Soc Choice Welfare (2002) 19: 921–940
Sustainability when capital management has stochastic
consequences
Geir B. Asheim1, Kjell Arne Brekke2
1 Department of Economics, University of Oslo, P.O. Box 1095 Blindern,
N-0317 Oslo, Norway (e-mail: g.b.asheim@econ.uio.no)
2 Centre for Development and the Environment, University of Oslo, P.O. Box 1116,
Blindern, N-0317 Oslo, Norway (e-mail: k.a.brekke@sum.uio.no)
Received: 26 April 1994/Accepted: 15 April 2002
Abstract. Sustainability is usually defined as a requirement of each generation
to manage its stocks of man-made and natural capital such that the utility that
it ensures itself can be shared by all future generations. Here we extend this
definition to the case where capital management does not have deterministic
consequences. A characterization is o¤ered where the sustainability of one
generation’s behavior can be determined by comparing its utility with the
utility of the succeeding generation, provided that the latter behaves in a sustainable manner. The properties of the definition are investigated and illustrated.
1 Introduction
The notion of ‘sustainable development’ was introduced into the political
agenda by the World Commission on Environment and Development through
its report (WCED 1987), also called the Brundtland Report. The Brundtland
Report does not give a precise definition of the notion of ‘sustainable development’. The quotation that is usually taken as a point of departure is the
following: ‘‘Sustainable development is a development that meets the needs
of the present without compromising the ability of future generations to meet
their own needs’’ (WCED 1987, p. 43). Thus, the Brundtland Report looks at
sustainability both as a requirement for intragenerational justice and as a requirement for intergenerational justice. We here choose to limit the discussion
by considering sustainability to be a requirement for intergenerational justice
We would like to thank Joseph Greenberg, two anonymous referees and seminar
participants at the EEAC, Dublin; EAERE, Fontainebleau, and Queens University,
Kingston, for helpful comments.
922
G. B. Asheim, K. A. Brekke
only. Then, sustainability requires from our generation to manage our stocks
of man-made and natural capital in a way that does not undermine the wellbeing of future generations. How can such a loosely phrased requirement be
made precise?
Solow’s (1974) application of the maximin principle (inspired by Rawls
1971) in a model of man-made capital accumulation and resource depletion
is an early analysis of sustainability within economics. Solow’s (1974) lead was
subsequently followed by Hartwick (1977), Dixit et al. (1980) and many others.
When the maximin principle leads to an e‰cient utility path (i.e., what Burmeister and Hammond 1977 call a regular maximin path), then utility is at
any time equal to the maximal sustainable level. Measured by a money-metric
utility function such a utility level can be seen as a formalization of Hicksian
income (‘‘. . . we ought to define a man’s income as the maximum value which
he can consume during a week, and still expect to be as well o¤ at the end of
the week as he was in the beginning’’, Hicks 1946, p. 172).
To establish a normative justification for sustainability one can invoke two
rather uncontroversial ethical axioms – Strong Pareto and Weak anonymity –
that correspond to the Suppes-Sen grading principle (Suppes 1966; Sen 1970).
Strong Pareto ensures that the social preferences are sensitive to a utility increase of any generation, while Weak anonymity entails equal treatment of
generations by requiring that social evaluation remains unchanged when the
utility levels of any two generations are permuted. In their justification of sustainability, Asheim et al. (2001) combine these two axioms with the following
assumption on the technology: If one generation has a higher utility than the
next, then it is feasible to transfer the excess utility enjoyed by the earlier generation to the later one at a negative transfer cost.1 Under this assumption it
follows that an intergenerational distribution where one generation is better
o¤ than the next cannot be maximal (i.e., not dominated) according to any
social preferences satisfying Strong Pareto and Weak anonymity. The reason
is that permuting the utility levels will – by Weak anonymity – leave social
evaluation unchanged, while the additional utility available due to the negative cost of such a transfer means – by Strong Pareto – that a preferred social
state can be reached. Hence, if a generation ensures itself a level of utility that
cannot be shared by its successors, then a transfer from an earlier and better
o¤ generation to a later and worse o¤ generation is feasible and should be
undertaken.
If a requirement of sustainability is not extended to later generations, it
does not rule out that some later generation ensures itself a level of utility that
cannot be shared by its successors. It seems, however, odd not to let sustainability be a requirement to later generations as well. In particular, it would be
unreasonable for our generation to have the welfare of distant generations in
1 This generalizes the usual assumption of positive net return on investment – in the
form of increased accumulation of man-made capital or diminished depletion of natural capital – to a setting where utility is an ordinal measure that is level comparable
between generations.
Sustainability and capital management
923
mind if we believed that the intermediate generations would not take part in
an e¤ort to give these generations a just level of utility. Extending the requirement of sustainability to later generations yields the following definition of
sustainable development:
A generation’s management of its stocks of man-made and natural capital is
sustainable if it constitutes a first part of a feasible development with nondecreasing utility.2
This definition does not entail that the realized development will have nondecreasing utility even when all generations act in accordance with sustainability. The reason is that sustainability does not preclude that one generation
makes a sacrifice – with future generations as beneficiaries – to such an extent
that its utility is lower than the utility of its predecessors.
A major deficiency with the above definition of sustainability is that it does
not take into account risk and uncertainty. This is unsatisfactory since the
long-term consequences of human activity are not deterministic; in particular, risk and uncertainty are present in the management of both man-made
and natural capital. The assumption of full certainty is, hence, inappropriate
for studying the most important issues relating to sustainability. E.g., we do
not know for sure whether future utility will be increasing or decreasing. The
crucial question is whether the risk of decreasing future utility is acceptable.
To raise and study this question, the definition of sustainability must be extended to the case where capital management does not have deterministic consequences. Also, methods for analyzing and characterizing sustainable capital
management policies must be developed. This is the purpose of the present
paper.
When the consequences of capital management is stochastic, present and
future development is characterized by an allowance-bequest strategy that for
any history determines the living generation’s level of utility allowance and the
stochastic bequest it leaves behind for its immediate successors. Sustainability
can then be defined as follows.
An allowance-bequest strategy is sustainable if, for any history, the living
generation’s utility does not exceed the certainty equivalent of the utility of
2 The interpretation of sustainability as the feasibility of non-decreasing utility has
been suggested in a number of references. It dates at least back to Tietenberg (1984)
and seems to have been fairly widely accepted; see, e.g., Repetto (1986) Mäler (1989);
Pezzey (1989, 1997); Krautkraemer (1998); and Asheim et al. (2001). A critical assessment of this interpretation of sustainability is given by Pearce et al. (1989, pp. 32 and
49). Brekke and Howarth (1996) argue that this interpretation of sustainability cannot
be justified without some assumption about the technology. Hammond (1994) gives an
interesting review of references relating to the notion of sustainability. See also Solow
(1993). Chichilnisky (1996) defines sustainable development through an axiomatic approach which results in a class of complete and transitive binary relations. In contrast,
we consider, in line with the above references, sustainability to be justified by a reflexive and transitive – but not necessarily complete – binary relation, ruling out developments that are not acceptable candidates for a social choice.
