoc’ (Rawls 1971, p. 298), is not needed to
obtain appealing allocations in Ramsey technologies.
5. Following (partly) the same line of argument, Schelling (1995, p. 401) concludes: ‘The discountrate question should disappear. In its place is what utility function to use in valuing future
increments in other people’s consumption. This is a real question, not a matter of mathematical
convenience.’
REFERENCES
ARROW, K. J. (1999). Discounting, morality, and gaming. In P. R. Portney and J. P. Weyant (eds.),
Discounting and Intergenerational Equity. Washington, DC: Resources for the Future.
ATKINSON, A. (1970). On the measurement of inequality. Journal of Economic Theory, 2, 244 –63.
ASHEIM, G. B., BUCHHOLZ, W. and TUNGODDEN, B. (2001). Justifying sustainability. Journal of
Environmental Economics and Management, 41, 252 –68.
CASS, D. and MITRA, T. (1991). Indefinitely sustained consumption despite exhaustible natural
resources. Economic Theory, 1, 119– 46.
CHAKRAVARTY, S. (1969). Capital and Development Planning. Cambridge, Mass.: MIT Press.
# The London School of Economics and Political Science 2003
{Journals}Ecca/70_3/g212/makeup/g212.3d
422
ECONOMICA
[AUGUST
COLLARD, D. (1994). Inequality aversion, resource depletion and sustainability. Economics Letters,
45, 513–15.
COWELL, D. (2000). Measurement of inequality. In A. B. Atkinson and F. Bourguignon (eds.),
Handbook of Income Distribution, Vol. I. Amsterdam: Elsevier.
DASGUPTA, P. S. and HEAL, G. M. (1974). The optimal depletion of exhaustible resources. Review
of Economic Studies, Symposium, 3–28.
—— and —— (1979). Economic Theory and Exhaustible Resources. Cambridge: Cambridge
University Press.
—— and MÄLER, K. G. (1995). Poverty, institutions, and the environmental resource-base. In J.
Behrman and T. N. Srinavasan (eds.), Handbook of Development Economics, Vol. III.
Amsterdam: Elsevier.
DASGUPTA, S. and MITRA, T. (1983). Intergenerational equity and efficient allocation of
exhaustible resources. International Economic Review, 24, 133 –53.
DIAMOND, P. (1965). The evaluation of infinite utility streams. Econometrica, 33, 170–7.
FLEURBAEY, M. and MICHEL, P. (1994). Optimal growth and transfers between generations.
Recherches Economiques de Louvain, 60, 281–300.
—— and —— (1999). Quelques réflexions sur la croissance optimale. Revue e´conomique, 50,
715–32.
—— and —— (2001). Transfer principles and inequality aversion, with an application to optimal
growth. Mathematical Social Sciences, 42, 1–11.
HARSANYI, J. C. (1953). Cardinal utility in welfare economics and in the theory of risk-taking.
Journal of Political Economy, 61, 434 –5.
HEAL, G. M. (1998). Valuing the Future. New York: Columbia University Press.
JOSHI, S. (1994). Duality between discounted and undiscounted models of economic growth.
Economics Letters, 44, 403–6.
KOOPMANS, T. C. (1960). Stationary ordinal utility and impatience. Econometrica, 28, 287 –309.
KREPS, D. and PORTEUS, E. L. (1979). Temporal von Neumann–Morgenstern and induced
preferences. Journal of Economic Theory, 20, 81–109.
LAUWERS, L. (1997). Continuity and equity with infinite horizons. Social Choice and Welfare, 14,
345–56.
MIRRLEES, J. A. (1967). Optimal growth when technology is changing. Review of Economic Studies,
34, 95–124.
MITRA, T. (1978). Efficient growth with exhaustible resources in a neoclassical model. Journal of
Economic Theory, 17, 114–29.
PAGE, T. (1977). Conservation and Economic Efficiency. Baltimore: Johns Hopkins University
Press.
PIGOU, A. C. (1932). The Economics of Welfare. London: Macmillan.
RAMSEY, F. (1928). A mathematical theory of saving. Economic Journal, 38, 543–59.
RAWLS, J. (1971). A Theory of Justice. Oxford: Oxford University Press.
SCHELLING, T. (1995). Intergenerational Discounting. Energy Economics, 23, 395 –401.
SIDGWICK, H. (1890). The Methods of Ethics. London: Macmillan.
SOLOW, R. M. (1974). Intergenerational equity and exhaustible resources. Review of Economic
Studies, Symposium, 29–45.
SVENSSON, L. G. (1980). Equity among generations. Econometrica, 48, 1251–6.
SZPILRAJN, E. (1930). Sur l’extension de l’ordre partial. Fundamenta Mathematicae, 16, 386–9.
WEITZMAN, M. L. (1998). Why the far-distant future should be discounted at the lowest possible
rate. Journal of Environmental Economics and Management, 36, 201–8.
—— (2001). Gamma discounting. American Economic Review, 91, 260–71.
# The London School of Economics and Political Science 2003