924
G. B. Asheim, K. A. Brekke
its immediate successors. A generation’s management of its stocks of manmade and natural capital is sustainable if it is in accordance with a sustainable consumption-bequest strategy.3
In order to analyze sustainable capital management, we will provide the
following characterization:
A generation’s management of its stocks of man-made and natural capital is
sustainable if and only if its utility level can be shared by the next generation
(in the sense of certainty equivalents) even if the latter abides by the requirement of sustainability.
Through this characterization we will under given assumptions be able to
prove the existence of a maximal sustainable utility allowance, provided that
the stocks of man-made and natural capital are such that sustainable capital
management is feasible. The characterization will also be used to analyze two
examples that aim at illustrating the concept of sustainability when capital
management has stochastic consequences.
The term ‘utility’ is used in the present paper as the scalar that indicates
the average quality of life of a generation. Hence, utility includes everything
that influences the situation in which people live; in particular, it includes
much more than material consumption. It is intended to capture the importance of health, culture, and nature. One limitation is of importance, though:
It does not include the welfare that people derive from their children’s utility.
Likewise, only the instrumental value in nature (i.e., recognized value to
humans) is included in the quality of life, not the intrinsic value in nature (i.e.,
value in its own right regardless of human experience).4 The rationale behind
these limitations is that it is desirable to separate the definition of sustainability from the forces that can motivate our generation to act in accordance
with the requirement of sustainability.
Development cannot be sustainable if consumption falls below the subsistence level. Therefore, throughout the present analysis, it is held as a necessary
condition for sustainable capital management that, with probability 1, present
and future utility is as large as the subsistence level.
The paper is organized as follows: Sect. 2 presents the model. In Sect. 3 we
discuss a justification for the notion of sustainability, based on the axioms of
Strong Pareto and Weak anonymity. In Sect. 4 we derive the main characterization result showing that sustainability of one generation can be determined
by comparing its utility with (the certainty equivalent of ) the utility of the succeeding generation, provided that the latter behaves in a sustainable manner.
3 This definition requires that certainty equivalents are well defined. The independence
axiom of expected utility may be added as an extra assumption, but is not needed in
the following analysis. Hence, our approach allows for evaluations of risk and uncertainty where negative catastrophic events with small probabilities are given a higher
weight than expected utility allows.
4 See e.g., Pearce and Turner (1990, pp. 21–22) for an elaboration on the di¤erence
between instrumental and intrinsic value.
Sustainability and capital management
925
Section 5 introduces the concept of maximal sustainable utility allowance, and
shows that this can be used to characterize the set of sustainable policies.
Finally, Sect. 6 provides two examples applying the results of the paper to
specific models.
2 The model
Any generation inherits a state k A K, where K J < n , and chooses a consumption vector c A <þm and a gross investment vector i A < l subject to ðc; iÞ A fðkÞ
for some feasible, compact set fðkÞ that depends on k. A generation’s average
quality of life is measured by a continuous utility function u : <þm ! <, where
uð0Þ ¼ 0 and, Ec A <þm , uðcÞ b 0. By increasing the dimension of c, this formulation allows for the possibility that utility depends directly on k and i. The
next generation’s state k 0 ¼ f ði; oÞ A K depends on i and o, where f is some
continuous function, and where o is a random variable. Assume that o has
finite support and is independently and indentically distributed across di¤erent
~ denote the distribution of this random variable, and let
generations. Let o
~ Þ A DðKÞ denote the distribution of the next generation’s state k 0 given
f~ði; o
the investment vector i, where DðKÞ denotes the set of simple5 probability distributions over K. Let DðKÞ be equipped with the weak topology of probabil~ Þ as the stochastic bequest that the generity measures. We refer to b ¼ f~ði; o
ation leaves behind.
Consider the set feasible consumption vectors given the inheritance k A K
and the bequest b A DðKÞ:
~ Þg:
Cðk; bÞ :¼ fc A < m j bi such that ðc; iÞ A fðkÞ and b ¼ f~ði; o
þ
If the bequest b is infeasible given the inheritance k, then Cðk; bÞ is empty.
If Cðk; bÞ is non-empty, then it follows from the compactness of fðkÞ and the
continuity of f that Cðk; bÞ is compact, implying by the continuity of u that
maxc A Cðk; bÞ uðcÞ exists. Define F : K DðKÞ ! < by
F ðk; bÞ :¼ max uðcÞ
c A Cðk; bÞ
if Cðk; bÞ is non-empty, and let F ðk; bÞ be an arbitrary negative number if
Cðk; bÞ is empty. Assuming free disposal of utility, it follows that the pair
ða; bÞ A <þ DðKÞ of a utility allowance a and a bequest b is feasible given k
i¤ 0 a a a F ðk; bÞ, while the bequest b is not feasible given k i¤ F ðk; bÞ < 0.
Lemma 1. The set of feasible bequests, fb A DðKÞ : 0 a F ðk; bÞg is compact.
Proof. Note that the corresponding set of feasible investments is the projection
of fðkÞ on < l , and hence is compact. It follows that set of feasible bequests,
fb A DðKÞ : 0 a F ðk; bÞg, is compact, since it is the continuous image of the
compact set of feasible investments.
r
That F is not time-dependent does not mean that the technology is sta5 I.e., with finite support.
926
G. B. Asheim, K. A. Brekke
tionary, since the time-dependency may be included as a component of k; see
Footnote 6. We assume that there always exists a feasible allowance-bequest
pair: Ek A K, bb A DðKÞ such that 0 a F ðk; bÞ. Let a denote the subsistence
level, where a b 0. If a~ is a simple probability distribution over <þ , let mð~
aÞ
denote the certainty equivalent of a~ where m : Dð<þ Þ ! <þ .
A bequest policy b is defined as a function b : K ! DðKÞ, where b is feasible i¤ Ek A K, 0 a F ðk; bðkÞÞ. An allowance policy a is defined as a function
a : K ! <þ , where ða; bÞ is feasible i¤ Ek A K, 0 a aðkÞ a F ðk; bðkÞÞ.
Let the set of histories H be defined inductively as follows: H1 :¼
fðh0 ; k1 Þ j k1 A Kg, where h0 is the empty prehistory. Furthermore, Et b 1,
h ¼ ðht1 ; kt Þ A Ht ; and ktþ1 A supp btþ1
Htþ1 :¼ ðht ; ktþ1 Þ t
:
for some btþ1 satisfying 0 a F ðkt ; btþ1 Þ
y
Then H :¼ 6t¼1 Ht . Note that every history is of the form h 0 ¼ ðh; kÞ; we may
thus define k : H ! K by, Eh 0 A H, kðh 0 Þ ¼ kðh; kÞ ¼ k.
A bequest strategy B maps any history h into a simple probability distribution b A DðKÞ and thus into a simple probability distribution in DðHÞ where
any element of the support is of the form ðh; kÞ; k A supp b. Hence, a bequest
strategy is a function B : H ! DðHÞ, where B is feasible i¤ Eh A H, 0 a
F ðkðhÞ; kðBðhÞÞÞ. An allowance strategy A is defined as a function A : H ! <þ .
The allowance-bequest strategy ðA; BÞ is feasible i¤ Eh A H, 0 a AðhÞ a
F ðkðhÞ; kðBðhÞÞÞ. Let B t ðhÞ be defined in the obvious way, as the probability
distribution in DðHÞ derived by applying B t times. Write B 0 ðhÞ ¼ h.6
The present analysis needs to consider history-dependent capital management (as formalized through the strategies A and B) in order to accommodate
that some generation
. behaves in an idiosyncratic way rather than implements an intergenerational plan,
. conditions its behavior not only on the realized state, but also on the manner that this state has come about.
We can now state a primitive definition of sustainability when capital management has stochastic consequences.
Definition. The feasible allowance-bequest strategy ðA; BÞ is sustainable given
y
h if Eh 0 A 6t¼0 supp B t ðhÞ, a a Aðh 0 Þ a mðAðBðh 0 ÞÞÞ. The allowance-bequest
pair ða; bÞ is sustainable given k if there exist h with kðhÞ ¼ k and a sustainable
allowance-bequest strategy ðA; BÞ given h with ða; bÞ ¼ ðAðhÞ; kðBðhÞÞÞ.
3 Justifying sustainability
While sustainability has a normative appeal on it own, Brekke and Howarth
(1996) demonstrate that taken as a primitive normative principle, it would
6 If time-dependency is included as a component of k, then kðhs Þ 0 kðht Þ whenever
hs A B s ðhÞ, ht A B t ðhÞ, and s 0 t.
Sustainability and capital management
927
violate many reasonable axioms of intertemporal preferences. On the other
hand, Asheim (1991) and Asheim et al. (2001) show that sustainability can be
derived from uncontroversial axioms on social preferences over intergenerational distributions, provided that weak productivity assumptions are imposed
on the technology. In this section we extend the analysis of the latter paper to
the case of stochastic bequest.
Adopt a purely consequentialistic approach where the social preferences
correspond to a reflexive and transitive binary relation R over intergenerational distributions, with I and P denoting the symmetric and asymmetric
parts respectively. Formally we apply R for comparing di¤erent allowancebequest strategies given some initial history h, as any allowance-bequest strategy combined with a history h determine an intergenerational distribution.
Consider the following axioms:
SP (Strong Pareto). A binary relation R satisfies SP if, Eh A H, ðA^; B^ÞPðA; BÞ
y
given h whenever B^ðh 0 Þ ¼ Bðh 0 Þ for all h 0 A 6t¼0 supp B t ðhÞ (implying
y
y
that 6t¼0 supp B^ t ðhÞ ¼ 6t¼0 supp B t ðhÞ), and A^ðh 0 Þ b Aðh 0 Þ for all h 0 A
y
y
t
6t¼0 supp B ðhÞ, with strict inequality for some h 0 A 6t¼0 supp B t ðhÞ.
WA (Weak anonymity). A binary relation R satisfies WA if, Eh A H,
y
ðA^; B^ÞI ðA; BÞ given h whenever bh 0 A 6t¼0 supp B t ðhÞ such that A^ðh 0 Þ ¼
mðAðBðh 0 ÞÞÞ, mðA^ðB^ðh 0 ÞÞÞ ¼ Aðh 0 Þ, and kðB^ 2 ðh 0 ÞÞ ¼ kðB 2 ðh 0 ÞÞ, while keeping
the probability distributions of utility allowance unchanged otherwise.
The SP axiom ensures that the social preferences are sensitive to a utility increase of a generation in any history that can be reached with positive probability. The WP axiom imposes equal treatment of generations by requiring that
the social preferences must leave social valuation unchanged when the utility
level of one generation is permuted with the certainty equivalent utility level
of the next generation. We apply a weak version of WA which only considers
permutations between one generation and the next, but this specification is
su‰cient for our purposes.
We want to show that with a reasonable productivity assumption on the
technology, then for any social preferences R satisfying SP and WA, a feasible
allowance-bequest strategy is maximal (i.e. not dominated) according to R
given some initial history h only if it is sustainable given h. The productivity
assumption, which is given below, is needed only for the next proposition.
Assumption 1 (Productivity). If ðA; BÞ is feasible and satisfies AðhÞ >
mðAðBðhÞÞ for some h A H, then there exists a feasible ðA^; B^Þ such that A^ðhÞ >
mðAðBðhÞÞ, mðA^ðB^ðhÞÞ ¼ AðhÞ, and kðB^ 2 ðhÞÞ ¼ kðB 2 ðhÞÞ.
Hence, if the utility level of the current generation exceeds the certainty equivalent of the next, then by this assumption it is feasible for the later generation
to receive the excess utility enjoyed by the earlier generation without having
the earlier one sacrifice all of this excess utility.
If the above assumption holds, then the following normative justification
for the definition of sustainability can be established.
928
G. B. Asheim, K. A. Brekke
Proposition 2. Assume that the allowance-bequest strategy ðA; BÞ is feasible and
y
satisfies Eh 0 A 6t¼0 supp B t ðhÞ, a a Aðh 0 Þ. If Assumption 1 holds and the social
preferences R satisfy SP and WA, then ðA; BÞ is maximal according to R given
h only if ðA; BÞ is sustainable given h.
y
Proof. If ðA; BÞ is not sustainable given h, then there exists h 0 A 6t¼0 supp B t ðhÞ
such that either a > Aðh 0 Þ, or Aðh 0 Þ > mðAðBðh 0 ÞÞÞ. Since a a Aðh 0 Þ by
assumption, the current generation must be better o¤ than the next: Aðh 0 Þ >
mðAðBðh 0 ÞÞÞ. Since by the assumption of productivity it is feasible for the later
generation to receive the excess utility enjoyed by the earlier generation without having the earlier one sacrifice all of this excess utility, the axioms of WA
and SP, combined with the transitivity of R, imply that ðA; BÞ is not maximal
according to R given h.
r
Note that the axioms of SP and WA and the assumption of productivity
require only that utility is an ordinal measure that is level comparable between
generations. Note also that while the subsistence constraint a a Aðh 0 Þ is
assumed and thus not justified by Proposition 2, the constraint will be nonbinding if a is set equal to 0. Hence, the definition of sustainability is su‰ciently general to include both the cases with and without a subsistence constraint.
4 Characterizing sustainability
The definition of sustainability in terms of strategies gives no indication of
how to find sustainable strategies. In this section we derive a complete characterization, which in later sections will be used for developing methods to find
sustainable strategies.
A standard of behavior (SB) s is a correspondence s : K ! <þ DðKÞ.
Associate sðkÞ with a set of sustainable allowance-bequest pairs given the inheritance k. In the following propositions we establish the characterization of
sustainability that was announced in the introduction: A generation’s behavior is sustainable if and only if its utility level can be shared by the next generation even if the latter behaves in a sustainable manner. We introduce the
operator T defined on SBs, where TðsÞ is the SB TðsÞ : K ! <þ DðKÞ determined by ða; bÞ A TðsÞðkÞ i¤ (i) ða; bÞ is feasible given k and (ii) a a a a
mðaðbÞÞ for some allowance-bequest policy ða; bÞ satisfying Ek 0 A supp b,
ðaðk 0 Þ; bðk 0 ÞÞ A sðk 0 Þ. Now the ‘‘only if ’’ part of the above characterization of
sustainability becomes
sðkÞ J TðsÞðkÞ:
ð1Þ
This follows since TðsÞðkÞ is the set of allowance-bequest pairs available to
the current generation provided the utility does not fall below the subsistence
level and can be shared with the next generation even if the latter abide by the
SB s. If for any SBs s 0 and s 00 , we write s 0 J s 00 if, Ek A K, s 0 ðkÞ J s 00 ðkÞ
(and similarly for s 0 K s 00 and s 0 ¼ s 00 ), and (1) holds for all k A K, we obtain
Sustainability and capital management
929
s J TðsÞ:
Say that an SB s is sustainable if s J TðsÞ.
Proposition 3. An allowance-bequest pair ða; bÞ is sustainable given k i¤ there
exists a sustainable SB s such that ða; bÞ A sðkÞ.
Proof. (Only if ) Since ða; bÞ is sustainable given k, there exist h with kðhÞ ¼ k
and a sustainable ðA; BÞ given h with ða; bÞ ¼ ðAðhÞ; kðBðhÞÞÞ. To prove the
only if part we have to construct a sustainable SB s, such that ða; bÞ A sðkÞ.
y
Let K 0 ¼ fkðh 0 Þ j h 0 A 6t¼0 supp B t ðhÞg denote the states that can be derived
by repeated use of the strategy ðA; BÞ. Define s by
(
)
y
sðk 0 Þ ¼ ðAðh 0 Þ; kðBðh 0 ÞÞÞ j h 0 A 6 supp B t ðhÞ and k 0 ¼ kðh 0 Þ
if k 0 A K 0
t¼0
and sðk 0 Þ ¼ q otherwise. Since ða; bÞ A sðkðhÞÞ, it remains to be shown that s
is sustainable.
Let ða 0 ; b 0 Þ A sðk 0 Þ for some k 0 A K 0 . To prove that s is sustainable, i.e.
that s J TðsÞ, it is su‰cient to prove that ða 0 ; b 0 Þ A TðsÞðk 0 Þ since ða 0 ; b 0 Þ is
y
arbitrary. Since k 0 A K 0 there exists h 0 A 6t¼0 supp B t ðhÞ such that k 0 ¼ kðh 0 Þ
0
0
0
0
and ða ; b Þ ¼ ðAðh Þ; kðBðh ÞÞÞ. Let ða; bÞ satisfy ðaðkðh 00 ÞÞ; bðkðh 00 ÞÞÞ ¼
ðAðh 00 Þ; kðBðh 00 ÞÞÞ for all h 00 A supp Bðh 0 Þ. It follows that ða 0 ; b 0 Þ is feasible
given k 0 and that a a a 0 a mðaðbÞÞ, since ðA; BÞ is sustainable. Hence ða 0 ; b 0 Þ A
TðsÞðk 0 Þ since ðaðk 00 Þ; bðk 00 ÞÞ A sðk 00 Þ for all k 00 A supp b 0 .
(If ) Assume that there exists a sustainable SB s such that ða; bÞ A sðkÞ.
To prove the if part, we have to construct a sustainable allowance-bequest
strategy ðA; BÞ, such that ða; bÞ ¼ ðAðhÞ; kðBðhÞÞÞ and kðhÞ ¼ k. Let h satisfy
kðhÞ ¼ k. We will proceed by constructing inductively a sustainable strategy
ðA; BÞ given h with ða; bÞ ¼ ðAðhÞ; kðBðhÞÞÞ. Since ða; bÞ ¼ ðAðhÞ; kðBðhÞÞÞ,
Eh 0 A supp B 0 ðhÞ ¼ fhg, ðAðh 0 Þ; kðBðh 0 ÞÞÞ A sðkðh 0 ÞÞ.
t
Let t b 0, and assume ðA; BÞ has been constructed on 6s¼0 supp B s ðhÞ
t
0
s
0
0
0
such that, Eh A 6s¼0 supp B ðhÞ, ðAðh Þ; kðBðh ÞÞÞ A sðkðh ÞÞ. Let h 0 A
supp B t ðhÞ. Since s is sustainable, ðAðh 0 Þ; kðBðh 0 ÞÞÞ is feasible, and a a
Aðh 0 Þ a mðaðBðh 0 ÞÞÞ for some allowance-bequest policy ða; bÞ satisfying,
Ek 00 A supp kðBðh 0 ÞÞ, ðaðk 00 Þ; bðk 00 ÞÞ A sðk 00 Þ. By letting, Ek 00 A supp kðBðh 0 ÞÞ,
ðAðh 0 ; k 00 Þ; kðBðh 0 ; k 00 ÞÞÞ ¼ ðaðk 00 Þ; bðk 00 ÞÞ, the strategy ðA; BÞ has been contþ1
tþ1
structed on 6s¼0 supp B s ðhÞ such that, Eh 00 A 6s¼0 supp B s ðhÞ, ðAðh 00 Þ;
y
00
00
00
kðBðh ÞÞÞ A sðkðh ÞÞ. By induction, Eh A 6t¼0 supp B t ðhÞ, ðAðh 00 Þ; kðBðh 00 ÞÞÞ
y
is feasible and a a Aðh 00 Þ a mðAðBðh 00 ÞÞÞ. By letting, Eh 00 A Hn6t¼0 supp B t ðhÞ,
00
00
ðAðh Þ; kðBðh ÞÞÞ be an arbitrary feasible allowance-bequest pair, ðA; BÞ becomes sustainable given h.
r
We proceed to show that there exists a largest sustainable standard of
behavior, s – implying by the previous proposition that s ðkÞ can be identified with the set of sustainable allowance-bequest pairs given the inheritance
k – and we establish that not only s J Tðs Þ holds, but also s K Tðs Þ.
Thereby the desired characterization follows: A generation’s behavior is sus-
930
G. B. Asheim, K. A. Brekke
tainable if and only if its utility level can be shared by the next generation even
if the latter behaves in a sustainable manner.
Proposition 4. There exists a largest sustainable standard of behavior, s . Thus,
s is sustainable and s J s whenever s is sustainable.
Proof. Let S be the set of sustainable SBs. Note that s q , defined by
s q ðkÞ ¼ q for all k is sustainable since Tðs q ÞðkÞ ¼ q. Hence the set S of
sustainable SBs is non-empty. Define a SB s by, for all k A K,
s ðkÞ :¼ 6 sðkÞ
sAS
Note that s is sustainable since for any s A S, sðkÞ J TðsÞðkÞ J Tðs ÞðkÞ
for all k. Hence, s is sustainable, and if s is sustainable, then s J s .
r
Proposition 5. s ¼ Tðs Þ.
Proof. We know by construction that s J Tðs Þ. To prove the theorem we
need to prove that it is not a proper subset. Suppose contrary that there exists
k 0 such that s ðk 0 Þ H Tðs Þðk 0 Þ. Construct s by
sðk 0 Þ ¼ Tðs Þðk 0 Þ
and sðkÞ ¼ s ðkÞ otherwise. Since s J s, then for k 0 , sðk 0 Þ ¼ Tðs Þðk 0 Þ J
TðsÞðk 0 Þ, while sðkÞ ¼ s ðkÞ J Tðs ÞðkÞ J TðsÞðkÞ otherwise, implying that
s is sustainable. A contradiction to s being the largest sustainable SB is
established since s J s and s 0 s.
r
5 Maximal sustainable utility allowance
In the present section we apply the characterization of the previous propositions to show existence of and characterize a maximal sustainable utility
allowance. For this we need some topological structure. The following
assumptions are needed for all the remaining propositions in this section.
Assumption 2. F ðk; bÞ is continuous in k and b.
Assumption 3 (Monotonicity). m : Dð<Þ 7! < is continuous in the weak topology and monotone, i.e. if the random allowance a~0 stochastically dominates a~,
then mð~
a 0 Þ b mð~
aÞ.
Define the correspondence s 0 : K ! <þ DðKÞ by, Ek A K, s 0 ðkÞ is the
set of all feasible allowance-bequest pairs given k: ða; bÞ A s 0 ðkÞ i¤ 0 a a a
F ðk; bÞ. Define s m inductively as s m ¼ Tðs m1 Þ for all m b 1.
Lemma 6. For all m b 1, s J s m J s m1 .
Proof. Suppose that s J s m1 , then s ¼ Tðs Þ J Tðs m1 Þ ¼ s m . Obviously, s J s 0 , hence the first part of the claim follows by induction. The
Sustainability and capital management
931
second inclusion is proven similarly. If s m J s m1 , then s mþ1 ¼ Tðs m Þ J
Tðs m1 Þ ¼ s m . Since s 1 J s 0 the claim follows by induction.
r
y
Lemma 7. Let s^ ¼ 7m¼0 s m . There exist a policy ða ; b Þ such that for all
ða; bÞ A s^ðkÞ, a a a a mða ðbÞÞ and for all k 0 A supp b, ða ðk 0 Þ; b ðk 0 ÞÞ A s^ðk 0 Þ.
Moreover, for s^ðkÞ 0 q, a ðkÞ ¼ maxfa 0 j bb 0 : ða 0 ; b 0 Þ A s^ðkÞg.
Proof. Suppose ða; bÞ A s^ðkÞ, then ða; bÞ A s mþ1 ðkÞ ¼ Tðs m ÞðkÞ for all m,
and thus we know that ða; bÞ is feasible and that there is an allowancebequest policy ða m ; b m Þ such that a a a a mða m ðbÞÞ and for all k 0 A supp b,
ða m ðk 0 Þ; b m ðk 0 ÞÞ A s m ðk 0 Þ. We first claim that a m can be chosen to satisfy
a m ðk 0 Þ ¼ maxfa 0 j bb 0 : ða 0 ; b 0 Þ A s m ðk 0 Þg
ð2Þ
on K m :¼ fk 0 A K : s m ðk 0 Þ 0 qg. By the monotonicity of m, a a a a mða m ðbÞÞ
and we need to prove that there exists b m such that for all k 0 A supp b,
ða m ðk 0 Þ; b m ðk 0 ÞÞ A s m ðk 0 Þ. Moreover, we claim that a m is continuous on K m .
Consider first m ¼ 0. We know that, Ek 0 A K 0 ¼ K, ða 0 ; b 0 Þ A s 0 ðk 0 Þ i¤
0 a a 0 a F ðk 0 ; b 0 Þ. Furthermore, Ek 0 A K 0 , fb 0 A DðKÞ j 0 a F ðk 0 ; b 0 Þg is nonempty and compact. Hence, Ek 0 A K 0 , the continuous function F ðk 0 ; b 0 Þ
achieves a maximum for some b 0 , denoted b 0 ðk 0 Þ. Let Ek 0 A K 0 , a 0 ðk 0 Þ :¼
F ðk 0 ; b 0 ðk 0 ÞÞ. By construction, a 0 , and b 0 have the required properties.
Suppose the claim holds for some m b 0. We want to prove that it also
holds for m þ 1. Since s mþ1 ¼ Tðs m Þ, we know that Ek 0 A K mþ1 J K m ,
ða 0 ; b 0 Þ A s mþ1 ðk 0 Þ, i¤ 0 a a 0 a F ðk 0 ; b 0 Þ and a a a 0 a mða m ðb 0 ÞÞ. Since m and
a m are continuous, Ek 0 A K mþ1 , fb 0 A DðKÞ j 0 a F ðk 0 ; b 0 Þ; a a mða m ðb 0 ÞÞg is
non-empty and compact and hence the continuous function minfF ðk 0 ; b 0 Þ;
mða m ðb 0 ÞÞg achieves a maximum, for some b 0 denoted b mþ1 ðk 0 Þ. Let
Ek 0 A K mþ1 ,
a mþ1 ðk 0 Þ ¼ minfF ðk 0 ; ðb mþ1 ðk 0 ÞÞÞ; mða m ðb mþ1 ðk 0 ÞÞÞg:
By construction a mþ1 and b mþ1 have the required properties, and hence, by
induction, the claim holds for all m.
Note that supp b J K m . Since the SBs are shrinking, Ek 0 A supp b, a m ðk 0 Þa
m1
a
ðk 0 Þ, there exist an a satisfying Ek 0 A supp b, a ðk 0 Þ ¼ lim a m ðk 0 Þ. Since b
has finite support, a m ðbÞ converges to a ðbÞ in the weak topology. Moreover,
as a a a a mða m ðbÞÞ, it follows by the continuity of m, that a a a a mða ðbÞÞ.
To construct b , note that Ek 0 A supp b, and the sequence b m ðk 0 Þ is in a compact set by Lemma 1, and hence the sequence b m ðk 0 Þ has a convergent subsequence. Let b ðk 0 Þ denote the limit of this convergent subsequence. By
continuity, Ek 0 A supp b, ða ðk 0 Þ; b ðk 0 ÞÞ A s m ðk 0 Þ. Since m is arbitrary this
concludes the proof of the first claim. The final claim immediately follows
from the construction of a .
r
Proposition 8. s ¼ s^.
Proof. Note first that s J s^ by Lemma 6. If s^ J Tð^
sÞ, then s^ J s since s is
the largest sustainable SB, and thus the claim follows.
932
G. B. Asheim, K. A. Brekke
To prove s^ J Tð^
sÞ, suppose ða; bÞ A s^ðkÞ. By Lemma 7 there is a policy
ða ; b Þ such that a a a a mða ðbÞÞ and for all k 0 A supp b, ða ðk 0 Þ; b ðk 0 ÞÞ A
s^ðk 0 Þ, implying that ða; bÞ A Tð^
sÞðkÞ.
r
Define K as K :¼ fk A K j s ðkÞ 0 qg. Hence, by Proposition 3, a sustainable allowance-bequest pair exists given k i¤ k A K .
Proposition 9. If k A K , then a ðkÞ ðbaÞ is the maximal sustainable utility
allowance. Moreover,
s ðkÞ ¼ fða; bÞ j 0 a a a F ðk; bÞ; b A DðK Þ; and a a a a mða ðbÞÞg
and
a ðkÞ ¼ maxfa j bb A DðK Þ such that 0 a a a F ðk; bÞ;
and a a a a mða ðbÞÞg:
Proof. By construction, if k A K , a ðkÞ as derived in Lemma 7 is the maximal
sustainable utility allowance. To show the second claim, if ða; bÞ A s ðkÞ J
Tðs ÞðkÞ, then 0 a a a F ðk; bÞ, b A DðK Þ and a a a a mða ðbÞÞ by the
monotonicity of m. Conversely, by Lemma 7 and Propsition 8, Ek 0 A supp b,
ða ðk 0 Þ; b ðk 0 ÞÞ A s ðk 0 Þ and hence, if 0 a a a F ðk; bÞ, b A DðK Þ and a a a a
mða ðbÞÞ, then ða; bÞ A Tðs ÞðkÞ J s ðkÞ by Proposition 5. The last part now
follows since a ðkÞ is the maximal sustainable utility allowance.
r
Proposition 10. k A K i¤ there exists a bequest b A DðK Þ such that a a
F ðk; bÞ.
Proof. The only if part follows since s ðkÞ J Tðs ÞðkÞ. To prove the if part,
if a a F ðk; bÞ and b A DðK Þ, then for all k 0 A supp b, a ðk 0 Þ b a and hence
a a mða ðbÞÞ, and for all k 0 A supp b, ða ðk 0 Þ; b ðk 0 ÞÞ A s ðk 0 Þ, implying that
ða; bÞ A Tðs ÞðkÞ J s ðkÞ by Proposition 5.
r
Proposition 11. If a ¼ 0 and both F and m are homogeneous of degree one, so
is s .
Proof. Clearly ls 0 ðkÞ ¼ s 0 ðlkÞ. Similarly if s m is homogeneous of degree 1,
so is s mþ1 ¼ Tðs m Þ. It thus follows by induction that for all m, s m is homogeneous of degree 1, and hence so is s .
r
The definition of sustainability requires that it must be possible to stay
above the subsistence level with certainty. Thus, even if the worst outcome is
always realized, it should be possible to stay above the subsistence level. To
formalize this observation, we introduce the maximin certainty equivalent
m 0 ð~
aÞ ¼ min supp a~
Let s 0 denote the largest sustainable SB corresponding to m 0 , and let K 0 ¼
fk j s 0 ðkÞ 0 qg denote the set of states that allows a sustainable pair.
Proposition 12. K 0 ¼ K Sustainability and capital management
933
Proof. Obviously K 0 J K . To prove the opposite, let s 00 ðkÞ, be defined by
Ek A K ;
s 00 ðkÞ ¼ fða; bÞ A s ðkÞ j a ¼ ag
Then fk j s 00 ðkÞ 0 qg ¼ K by Proposition 9, and s 00 is a sustainable SB
given m 0 since s is sustainable given m. Hence s 00 J s 0 , and it follows that
K J K 0.
r
6 Two examples
The following two examples demonstrate the use of the results above. In the
first example the question is to what extent extra capital productivity can compensate for increased risk. It is shown that a risky technology may give the
highest sustainable allowance. In the second example we consider the possibility of compensating a risk of environmental degradation through increasing
the stock of man-made capital. This example also illustrates the importance of
the subsistence level.
6.1 Risky or safe technology
Consider a situation where each generation must choose between a safe and a
risky technology. The bequest b is determined by the amount of capital that is
not consumed, i, and the choice of technology, i0 , where i0 ¼ 0 means choosing
the safe technology and i0 ¼ 1 means choosing the risky technology. Hence,
the triple ðc; i; i0 Þ is feasible given k i¤ c þ i a k, c b 0, i b 0 and i0 A f0; 1g.
The next generation’s state k 0 ¼ f ði; i0 ; oÞ depends on the random variable o,
which takes the values 0 and 1 with probability 12 each. If the risky technology
is chosen, i.e., if i0 ¼ 1, then f ði; 1; 0Þ ¼ i and f ði; 1; 1Þ ¼ 4i. If the safe technology is chosen, i.e. if i0 ¼ 0, then f ði; 0; oÞ ¼ 2i independently of o. By
having a ¼ uðcÞ ¼ c, this means that if i a k, then
F ðk; bÞ¼ k i, with b ¼
1 2i if the safe technology is chosen and b ¼ 12 i; 12 4i if the risky technology is chosen, while F ðk; bÞ < 0 if i > k or b is specified otherwise.7 The
subsistence level a is assumed to be equal to 0.
Restrict attention to the class of certainty equivalence functions with constant relative risk aversion implying that m is homogeneous of degree 1 and
mð~
aÞ is given by
( PJ
ð j¼1 pj aj1r Þ 1=ð1rÞ for r 0 1; r b 0
mð~
aÞ ¼
PJ
pj ln aj Þ
for r ¼ 1
expð j¼1
if a~ ¼ ðp1 a1 ; . . . ; pJ aJ Þ, where r is the Arrow-Pratt measure of relative
risk aversion.
It can be shown8 that the technology satisfies Assumption 1 (Productivity),
7 The notation ð p1 k10 ; . . . ; pJ kJ0 Þ means that k 0 takes values ðk10 ; . . . ; kJ0 Þ with
probabilities ð p1 ; . . . ; pJ Þ.
8 A proof can be obtained from the authors.
934
G. B. Asheim, K. A. Brekke
and hence, assuming Strong Pareto and Weak anonymity, sustainability is justified. To characterize the sustainable policies, note that from Proposition 9 we
know that the maximal non-decreasing SB satisfies
fa j bb such that ða; bÞ A s ðkÞg ¼ ½0; a ðkÞ:
Since the subsistence level is equal to 0, and both F and m are homogeneous of
degree one, the maximal sustainable allowance level must, according to Proposition 11, be of the form
a ðkÞ ¼ x k:
Hence the problem of finding a may be solved by maximizing x subject to
x a F ð1; bÞ and x a mðxbÞ.
If only the safe technology were available, then x a 1 i and x a 2xi,
implying that x ¼ 12 and i ¼ 12 . If only the risky technology were available,
then
1 x1a 1 i and x a mðbxÞ ¼ m~ix,1 where m~ is the 1certainty equivalent of
2 1; 2 4 . This implies that i ¼ m~ , and x ¼ 1 m~ , where the certainty
equivalent is
8h
< 1þ4 1r i1=ð1rÞ
for r b 0; r 0 1
2
m~ ¼
:
2
for r ¼ 1:
Since m~ a 2 for r b 1; it follows that with the risky technology x a 12 if and
only if r b 1. It follows that if both the safe and the risky technologies are
available, then the safe technology maximizes sustainable allowance if the
Arrow-Pratt measure of relative risk aversion exceeds 1, and the risky technology maximizes sustainable allowance if the relative risk aversion is lower
than 1. Thus
8
1=ð1rÞ >
2
< 1
k for 1 > r b 0
1þ4 1r
a ðkÞ ¼
>
:k
for r b 1:
2
6.2 Compensating possible irreversible changes
Consider a society that produces a single commodity and is dependent upon
environmental services. The commodity can either be used for material consumption or investment, and is produced from man-made capital with a linear
technology. Suppose that there is a positive probability of a significant degradation of environmental quality. For simplicity we will consider the case where
this probability is independent of investments, thus the flow of environmental
services will be exogenously given.
In this model the capital stock is two-dimensional k ¼ ðk1 ; k2 Þ, consisting
of man-made capital k1 and environmental capital k2 . Similarly, the bequest is
two-dimensional b ¼ ðb1 ; b2 Þ. The dynamics of man-made capital expansion is
given by
c1 þ i1 a k1 ð1 þ rÞ
and
b1 ¼ 1 i 1 ;
ð3Þ
Sustainability and capital management
935
where c1 ðb0Þ is material consumption and i1 ðb0Þ is gross investment in manmade capital. Hence, the bequest of man-made capital is deterministic. In constrast, the bequest of environmental capital is stochastic, but does not depend
on the investment activities of the current generation:
c2 a k 2 ;
i2 ¼ k2 ;
and
b2 ¼ ðð1 pÞ i2 ; p i2 zÞ:
ð4Þ
That is, environment capital k2 ðb0Þ limits the availability of environmental
services c2 ðb0Þ and is degraded to a fraction z < 1 of its previous positive size
with probability p. The quality of life depends on both material consumption
and environmental services,
a ¼ uðc1 ; c2 Þ ¼ c1r c21r ;
ð5Þ
where u is assumed to be a von Neumann-Morgenstern utility function, and
hence the certainty equivalent of any lottery over a is equal to the expected
value. The above implies that the technology can be characterized by
F ðk; bÞ ¼ ½k1 ð1 þ rÞ b1 r k21r
if the deterministic bequest of man-made capital b1 satisfies 0 a b1 a k1 ð1 þ rÞ
and the stochastic bequest of environmental capital b2 is given by (4);
F ðk; bÞ < 0 otherwise. Note that both the technology and m are homogeneous
of degree one. It can be shown9 that the technology satisfies Assumption 1
(Productivity) if and only if 1 þ r > A1=r . Hence, the justification of sustainability in Proposition 2 applies only in this case.
Proposition 13. Suppose a ¼ 0, and let A ¼ 1 p þ pz 1r . If 1 þ r > A1=r ,
then the maximal sustainable allowance level is
a ðkÞ ¼ uðdk1 ; k2 Þ;
ð6Þ
where d is given as
d ¼ 1 þ r A1=r
ð7Þ
If 1 þ r a A1=r , then a ðkÞ ¼ 0 for all k.
Proof. Note first that for all k, c1 ¼ 0 is feasible, and hence, for all k, ða; bÞ
with a ¼ 0 is feasible for some b. Thus, for all k, there exists a sustainable
allowance-bequest pair with a ¼ 0. To prove the proposition, we need to
prove that if 1 þ r a A1=r , an allowance-bequest pair is sustainable only if
a ¼ 0, while for 1 þ r > A1=r , the maximal sustainable allowance policy is
as given in the first part of the proposition.
We know from Proposition 9 that there is a maximal sustainable allowance
a ðkÞ. This policy uniquely determines material consumption, denoted g1 ðkÞ,
and the corresponding bequest of man-made capital b1 ðkÞ ¼ ð1 þ rÞk1 g1 ðkÞ.
By Proposition 11, a ðkÞ is homogeneous of degree 1, and it follows that so is
g1 ðkÞ. We first prove the stronger claim that g1 ðkÞ is independent of k2 and
hence of the form g1 ðkÞ ¼ dk1 .
9 A proof that can be obtained from the authors.
936
G. B. Asheim, K. A. Brekke
Let g1m ðkÞ be the material consumption policy corresponding to a m ðkÞ.
Clearly, g10 ðkÞ is independent of k2 and it is easy to show that if g1m ðkÞ is independent of k2 then so is g1mþ1 ðkÞ. It follows by induction that for every
m b 0, g1m ðkÞ is independent of k2 , and hence, this also applies to g1 ðkÞ.
Sustainability requires a ðkÞ a mða ðbÞÞ. If this inequality was strict, it
would be possible to increase allowance slightly by reducing the deterministic
bequest of man-made capital only slightly. The inequality would still be valid,
violating the maximality of a ðkÞ. We thus conclude that the requirement has
to be an equality,
a ðkÞ ¼ mða ðbÞÞ:
ð8Þ
Combining this with the linear form g1 ðkÞ ¼ dk1 , the equality can be written
d r k1r k21r ¼ ððdð1 þ r dÞÞ r k1r ÞAk21r
ð9Þ
Note that d ¼ 0 solves this equation, but it may not be the only solution.
Suppose that d 0 0, then this equation simplifies to
ð1 þ r dÞ r A ¼ 1;
ð10Þ
which gives the claimed value of d for 1 þ r > A1=r . For 1 þ r a A1=r , we
see that d is non-positive, and hence d ¼ 0 is the maximum.
r
We next characterize the set of states where a sustainable allowancebequest pair exists with a > 0. Note that the minimax preferences can be
derived by setting p ¼ 1. Let a 0 ðkÞ denote the maximal sustainable allowance
as given in Proposition 13, with p ¼ 1. Since this would give a deterministic
model, a 0 ðkÞ is the maximal sustainable allowance. From Proposition 13
a 0 ðk1 ; k2 Þ ¼ uðd 0 k1 ; k2 Þ;
where d 0 is given as
1 þ r zðr1Þ=r
d0 ¼
0
ð11Þ
for 1 þ r > zðr1Þ=r
otherwise
ð12Þ
If a > 0 and d 0 ¼ 0 there is no sustainable allowance-bequest pair for any
k, since a 0 ðkÞ ¼ 0 < a, while if a > 0 and d 0 > 0 there exists a sustainable
allowance-bequest pair provided a 0 ðkÞ b a. Using Proposition 12, we derive
the set of stocks where a sustainable allowance-bequest pair exists, even when
preferences are not maximin:
Corollary. Given k, a sustainable allowance-bequest pair exists for a > 0, i¤
1 þ r > zðr1Þ=r
ð13Þ
and
a 0 ðkÞ b a
ð14Þ
Sustainability and capital management
937
Note that zðr1Þ=r > A1=r ¼ ð1 pð1 z 1r ÞÞ1=r , and hence the requirement 1 þ r > zðr1Þ=r is stricter than the requirement in the previous proposition that 1 þ r > A1=r . Thus the mere existence of a positive subsistence level,
however low, makes sustainability harder to achieve. We also note that a is
not continuous as a function of a, at the point a ¼ 0. If d in (7) is positive, but
the requirement in (13) is not met, then a slight increase in a will leave all
policies unsustainable, even at states where a ðkÞ is very high for a ¼ 0.
It is interesting to compare the requirement of sustainability with discounted utilitarianism. The next proposition gives the material consumption
policy that maximizes the sum of discounted utilities, with discount factor d
satisfying 0 < d < 1. We introduce subscript t for time on the relevant variables in the theorem.
Proposition 14. Let d ¼ ðAdð1 þ rÞÞ 1=ð1rÞ 1. Assume that r > d. Consider
the problem of finding the allowance maximizing
"
#!
y
X
st
Jðk1t ; k2t Þ ¼ max E
uðc1t ; k2t Þd
ð15Þ
s¼t
subject to (3) and the intertemporal budget constraint
lim k1t ð1 þ rÞt b 0:
ð16Þ
t!y
Then the optimal material consumption is c1t ¼ ðr dÞk1t .
Proof. Using the optimality equation from dynamic programming, we only
have to consider two successive time period, we thus drop the time subscript.
We first prove that J given as
rd
Jðk1 ; k2 Þ ¼
1þd
r1
1
ðk r k 1r Þ
Ad 1 2
ð17Þ
solves the optimality equation
Jðk1 ; k2 Þ ¼ max½uðc1 ; k2 Þ þ dEfJðb1 ; b2 Þg
c1
Inserting for J, b1 and b2 , the term maximization is
"
#
r1
rd
1
1r
r
r
ðk1 ð1 þ rÞ c1 Þ
max k2
c1 þ Ad
c1
Ad
1þd
with first order conditions
rd
c1 ¼
ðk1 ð1 þ rÞ c1 Þ
1þd
The solution to this equation is c1 ¼ ðr dÞk1 .
ð18Þ
938
G. B. Asheim, K. A. Brekke
Fig. 1. Maximum allowable material consumption under sustainability and discounted
utilitarianism; material consumption c1 relative to man-made capital k1 as a function
of marginal productivity r
Reinserting the optimal policy into the optimality equation using (18) and
simplifying, we get
"
#
r1
rd
1r
r
r
ððr dÞk1 Þ þ
ðk1 ð1 þ dÞÞ
Jðk1 ; k2 Þ ¼ k2
1þd
¼ ½ð1 þ rÞð1 þ dÞ
¼
r1
r1
rd
k1r k21r
1þd
r1
1 rd
k1r k21r
Ad 1 þ d
and thus the optimality equation is satisfied.
In Fig. 1, the upper limit for material consumption under the requirement
of sustainability is given as a function of marginal productivity of capital.10
We consider two cases, a ¼ 0, and a # 0. This is compared to the optimal
allowance.
Consider first the case a ¼ 0. We see that for very low returns to capital, any sustainable allowance-bequest pair entails that material consumption
equals zero, but on the other hand, the productivity assumption used to justify
10 In this figure we have used the following parameters: d ¼ 0:97, z ¼ 0:95, p ¼ 0:15
and r ¼ 0:5, where productivity applies for r > 0:0075
Sustainability and capital management
939
sustainability will not apply. Above this level, there is an interval (from about
0.7% to 3.5% return with the chosen parameters) where the optimal material
consumption exceeds the maximal allowable level under sustainability, even
though the economy is productive. Note that discounted utilitarianism is not
consistent with the Weak Anonymity principle, and hence, the discounted utilitarian optimum may not be sustainable.
Consider then the case a > 0. The dotted line is the maximal allowable
material consumption under sustainability with maximin certainty equivalent.
The dotted line is thus not the maximal allowable material consumption under
sustainability with the actual certainty equivalent. Still, by Proposition 12, it
follows that when the dotted line is not above zero, sustainability is impossible
however low we choose a > 0. We note that with a > 0 sustainability is much
harder to achieve than without the subsistence requirement.
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